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We propose a metapopulation version of the Schelling model where two kinds of agents relocate themselves, with unconstrained destination, if their local fitness is lower than a tolerance threshold. We show that, for small values of the latter, the population redistributes highly heterogeneously among the available places. The system thus stabilizes on these heterogeneous skylines after a long quasi-stationary transient period, during which the population remains in a well mixed phase.\
Varying the tolerance passing from large to small values, we identify three possible global regimes: microscopic clusters with local coexistence of both kinds of agents, macroscopic clusters with local coexistence (soft segregation), macroscopic clusters with local segregation but homogeneous densities (hard segregation). The model is studied numerically and complemented with an analytical study in the limit of extremely large node capacity.
author:
- 'F. Gargiulo$^{1}$, Y. Gandica$^{1}$, T. Carletti$^{1}$'
title: Urban skylines from Schelling model
---
Introduction
============
Modern societies are often faced to segregation; dictated by race, religion, social status or incomes differences, it represents a major issue whose outcome can range from social unrest, to riots and possibly to civil wars. The understanding of the rise of such phenomenon has thus attracted a lot of attention from economists, politicians and sociologists [@CutlerGlaeser1997; @FagioloEtAl2007; @FreyFarley1996; @ConradtEtAl2009].
In a couple of papers written in the late 60s, Thomas Schelling [@Schelling1969; @Schelling1971] proposed a stylized model to describe the unset of segregation and pointed out a counterintuitive but widely observed result: a well integrated society can evolve into a rather segregated one even if at individual level nobody strictly prefers this final outcome. Even when individuals are quite tolerant to neighbors of their opposite kind, allowing them to relocate themselves to satisfy their preferences - namely maximize their perceived local fitness/utility - will make segregation to emerge as a global aggregated phenomenon not directly foreseen from the individual choices.
Since the pioneering works of Schelling the model has attracted the attention of the community of physicists and mathematicians, interested in its simplicity and in the emergent behaviors recalling models such as Ising and Potts ones [@VinkovicKirman2006; @DallAstaEtAl2008; @Schulze2005; @StaufferSolomon2007; @GrauwinEtAl2009; @RogersMcKane2012; @RogersMcKane2011]. In its simplest form the model proposed by Schelling considers a population composed by two kinds of agents sitting on the top of a regular lattice (or a 1 dimensional ring), each site being able to receive at most one agent. The latter being allowed to hop to a new empty lattice site once unhappy, that is once the fraction of agents of her opposite kind in her Moore neighborhood, is larger than a given [*tolerance threshold*]{}. The striking result by Schelling is that segregation will emerge for tolerances slightly larger than $1/3$, well below the more natural bound$1/2$.
Several variations have been performed onto the original Schelling model (see [@PancsVriend2007] for a review), for instance different ways of computing the agent utility [@RogersMcKane2011] and rules according which agents do move (swap agents positions, restrict moves to increasing fitness locations [@VinkovicKirman2006; @DallAstaEtAl2008], to fixed distance places or following the path of an underlying network [@FagioloEtAl2007]). All such models contains essentially two parameters, the tolerance threshold and the fraction of empty spaces; the latter having received much more attention because it is responsible for a phase transition in the model [@VinkovicKirman2006; @DallAstaEtAl2008; @GrauwinEtAl2009; @RogersMcKane2011; @RogersMcKane2012].
Recently the Schelling model has been improved in realism by considering a metapopulation scheme [@GrauwinEtAl2009; @RogersMcKane2012; @DurettYuan2014]. Nodes have a large but finite carrying capacity representing thus a bunch of residences in a district whose spatial extension can be neglected (well mixed population); happiness and thus willingness to move is thus conditioned by the fraction of agents of the opposite kind inside a given node, with respect to the nodes occupancy ($1$-node fitness) [@DurettYuan2014].
The Model
=========
In this paper we elaborate further in this direction by considering a metapopulation version of the Schelling model where two kinds of agents, say Red and Blue, can move across $N$ nodes, each of which can receive at most $L$ agents [^1] (carrying capacity) at any given time. We assume there are in the model $\rho LN$ vacancies, i.e. empty spaces, and an equal number of Red and Blue agents.
The considered model resembles to the one proposed in [@DurettYuan2014], the main difference being that in our case agents don’t have any information about the selected destination, therefore also moves that decrease or leave invariant the fitness are allowed, we define such case *weak liquid* version of the Schelling model, the liquid case referring in the literature to agents’ moves for which the fitness does not decrease [@VinkovicKirman2006]. A second difference is that in [@DurettYuan2014] the agent fitness is computed using single node informations, namely the number of agents in a given sites, on the contrary we reintroduce as Schelling originally did, the concept of spatial proximity [@Schelling1971], agent fitness takes into account the number of agents in a given node and in the neighboring ones, namely nodes at distance $1$ from the current node (local fitness).
Nodes are assumed to be arranged in regular lattices, as initially assumed by Schelling, but the model can be easily extended to complex networks. The local update rules is defined as follows. The neighborhood of an agent is given by the topological neighborhood of the node where she lives, including the latter. At each time step an agent is selected and her fitness is computed as the fraction of agents of the opposite kind of her, living in her neighborhood with respect to the total number of agents living in the same neighborhood. Mathematically, assuming she is a Blue agent living in node $i$, then her fitness is given by: $$\label{eq:fitness}
{f}_i^B=\frac{\sum_{j\in i}n_j^A}{\sum_{j\in i}(n_j^B+n_j^A)}\, ,$$ where $n_j^X$, $X=A,B$, is the number of agent of $X$-kind in node $j$, and we used the notation $j\in i$ to denote all nodes $j$ belonging to a neighborhood of node $i$, including the latter, that is the set of nodes at distance smaller or equal to $1$ from $i$.
As in the original Schelling model, agents are unhappy if their fitness is larger than a given [*tolerance threshold*]{}, $\epsilon\in(0,1)$, hereby assumed to be the same for all agents of both kinds; to reduce their uneasiness agents move by choosing uniformly at random another node $k$ not completely full, i.e. $n_k^A+n_k^B<L$: $$\label{eq:faction}
\text{if ${f}_i^B>\epsilon$} \Rightarrow \text{agent $A$ leaves node $i$.}$$ The case for a Red agent is similar.
One time step is the random selection with reinsertion of $(1-\rho)NL$ agents. We define the [*convergence time*]{}, to be the time needed for the system to reach the equilibrium, namely once no agents will move anymore.
Results {#sec:res}
=======
We hereby present the numerical analysis of the proposed model once the underlying network is a regular lattice with periodic boundary conditions and each node has $4$ neighboring nodes. The system is initialized with $\rho NL$ vacancies, $(1-\rho)NL/2$ Red agents and the same number of Blue ones, uniformly random distributed among the $N=400$ nodes. The carrying capacity has been fixed to $L=100$ and we check that the initial conditions satisfy the local constraint $n_i^A+n_i^B\leq L$ for all $i$. Throughout the paper the emptiness has been fixed to $\rho=0.9$.
Single node properties
----------------------
The aim of this section is to present the local properties of the system, namely at the level of single nodes. The metric we used is the [*average value*]{}, over all the nodes, of the node [*magnetization*]{}: $$\label{eq:magnetization}
\langle\mu\rangle=\frac{1}{N}\sum_i\frac{|n_i^B-n_i^A|}{n_i^B+n_i^A}\, ,$$ small values of $\langle\mu\rangle$ mean that, on average, each node is populated by the same number of agents of both kinds, while large values are associated to nodes filled with agents of only one kind.
For $\epsilon\leq 0.5$ we observe (see Fig. \[local\] panels A and B upper plot) that asymptotically $\langle\mu\rangle\rightarrow1$, meaning that the system stabilizes into a frozen state where [*local segregation*]{} is present in all nodes: each node contains only agents of one kind. Let us observe (see Fig. \[local\] B upper plot) that the same behavior is also present in the simplified model where the fitness is calculated on the single node, as done in [@DurettYuan2014], and thus it is intrinsic to the displacement dynamics and not to the way the agent fitness is computed. We also notice (see Fig. \[local\] panels A and Fig. 6 that as $\epsilon$ decreases toward zero, the time needed to reach the frozen state gets longer, going to infinity in the limit $\epsilon\rightarrow 0$ . The system exhibits thus a transient phase where it remains stuck for very long time into a [*quasi-stationary non-segregated state*]{} (see red circles and orange stars curves in Fig. \[local\]A).
A second fundamental self-organized phenomenon emerges for $\epsilon\leq 0.5$, the initial homogeneously distributed population organizes itself into an heterogeneous state across the network nodes (Fig. \[local\] panel C lower plots): most of the nodes contain $\sim 10$ agents, while very few nodes have as much as $\sim 100$ agents, recall that $L=100$ is the maximum node capacity.
Observe that such asymptotic distribution is correlated with the time the system spends in the quasi-stationary non-segregated state, the longer this time the more the population distribution across nodes moves from a Poissonian distribution (Fig.\[local\] panel C upper plots) to a power law (Fig. \[local\] panel C lower plots). The maximal node population $n_{max}=\max_i(n_i^A+n_i^B)$ increases for $\epsilon\rightarrow 0$ (Fig. \[local\] panel B middle plot). Notice that the maximal node capacity ($L=100$) is never reached. At the same time we observe the formation of a relevant fraction of completely empty nodes, i.e. nodes for which $n_i^A+n_i^B=0$ (Fig. \[local\] panel B lower plot).
{width="100.00000%"}
The transition to the local segregation at $\epsilon=0.5$ is an important issue of the metapopulation version of the Schelling model that can be easily explained once the fitness depends only on one node, with the following argument. A mixture of agents of both kinds in each node, determines a global happy configuration if and only if: $$\forall i\quad \frac{n_A^i}{n_A^i+n_B^i}\leq\epsilon \qquad \frac{n_B^i}{n_A^i+n_B^i}\leq\epsilon\, ,$$ but it is straightforward to observe that this equation admits a solution, for which $n_A^i>0$ and $n_B^i>0$, only for $\epsilon>1/2$. Hence for $\epsilon \leq1/2$ the above relations cannot be satisfied and thus unhappy agents start to move, increasing the happiness of agents of the same kind and decreasing the happiness of agents of opposite kind. The net result of such behavior is segregation.
Global properties {#ssec:global}
-----------------
To analyze the spatial structures, thus beyond the single node, we define a [*cluster*]{} based on node’s majority, more precisely two linked nodes belong to the same cluster if they both are characterized by the majority of agents of the same kind [^2]: $$\label{eq:cluster}
\text{$i,j\in C$ if \{$n_i^A>n_i^B$ and $n_j^A>n_j^B$\} or \{$n_i^A<n_i^B$ and $n_j^A<n_j^B$\}}\, .$$ If a node share the same (positive) number of Red and Blue agents, then it will be considered part of the [*interface*]{}. An edge between two neighboring nodes $i$ and $j$ is considered interface if $(n^A_i-n^B_i)(n^A_j-n^B_j)\leq 0$. Hence a cluster is made by nodes while an interface can contains both nodes and edges among them.
This indicator allows us to show that (Fig. \[global\] panels B and C) for $\epsilon\leq 0.6$ macroscopic percolating clusters are always formed, the average size of the largest cluster oscillating between $0.4N$ and $0.5N$ and at the same time the average sizes of the first and second largest cluster added together cover more than $75\%$ of the available nodes. This critical threshold is the same observed in the original Schelling model. Notice that for $\epsilon=0.5$ the first and second cluster cover almost completely the lattice, containing more than $95\%$ of nodes. From the behavior of $\langle S_{max}\rangle$ for $0.5<\epsilon\leq 0.6$ and the results of Fig. \[local\] Panel B, one can conclude that the largest clusters contain a majority of agents of the same kind, but different agents can coexist in the same node. We call this scenario *soft segregation* because it allows a small mixing in the population. For $\epsilon\leq 0.5$ each spatial cluster contains only one kind of agent, resulting in a *hard segregation* of the population. For $\epsilon\rightarrow 0$ clusters of empty cells are created increasing the distance between the two monochromatic structures by interposing interfaces (Fig. \[local\] panels B lower plot and white nodes in Fig. \[global\] panels A3, A4 and A5).
Looking at the temporal growth of the interfaces (Fig. \[global\] panels D), we can observe another remarkable self-organized phenomenon: for $0.3<\epsilon\leq0.6$ the size of the interface, monotonically decreases in time as $t^{-1/z}$ ($z=4$ for $\epsilon=0.6$ and $z=3$ for $\epsilon\geq 0.5$). This is the typical signature of a coarsening phenomenon, that has already been observed in the classical Schelling model [@DallAstaEtAl2008]. On the other side, for low values of the tolerance where the system remains for long time in the quasi-stationary state (for instance $\epsilon\leq 0.3$), the size of the interface has a slowly increasing phase (red curve Fig. \[global\] panel D), corresponding to the formation of temporal segregated domains followed by an abrupt decrease when the system reaches the local segregation equilibrium. In this case the domain formation mechanism cannot be ascribed to the coarsening framework: the spatial segregation indeed is reached through a mechanism of aggregation of agents around the high density instabilities that allow the system to exit the quasi-stationary state. Once the clusters are formed, the complete equilibrium is reached as the interface between Blue and Red zones becomes empty (See Fig. 7 for a detailed explanation).
While the creation of heterogeneity distribution of agents among nodes is due to the metapopulation mechanism, the formation of the monochromatic clusters was already present in the classical Schelling model and thus is mainly due to the behavioral rules of the latter. We can therefore imagine the process at play in our model as the combined outcome of a local birth and death process (migration in-out) on the single node and a coarse-grained node dynamics on the lattice.
{width="100.00000%"}
An analytical simplified model
==============================
To gain insight into the behavior of the previously presented model, we hereby introduce a model ables to capture the main behavior of the metapopulation Schelling model, but simple enough to be analytically tractable. Because we consider the case for extremely large $L$, our model complements the one proposed in [@DurettYuan2014] devoted to the case $L=2$.
Using the notations introduced before the state of the system is thus completely characterized by the knowledge of $(\vec{n}^A(t),\vec{n}^B(t))$, being $\vec{n}^A(t)=(n^A_1(t),\dots,n^A_N(t))$ and $\vec{n}^B(t)=(n^B_1(t),\dots,n^B_N(t))$. For a sake of simplicity we decided to compute the fitness using only the information from a single node, Eq. with $j=i$. The system evolution is done as previously and we still use the weak liquid version, an unhappy agent will move to an uniformly randomly chosen new node, provide there is enough space there.
The model is thus intrinsically stochastic and hence it can be described by the probability $P(\vec{n}^A,\vec{n}^B,t)$ to be at time $t$ in state $(\vec{n}^A,\vec{n}^B)$, whose evolution is dictated by the master equation: $$\begin{aligned}
\label{eq:ME}
P(\vec{n}^A,\vec{n}^B,t+1)&=&P(\vec{n}^A,\vec{n}^B,t)+\\&+&\hspace{-2em}\sum_{(\vec{n}^{A^\prime},\vec{n}^{B^\prime})} \hspace{-1em}\big[ T(\vec{n}^A,\vec{n}^B\vert\vec{n}^{A^\prime},\vec{n}^{B^\prime})P(\vec{n}^{A^\prime},\vec{n}^{B^\prime},t)\notag\\
&-&T(\vec{n}^{A^\prime},\vec{n}^{B^\prime}\vert\vec{n}^A,\vec{n}^B)P(\vec{n}^A,\vec{n}^B,t)\big]\notag\, ,\end{aligned}$$ being $T(\vec{n}^{A^\prime},\vec{n}^{B^\prime}\vert\vec{n}^A,\vec{n}^B)$ the [*Transition probability*]{} to pass from state $(\vec{n}^A,\vec{n}^B)$ to the new compatible one $(\vec{n}^{A^\prime},\vec{n}^{B^\prime})$. For incompatible states we set $T(\vec{n}^{A^\prime},\vec{n}^{B^\prime}\vert\vec{n}^A,\vec{n}^B)=0$.
The non zero transition probabilities can be computed using the rules previously defined (see Appendix\[sec:analytical\] for a detailed account of the transition probabilities calculation), for instance the transition probability that an $A$ agent moves from node $i$–th to node $j$–th is given by: $$\begin{aligned}
&&T_1(n^A_i-1,n^A_j+1\vert n^A_i,n^A_j)= \\&=&\frac{1}{N}\frac{n^A_i}{n^A_i+n^B_i}\Theta\left(\frac{n^B_i}{n^A_i+n^B_i}-\epsilon\right)\frac{L-n^A_j-n^B_j}{L}\, .\end{aligned}$$ The function $\Theta(x)$, defined to be $1$ if $x>0$ and $0$ otherwise, translates the willingness of an unhappy agent to move; smoother versions can be used as well. Let us observe that to lighten the notation we wrote only the variables whose values change because of the transition.
From the master equation one can compute the time evolution of some relevant quantities, for instance the average number of $X$–agents in every node $i$ at time $t$, $\langle n^X_i\rangle(t)=\sum_{\vec{n}^X}n^X_iP(\vec{n}^A,\vec{n}^B,t)$, $X=A,B$.
Using the expression for the transition probabilities Eq. \[B3\] and assuming correlations can be neglected, i.e. $\langle (n^A_i)^2\rangle\sim \langle n^A_i\rangle^2$, we obtain a system of finite differences describing the evolution of $\langle n^A_i\rangle$ and $\langle n^B_i\rangle$.
To go one step further, we introduce the average fraction of $A$ and $B$ in each node, $\alpha_i={\langle n^A_i\rangle}/{L}$ and $\beta_i={\langle n^B_i\rangle}/{L}$, we rescale time by defining $s=t/L$ and finally we assume each node to have an infinite large carrying capacity (see Appendix\[sec:analytical\] for a more detailed discussion): $$\begin{aligned}
\label{eq:alphat1}
\frac{d\alpha_i(s)}{ds}&=&-{\rho}\frac{\alpha_i}{\alpha_i+\beta_i}\Theta\left(\frac{\beta_i}{\alpha_i+\beta_i}-\epsilon\right)\notag\\&+&\frac{1-\alpha_i-\beta_i}{N}\sum_{j}\frac{\alpha_j}{\alpha_j+\beta_j}\Theta\left(\frac{\beta_j}{\alpha_j+\beta_j}-{\epsilon}\right)\end{aligned}$$ and $$\begin{aligned}
\label{eq:betat1}
\frac{d\beta_i(s)}{ds}&=&-{\rho}\frac{\beta_i}{\alpha_i+\beta_i}\Theta\left(\frac{\alpha_i}{\alpha_i+\beta_i}-\epsilon\right)\notag\\&+&\frac{1-\alpha_i-\beta_i}{N}\sum_{j}\frac{\beta_j}{\alpha_j+\beta_j}\Theta\left(\frac{\alpha_j}{\alpha_j+\beta_j}-{\epsilon}\right)\end{aligned}$$ where $\rho$ is the fraction empty nodes, $\sum_i \left(1-\alpha_i(s)-\beta_i(s)\right)=\rho N$.
To disentangle the evolution of $\alpha_i$ and $\beta_i$ we introduce a new set of variables, the [*node emptiness*]{}, $\gamma_i=1-\alpha_i-\beta_i$, and the [*difference of fractions*]{} of $A$ and $B$, $\zeta_i=\alpha_i-\beta_i$. The new variables range in $\zeta_i\in[-1,1]$ and $\gamma_i\in[0,1]$. Introducing the functions $$\begin{aligned}
\label{eq:FG}
F(x)&=&x\Theta(1-x-\epsilon)+(1-x)\Theta(x-\epsilon)\\
G(x)&=&x\Theta(1-x-\epsilon)-(1-x)\Theta(x-\epsilon)\, ,\end{aligned}$$ we can rewrite Eqs. and as follows $$\label{eq:gammat}
\frac{d\gamma_i(s)}{ds}=\rho F\left(\frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}\right)-\frac{\gamma_i}{N}\sum_jF\left(\frac{1}{2}+\frac{\zeta_j}{2(1-\gamma_j)}\right)$$ and $$\label{eq:zetat}
\frac{d\zeta_i(s)}{ds}=-\rho G\left(\frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}\right)+\frac{\gamma_i}{N}\sum_jG\left(\frac{1}{2}+\frac{\zeta_j}{2(1-\gamma_j)}\right)\, .$$
The agents initialization used in the previous section, namely $\rho N$ density of vacancies and an equal density $(1-\rho)N/2$ of $A$ and $B$ agents, translate into $(\zeta_i,\gamma_i)$ uniformly distributed in a neighborhood of $(0,\rho)$. We thus divide the domain of definition of $(\zeta_i,\gamma_i)$ into four zones (see Fig. \[fig:diagramma\]) and we will look closely to the dynamics in the $Z_2$ zone if $\epsilon<1/2$ and in the $Z_4$ zone if $\epsilon>1/2$.
![The four zones $Z_i$ used to study the solutions of system , .[]{data-label="fig:diagramma"}](DiagrammaZetaGamma.pdf){width="9cm"}
Let $\epsilon<1/2$, then one can easily prove that if $(\zeta,\gamma)\in Z_2$ one has $F\left(\frac{1}{2}+\frac{\zeta}{2(1-\gamma)}\right)=1$ and $G\left(\frac{1}{2}+\frac{\zeta}{2(1-\gamma)}\right)=\frac{\zeta}{1-\gamma}$, so assuming that for all $i$ one has $(\zeta_i(0),\gamma_i(0))\in Z_2$, then Eqs. and rewrite: $$\label{eq:zetagammatZ2}
\begin{cases}
\frac{d\gamma_i(s)}{ds}=\rho -\gamma_i\\
\frac{d\zeta_i(s)}{ds}=-\rho \frac{\zeta_i}{1-\gamma_i}+\frac{\gamma_i}{N}\sum_j \frac{\zeta_j}{1-\gamma_j}\, ,
\end{cases}$$ as long as $(\zeta_i(t),\gamma_i(t))$ will not leave $Z_2$. Assume for a while this statement to hold, then the first equation can be straightforwardly solved to give $\gamma_i(t)=\rho+e^{-t}(\gamma_i(0)-\rho)$, that is for all $i$, $\gamma_i(t)\rightarrow \rho$ when $t\rightarrow\infty$. The second equation can also be solved (see Eq. \[S13\]) and thus to prove that $\zeta_i(t)\rightarrow 0$ for all $i$ when $t\rightarrow 0$. This proves also [*a posteriori*]{} that $(\zeta_i(t),\gamma_i(t))$ will never leave $Z_2$. The average magnetization rewrites in such variables as: $$\label{eq:magnet}
\langle\mu \rangle=\frac{1}{N}\sum_i\frac{\lvert \zeta_i\rvert}{1-\gamma_i}\, ,$$ we have hence proved that for initial conditions in $Z_2$ the magnetization asymptotically vanishes (see Fig. \[fig:Mu\]).
The remaining case, $\epsilon>1/2$, can be handle as well but it is more cumbersome (see Appendix\[appC\]). Let us only observe there that for $(\zeta,\gamma)\in Z_4$ one has $F\left(\frac{1}{2}+\frac{\zeta}{2(1-\gamma)}\right)=G\left(\frac{1}{2}+\frac{\zeta}{2(1-\gamma)}\right)=0$, so assuming that for all $i$ one has $(\zeta_i(0),\gamma_i(0))\in Z_4$, then the right hand side of Eqs. and identically vanishes and thus the average magnetization depends on the domain where initial conditions have been set. However for a fixed size of the latter, one cannot satisfy the hypothesis $(\zeta_i(0),\gamma_i(0))\in Z_4$ if $\epsilon$ is closer enough but larger then $0.5$, indeed the $Z_4$ zone shrinks to zero in this case (see Fig. \[fig:diagramma\]). In this case one should take into account the dynamics of orbits whose initial conditions are set in $Z_1$ and $Z_3$ (see Appendix\[sec:analytical\] and Figs. 8 and 9). In conclusion the analytical model perfectly fits with the ABM for $\epsilon\geq 0.5$ while it has a different behaviour for $\epsilon<0.5$, because in this range the ABM dynamics is strongly dictated by the stochasticity of the model, orbits tending to converge toward the equilibrium $(0,\rho)$ ($\langle \mu\rangle\sim 0$) are destabilized by fluctuations and thus sent into the zones $Z_1$ and $Z_3$ ($\langle \mu\rangle\sim 1$).
![The average asymptotic magnetization as a function of $\epsilon$. Each point is the average over $50$ simulations whose initial conditions are close to the equilibrium point $\gamma_i=\rho$ and $\zeta_i=0$. Black circles represent the results of the analytical model while grey diamonds to the ABM with $1$–node fitness. Parameters are: $N=100$ and $\rho=0.9$.[]{data-label="fig:Mu"}](finalConv.pdf){width="8cm"}
Conclusions
===========
We proposed and analyzed an extension of the classical Schelling model to a metapopulation framework, whose main outcome is the spontaneous emergence, for low values of the tolerance threshold, of heterogeneously populated nodes without any exogenous preferential attachment mechanism (for 1D lattice this phenomenon induces the formation of urban skylines, Fig. \[regimesSchelling\]). This behavior is connected to the permanence for long time of the system in a quasi-stationary non segregated state, where in each node the two populations are equally distributed. This quasi-stationary state can be recovered as stable equilibrium of the simplified analytical model. We can evince that the system stabilization toward the magnetized state (for the ABM) passes through the creation (by random agents moves) of highly populated nodes, hereby named *towers*. At the same time, global patterns emerge as in the classical Shelling model. Figure \[regimesSchelling\] summarizes the possible behaviors of the system. For $\epsilon<0.5$ the global clusters are described as neighborhoods formed by a strong majority of individuals of the same color (soft segregation). For $\epsilon\geq 0.5$ we have the formation of ghettos (hard segregation).\
The mechanism of formation of the clusters is a typical coarsening phenomenon for low values of $\epsilon$. On the contrary for low tolerance cases the clusters are formed around the towers that become stable points for a certain type of nodes (once an agent, whose kind corresponds to the majority already inside the tower, enters she never gets out). Once a higher density zone starts to exist, this mechanism reinforces the (majority color) population growth in this node and in the neighborhood. The global patterns start therefore to stabilize around the towers.\
Local magnetization phenomenon has been explained using an analytical approach able to describe the different equilibria of the model and the origin of the quasi-stationary state for low tolerances.
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Acknowledgments {#acknowledgments .unnumbered}
===============
The work of F.G., Y.G. and T.C. presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office.\
T.C. is grateful to Cyril Vargas, student at the ENSEEIHT - INP Toulouse (France), involved in a preliminary study of this subject during his Master Degree.
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Patterns dynamics {#sec:convergence}
=================
The aim of this section is to present some details concerning the convergence of the agent based model to the asymptotic equilibrium pattern. We start by considering the [*convergence time*]{}, defined as the time needed for the system to reach such equilibrium where no agents has incentive to move anymore, and its dependence of the tolerance threshold $\epsilon$. Intuitively, if $\epsilon$ is large then most agents are happy and thus they will not move, hence the equilibrium will be reached quite soon. On the other hand if $\epsilon$ is small, agents will be very often unhappy and thus they will relocate themselves to reduce this uneasiness, this will increase the convergence time. In the limit $\epsilon\rightarrow 0$ this equilibrium will be never achieved and thus the convergence time will diverge.
![\[convTimes\] Convergence time as a function of $\epsilon$. The blue line represents the average convergence time as a function of $\epsilon$ for $100$ replicas of the system with same initial conditions and parameters values. The inset shows the boxplots for the convergence times for some representative values of $\epsilon=0.3,0.4,0.5,0.6,0.7$.](convTimes.pdf){width="60.00000%"}
In Fig. \[convTimes\] we report the results obtained by the simulation of the metapopulation model and we can observe that the convergence time does not have a monotonic behavior as a function of $\epsilon$. Passing from from $\epsilon=1$ to $\epsilon\sim 0.6$ we observe an exponential growth of the convergence time (note the logarithmic vertical scale), these values of $\epsilon$ correspond to the absence of macroscopic clusters (see Fig. 5 main text). Let us notice that the time to converge increase between $\epsilon\sim 0.5$ and $\epsilon\sim 0.6$ is due to a slower dynamics for the latter case (to move is less probable). Then for $0.3\leq\epsilon< 0.6$ the convergence time exhibits an almost stable value - horizontal plateau - with a local minimum at $\epsilon=0.5$, this is the phase where macroscopic clusters are formed through a coarsening process (see Fig. 5 main text). Finally for $\epsilon\leq 0.3$ the convergence times start to increase faster than exponentially, the dynamics exhibits quasi-stationary states and the system reaches the equilibrium through the formation of towers (high densely populated set of contiguous nodes). Let us observe that asymptotic equilibrium is affected by the intrinsic stochasticity of the ABM, the convergence time will thus reflect this fact by showing a large variance, mainly for small $\epsilon$ (see inset of Fig. \[convTimes\]).
The patterns arising for small tolerance threshold are intriguing and peculiar to our model. As already observed the system spends quite a long time in a quasi-stationary (almost) homogeneous state and then suddenly jumps to a macroscopic segregated one. Such behavior is schematically represented in Fig. \[timeEvol\] in the case of the $1D$-lattice for $\epsilon=0.25$. The system stabilization toward the magnetized state passes through the creation (by random agents moves) of a highly populated nodes, hereby named *towers* corresponding to monochromatic clusters in the $2$D model presented before. Such towers become attracting selective places for each kind of agents: once an agent, whose kind corresponds to the one of the majority already inside the tower, enters she never gets out from the tower because she will be happy there and this will increase the unhappiness of agents of the opposite kind still inside the tower. The net result is that once a highly dense zone starts to exist, this mechanism reinforces the (majority color) population growth in this node and in the neighborhood, because of the way the fitness is computed, using nodes at distance $1$. The global patterns start therefore to stabilize around the towers. As the first kind of agents stabilizes, with a certain delay also the stabilization of the second population is reached. Once the towers/clusters are formed, the complete equilibrium is reached as the interface between Blue and Red zones becomes empty. The lower is the tolerance value $\epsilon$, the higher is the initial density unbalance needed to start the stabilization process.
![\[timeEvol\]Generic time evolution toward a hard segregated pattern. Upper main plot: first and second cluster size as a function of time. Lower main plot: average magnetization of the system as a function of time. Left and right panels: System snapshots at different times. Each box represents a node of the $1$-dimensional lattice, the height of the box is given by the total blue and red population of the node. The colored rectangle under the cell represents the majority color of the cell.](timeEvolution){width="100.00000%"}
More details about the analytical model {#sec:analytical}
=======================================
The goal of this section is to present more details of the analytical model we introduced to capture the main behavior of the metapopulation Schelling model presented in the main text.
Let us denote by $n^A_i(t)$, respectively $n^B_i(t)$, the number of agents of type $A$, respectively $B$, at node $i$–th at time $t$. Each node is also characterized by a number of vacancies $n^E_i(t)$ at time $t$, however for all $t$ and all $i$ one has $n^A_i(t)+n^B_i(t)+n^E_i(t)=L$ and moreover the number of agents of each kind is a preserved quantity, i.e. the system is closed. The state of the system is thus completely characterized by knowledge of $(\vec{n}^A(t),\vec{n}^B(t))$, being $\vec{n}^A(t)=(n^A_1(t),\dots,n^A_N(t))$ and $\vec{n}^B(t)=(n^B_1(t),\dots,n^B_N(t))$.
An agent $A$ in node $i$–th is unhappy - or her fitness is low - if the fraction of $B$ in the same node, that is $n^B_i/(n^A_i+n^B_i)$, is larger than a given [*tolerance threshold*]{} $\epsilon\in(0,1)$: $$\label{eq:Aunhappy}
\text{$A$ is unhappy at $i$ if }\frac{n^B_i}{n^A_i+n^B_i}\geq \epsilon\, ,$$ and similarly for $B$.
We assume that once an agent decides to move because unhappy, she will move to an uniformly randomly chosen new node, provide there is enough space there (that we named [*weak liquid Schelling model*]{}). The model is intrinsically stochastic and hence it can be described by the probability to be at time $t$ in state $(\vec{n}^A,\vec{n}^B)$, that is $P(\vec{n}^A,\vec{n}^B,t)$. The evolution of such probability can be obtained using the master equation: $$\begin{aligned}
\label{eq:ME}
P(\vec{n}^A,\vec{n}^B,t+1)=P(\vec{n}^A,\vec{n}^B,t)+\sum_{(\vec{n}^{A^\prime},\vec{n}^{B^\prime})} \left[ T(\vec{n}^A,\vec{n}^B\vert\vec{n}^{A^\prime},\vec{n}^{B^\prime})P(\vec{n}^{A^\prime},\vec{n}^{B^\prime},t)-T(\vec{n}^{A^\prime},\vec{n}^{B^\prime}\vert\vec{n}^A,\vec{n}^B)P(\vec{n}^A,\vec{n}^B,t)\right]\, ,\end{aligned}$$ being $T(\vec{n}^{A^\prime},\vec{n}^{B^\prime}\vert\vec{n}^A,\vec{n}^B)$ the [*Transition probability*]{} to pass from state $(\vec{n}^A,\vec{n}^B)$ to the new compatible one, i.e. the system can pass from the former to the latter, $(\vec{n}^{A^\prime},\vec{n}^{B^\prime})$. Non compatible states cannot be linked and thus we will set $T(\vec{n}^{A^\prime},\vec{n}^{B^\prime}\vert\vec{n}^A,\vec{n}^B)=0$.
The non zero transition probabilities can be computed using the behavioral rules of the model: $$\begin{aligned}
\label{eq:transprob}
\text{$A$ moves from node $i$ to node $j$: }T_1(n^A_i-1,n^A_j+1\vert n^A_i,n^A_j)&=& \frac{1}{N}\frac{n^A_i}{n^A_i+n^B_i}\Theta\left(\frac{n^B_i}{n^A_i+n^B_i}-\epsilon\right)\frac{n^E_j}{L}\notag\\
\text{$A$ moves from node $j$ to node $i$: }T_2(n^A_i+1,n^A_j-1\vert n^A_i,n^A_j)&=& \frac{1}{N}\frac{n^A_j}{n^A_j+n^B_j}\Theta\left(\frac{n^B_j}{n^A_j+n^B_j}-\epsilon\right)\frac{n^E_i}{L}\notag\\
\text{$B$ moves from node $i$ to node $j$: }T_3(n^B_i-1,n^B_j+1\vert n^B_i,n^B_j)&=& \frac{1}{N}\frac{n^B_i}{n^A_i+n^B_i}\Theta\left(\frac{n^A_i}{n^A_i+n^B_i}-\epsilon\right)\frac{n^E_j}{L}\notag\\
\text{$B$ moves from node $j$ to node $i$: }T_4(n^B_i+1,n^B_j-1\vert n^B_i,n^B_j)&=& \frac{1}{N}\frac{n^B_j}{n^A_j+n^B_j}\Theta\left(\frac{n^A_j}{n^A_j+n^B_j}-\epsilon\right)\frac{n^E_i}{L}\, .\end{aligned}$$ Let us observe that to lighten the notation we wrote only the variables whose values change because of the transition. The function $\Theta(x)$ is defined to be $1$ if $x>0$ and $0$ otherwise; smoother versions can be used as well. Being the structure of such transition probabilities very similar we will detail the way we compute the first one:
- $\frac{1}{N}$ : probability to draw the $i$–th node with uniform random probability;
- $\frac{n^A_i}{n^A_i+n^B_i}$ : probability to draw one $A$ agent among the $n^A_i$ present over the total node population $n^A_i+n^B_i$;
- $\Theta\left(\frac{n^B_i}{n^A_i+n^B_i}-\epsilon\right)$: probability the selected $A$ agent is unhappy;
- $\frac{n^E_j}{L}$: probability to select a vacancy in node $j$.
The master equation is unmanageable but one can go one step further by computing the time evolution of relevant quantities, for instance the average number of agents $A$ and $B$ in every node $i$ at time $t$, $\langle n^A_i\rangle(t)=\sum_{\vec{n}^A}n^A_iP(\vec{n}^A,\vec{n}^B,t)$: $$\begin{aligned}
\langle n^A_i\rangle(t+1)&-&\langle n^A_i\rangle(t)=\sum_{\vec{n}^A}\sum_{j\neq i} \Big[n^A_iT_1(n^A_i,n^A_j\vert n^A_i+1,n^A_j-1)P(n^A_i+1,n^A_j-1,t)\\
&+&n^A_iT_2(n^A_i,n^A_j\vert n^A_i-1,n^A_j+1)P(n^A_i-1,n^A_j+1,t)-n^A_iT_1(n^A_i-1,n^A_j+1\vert n^A_i,n^A_j)P(n^A_i,n^A_j,t)\\&-&n^A_iT_2(n^A_i+1,n^A_j-1\vert n^A_i,n^A_j)P(n^A_i,n^A_j,t)\Big]\\
%&=&\sum_{\vec{a}}\sum_{j\neq i} \Big[(n^A_i-1)T_1(n^A_i-1,n^A_j+1\vert n^A_i,n^A_j)+(n^A_i+1)T_2(n^A_i+1,n^A_j-1\vert n^A_i,n^A_j)-n^A_iT_1(n^A_i-1,n^A_j+1\vert n^A_i,n^A_j)\\&-&n^A_iT_2(n^A_i+1,n^A_j-1\vert n^A_i,n^A_j)\Big]P(n^A_i,n^A_j,t)\\
%&=&\sum_{\vec{a}}\sum_{j\neq i} \Big[-T_1(n^A_i-1,n^A_j+1\vert n^A_i,n^A_j)+T_2(n^A_i+1,n^A_j-1\vert n^A_i,n^A_j)\Big]P(n^A_i,n^A_j,t)\\
&=&-\sum_{j\neq i}\langle T_1(n^A_i-1,n^A_j+1\vert n^A_i,n^A_j)\rangle+\sum_{j\neq i}\langle T_2(n^A_i+1,n^A_j-1\vert n^A_i,n^A_j)\rangle\, .\end{aligned}$$ And similarly for $B$: $$\begin{aligned}
\langle n^B_i\rangle(t+1)-\langle n^B_i\rangle(t)=-\sum_{j\neq i}\langle T_3(n^B_i-1,n^B_j+1\vert n^B_i,n^B_j)\rangle+\sum_{j\neq i}\langle T_4(n^B_i+1,n^B_j-1\vert n^B_i,n^B_j)\rangle\, .\end{aligned}$$
Using the expression for the transition probabilities and assuming correlations can be neglected, i.e. $\langle (n^A_i)^2\rangle\sim \langle n^A_i\rangle^2$, we get: $$\begin{aligned}
\label{eq:at}
\langle n^A_i\rangle(t+1)-\langle n^A_i\rangle(t)&=&-\frac{1}{N}\sum_{j\neq i}\frac{\langle n^A_i\rangle}{\langle n^A_i\rangle+\langle n^B_i\rangle}\Theta\left(\frac{\langle n^B_i\rangle}{\langle n^A_i\rangle+\langle n^B_i\rangle}-\epsilon\right)\frac{L-\langle n^A_j\rangle-\langle n^B_j\rangle}{L}\notag\\&+&\frac{1}{N}\sum_{j\neq i}\frac{\langle n^A_j\rangle}{\langle n^A_j\rangle+\langle n^B_j\rangle}\Theta\left(\frac{\langle n^B_j\rangle}{\langle n^A_j\rangle+\langle n^B_j\rangle}-\epsilon\right)\frac{L-\langle n^A_i\rangle-\langle n^B_i\rangle}{L}\, .\end{aligned}$$ and $$\begin{aligned}
\label{eq:bt}
\langle n^B_i\rangle(t+1)-\langle n^B_i\rangle(t)&=&-\frac{1}{N}\sum_{j\neq i}\frac{\langle n^B_i\rangle}{\langle n^A_i\rangle+\langle n^B_i\rangle}\Theta\left(\frac{\langle n^A_i\rangle}{\langle n^A_i\rangle+\langle n^B_i\rangle}-\epsilon\right)\frac{L-\langle n^A_j\rangle-\langle n^B_j\rangle}{L}\notag\\&+&\frac{1}{N}\sum_{j\neq i}\frac{\langle n^B_j\rangle}{\langle n^A_j\rangle+\langle n^B_j\rangle}\Theta\left(\frac{\langle n^A_j\rangle}{\langle n^A_j\rangle+\langle n^B_j\rangle}-\epsilon\right)\frac{L-\langle n^A_i\rangle-\langle n^B_i\rangle}{L}\, .\end{aligned}$$ Observe that the right hand sides of and remain unchanged if we allow both sums to run over all node indexes, namely include also $j=i$.
To go one step further, let us define the fraction of $A$ and $B$ in each node, that is $$\label{eq:fracAB}
\alpha_i=\frac{\langle n^A_i\rangle}{L}\text{ and }\beta_i=\frac{\langle n^B_i\rangle}{L}\, ,$$ then we can rewrite [^3] the previous equations and in terms of $\alpha_i$ and $\beta_i$. Finally rescaling time by $s=t/L$, dividing the equations for $\alpha_i$ and $\beta_i$ by $1/L$ and passing to the limit $L\rightarrow +\infty$ we get: $$\begin{aligned}
\label{eq:alphat1}
\frac{d\alpha_i(s)}{ds}&=&\lim_{L\rightarrow+\infty}\frac{\alpha_i(s+1/L)-\alpha_i(s)}{1/L}\\&=&-\frac{1}{N}\frac{\alpha_i}{\alpha_i+\beta_i}\Theta\left(\frac{\beta_i}{\alpha_i+\beta_i}-\epsilon\right)\sum_{j}(1-\alpha_j-\beta_j)+\frac{1-\alpha_i-\beta_i}{N}\sum_{j}\frac{\alpha_j}{\alpha_j+\beta_j}\Theta\left(\frac{\beta_j}{\alpha_j+\beta_j}-{\epsilon}\right)\notag\, ,\end{aligned}$$ and $$\begin{aligned}
\label{eq:betat1}
\frac{d\beta_i(s)}{ds}&=&\lim_{L\rightarrow+\infty}\frac{\beta_i(s+1/L)-\beta_i(s)}{1/L}\\&=&-\frac{1}{N}\frac{\beta_i}{\alpha_i+\beta_i}\Theta\left(\frac{\alpha_i}{\alpha_i+\beta_i}-\epsilon\right)\sum_{j}(1-\alpha_j-\beta_j)+\frac{1-\alpha_i-\beta_i}{N}\sum_{j}\frac{\beta_j}{\alpha_j+\beta_j}\Theta\left(\frac{\alpha_j}{\alpha_j+\beta_j}-{\epsilon}\right)\notag\, .\end{aligned}$$
Let us observe that the model correctly preserves the total fractions of $A$ and $B$ agents in time and thus the total vacancies $\sum_i \left(1-\alpha_i(s)-\beta_i(s)\right)=\sum_i \left(1-\alpha_i(0)-\beta_i(0)\right)=\rho N$, where $\rho$ is the emptiness defined previously, i.e. the total fraction of vacancies.
To prove this statement is enough to take the time derivative of $\sum_i \left(1-\alpha_i(s)-\beta_i(s)\right)$ and observe that using Eqs. and one gets: $$\frac{d}{ds}\sum_i \left(1-\alpha_i(s)-\beta_i(s)\right)=0\, .$$
One can similarly prove the statement about the total fraction of $A$ and $B$ agents
So in conclusion the system is ruled by the following system of differential equations: $$\label{eq:alphabetat}
\begin{cases}
\frac{d\alpha_i(s)}{ds}=-\rho \frac{\alpha_i}{\alpha_i+\beta_i}\Theta\left(\frac{\beta_i}{\alpha_i+\beta_i}-\epsilon\right)+\frac{(1-\alpha_i-\beta_i)}{N}\sum_{j}\frac{\alpha_j}{\alpha_j+\beta_j}\Theta\left(\frac{\beta_j}{\alpha_j+\beta_j}-{\epsilon}\right)\\
\frac{d\beta_i(s)}{ds}=-\rho\frac{\beta_i}{\alpha_i+\beta_i}\Theta\left(\frac{\alpha_i}{\alpha_i+\beta_i}-\epsilon\right)+\frac{(1-\alpha_i-\beta_i)}{N}\sum_{j}\frac{\beta_j}{\alpha_j+\beta_j}\Theta\left(\frac{\alpha_j}{\alpha_j+\beta_j}-{\epsilon}\right)\, .
\end{cases}$$
The average magnetization {#appC}
=========================
To better analyze the system and in particular be able to describe the dependence of the average magnetization on $\epsilon$, we introduce a new set of variables, the [*local emptiness*]{}, $\gamma_i=1-\alpha_i-\beta_i$, and the [*local difference of fractions*]{} of $A$ and $B$, $\zeta_i=\alpha_i-\beta_i$. The original coordinates can be obtained back using $\alpha_i=(1-\gamma_i+\zeta_i)/2$ and $\beta_i=(1-\gamma_i-\zeta_i)/2$. The new variables ranges are $\zeta_i\in[-1,1]$ and $\gamma_i\in[0,1]$ and the magnetization rewrites as $$\label{eq:magnezg}
\langle \mu\rangle =\frac{1}{N}\sum_i\frac{|\zeta_i|}{1-\gamma_i}\, .$$
Let us introduce the functions $$\label{eq:FG}
F(x):=x\Theta(1-x-\epsilon)+(1-x)\Theta(x-\epsilon)\text{ and }G(x):=x\Theta(1-x-\epsilon)-(1-x)\Theta(x-\epsilon)\, ,$$ then observing that $$\frac{\alpha_i}{\alpha_i+\beta_i}=\frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)} \text{ and }\frac{\beta_i}{\alpha_i+\beta_i}=\frac{1}{2}-\frac{\zeta_i}{2(1-\gamma_i)}\, ,$$ we can rewrite Eq. as follows $$\label{eq:zetagammat}
\begin{cases}
\frac{d\gamma_i(s)}{ds}=\rho F\left(\frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}\right)-\frac{\gamma_i}{N}\sum_jF\left(\frac{1}{2}+\frac{\zeta_j}{2(1-\gamma_j)}\right)\\
\frac{d\zeta_i(s)}{ds}=-\rho G\left(\frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}\right)+\frac{\gamma_i}{N}\sum_jG\left(\frac{1}{2}+\frac{\zeta_j}{2(1-\gamma_j)}\right)\, .
\end{cases}$$
To qualitatively study the solutions of the latter system we define four zones in the $(\zeta,\gamma)$ plane as follows (see Fig. 3 main text): $$\begin{aligned}
Z_1&=&\{(\zeta,\gamma)\in[-1,1]\times [0,1]: \frac{1}{2}-\frac{\zeta_i}{2(1-\gamma_i)}-\epsilon \geq 0 \text{ and } \frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}-\epsilon < 0\}\\
Z_2&=&\{(\zeta,\gamma)\in[-1,1]\times [0,1]: \frac{1}{2}-\frac{\zeta_i}{2(1-\gamma_i)}-\epsilon \geq 0 \text{ and } \frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}-\epsilon \geq 0\}\\
Z_3&=&\{(\zeta,\gamma)\in[-1,1]\times [0,1]: \frac{1}{2}-\frac{\zeta_i}{2(1-\gamma_i)}-\epsilon < 0 \text{ and } \frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}-\epsilon \geq 0\}\\
Z_4&=&\{(\zeta,\gamma)\in[-1,1]\times [0,1]: \frac{1}{2}-\frac{\zeta_i}{2(1-\gamma_i)}-\epsilon < 0 \text{ and } \frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}-\epsilon < 0\}\, .\end{aligned}$$ Let us observe that if $\epsilon<1/2$ then $Z_4$ is empty and if $\epsilon>1/2$ then $Z_2$ is empty.
The initial condition of the Shelling model presented in the main text translate into $(\zeta_i,\gamma_i)$ distributed close to $(0,\rho)$. We will thus be interested in the behavior of the solutions of Eq. in the $Z_2$ zone if $\epsilon<1/2$ and in the $Z_4$ zone if $\epsilon>1/2$.
Let us consider the case $\epsilon<1/2$, then one can easily prove that for all $(\zeta,\gamma)\in Z_2$ one has $F\left(\frac{1}{2}+\frac{\zeta}{2(1-\gamma)}\right)=1$ and $G\left(\frac{1}{2}+\frac{\zeta}{2(1-\gamma)}\right)=\frac{\zeta}{1-\gamma}$, so assuming that for all $i$ one has $(\zeta_i(0),\gamma_i(0))\in Z_2$, then: $$\label{eq:zetagammatZ2}
\begin{cases}
\frac{d\gamma_i(s)}{ds}=\rho -\gamma_i\\
\frac{d\zeta_i(s)}{ds}=-\rho \frac{\zeta_i}{1-\gamma_i}+\frac{\gamma_i}{N}\sum_j \frac{\zeta_j}{1-\gamma_j}\, ,
\end{cases}$$ as long as $(\zeta_i(t),\gamma_i(t))$ will not leave $Z_2$. Assume for a while this statement, then the first equation can be straightforwardly solved to give $\gamma_i(t)=\rho+e^{-t}(\gamma_i(0)-\rho)$, that is for all $i$, $\gamma_i(t)\rightarrow \rho$ when $t\rightarrow\infty$.
The solution for the second is more cumbersome, but one prove the existence of a lower (upper) solution $\zeta_i^-(t)$ ($\zeta_i^+(t)$) such that $\zeta_i^-(t)<\zeta_i(t)<\zeta_i^+(t)$ where: $$\label{eq:explsol}
\zeta^{\pm}_i(t)=\left(\frac{e^{-t}(1-\gamma_i(0))}{1-\rho-e^{-t}(\gamma_i(0)-\rho)}\right)^{\frac{\rho}{1-\rho}}\left[\zeta_i(0)\mp(1-2\epsilon)\left(\frac{\gamma_i(0)-1}{1-\gamma_i(0)}\right)^{\frac{\rho}{1-\rho}}\int_{\gamma_i(0)}^{\rho+e^{-t}(\gamma_i(0)-\rho)}\frac{x(1-x)^{\frac{\rho}{1-\rho}}}{(x-\rho)^{\frac{\rho}{1-\rho}}}dx\right]\, ,$$ from which one can get $\zeta_i(t)\rightarrow 0$ for all $i$ when $t\rightarrow 0$.
Because the point $(0,\rho)$ belongs to $Z_2$ we have [*a posteriori proved*]{} that $(\zeta_i(t),\gamma_i(t))$ will never leave $Z_2$. A simpler but also weaker statement can be obtained by observing that the equilibrium $(\zeta_i,\gamma_i)=(0,\rho)$ for all $i$ is stable being its eigenvalues $-\rho/(1-\rho)$ and $-1$, each one with multiplicity $N$, thus there exists a neighborhood of $(0,\rho)$ such that all orbits whose initial conditions are inside never leave it. Using the definition of $\langle \mu\rangle$ given by Eq. , we have thus proven that for $\epsilon<0.5$ the average magnetization converges asymptotically to $0$ (see Fig. 4 main text).
Assuming now $\epsilon>1/2$, then one can easily prove that for all $(\zeta,\gamma)\in Z_4$ one has $F\left(\frac{1}{2}+\frac{\zeta}{2(1-\gamma)}\right)=G\left(\frac{1}{2}+\frac{\zeta}{2(1-\gamma)}\right)=0$, so assuming that for all $i$ one has $(\zeta_i(0),\gamma_i(0))\in Z_4$, then: $$\label{eq:zetagammatZ4}
\begin{cases}
\frac{d\gamma_i(s)}{ds}=0\\
\frac{d\zeta_i(s)}{ds}=0\, ,
\end{cases}$$ and thus $(\zeta_i(t),\gamma_i(t))$ will not evolve and thus remain in $Z_4$. However fixed the size of the domain centred at $(\zeta,\gamma)=(0,\rho)$, where initial conditions are taken, the above assumption cannot be satisfied if $\epsilon >0.5$ but close to it. Indeed looking at the right panel of Fig. 3 in the main text, we realize that the zone $Z_4$ shrinks to zero as $\epsilon\rightarrow 0.5$ from above. The aim of the rest of this section is to prove that we can take this fact into account and explain the behavior of $\langle \mu\rangle$ also for $\epsilon >0.5$.
As already remarked the functions $F$ and $G$ vanish for $(\zeta,\gamma)\in Z_4$, thus Eq. can be rewritten as: $$\label{eq:zetagammat2}
\begin{cases}
\frac{d\gamma_i(s)}{ds}=\rho F\left(\frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}\right)-\frac{\gamma_i}{N}\left[\sum_{j\in Z_3}F\left(\frac{1}{2}+\frac{\zeta_j}{2(1-\gamma_j)}\right)+\sum_{j\in Z_1}F\left(\frac{1}{2}+\frac{\zeta_j}{2(1-\gamma_j)}\right)\right]\\
\frac{d\zeta_i(s)}{ds}=-\rho G\left(\frac{1}{2}+\frac{\zeta_i}{2(1-\gamma_i)}\right)+\frac{\gamma_i}{N}\left[\sum_{j\in Z_3}G\left(\frac{1}{2}+\frac{\zeta_j}{2(1-\gamma_j)}\right)+\sum_{j\in Z_1}G\left(\frac{1}{2}+\frac{\zeta_j}{2(1-\gamma_j)}\right)\right]\, ,
\end{cases}$$ where we used the shortened notation $j\in Z_k$, to mean $(\zeta_j,\gamma_j)\in Z_k$ and where the zones $Z_1$ and $Z_3$ have been defined in the Fig. 3 of the main text.
The initialization of the metapopulation model, translates into original variables $(\alpha_i,\beta_i)$ initially uniformly randomly distributed in $(1-\rho)/2+\delta U[-1,1]$, for some positive small $\delta$. Such domain is distorted passing to the new variables $(zeta_i,\gamma_i)$, in particular it is translated into $(0,\rho)$, rotated by $45^{\circ}$ and expanded by a factor $\sqrt{2}$ in both directions (see left panel of Fig. \[dynZ1Z3\]).
Let us observe that $F(0)=G(0)=F(1)=G(1)=0$ and thus Eqs. admit as equilibrium points $\zeta_i=\pm (1-\gamma_i)$. In conclusion, initial conditions inside $Z_4$ will remain there while orbits originated from initial conditions in $Z_1$ and $Z_3$ will converge somewhere onto the straight lines $\zeta=\pm(1-\gamma)$. In the left panel of Fig. \[dynZ1Z3\] we report $5000$ initial conditions built using the above described procedure for $\delta=0.02$, points in $Z_1$ are marked in green, points in $Z_4$ in blue and points in $Z_3$ in red. The Eqs. are then numerically solved and the asymptotic positions are drawn in the right panel of Fig. \[dynZ1Z3\], giving to any points the color corresponding to the zone from where is started. One can clearly observe that all blue points remain in the $Z_4$ zone, while green (respectively red) points converge to $\zeta=\gamma-1$ (respectively to $\zeta=1-\gamma$ ).
![\[dynZ1Z3\] Dynamics for $\epsilon>0.5$. Left panel: $5000$ initial conditions are uniformly drawn $(\alpha_i,\beta_i)\in(1-\rho)/2+\delta U[-1,1]$ and then transformed into the new variables $(\zeta_i,\gamma_i)$; points in $Z_1$ are colored in green, points in $Z_4$ in blue and points in $Z_3$ in red. Right panel: asymptotic configuration of the orbits corresponding to the initial conditions given in the left panel, each point is drawn using its initial color. The black solid line is the straight line $\zeta=(2\epsilon -1)(1-\gamma)$, the dot-dashed line is the straight line $\zeta=(1-2\epsilon)(1-\gamma)$ and the dashed lines (right panel) are the straight lines $\zeta=\pm(1-\gamma)$.](Schelling26042015_001InicCond "fig:"){width="45.00000%"}![\[dynZ1Z3\] Dynamics for $\epsilon>0.5$. Left panel: $5000$ initial conditions are uniformly drawn $(\alpha_i,\beta_i)\in(1-\rho)/2+\delta U[-1,1]$ and then transformed into the new variables $(\zeta_i,\gamma_i)$; points in $Z_1$ are colored in green, points in $Z_4$ in blue and points in $Z_3$ in red. Right panel: asymptotic configuration of the orbits corresponding to the initial conditions given in the left panel, each point is drawn using its initial color. The black solid line is the straight line $\zeta=(2\epsilon -1)(1-\gamma)$, the dot-dashed line is the straight line $\zeta=(1-2\epsilon)(1-\gamma)$ and the dashed lines (right panel) are the straight lines $\zeta=\pm(1-\gamma)$.](Schelling26042015_001FinalState "fig:"){width="45.00000%"}
We are now able to provide a good approximation of the average magnetization. First of all let us rewrite Eq. as: $$\langle \mu\rangle =\frac{1}{N}\left(\sum_{i\in Z_4}\frac{|\zeta_i|}{1-\gamma_i}+\sum_{i\in Z_1\cup Z_3}\frac{|\zeta_i|}{1-\gamma_i}\right)=\frac{N_4}{N}\frac{1}{N_4}\sum_{i\in Z_4}\frac{|\zeta_i|}{1-\gamma_i}+\frac{N_{13}}{N}\frac{1}{N_{13}}\sum_{i\in Z_1\cup Z_3}\frac{|\zeta_i|}{1-\gamma_i}\, ,$$ where $N_4$ is the number of points in the $Z_4$ zone and $N_{13}$ the total number of points in the $Z_1$ and $Z_3$ zones. Because points in $Z_1$ and $Z_3$ converge to $\zeta=\pm(1-\gamma)$ the right most term in the previous equation is trivially equal to $N_{13}/N$.
To compute the contribution arising from points in $Z_4$ is more cumbersome, let us hereby stress the main ideas. One can easily show that if $\epsilon >0.5+\delta/(1-\rho)$ then the lines $\zeta=(2\epsilon -1)(1-\gamma)$ and $\zeta=(1-2\epsilon)(1-\gamma)$ do not intersect the diamond like domain (see left panel Fig. \[domains\]) and thus all points are initially in the $Z_4$ zones, this means that $N_4=N$ and $N_{13}=0$ so the magnetization has only a contribution from the $Z_4$ zone. On the contrary if $\epsilon <0.5+\delta/(1-\rho)$ some initial conditions are in the $Z_1$ and $Z_3$ zones (see right panel Fig. \[domains\]) and thus the magnetization has both contributions.
![\[domains\] The geometries used to compute an approximation to the average magnetization in the case $\epsilon >0.5$.](Domains){width="80.00000%"}
In the case $\epsilon >0.5+\delta/(1-\rho)$ one can estimate the contribution to the magnetization as: $$\label{eq:muint1}
\mu_{int,1}=\frac{1}{N_4}\sum_{i\in Z_4}\frac{|\zeta_i|}{1-\gamma_i}\sim\frac{2}{8\delta^2}\int_0^{2\delta}d\zeta\, \zeta\int_{\zeta+\rho-2\delta}^{-\zeta+\rho+2\delta}d\gamma\frac{1}{1-\gamma}=\frac{1}{4\delta^2}\int_0^{2\delta}d\zeta\, \zeta \log\frac{1-\zeta-\rho+2\delta}{1+\zeta-\rho-2\delta}\, ,$$ being $8\delta^2$ the measure of the blue diamond domain in the left panel of Fig. \[domains\]. While for $\epsilon <0.5+\delta/(1-\rho)$ one can find the following estimation $$\label{eq:muint2}
\mu_{int,2}=\frac{1}{N_4}\sum_{i\in Z_4}\frac{|\zeta_i|}{1-\gamma_i}\sim\frac{2}{\mathcal{P}}\int_0^{(\zeta_P+\zeta_Q)/2}d\zeta\, \zeta\int_{\zeta+\rho-2\delta}^{-\zeta+\rho+2\delta}d\gamma\frac{1}{1-\gamma}=\frac{2}{\mathcal{P}}\int_0^{(\zeta_P+\zeta_Q)/2}d\zeta\, \zeta \log\frac{1-\zeta-\rho+2\delta}{1+\zeta-\rho-2\delta}\, ,$$ where $\mathcal{P}$ is the measure of the polygon $TPQRQ'P'$ (in blue in the right hand side of Fig. \[domains\]). Let us observe that in the last integral we make the approximation that points $P$ and $Q$ can be replaced by an average point whose $\zeta$ coordinate is the average of the ones for $P$ and $Q$; this is a minor assumption that helps to compute the integral and will not influence the final result as show below. Now both integrals can be exactly computed.
To get the final estimate for $\langle \mu\rangle$ we need to compute $N_4/N$ and $N_{13}/N$ in the case $\epsilon <0.5+\delta/(1-\rho)$, the other case being trivial and already considered. Assuming the number of points sufficiently large we can affirm that such fractions are well approximated by the ratio of the corresponding polygons, that is $$\frac{N_4}{N}\sim \frac{\mathcal{P}}{8\delta^2}\quad\text{and}\quad \frac{N_{13}}{N}\sim 1-\frac{\mathcal{P}}{8\delta^2}\, .$$
In conclusion we can approximate the average magnetization for all $\epsilon>0.5$ with the formula: $$\label{eq:muapprox}
\langle \mu \rangle^{approx} = \frac{\mathcal{P}}{8\delta^2} \mu_{int,k}+ (1-\frac{\mathcal{P}}{8\delta^2})\, ,$$ where $\mu_{int,1}$ is given by Eq. valid for $\epsilon >0.5+\delta/(1-\rho)$, and $\mu_{int,2}$ by Eq. for the case $\epsilon <0.5+\delta/(1-\rho)$.
In Fig. \[comparemu\] we compare the average magnetization obtained using its very first definition Eq. and the numerical integration of the system , with the approximation given by Eq. for several values of $\delta$ and $\rho=0.9$. One can observe the very good agreement, providing thus an a posteriori validation of the assumptions made so far. Comparing with Fig. 4 of the main text, this provide also a very good approximation to the average magnetization computed using the metapopulation model.
![\[comparemu\]Averaged magnetization $\langle \mu\rangle$ and the approximation $\langle \mu \rangle^{approx}$ as a function of $\epsilon >0.5$. Symbols correspond to numerical integrations of the system and the use of the definition for the magnetization, lines to the calculation of the approximation Eq. . Blue circles correspond to $\delta=0.02$, black squares to $\delta=0.01$ and red diamonds to $\delta=0.005$. Each point correspond to the average over $50$ replicas. The emptiness has been fixed to $\rho=0.9$.](EpsVsMu27042015){width="80.00000%"}
[^1]: To make things simple we assumed a constant carrying capacity for each node, but of course one can improve the mode by considering a different value of $L_i$ for each node $i$.
[^2]: Excluding the case where there is not strict majority in a node, this definition can be restated as: $i$ and $j$ belong to the same cluster if $(n^A_i-n^B_i)(n^A_j-n^B_j)>0$.
[^3]: Let us observe that $\langle n^A_i\rangle/(\langle n^A_i\rangle+\langle n^B_i\rangle)-\epsilon$ is positive if and only if $\alpha_i/(\alpha_i+\beta_i)-{\epsilon}>0$, and similarly for the other term.
|
---
abstract: 'Autoencoders have been successful in learning meaningful representations from image datasets. However, their performance on text datasets has not been widely studied. Traditional autoencoders tend to learn possibly trivial representations of text documents due to their confounding properties such as high-dimensionality, sparsity and power-law word distributions. In this paper, we propose a novel k-competitive autoencoder, called [<span style="font-variant:small-caps;">KATE</span>]{}, for text documents. Due to the competition between the neurons in the hidden layer, each neuron becomes specialized in recognizing specific data patterns, and overall the model can learn meaningful representations of textual data. A comprehensive set of experiments show that [<span style="font-variant:small-caps;">KATE</span>]{}can learn better representations than traditional autoencoders including denoising, contractive, variational, and k-sparse autoencoders. Our model also outperforms deep generative models, probabilistic topic models, and even word representation models (e.g., Word2Vec) in terms of several downstream tasks such as document classification, regression, and retrieval.'
author:
- Yu Chen
- 'Mohammed J. Zaki'
bibliography:
- 'reports.bib'
title: 'KATE: K-Competitive Autoencoder for Text'
---
<ccs2012> <concept> <concept\_id>10002951.10003227.10003351</concept\_id> <concept\_desc>Information systems Data mining</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003318</concept\_id> <concept\_desc>Information systems Document representation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003347</concept\_id> <concept\_desc>Information systems Retrieval tasks and goals</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010178.10010179.10003352</concept\_id> <concept\_desc>Computing methodologies Information extraction</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
Introduction
============
An autoencoder is a neural network which can automatically learn data representations by trying to reconstruct its input at the output layer. Many variants of autoencoders have been proposed recently [@vincent2010stacked; @rifai2011contractive; @kingma2013auto; @makhzani2015adversarial; @makhzani2013k; @makhzani2015winner]. While autoencoders have been successfully applied to learn meaningful representations on image datasets (e.g., MNIST [@lecun1998gradient], CIFAR-10 [@krizhevsky2009learning]), their performance on text datasets has not been widely studied. Traditional autoencoders are susceptible to learning trivial representations for text documents. As noted by Zhai and Zhang [@zhai2015semisupervised], the reasons include that fact that textual data is extremely high dimensional and sparse. The vocabulary size can be hundreds of thousands while the average fraction of zero entries in the document vectors can be very high (e.g., 98%). Further, textual data typically follows power-law word distributions. That is, low-frequency words account for most of the word occurrences. Traditional autoencoders always try to reconstruct each dimension of the input vector on an equal footing, which is not quite appropriate for textual data.
Document representation is an interesting and challenging task which is concerned with representing textual documents in a vector space, and it has various applications in text processing, retrieval and mining. There are two major approaches to represent documents: 1) [ **Distributional Representation**]{} is based on the hypothesis that linguistic terms with similar distributions have similar meanings. These methods usually take advantage of the co-occurrence and context information of words and documents, and each dimension of the document vector usually represents a specific semantic meaning (e.g., a topic). Typical models in this category include Latent Semantic Analysis (LSA) [@deerwester1990indexing], probabilistic LSA (pLSA) [@hofmann1999probabilistic] and Latent Dirichlet Allocation (LDA) [@blei2003latent]. 2) [**Distributed Representations**]{} encode a document as a compact, dense and lower dimensional vector with the semantic meaning of the document distributed along the dimensions of the vector. Many neural network-based distributed representation models [@larochelle2012neural; @srivastava2013modeling; @maaloe2015deep; @cao2015novel; @miao2015neural] have been proposed and shown to be able to learn better representations of documents than distributional representation models.
In this paper, we try to overcome the weaknesses of traditional autoencoders when applied to textual data. We propose a novel autoencoder called [<span style="font-variant:small-caps;">KATE</span>]{}(for [**K**]{}-competitive [ **A**]{}utoencoder for [**TE**]{}xt), which relies on competitive learning among the autoencoding neurons. In the feedforward phase, only the most competitive $k$ neurons in the layer fire and those $k$ “winners” further incorporate the aggregate activation potential of the remaining inactive neurons. As a result, each hidden neuron becomes better at recognizing specific data patterns and the overall model can learn meaningful representations of the input data. After training the model, each hidden neuron is distinct from the others and no competition is needed in the testing/encoding phase. We conduct comprehensive experiments qualitatively and quantitatively to evaluate [<span style="font-variant:small-caps;">KATE</span>]{}and to demonstrate the effectiveness of our model. We compare [<span style="font-variant:small-caps;">KATE</span>]{}with traditional autoencoders including basic autoencoder, denoising autoencoder [@vincent2010stacked], contractive autoencoder [@rifai2011contractive], variational autoencoder [@kingma2013auto], and k-sparse autoencoder [@makhzani2013k]. We also compare with deep generative models [@maaloe2015deep], neural autoregressive [@larochelle2012neural] and variational inference [@miao2015neural] models, probabilistic topic models such as LDA [@blei2003latent], and word representation models such as Word2Vec [@mikolov2013distributed] and Doc2Vec [@le2014distributed]. [<span style="font-variant:small-caps;">KATE</span>]{}achieves state-of-the-art performance across various datasets on several downstream tasks like document classification, regression and retrieval.
Related Work
============
**Autoencoders.** The basic autoencoder is a shallow neural network which tries to reconstruct its input at the output layer. An autoencoder consists of an encoder which maps the input ${{\boldsymbol{x}}}$ to the hidden layer: ${{\boldsymbol{z}}}= g({{\boldsymbol{W}}}{{\boldsymbol{x}}}+ {{\boldsymbol{b}}})$ and a decoder which reconstructs the input as: ${\hat{{{\boldsymbol{x}}}}}= o({{\boldsymbol{W}}}'{{\boldsymbol{z}}}+ {{\boldsymbol{c}}})$; here ${{\boldsymbol{b}}}$ and ${{\boldsymbol{c}}}$ are bias terms, ${{\boldsymbol{W}}}$ and ${{\boldsymbol{W}}}'$ are input-to-hidden and hidden-to-output layer weight matrices, and $g$ and $o$ are activation functions. Weight tying (i.e., setting ${{\boldsymbol{W}}}' = {{\boldsymbol{W}}}^T$) is often used as a regularization method to avoid overfitting. While plain autoencoders, even with perfect reconstructions, usually only extract trivial representations of the data, more meaningful representations can be obtained by adding appropriate regularization to the models. Following this line of reasoning, many variants of autoencoders have been proposed recently [@vincent2010stacked; @rifai2011contractive; @kingma2013auto; @makhzani2015adversarial; @makhzani2013k; @makhzani2015winner]. The denoising autoencoder (DAE) [@vincent2010stacked] inputs a corrupted version of the data while the output is still compared with the original uncorrupted data, allowing the model to learn patterns useful for denoising. The contractive autoencoder (CAE) [@rifai2011contractive] introduces the Frobenius norm of the Jacobian matrix of the encoder activations into the regularization term. When the Frobenius norm is 0, the model is extremely invariant to perturbations of input data, which is thought as good. The variational autoencoder (VAE) [@kingma2013auto] is a generative model inspired by variational inference whose encoder $q_\phi({{\boldsymbol{z}}}|{{\boldsymbol{x}}})$ approximates the intractable true posterior $p_\theta({{\boldsymbol{z}}}|{{\boldsymbol{x}}})$, and the decoder $p_\theta({{\boldsymbol{x}}}|{{\boldsymbol{z}}})$ is a data generator. The k-sparse autoencoder (KSAE) [@makhzani2013k] explicitly enforces sparsity by only keeping the $k$ highest activities in the feedforward phase.
We notice that most of the successful applications of autoencoders are on image data, while only a few have attempted to apply autoencoders on textual data. Zhai and Zhang [@zhai2015semisupervised] have argued that traditional autoencoders, which perform well on image data, are less appropriate for modeling textual data due to the problems of high-dimensionality, sparsity and power-law word distributions. They proposed a semi-supervised autoencoder which applies a weighted loss function where the weights are learned by a linear classifier to overcome some of these problems. Kumar and D’Haro [@kumar2015deepae] found that all the topics extracted from the autoencoder were dominated by the most frequent words due to the sparsity of the input document vectors. Further, they found that adding sparsity and selectivity penalty terms helped alleviate this issue to some extent.
**Deep generative models.** Deep Belief Networks (DBNs) are probabilistic graphical models which learn to extract a deep hierarchical representation of the data. The top 2 layers of DBNs form a Restricted Boltzmann Machine (RBM) and other layers form a sigmoid belief network. A relatively fast greedy layer-wise pre-training algorithm [@hinton2006fast; @hinton2006reducing] is applied to train the model. Maaloe et al. [@maaloe2015deep] showed that DBNs can be competitive as a topic model. DocNADE [@larochelle2012neural] is a neural autoregressive topic model that estimates the probability of observing a new word in a given document given the previously observed words. It can be used for extracting meaningful representations of documents. It has been shown to outperform the Replicated Softmax model [@hinton2009replicated] which is a variant of RBMs for document modeling. Srivastava et al. [@srivastava2013modeling] introduced a type of Deep Boltzmann Machine (DBM) that is suitable for extracting distributed semantic representations from a corpus of documents; an Over-Replicated Softmax model was proposed to overcome the apparent difficulty of training a DBM. NVDM [@miao2015neural] is a neural variational inference model for document modeling inspired by the variational autoencoder.
**Probabilistic topic models.** Probabilistic topic models, such as probabilistic Latent Semantic Analysis (pLSA) and Latent Dirichlet Allocation (LDA) have been extensively studied [@hofmann1999probabilistic; @blei2003latent]. Especially for LDA, many variants have been proposed for non-parametric learning [@teh2012hierarchical; @blei2010nested], sparsity [@wang2009decoupling; @eisenstein2011sparse; @zhu2012sparse] and efficient inference [@teh2007collapsed; @canini2009online]. Those models typically build a generative probabilistic model using the bag-of-words representation of the documents.
**Word representation models.** Distributed representations of words in a vector space can capture semantic meanings of words and help achieve better results in various downstream text analysis tasks. Word2Vec [@mikolov2013distributed] and Glove [@pennington2014glove] are state-of-the-art word representation models. Pre-training word embeddings on a large corpus of documents and applying learned word embeddings in downstream tasks has been shown to work well in practice [@kim2014convolutional; @das2015gaussian; @nguyen2015improving]. Doc2Vec [@le2014distributed] was inspired by Word2Vec and can directly learn vector representations of paragraphs and documents. NTM [@cao2015novel], which also uses pre-trained word embeddings, is a neural topic model where the representations of words and documents are combined into a uniform framework.
With this brief overview of existing work, we now turn to our competitive autoencoder approach for text documents.
K-Competitive Autoencoder
=========================
Although the objective of an autoencoder is to minimize the reconstruction error, our goal is to extract meaningful features from data. Compared with image data, textual data is more challenging for autoencoders since it is typically high-dimensional, sparse and has power-law word distributions. When examining the features extracted by an autoencoder, we observed that they were not distinct from one another. That is, many neurons in the hidden layer shared similar groups of input neurons (which typically correspond to the most frequent words) with whom they had the strongest connections. We hypothesized that the autoencoder greedily learned relatively trivial features in order to reconstruct the input.
To overcome this drawback, our approach guides the autoencoder to focus on important patterns in the data by adding constraints in the training phase via mutual competition. In competitive learning, neurons compete for the right to respond to a subset of the input data and as a result, the specialization of each neuron in the network is increased. Note that the specialization of neurons is exactly what we want for an autoencoder, especially when applied on textual data. By introducing competition into an autoencoder, we expect each neuron in the hidden layer to take responsibility for recognizing different patterns within the input data. Following this line of reasoning, we propose the k-competitive autoencoder, [<span style="font-variant:small-caps;">KATE</span>]{}, as described below.
Training and Testing/Encoding
-----------------------------
Feedforward step: ${{\boldsymbol{z}}}= tanh({{\boldsymbol{W}}}{{\boldsymbol{x}}}+ {{\boldsymbol{b}}})$ Apply k-competition: ${\hat{{{\boldsymbol{z}}}}}= \text{k-competitive\_layer}({{\boldsymbol{z}}})$ Compute output: ${\hat{{{\boldsymbol{x}}}}}= sigmoid({{\boldsymbol{W}}}^T{\hat{{{\boldsymbol{z}}}}}+ {{\boldsymbol{c}}})$ Backpropagate error (cross-entropy) and iterate
[ ]{} Encode input data: ${{\boldsymbol{z}}}= tanh({{\boldsymbol{W}}}{{\boldsymbol{x}}}+ {{\boldsymbol{b}}})$
The pseudo-code for our k-competitive autoencoder [<span style="font-variant:small-caps;">KATE</span>]{}is shown in Algorithm \[alg:kcae\]. [<span style="font-variant:small-caps;">KATE</span>]{}is a shallow autoencoder with a (single) competitive hidden layer, with each neuron competing for the right to respond to a given set of input patterns. Let ${{\boldsymbol{x}}}\in {{\mathord{\mathbb R}}}^d$ be a $d$-dimensional input vector, which is also the desired output vector, and let $h_1, h_2, ..., h_m$ be the $m$ hidden neurons. Let ${{\boldsymbol{W}}}\in {{\mathord{\mathbb R}}}^{d \times m}$ be the weight matrix linking the input layer to the hidden layer neurons, and let ${{\boldsymbol{b}}}\in{{\mathord{\mathbb R}}}^m$ and ${{\boldsymbol{c}}}\in {{\mathord{\mathbb R}}}^d$ be the bias terms for the hidden and output neurons, respectively. Let $g$ be an activation function; two typical functions are $tanh(x) = \frac{e^{2x}-1}{e^{2x}+1}$ and $sigmoid(x) = \frac{1}{1+e^{-x}}$. In each feed-forward step, the activation potential at the hidden neurons is then given as ${{\boldsymbol{z}}}= g({{\boldsymbol{W}}}{{\boldsymbol{x}}}+{{\boldsymbol{b}}})$, whereas the activation potential at the output neurons is given as ${\hat{{{\boldsymbol{x}}}}}= g({{\boldsymbol{W}}}^T{{\boldsymbol{z}}}+{{\boldsymbol{c}}})$. Thus, the hidden-to-output weight matrix is simply ${{\boldsymbol{W}}}^T$, being an instance of [*weight tying*]{}.
In [<span style="font-variant:small-caps;">KATE</span>]{}, we represent each input text document as a log-normalized word count vector ${{\boldsymbol{x}}}\in {{\mathord{\mathbb R}}}^d$ where each dimension is represented as $$x_i = \frac{log (1 + n_i)}{\max_{i \in V}{log (1+ n_i)}}, \text{ for } i \in V$$ where $V$ is the vocabulary and $n_i$ is the count of word $i$ in that document. Let ${\hat{{{\boldsymbol{x}}}}}$ be the output of [<span style="font-variant:small-caps;">KATE</span>]{}on a given input ${{\boldsymbol{x}}}$. We use the binary cross-entropy as the loss function, which is defined as $$l({{\boldsymbol{x}}}, {\hat{{{\boldsymbol{x}}}}}) = -\sum_{i \in V}{x_i log({\hat{x}}_i) + (1-x_i) log(1-{\hat{x}}_i)}$$ where ${\hat{x}}_i$ is the reconstructed value for $x_i$.
Let $H$ be some subset of hidden neurons; define the [**energy**]{} of $H$ as the total activation potential for $H$, given as: $E(H) = \sum_{h_i \in H} |z_i|$, i.e., sum of the absolute values of the activations for neurons in $H$. In [<span style="font-variant:small-caps;">KATE</span>]{}, in the feedforward phase, after computing the activations ${{\boldsymbol{z}}}$ for a given input ${{\boldsymbol{x}}}$, we select the most competitive $k$ neurons as the “winners” while the remaining “losers” are suppressed (i.e., made inactive). However, in order to compensate for the loss of energy from the loser neurons, and to make the competition among neurons more pronounced, we amplify and reallocate that energy among the winner neurons.
sort positive neurons in ascending order $z^+_1 ... z^+_P$ sort negative neurons in descending order $z^-_1 ... z^-_N$
$E_{pos} = \sum_{i=1}^{P - \lceil k/2 \rceil}{z^+_i}$ `z^+_i := z^+_i + \alpha \cdot E_{pos}`
`z^+_i := 0`
$E_{neg} = \sum_{i=1}^{N - \lfloor k/2 \rfloor}{z^-_i}$ `z^-_i := z^-_i + \alpha \cdot E_{neg}`
`z^-_i := 0`
updated $z^+_1...z^+_P, z^-_1 ... z^-_N$
in [1,missing,2]{} (input-) at (-1,-1.25) ;
in [1,2, 3, 4, 5,6]{} (hidden-) at (2,2-1.25) ;
in [1,2]{}[ (adder-) at (1,-1-1.25) ; (1,-1-1.25) node\[cross=5pt,rotate=45\]; ]{}
in [1,missing,2]{} (output-) at (4,-1.25) ;
at (input-1.north) [$i_1$]{}; at (input-2.north) [$i_d$]{};
at (output-1.north) [$o_1$]{}; at (output-2.north) [$o_d$]{};
at (hidden-1.north) [$h_1$]{}; at (hidden-2.north) [$h_2$]{}; at (hidden-3.north) [$h_3$]{}; at (hidden-4.north) [$h_4$]{}; at (hidden-5.north) [$h_5$]{}; at (hidden-6.north) [$h_6$]{};
at (hidden-1.east) [0.8 $+0.3\alpha$]{}; at (hidden-2.east) [0.2 $\rightarrow 0$]{}; at (hidden-3.east) [0.1 $\rightarrow 0$]{}; at (hidden-4.east) [-0.1 $\rightarrow 0$]{}; at (hidden-5.east) [-0.3 $\rightarrow 0$]{}; at (hidden-6.east) [-0.6 $-0.4\alpha$]{};
at (adder-1.west) [$PosAdder$]{}; at (adder-2.west) [$NegAdder$]{}; at (adder-1.north) [0.3]{}; at (adder-2.north) [-0.4]{};
(input-1) – (hidden-1); (input-1) – (hidden-6); (input-2) – (hidden-1); (input-2) – (hidden-6);
(adder-1) – (hidden-1) node \[midway, above\] (TextNode) [$\alpha$]{}; (adder-2) – (hidden-6) node \[midway, below\] (TextNode) [$\beta$]{}; (hidden-2) – (adder-1); (hidden-3) – (adder-1); (hidden-4) – (adder-2); (hidden-5) – (adder-2);
(hidden-1) – (output-1); (hidden-1) – (output-2); (hidden-6) – (output-1); (hidden-6) – (output-2);
; at (1\*-1,-7.2) [Input\
layer]{}; at (1\*2,-7.2) [k-competitive\
layer]{}; at (1\*4,-7.2) [Output\
layer]{};
[<span style="font-variant:small-caps;">KATE</span>]{}uses [*tanh*]{} activation function for the k-competitive hidden layer. We divide these neurons into positive and negative neurons based on their activations. The most competitive $k$ neurons are those that have the largest absolute activation values. However, we select the $\lceil k/2 \rceil$ largest positive activations as the positive winners, and reallocate the energy of the remaining positive loser neurons among the winners using an $\alpha$ amplification connection, where $\alpha$ is a hyperparameter. Finally, we set the activations of all losers to zero. Similarly, the $\lfloor k/2 \rfloor$ lowest negative activations are the negative winners, and they incorporate the amplified energy from the negative loser neurons, as detailed in Algorithm \[alg:kcl\]. We argue that the $\alpha$ amplification connections are a critical component in the k-competitive layer. When $\alpha=0$, no gradients will flow through loser neurons, resulting in a regular k-sparse autoencoder (regardless of the activation functions and k-selection scheme). When $\alpha > 2/k$, we actually boost the gradient signal flowing through the loser neurons. We empirically show that amplification helps improve the autoencoder model (see Sec. \[sec:mscd\] and \[sec:effects\_of\_params\]). As an example, consider Figure \[fig:competition\], which shows an example feedforward step for $k=2$. Here, $h_1$ and $h_6$ are the positive and negative winners, respectively, since the absolute activation potential for $h_1$ is $|z_1| = 0.8$, and for $h_6$ it is $|z_6| = 0.6$. The positive winner $h_1$ takes away the energy from the positive losers $h_2$ and $h_3$, which is $E(\{h_2,h_3\}) = 0.2+0.1 = 0.3$. Likewise, the negative winner $h_6$ takes away the energy from the negative losers $h_4$ and $h_5$, which is $-E(\{h_4, h_5\}) = -(|-0.1| + |-0.3|) = -0.4$. The hyperparameter $\alpha$ governs how the energy from the loser neurons is incorporated into the winner neurons, for both positive and negative cases. That is $h_1$’s net activation becomes $z_1 = 0.8+0.3\alpha$, and $h_6$’s net activation is $z_6 = -0.6-0.4\alpha$. The rest of the neurons are set to zero activation.
Finally, as noted in Algorithm \[alg:kcae\] we use weight tying for the hidden to output layer weights, i.e., we use ${{\boldsymbol{W}}}^T$ as the weight matrix, with different biases ${{\boldsymbol{c}}}$. Also, since the inputs ${{\boldsymbol{x}}}$ are non-negative for document representations (e.g., word counts), we use the [*sigmoid*]{} activation function at the output layer to maintain the non-negativity. Note that in the back-propagation procedure, the gradients will first flow through the winner neurons in the hidden layer and then the loser neurons via the $\alpha$ amplification connections. No gradients will flow directly from the output neurons to the loser neurons since they are made inactive in the feedforward step.
### Testing/Encoding {#testingencoding .unnumbered}
Once the k-competitive network has been trained, we simply encode each test input as shown in Algorithm \[alg:kcae\]. That is, given a test input ${{\boldsymbol{x}}}$, we map it to the feature space to obtain ${{\boldsymbol{z}}}= tanh({{\boldsymbol{W}}}{{\boldsymbol{x}}}+ {{\boldsymbol{b}}})$. No competition is required for the encoding step since the hidden neurons are well trained to be distinctive from others. We argue that this is one of the superior features of [<span style="font-variant:small-caps;">KATE</span>]{}.
Relationship to Other Models
----------------------------
### KATE vs. K-Sparse Autoencoder {#kate-vs.k-sparse-autoencoder .unnumbered}
The k-sparse autoencoder [@makhzani2013k] is closely related to our model, but there are several differences. The k-sparse autoencoder explicitly enforces sparsity by only keeping the $k$ highest activities at training time. Then, at testing time, in order to enforce sparsity, only the $\alpha
k$ highest activities are kept where $\alpha$ is a hyperparameter. Since its hidden layer uses a linear activation function, the only non-linearity in the encoder comes from the selection of the $k$ highest activities. Instead of focusing on sparsity, our model focuses on competition to drive each hidden neuron to be distinct from the others. Thus, at testing time, no competition is needed. The non-linearity in KATE’s encoding comes from the [*tanh*]{} activation function and the winner-take-all operation (i.e., top k selection and amplifying energy reallocation).
It is important to note that for the k-sparse autoencoder, too much sparsity (i.e., low $k$) can cause the so-called “dead” hidden neurons problem, which can prevent gradient back-propagation from adjusting the weights of these “dead” hidden neurons. As mentioned in the original paper, the model is prone to behaving in a manner similar to k-means clustering. That is, in the first few epochs, it will greedily assign individual hidden neurons to groups of training cases and these hidden neurons will be re-enforced but other hidden neurons will not be adjusted in subsequent epochs. In order to address this problem, scheduling the sparsity level over epochs was suggested. However, by design our approach does not suffer from this problem since the gradients will still flow through the loser neurons via the $\alpha$ amplification connections in the k-competitive layer.
### KATE vs. K-Max Pooling {#kate-vs.k-max-pooling .unnumbered}
Our proposed k-competitive operation is also reminiscent of the k-max pooling operation [@blunsom2014convolutional] applied in convolutional neural networks. We can intuitively regard k-max pooling as a global feature sampler which selects a subset of k maximum neurons in the previous convolutional layer and uses only the selected subset of neurons in the following layer. Unlike our k-competitive approach, the objective of k-max pooling is to reduce dimensionality and introduce feature invariance via this downsampling operation.
### KATE as a Regularized Autoencoder {#kate-as-a-regularized-autoencoder .unnumbered}
We can also regard our model as a special case of a fully competitive autoencoder where all the neurons in the hidden layer are fully connected with each other and the weights on the connections between them are fully trainable. The difference is that we restrict the architecture of this competitive layer by using a positive adder and a negative adder to constrain the energy, which serves as a regularization method.
Experiments
===========
In this section, we evaluate our k-competitive autoencoder model on various datasets and downstream text analytics tasks to gauge its effectiveness in learning meaningful representations in different situations. All experiments were performed on a machine with a 1.7GHz AMD Opteron 6272 Processor, with 264G RAM. Our model, [<span style="font-variant:small-caps;">KATE</span>]{}, was implemented in Keras ([github.com/fchollet/keras](github.com/fchollet/keras)) which is a high-level neural networks library, written in Python. The source code for KATE is available at **[github.com/hugochan/KATE](github.com/hugochan/KATE)**.
dataset 20 news reuters wiki10+ mrd
-------------- ------------ --------- --------- ------------
train.size 11,314 554,414 13,972 3,337
test.size 7,532 250,000 6,000 1,669
valid.size 1,000 10,000 1,000 300
vocab.size 2,000 5,000 2,000 2,000
avg.length 93 112 1,299 124
classes/vals 20 103 25 $[0,1]$
task class & DR MLC MLC regression
: Datasets: Tasks include classification (class), regression, multi-label classification (MLC), and document retrieval (DR).[]{data-label="table:datasets"}
Datasets
--------
For evaluation, we use datasets that have been widely used in previous studies [@Lang95; @lewis2004rcv1; @zubiaga2009getting; @pang2002thumbs; @pang2004sentimental; @pang2005seeing]. Table \[table:datasets\] provides statistics of the different datasets used in our experiments. It lists the training, testing and validation (a subset of training) set sizes, the size of the vocabulary, average document length, the number of classes (or values for regression), and the various downstream tasks we perform on the datasets.
The 20 Newsgroups [@Lang95] ([www.qwone.com/\~jason/20Newsgroups](www.qwone.com/~jason/20Newsgroups)) data consists of 18846 documents, which are partitioned (nearly) evenly across 20 different newsgroups. Each document belongs to exactly one newsgroup. The corpus is divided by date into training (60%) and testing (40%) sets. We follow the preprocessing steps utilized in previous work [@larochelle2012neural; @srivastava2013modeling; @miao2015neural]. That is, after removing stopwords and stemming, we keep the most frequent 2,000 words in the training set as the vocabulary. We use this dataset to show that our model can learn meaningful representations for classification and document retrieval tasks.
The Reuters RCV1-v2 dataset [@lewis2004rcv1] ([www.jmlr.org/papers/volume5/lewis04a](www.jmlr.org/papers/volume5/lewis04a)) contains 804,414 newswire articles, where each document typically has multiple (hierarchical) topic labels. The total number of topic labels is 103. The dataset already comes preprocessed with stopword removal and stemming. We randomly split the corpus into 554,414 training and 25,000 test cases and keep the most frequent 5,000 words in the training dataset as the vocabulary. We perform multi-label classification on this dataset.
The Wiki10+ dataset [@zubiaga2009getting] ([www.zubiaga.org/datasets/wiki10+/](www.zubiaga.org/datasets/wiki10+/)) comprises English Wikipedia articles with at least 10 annotations on delicious.com. Following the steps of Cao et al. [@cao2015novel], we only keep the 25 most frequent social tags and those documents containing any of these tags. After removing stopwords and stemming, we randomly split the corpus into 13,972 training and 6,000 test cases and keep the most frequent 2,000 words in the training set as the vocabulary for use in multi-label classification.
The Movie review data (MRD) [@pang2002thumbs; @pang2004sentimental; @pang2005seeing] ([www.cs.cornell.edu/people/pabo/movie-review-data/](www.cs.cornell.edu/people/pabo/movie-review-data/)) contains a collection of movie-review documents, with a numerical rating score in the interval $[0,1]$. After removing stopwords and stemming, we randomly split the corpus into 3,337 training and 1,669 test cases and keep the most frequent 2,000 words in the training set as the vocabulary. We use this dataset for regression, i.e., predicting the movie ratings.
Note that among the above datasets, only the 20 Newsgroups dataset is balanced, whereas both the Reuters and Wiki10+ datasets are highly imbalanced in terms of class labels.
Comparison with Baseline Methods
--------------------------------
We compare our k-competitive autoencoder [<span style="font-variant:small-caps;">KATE</span>]{}with a wide range of other models including various types of autoencoders, topic models, belief networks and word representation models, as listed below.
**LDA** [@blei2003latent]: a directed graphical model which models a document as a mixture of topics and a topic as a mixture of words. Once trained, each document can be represented as a topic proportion vector on the topic simplex. We used the gensim [@rehurek_lrec] LDA implementation in our experiments.
**DocNADE** [@larochelle2012neural]: a neural autoregressive topic model that can be used for extracting meaningful representations of documents. The implementation is available at [www.dmi.usherb.ca/\~larocheh/code/DocNADE.zip](www.dmi.usherb.ca/~larocheh/code/DocNADE.zip).
**DBN** [@maaloe2015deep]: a direct acyclic graph whose top two layers form a restricted Boltzmann machine. We use the implementation available at [github.com/larsmaaloee/deep-belief-nets-for-topic-modeling](github.com/larsmaaloee/deep-belief-nets-for-topic-modeling).
**NVDM** [@miao2015neural]: a neural variational inference model for document modeling. The authors have not released the source code, but we used an open-source implementation at [github.com/carpedm20/variational-text-tensor-flow](github.com/carpedm20/variational-text-tensor-flow).
**Word2Vec** [@mikolov2013distributed]: a model in which each document is represented as the average of the word embedding vectors for that document. We use Word2Vec$_{pre}$ to denote the version where we use Google News pre-trained word embeddings which contain 300-dimensional vectors for 3 million words and phrases. Those embeddings were trained by state-of-the-art word2vec skipgram model. On the other hand, we use Word2Vec to denote the version where we train word embeddings separately on each of our datasets, using the gensim [@rehurek_lrec] implementation.
**Doc2Vec** [@le2014distributed]: a distributed representation model inspired by Word2Vec which can directly learn vector representations of documents. There are two versions named Doc2Vec-DBOW and Doc2Vec-DM. We use Doc2Vec-DM in our experiments as it was reported to consistently outperform Doc2Vec-DBOW in the original paper. We used the gensim [@rehurek_lrec] implementation in our experiments.
**AE**: a plain shallow (i.e., one hidden layer) autoencoder, without any competition, which can automatically learn data representations by trying to reconstruct its input at the output layer.
**DAE** [@vincent2010stacked]: a denoising autoencoder that accepts a corrupted version of the input data while the output is still the original uncorrupted data. In our experiments, we found that masking noise consistently outperforms other two types of noise, namely Gaussian noise and salt-and-pepper noise. Thus, we only report the results of using masking noise. Basically, masking noise perturbs the input by setting a fraction $v$ of the elements $i$ in each input vector as 0. To be fair and consistent, we use a shallow denoising autoencoder in our experiments.
**CAE** [@rifai2011contractive]: a contractive autoencoder which introduces the Frobenius norm of the Jacobian matrix of the encoder activations into the regularization term.
**VAE** [@kingma2013auto]: a generative autoencoder inspired by variational inference.
**KSAE** [@makhzani2013k]: a competitive autoencoder which explicitly enforces sparsity by only keeping the $k$ highest activities in the feedforward phase.
We implemented the AE, DAE, CAE, VAE and KSAE autoencoders on our own, since their implementations are not publicly available. [**Training Details:**]{} For all the autoencoder models (including AE, DAE, CAE, VAE, KSAE, and [<span style="font-variant:small-caps;">KATE</span>]{}), we represent each input document as a log-normalized word count vector, using binary cross-entropy as the loss function and Adadelta [@zeiler2012adadelta] as the optimizer. Weight tying is also applied. For CAE and VAE, additional regularization terms are added to the loss function as mentioned in the original papers. As for VAE, we use [*tanh*]{} as the nonlinear activation function while as for AE, DAE and CAE, [*sigmoid*]{} is applied. As for KSAE, we found that omitting sparsity in the testing phase gave us better results in all experiments. When training models, we randomly extract a subset of documents from the training set as a validation set, as noted in Table \[table:datasets\], which is used for tuning hyperparameters and early stopping. Early stopping is a type of regularization used to avoid overfitting when training an iterative algorithm. We stop training after 5 successive epochs with no improvement on the validation set. All baseline models were optimized as recommended in original sources. For [<span style="font-variant:small-caps;">KATE</span>]{}, we set $\alpha$ as 6.26, learning rate as 2, batch size as 100 (for the Reuters dataset) or 50 (for other datasets) and $k$ as 6 (for the 20 topics case), 32 (for the 128 topics case) or 102 (for the 512 topics case), as determined from the validation set.
Qualitative Analysis
--------------------
In this set of qualitative experiments, we compare the topics generated by [<span style="font-variant:small-caps;">KATE</span>]{}to other representative models including AE, KSAE, and LDA. Even though [<span style="font-variant:small-caps;">KATE</span>]{}is not explicitly designed for the purpose of word embeddings, we compared word representations learned by [<span style="font-variant:small-caps;">KATE</span>]{}with the Word2Vec model to demonstrate that our model can learn semantically meaningful representations from text. We evaluate the above models on the 20 Newsgroups data. Matching the number of classes, the number of topics is set to exactly 20 for all models. For both KSAE and [<span style="font-variant:small-caps;">KATE</span>]{}, $k$ (the sparsity level/number of winning neurons) is set as 6.
-- -------------------------------------------------------- --
AE
KSAE
[<span style="font-variant:small-caps;">KATE</span>]{}
LDA
[<span style="font-variant:small-caps;">KATE</span>]{}
LDA
[<span style="font-variant:small-caps;">KATE</span>]{}
LDA
-- -------------------------------------------------------- --
: Topics learned by various models.[]{data-label="table:topics"}
### Topics Generated by Different Models
Table \[table:topics\] shows some topics learned by various models. As for autoencoders, each topic is represented by the 10 words (i.e., input neurons) with the strongest connection to that topic (i.e., hidden neuron). As for LDA, each topic is represented by the 10 most probable words in that topic. The basic AE is not very good at learning distinctive topics from textual data. In our experiment, all the topics learned by AE are dominated by frequent common words like *line*, *subject* and *organ*, which were always the top 3 words in all the 20 topics. KSAE learns some meaningful words but only alleviates this problem to some extent, for example, *line*, *subject*, *organ* and *white* still appears as top 4 words in 6 topics. For this reason, the output of AE and KSAE is shown for only one of the newsgroups ([*soc.religion.christian*]{}). On the other hand, we find that [<span style="font-variant:small-caps;">KATE</span>]{}generates 20 topics that are distinct from each other, and which capture the underlying semantics very well. For example, it associates words such as *god, christian, jesu, moral, bibl, exist, religion, christ* under the topic *soc.religion.christian*. It is worth emphasizing that [<span style="font-variant:small-caps;">KATE</span>]{}belongs to the class of distributed representation models, where each topic is “distributed” among a group of hidden neurons (the topics are therefore better interpreted as “virtual” topics). However, we find that [<span style="font-variant:small-caps;">KATE</span>]{}can generate competitive topics compared with LDA, which explicitly infers topics as mixture of words.
Model **weapon** **christian** **compani** **israel** **law** **hockey** **comput** **space**
------- ------------ --------------- ------------- ------------- ----------- ------------ ------------ -----------
effort close hold cost made plane inform studi
muslim test simpl isra live tie run data
sort larg serv arab give sex program answer
america result commit fear power english base origin
escap answer societi occupi reason intel author unit
qualiti god commit occupi back int run data
challeng power student enhanc govern cco inform process
tire lie age azeri reason monash part answer
7u logic hold rpi call rsa case heard
learn simpl consist sleep answer pasadena start version
arm belief market arab citizen playoff scienc launch
crime god dealer isra constitut nhl dept orbit
gun believ manufactur palestinian court team cs mission
firearm faith expens occupi feder wing math shuttl
handgun bibl cost jew govern coach univ flight
assault understand insur lebanon court sport engin launch
militia belief feder isra prohibit nhl colleg jpl
possess believ manufactur lebanes ban playoff umich nasa
automat god industri arab sentenc winner subject moon
gun truth pay palestinian legitim cup perform gov
### Word Embeddings Learned by Different Models
In AE, KSAE and [<span style="font-variant:small-caps;">KATE</span>]{}, each input neuron (i.e., a word in the vocabulary set) is connected to each hidden neuron (i.e., a virtual topic) with different strengths. Thus, each row $i$ of the input to hidden layer weight matrix ${{\boldsymbol{W}}}\in {{\mathord{\mathbb R}}}^{d\times m}$ is taken as an $m$-dimensional word embedding for word $i$. In order to evaluate whether [<span style="font-variant:small-caps;">KATE</span>]{}can capture semantically meaningful word representations, we check if similar or related words are close to each other in the vector space. Table \[table:similar\_words\] shows the five nearest neighbors for some query words in the word representation space learned by AE, KSAE, [<span style="font-variant:small-caps;">KATE</span>]{}and Word2Vec. [<span style="font-variant:small-caps;">KATE</span>]{}performs much better than AE and KSAE. For example [<span style="font-variant:small-caps;">KATE</span>]{}lists words like [*arm, crime, gun, firearm, handgun*]{} among the nearest neighbors of query word [*weapon*]{} while neither AE nor KSAE is able to find relevant words. One can observe that [<span style="font-variant:small-caps;">KATE</span>]{}can learn competitive word representations compared to Word2Vec in terms of this word similarity task.
### Visualization of Document Representations
A good document representation method is expected to group related documents, and to separate the different groups. Figure \[fig:PCA\] shows the PCA projections of the document representations taken from the six main groups in the 20 Newsgroups data. As we can observe, neither AE nor the KSAE methods can learn good document representations. On the other hand, [<span style="font-variant:small-caps;">KATE</span>]{}successfully extracts meaningful representations from the documents; it automatically clusters related documents in the same group, and it can easily distinguish the six different groups. In fact, [<span style="font-variant:small-caps;">KATE</span>]{}is very competitive with LDA (arguably even better on this dataset, since LDA confuses some categories), even though the latter explicitly learns documents representations as mixture of topics, which in turn are mixture of words. Figure \[fig:TSNE\] shows the T-SNE based visualization [@maaten2008visualizing] of the above document representations and we can draw a similar conclusion.
Quantitative Experiments
------------------------
We now turn to quantitative experiments to measure the effectiveness of [<span style="font-variant:small-caps;">KATE</span>]{}compared to other models on tasks such as classification, multi-label classification (MLC), regression, and document retrieval (DR). For classification, MLC and regression tasks, we train a simple neural network that uses the encoded test inputs as feature vectors, and directly maps them to the output classes or values. A simple softmax classifier with cross-entropy loss was applied for the classification task, and multi-label logistic regression classifier with cross-entropy loss was applied for the MLC task. For the regression task we used a two-layer neural regression model (where the output layer is a sigmoid neuron) with squared error loss. The same architecture is used for all methods to ensure fairness. Note that when comparing various methods, the same number of features were learned for all of them except for Word2Vec$_{pre}$ which uses 300-dimensional pre-trained word embeddings and thus its number of features was fixed as 300 in all experiments.
### Mean Squared Cosine Deviation among Topics {#sec:mscd}
We first quantify how distinct are the topics learned via different methods. Let ${{\boldsymbol{v}}}_i$ denote the vector representation of topic $i$, and let there be $m$ topics. The cosine of the angle between ${{\boldsymbol{v}}}_i$ and ${{\boldsymbol{v}}}_j$, given as $\cos({{\boldsymbol{v}}}_i, {{\boldsymbol{v}}}_j) =
\tfrac{{{\boldsymbol{v}}}_i^T{{\boldsymbol{v}}}_j}{\|{{\boldsymbol{v}}}_i\| \cdot \|{{\boldsymbol{v}}}_j\|}$, is a measure of how similar/correlated the two topic vectors are; it takes values in the range $[-1,1]$. The topics are most dissimilar when the vectors are orthogonal to each other, i.e., with the angle between them is $\pi/2$, with the cosine of the angle being zero. Define the pair-wise mean squared cosine deviation among $m$ topics as follows $$MSCD = \sqrt{\frac{2}{m(m-1)}\sum_{i, j > i}
{\cos^2({{\boldsymbol{v}}}_i, {{\boldsymbol{v}}}_j)}}$$ Thus, MSCD $ \in [0,1]$, and smaller values of MSCD (closer to zero) imply more distinctive, i.e., orthogonal topics.
--------------------------------------------------------------------- ----------- ----------- -----------
**20** **128** **512**
AE 0.976 0.722 0.319
KSAE 0.268 0.198 0.056
LDA 0.249 0.059 0.028
[<span style="font-variant:small-caps;">KATE</span>]{}$_{no\_amp.}$ 0.154 0.069 0.037
[<span style="font-variant:small-caps;">KATE</span>]{} **0.097** **0.024** **0.014**
--------------------------------------------------------------------- ----------- ----------- -----------
: Mean squared cosine deviation among topics; smaller means more distinctive topics.[]{data-label="table:mean_squared_deviation"}
We evaluate MSCD for topics generated by AE, KSAE, LDA, and [<span style="font-variant:small-caps;">KATE</span>]{}. We also evaluate [<span style="font-variant:small-caps;">KATE</span>]{}without amplification. In LDA, a topic is represented as its probabilistic distribution over the vocabulary set, whereas for autoencoders, it is defined as the weights on the connections between the corresponding hidden neuron and all the input neurons. We conduct experiments on the 20 Newsgroups dataset and vary the number of topics from 20 to 128 and 512. Table \[table:mean\_squared\_deviation\] shows these results. We find that [<span style="font-variant:small-caps;">KATE</span>]{}has the lowest MSCD values, which means that it can learn more distinctive (i.e., orthogonal) topics than other methods. Our results are much better than LDA, since the latter does not prevent topics from being similar. On the other hand, the competition in [<span style="font-variant:small-caps;">KATE</span>]{}drives topics (i.e., the hidden neurons) to become distinct from each other. Interestingly, [<span style="font-variant:small-caps;">KATE</span>]{}with amplification (i.e., here we have $\alpha=6.26$) consistently achieves lower MSCD values than [<span style="font-variant:small-caps;">KATE</span>]{}without amplification, which verifies the effectiveness of the $\alpha$ amplification connections in terms of learning distinctive topics.
-------------------------------------------------------- ----------- -----------
**128** **512**
LDA 0.657 0.685
DBN 0.677 0.705
DocNADE 0.714 0.735
NVDM 0.052 0.053
Word2Vec$_{pre}$ 0.687 0.687
Word2Vec 0.564 0.586
Doc2Vec 0.347 0.399
AE 0.084 0.516
DAE 0.125 0.291
CAE 0.083 0.512
VAE 0.724 0.747
KSAE 0.486 0.675
[<span style="font-variant:small-caps;">KATE</span>]{} **0.744** **0.761**
-------------------------------------------------------- ----------- -----------
: Classification accuracy on 20 Newsgroups dataset.[]{data-label="table:mcc_results"}
### Document Classification Task
In this set of experiments, we evaluate the quality of learned document representations from various models for the purpose of document classification. Table \[table:mcc\_results\] shows the classification accuracy results on the 20 Newsgroups dataset (using 128 topics). Traditional autoencoders (including AE, DAE, CAE) do not perform well on this task. We observed that the validation set error was oscillating when training these classifiers (also observed in the regression task below), which indicates that the extracted features are not representative and consistent. KSAE consistently achieves higher accuracies than other autoencoders and does not exhibit the oscillating phenomenon, which means that adding sparsity does help learn better representations. VAE even performs better than KSAE on this dataset, which shows the advantages of VAE over other traditional autoencoders. However, as we will see later, VAE fails to consistently perform well across different datasets and tasks. Word2Vec$_{pre}$ performs on par with DBN and LDA even though it just averages all the word embeddings in a document, which suggests the effectiveness of pre-training word embeddings on a large external corpus to learn general knowledge. Not surprisingly, DocNADE works very well on this task as also reported in previous work [@larochelle2012neural; @srivastava2013modeling; @cao2015novel]. Our [<span style="font-variant:small-caps;">KATE</span>]{}model significantly outperforms all other models. For example, [<span style="font-variant:small-caps;">KATE</span>]{}obtains 74.4% accuracy which is significantly higher than the 72.4% accuracy achieved by VAE.
Table \[table:reuters\_mlc\_results\] shows multi-label classification results on Reuters and Wiki10+ datasets. Here we show both the Macro-F1 and Micro-F1 scores (reflecting a balance of precision and recall) for different number of features. Micro-F1 score biases the metric towards the most populated labels, while Macro-F1 biases the metric towards the least populated labels. Both Reuters and Wiki10+ are highly imbalanced. For example in Wiki10+, the documents belonging to ‘wikipedia’ or ‘wiki’ account for 90% of the corpus while only around 6% of the documents are relevant to ‘religion’. Similarly, in Reuters, the documents belonging to ‘CCAT’ account for 47% of the corpus while there are only 5 documents relevant to ‘GMIL’. DocNADE works the very well on this task, but the sparse and competitive autoencoders also perform well. [<span style="font-variant:small-caps;">KATE</span>]{}outperforms KSAE on Reuters and remains competitive on Wiki10+. We don’t report the results of DBN on Reuters since the training did not end even after a long time.
-------------------------------------------------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------
**Macro-F1** **Micro-F1** **Macro-F1** **Micro-F1** **Macro-F1** **Micro-F1** **Macro-F1** **Micro-F1**
LDA 0.408 0.703 0.576 0.766 0.442 0.584 0.305 0.441
DBN - - - - 0.330 0.513 0.339 0.536
DocNADE [**0.564**]{} [**0.768**]{} [**0.667**]{} [**0.831**]{} [**0.451**]{} 0.585 0.423 0.561
NVDM 0.215 0.441 0.195 0.452 0.187 0.461 0.036 0.375
Word2Vec$_{pre}$ 0.549 0.712 0.549 0.712 0.312 0.454 0.312 0.454
Word2Vec 0.458 0.648 0.595 0.761 0.205 0.318 0.234 0.325
Doc2Vec 0.004 0.082 0.000 0.000 0.289 0.486 0.344 0.524
AE 0.025 0.047 0.459 0.651 0.016 0.040 0.382 0.569
DAE 0.275 0.576 0.489 0.685 0.359 0.560 0.375 0.534
CAE 0.024 0.045 0.549 0.726 0.091 0.168 0.404 0.547
VAE 0.325 0.458 0.490 0.594 0.342 0.497 0.373 0.511
KSAE 0.457 0.660 0.605 0.766 0.449 [**0.594**]{} [**0.471**]{} [**0.614**]{}
[<span style="font-variant:small-caps;">KATE</span>]{} 0.539 0.716 0.615 0.767 0.445 0.580 0.446 0.580
-------------------------------------------------------- --------------- --------------- --------------- --------------- --------------- --------------- --------------- ---------------
\[table:wiki\_mlc\_results\]
-------------------------------------------------------- ----------- -----------
**128** **512**
LDA 0.287 0.226
DBN 0.277 0.369
DocNADE 0.404 0.424
NVDM 0.199 0.191
Word2Vec$_{pre}$ 0.409 0.409
Word2Vec 0.143 0.136
Doc2Vec 0.052 0.032
AE -0.001 0.203
DAE 0.067 0.100
CAE 0.018 0.118
VAE 0.111 0.355
KSAE 0.152 0.365
[<span style="font-variant:small-caps;">KATE</span>]{} **0.463** **0.516**
-------------------------------------------------------- ----------- -----------
: Comparison of regression $r^2$ score on MRD dataset.[]{data-label="table:regression_results"}
### Regression Task
In this set of experiments, we evaluate the quality of learned document representations from various models for predicting the movie ratings in the MRD dataset, as shown in Table \[table:regression\_results\] (using 128 features). The coefficient of determination, denoted $r^2$, from the regression model was used to evaluate the methods. The best possible $r^2$ statistic value is 1.0; negative values are also possible, indicating a poor fit of the model to the data. In general, other autoencoder models perform poorly on this task, for example, AE even gets a negative $r^2$ score. Interestingly, Word2Vec$_{pre}$ performs on par with DocNADE, indicating that word embeddings learned from a large external corpus can capture some semantics of emotive words (e.g., good, bad, wonderful). We observe that [<span style="font-variant:small-caps;">KATE</span>]{}significantly outperforms all other models, including Word2Vec$_{pre}$, which means it can learn meaningful representations which are helpful for sentiment analysis.
![Document retrieval on 20 Newsgroups dataset (128 features).[]{data-label="fig:20News_Retrieval"}](resources/20news_doc_retrieval_128D.png){width="3.5in" height="2in"}
### Document Retrieval Task
We also evaluate the various models for document retrieval. Each document in the test set is used as an individual query and we fetch the relevant documents from the training set based on the cosine similarity between the document representations. The average fraction of retrieved documents which share the same label as the query document, i.e., precision, was used as the evaluation metric. As shown in Figure \[fig:20News\_Retrieval\], VAE performs the best on this task followed by DocNADE and [<span style="font-variant:small-caps;">KATE</span>]{}. Among the other models, DBN and LDA also have decent performance, but the other autoencoders are not that effective.
### Timing
Model LDA DBN DocNADE NVDM Word2Vec Doc2Vec AE DAE CAE VAE KSAE [<span style="font-variant:small-caps;">KATE</span>]{}
---------- ----- -------- --------- ------ ---------- --------- ----- ----- ----- ----- ------ --------------------------------------------------------
Time (s) 399 15,281 4,787 645 977 992 566 361 729 660 489 1,214
Finally, we compare the training time of various models. Results are shown in Table \[table:training\_time\] for the 20 Newsgroups dataset, with 20 topics. Our model is much faster than deep generative models like DBN and DocNADE. It is typically slower than other autoencoders since it usually takes more epochs to converge. Nevertheless, as demonstrated above, it significantly outperforms other models in various text analytics tasks.
[<span style="font-variant:small-caps;">KATE</span>]{}: Effects of Parameter Tuning {#sec:effects_of_params}
-----------------------------------------------------------------------------------
Having demonstrated the effectiveness of [<span style="font-variant:small-caps;">KATE</span>]{}compared to other methods, we study the effects of various hyperparameter choices in [<span style="font-variant:small-caps;">KATE</span>]{}, such as the number of topics (i.e., hidden neurons), the number of winners $k$ and the energy amplification parameter $\alpha$. The default values for the number of topics is 128, with $k=32$ and $\alpha=6.26$. Note when exploring the effect of the number of topics, we also vary $k$ to find its best match to the given number of topics. Figure \[fig:effects\_hyperparam\] shows the classification accuracy on the 20 Newsgroups dataset, as we vary these parameters. We observe that as we increase the number of topics or hidden neurons (in Figure \[fig:effects\_topics\]), the accuracy continues to rise, but eventually drops off. We use 128 as the default value since it offers the best trade-off in complexity and performance; only relatively minor gains are achieved in increasing the number of topics beyond 128. Considering the number of winning neurons (see Figure \[fig:effects\_k\]), the main trend is that the performance degrades when we make $k$ larger, which is expected since larger $k$ implies lesser competition. In practice, when tuning $k$, we find that starting by a value close to around a quarter of the number of topics is a good strategy. Finally, as we mentioned, the $\alpha$ amplification connection is crucial as verified in Figure \[fig:effects\_alpha\]. When $\alpha=2/k=0.0625$, which means there is no amplification for the energy, the classification accuracy is 71.1%. However, we are able to significantly boost the model performance up to 74.6% accuracy by increasing the value of $\alpha$. We use a default value of $\alpha=6.26$, which once again reflects a good trade-off across different datasets. It is also important to note that across all the experiments, we found that using the [*tanh*]{} activation function (instead of [*sigmoid*]{} function) in the k-competitive layer of [<span style="font-variant:small-caps;">KATE</span>]{}gave the best performance. For example, on the 20 Newsgroups data, using 128 topics, [<span style="font-variant:small-caps;">KATE</span>]{}with [*tanh*]{} yields 74.4% accuracy, while with [*sigmoid*]{} it was only 56.8%.
Conclusions
===========
We described a novel k-competitive autoencoder, [<span style="font-variant:small-caps;">KATE</span>]{}, that explicitly enforces competition among the neurons in the hidden layer by selecting the $k$ highest activation neurons as winners, and reallocates the amplified energy (aggregate activation potential) from the losers. Interestingly, even though we use a shallow model, i.e., with one hidden layer, it outperforms a variety of methods on many different text analytics tasks. More specifically, we perform a comprehensive evaluation of [<span style="font-variant:small-caps;">KATE</span>]{}against techniques spanning graphical models (e.g., LDA), belief networks (e.g., DBN), word embedding models (e.g., Word2Vec), and several other autoencoders including the k-sparse autoencoder (KSAE). We find that across tasks such as document classification, multi-label classification, regression and document retrieval, [<span style="font-variant:small-caps;">KATE</span>]{}clearly outperforms competing methods or obtains close to the best results. It is very encouraging to note that [<span style="font-variant:small-caps;">KATE</span>]{}is also able to learn semantically meaningful representations of words, documents and topics, which we evaluated via both quantitative and qualitative studies. As part of future work, we plan to evaluate [<span style="font-variant:small-caps;">KATE</span>]{}on more domain specific datasets, such as bibliographic networks, for example for topic induction and scientific publication retrieval. We also plan to improve the scalability and effectiveness of our approach on much larger text collections by developing parallel and distributed implementations.
This work was supported in part by awards and .
|
---
abstract: 'The interpolated bounce-back scheme and the immersed boundary method are the two most popular algorithms in treating a no-slip boundary on curved surfaces in the lattice Boltzmann method. While those algorithms are frequently implemented in the numerical simulations involving complex geometries, such as particle-laden flows, their performances are seldom compared systematically over the same local quantities within the same context. In this paper, we present a systematic comparative investigation on some frequently used and most state-of-the-art interpolated bounce-back schemes and immersed boundary methods, based on both theoretical analyses and numerical simulations of four selected 2D and 3D laminar flow problems. Our analyses show that immersed boundary methods (IBM) typically yield a first-order accuracy when the regularized delta-function is employed to interpolate velocity from the Eulerian to Lagrangian mesh, and the resulting boundary force back to the Eulerian mesh. This first order in accuracy for IBM is observed for both the local velocity and hydrodynamic force/torque, apparently different from the second-order accuracy sometime claimed in the literature. Another serious problem of immersed boundary methods is that the local stress within the diffused fluid-solid interface tends to be significantly underestimated. On the other hand, the interpolated bounce-back generally possesses a second-order accuracy for velocity, hydrodynamic force/torque, and local stress field. The main disadvantage of the interpolated bounce-back schemes is its higher level of fluctuations in the calculated hydrodynamic force/torque when a solid object moves across the grid lines. General guidelines are also provided for the necessary grid resolutions in the two approaches in order to accurately simulate flows over a solid particle.'
address:
- 'Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China'
- '126 Spencer Lab, Department of Mechanical Engineering, University of Delaware, Newark, DE, USA, 19716'
- '111A Kaufman Hall, Department of Engineering Technology, Old Dominion University, Norfolk, VA, 23529, USA'
author:
- Cheng Peng
- 'Orlando M. Ayala'
- 'Lian-Ping Wang'
bibliography:
- 'elsarticle-template.bib'
title: 'A comparative study of immersed boundary method and interpolated bounce-back scheme for no-slip boundary treatment in the lattice Boltzmann method: Part I, laminar flows'
---
lattice Boltzmann method ,interpolated bounce-back schemes ,immersed boundary methods ,no-slip boundary
|
---
author:
- |
\
The Netherlands Foundation for Radio Astronomy (ASTRON)\
E-mail:
- |
Robert Braun\
CSIRO-ATNF\
E-mail:
title: Magnetic fields in nearby galaxies
---
The WSRT-SINGS Survey
=====================
Magnetic fields are a critical component of the interstellar medium (ISM) in galaxies, yet major questions still remain about their properties. One of the unknowns is the three-dimensional structure of galactic magnetic fields. The most effective way to measure magnetic fields in galaxies is through radio measurements of synchrotron emission (using the polarized emission to study the ordered fields perpendicular to the line of sight, and its associated Faraday rotation to trace fields parallel to the line of sight). Observations of face-on spiral galaxies have shown that the planar component of magnetic fields traces the spiral arms (e.g. [@beck_2009]), while observations of edge-on disks tend to show a characteristic [X]{}-shaped morphology (e.g. [@krause_2008]). Galactic magnetic fields are expected to take either a dipolar or quadrupolar form (e.g. [@widrow_2002]), but previous observations of external galaxies have not been able to strongly constrain which type of field might be present.
The Westerbork Synthesis Radio Telescope (WSRT) was recently used to observe a sample of several Spitzer Infrared Nearby Galaxies Survey (SINGS; [@kennicutt_etal_2003]) galaxies, in full polarization, in order to provide supplementary data on the radio continuum emission. The WSRT-SINGS survey itself is described by [@braun_etal_2007], and the polarization analysis by [@heald_etal_2009]. Briefly, each of the targets was observed for a total of 12 hours. Polarization data were obtained in two wide bands (effective bandwidth $\approx132\,\mathrm{MHz}$ in 512 channels) centered at frequencies of 1366 and 1697MHz. A band-switching technique was employed which provided an effective integration time of 6 hours in each band, while retaining the full $uv$ coverage in both. Following calibration, each channel was imaged individually. The Rotation Measure Synthesis (RM-Synthesis) technique [@brentjens_debruyn_2005; @heald_2009] was used to coherently detect the polarized emission, and its associated RM, across the two observing bands. The key strengths of this method are that it avoids the $n\pi$ ambiguity problem that plagues traditional RM determinations; and that it allows the detection of polarized flux, and its associated RM, even at very low signal-to-noise levels. The typical rms noise level achieved in the WSRT-SINGS polarization maps is $\approx10\,\mu\mathrm{Jy\,beam}^{-1}$. Of the 28 galaxies studied in the polarization survey, 21 were detected in polarized emission. All 21 detections were of spiral galaxies; the few Magellanic and elliptical galaxies in the sample (three and one of these, respectively) were all undetected in polarized emission. The output of the RM-Synthesis step was used to generate maps of polarized intensity, polarization angle, and Faraday rotation measure. Image galleries are presented by [@heald_etal_2009].
Observational Trends and Interpretation
=======================================
Two major observational trends are observed when considering the resulting data for the entire WSRT-SINGS sample:
1. All galaxies with extended polarized flux show a clear pattern in its azimuthal distribution. In face-on targets, the minimum in polarized flux always occurs near the kinematically receding major axis. Two of the clearest examples of this behavior, NGC 6946 and NGC 4321, are shown in Figure \[fig:n6946n4321\]. Highly inclined galaxies have a minimum level of polarized flux along both sides of the major axis, and maxima near the minor axis. The shifting of the peak polarized flux from the approaching major axis to the minor axis appears to occur smoothly as the inclination of the galaxy increases.
2. All galaxies with compact nuclear radio emission have complicated nuclear Faraday spectra. Examples are shown in Figure \[fig:nuclei\]. The limitations of our observing setup (specifically, the relatively low observing frequencies) make it unclear whether these features are indicative of a detection of multiple Faraday thin media in the galaxy nuclei, or rather a single Faraday thick medium which causes substantial depolarization (at RM values between the main peaks in Figure \[fig:nuclei\]). Further measurements at higher spatial resolution and higher frequency will be required to clarify the physical situation in these nuclei.
![Two galaxies observed in the WSRT-SINGS survey: NGC 6946 (left panels) and NGC 4321 (right panels). The optical pictures (top panels) were composed using images from the Digitized Sky Survey. The location of the kinematically receding major axis is indicated with a white arrow. Polarized flux is shown with red contours, starting at $50\mu\mathrm{Jy\,beam}^{-1}$ and increasing by powers of two. Magnetic field orientations, derived from the Faraday rotation corrected polarization angles, are shown with orange lines. The bottom panels show the mean azimuthal variation in polarized flux. The receding major axis corresponds to azimuth $0^{\circ}$.[]{data-label="fig:n6946n4321"}](n6946n4321_v2.pdf){width="\textwidth"}
![Faraday dispersion functions for four of the WSRT-SINGS galaxies, showing the polarized emission as a function of RM in the nucleus of each target. The dotted lines show the estimated local foreground RM value. Polarized emission is clearly detected at two different RM values (perhaps three in the case of NGC 6946), which appear symmetrically distributed about the zero-level. The numbers indicate the polarization angle (in degrees) at the local peak of the Faraday dispersion function.[]{data-label="fig:nuclei"}](centralRM.pdf){width="75.00000%"}
That the polarized flux is always minimized along the [*receding*]{} major axis in the more face-on targets implies that this feature is a key handle on the fundamental relationship between the structure of the magnetic field, and that of the galaxy itself. In [@braun_etal_2009], we model several three-dimensional forms of magnetic field geometry in an attempt to match the observations of both polarized flux and Faraday rotation measure. A key feature of the models is that midplane depolarization prevents us from observing the backside of galaxy disks at frequencies near 1.4 GHz. In fact, high frequency observations of NGC 6946 (see [@beck_2007]) show that the strong azimuthal variation in the polarized intensity vanishes at wavelengths of 3 cm and 6 cm. We interpret this as a reduction in the midplane depolarization at high frequencies. Using this assumption, we are able to constrain the vertical morphology of the large-scale magnetic field, finding that the observations prefer an axisymmetric spiral plus a quadrupolar component. The model is also consistent with the disappearance of the azimuthal dependence of polarized intensity at higher observing frequency. Details of the modeling results are provided by [@braun_etal_2009].
Future observations are needed in order to confirm this trend in a larger sample of galaxies. A larger sample will also provide leverage on any variation in the three-dimensional magnetic field geometry with other galaxy properties such as Hubble type, star formation rate, and rotation speed. Additional high-frequency measurements are also required in order to confirm the prediction that the azimuthal variation in polarized intensity vanishes above $\sim5\,\mathrm{GHz}$.
Future Facilities
=================
Polarization observations are expected to flourish with the advent of future radio facilities such as the planned WSRT upgrade Aperture Tile in Focus (APERTIF; [@oosterloo_tv]) and the Australian Square Kilometre Array Pathfinder (ASKAP; [@braun_tv]). These radio telescopes in particular, which are both based on dense Focal Plane Array (FPA) technology, are by their nature expected to provide excellent data sets for making progress in understanding the issues described above. The key desired observational characteristics for future polarization surveys are (1) large field-of-view; (2) excellent control and/or suppression of instrumental polarization; and (3) wide observing bandwidth.
Both APERTIF and ASKAP are expected to perform radio continuum surveys of large areas of the sky; in tandem, it is possible that they will together survey the *entire* sky. The expected noise levels resulting from large-area continuum surveys with both telescopes are comparable to the sensitivity achieved with the WSRT-SINGS survey. With their wide bandwidth, the precision with which RM values can be determined using the RM-Synthesis technique is excellent (for example, using a nominal 1000-1700MHz bandwidth, the expected native RM resolution is approximately $\sigma\approx25$radm$^{-2}$, and the effective RM resolution improves with increasing signal-to-noise; see e.g. [@brentjens_debruyn_2005]). By comparison, the native RM resolution of the WSRT-SINGS observations described above was $\sigma\approx\,61\,\mathrm{rad\,m}^{-2}$.
Polarized radio continuum maps will be one of the data products produced by the polarization surveys to be performed with APERTIF and ASKAP. One of the main science drivers for such a massive polarization survey is the development of an all-sky Rotation Measure grid (RM-grid), which will be a powerful intermediate step as we work towards the incredible RM-grid which will be provided by the Square Kilometre Array (SKA) [@gaensler_etal_2004]. During such a survey, a large number of nearby (i.e., spatially resolved) galaxies will also be observed. To roughly estimate the number of resolved galaxies which will be detected in such a survey using APERTIF, the Nearby Galaxy Catalog [@tully_1988] was queried for spiral galaxies above a declination of $30^{\circ}$ and with angular size sufficient for $\gtrsim10$ resolution elements at the nominal angular resolution of the WSRT. Such a query results in 226 galaxies. If, as in the WSRT-SINGS sample, $\sim40\%$ have detectable extended polarized emission at 1.4GHz, a much larger version of WSRT-SINGS in the northern hemisphere would be able to place about 100 galaxies in a future version of Figure \[fig:n6946n4321\]. Extending to the entire sky, a huge sample of about 350 galaxies can be built up. Of course kinematic information will also be available for all targets, since H$\,\textsc{i}$ surveys will also be performed by both instruments.
One of the main reasons that FPA designs are expected to be so powerful for large-area polarization surveys is a side-effect of the digital beamforming. To provide such a large field of view, FPAs form multiple, angularly separated but densely packed beams simultaneously on the sky. This is achieved by electronically applying the proper complex weights to the signals coming from the individual FPA elements. It is expected (but remains to be shown with a prototype system) that this same weighting procedure can be used to control the polarization response of the system, with the end result that instrumental polarization can be greatly reduced without a significant impact on the overall sensitivity of the telescope (e.g. [@capellen_bakker_2009]).
Of crucial importance in the interpretation of the polarization data obtained for each galaxy is an estimate of the foreground RM contribution. The best way to estimate this quantity is to investigate the RM of polarized background sources surrounding the target galaxy. By utilizing such a “local RM grid,” rather than relying on the mean RM value of the target itself, information about a zero-level offset (as might result from a vertical magnetic field in a face-on target) is not lost. The current field of view of the WSRT is sufficient to detect a handful of such surrounding background sources. With future facilities and their large fields of view, the ability to estimate the foreground RM level will be greatly enhanced.
Future radio facilities with smaller instantaneous fields of view, such as the Expanded Very Large Array (EVLA; [@rupen_tv]) and the Karoo Array Telescope (MeerKAT; [@jonas_tv]) have the benefits of higher sensitivity in a single pointing, and also being able to operate at higher observing frequencies than either APERTIF or ASKAP. They will be able to provide sensitive polarization observations of local galaxies both at $\sim\,1.4\,\mathrm{GHz}$ and at higher frequencies, allowing not only observations of weaker magnetic fields and increased RM resolution due to the increased sensitivity, but also crucial tests of midplane depolarization and the polarization properties of galactic nuclei at the higher frequencies.
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R. Beck, *Measuring interstellar magnetic fields by radio synchrotron emission*, *IAUS* [**259**]{} (2009) 3.
R. Braun, T.A. Oosterloo, R. Morganti, U. Klein & R. Beck, *The Westerbork SINGS survey. I. Overview and image atlas*, [*A&A*]{} [**461**]{} (2007) 455.
R. Braun, G. Heald & R. Beck, *The Westerbork SINGS Survey. III. Global Magnetic Field Topology*, 2009, submitted
R. Braun, *Panoramic Surveys of the Radio Sky with the Australian SKA Pathfinder*, in proceedings of *PRA2009*, .
M.A. Brentjens & A.G. de Bruyn, *Faraday rotation measure synthesis*, [*A&A*]{} [**441**]{} (2005) 1217.
W.A. van Cappellen & L. Bakker, *Experimental Results of a 112 Element Phased Array Feed for the Westerbork Synthesis Radio Telescope*, *IEEE Symp. on Ant. and Propagat.* 2009.
B.M. Gaensler, R. Beck & L. Feretti, *The origin and evolution of cosmic magnetism*, *NewAR* [**48**]{} (2004) 1003.
G. Heald, *The Faraday rotation measure synthesis technique*, [*IAUS*]{} [**259**]{} (2009) 591.
G. Heald, R. Braun & R. Edmonds, *The Westerbork SINGS Survey. II. Polarization, Faraday rotation, and magnetic fields*, [*A&A*]{} [**503**]{} (2009) 409.
J. Jonas, *The MeerKAT SKA precursor telescope*, in proceedings of *PRA2009*, .
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T. Oosterloo, *The latest on Apertif*, in proceedings of *PRA2009*, .
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L.M. Widrow, *Origin of galactic and extragalactic magnetic fields*, *RvMP* [**74**]{} (2002) 775.
|
---
abstract: 'We present a polynomial Hybrid Monte Carlo (PHMC) algorithm as an exact simulation algorithm with dynamical Kogut-Susskind fermions. The algorithm uses a Hermitian polynomial approximation for the fractional power of the KS fermion matrix. The systematic error from the polynomial approximation is removed by the Kennedy-Kuti noisy Metropolis test so that the algorithm becomes exact at a finite molecular dynamics step size. We performed numerical tests with $N_f$$=$$2$ case on several lattice sizes. We found that the PHMC algorithm works on a moderately large lattice of $16^4$ at $\beta$$=$$5.7$, $m$$=$$0.02$ ($m_{\mathrm{PS}}/m_{\mathrm{V}}$$\sim$$0.69$) with a reasonable computational time.'
address:
- '[Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan]{}'
- '[Center for Computational Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan]{}'
- '[Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan]{}'
- '[High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan]{}'
- '[Department of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan]{}'
author:
- 'JLQCD Collaboration: K-I. Ishikawa$^{,}$, M. Fukugita, S. Hashimoto, N. Ishizuka$^{\mathrm{a,b}}$, Y. Iwasaki$^{\mathrm{a,b}}$, K. Kanaya$^{\mathrm{a}}$, Y. Kuramashi$^{\mathrm{c}}$, M. Okawa, N. Tsutsui$^{\mathrm{c}}$, A. Ukawa$^{\mathrm{a,b}}$, N. Yamada$^{\mathrm{c}}$, T. Yoshié$^{\mathrm{a,b}}$'
title: 'An exact Polynomial Hybrid Monte Carlo algorithm for dynamical Kogut-Susskind fermions [^1]'
---
Introduction
============
The low energy QCD dynamics in the real world will be understood by lattice QCD with three-flavors of dynamical quarks. Several efforts have been spent to develop exact numerical algorithms with an odd-numbers of the Wilson type quarks [@Nf3].
The Kogut-Susskind (KS) fermion is an attractive formalism since the numerical simulation with much lighter quark masses are possible thanks to the remnant chiral symmetry. Although lattice QCD with the two- or single-flavor KS fermions can be defined by taking the fractional power of the KS fermion, efficient exact algorithms are not still known. Approximate algorithms such as the $R$-algorithm [@R_Algorithm] have been used in these cases.
Several exact algorithms are proposed for two- or single-flavor dynamical KS fermions [@RHMC; @Hasenfratz_Knechtli]. In this paper, we further study the idea by Horváth *et al.* [@RHMC] in the case of the polynomial Hybrid Monte Carlo (PHMC) algorithm. We develop two types of the PHMC algorithm depending on the choice of the effective action derived through the polynomial approximation. We compare the computational cost of these two PHMC algorithms. We investigate the property of the algorithm on several lattice sizes in $N_f$$=$$2$ case. We found that our algorithm shows satisfactory efficiency on a $16^4$ lattice with $\beta$$=$$5.7$, $m$$=$$0.02$ $(m_{\mathrm{PS}}/m_{\mathrm{V}}$$\sim$$0.69)$.
Algorithm
=========
We construct two types of the PHMC algorithm, which are refereed to as case A and case B. Introducing a polynomial approximation and pseudo-fermion field, the partition function can be generally rewritten in the following form: $$\begin{aligned}
Z=&&\!\!\!\!\!\!\!\!\!\!\!
\int {\cal D}U{\cal D}\phi_{o}^{\dag}{\cal D}\phi_{o}
\ \det[W^{(X)}[{\hat{D}_{oo}}]]^{N_{f}/4}
\nonumber\\
&& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times e^{-S_{g}[U]-S^{(X)}_{q}[U,\phi_{o}^{\dag},\phi_{o}]},
\label{eq:Effective_Action1}\end{aligned}$$ where $S_{g}$ is a lattice gauge action, $\phi_{o}$ is pseudo-fermion field living only on odd sites. $S^{(X)}_{q}$ and $W^{(X)}$ are the pseudo-fermion action and the correction matrix respectively. The superscript $(X)$ takes $(A)$ or $(B)$ depending on the type of the PHMC algorithm as follows.
**Case A:**We approximate ${\hat{D}_{oo}}^{-N_f/8}$ by a ${N_{\mathit{poly}}}^{(A)}$ order polynomial $P_{{N_{\mathit{poly}}}^{(A)}}[{\hat{D}_{oo}}]$. The pseudo-fermion action and the correction term become $$\begin{aligned}
S^{(A)}_{q}[U,\phi_{o}^{\dag},\phi_{o}]&=&
|P_{{N_{\mathit{poly}}}^{(A)}}[{\hat{D}_{oo}}]\phi_{o}|^{2},\nonumber\\
W^{(A)}[{\hat{D}_{oo}}]&=&{\hat{D}_{oo}}(P_{{N_{\mathit{poly}}}^{(A)}}[{\hat{D}_{oo}}])^{8/N_{f}}.
\label{eq:CaseA_Action}\end{aligned}$$
**Case B:**We approximate ${\hat{D}_{oo}}^{-N_f/4}$ by an even-order ${N_{\mathit{poly}}}^{(B)}$ polynomial $P_{{N_{\mathit{poly}}}^{(B)}}[{\hat{D}_{oo}}]$. The pseudo-fermion action and the correction term can be written as $$\begin{aligned}
S^{(B)}_{q}[U,\phi_{o}^{\dag},\phi_{o}]&=&
|Q_{{N_{\mathit{poly}}}^{(B)}}[{\hat{D}_{oo}}]\phi_{o}|^{2},\nonumber\\
W^{(B)}[{\hat{D}_{oo}}]&=&{\hat{D}_{oo}}(P_{{N_{\mathit{poly}}}^{(B)}}[{\hat{D}_{oo}}])^{4/N_{f}},
\label{eq:CaseB_Action}\end{aligned}$$ where $Q_{{N_{\mathit{poly}}}^{(B)}}$ is the ${N_{\mathit{poly}}}^{(B)}/2$ order polynomial defined by $P_{{N_{\mathit{poly}}}^{(B)}}[{\hat{D}_{oo}}]$$=$$|Q_{{N_{\mathit{poly}}}^{(B)}}[{\hat{D}_{oo}}]|^{2}$.
The KS-fermion operator ${\hat{D}_{oo}}$ is even-odd preconditioned as ${\hat{D}_{oo}}$$=$$\mathbf{1}_{oo}$$-$$\lambda^2 \hat{M}_{oo}$ with $\lambda^{2}$$=$$2 \Lambda_{\mathrm{max}}$$/$ $(4m^2$$+$$2\Lambda_{\mathrm{max}}^{2})$ and $\hat{M}_{oo}$$=$$2 M_{oe}M_{eo}/\Lambda_{\mathrm{max}}^2$$+$$\mathbf{1}_{oo}$. $\Lambda_{\mathrm{max}}$ is chosen so that the all eigenvalues of $\hat{M}_{oo}$ fall into the region $[-1,1]$. $M_{oe}$ ($M_{eo}$) is the usual KS hopping matrix from even (odd) to odd (even) sites.
For both cases, the algorithm takes the following two steps; (i) perform the HMC algorithm according to the effective action Eq. (\[eq:CaseA\_Action\]) or (\[eq:CaseB\_Action\]), (ii) when the HMC Metropolis test is accepted, apply the Kennedy-Kuti noisy Metropolis test to incorporate the correction term $W^{(X)}[{\hat{D}_{oo}}]]^{N_f/4}$. Thus we obtain two types of the PHMC algorithm depending on the choice of the effective action.
The acceptance probability of the noisy-Metropolis test is defined by $$\begin{aligned}
&&P_{\mathrm{NMP}}[U\rightarrow U']=\min[1,e^{-dS}],\nonumber\\
&&dS=\zeta_{o}^{\dag}W^{(X)}[{\hat{D}_{oo}}']^{-N_f/4}\zeta_{o}-|\eta_{o}|^2,
\label{eq:dS}\end{aligned}$$ where $\zeta_{o}$$=$$W^{(X)}[{\hat{D}_{oo}}]^{N_f/8}\eta_{o}$ with a Gaussian noise vector $\eta_{o}$. $W^{(X)}[{\hat{D}_{oo}}]$ is calculated on an initial configuration and $W^{(X)}[{\hat{D}_{oo}}']$ is on a trial configuration generated by the preceding HMC algorithm.
The fractional power of the correction matrix $W^{(X)}$ is taken by the Lanczos based Krylov subspace method proposed by Boriçi [@Borici]. We modified his algorithm suitable to our purpose. As indicated by Boriçi we employ CG based stopping criterion for the Lanczos based method.
Cost estimate
=============
The computational cost is counted as the number of multiplication of the hopping matrix to evolve the algorithm unit trajectory. We employ single leapfrog integration scheme for the molecular dynamics (MD) step. We roughly estimate it as $$\begin{aligned}
&&\!\!\!\!\!\!\!\!\!\!\!
N_{\mathrm{Cost A}}
= (2 {N_{\mathit{poly}}}^{(A)}-1)\times N^{(A)}_{\mathrm{MD}} \nonumber\\
&& \ \ \ \ \ + 3\times ((8/N_f)\times {N_{\mathit{poly}}}^{(A)} +1 ) \times N^{(A)}_{\mathrm{CG}},
\nonumber\\
&&\!\!\!\!\!\!\!\!\!\!\!
N_{\mathrm{Cost B}}
= ( {N_{\mathit{poly}}}^{(B)}-1)\times N^{(B)}_{\mathrm{MD}} \nonumber\\
&& \ \ \ \ \ + 3\times ((4/N_f)\times {N_{\mathit{poly}}}^{(B)} +1 ) \times N^{(B)}_{\mathrm{CG}},
\nonumber\end{aligned}$$ where $N_{\mathrm{MD}}$ is the number of MD step and $N^{(X)}_{\mathrm{CG}}$ the number of iteration of CG algorithm. The CG algorithm is used to generate $\phi_{o}$ with the global heat-bath method and in the Lanczos based algorithm for the noisy Metropolis test.
Now we compare the costs by specifying ${N_{\mathit{poly}}}^{(A)}$ and ${N_{\mathit{poly}}}^{(B)}$. For this purpose we employ the Chebyshev polynomial approximation; $$x^{-s}=(1+\lambda^{2} y)^{-s}\sim
P_{{N_{\mathit{poly}}}}[x]=\sum_{i=0}^{{N_{\mathit{poly}}}} c_{k}T_{k}[y],$$ where $T_{k}$ is the $k$-th order Chebyshev polynomial, $x$$=$${\hat{D}_{oo}}$ and $y$$=$$-\hat{M}_{oo}$ can be read of. The coefficients $c_{k}$ are calculated as usual.
Figure \[fig:1\] shows the cost ${N_{\mathit{poly}}}/s$ dependence of the integrated residual defined by $\sqrt{R^2}$$=$$(
\int_{-1}^{1}dy | x (P_{{N_{\mathit{poly}}}}[x])^{1/s}-1 |^{2}
)^{1/2}$ for each $s$. We observe that as decreasing $s$ by factor $1/2$ the cost ${N_{\mathit{poly}}}/s$ increases by factor $2$ at a constant $\sqrt{R^{2}}$. This is nothing but ${N_{\mathit{poly}}}$ does not depend on the choice of $s$ and we obtain ${N_{\mathit{poly}}}^{(A)}$$\sim$${N_{\mathit{poly}}}^{(B)}$ at a constant approximation level.
Using this relation and assuming $N_{\mathrm{CG}}^{(A)}$$=$$N_{\mathrm{CG}}^{(B)}$ and $N_{\mathrm{MD}}^{(A)}$$=$$N_{\mathrm{MD}}^{(B)}$, we find $N_{\mathrm{Cost A}}$$\sim$$ 2 N_{\mathrm{Cost B}}$. We employ the Chebyshev polynomial and the plaquette gauge action, and apply the case B PHMC algorithm for numerical simulations.
\[fig:1\]
Results
=======
Figure \[fig:2\] shows the MD step size dependence of the averaged plaquette on a $8^3\times 4$ lattice at $\beta$$=$$5.26$ and $m$$=$$0.025$. The results with the PHMC algorithm of ${N_{\mathit{poly}}}$$=$$200$ do not depend on $dt$ as it should be and produce the consistent result to that in the zero MD step size limit of the $R$-algorithm. Although we do not show the results on the ${N_{\mathit{poly}}}$ dependence, the results are independent of the choice of ${N_{\mathit{poly}}}$.
In Table \[tab:1\] we show the numerical results on a $16^4$ lattice at $\beta$$=$$5.7$ and $m$$=$$0.02$. The averaged plaquette are independent of ${N_{\mathit{poly}}}$ and consistent with each other as expected. On the other hand, the $R$-algorithm yields $\langle P\rangle$$=$$0.577261(49)$ [@OKAWA], which differs from ours by $\sim$$2\sigma$. This indicates a potential systematic error for the $R$-algorithm at finite MD step size. The computational time for ${N_{\mathit{poly}}}$$=$$300$ with $m$$=$$0.02$ (which corresponds to $m_{\mathrm{PS}}/m_{\mathrm{V}}$$\sim$$0.69$ [@OKAWA]) was measured as 112 sec. to achieve unit trajectory with 14 GFlops sustained speed of SR8000 at KEK.
\[fig:2\]
------------------------ -------------- -------------- --------------
$N_{\mathrm{poly}}$ 300 400 500
$[dt,N_{\mathrm{MD}}]$ $[0.02,50]$ $[0.02,50]$ $[0.02,50]$
Traj. 1700 1050 800
$\langle P\rangle$ 0.577099(46) 0.577130(46) 0.577023(43)
------------------------ -------------- -------------- --------------
: Numerical results on a $16^4$ lattice at $\beta$$=$$5.7$ and $m$$=$$0.02$. $\Lambda_{\mathrm{max}}$$=$ $2.28$ is employed. []{data-label="tab:1"}
Consequently we conclude that the PHMC algorithm we constructed works on a moderately large lattice size with rather heavy quark masses with reasonable computational cost. The algorithm also works with a single-flavor fermion. We emphasize that because the PHMC algorithm is exact one, it must be a promising algorithm for future realistic simulations. This work is supported by the Supercomputer Project No. 79 (FY2002) of High Energy Accelerator Research Organization (KEK), and also in part by the Grant-in-Aid of the Ministry of Education (Nos. 11640294, 12304011, 12740133, 13135204, 13640259, 13640260). N.Y. is supported by the JSPS Research Fellowship.
[99]{}
T. Takaishi and Ph. de Forcrand, Int. J. Mod. Phys. C13 (2002) 343; Nucl. Phys. (Proc. Suppl.) 94 (2001) 818; hep-lat/0009024; JLQCD collaboration, S. Aoki *et al.*, Phys. Rev. D 65 (2002) 094507; F. Farchioni, C. Gebert, I. Montvay, and W. Schroers, Nucl.Phys.Proc.Suppl. 106 (2002) 215.
M. Clark, in these proceedings.
I. Horváth, A. D. Kennedy, and S. Sint, Nucl. Phys. B (Proc. Suppl.) 73 (1999) 834.
A. Hasenfratz and F. Knechtli, hep-lat/0106014; Nucl. Phys. B (Proc. Suppl.) 106 (2002); hep-lat/0203010; in these proceedings.
A. D. Kennedy and J. Kuti, Phys. Rev. Lett. 54 (1985) 2473.
A. Boriçi, Phys. Lett. B453 (1999) 46; J. Comput. Phys. 162 (2000) 123; hep-lat/0001019.
M. Fukugita *et al.*, Phys. Rev. D 47 (1993) 4739.
[^1]: Presented by K-I. Ishikawa
|
---
abstract: 'The Einstein Equivalence Principle has as one of its implications that the non-gravitational laws of physics are those of special relativity in any local freely-falling frame. We consider possible tests of this hypothesis for systems whose energies are due to radiative corrections, [*ie.*]{} which arise purely as a consequence of quantum field theoretic loop effects. Specifically, we evaluate the Lamb shift transition (as given by the energy splitting between the $2S_{1/2}$ and $2P_{1/2}$ atomic states) within the context of violations of local position invariance and local Lorentz invariance, as described by the $T H \epsilon\mu$ formalism. We compute the associated red shift and time dilation parameters, and discuss how (high-precision) measurements of these quantities could provide new information on the validity of the equivalence principle.'
---
6.0in 8.5in -0.25truein 0.30truein 0.30truein
=0.6cm
[**Testing the Equivalence Principle\
by Lamb shift Energies**]{}\
C. Alvarez\
and\
R.B. Mann\
Department of Physics\
University of Waterloo\
Waterloo, ONT N2L 3G1, Canada\
\
Introduction
============
The Einstein Equivalence Principle (EEP) is foundational to our understanding of gravity. It states that (i) all test bodies fall with the same acceleration regardless of their composition (the weak equivalence principle, or WEP) and (ii) the outcome of any local nongravitational test experiment is independent of the velocity and the spacetime orientation and location of the (freely-falling) apparatus [@will1]. Theories which obey the EEP, such as general relativity and Brans-Dicke Theory are called metric theories because they endow spacetime with a metric $g_{\mu\nu}$ that couples universally to all non-gravitational fields. Non-metric theories do not have this feature: they break universality by coupling auxiliary gravitational fields directly to matter. In this context a violation of the EEP means the breakdown of either Local Position Invariance (LPI) or Local Lorentz Invariance (LLI) (or both) so that observers performing local experiments could detect effects due to their position (if LPI is violated) or their velocity (if LLI is violated) in an external gravitational environment by using clocks and rods of differing composition. Limits on LPI and LLI are set by gravitational red-shift and atomic physics experiments respectively [@redshift; @PLC1; @PLC], each of which compares relative frequencies of transitions between particular energy levels that are sensitive to any potential LPI/LLI-violating effects.
The next generation of gravitational experiments will significantly extend our current understanding of the empirical foundations of the EEP. A proposed Eötvös experiment in space, known as the Satellite Test of the Equivalence Principle (STEP) attempts to test WEP to one part in $10^{17}$. The precision of gravitational red shift experiments could be improved to one part in $10^{9}$ by placing a hydrogen maser clock on board Solar Probe, a proposed spacecraft (see ref. [@will1] and references therein).
The dominant form of energy governing the transitions these experiments probe is nuclear electrostatic energy, although violations of WEP/EEP due to other forms of energy (virtually all of which are associated with baryonic matter) have also been estimated [@HW]. However there exist many other physical systems, dominated by primarily non-baryonic energies, for which the validity of the EEP is comparatively less well understood [@Hughes]. Such systems include photons of differing polarization [@Jody], antimatter systems [@anti], neutrinos [@utpal], mesons [@Kenyon], massive leptons, hypothesized dark matter, second and third generation matter, and quantum vaccum energies. Indeed, potential violations of the EEP due to vacuum energy shifts, which are peculiarly quantum-mechanical in origin ([*i.e.*]{} do not have a classical or semi-classical description) provide an interesting empirical regime for gravitation and quantum mechanics.
In this paper we investigate the effects that EEP-violating couplings have on Lamb-shift transition energies. Such transitions arise solely due to the radiative corrections inherent in quantum electrodynamics. A test of the EEP for this form of energy therefore provides us with a qualitatively new empirical window of the foundations of gravitational theory.
The Lamb shift is the shift in energy levels of a Hydrogenic atom due to radiative corrections. Such energy shifts break the degeneracy between states of with the same principal quantum number and total angular momentum, but differing orbital and spin angular momenta. The best known example is the energy shift between the $2S_{1/2}$ and $2P_{1/2}$ states in a Hydrogen-like atom, which arises due to interactions of the electron with the quantum-field-theoretic fluctuations of the electromagnetic field. For metric theories, the lowest order contribution for the Lamb shift is $1052$ MHz for hydrogen atoms. There is a $5$ MHz discrepancy with the experimental value of$1057.845(9)$ MHz [@explamb1] or $ 1057.851(2)$ MHz [@explamb2], that can be improved with the inclusion of higher order terms and corrections coming from the structure and recoil of the nucleus.
Any breakdown of LPI/LLI is determined entirely by the form of the couplings of the gravitational field to matter since local, nongravitational test experiments simply respond to their external gravitational environment. To explore such effects it is necesssary to develop a formalism capable of representing such couplings for as wide a class of gravitational theories as possible. We consider in this paper Lagrangian-based theories in which the dynamical equations governing the evolution of the gravitational and matter fields can be derived from the action principle \[action0\] d\^4x[L]{}d\^4x([L]{}\_G+[L]{}\_[NG]{})=0 . The gravitational part ${\cal L}_G$ of the Lagrangian density contains only gravitational fields; it determines the dynamics of the free gravitational field. The nongravitational part ${\cal L}_{NG}$ contains both gravitational and matter fields and defines the couplings between them. The dynamics of matter in an external gravitational field follow from the action principle \[action1\] d\^4x[L]{}\_[NG]{}=0 by varying all matter fields in an external gravitational environment.
We work in the context of a wide class of non-metric theories of gravity as described by the formalism [@tmu]. Phenomenological models of ${\cal L}_{NG}$ provide a general framework for exploring the range of possible couplings of the gravitational field to matter and, thus, the range of mechanisms that might conceivably break LPI or LLI. The formalism is one such model. It deals with the dynamics of charged particles and electromagnetic fields in a static, spherically symmetric gravitational field. In addition to all metric theories of gravitation, the formalism encompasses a wide class of non-metric theories.
A quantum-mechanical extension of the original classical formalism was developed by Will [@will] to calculate the energy shifts (due to [*e.g.*]{} hyperfine effects) in hydrogenic atoms at rest in a gravitational field. Since the ticking rate of a hydrogen-maser clock is governed by the transition between a pair of these atomic states, this extension can be used to determine the effect of the gravitational field on the ticking rate of such clocks. This provides a basis for a quantitative interpretation of gravitational redshift experiments which employ hydrogen-maser clocks, for example, the gravity probe A rocket-redshift experiment [@redshift]. Such experiments are a direct test of LPI.
This formalism was further extended by Gabriel and Haugan [@gabriel] who calculated the effects the motion of an atomic system through a gravitational field would have on the ticking rate of hydrogen-maser and other atomic clocks. Their extension can be used to compute energies of hyperfine and other energy shifts of hydrogen atoms in motion through a field. Here the physical effect under consideration is time dilation rather than the gravitational redshift. When LLI is broken, the rates of clocks of different types that move together through the gravitational field are slowed by different time-dilation factors. This nonuniversal behavior is a characteristic symptom of the breakdown of LLI [@haugan], just as nonuniversal gravitational redshift is the hallmark of LPI violation [@will].
We are concerned in this paper with extending this analysis to the Lamb shift, an energy shift whose origin is due to radiative corrections. We compute the Gravitationally Modified (GM) Lamb Shift in a field, and then discuss experiments which could potentially measure such effects. We find both EEP-violating contributions to the Lamb shift from the semiclassical Hamiltonian and its radiative corrections. The semiclassical contribution violates LLI only and is isotropic; the radiative corrections violate both LLI and LPI and are not isotropic. These contributions are functions of non-metric parameters which arise in the leptonic sector of the standard model, and so are not constrained by previous high-precision experiments which have set stringent bounds for analogous parameters in the baryonic sector [@PLC]. Of course all such contributions vanish for metric theories.
In order to calculate the (GM) radiative corrections, we shall modify the Feynman rules of Quantum Electrodynamics (QED) within the context of the formalism. Although we cannot use LPI/LLI symmetries, the gauge invariance of the theory is still present. We shall be concerned with the one photon contribution to the (GM) Lamb shift up to order $m\al(Z\al)^4$, with the nucleus treated as a fixed point charge. We do not include further (higher-order) refinements, since we are interested in the role of Lamb shift energies in the investigation of possible LPI/LLI violations and so expect any such violations to be qualitatively different from higher order corrections.
Our paper is organized as follows. In Sec. II the action is introduced and extended to frames moving with respect to the preferred frame defined by the gravitational field. This formalism is then used to calculate the electromagnetic fields produced by a point-like charge and to formulate (GM)QED. In Sec. III the (GM) Dirac equation is used to find the energy levels of hydrogenic atoms, and we compute the radiative corrections for those states in Sec. IV. In Sec. V the GM Lamb shift is related to redshift and time dilation parameters to study possible LPI and LLI violations respectively. Final conclusions are presented in Sec. VI. Several appendices summarize details of our calculations.
(GM) Action
============
The formalism was constructed to study electromagnetically interacting charged structureless test particles in an external, static, spherically symmetric (SSS) gravitational field, encompassing a wide class of non-metric (and all metric) gravitational theories. Originally employed as a computational framework designed to test Schiff’s conjecture [@will1], it permits one to extract quantitative information about the implications of EEP-violation that can be compared to experiment. It assumes that the non-gravitational laws of physics can be derived from an action: $$\begin{aligned}
\label{1}
S_{NG}&=&-\sum_a m_a\int dt\, (T-Hv_a^2)^{1/2}+\sum e_a \int dt\, v_a^\mu
A_\mu(x_a^\nu)\nonumber\\
& &+ \half\int d^4x\,(\epsilon E^2- B^2/\mu),\end{aligned}$$ where $m_a$, $e_a$, and $x_a^\mu(t)$ are the rest mass, charge, and world line of particle $a$, $x^0\equiv t$, $v_a^\mu\equiv dx_a^\mu/dt$, $\vec E\equiv-\vec\gr A_0-\pd\vec A/\pd t$, $\vec B \equiv \vec\gr\times\vec A$. The parameters $\epsilon$, and $\mu$ are arbitrary functions of the Newtonian gravitational potential $U= GM/r$ (which approaches unity as $U\to 0$), as are $T$ and $H$ which in general will depend upon the species of particles within the system (leptons in the present case).
A quantum mechanical extension of the action (\[1\]) which incorporates the Dirac Lagrangian was used by Will [@will] to study the energy levels of hydrogen atoms. In that case a local approximation to the action is employed. The spacetime scale of atomic systems allows one to ignore the spatial variations of $T$, $H$, $\epsilon$, $\mu$, and evaluate them at the center of mass position of the system, $\vec X=0$. This work was further extended by Gabriel and Haugan [@gabriel] who showed that after rescaling coordinates, charges, and electromagnetic potentials, the field theoretic extension of the action (\[1\]) can be written in the form $$\label{act}
S=\int d^4x\, \sp(i\nn\pd+e\nn\! A-m)\psi
+ \half\int d^4x\,(E^2-c^2B^2),$$ where local natural units are used, $\nn\! A=\gamma_\mu
A^\mu$, and $c^2=H_0/T_0\epsilon_0\mu_0$ with the subindex “0” denoting the functions evaluated at $\vec X=0$. The parameter $c$ is the ratio of the local speed of light to the limiting speed of the species of massive particle under consideration.
The action (\[1\]) (or (\[act\])) has been widely used in the study of LPI/LLI violating effects such as the effect of non-metric gravitational fields on the differential ticking rates of different types of atomic clocks, a violation of LPI [@will]. An analysis of the electrostatic structure of atoms and nuclei in motion through a gravitational field using (\[1\]) shows that the non-metric couplings encompassed by the formalism can also break LLI [@haugan]. This symmetry is broken when the local speed of light $c_*\equiv (\mu_0\epsilon_0)^{-1/2}$ differs from the limiting speed of a given species of massive particle $c_0\equiv (T_0/H_0)^{1/2}$, the latter being normalized to unity in (\[act\]). Further implications of the breakdown of LLI on various aspects of atomic and nuclear structure have also been investigated. Shifts in energy levels (including the hyperfine splitting) of hydrogenic atoms in motion through a gravitational field have been calculated [@gabriel] by transforming the representation of the action (\[act\]) to a local coordinate system in which the atom is initially at rest and then analyzing the atom’s structure in that frame. The local coordinate system in which the action is represented by Eq. (\[act\]), is called the preferred frame; moving frames are those systems of local coordinates that move relative to the preferred frame.
In the present work we generalize this analysis by using the action (\[act\]) to study radiative corrections to bound state energy levels in hydrogenic atoms. We follow the scheme given in Ref. [@gabriel], and analyze the atomic states in moving frames whose velocity is $\vec u$.
Consider an atom that moves with velocity $\vec u$ relative to the preferred frame. The moving frame in which this atom is initially at rest is defined by means of a standard Lorentz transformation. A convenient representation [@gabriel] of the action in this new coordinate system if the nongravitational fields $\psi$, $\vec A$, $\vec
E$, and $\vec B$ transform via the corresponding Lorentz transformations laws for Dirac, vector, and electromagnetic fields is, to $ O(\vec u^2)$, $$\begin{aligned}
\label{3}
S&=&\int d^4x\, \sp(i\nn\pd+e\nn\! A-m)\psi + \int d^4x\, J_\mu A^\mu
\nonumber\\
&+&\half\int d^4x\,\left[(E^2-B^2)\right.\\
&+& \left.\xi\left(\vec u^2 E^2-(\vec u\cdot\vec E)^2+(1+\vec u^2)B^2-(\vec u\cdot\vec B)^2
+2\vec u\cdot(\vec E\times\vec B)\right)\right]\nonumber.\end{aligned}$$ where $J^\mu$ is the electromagnetic 4-current associated with some external source (taken here to be a pointlike spinless nucleus). In our formulation, all non metric effects arise from the inequality between $c_0$ and $c_*$ in the electromagnetic sector of the action. The dimensionless parameter $\xi\equiv 1-(c_*/c_0)^2 =1-c^2 $ measures the degree to which LPI/LLI is broken for a given species of particle. The natural scale for $\xi$ in theories that break local Lorentz invariance is set by the magnitude of the dimensionless Newtonian potential, which empirically is much smaller than unity in places we can imagine performing experiments [@will1]. We are therefore able to compute effects of the terms in Eq. (\[3\]) that break local Lorentz invariance via a perturbative analysis about the familiar and well-behaved $c\to 1$ or $\xi\to 0$ limit.
The fermion sector of the action (\[3\]) implies that the equation of motion for the $\psi$ field is simply the Dirac equation coupled in the usual fashion to the potential $A_\mu$. On the other hand, the pure electromagnetic part of the action is modified with an extra term proportional to the small (species-dependent) parameter $\xi$. This will affect the electromagnetic field equations, and the photon propagator. In both cases we can calculate effects of the additional terms perturbatively.
The field equations coming from the action (\[3\]) are [@gabriel] $$\begin{aligned}
\label{4}
\vec\gr\cdot\vec E&=&\rho+\xi\left[\vec u\cdot\vec\gr(\vec u\cdot\vec E)-\vec
u\cdot\vec\gr\times
\vec B-\vec u^2\vec\gr\cdot\vec E\right],\\
\vec\gr\times\vec B-\dot{\vec E}&=&\vec j+\xi\left[\vec\gr\times(\vec u\times \vec E)+
\vec u\times\vec\gr(\vec u\cdot \vec B)+(1+\vec u^2)\vec\gr\times \vec B\right.\nonumber\\
&+&\left.\vec u^2\dot{\vec E}-\vec u(\vec u\cdot\dot{\vec E})
-\vec u\times\dot{\vec B}\right]\nonumber\end{aligned}$$ where $\rho$ and $\vec j$ are the charge density and current associated with the fermion field plus and external source (such as a nucleus.) Perturbatively solving these equations for electromagnetic potentials produced by a pointlike nucleus of charge $Ze$ at rest in the moving frame yields $$\begin{aligned}
\label{po}
A_0 &=& [1-\frac{\xi}{2}(\vec u^2+(\vec u\cdot\hat n)^2)]\phi\equiv\phi+\xi\phi'
\nonumber\\
\vec A&=&\frac{\xi}{2}[\vec u+\hat n(\vec u\cdot\hat n)]\phi\equiv\xi\vec A\,'\end{aligned}$$ where $ \hat n=\vec x/|\vec x|$, $\phi=Ze/4\pi|\vec x|$, and $\vec\gr\cdot \vec A=0$. Note that Eq. (\[po\]) agrees with the corresponding result from Ref. [@gabriel].
The primed fields in Eq. (\[po\]) signal a breakdown of LLI. Consequently we expect that this electromagnetic potential will modify the energy states of hydrogenic atoms prior to the inclusion of radiative corrections. We shall calculate these effects for the Lamb shift in the next section. In order to find the radiative corrections to these energy levels we must reformulate Quantum Electrodynamics (QED) according to the action (\[3\]). Since the fermion sector of the action does not change, the fermion propagator is unaltered; only the photon propagator needs to be modified.
To find the photon propagator, we go back to the action (\[act\]) and add a gauge fixing term of the form \[6\] S\_[GF]{}=-d\^4x, after which the resulting electromagnetic part can be written as $$\begin{aligned}
\label{7}
S_{EM}&=&\int d^4x\,\left[\half A_\mu\pd^\nu\pd_\nu A^\mu\right.\\
&+&\left.\frac{\xi}{2}(A_\mu\pd_0\pd^0 A^\mu+A_0\pd^\mu\pd_\mu A^0
-A_\mu\pd^\nu\pd_\nu A^\mu)\right]\nonumber\end{aligned}$$ where we have integrated by parts and neglected surface terms.
This action is still given in preferred frame coordinates. We can go the moving frame by performing the Lorentz transformations $$\begin{aligned}
\label{8} A_0\to A_0'&=&\gamma(A_0-\vec u\cdot \vec
A)\equiv\gamma\beta\cdot A\\ \pd_0\to\pd_0'&=&\gamma(\pd_0-\vec
u\cdot\vec\gr)\equiv\gamma\beta\cdot \pd\nonumber \end{aligned}$$ where $\gamma^2\equiv 1/(1-\vec u^2)$ and $\beta^\mu\equiv(1,\vec u)$; henceforth $\beta^2\equiv 1-\vec u^2$. Transforming Eq. (\[7\]) by using Eq.(\[8\]) gives \[9\] S\_[EM]{}=d\^4xA\^\_A\^where (in momentum space) $$\begin{aligned}
\label{10}
{\cal K}_{\mu\nu}=&-&\eta_{\mu\nu}k^2(1-\xi)\\
&-&\xi\gamma^2\left[\eta_{\mu\nu}(\beta\cdot k)^2+
\beta_\mu\beta_\nu k^2\right]\nonumber\end{aligned}$$ where $\eta_{\mu\nu}$ is the Minkowski tensor with a signature (+ - - -) and ${\cal K}_{\mu\nu}$ is the inverse of the photon propagator $G_{\mu\nu}$. Therefore after solving \_G\^=\_\^, we find up to first order in $\xi$ $$\label{11}
G_{\mu\nu}=-(1+\xi)\frac{\eta_{\mu\nu}}{k^2}
+\xi\frac{\gamma^2}{k^2}\left[\eta_{\mu\nu}\frac{(\beta\cdot k)^2}{k^2}
+\beta_\mu\beta_\nu\right] \quad .$$ The terms proportional to $\xi$ in Eq. (\[11\]) signal the breakdown of both LPI and LLI, since those terms are still present even if $\vec u=0$. The (GM) QED then differs from standard QED only in the expression for the photon propagator; the fermion propagator and Feynman rules are unchanged.
As the Lamb Shift is the shift between the $2S_{1/2}$ and $2P_{1/2}$ states, and since the Dirac equation for a Coulomb potential predicts those states to be degenerate, the difference between them in metric theories comes only from radiative corrections. For non-metric theories which can be described by the formalism these energy levels will be modified by the EEP-violating terms introduced in the source (Eq. (\[po\])), removing this degeneracy before introducing radiative corrections. Note that the fermion sector of the action does not change and therefore neither does the Dirac equation. The preferred frame effects appear only in the expression for the electromagnetic source produced by the nucleus. We shall now evaluate this contribution.
(GM) Dirac States
=================
The Dirac equation in the presence of an external electromagnetic field still reads as in the metric case: \[diraceq\] H|n=(p+m-e A\^0+eA)|n=E\_n|nwhere the various symbols have their usual meaning.
The (GM) energy levels of hydrogenic atoms are found by solving (\[diraceq\]) in the presence of the electromagnetic field (\[po\]) produced by the nucleus which entirely accounts for the preferred frame effects. If we replace Eq. (\[po\]) in (\[diraceq\]), the Hamiltonian can be written as \[eqq\] H=H\_0+H’,H’=-e’+eA’ where $H_0$ corresponds to the standard Hamiltonian (with Coulomb potential only), and the primed fields are defined as in Eq. (\[po\]). In terms of the known solutions for $H_0|n\rangle^0=E_n^0|n\rangle^0$, we can perturbatively solve Eq. (\[diraceq\]) by writing \[solution\] E\_n=E\_n\^0+E\_n’|n=|n\^0+|n’ with \[E’\] E\_n’&=&\^0n |H’|n\^0\^[(E)]{}+[E’\_n]{}\^[(M)]{}\
\[n’\] |n’&=&\_[r=n]{}|r\^0 where ${E'_n}^{(E)}$ and ${E'_n}^{(M)}$ account for the contributions coming from the respective electric and magnetic potentials.
We now proceed to calculate the energy levels related to the Lamb shift states. To obtain these, we find it convenient to use the exact solution for the Dirac spinor $|n\rangle^0$, expanding the final answer in powers of $Z\al$ to $ O((Z\al)^4)$. The relationship between this approach and an alternate one in which the Hamiltonian is first expanded in powers of $Z\al$ using a Foldy-Wouthuysen transformation is discussed in appendix \[Adirac\].
The unperturbed Dirac state $|n\rangle^0$ can be expressed as: \[spinor\] |n\^0=(
[r]{} G\_[lj]{}(r)|l;jm\
-iF\_[lj]{}(r)|l;jm
)where $|l;jm\rangle$ is the spinor harmonic eigenstate of $J^2$,$L^2$ and $J_z$, with respective quantum numbers $j,l$ and $m$. The functions $F$ and $G$ can be written in terms of confluent hypergeometric functions that depend in a non-trivial way on $Z\al$ for a given $l$ and $j$ [@exdirac].
Inserting the fields from (\[po\]) and (\[spinor\]) in $E_n'$, we write \[coul\] [E’\_n]{}\^[(E)]{}& =&(R\_[GG]{}+R\_[FF]{})jm;l|u\^2+(u)\^2| l; jm\
\[mag\] [E’\_n]{}\^[(M)]{}&=&-iR\_[GF]{}jm;l|()(u) +u|l;jm+ where “h.c.” means Hermitian conjugate and where \[rggo\] R\_[GG]{}=GGr\^2dr with $R_{FF}$ and $R_{GF}$ defined in an analogous manner.
We now evaluate this energy for the $2S_{1/2}$ and $2P_{1/2}$ states in this semiclassical approximation, prior to the inclusion of any radiative corrections. Since the angular operator in (\[mag\]) has odd parity (as given by $\nv$), it is straightforward to show that the magnetic contribution ${E'_n}^{(M)}=0$, so $E_n'={E'_n}^{(E)}$ for any state. Using the corresponding expressions for the harmonic spinors and the $F$, $G$ functions in (\[coul\]) for each Lamb state [@exdirac], we find \[elevels\] E’\_[2S\_[1/2]{}]{}&=&u\^2m(Z)\^2+\
E’\_[2P\_[1/2]{}]{}&=&u\^2m(Z)\^2+where we have expanded the exact solutions for $R_{GG}$ and $R_{FF}$ in powers of $(Z\alpha)^2$, and kept the first relativistic correction only. The angular integration and the $R_{GG}$ term are the same for both states, and so the non-relativistic limit is still degenerate for them. However the first relativistic correction coming from the $R_{FF}$ factor breaks the degeneracy, yielding \[ED\] E\_L\^[(D)]{}=E\_[2S\_[1/2]{}]{}-E\_[2P\_[1/2]{}]{}=m(Z)\^4+ O((Z)\^6)
We obtain the result that the $2S_{1/2}$–$2P_{1/2}$ degeneracy is lifted before radiative corrections are introduced. This ‘semiclassical’ nonmetric contribution to the Lamb shift is isotropic in the 3-velocity $\vec u$ of the moving frame and vanishes when $\vec{u} = 0$. Hence it violates LLI but not LPI.
In order to proceed to a computation of the relevant radiative corrections, we need to find the perturbative corrections for the energies and spinor states given by (\[E’\]) and (\[n’\]) respectively. The radiative correction $\delta E_n$ to the Dirac energy $E_n$ can be formally expressed as \[den\] E\_n=n|H|nwhere $\delta H$ accounts for the loop contributions as given by the gravitationally modified QED. Since EEP violating effects appear in both the photon propagator and the classical electromagnetic field, we expect \[h’\] H=H\^0+H’ In addition, the state $|n\rangle$ may be analogously expanded. Up to first order in $\xi$, we can therefore write (\[den\]) in the form \[den1\] E\_n=\^0n|H\^0|n\^0+
The contributions from the $|n\rangle'$ states are of the same order of magnitude (in terms of powers of $Z\al$) as the $\delta H'$ terms and so cannot be neglected. This may be seen by noting that, apart from the $\vec u$ dependance, $\phi'\sim\phi$ and so $^0\langle n|H'|r\rangle^0\sim E^0_n-E^0_r$. Inserting this in (\[n’\]) proves the statement. Note that the effect of the $|n\rangle'$ states was overlooked in Ref. [@gabriel]. If we identify $\delta H\rightarrow H_{(hf)}$,where $H_{(hf)}$ represents the perturbation to the Dirac Hamiltonian due to the spin of the nucleus, then by the same arguments as before, we can show that the term $\{^0\langle n| {H_{(hf)}}^0|n\rangle'+\hbox{h.c.}\}$ was omitted in the corresponding expression for the hyperfine energy.
(GM) Radiative Corrections
===========================
To lowest order in QED there are two types of radiative corrections to the energy levels of an electron bound in an external electromagnetic potential: the vacuum polarization ($\Pi$) and self-energy ($\Sigma$), along with a counterterm ($\delta C$) that subtracts the analogous processes for a free electron. These contributions are illustrated in Fig.1.
The energy shift due to these contributions for the state $|n\rangle$ can then be written as \[a1\] E\_n=E\_S+E\_P where \[as\] E\_S=n|-C|n, which corresponds to the self-energy contribution in Fig. 1(a) minus the corresponding counterterm, and \[app\] E\_P=n||n, which is the vacuum polarization contribution illustrated in Fig. 1(b).
In Fig. 1 the bold line represents the bound electron propagator. This propagator can be written in operator form as $(\nn p-\nn V-m)^{-1}$, with $$V^\mu(\vec x)\equiv-eA^\mu(\vec x) \quad\mbox{\rm and}\quad
p^\mu\equiv(E_n, \vec p)$$ where $A^\mu$ is the external electromagnetic potential. Here $E_n$ is the total energy of the state $|n\rangle $, which satisfies the Dirac equation $(\nn p-\nn V-m)|n\rangle =0$
Eq. (\[a1\]) represents the one loop correction (one power of $\alpha$) to the atomic energy levels as given by $E_n$. We are interested in obtaining the “lowest order" Lamb shift, which is the $\al(Z\al)^4$ contribution. (There are still more approximations that come after expanding the bound propagator, which introduce additional nonanalytic terms in the expression for the Lamb shift that behave like $\al(Z\al)^4\ln (Z\al)$).
The GM radiative corrections are found by evaluating (\[a1\]) where the external electromagnetic potential and the photon propagator are respectively given by Eqs. (\[po\]) and (\[11\]). All expressions will be expanded in terms of the LPI/LLI violating parameter $\xi$, and the velocity of the moving frame $\vec u$ up to $O(\xi)$ and $O(\vec u^2)$ as implied by (\[po\]) and (\[11\]). EEP-violating effects are all contained in the terms proportional to these quantities.
A variety of methods are available for evaluating the corrections in (\[a1\]), each differing primarily in the manner in which the bound electron propagator is treated. We shall follow the method of Baranger, Bethe and Feynman [@BBF] (hereafter referred to as BBF), in which the corrections in (\[as\]) are separated into a term in which the external potential acts only once, and another term in which it acts at least twice. This latter ‘many-potential’ term can be further separated into a nonrelativistic part, and a relativistic part which can be calculated by considering the intermediate states as free. This approach is sufficient for the lowest order calculation we consider here. We now proceed to outline the main steps of this method.
The self-energy term in Eq. (\[a1\]) can be written as $$\begin{aligned}
\label{a4}
\delta E_S&=&\frac{\alpha}{4\pi^3}\int d^4k\;iG^{\mu\nu}(k)
\langle n|\gamma_\mu\frac{1}{\nn p-\nn V -\nn k-m}\gamma_\nu|n\rangle -
\langle n|\delta C|n\rangle \quad .\end{aligned}$$ This expression gives a complex result for the level shift, since the denominators in the integral each have a small positive imaginary part. The resulting imaginary part of $\delta E_S$ represents the decay rate of the state $|n\rangle$ through photon emission. The Lamb shift refers to the real part of the shift, and only that part will be retained in the computation of Eq. (\[a4\]).
The difficulty in evaluating Eq. (\[a4\]) arises entirely from choosing a convenient expression for the bound propagator. The integrand in (\[a4\]) is rearranged in order to obtain one part which is of first order in the potential ($\delta E_1$), and another part ($\delta E_2$) which contains the potential at least twice. Using the identity[@BBF] O(p\_b-m)- (p\_a-m), to re-express $\gamma_\mu$ and $\gamma_\nu$ in (\[a4\]) and respectively identifying $p_b=p$, $p_a=p-k$, and $p_b=p-k$, $p_a=p$ yields after some manipulation \[a5\] E\_S=E\_1+E\_2, where \[a6\] E\_1=d\^3pd\^3p’\_n(p’){I\_1+I\_2+I\_3}\_n(p), with
\[a7\] I\_1&=& VG\^(k)d\^4k\
I\_2&=&V G\^(k)d\^4k\
I\_3&=& \_G\^(k)d\^4k-C and where \[a8\]E\_2&=&\_n(p’)M\_(p’,p’-s’-k)\
&&K\_+\^V(E\_0-k\_0;p’-s’-k,p+s-k)\
&&M\_\^(p+s-k,p)\_n(p)G\^(k)d\^4kd\^3pd\^3p’d\^3sd\^3s’\
&& M K\_+\^V M with \[a9\] M\_(p’,p-k)&=&V(p’-p) -V(p’-p)\
M\_\^(p’-k,p)&=&V(p’-p) -V(p’-p)The quantity $K_+^V$ is defined as $ -iK_+^V\equiv(\nn p-\nn V-m)^{-1}$,where in momentum space $K_+^{V}=\delta(E'-E)K_+^V(E;\vec p\,',\vec p)$.
In Eqs.(\[a6\]) and (\[a8\]) the $p$’s have time component $E_n$ and the $s$’s have time component 0. Note that the above derivations are independent of the specific form of the photon propagator $G_{\mu\nu}$.
Further evaluation entails a lengthy computation which in principle is analogous to that of BBF. In practice though, the calculation is substantially more complicated than in the metric case due to the additional non metric terms present in the photon propagator and the electromagnetic source related to a charged point particle. Regularization and renormalization procedures have to be modified accordingly as well. Details involving the subsequent computation of the self energy (and vacuum polarization) term are given in appendix \[Aloop\].
The final result for the loop corrections related to the Lamb shift is of the form \[EQ\] E\^[(Q)]{}\_L=E\_[2S\_[1/2]{}]{}-E\_[2P\_[1/2]{}]{} where each term is obtained from eq. (\[efinal\]) (and its relevant subsidiary equations) as calculated for the corresponding atomic state. By adding the “semiclasical" correction coming from the Dirac level (labeled by $(D)$ in Sec. III), the total Lamb shift reads \[EDQ\]E\_L&=&E\^[(D)]{}\_L+E\^[(Q)]{}\_L\
&=&(Z)\^4{-2.084+ +} where we have introduced the dimensionless parameter $\Delta\hat\epsilon^{ij}\equiv
2\Delta\hat E^{ij}/((Z\al)^4m^3)$ (see (\[Eij\])), and used Eqs. (\[refener\]) and (\[E0\]) in the evaluation of (\[EQ\]) through Eq. (\[efinal\]).
The former result is the energy shift associated with the particular states in (\[EQ\]). However in Eq. (\[efinal\]) we have derived a general expression for the one-loop radiative corrections related to any atomic state. These are \[en0\] E\_[n0]{}=mfor $l=0$, and \[enl\] E\_[nl]{}=mfor $l\not=0$; where we have not explicilty written the terms proportional to the moving frame velocity. Here C\_[lj]{}={
[ll]{} 1/(l+1)&\
-1/l&
. and $E_*$ is defined by (\[erefer\]). Values for this reference energy can be obtained from Ref.[@harriman] up to states with $n=4$.
Note that in addition to the explicit dependence on the frame velocity in Eq. (\[EDQ\]), there exists a position dependence hidden by the rescaling of the original action (Eq. (\[3\])), which was considered locally constant throughout the computation. The full parameter dependence in Eq. (\[EDQ\]) can be recovered by replacing \[rede\] ,mm,E\_LE\_L in the preceding equations.
Note that $\xi$ in Eq. (\[EDQ\]) accounts for any EEP violation coming from a non-universal gravitational coupling between photons and leptons. A further distinction can still be made between leptons and antileptons. In principle a matter/antimatter violation of the EEP could be measured in a Lamb shift transition, through the appearance of virtual positron/electron pairs in the vacuum polarization loop contribution [@Schiff]. This will add a non metric term to Eq. (\[EDQ\]), of the form (see appendix \[Apol\] for more details): \[posivac\] E\_L\^[(+)]{}=-\_[e\_+]{} (Z)\^4(1+2|u|\^2) where $\xi_{e_+}=1-c_{e_-}/c_{e_+}$ accounts for the difference between the limiting speed of electrons ($c_{e_-}=c_0$) and positrons ($c_{e_+}$).
We turn next to the question of relating the Lamb shift to observable quantities in order to parameterize possible violations. of the EEP.
Test for LPI/LLI Violations
===========================
We begin by considering a general idealized composite body made up of structureless test particles that interact by some nongravitational force to form a bound system. The conserved energy function of the body $E$ is assumed to have the quasi-Newtonian form [@haugan] \[pp1\] E=M\_Rc\_0\^2-M\_RU(X)+M\_R||\^2+.. where $\vec X$ and $\vec V$ are respectively the quasi-Newtonian coordinates and velocity of the center of mass of the body, $M_R$ is the rest energy of the body and $U$ is the external gravitational potential. Potential violations of the EEP arise when the rest energy $M_R$ has the form M\_Rc\_0\^2=M\_0c\_0\^2-E\_B(X,V) where $M_0$ is the sum of the rest masses of the structureless constituent particles and $E_B$ is the binding energy of the body. It is the position and velocity dependence of $E_B$, which signals the breakdown of the EEP. Expanding $E_B$ in powers of $U$ and $V^2$ to an order consistent with (\[pp1\]) we have \[bin\] E\_B(X,V)=E\_B\^0+m\_P\^[ij]{}U\^[ij]{}-m\_I\^[ij]{}V\^iV\^j where $ U^{ij}$ is the external gravitational potential tensor, satisfying $U^{ii}=U$. The quantities $\delta m_P^{ij}$ and $\delta m_I^{ij}$ are respectively called the anomalous passive gravitational and inertial mass tensors. They depend upon the detailed internal structure of the composite body. In an atomic system they can be expected to consist of terms proportional to the electrostatic, hyperfine, Lamb shift, and other contributions to the binding energy of an atomic state.
In a gravitational redshift experiment one compares the local energies at emission $E_{em}$ and at reception $E_{rec}$ of a photon transmitted between observers at different points in an external gravitational field. The measured redshift is defined as $$Z=\frac{E_{em}-E_{rec}}{E_{em}}$$ Using (\[pp1\]) (with $\vec V=0$) to relate the transition energies at the two different points, this parameter can be expressed as [@haugan] \[redz\] Z=U(1-), = . Clearly $Z$ depends (through $\delta m_P^{ij}$) upon the specific test system used in the experiment. An absence of LPI violations will mean $\Xi=0$, and so $Z$ will be independent of the detailed physics underlying the energy transition .
The LLI violations may be empirically probed through time dilation experiments. These experiments compare atomic energy transitions as measured by the moving frame ($\Delta E_B$) and preferred frame ($\Delta E_B^0$), which can be related via [@gabriel] E\_B=E\_B\^0(1-\[A-1\]) with the time dilation coefficient $A$ defined by \[timedi\] A=1- .
Here $\delta m_I^{ik}$ represents the difference between the anomalous inertial tensors related to the atomic states involved in the transition. The coefficient $A$ represents the dilation of the rate of a moving atomic clock whose frequency is governed by the transition. Since the anomalous mass tensor is not isotropic, $A$ depends upon the orientation of the atom’s quantization axis relative to its velocity through the preferred frame. Note that if LLI is valid the anomalous inertial mass tensor associated with every atomic state vanishes, so that $A=1$.
Here we consider the possibility of employing the Lamb shift as the atomic transition governing the appropriate experiment. To do so we must compute the relevant $\Xi$ and $A$ coefficients respectively.
In order to calculate the corresponding $\delta m_p^{ij}$ related to the Lamb shift, we must find the manner in which $\Delta E_L$ varies as the location of the atom is changed. Setting $\vec u=0$ in (\[EDQ\]) and performing the rescaling given in (\[rede\]), we obtain \[lpi\]E\_L =[E]{}\_L()\^[5/2]{} {1+a+b(1+)(\^2)} with ${\cal E}_L=\frac{m}{6\pi}(Z\al)^4\al /b$, and $$a=b(-4.534+\frac{3}{2}\ln{1\over\al^2})\qquad
b=1/(-2.084+\ln{1\over\al^2})$$ where $ {\cal E}_L$ represents the metric value (within the given approximations) for the Lamb shift. Note that there is still a position dependence in (\[lpi\]) through the definition of 1- .
We recall that the total energy of the system can be expressed in term of \[etotal\] E=m+E\_L+where $\cdots$ represents other contributions for the binding energy of the system.
The functions $T$, $H$, $\epsilon$ and $\mu$, considered to be functions of $U$ and evaluated at the instantaneous center of mass location $\vec X=0$ for purposes of the calculation of $\Delta E_L$, are now expanded in the form \[tu\] T(U)=T\_0+T\_0’g\_0X+O(g\_0X)\^2 where $\vec g_0=\vec\gr U|_{\vec X=0}$, $T_0=T|_{\vec X=0}$, and $T_0'=dT/dU|_{\vec X=0}$. It is useful to redefine the gravitational potential $U$ by \[u\] U-g\_0X whose gradient yields the test-body acceleration $\vec g$.
If the above is used to expand (\[etotal\]), we get \[eee\] E=(m+[E]{}\_L)(1-U)+[E]{}\_L U{(5-a-2b)\_0 -a\_0} where we have used (\[rede\]); and neglected terms proportional to $\xi$, since the main position dependence parameterization is given in terms of: \[para\] \_0=(+ -), \_0=(+ -)
If we now identify (\[eee\]) with Eqs. (\[pp1\]) and (\[bin\]), we can obtain the corresponding Lamb shift contributions to the binding energy and anomalous passive mass tensor as \[massten\] E\_B\^[0(L)]{}&=&-[E]{}\_L\
m\_P\^[ij(L)]{}&=&E\_B\^[0(L)]{}{(5-a-2b)\_0 -a\_0} .This result was first presented in Ref. [@cm], where in (\[massten\]) we have corrected the latter for a sign error in the coefficient multiplying $\Lambda_0$ and a missing factor $b$ in the $\Gamma_0$ term.
Inserting (\[massten\]) in (\[redz\]), we obtain \^L=3.424\_0-1.318\_0 \[lamlpi\] as the LPI violating parameter associated with the Lamb shift transition. Note that if LPI is valid then $\Gamma_0=\Lambda_0=0$.
In comparing the result (\[lamlpi\]) to anomalous redshift parameters computed for other systems, it is important to note that we are working with units that are species dependent. Recall that the choice of $c_0=1$, and the redefinition of the gravitational potential (\[u\]) involves the $T$ and $H$ functions associated with electrons (or more generally a given species of lepton).
Consider, for example, hyperfine transitions (maser clocks). In this case the leptonic and baryonic gravitational parameters appear simultaneously. This atomic splitting comes from the interaction between the magnetic moments of the electron and proton (nucleus). The proton metric appears only in the latter, and so it does not affect the principal and fine structure atomic energy levels. It is simple to check that the hyperfine splitting scales as \[hf\] E\_[hf]{}=[E]{}\_[hf]{} where the label $B$ is added to distinguish baryonic related functions from leptonic ones; and ${\cal E}_{hf}$ depends only on atomic parameters.
In expanding (\[hf\]) according to (\[tu\]), we obtain \[hf1\] E\_[hf]{}=[E]{}\_[hf]{}(1-U\_B)+[E]{}\_[hf]{} U\_B\^[hf]{} with \^[hf]{}=3\_B-\_B+where $U_B$, $\Gamma_B$ and $\Lambda_B$ are the baryonic analogues of (\[u\]), and (\[para\]) respectively. In (\[hf1\]) we rescaled the atomic parameters to absorb the functions and chose units such that $c_B=1$. The quantity $\Delta$ is given by =2and would vanish under the assumption that the leptonic and baryonic parameters were the same.
Turning next to experiments which test LLI, we need to obtain the tensor $\delta m_I^{ij}$ appropriate to the Lamb shift. This tensor is obtained after taking partial derivatives of $\Delta E_L$ with respect to $u_i$ and $u_j$ (note $\vec
V\equiv\vec u$). Substituting the result into (\[timedi\]) yields \[al\] 1-A\^L={+ 3.074-0.011\^2+ \^[ij]{}} for the Lamb shift time dilation coefficient, where $\theta$ is the angle between the atom’s quantization axis and its velocity with respect to the preferred frame.
Note that the coefficient $A_L$ depends upon $\Delta\hat
\epsilon^{ij}$, the evaluation of which involves the computation of an infinite sum as given by (\[Eij\]). The dominant contribution in Eq. (\[al\]) comes from the Dirac part of the energy (proportional to $\frac{1}{\al}$ ), which produces an overall shift only. Non-isotropic effects arise solely due to radiative corrections.
In general, an experimental test of LLI involves a search for the effects of motion relative to a preferred frame such as the rest frame of the cosmic microwave background. A detailed analysis about the interpretation of LLI violating experiments is presented in Ref. [@gabriel], which analyzed experiments concerned with hyperfine transitions, obtaining an expression for the time dilation parameter corresponding to that kind of transition. This parameter is negligible in comparison with other sources of energy, such as nuclear electrostatic energy in the case of the $^9$Be$^+$ clock experiment [@PLC1].
In summary, we have been able to parameterize EEP violations arising from Lamb shift transitions associated with redshift and time dilation experiments. In these types of EEP violating experiments one typically looks for variations of the energy shift due to changes in either the gravitational potential or the direction of the preferred frame velocity. The feasibility of such experiments is hindered by the present level of precision of Lamb shift transitions (one part in $10^6$) in comparison to the magnitudes of such changes. In the first case, any Earth based experiments will be limited by the small size of the Earth’s gravitational potential ($\approx 10^{-9}$), which is well beyond any foreseeable improvement in Lamb shift precision. Similar problems appear in the second case, where the known upper bound $|\vec{u}| < 10^{-3}$ [@will1] for the preferred frame velocity, leaves no room for any improvement on the EEP violating parameter $\xi$, since anisotropic effects go as $\xi |\vec{u}|^2$.
However useful information can still be extracted from Eq. (\[EDQ\]) if we use the current level of discrepancy between the experimental result [@explamb1] and the theoretical (metric) value [@Eides] to bound the nonmetric contributions for the Lamb shift. This constrains $\xi< 1(1)\times10^{-5}$. Similar bounds can be obtained by considering empirical information about other atomic states. In this context, the indirect measurement of the $1S$ Lamb shift [@weitz] gives a limit $\xi< 1.4(1) \times10^{-5}$, and the measurement of the $2S_{1/2}-2P_{3/2}$ fine structure interval [@hagley]: $\xi < 0.7 (1.4)\times 10^{-5}$. If we drop the assumption that positrons and electrons have equivalent couplings to the gravitational field [@Schiff], we find that there is an additional contribution to (\[EDQ\]) due to $\xi_{e^+}
\neq \xi_{e^-}$. This contribution arises entirely from radiative corrections and is given by eq. (\[posivac\]). Making the same comparisons as above, we find the most stringent bound on this quantity to be $|\xi_{e^+} | < 10^{-3}$.
The previous bounds were obtained by using (\[coul\]) and (\[en0\]) or (\[enl\]) to calculate the corresponding nonmetric Dirac and radiative corrections contributions respectively. The $1S$ Lamb shift experiment, actually measures the transition: $(E_{4S}-E_{2S})-\frac{1}{4}(E_{2S}-E_{1S})$, and so we use this one to make the comparison, where experimental and theoretical values are given in ref.[@weitz]. In the other experiment we need to use the non metric part of $E_{2S_{1/2}}-E_{2P_{3/2}}$ ($\equiv \Delta_\xi$), namely: \[fine\] \_=(Z)\^2mwhere the first term comes from the Dirac contributions (here + and - label the transition coming from the $2P_{3/2}$ state with $|M|=3/2$ and $|M|=1/2$ respectively) and the second one from radiative corrections. Note that the leading anisotropic effects stem from the non relativistic contributions, and so their ratio with the metric value, $O(m( Z\al)^4)$, is $O(\xi u^2 /(Z\al)^2)$ , instead of $O(\xi u^2)$ as for the [*classical*]{} Lamb shift. Time dilation experiments will look for changes on the $E_{2S_{1/2}}-E_{2P_{3/2}}$ splitting as the Earth rotates, which would single out only the preferred frame contributions. Current experiments [@hagley] measure a value of $9911.200(12)$ MHz for that transition, which gives a nominal bound (coming from the experimental error) of $\frac{3}{2}\xi\cos^2\theta<1\times 10^{-4}$ for the preferred frame part. This bound should improve once appropriate experiments are carried out, since these will look for periodic behaviour which can be isolated and measured with high precision.
Note that an empirical value for the Lamb shift is obtained from Ref.[@hagley] by subtracting the theoretical result of the fine splitting $2P_{1/2}-2P_{3/2}$. Now by following the previous formalism we can parameterize the LPI violation in the former experimental result through: \[fineb\] E\_[2S\_[1/2]{}-2P\_[3/2]{}]{}\^[exp]{}=([E]{}\_f+[E]{}\_L)(1-U) +U([E]{}\_f\^f+[E]{}\_L\^L) where we have added the corresponding parameters related to the fine transition [@will1]: ${\cal E}_f$ and $\Xi^f$. Constraining the ratio of this quantity to a direct measurement of the Lamb shift [@explamb1] to lie within experimental/theoretical error, we obtain the bound $|U(\Xi^L-\Xi^f)|=|U(0.576\Gamma_0+1.318\Lambda_0)| < 10^{-5}$. This result is sensitive to the absolute value of the total local gravitational potential [@Hughes; @Good], whose magnitude has recently been estimated to be as large as $3\times 10^{-5}$ due to the local supercluster [@Kenyon]. Hence measurements of this type can provide us with empirical information sensitive to radiative corrections that constrains the allowed regions of $(\Gamma_0,\Lambda_0)$ parameter space. Unfortunately the present level of precision in measuring the Lamb shift allows only a rather weak constraint.
Discussion
==========
We have computed for the first time radiative corrections to a physical process, namely the energy shift between two hydrogenic energy levels that are semi-classically degenerate, within the context of the formalism. The corresponding (GM) QED was derived, and the (GM) expressions for the propagators were obtained. The nonmetric aspects of a theory describable by the formalism can be all included in the photon propagator, given an appropriate choice of coordinates, leaving the fermion propagator unchanged. The addition of more parameters to the theory (by the functions) entail new renormalizations, where not only charge and mass need to be redefined but also the parameters.
The approach we took to solve for the semi-classical Dirac energies (sec. III) differs from the one given in Ref. [@gabriel], in which the Dirac Hamiltonian was expanded using Foldy-Wouthuysen transformations yielding the first relativistic correction to the Schrödinger Hamiltonian (as introduced for example, for the Darwin and spin-orbit terms), and subsequently the energies. Instead we began from the fully relativistic expression, where the perturbations come only from the preferred frame terms of the electromagnetic potential. Our approach involved evaluating expectation values with respect to the relativistic spinors instead of their nonrelativistic extensions (or Pauli states). The effects of relativistic corrections such as spin-orbit coupling are therefore included exactly in this approach. Once this is done, the final result is expanded to keep it within the desired order. The semi-relativistic approach is not suitable when preferred frame effects are studied.
Qualitatively new information on the validity of the EEP will be obtained by setting new empirical bounds on the parameters $\xi$, $A_L$ and $\Xi_L$ which are associated with purely [*leptonic*]{} matter. Relatively little is known about empirical limits on EEP-violation in this sector [@Hughes]. Previous experiments have set the limits [@PLC] $|\xi_B| \equiv |1 -c_B^2|\,<\,6\,\times \,10^{-21}$ where $c_B$ is the ratio of the limiting speed of baryonic matter to the speed of light. In our case we obtain an analogous bound on $\xi$ for electrons from the difference between current experimental and theoretical values, giving $|\xi| < 10^{-5}$. Although much weaker than the bounds on $\xi_B$, it is comparable to that noted in a different context by Greene [*et. al.*]{} [@Greene]. They considered a similar formalism (with $\vec u=0$) for analyzing the measurement of the photon wavelength emitted in a transition where a mass $\Delta m$ is converted into electromagnetic radiation, thereby providing an empirical relationship between the limiting speed of massive particles (electrons) and light.
The breakdown of LPI for the Lamb shift in the context of a nonmetric theory of gravity describable by the formalism is embodied in the the anomalous gravitational redshift parameter (\[lamlpi\]). Recall that $\Xi$ depends on the nature of the atomic transition through the evaluation of the anomalous passive tensor. This tensor will have differing expressions for differing types of atomic transitions [@will1]. An atomic clock based on the Lamb shift transition will, in a non-metric theory, exhibit a ticking rate that is dependent upon the location of the spacetime frame of reference and that differs from frequencies of clocks of differing composition. For example, the gravity probe A experiment [@redshift] employed hydrogen-maser clocks, and was able to constrain the corresponding LPI violating parameter related to hyperfine transitions: $$|\Xi^{Hf}|=|3\Gamma_B-\Lambda_B +\Delta|<2\;\times\;10^{-4}
\label{hyplpi}$$ This experiment involves interactions between nuclei and electrons and so does not (at least to the leading order to which we work) probe the leptonic sector in the manner that Lamb-shift experiments would. In general Eq. (\[redz\]) will describe the gravitational redshift of a photon emitted due to a given transition in a hydrogenic atom; for a hyperfine transition the redshift parameter is (\[hyplpi\]), whereas it is (\[lamlpi\]) for the Lamb shift transition.
An analogous experiment to test for LPI violations based on Lamb shift transition energies poses a formidable experimental challenge because of the intrinsic uncertainties of excited states of Hydrogenic atoms. Setting empirical bounds on $\Xi_L$ by precisely comparing two identical Lamb shift transitions at different points in a gravitational potential would appear unfeasible since the anticipated redshift in the background potential of the earth ($\approx 10^{-9}$) is much smaller than any foreseeable improvement in the precision of Lamb-shift transition measurements [@Eides]. One would at least need to perform the experiment in a stronger gravitational field (such as on a satellite in close solar orbit) with 1-2 orders-of-magnitude improvement in precision. A ‘clock-comparison’ type of experiment between a ‘Lamb-shift clock’ and some other atomic frequency standard [@will1] is, in principle, sensitive to the absolute value of the total local gravitational potential [@Hughes; @Good], as noted earlier. With this interpretation, comparitive transition measurements of the type discussed in the previous section can more effectively constrain the allowed regions of $(\Gamma_0,\Lambda_0)$ parameter space than can measurements which depend upon changes in the gravitational potential. Of course exploiting anticipated improvements in precision of measurements of atomic vacuum energy shifts [@Eides] will yield better bounds on $\xi_{e^-}$ and $\xi_{e^+}$ via (\[EDQ\]).
Violations of LLI single out a preferred frame of reference. In fact, the search for a preferred direction motivated the most precise tests of LLI performed so far [@PLC1; @PLC]. We have extended the analysis of the effects of motion relative to a preferred frame to account for the radiative correction for the atomic energies associated with the Lamb shift, as embodied in the expression (\[al\]). This non-universality reflects the breakdown of spatial isotropy for quantum-mechanical vacuum energies. The coefficient $A_L$ depends upon $\Delta\hat \epsilon^{ij}$, the evaluation of which involves the numerical computation of the sum in (\[Eij\]). Unfortunately, the intrinsic linewidths of the relevant states render direct measurement of such effects unfeasible. More precise empirical information on the value of $\xi$ can be obtained by precisely measuring changes in the $E_{2S_{1/2}}-E_{2P_{3/2}}$ splitting as functions of terrestrial or solar motions. However these effects are insensitive to radiative corrections, depending instead upon the semi-classical non-metric effects discussed in section III.
Finally, we note that our formalism could be applied for muonic atoms. For a muon-proton bound system, we will obtain an expression similar to that of (\[efinal\]), but where all parameters refer to muons. For an anti-muon electron bound system (a muonic atom) a similar analysis would apply. However in both cases the mass and spin of the muon could not be neglected.
We expect that the intrinsically quantum-mechanical character of the radiative corrections will motivate the development of new LPI/LLI experiments based on the Lamb shift transition. In so doing we will extend our understanding of the validity of the equivalence principle into the regime of quantum-field theory.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. We are grateful to C.M. Will for his initial encouragement in this work, and to M. Haugan and R. Moore for helpful discussions.
Appendices
===========
Semi-Relativistic Calculation of Hydrogenic Energy Levels \[Adirac\]
====================================================================
Consider a hydrogenic atom immersed in an external gravitational field, moving with velocity $\vec u$ relative to the preferred frame. In Sec. III we follow a fully relativistic approach to solve for the atomic energy levels. That is, we perturbatively solve the Dirac equation in the presence of the electromagnetic field of the nucleus, where the unperturbed states correspond to the Dirac solution in the presence of a Coulomb potential only (the metric case).
We consider here the use of the Foldy-Wouthuysen transformation in solving (\[diraceq\]). In this approach, we write H=H\_[c]{}+H\_[mag]{}+H\_[mv]{}+H\_[SO]{}+H\_[D]{} with \[semi\] H\_[c]{}&=&m+-eA\_0\
H\_[mag]{}&=&(pA+Ap)+ B\
H\_[mv]{}&=&-\
H\_[SO]{}&=&E +Ep\
H\_D&=&Ewhere $A_\mu$ is given by Eq. (\[po\]).
As shown in section III, we can take $H_{mag}\rightarrow 0$, since the magnetic field does not contribute to the atomic energy levels. We can then group the terms in the Hamiltonian as H&=&H\_[c]{}+H\_f\
H\_f&=&H\_[mv]{}+H\_[SO]{}+H\_D where we have defined the fine contribution to the Hamiltonian ($H_f$), in order to account for the first relativistic correction $O((Z\al)^4)$ to the atomic energy levels.
We start writting a formal solution for $H|n\rangle=E_n|n\rangle$, in term of its non-relativistic limit: \[12\] H\_[c]{}|n\_[c]{}=E\_n\^[c]{}|n\_[c]{}, as \[13\] |n =|n \_[c]{}+|n\_f,E\_n=E\_n\^[c]{}+\_[c]{}n|H\_f|n \_[c]{} where the index “$f$" accounts for the first relativistic correction to the states and energies.
Since $A_0=\phi+\xi\phi'$, and so $H_{c}=H_{c}^0+\xi H_{c}'$, we do not know the exact solution for (\[12\]), but the perturbative expansion: \[17\] |n\_[c]{}=[|n\_[c]{}]{}\^0+|n\_[c]{}’\^[c]{}=[E\_n]{}\^[c(0)]{}+\^0\_[c]{}n|H\_[c]{}’|n \_[c]{}\^0 where \[17a\] H\_[c]{}\^0|n\_[c]{}\^0=(m+-e)|n\_[c]{}\^0= E\_n\^[c(0)]{}|n\_[c]{}\^0
If we use (\[17\]) along with $H_f=H_f^0+\xi H_f'$ in (\[13\]), we can finally write up to $O(\xi)$, \[222\] E\_n&=&E\_n\^0+E\_n’ =\^0\_[c]{}n|(H\_c\^0+H\_f\^0)|n \_[c]{}\^0\
&+&+O((Z)\^6)
We see then that under this semi-relativistic approach, we must address the problem of finding the states $|n \rangle_{c}'$, whose contribution to (\[222\]) is between the brace brackets. This is equivalent to include the first relativistic correction coming after solving H\^0|n\^0=(H\_[c]{}\^0+H\_f\^0+)|n\^0 as |n\^0=|n\^0\_[c]{}+|n\^0\_f+, since, we can show \[hc\] {\^0\_[c]{}n|H\_f\^0|n \_[c]{}’+}={\_f\^0n|H\_[c]{}’|n \_[c]{}\^0 +}
This relation allows us to rewrite part of (\[222\]) as \[26\] [E\_n]{}’&=&(\_[c]{}\^0n|+\_f\^0n|+)(H\_[c]{}’+H\_f’+) (|n\_[c]{}\^0+|n\_[f]{}\^0+)\
&=&\^0n|H’|n\^0.
It is clear then that if we start with the exact solution for the Dirac equation in the presence of a Coulomb potential, we can avoid working with the states $|n \rangle_{c}'$. Note that since we are interested only in the first relativistic correction, the result (\[26\]) must be expanded to $O((Z\al)^4)$.
Unfortunately for hyperfine or Lamb shift energies, the effect of the primed states cannot be removed, since they both come from perturbations to the (known) relativistic solution of the Dirac equation in the presence of a Coulomb potential only.
A semi-relativistic expression for the Hamiltonian of a hydrogenic system was worked out in Ref. [@gabriel], where the effects of nuclear spin (hyperfine effect) were also included within the context of LLI violations. The result presented there for the atomic energy levels is incomplete though, since the contribution of the prime states was overlooked, as discussed at the end of Sec. III.
Loop calculations\[Aloop\]
==========================
Given the form of the photon propagator (\[11\]), it is convenient to divide the calculation into two parts E\_S= E\_S\^[(A)]{}+E\_S\^[(B)]{} where $\delta E_S^{(A)}$ groups the contributions of the terms proportional to $\eta_{\mu\nu}$ in $G_{\mu\nu}$, whereas $\delta E_S^{(B)}$ contains those proportional to $\gamma^2 = 1/(1-\vec u^2)$ and $\xi$. We are interested in solving for the shift in energy levels up to first order in $\xi$, so it is enough to consider a Coulomb potential as the source for part B, while for part A the full source as defined in Eq. (\[po\]) needs to be included.
We mention again that we are interested in calculating the GM Lamb shift to lowest nontrivial order in $\alpha$, [*i.e.*]{} up to $O(\al(Z\al)^4)$. To this order, we can use the nonrelativistic expressions for both the large and small component of the electron spinor $\psi$. So for example, if we make the substitution \[a20\] (p)=(Zm)\^[-3/2]{} w(t), where $w(\vec t)$ is a dimensionless spinor whose first two components are of order unity, and the last two are of order $Z\alpha$, we can assign orders to the various terms according to \[a20a\] & &p\_i\~Zm, E\_0-m\~(Z)\^2m\
& &eA\_0d\^3p’\~eA\_id\^3p’\~(Z)\^2m\
& & \_i\_nd\^3p\~Zm.These approximations will be used in the sequel to simplify the expressions we obtain.
Type A Contributions to the Self-energy
---------------------------------------
Here we will consider \[a12\] G\_\^[(A)]{}=-(1+) and $\nn V=-e A_\mu\gamma^{\mu}$, with $ A_\mu$ given by Eq.(\[po\]). This part of the calculation is almost identical to that of BBF[@BBF]; the only difference is that now we have to consider a source that contains a magnetic part in addition to the electric one.
We begin by computing $\delta E_1$. Relating the counterterm $\delta C$ to the renormalization of the electron mass and regularizing the photon propagator via \[a14\] -\_[\^2]{}\^[\^2]{}. we find that $I_2$ and $I_3$ in (\[a6\]) become \[i2i3\] I\_2&=(1+)&V{(p\^2/\^2)-(\^2/p\^2)}\
I\_3&=(1+)&{ V ((\^2/p\^2)+)+m (m\^2/p\^2)} . On the other hand, we obtain for $I_1$ I\_1&=&(1+){-V -p’pV\_0\^1 (p\_x\^2/\^2)+V\_0\^1dx(\^2/p\_x\^2)\
&+&\_0\^1{[(1-x)p\^2+xp’\^2+2p’]{}V + p’Vp\
&-&2Vp’(1-x)p-2Vp xp’+Vp\_xp\_x}},where $p_x=xp'+(1-x)p$.
We can simplify this expression by letting the momentum operators $\nn p'$ and $\nn p$ respectively act on the spinors $\sp (\vec{p}\,')$ and $\psi(\vec p)$, using the Dirac equation and (\[a20a\]) to keep terms up to the desired order.
Adding together $I_1$, $I_2$, and $I_3$ we obtain a result correct to order $\alpha(Z\alpha)^4$: \[a17\] E\_1\^[(A)]{}&=&(1+)(p’)(p)d\^3p’d\^3p\
&-&(1+)n||n, with $q=p'-p$, and $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^\mu,\gamma^\nu]$. Note that the term proportional to $q^2$ in Eq. (\[a17\]) needs to be evaluated with only the large component of $\psi$ and $\nn V\simeq V_0$ ($\gamma_0\sim 1$).
We point out that the initial ultraviolet divergence in (\[i2i3\]) is cancelled after the addition of the $I$’s in (\[a17\]). The remaining infrared divergence will be cancelled by a similar term which comes from the many-potential part of the level shift. A similar cancellation occurs in the non-gauge invariant term present in Eq. (\[a17\]). These cancellations are non-trivial, and provide useful cross checks to our calculation.
Consider next the evaluation of $\delta E_2$. Since the operator $M_\mu$ satisfies the transversality condition kM=kM\^=0 we can write $M_0= \vec k\cdot\vec M/k_0$.
Using Vk\_=2Vk\_-2V\_k +k\_V, in the first term of Eq.(\[a9\]) the operator $M_\mu^{\dag}$can be decomposed into \[Msplit\] [M\_]{}\^=[M\_]{}\^[I]{}+[M\_]{}\^[II]{} with \[mIe\] M\_\^[I]{}&=&{-\
&+&k\_(-)}V\
M\_\^[II]{}&=&2(V\_k-Vk\_)/(k\^2-2pk), \[mIIe\] each of which still satisfies \[a25\] M\^[I]{}k=M\^[II]{}k=0.
In terms of these operators we now have \[a23a\] E\_2= M\^IK\_+\^VM\^I+ M\^[II]{}K\_+\^VM\^[II]{}+ M\^IK\_+\^VM\^[II]{}+M\^[II]{}K\_+\^VM\^[I]{}, where each term represents a contribution to Eq. (\[a8\]) involving the products of only $M^I$ or $M^{II}$ or cross terms operators. The simplification of these terms is quite analogous to that shown in BBF [@BBF]. The decomposition of the M operator in (\[Msplit\]) allows one to use simpler expressions for the bound propagator $K_+^V$. In appendix \[Amany\] it is shown that only in the part $\langle M^IK_+^VM^I\rangle $ will it be necessary to use the bound electron propagator; in all other contributions it is sufficient to replace $K_+^V$ by the propagator for free electrons, $K_+^0$. Moreover the main contribution to $\langle M^IK_+^VM^I\rangle $ arises from intermediate states of the electron with nonrelativistic energy so that both $K_+^V$ and $M^I$ can be replaced by their simpler nonrelativistic approximations. It is also shown that the cross term in Eq. (\[a23a\]) gives a contribution of order $\al(Z\al)^5$ and is therefore not relevant in our calculation. According to the above considerations we can then approximate Eq. (\[a23a\]) by \[Ene2\] E\_2M\_[NR]{}\^IK\_[NR]{}\^VM\_[NR]{}\^I+M\^[II]{}K\_+\^0M\^[II]{} M\^I+M\^[II]{}.
We start evaluating the first term of Eq. (\[Ene2\]). The nonrelativistic prescription for $K_+^V$ is given by K\_[NR]{}\^V(x’,x)= { . or in momentum space \[a28\] K\_[NR]{}\^V(E\_n-k\_0;p’,p)= -i\_r\_r(p’)\_r\^\* (p)(E\_r-E\_n+k\_0)\^[-1]{} where $\varphi_r$ represents the large component of the Dirac spinor.
In the same nonrelativistic approach $M_\mu^I$ reduces to \[a29\] M\_\^[I(NR)]{}(p\_’-p\_)R\_, where we have approximated $\nn V\simeq V_0$, because although the magnetic and electric potential have the same order of magnitude (as powers of $Z\al$), the $\vec\gamma$ matrix mixes large components of the intermediate states with small ones and therefore introduces corrections one order higher in $Z\al$.
Therefore, after replacing Eq. (\[a28\]) and (\[a29\]) in Eq. (\[a8\]) we obtain \[a29.a\] M\^I= d\^4kG\^(k) \_r where we have neglected the contribution of the photon momentum $k$ to the momentum of the intermediate electron states. This is equivalent to leaving out the factor $\exp(i\vec k\cdot\vec x)$ in the spatial integration. This can be done because $k\sim E_n-E_r\sim m(Z\al)^2$, which is small compared with the electron momentum $\vec p\sim mZ\al$ for nonrelativistic states.
Inserting (\[a12\]) into (\[a29.a\]), and using Eq. (\[a25\]) to relate the temporal component of $R$ with its spatial components, which satisfy \[p60\] n|R |r=(E\_n-E\_r)n|p|r, we find, after integration \[a30\] M\^I=(1+)\_r|n|p|r|\^2\
(E\_r-E\_n)where all the states and energies represent the non relativistic limit of the Dirac solution.
Eq.(\[a30\]) can be simplified by using \[p62\] \_r|n|p|r|\^2(E\_r-E\_n)=n|\^2 V\_0|n, which finally gives \[a32a\] M\^I=(1+) with \[cij\] CC\^[ii]{},C\^[ij]{}=2\_rr|p\_i |nn|p\_j |r (E\_r-E\_n)|| where $E_*$ is a reference energy to be defined, and $\hat C^{ij}$ has been introduced for later convenience. To obtain this result we have neglected the imaginary part of $\langle
M^I\rangle $ retaining only the leading terms of $\langle M^I\rangle $ in the limit $\mu\to 0$.
In computing $\langle M^{II}\rangle $, we can take $K_+^V$ to be the free electron propagator, which is K\_+\^V(E\_n-k\_0;p’-s’-k,p+s-k)=, where r\^=(m,s\_\*),s\_\*=p’-s’=p+supon which $\langle M^{II}\rangle $ becomes \[a38\] M\^[II]{}=d\^3p’d\^3pd\^3s\_\*(p’)V\_(p’-s\_\*) N\^\_(p\_\*,s\_\*)V\^(s\_\*-p)\_n(p), with \[a39\] N\^\_(p\_\*,s\_\*)=-G\_(k)d\^4k.
In the nonrelativistic domain $\int d^3p V_{\al} \approx (Z\al)^2m$ and so the constant value of $N_{\al}^{\beta}$ (independent of the momentum and energy of the intermediate states) will already yield an overall contribution to Eq. (\[a38\]) of the desired order $\al(Z\al)^4$. Note that $N_{\al}^{\beta}$ can be expanded in powers of the momentum $\vec p\,'$, $\vec p$ or $\vec s_*$, which are of order $mZ\al$, and therefore any contribution beyond the constant, $Z\al$-independent term will be of higher order. The same argument can be used to neglect the binding energy of the intermediate states. We can therefore evaluate (\[a39\]) by approximating $p\sim p_*$ and $p'\sim p_*$ in the denominator of $M^{I\dag}$ and $M^I$ respectively, so that $p_*\approx (m,0)$ and $s_*\approx 0$.
Evaluating $N$ as in reference [@BBF] we find that (\[a38\]) becomes \[a40\] M\^[II]{}=(1+)n||n. Note that this term will exactly cancel the non-gauge invariant term present in Eq. (\[a17\]).
Finally we add Eq.(\[a32a\]) to Eq.(\[a40\]) to obtain $\delta E_2^{(A)}$, and then add it to Eq. (\[a17\]) to give the final result for the type-A contribution to the self-energy: \[a42\] E\_S\^[(A)]{}&=&(1+).
Apart from the constant $(1+\xi)$ factor, there is no formal difference between the result (\[a42\]) for this contribution to the level shift and the standard one [@BBF]. However there are implicit differences which appear in the expression for $V^\mu$ and the solution for the Dirac states $|n\rangle$ (in the non-relativistic approach here) in the presence of that source.
Type B Contributions to the Self-energy
---------------------------------------
To solve the type-B contributions we have to consider the photon progator \[a64\] G\_\^[(B)]{}=and a source $A_\mu\simeq\eta_{\mu 0}\phi$.
The evaluation of $\delta E_S^{(B)}$ is achieved by the same procedure as for part A, where now we use Eq. (\[a64\]) in (\[a6\]) and (\[a8\]) to solve for $\delta E_1^{(B)}$ and $\delta E_2^{(B)}$ respectively. This computation is somewhat more laborious than that in part A, due to the $\beta_\mu\beta_\nu$ tensorial dependence and the factor $\frac{(\beta\cdot k)^2}{k^2}$ present in this part of the (GM) photon propagator.
To evaluate $I_1$, $I_2$, and $I_3$ we need to modify the BBF technique by using (\[a14\]) along with \[a69\] -2\_[\^2]{}\^[\^2]{}. to regulate (\[a64\]). The expressions for the $I$’s are somewhat more complicated than those for $\delta E_S^{(A)}$ (as expected); but their manipulation and further algebra follow from BBF [@BBF]. The relevant details are in appendix \[Adetail\]; the result for the one potential part is \[a70\] E\_1\^[(B)]{}&=&\^2(p’){ V q\^2\
&+&V (q)\^2\
&+&(V-Vm)i\^[ij]{}u\_i q\_j -mq i\^ V\_\_\
&+&m(-)i\^V\_q\_} (p)d\^3p’d\^3p\
&-&\^2(1+\^2)n||n which is good up to order $\al(Z\al)^4$, and we have retained only the leading terms as $\mu\to 0$.
The evaluation of $\delta E_2^{(B)}$ is quite analogous to that for $\delta E_2^{(A)}$. The starting point is Eq. (\[Ene2\]), where $\langle M^I\rangle$ and $\langle M^{II}\rangle$ are still defined by (\[a29.a\]) and (\[a38\]) respectively. We give calculational details in appendix \[Adetail\], and quote here only the final result: \[a72\] E\_2\^[(B)]{}&=&\^2{n|\^2 V\_0|n\
&+&n|()\^2 V\_0|n+u\_iu\_jC\^[ij]{}+(+)C}\
&+&\^2(1+\^2)n||n
We now add (\[a70\]) to (\[a72\]) to obtain \[a74\] E\_S\^[(B)]{}&=&{n|\^2 V\_0|n\
&+&()n|()\^2 V\_0|n+u\_iu\_jC\^[ij]{}+(+u\^2)C\
&+&(p’)(p)d\^3p’d\^3p} where we approximated $\gamma^2\simeq1+\vec u^2$ in order to keep terms only up to order $\vec u^2$. As a cross-check on the above result we note that, before expanding $\gamma^2$, the limit $\beta_\mu\beta_\nu\rightarrow
\eta_{\mu\nu}$, yields $\delta E_S^{(B)}\rightarrow -2\xi\gamma^2\delta E_S^0$. This is as expected since according to (\[a64\]), $G_{\mu\nu}^{(B)}
\rightarrow-2\xi\gamma^2G_{\mu\nu}^0$, where $G_{\mu\nu}^0$ is the standard (metric) propagator.
We close this section with a comment on the renormalization procedure. For $\delta E_S^{(A)}$, the counterterm $\delta C$ was related to mass renormalization. However in this part of the calculation we must also account for the renormalization of the parameters, which show up as functions of: the limiting speed for massive particles, $c_0^2\equiv T_0/H_0$, and the photon velocity, $c_*^2\equiv1/\mu_0
\epsilon_0$. Charge renormalization is not necessary here because the Ward Identity forces a cancellation between the divergences coming from the one potential part and many potential part of the self energy. Details of this process are shown in appendix \[Arenor\].
Vacuum Polarization
-------------------
We now need to obtain the vacuum polarization contribution. To the desired approximation, the electrons forming the loop in diagram 1(b) can be considered free. This is because Furry’s theorem implies that the next-order correction to this is a diagram which contains a loop with 4 vertices, which is expected to be of order $\al(Z\al)^6$. In that case the result is known to be \[ap\] E\_P=(p’)i\^(q)iG\_(q)\^V\_(q)(p)d\^3p’d\^3p, The evaluation of $\Pi^{\mu\nu}$ is identical to the standard (metric) case, since it only involves the product of fermion propagators, which are unchanged by the action. The differences appear in the renormalization process, where both the charge and the parameters must be renormalized, the details of which are shown in appendix \[Arenor\]. The result is \[pol\] \^(q)-(q\^2\^-q\^q\^) If we substitute Eqs. (\[11\]) and (\[pol\]) in (\[ap\]), we obtain after some manipulation \[pola\] E\_P={n|\^2 V\_0|n(- +)-n|(u)\^2 V\_0)|n}
We next proceed to add together the self energy and vacuum polarization contributions to the level shift.
The total GM Radiative Correction
---------------------------------
Up to this point we have been able to solve the level shift in terms of \[abp\] E\_n=E\_S\^[(A)]{}+E\_S\^[(B)]{}+E\_P where each term has been defined in Eqs. (\[a42\]), (\[a74\]) and (\[pola\]).
We note that in $\delta E_S$ there are terms proportional to $\vec\gamma$, which mix large ($\varphi$) and small component ($\chi$) of $\psi$. Within the accuracy required we can relate them by $\chi=-i\frac{\vec\sigma\cdot\vec \gr}{2m}\varphi$, and so write everything in terms of the large component only.
Replacing the expression for the external source (\[po\]) in (\[abp\]), we obtain after some algebra \[Elamb\] E\_n=where $\hat C$ and $\hat C^{ij}$ are defined by Eq. (\[cij\]), and \[hate\] E&=&4Z(x)\
&+&3L\
&-& \[3(u)\^2-u\^2\]\
&+&We have omitted operators with odd parity (such as $\vec u\times\nv\cdot\vec\sigma$) in (\[hate\]), since their expectation values vanish for states of definite parity.
There is still an implicit dependence on $\xi$ and $\vec u$ in (\[Elamb\]), which comes from the Dirac states (as seen at the end of Sec. III). Note that up to this order all atomic states and energies refered in Eqs. (\[Elamb\]) and (\[cij\]) are considered within a non relativistic approach.
In terms of the formal solution for the Dirac equation (\[solution\]), we can single out the complete $\xi$ dependence in (\[Elamb\]), and write \[efinal\] E\_n={ (1+(+u\^2))C\^0 +u\_iu\_j E\^[ij]{}+\^0n|E|n\^0} with \[Eij\] u\_iu\_j E\^[ij]{}=u\_iu\_j C\^[ij]{}+C’+(\^0n|E\_[=0]{}|n’+) where $\hat C'$ groups all the terms in Eq. (\[cij\]) depending on the perturbative states ($|n\rangle'$) or energies ($E_n'$) as introduced in Eq. (\[solution\]). These perturbative states are needed not only for the $|n\rangle$ state related to the level shift, but for all the intermediate states introduced by (\[cij\])as well. Eq. (\[efinal\]) is valid up to $O(\xi)O(\vec u^2)O(\al(Z\al)^4 )$.
We can define the reference energy $E_*$ as in the metric case by [@IZ] \[erefer\] .
[ll]{} (E\_\*\^[n0]{}) = &\
&\
2(Z)\^4()=\_r|r|p|n|\^2 (E\_r-E\_n)|| &
. where the subscript $0$ has been omitted in the energies and states. This definition reduces \[C0\]C\^0={
[ll]{} 0&\
4(Z)\^4()&
. which provides an elegant way to write the “Bethe-sum". The presence of prefered frame effects will induce more “Bethe-sum"–like terms in $\hat C^{ij}$ which, along with the contribution from the perturbative states (both ones counted by $\delta \hat E^{ij}$) will have to be evaluated numerically for any particular state.
For the Lamb shift states we can use [@IZ] : \[refener\] E\_\*\^[2S]{}=16.640E\_\*\^[2P]{}=0.9704 and simplify the last term in Eq. (\[efinal\]) as \[E0\] \^0E\^0\_[2S\_[1/2]{}]{}&=&m\^3{ +()- }\
\^0E\^0\_[2P\_[1/2]{}]{}&=&m\^3{ --} where $\theta$ represents the angle between the atom’s quantization axis and the frame velocity $\vec u$.
Many potential part approximations\[Amany\]
-------------------------------------------
In this appendix we justify the following approximations: \[ap1\] M\^IK\_+\^VM\^I&&M\_[NR]{}\^IK\_[NR]{}\^VM\_[NR]{}\^I\
\[ap2\] M\^[II]{}K\_+\^VM\^[II]{}&&M\^[II]{}K\_+\^0M\^[II]{}\
\[ap3\] M\^I K\_+\^V M\^[II]{}&&O((Z)\^5) following arguments similar to those presented by BBF [@BBF].
We first note that, as powers of $Z\al$, the orders of magnitude of the different terms involved in the expressions in (\[ap1\]) are equivalent to those for the metric case. For example, if we look at the source, we see that $eA_\mu\sim e\phi $, where $A_\mu$ is given by Eq. (\[po\]) and $\phi$ is the ordinary Coulomb potential, and so the relative order between the nonmetric and metric case is the same. Furthermore, as discussed at the end of Sec. III, the states $|n\rangle$ and $|n\rangle ^0$ also have the same order of magnitude, as do the quantities $E_n$ and $E_n^0$. Discrepancies that could be expected from the photon propagator, particularly from the part proportional to $\beta^\mu\beta^\nu$ (in contrast to the $\eta_{\mu\nu}$ dependence for the standard case), are not important as long as the transversality condition is satisfied for the $M$ operators, since this condition relates the differing components with the appropriate orders of magnitude. Finally, unlike the photon propagator, the bound propagator retains the same form as in the standard case, with differences arising only from the expression for the external source. As a consequence its further simplification is analogous to the metric (BBF) case.
Let us look at the many potential part. From (\[a8\]) we get \[a88\] M K\_+\^V M &=&\_n(p’)M\_(p’,p’-s’-k)\
&&K\_+\^V(E\_n-k\_0;p’-s’-k,p+s-k)\
&&M\_\^(p+s-k,p)\_n(p)G\^(k)for the generic structure of the terms on the left hand sides of (\[ap1\])–(\[ap3\]), where the constant factors and integrations over $p_i$ and $s_i$ have been omitted. The nonrelativistic and relativistic regions are defined according to $|\vec{k}|\sim(Z\al)^2m<<m$ and $|\vec{k}|>m$,respectively. In considering the relevant orders of magnitude in each of the expressions (\[ap1\])–(\[ap3\]) that follow from (\[a88\]), we note that, to lowest order in $Z\al$, the relevant contribution from $G^{\mu\nu}$ comes when $k_0\sim |\vec{k}|$, and that we can employ the nonrelativistic expressions for the $\psi_n$, making use of the approximations given by (\[a20a\]).
Turning now to the relation (\[ap1\]), we can prove it by showing that the contribution of relativistic states for $M^I$ is of a higher order of magnitude than for $M^{II}$. We can see from (\[mIe\]) and (\[mIIe\]) that $M^I$ differs from $M^{II}$ by a factor (leaving aside the temporal component) $(\vec{p}^\prime - \vec{p})/k_0$, which in the relativistic region ($k_0 \sim m$) is of order $Z\al$. Therefore the contribution of $M^I$ in that domain will be of at least one order higher than that of $M^{II}$. Since the latter is already of the desired order (assuming the validity of (\[ap2\]) ) we can neglect the contribution of the relativistic states for $M^I$, and consider it, along with the bound propagator, in its nonrelativistic limit.
To prove the relation (\[ap2\]) we evaluate the error due to the neglect of the electromagnetic potential in the intermediate states. We imagine that one extra potential ($\nn V$) acts between $M^{II\dag}$ and $M^{II}$. This introduces an extra factor of order d\^3r’\~d\^3r’k \~(Z)\^2 which is negligible within the accuracy required. We have then shown that, in the evaluation of $M^{II}$, the intermediate states may indeed be regarded as free.
The relation (\[ap3\]) follows from arguments similar to those used to justify (\[ap1\]). Since in the relativistic region $M^{I}$ is one order higher than $M^{II}$, the cross term in that region will also be one order higher than $\langle M^{II}\rangle$, and so is negligible. On the other hand in the nonrelativistic region $M^{I}$ will be dominant (note the factor $k_0$ in its denominator) over $M^{II}$. That is ||\~||\~Zand so the product of these terms will be negligible in comparison with $\langle M^{I}\rangle$. Hence the cross terms yield results that are at least one order higher than the desired order, and so they do not need to be included.
Renormalization\[Arenor\]
-------------------------
Just as in the standard (metric) case, we need to renormalize the various parameters of the theory in order to get rid of the divergences. In the standard case, those parameters are the mass and charge, although the latter only needs to be renormalized for the vacuum polarization contribution. The self energy part has no need for such a renormalization, since the divergences coming from the one potential and many potential parts cancel each other. In the nonmetric case, we have also to include the renormalization of the parameters, which show up as functions of $c_0^2\equiv T_0/H_0$ and $c_*^2\equiv 1/\mu_0\epsilon_0$.
In part A of the calculation, renormalization is identical to the standard case. The counterterm $\delta C$ is just related to mass renormalization. In part B, we need to consider additional counterms, since $\delta C$ should also account for the renormalization of the parameters.
In units where $c_0\equiv 1$ ($c_*=c$), EEP-violating corrections only appear in the electromagnetic sector of the action (as terms proportional to $\xi$). However we could choose more generally $c_0\not =1$, for which the particle sector of the Lagrangian density is of the form \_D=(p-V-m)+\_0(p\_0-A\_0)\^0with $\xi_0\equiv 1-c_0^{-1}$; or in the moving frame (after using (\[8\])) is \[apbD\] [L]{}’\_D&=&(p-V-m)\
&+&\_0\^2(p-V)up to a constant.
From (\[apbD\]) we see that quantum corrections of the form \_D=(\_0\^[(1)]{}p-\_0\^[(2)]{} V)can still be expected. Note that gauge invariance will guarantee $\delta\xi_0^{(1)}=
\delta\xi_0^{(2)}=\delta\xi_0$. Hence, in order to renormalize the mass and the parameters, we have to include counterterms of the form C=m+\_0(p-V) where $\delta m$ and $\delta\xi_0$ are chosen such that $\delta E_S$ gives zero contribution as the source is turned off. This condition forces $I_3=0$ when acting on free spinors.
Finally, for the vacuum polarization contribution the charge has to be renormalized along with the parameters. Charge renormalization is identical to the standard case. For the parameters the procedure is equivalent to the self energy part, where now, given the form of the electromagnetic action (see Eq. (\[9\])), we expect quantum fluctuations of the form \_[EM]{}=A\^{(k\^2-(k)\^2)\_ -\_\_k\^2}A\^to occur. Hence a counter term of that form it is needed to renormalize the parameters, or equivalently $\xi\equiv 1-H_0/T_0\mu_0\epsilon_0$.
Calculational Details of Type B Contributions\[Adetail\]
--------------------------------------------------------
We present here further details underlying the computation leading to Eqs. (\[a70\]) and (\[a72\]), which are referred as the type-B contributions to the self energy. In this part the photon propagator to be considered is given by (\[a64\]), where the first and second terms have respectively a tensor dependence like $\beta_\mu\beta_\nu$ and $\eta_{\mu\nu}$, and need to be regularized according to (\[a14\]) and (\[a69\]). We show the relevant details involving the first term of the propagator only, since the remainder can be computed in a similar way.
We begin then with the one potential part by simplifying $I_1$. After replacing (\[a64\]) in (\[a7\]), we get \[c1\] I\_1=-\^2 VdL+where from now on $\cdots$ stands for the contributions coming from the second term of (\[a64\]).
If we use =6\_0\^1dx\_0\^1 we can rewrite Eq. (\[c1\]) as I\_1=&-&4pp’V J\_0+2p\^V J\_\
&+&2p’V\^J\_-\^V\^J\_ +where \[c3\] J\_[{0;;}]{}=-\^2\_0\^1dx\_0\^1z(1-z)dz{1;k\_;k\_k\_} with p\_x=xp’+(1-x)p\_L=p\_x\^2(1-z)\^2+Lz After evaluating (\[c3\]), we can express I\_1&=&\^2{V\
&-&x(pV+p’V) -(pV+p’V)(1-x)\
&+&p\_x(V(p +p’)-p\_xV)\
&+&(p\_xp\_xV- Vp\_x\^2(1+ -\
&-&(1-x)p Vp-x p’Vp’)}+
The evaluation of the remaining $I$’s is analogous, and so I\_2&=&V\^2{(-1)+ (1+)}+\
I\_3&=&-p\^2(+ )-p\^2\^2(-)++ C
From appendix \[Arenor\], we know $\delta C=\delta m+\delta\xi_0\nn\beta(\beta\cdot p-\beta\cdot V)$, where in this case m=\^2(-)+\_0=\^2(+)+
Since here $V^\mu=\eta^{\mu 0} V_0$, we can rewrite after some manipulation I\_1+I\_2+I\_3=\^2(K\_1+K\_2+K\_3)+where K\_1&=&V\
K\_2&=&(m+p)\
K\_3&=&-{ xp’p’V+(1-x)pVp\
&+&Vp(p\_x+p\_x) -2V(p+p’)p\_x+xpp’V\
&+&(1-x)p’Vp-p\_x\^2}
We want a result good to $\al(Z\al)^4$, and so we can simplify the above expressions by using the assigned order given by (\[a20a\]), from which we can relate
[l]{} qp’-p\~Zm\
p’\^2-p\^2\~(p)\^2-p\^2\~p\_x\^2-m\^2\~(Z)\^2 m\^2
and then reduce $K_1$ to K\_1{() -q\^2(+\^2)+(q)\^2} where antisymmetric terms under $p'\leftrightarrow p$ vanish.
To simplify $K_2$ we follow BBF and use (p-m)\^2&=&V(p-m)=2pV-2mV -VV\
&&2(Vp-mV)-qV- V\^2 where we have assumed the operator is acting on Dirac spinors of momentum $p$ and omitted the integration coming from (p)(p-m)=(q\_0)(p’)V(q)d\^3p’
Note that $\nn V\nn\beta\nn V\simeq V^2$, since the square of the potential (after factoring out the spinors and integration variables) is already of the desired order $(Z\al)^4$ (see (\[a20a\])) and so $\nn\beta\simeq\gamma_0
\simeq 1$.
The final result is K\_2-(+5)+(Vp-mV) -qV
Following a similar approach we reduce K\_3&&-(Vp- mV)++V -Vp\
&-&V-q+qV-(-1)V
We can make further simplifications by using (p’)B(p’,p)(p)d\^3p’d\^3p=0, provided $\gamma^0 B^{\dag}(p',p)\gamma^0=-B(p,p')$, where $B$ represents any operator as a function of $p'$ and $p$, as for example, $\beta\cdot q\nn V$. Note that we are interested only in the real part of the level shift.
Putting everything together, we obtain after some manipulation
\[aE1\] E\_1\^[(B)]{}&=&\^2(p’){ V (q)\^2-V q\^2\
&-&(V+V)i\^[ij]{}u\_i q\_j -q i\^ V\_\_\
&+&m(-)i\^V\_q\_} (p)d\^3p’d\^3p -\^2(1+)n||n+
Note again that this represents the calculation involving only the first term of Eq. (\[a64\]).
Now to evaluate the many potential part contribution we need to solve Eq. (\[Ene2\]), with $\langle M^I\rangle$ and $\langle M^{II}\rangle$ given by Eqs. (\[a29.a\]) and (\[a38\]) respectively.
So, after substituting (\[a64\]) in (\[a29.a\]) \[dd1\] M\^I=\^2\_r +with $${\cal M}_\mu\equiv\langle n |R_\mu|r\rangle,$$
Using the transversality condition, we relate $${\cal M}_0=\frac{\vec k\cdot\vec{\cal M}}{k_0}=\frac{|\vec k|}{k_0}|\vec{\cal M}|
\cos\theta$$ which reduces the integral on the angles of $\vec k$ to d||\^2=4(||\^2 +|u|\^2)
We evaluate the remaining $k_0$ and $|\vec k|$ integrations in (\[dd1\]), by using (\[p60\]), (\[p62\]) along with the analogous relations u&=&(E\_r-E\_n)n |up|r\
\_r|n|up|r|\^2(E\_r-E\_n)&=&n|uV\_0|nto finally obtain \[mm1\] M\^I=\^2&{&C+u\_i u\_j C\^[ij]{}+n|\^2V\_0|n\
&+& n|(u) \^2V\_0|n}+where we have kept only the leading terms as $\mu\rightarrow 0$ and neglected the imaginary part.
The computation of $\langle M^{II}\rangle$ is straightforward. Here we need to replace (\[a64\]) in (\[a38\]), and use $V_\alpha=\eta_{\al 0} V^0$. Further simplifications follow from BBF and the assigned order of magnitude given before. The final result is \[mm2\] M\^[II]{}=\^2(+1)n | |n+
Adding together (\[aE1\]), (\[mm1\]), and (\[mm2\]) will give us then the final expression for the self energy contribution for this part of the calculation. Note that the above results can be verified by taking the limit $\beta_\mu\beta_\nu\rightarrow\eta_{\mu\nu}$, which reduces $$G_{\mu\nu}^{(B)}\rightarrow-\gamma^2\xi G_{\mu\nu}^0+\cdots,$$ and therefore the former expressions should reduce up to a constant, to the metric case.
Virtual non metric anomaly\[Apol\]
==================================
In the formalism, gravity interacts with matter through the $T$ and $H$ functions, which are assumed locally constant within atomic scales. A priori they do not need to be the same for differents type of matter (like baryons and leptons), or furthermore for matter and antimatter. In this context for example, a non metric anomaly related to electron/ positron difference will modified the Lagrangian density related to fermions by \[posi1\] [L]{}\_D=(p-V-m)+\_+\^+(p\_0-A\_0)\^0\^+ where $\xi_+\equiv 1-c_-/c_+$ and $c_\mp=(T_\mp/H_\mp)^{1/2}$, with $-$ and $+$ labeling electrons and positrons respectively. After using (\[8\]), we can refer (\[posi1\]) to the moving frame as \[posi2\] [L]{}’\_D=(p-V-m)+\_+\^2\^+(p-V)\^+
The imposed broken symmetry between particle and antiparticle changes the fermion propagator (in the positron case) to (up to $O(\xi_+)$): \[posi3\] S\_F\^[+]{}=(p-m)\^[-1]{}+\_+(p-m)\^[-1]{}\^2p(p-m)\^[-1]{}
where the first term represents the unchanged electron propagator $S_F^-$.
The positron-electron pairs produced in the electric field of the atomic nucleus, are seen in the Lamb shift transition via the vacuum polarization contribution given by (\[ap\]), where in this case: \[posi4\] i\^(q)=(-1) Trd\^4p\^i S\_F\^-(p+q) \^i S\_F\^+(p)
After using eq. (\[posi3\]) along with standar technics [@IZ], we obtain that the non metric part of (\[posi4\]) is up to $O(q^2)$ \[posi5\] i\^(q)\^+=-\^2\^{ q\^2\^2-(q)\^2}+where $\cdots$ accounts for the gauge dependent terms which give no contribution to (\[ap\]). Eq (\[posi5\]) also comes after proper regularization and renormalization processes, which follow from previous sections.
In this EEP violating context, the radiative corrections related to atomic energy levels are modified by (up to $O(\al(Z\al)^4\,O(u^2)$) E\_L\^+=E\_P\^+=-\_+{n|\^2 V\_0|n+n|(u)\^2 V\_0)|n} where we have replaced (\[posi5\]) in (\[ap\]) and simplified afterwards. By taking the Lamb atomic states, we finally obtain E\_L\^+=-\_+ (Z)\^4(1+2u\^2)
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|
---
author:
- 'K. Biazzo'
- 'A. Frasca'
- 'E. Marilli'
- 'E. Covino'
- 'J. M. Alcalà'
- '${\rm \ddot{O}}$. Çakirli'
title: Magnetic activity in the young star SAO 51891
---
Introduction
============
Stars just arrived on the Zero Age Main Sequence (ZAMS) or on their way to reach it are in an important evolutionary phase because they start to spin up getting free from their circumstellar disks which can begin to condensate giving rise to proto-planetary systems. At the same time, they start to loose angular momentum via magnetic braking. SAO51891 is in this evolutionary stage; it is indeed a young star counterpart of an EUV source with an IR excess attributed to dust around the star (@Najita2005).
The spectra were acquired in August 2006 with FOCES@CAHA at a spectral resolution $R\approx40000$ in the wavelength range 3720-8850 Å with a signal-to-noise ratio higher than 200. The contemporaneous photometry was performed at OACt in the $BV$ Johnson bands.
Stellar Parameters
==================
Applying the ROTFIT code (@Frasca03) to the yellow-red portion of the FOCES spectra, we derive a spectral type of K0-1V and a $v\sin i$ of 19 km s$^{-1}$. From its position on the HR diagram, with the effective temperature $T_{\rm eff}$=5260 K derived through the line-depth ratio (LDR) method, we obtain a mass of 0.8$\pm$0.2 $M_\odot$ by comparison with the evolutionary tracks of [@dant_mazz1997] and [@palla1999].
The lithium abundance of $\log N({\rm Li})\approx3.2$ dex deduced from the equivalent width (EW) of the Li line at 6708 Å is somewhat lower than the Pleiades upper envelope, indicating an age of $\sim100$ Myr corresponding to a Post T Tauri (PTT) or ZAMS star. Thus, the magnetic activity detected at photospheric (@Henry95), chromospheric (@Mulliss94) and coronal (@Voges1999) levels should be essentially the effect of its young age.
Magnetic activity {#sec:phot}
=================
![ Observed (thick lines) spectrum in three spectral regions, together with the non-active template spectrum (thin lines). []{data-label="fig:spectra"}](SAO51891.16aug0011n_CaIIK.ps "fig:"){width="3.39cm"} ![ Observed (thick lines) spectrum in three spectral regions, together with the non-active template spectrum (thin lines). []{data-label="fig:spectra"}](SAO51891.16aug0011n_Halpha.ps "fig:"){width="2.27cm"} ![ Observed (thick lines) spectrum in three spectral regions, together with the non-active template spectrum (thin lines). []{data-label="fig:spectra"}](SAO51891.16aug0011n_CaIRT8498_8542.ps "fig:"){width="3.39cm"}
The spectroscopic method based on LDRs (@Gray1991 [@Cata02; @Biazzo07a]) allows us to detect a $T_{\rm eff}$ variation with an amplitude of 90K, which is intermediate between the value of $\sim\,$40 K found in stars with moderate activity (e.g., $\kappa$1 Cet; @Biazzo07b) and 130 K found in stars with a very high activity level (e.g., II Peg; @Frasca08). Moreover, the $T_{\rm eff}$ curve is in phase with the $BV$ photometry, confirming the hypothesis of cool spots as the primary cause of the observed variations.
As diagnostics of chromospheric emission we used H&K, -D3, H$\alpha$, and IRT lines, formed at different atmospheric levels. Using the spectral subtraction technique (@Fra94) we obtain the chromospheric radiative losses in these lines. In Fig. \[fig:spectra\] we show portions of spectrum in the H&K, H$\alpha$, and IRT spectral regions, with the non-active template superimposed. The H$\alpha$ and the IRT profiles are filled-in by emission, with the latter displaying a central reversal nearly reaching the continuum and suggesting a strong contribution to the total chromospheric losses (@Busa07). The H&K lines show a strong core emission typical of cool magnetically active stars. Measuring the EW of residual emission profiles in the difference spectrum, we find that the net H$\alpha$ chromospheric emission does not show any detectable variation with phase, while the IRT displays a fair modulation with a possible phase shift with respect to the light curve.
Conclusions
===========
From the study of photospheric and chromospheric inhomogeneities based on spectroscopic and photometric monitoring of SAO 51891, we find a clear light and $T_{\rm eff}$ rotational modulation due to spots and a modulation of the total IRT emission due to plages. This chromospheric diagnostics seems to indicate a possible shift between spots and plages. Thus, as a follow/up of our previous works (@Biazzo07b [@Frasca05; @Frasca08]), we aim to develop a spot/plage model for reproducing the observed behaviors at photospheric and chromospheric levels and for deriving spot/plage parameters. SAO51891, and other weak-line T Tauri and PTT, and ZAMS stars already observed by us with FOCES@CAHA and SARG@TNG, are important to explore the correlations between global stellar parameters (e.g., $\log g$, $T_{\rm eff}$) and spot characteristics (e.g., filling factor and temperature) in stars with different evolutionary stage and activity level.
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|
---
abstract: 'We study non-adiabatic transitions among energy levels of a slowly deforming nuclei so that the system evolves to states which are out of thermodynamic equilibrium, and phenomena like heating and dissipation occur. Such states verify well-known results of non-equilibrium statistical mechanics given by Jarzynski and Bochkov - Kuzovlev equalities. We study the validity of these equalities for various nuclei.'
author:
- |
Nishchal R. Dwivedi$^{*1,2}$, and Sudhir R. Jain$^{1,3,4}$\
$^1$Nuclear Physics Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400 085, India\
$^2$Department of Physics, University of Mumbai, Vidyanagari Campus, Mumbai 400 098, India\
$^3$Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400 094, India\
$^4$UM-DAE Centre for Excellence in Basic Sciences, University of Mumbai, Mumbai 400 098, India\
$^*$ `dwivedi.nishchal@gmail.com`
title: 'Non-adiabatic transitions and non-equilibrium statistics of deforming nuclei'
---
One of the central themes of many-body quantum physics is to relate the relaxation of a system to the evolution of its single-particle energy levels. Given a many-body system of interacting particles in thermal equilibrium at temperature, $T$, the occupation of the particles is decided by an equilibrium distribution function with a mean-field Hamiltonian [@blaizot1986quantum]. There are very well-known constructions describing nuclei and metallic clusters like the Nilsson model [@nilsson1955binding] and the Clemenger-Nilsson model [@brack1993physics], billiard models for quantum dots and electromagnetic cavities [@jain2017nodal], and so on. It is also well-known that in this description, when a system evolves in time, the energy levels evolve - they cross, or, avoid to cross, depending on the quantum numbers of their corresponding states [@von1993merkwurdige]. For such systems, it has been shown that the energy diffuses in a rigorous manner [@jain1999diffusion; @jain2000dissipation]. However, for specific case of nuclei, the effect of non-adiabatic transitions on the many-body system, as it is thrown out of canonical equilibrium, has not been understood. It is clear that such a study is broadly relevant to many-body physics, but it is particularly significant as we are engaging in experiments where quantum systems are manipulated in the context of ultracold atomic physics or quantum computing and communication.
Here, we consider a complex enough case of nuclei where we are able to understand the nature of state out of equilibrium and see how it emerges as a result of non-adiabatic transitions. We do not claim that our conclusions and results are applicable, in letter, to all many-body systems, but we do believe that they are relevant and useful in spirit. Nilsson [@nilsson1955binding] presented a model in terms of a three-dimensional oscillator with spin-orbit interaction and a correction to the oscillator potential for higher angular momentum values. This well-known Hamiltonian is given by: $$H= H_0 + C \bar{l}.\bar{s} +D \bar{l}^2$$ where, $$\begin{aligned}
H_0= -\frac{\hbar ^2}{2m} \Delta + \frac{m}{2} (\omega_x^2 x^2+\omega_y^2 y^2+\omega_z^2 z^2).\end{aligned}$$ Here, $x,y,z$ are the position coordinates of a particle in a frame fixed with respect to the nucleus. $C$ and $D$ are constants which are fixed as discussed in [@nilsson1955binding], $l$ is the orbital quantum number, $s$ is the spin quantum number. For the case of spherical nuclei, $\omega_x=\omega_y=\omega_z=\omega_0=\frac{41}{A^{1/3}\hbar}$. The $\bar{l}^2$ term gives correction to the oscillator potential at large distances and is important for large $l$-values. In order to study how these energy levels change with deformation, we take the case of $\omega_x\neq\omega_y\neq\omega_z$, and we introduce a deformation parameter [@jain1990intrinsic]. The Hamiltonian obtained thus, upon diagonalization, entails the energy levels which are obviously dependent on deformation in a complicated manner.
Non-adiabatic transitions have not only been observed in finite systems like metallic clusters [@xu2005magnetic] and quantum dots [@norris1996measurement] but also their studies have been instrumental in measurement of properties like decoherence for a quantum computer [@ashhab2006decoherence]. For studying non-adiabatic transitions in a nucleus, we have considered these ‘Nilsson’ levels, decorating the iconic, so-called Nilsson diagrams. Non-adiabatic transitions may occur at these avoided crossings whenever the levels come closer than the mean spacing at that deformation [@zener1932non]. The ensuing Landau-Zener-Stückelberg (LZS) probability between two adjacent levels is given by, $$\label{eq:lzs}
P_{\rm LZS} \sim \exp \left[- \frac{2 \pi}{\hbar} \frac{(\epsilon _{i}-\epsilon _{i-1})^2}{\bigg| \frac{d(\epsilon _i - \epsilon _{i-1})}{dt}\bigg|}\right]$$ where $\{\epsilon _i\}$ are the single-particle energy levels on the Nilsson diagram. These levels change with deformation, which is time-dependent. Typical time scales are that of thermalisation of a nucleus, where the nucleus loses its excitation energy (say, after a nuclear reaction) and comes back to its ground state. It is about $10^{-21}$ seconds.
We pose the problem of studying how an initially canonical distribution of a nucleus changes as the system is subjected to deformation. We would like to note that there are treatments involving more than two levels which play an important role in the case of complex molecules.
There is another motivation for carrying out this work, in addition to furthering our understanding of nuclear friction or heating and thermodynamics of such finite systems [@beck1995thermodynamics]. In a discussion on mass parameters in large amplitude collective motion, a relation was found between microscopic dynamics and diffusive modes for large amplitude collective motion [@jain2012origin]. Mass parameter was shown to originate due to chaotic single-particle motion, and a fractal dimension of the ensemble of paths in the deformation space. Thus, it is of interest to consider a collection of paths in deformation space and let the system evolve; eventually of course we extract average quantities.
Initially, we distribute nucleons among the energy levels according to the Maxwell - Boltzmann distribution for some temperature. We then perform a Gaussian random [@kusnezov2000robust] walk of 1000 steps in the deformation space. At each point in the deformation space, the LZS transition probability is calculated and the nucleons are redistributed by these probabilities. At the end of the random walk, a new distribution is obtained, from which a Boltzmann-like temperature is extracted by fitting. We collect 1000 such random walks for various mass numbers ($A$) at 2 and 5 $MeV$. It should be noted that a case with more number of random walks shows similar results, but more number of random walks are computationally expensive. So, an optimal number of 1000 random walks have been chosen.
The total energy is calculated by, $$E_{total}=\sum_{i=1}^k E_i N_i$$ where, $E_i$ is the energy of the $i^{th}$ level at a specific value of deformation and $N_i$ is the number of particles in the $i^{th}$ level. $k$ is the total number of levels between which these nonadiabatic transitions are being studies.
Here, we consider the levels in the state $\frac{1}{2}^+$. These levels are populated with $A$ number of particles corresponding to the number of nucleons in the nucleus. We then take the extracted Boltzmann- like temperatures and plot a normalized histogram, which is shown in Fig. \[fig:2mev\] and \[fig:5mev\], for $A=20,40,60,100,200$ number of particles in a nucleus. It can be seen that a clear heating of the nucleus is observed due to the LZS transitions occurring during the deformation of the nucleus.
![An initial Boltzmann distribution for various $A$ values is prepared at $k_b T=2~MeV$ and subjected to a random walk in deformation space. At the end of the deformation, a Boltzmann-like temperature is extracted. Such 1000 realisations are done and are represented as a normalized histogram. The mid points of the histogram are joined to obtain the above figure. It can be seen that the final temperature is always greater than the initial temperature of $2~MeV$.[]{data-label="fig:2mev"}](1.pdf)
![An initial Boltzmann distribution for various $A$ values is prepared at $k_b T=5~MeV$ and subjected to a random walk in deformation space. At the end of the deformation, a Boltzmann-like temperature is extracted. Such 1000 realisations are done and are represented as a normalized histogram. The mid points of the histogram are joined to obtain the above figure. It can be seen that the final temperature is always greater than the initial temperature of $5~MeV$.[]{data-label="fig:5mev"}](2.pdf)
Jarzynski and Bochkov-Kuzovlev equalities
=========================================
Consider a thermally isolated system on which external forces deliver work. Then, the Hamiltonian will depend on a time-dependent parameter $\lambda(t)$, which will lead to the evolution of the Hamiltonian. At temperature $T=1/(k_b \beta)$, the average work is given by, $$\langle W \rangle = \langle H(p_f, q_f, \lambda(t_f)) - H(p_i, q_i, \lambda (t_i))\rangle$$ where the average, $\langle \rangle$, is performed over repeated experiments from an initial $i$ to final $f$ epochs.
The Jarzynski equality [@jarzynski1997nonequilibrium] states that, $$\langle \exp{(-\beta W)} \rangle = \exp{(-\beta \Delta F)}
\label{jar}$$ where, $\Delta F = F_f - F_i$, where $F$ is the equilibrium free energy of the system.
If there is a rapid change in the parameter $\lambda(t)$, then the entropy increases. In such a case,
$$\langle W \rangle > \Delta F.
\label{Weq}$$
For a cyclic process, i.e. when the final and the initial values of the parameter $\lambda(t)$ are the same, then $\Delta F = 0$. In such cases equation \[jar\] becomes Bochkov - Kuzovlev (BK) equality [@bochkov1981-I; @bochkov1981-II], $$\langle \exp{(- \beta W)} \rangle = 1
\label{bk}$$
To be sure that our study is consistent with fundamental ideas of non-equilibrium statistical mechanics, we now study Jarzynski and Bochkov-Kuzovlev equalities. It is difficult to overstate the significance of these inequalities [@pitaevskii2011rigorous], thus their verification is of importance for validity of our ideas.
$\Delta F$ is given by, $$\Delta F = -k_b T \log \langle \exp{-\beta W} \rangle.$$
If we write work as $W=\langle W \rangle + \delta W$ , and expand in terms of $\delta W$, we arrive at the expression [@hermans1991simple], $$\Delta F \approx \langle W \rangle - \frac{\beta \langle (\delta W)^2 \rangle}{2}.$$
The values of $ \langle {\exp{(-\beta W)}} \rangle $ and $ \exp{(-\beta \Delta F)} $ for two temperatures are shown in the Table \[tab\]. It can be seen that for temperature $2~MeV$, the Jarzynski equality is followed. The Jarzynski equality is also followed for $5~MeV$ till $A=60$. For high temperature and high $A$ values, the inequality (\[Weq\]) is followed, hinting that due to the increase in entropy Jarzynski equality is not followed.
$~$
----- ---------------------------------------- ------------------------------ ---------------------------------------- ------------------------------
$A$ $ \langle {\exp{(-\beta W)}} \rangle $ $ \exp{(-\beta \Delta F)} $ $ \langle {\exp{(-\beta W)}} \rangle $ $ \exp{(-\beta \Delta F)} $
20 1.0001 1.0011 1.0077 1.0117
40 1.0002 1.0002 1.0513 1.1480
60 1.0008 1.0083 1.1239 1.7179
100 1.0040 1.0042 995.0000 50.8406
200 1.0214 1.0368 989.0000 $1.01 \times 10^{18}$
: The Jarzynski equality is tested for various $A$ values for 2 and 5 $MeV$. 1000 instances of a Gaussian random walk with 1000 steps each in the deformation space are considered. The work done and free energy due to LZS transitions are calculated and their values are compared. The Jarzynski equality is valid for all cases except for $5~MeV$ for $A=100,200$. For high excitation and high number of particles, the entropy is high, which leads to the system following equation \[Weq\] and hence leading in the higher value of $\exp{(-\beta \Delta F)}$.[]{data-label="tab"}
We construct a set of deformation paths which are governed by Gaussian random distribution and return to the initial condition. This is the case when a deforming nucleus returns to its original shape. In this case (\[bk\]) is valid, as seen in Table \[tab2\].
$~$ 2 $MeV$ 5 $MeV$
----- ---------------------------------------- ---------------------------------------- -- --
$A$ $ \langle {\exp{(-\beta W)}} \rangle $ $ \langle {\exp{(-\beta W)}} \rangle $
20 1.0000 0.9999
40 1.0000 0.9999
60 0.9999 0.9998
100 0.9998 0.9971
200 0.9895 0.9176
: For a cyclic process with the deformations being governed by Gaussian random process, the $\Delta F$ value is equal to zero. For the validity of the BK equality, the value of $ \langle {\exp{(-\beta W)}} \rangle $ should be one. With 1000 random walks of 1000 steps each, this W is calculated for the LZS transitions in the nucleus. The values of the average of $\exp{-\beta W}$ are tabulated. It can be seen that for cyclic deformation process guided by the Nilsson Hamiltonian, the BK equality is fulfilled.[]{data-label="tab2"}
Figures \[fig:2mevW\] and \[fig:5mevW\] show the probability distribution for work done by LZS transitions along random paths. It can be seen that for heavier nuclei, the distribution of work done is broader as compared to that for lighter nuclei. This is perhaps because in heavier nuclei, the energy levels are closer than that in lighter nuclei. The closeness of the energy levels increases the LZS transitions, leading to more changes in energy of the system and hence the work done. In that it becomes a faster process, thereby verifying (\[Weq\]).
![The normalized histogram of the distribution of $W$ is shown in the figure above for $2~MeV$ of initial temperature. This histogram is made by collecting 1000 instances of 1000 Gaussian random walks in deformation space each. At the end of each walk, the difference in the initial and final energies gives the work done.[]{data-label="fig:2mevW"}](3.pdf)
![The normalized histogram of the distribution of $W$ is shown in the figure above for $5~MeV$ of initial temperature. This histogram is made by collecting 1000 instances of 1000 Gaussian random walks in deformation space each. At the end of each walk, the difference in the initial and final energies gives the work done.[]{data-label="fig:5mevW"}](4.pdf)
Conclusion
==========
We study the Landau-Zener-Stückelberg transitions in a deforming nucleus with the number of nucleons as 20, 40, 60, 100 and 200. We observe nuclear heating due to these transitions. The probability density of the work done broadens as the number of particles increase due to closely spaced energy levels in heavier nuclei. We further verify the Jarzynski and Bochkov-Kuzovlev equalities for this system of a deforming nucleus undergoing transitions through LZS mechanism.
We would like to emphasize that the collective properties are emerging from the detailed re-organization of occupation of the particles on the energy levels. In usual treatments, these effects are not considered. We believe that self-consistent inclusion of energy level dynamics is crucial, and that this also serves to throw the system out of equilibrium. The initial distribution or density matrix is not important, as shown by the derivation in [@jain1999diffusion].
Understanding the relationship of non-equilibrium features like heating and dissipation and microscopic quantum systems like nuclei has been a topic of fundamental discussion for a long time [@jain1998adiabatic]. Here we have shown that one of the mechanisms is the LZS transition by which nuclei evolve to a state which is out of thermodynamic equilibrium. This state is characterized by Jarzynski and Bochkov - Kuzovlev equalities.
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author:
- Xijia Miao
date: 'Somerville, Massachusetts. Date: April 2006. '
title: 'The basic principles to construct a generalized state-locking pulse field and simulate efficiently the reversible and unitary halting protocol of a universal quantum computer'
---
A halting protocol is generally irreversible in the classical computation, but surprisingly it is usually also irreversible and non-unitary in the universal quantum computational models. The inherent incompatibility within the universal quantum computational models between the irreversible and non-unitary halting protocol and the unitary quantum computational process that obeys the Schrödinger equation in quantum physics has been known since the early set-up of the universal quantum computational models. The irreversibility and non-unitarity of the halting protocol is closely related to the inherent irreversibility and non-unitarity of quantum measurement in quantum mechanics. The unitary dynamics quantum mechanically implies that the halting protocol in the universal quantum computational models should be made reversible and unitary so as to eliminate the inherent incompatibility. It has been shown in Ref. \[24\] (Arxiv: quant-ph/0507236) that a universal quantum computer could be powerful enough to solve efficiently the quantum search problem in the cyclic-group state subspaces, and the reversible and unitary halting protocol is the key component to construct the efficient quantum search processes based on the unitary quantum dynamics, while the state-locking pulse field is the key component to generate the reversible and unitary halting protocol. In this paper the reversible and unitary halting protocol and the generalized state-locking pulse field have been extensively investigated theoretically. The basic principles to construct the state-locking pulse field and design the reversible and unitary halting protocol are described and studied in detail. A generalized state-locking pulse field is generally dependent upon the time and space variables. It could be a sequence of time- and space-dependent electromagnetic pulse fields and could also contain the time- and space-dependent potential fields. Thus, the reversible and unitary halting protocol built up out of the state-locking pulse field generally consists of a sequence of time- and space-dependent unitary evolution processes. It is shown how the quantum control process is constructed to simulate efficiently the reversible and unitary halting protocol. An improved subspace-reduction quantum program and circuit based on the reversible and unitary halting protocol, which is much simpler than that one in the previous paper \[24\], is proposed as the key component to construct further an efficient quantum search process. A simple atomic physical system which is an atomic ion or a neutral atom in the double-well potential field is proposed to show how the state-locking pulse field is generated and how to implement the reversible and unitary halting protocol.
A halting protocol (or a halting operation briefly) of a computational model such as the Turing machine is one of the key components in computation. It is well known that the halting operation generally is not a reversible operation in the classical computation and often is related to measurement of computational results. Though in the reversible computational model \[1\] and probably in the quantum Turing machine \[2\] the halting protocol could be made reversible, it is surprised that the halting protocol usually is not yet reversible and unitary in the universal quantum computational models including the universal quantum Turing machine \[3, 9\] and the universal quantum circuit model \[4\]. When the universal quantum Turing machine was proposed in the early day \[3\], the halting protocol was also introduced as one of the important components of the universal quantum computational model. According as the universal quantum Turing machine, in addition to the computational quantum system that is used to perform quantum computation there is also an extra quantum bit used to instruct what time a quantum computational process is halted on the universal quantum Turing machine. This extra quantum bit is named the halting quantum bit or briefly the halting qubit. The halting quantum bit should be observed periodically from the outside in a non-perturbation manner so that the quantum measurement on the halting quantum bit does not disturb the unitary evolution process of the quantum system during the quantum computational process. Once the halting qubit is found to be in the halting state, e.g., the state $%
|1\rangle ,$ the quantum computational process terminates. Therefore, the halting operation is performed to stop the quantum computational process only after a sequence of non-perturbation quantum measurements on the halting qubit were carried out during the quantum computational process. However, the halting protocol could not be compatible with unitary quantum computational processes within the universal quantum computational models. It is well known that a quantum computational process on a universal quantum computer obeys the quantum physical laws \[3, 4\], that is, the time evolution process of the quantum system of the quantum computer obeys the Schrödinger equation in quantum mechanics during the quantum computational process. It is also known that the quantum parallel principle \[3\] that the quantum computational process is performed on a superposition of the basis states of the quantum system is the characteristic feature of the universal quantum computation models. The halting protocol is essentially different from a unitary quantum computational process in the universal quantum computational models in that the halting protocol involved in the non-unitary quantum measurement is generally irreversible and non-unitary. This basic and inherent incompatibility within the universal quantum computational models between the facts that the quantum computational process obeys the unitary quantum dynamics and that the halting protocol is irreversible and non-unitary due to the non-unitary quantum measurement really originates from the basic quantum mechanical laws. It is well known that there is an inherent incompatibility in quantum mechanics \[5\] between the non-unitary quantum measurement and the unitary time evolution process of a closed quantum system that obeys the Schrödinger equation.
For many years since the early proposals of the universal quantum computational models \[3, 4\] the computational power of the universal quantum computational models has been investigated extensively and continuously. In the past two decades it has been shown that there are several possible candidates to fuel the quantum computational power, which include the superposition and the quantum parallel principle \[3, 4\], the quantum entanglement \[29\] and the multiple-quantum coherence \[17, 20\], the quantum coherence interference \[18, 20\], and the unitary dynamics quantum mechanically associated with the symmetry and structure of a quantum system \[17, 18, 20, 22, 23, 24\]. A large number of quantum algorithms \[13, 14\] including the Shor$^{\prime }s$ prime factorization and discrete logarithm algorithms \[6\] and the Grover$^{\prime }s$ quantum search algorithm \[7\] which outperform their classical counterparts have been discovered and developed in the past two decades. Most of these quantum algorithms are based on the universal quantum circuit model \[4\]. The computational results of these quantum algorithms generally are output at the end of the computational processes by the proper quantum measurement. Thus, these quantum algorithms are not related to the halting protocol of the universal quantum computational models. It has been believed extensively that power of the quantum computation that outperforms the classical computation could come mainly from the quantum parallel principle \[3\]. But it was suspected whether or not an arbitrary recursive function mathematically could be computed more efficiently on a universal quantum computer than a classical computer with the help of the quantum parallel principle \[8\]. The quantum parallel principle requires that a quantum computational process take a superposition state as its initial state. However, an incompatibility within the universal quantum computational models arises when a universal quantum computer computes a general recursive function in mathematics by starting at a superposition state. This incompatibility is really due to the conflict between the halting protocol and the quantum parallel principle \[8\]. It results in the question whether or not a universal quantum computer is capable of computing more efficiently an arbitrary recursive function when the quantum parallel principle is employed. A quantum computational process allows its initial state to be a superposition state of the quantum system of a universal quantum computer \[3, 4, 9\]. A superposition state may be expressed as a linear combination of the conventional computational bases of the quantum system. Each computational base of the superposition state could be thought of as one input state of a quantum algorithm running on the universal quantum computer if the superposition state is taken as the initial state of the quantum computational process of the quantum algorithm. This implies that the quantum algorithm can be performed in a parallel form on the universal quantum computer by taking at the same time all these different computational bases of the superposition state as its input states. From the point of view of the classical computational model a different input state corresponds to a different classical computational process. Then one could imagine that in effect the quantum computational process really performs simultaneously many different $^{\prime }$classical$%
^{\prime }$ computational processes with different input states when the initial state of the quantum computational process is a superposition state. Although these different $^{\prime }$classical$^{\prime }$ computational processes start at the same initial time, they could end at different times in computation, respectively. As a typical example, this situation will occur if the quantum algorithm is used to compute a general recursive function in mathematics and the end state of the computation is fixed. Because there is not the same end time for all these different $^{\prime }$classical$^{\prime }$ computational processes a conflict arises when the quantum computer decides what time the quantum computational process is halted. Thus, from the viewpoint of the mathematical logic principle of computational model the halting protocol is incompatible with the quantum parallel principle.
This conflict is apparently between the halting protocol and the quantum parallel principle, but it really related inherently to the non-unitarity of quantum measurement which is well known in quantum mechanics \[5\]. According as the halting protocol the halting quantum bit usually is set to the halting state $|n_{h}\rangle =|0\rangle $ at the initial time $t_{0}$ of the quantum computational process. Suppose that the initial superposition state of the whole quantum system including the halting qubit of the quantum computer is given by $$|\Psi (t_{0})\rangle =\underset{j}{\sum }a_{j}(t_{0})|n_{h}\rangle |\varphi
_{j}(t_{0})\rangle =\underset{j}{\sum }a_{j}(t_{0})|0\rangle |\varphi
_{j}(t_{0})\rangle .$$According as the halting protocol, when some $^{\prime }$classical$^{\prime
} $ computational processes arrive in their end states during the quantum computational process their halting state $|n_{h}\rangle $ becomes the state $|1\rangle .$ Since there is not the same end time for all these different $%
^{\prime }$classical$^{\prime }$ computational processes, there exists some time $t$ in the quantum computational process such that some $^{\prime }$classical$^{\prime }$ computational processes arrive in their end states ($%
\{|\varphi _{l}(t)\rangle \}$) and hence their halting state becomes the state $|1\rangle $, while the rest have not yet arrived in their end states and their halting state remains unchanged and is still the state $|0\rangle
. $ Then at the time $t$ the quantum state of the whole quantum system is written as $$|\Psi (t)\rangle =\underset{k}{\sum }a_{k}(t)|0\rangle |\varphi
_{k}(t)\rangle +\underset{l\neq k}{\sum }a_{l}(t)|1\rangle |\varphi
_{l}(t)\rangle .$$The state $|\Psi (t)\rangle $ is clearly a superposition state involved in the halting state $|n_{h}\rangle $ and the computational states $\{|\varphi
_{j}(t)\rangle \}$ of the quantum system. Again according as the halting protocol \[3\], the quantum measurement is carried out on the halting quantum bit periodically from the outside during the quantum computational process. Then the quantum measurement will change the state $|\Psi (t)\rangle $ and spoil the quantum computational process \[8\]. Actually, according to the quantum mechanics \[5\] the quantum-state collapse on the superposition state $%
|\Psi (t)\rangle $ occurs inevitably when the quantum measurement on the halting qubit is performed on the superposition state $|\Psi (t)\rangle .$ It is well known that a quantum-state collapse process is a non-unitary process in quantum mechanics \[5b\]. Thus, the halting protocol generally is irreversible and non-unitary in the universal quantum computational models in which the quantum parallel principle is a basic principle.
There are a number of works \[3, 8, 9, 10, 11, 12\] to discuss in detail the conflict and to propose schemes to avoid it for the universal quantum computational models. However, so far there is not any satisfactory and universal scheme to avoid the conflict when the initial state for a quantum computational process is a superposition state. On the other hand, the halting protocol may be made reversible in the reversible computational model \[54\] due to that any initial state of computation is generally a single basis state and hence there is not such a conflict in the reversible computational model. It has also been shown \[3, 8, 9, 10, 11, 12\] that this conflict could be avoided when the quantum computational process consists of a single $^{\prime }$classical$^{\prime }$ computational process or several different $^{\prime }$classical$^{\prime }$ computational processes which arrive in their end states at the same time. This is because in this case the state $|\Psi (t)\rangle $ above is either the first term with the halting state $|0\rangle $ or the second term with the halting state $%
|1\rangle $ but not a superposition of the two terms. Since an initial basis state corresponds one-to-one to a $^{\prime }$classical$^{\prime }$ computational process in the quantum computational process one may also say that the conflict could be avoided if the initial state of the quantum computational process is limited to a single basis state. Although from the viewpoint of the mathematical logic principle of computational model the halting protocol becomes reasonable if the initial state of the quantum computational process is limited to a single basis state, it could not be generally reversible and unitary as the halting protocol contains the non-unitary quantum measurement. The original halting protocol \[3\] uses the quantum measurement to achieve the halting operation, that is, if the halting qubit is found in the halting state $|1\rangle $ through the periodic quantum measurement the quantum computational process is stopped by the brute-force method from the outside. Therefore, even in the case that the initial state of the quantum computational process is restricted to be a basis state the original halting protocol generally is not reversible and unitary. The original halting protocol \[3\] also implies that there must be a conditional operation such that when the computational end state appears during the computational process this conditional operation converts immediately the initial halting state $|0\rangle $ into the state $|1\rangle
.$ Some improved halting protocols \[11, 12\] use explicitly the conditional halting operation to replace the original halting operation. It was also proposed in the halting protocol \[11\] that when the quantum computational process is stopped due to that the halting state $|0\rangle $ is changed to the state $|1\rangle $ it needs to start an extra unitary evolution process at the same time so that while both the computational end state and the halting state $|1\rangle $ are kept unchanged after the halting operation, the total evolution process can be unitary for the whole quantum system including the computational quantum system, the halting qubit, and the auxiliary qubits (the halting protocol \[11\] needs to use an extra auxiliary register). In the reversible computational model the reversible halting protocol could be achieved by executing successively a computational process, the conditional (logic) halting operation, and the inverse computational process \[54\]. However, the quantum measurement must be given up in these halting protocols \[3, 9, 10, 11, 12\] if one wants to make these halting protocols reversible and unitary thoroughly as the quantum measurement generally could lead to information loss of a quantum system even when the initial state is a single basis state. Therefore, the irreversibility and non-unitarity of these halting protocols in the universal quantum computational models are traced ultimately back to the irreversibility and non-unitarity of quantum measurement in quantum mechanics.
The halting protocol usually is not reversible and unitary in the universal quantum computational models due to the non-unitary quantum measurement, but for most of the present quantum algorithms based on these universal quantum computational models the computational results usually are not significantly affected by the non-unitarity of quantum measurement even though the non-unitarity could lead to loss of partial information of the computational results. There are also the incompatibilities within the universal quantum computational models between the quantum parallel principle and the halting protocol from viewpoint of the mathematical logic principle and between the unitary quantum computational process and the irreversible and non-unitary halting protocol from the viewpoint of the quantum physics, but up to now a number of powerful quantum algorithms \[13, 14\] that outperform their classical counterparts have been found within the universal quantum computational models and the halting protocol has not yet any significant effect on these powerful quantum algorithms. Therefore, on one hand, the quantum parallel principle is paid much attention, on the other hand, the halting protocol has a negligible influence on the quantum computational science in the past years. It has been explored in theory to eliminate the incompatibilities within the universal quantum computational models, but people are not clear whether or not the incompatibilities could lead to an essential impact on the power of quantum computation and do not yet know whether or not the reversibility and unitarity of the halting protocol is important to discover and develop new and efficient quantum algorithms. However, these powerful quantum algorithms \[13, 14\] can only treat successfully few special mathematical problems. Most of them are based on the unitary quantum circuit model \[4\] in which the irreversible and non-unitary halting protocol neither is used nor has any essential effect on the computational power of these quantum algorithms. It is also known that the incompatibilities could lead to that a universal quantum computer could not be more efficient than a classical computer in computing an arbitrary recursive function in mathematics. On the other hand, it is well known that a quantum computational process obeys the unitary quantum dynamics and is compatible with the mathematical logic principles used by the computational process. These facts show that it is worthwhile to investigate further how the halting protocol can be made reversible and unitary so that the incompatibilities could be eliminated in the universal quantum computation models.
The unitary dynamical principle of quantum mechanics plays a key role in the scalable quantum computation in a mixed-state quantum ensemble. This basic principle states simply that both a closed quantum system and its quantum ensemble obey the same unitary dynamics quantum mechanically \[17, 18, 20, 22, 23, 24\]. The unitary quantum dynamics associated with the symmetry and structure of a quantum system has been used to discover and develop new quantum search processes in a quantum ensemble \[17\] and a pure-state quantum system \[23, 24\]. Quantum search processes are extremely important in quantum computation as the unsorted quantum search process has an extensive application in computational science and can be used to solve the NP-hard problems. It has been shown that the square speedup for the standard quantum search algorithm \[7\] is optimal in a pure-state quantum system \[15\]. The first attempt to break through the quantum-searching square speedup limit was carried out in a nuclear magnetic resonance (NMR) spin ensemble \[16, 17a\]. Though these quantum search processes \[16, 17\] are not scalable in a spin ensemble, their speed is really exceed greatly the allowed value of any quantum search algorithm with the square speedup limit in a range of a few quantum bits. However, these quantum search processes do not achieve a real breakthrough of the square speedup limit because their output NMR signal intensities decay exponentially as the qubit number of the spin system increases and hence beyond a few quantum bits their quantum-searching speed falls off rapidly. So far a scalable quantum search algorithm working in a quantum ensemble has not yet been found. Both the unitary dynamics quantum mechanically and the quantum coherence interference play a key role in achieving the fast and scalable quantum computation in a spin ensemble \[17, 18, 20\]. It has been shown that many oracle-based quantum algorithms including the parity-determination algorithm are subjected to the polynomial speedup bounds in a pure-state quantum system \[19\]. Again the unitary quantum dynamics has been shown to play an important role in achieving a much fast computational speed to solve the parity-determination problem in a spin ensemble which is beyond the polynomial speedup bounds on the oracle-based quantum algorithms in a range of a few quantum bits \[18\]. Up to now, the scalable problem has not yet been solved for the quantum parity-determination algorithm in a quantum ensemble. These results obtained in a spin ensemble encourage one to explore further the potential ways to break through the quantum-searching square speedup limit and the polynomial speedup bounds upon the oracle-based quantum algorithms. The unitary quantum dynamics associated with the symmetry and structure of a spin system has also showed that the prime factorization for a large composite integer may be implemented in a scalable form in a spin ensemble \[20\]. The efficient factoring algorithm was first discovered in a pure-state quantum system \[6\]. In order to achieve the scalable quantum computation for the prime factorization in a spin ensemble it is necessary to exploit the symmetric property and structure of the spin system of the spin ensemble to help the unitary quantum dynamics to solve the prime factorization problem. Here both the time-reversal symmetry and the rotation symmetry in spin space \[5a\] of the spin system play a key role in the scalable factoring algorithm in a spin ensemble \[20\]. The factoring algorithm in the spin ensemble usually is divided into four time periods \[20\], which is similar to the conventional NMR experimental counterparts \[21\]. The first period is to generate the multiple-quantum coherences of the spin ensemble. In this period information of the order of the modular exponential function is loaded on the multiple-quantum coherences. The second is the time evolution process to carry out the frequency labeling for these different order multiple-quantum coherences generated in the first period. The third is the time-reversal process of the first period. The final is the quantum measurement to output the NMR multiple-quantum coherence signal that carries the information of the order of the modular exponential function. Here the output NMR multiple-quantum spectrum usually is used to determine the order of the modular exponential function. The multiple-quantum spectrum may be obtained by Fourier transforming the output NMR multiple-quantum coherence signal that is measured indirectly in experment. The time-reversal symmetry \[5a\] ensures that the dephased NMR multiple-quantum coherences in the first period can be refocused in the third period and hence the multiple-quantum coherences become inphase so that the multiple-quantum coherence interference can lead to coherent enhancement of the output NMR multiple-quantum coherence signal in the factoring algorithm. Therefore, it becomes possible that the output NMR multiple-quantum coherence signal does not decrease exponentially as the qubit number of the spin system increases. The time-reversal symmetry has been used extensively to obtain highly sensitive and inphase NMR multiple-quantum spectra in high-resolution nuclear magnetic resonance experiments in solid \[21\]. On the other hand, the rotation symmetry in spin space \[5a\] of a spin system is the basis of the multiple-quantum operator algebra spaces of the Liouville operator space of the spin ensemble \[22\]. Though the total number of independent NMR multiple-quantum transitions in an $n-$qubit spin system increases exponentially as the qubit number of the spin system, a large number of the independent multiple-quantum transitions are really degenerative or nearly degenerative in transition frequency due to the rotation symmetry in spin space, and the multiple-quantum transitions with significantly different transition frequencies are really very few in the spin system. For example, there may be only $(n+1)$ different order multiple-quantum transitions in the $n-$qubit spin system and each has its own transition frequency, while the total number of the independent multiple-quantum transitions of the spin system is $(4^{n}-2^{n})/2$. In theory the number of the multiple-quantum-transition spectral lines in the NMR multiple-quantum spectrum may be equal to the number of the independent multiple-quantum transitions of the spin system. Note that the total spectral intensity of the NMR multiple-quantum-transition spectrum of the spin system is not more than the total magnetization ($\thicksim n2^{n}$) of the spin system in the thermal equilibrium state \[55\], while the total number $(4^{n}-2^{n})/2$ of the NMR multiple-quantum-transition spectral lines increases exponentially ($%
\thicksim 4^{n}$) as the qubit number. The intensity of each multiple-quantum-transition spectral line therefore weakens exponentially ($%
\thicksim n/2^{n}$) as the qubit number increases, although all these multiple-quantum spectral lines are inphase due to the time-reversal symmetry. Then due to noise in the detected NMR signal each of these multiple-quantum-transition spectral lines will become unobservable even when the qubit number is moderate if most of these multiple-quantum-transition spectral lines have different resonance frequencies (i.e., transition frequencies). Fortunately, due to the rotation symmetry in spin space there are very few multiple-quantum transitions with significantly different transition frequencies in the spin system and hence very few observable multiple-quantum spectral lines with significantly different resonance frequencies in the multiple-quantum spectrum. A large number of the inphase multiple-quantum spectral lines overlap with each other in the multiple-quantum spectrum due to that they have the same resonance frequency. As a result, all these inphase multiple-quantum spectral lines with the same resonance frequency become really a single multiple-quantum spectral line, and the intensity of the single multiple-quantum spectral line is really the sum of all the intensities of these inphase multiple-quantum spectral lines. This intensity becomes so large that the noise in the detected NMR signal can not have any significant effect on the single multiple-quantum spectral line. As shown in Ref. \[20\], if now there are only the $(n+1)$ different order multiple-quantum transitions in the $n-$qubit spin system, each with a significantly different transition frequency, then each of the $(n+1)$ different order multiple-quantum transitions is composed of the degenerative multiple-quantum transitions which have the number $\thicksim $ $%
(4^{n}/2-2^{n-1})/(n+1)$ on average and its multiple-quantum-transition spectral intensity is proportional to the number $(4^{n}/2-2^{n-1})/(n+1)$ on average. Here the total spectral intensity of the $(4^{n}/2-2^{n-1})$ inphase multiple-quantum-transition spectral lines plus the spectral intensity of the longitudinal magnetization and spin order components is really equal to the total magnetization of the spin system \[55\] which can be observable in the conventional NMR experiments even for a very large qubit number. Obviously, the average intensity for each multiple-quantum-transition spectral line is inversely proportional to the qubit number and some of these $(n+1)$ multiple-quantum spectral lines do not weaken exponentially as the qubit number increases. Thus, both the time-reversal symmetry and the rotation symmetry in spin space ensure that the output NMR multiple-quantum-transition spectral intensities can be efficiently detected in the spin ensemble in the factoring algorithm. On the other hand, so far quantum entanglement has not yet been proven strictly to play a key role in speeding up a quantum computation, whereas the locally efficient and scalable factoring algorithm in the spin ensemble \[20\] shows that quantum entanglement could not be a key component to make quantum computation much more powerful than classical computation. Therefore, the unitary dynamics quantum mechanically associated with the symmetry and structure of a quantum system could be the key component to power the quantum computation and hence is a general guidance to discover and develop new and efficient quantum algorithms in quantum computational science. The unitary dynamical principle quantum mechanically implies that the irreversible and non-unitary halting protocol should be modified to be reversible and unitary so that the incompatibility between the irreversible and non-unitary halting protocol and the unitary quantum computational process can be eliminated within the universal quantum computational models.
The unitary dynamics quantum mechanically associated with the symmetry and structure of a quantum system also plays a key role in discovering the scalable and efficient quantum algorithms in a pure-state quantum system. It is well known that there are $2^{n}$ basis states in the Hilbert space of an $n-$qubit spin system. These basis states may be chosen as the eigenstates of the $z-$component operator $J_{z}$ of the total spin angular momentum of the spin system. But due to the rotation symmetry in spin space of the $n-$qubit spin system many of these $2^{n}$ basis states are degenerative and have the same eigenvalue. Thus, the whole Hilbert space span by these $2^{n}$ eigenstates of the $z-$component operator $J_{z}$ may be divided into $(n+1)$ different state subspaces according to the rotation symmetry in spin space of the spin system \[5a, 5c\]. The quantum search space which is just the whole Hilbert space of the $n-$qubit spin system for the standard quantum search algorithm therefore is reduced to the largest state subspace among these $(n+1)$ state subspaces. Hence the standard quantum search algorithm is improved \[23\], since the largest state subspace is still much smaller than the whole Hilbert space. Moreover, it has been shown \[23\] that any unknown quantum state can be efficiently transferred to a larger state subspace from a state subspace in the Hilbert space of the $n-$qubit spin system, while the inverse process of the state transfer is generally harder to be carried out. This general rule not only is useful for solving the quantum search problem but also helpful for understanding deeply non-equilibrium processes of a quantum ensemble from the viewpoint of the unitary quantum dynamics instead of the conventional probability theory. As shown in Ref. \[24\], this general rule is also closely related to that there exists a computational-power difference between a unitary evolution process and its inverse process in a quantum system. Of course, this inverse unitary process may exist or may not in the quantum system. A direct extension of the idea that the symmetric property and structure of a quantum system could help the unitary quantum dynamics to solve the quantum search problem is to exploit further the symmetric property and structure of a general group such as a cyclic group in a quantum system to help the unitary quantum dynamics to discover and develop new and efficient quantum search algorithms \[24\]. With the help of the reversible and unitary halting protocol based on the state-locking pulse field and the property and structure of a cyclic group in a quantum system it has been shown by the unitary quantum dynamics that a universal quantum computer could be enough powerful to solve efficiently the quantum search problem in a cyclic-group state space \[24\]. Here the important point to arrive at this conclusion is that the halting protocol of the universal quantum computational models is available and may be made reversible and unitary for the quantum search process if the initial state of the quantum search process is limited to a single computational basis state.
All these conventional halting protocols \[3, 9, 10, 11, 12\] of the universal quantum computational models generally can not be used to construct an efficient quantum search algorithm based on the unitary quantum dynamics \[24\]. This is because they are either irreversible and non-unitary or dependent sensitively upon initial states of the quantum computational process under study. For example, in the reversible computational model the step number of computational process in the reversible halting protocol \[54\] is dependent sensitively upon the initial state if the output state is fixed in the reversible halting protocol, while in the reversible and unitary halting protocol \[11\] the output state is dependent sensitively upon the initial state if the step number of computational process is fixed. Thus, both the reversible and unitary halting protocols are not suitable to construct an efficient quantum search process based on the unitary quantum dynamics. Only the specific reversible and unitary halting protocols that are based on the state-locking pulse field \[24\] could be useful for solving efficiently the quantum search problem in the cyclic group state space. This is because neither the output state nor the step number of computational process in such a reversible and unitary halting protocol is dependent sensitively upon any initial state of the computational process and this is the key point for the reversible and unitary halting protocol to be useful for solving efficiently the quantum search problem. While the reversible and unitary halting protocol is the key component of the quantum search process to solve efficiently the quantum search problem in the cyclic group state space, the state-locking pulse field plays a key role in constructing such a reversible and unitary halting protocol. This is because the state-locking pulse field could make both the output state and the step number of computational process in the reversible and unitary halting protocol insensitive to any unknown marked state of the quantum search problem. The computational complexity for the quantum search process in the cyclic group state space \[24\] could be mainly dependent on the performance of the state-locking pulse field used in the quantum search process. The reversible and unitary halting protocol not only plays an important role in solving efficiently a quantum search problem in the cyclic group state space but also has an extensive application in quantum computation. This is one of the key components to realize a universal quantum computer to replace fully a classical computer in future. Therefore, it is necessary to design a good-performance unitary quantum control unit (or circuit) to simulate faithfully and efficiently the reversible and unitary halting protocol of the universal quantum computational models. A unitary quantum control unit that consists of a trigger pulse, a state-locking pulse field, and a control state subspace has been proposed to simulate faithfully and efficiently the reversible and unitary halting protocol \[24\]. The key component of the unitary quantum control unit is the state-locking pulse field. A state-locking pulse field is able to keep a desired state almost unchanged in a unitary form for a long time in a quantum computational process, and this is the reason why the output state of the reversible and unitary halting protocol based on the state-locking pulse field does not depend sensitively upon any initial states. How to design a good-performance state-locking pulse field is a challenge and also an important research subject in quantum computation in future. In general, a general state-locking pulse field may be dependent upon the time variable and the space variables and even the quantum field variables. A state-locking pulse field generally could be a sequence of time- and space-dependent electromagnetic pulse fields such as the laser pulses and could also contain any time- and space-dependent potential fields.
In this paper a new quantum program and circuit is constructed explicitly to reduce the quantum search space which may be generally a cyclic-group state space or the Hilbert space of an $n-$qubit quantum system. This quantum program and circuit is much simpler than that one in the previous paper \[24\], where in order to show clearly that an ideal universal quantum computer could be powerful enough to solve efficiently the quantum search problem in the multiplicative-cyclic-group state space the quantum program and circuit used to reduce the quantum search space is designed in a more complex form. It can also be used to construct further a quantum search process to solve efficiently the unsorted quantum search problem in a general cyclic group state space or the Hilbert space of an $n-$qubit quantum system. The quantum program and circuit may be divided into two almost independent units: the unitary quantum computational unit and the unitary quantum control unit. The quantum control unit simulates efficiently the reversible and unitary halting protocol based on the state-locking pulse field, while the quantum computational unit is responsible for the reduction of the quantum search space. In the paper the quantum program and circuit is first analyzed completely. Then the basic properties of an ideal state-locking pulse field are described in detail, and it is shown how the quantum control unit simulates faithfully and efficiently the reversible and unitary halting protocol. The basic principles to construct a general state-locking pulse field and simulate efficiently the reversible and unitary halting protocol are suggested and then explained in detail. It is proposed in the paper that a simple atomic physical system which consists of an atomic ion or a neutral atom in the double-well potential field is used to realize the reversible and unitary halting protocol. In the atomic physical system a generalized state-locking pulse field used to build up the reversible and unitary halting protocol is also constructed explicitly.
As an important application the reversible and unitary halting protocol may be used to build up the reversible and unitary quantum program and circuit to compute some mathematical functions. Here by solving a simple problem given below one may illustrate how to use the reversible and unitary halting protocol to solve a general mathematical problem in quantum computation. Suppose that given a periodic function $f(x)=f(x+x_{T})$ with the period $%
x_{T}$ and the integer variable $x=0,1,...,x_{T}-1,$ there is a computational circuit $U_{f}$ to compute the functional value $f(x+1)$ from the functional value $f(x)$ for any integer $x=0,1,...,x_{T}-1$, where the functional value $f(x)$ is a distinct integer for every distinct integer $x$ for $0\leq x<x_{T}$. The functional operation $U_{f}$ could be expressed as $%
U_{f}f(x)=f(x+1)$ \[13, 14\] for $x=0,1,...,x_{T}-1.$ Now given an unknown functional value $f(x_{0})$ with the unknown integer $x_{0}\in
\{0,1,...,x_{T}-1\},$ one wants the unknown functional value $f(x_{0})$ to be changed to the desired functional value $f(x_{f})$ with the known integer $x_{f}\in \{0,1,...,x_{T}-1\},$ e.g., $f(x_{f})=0$ or $1.$ A simple scheme to solve this simple problem is that one computes one-by-one the functional value $f(x_{0}+x)$ for the integer $x=0,1,...,x_{T}-1$ by the computational circuit $U_{f}$ and checks the functional value for each computing step, and when the functional value $f(x_{0}+x)$ is found to be equal to the desired functional value $f(x_{f})$ the computational process is halted. Then the final result of the computational process is clearly the desired functional value $f(x_{f})$. Though this problem is very simple from the viewpoint of the computational science, it is surprising that the scheme that can solve efficiently this simple problem in a reversible and unitary form could also be used to solve efficiently the quantum search problem. Note that given any initial value $x_{0}$ and the final value $x_{f}$ there is the unique integer $x$ within the range $0\leq x<x_{T}$ such that $f(x_{0}+x)=f(x_{f}).$ Now a classical computational program $Q_{cl}$ for the scheme could be written down as $$n_{h}=0$$$$f(x)=f(x_{0})$$$$\text{For }i=1\text{ to }x_{T}$$$$\text{If }f(x)=f(x_{f})\text{ then }n_{h}=1$$$$\text{while }n_{h}=1,\text{ halting}$$$$\text{else }f(x)\rightarrow f(x+1)\text{ end if}$$$$\text{end for}$$where $n_{h}=0$ or $1$ is the halting bit that is used to indicate when the program terminates. As shown in the program, when the halting bit value $%
n_{h}=1$ the program terminates. The program consists of $x_{T}$ cycles with the cyclic index $i=1,2,...,x_{T}$. Evidently, this simple program outputs the desired functional value $f(x_{f})$ no matter what the initial functional value $f(x_{0})$ is with the possible integer $%
x_{0}=0,1,...,x_{T}-1$. Here the unknown functional value $f(x_{0})$ may be stored in memory on the computer in advance or is input from outside the program. The program first checks whether the initial functional value $%
f(x_{0})$ equals $f(x_{f}).$ If $f(x_{f})=f(x_{0})$, the halting bit value $%
n_{h}$ is changed to $1$ from the initial value $0$ and then program terminates due to $n_{h}=1,$ and the output result is $f(x_{f});$ otherwise the program computes the functional value $f(x_{0}+1)$ and checks again whether the functional value $f(x_{0}+1)$ is $f(x_{f})$ or not. This computing process is repeated until the computed functional value $%
f(x_{0}+i) $ is found to be equal to $f(x_{f}),$ here $(x_{0}+i)\func{mod}%
x_{T}=x_{f}.$ Then the halting bit value $n_{h}=0$ is changed to $1$ so that the program is halted. Thus, the output functional value for the computational program $Q_{cl}$ is always $f(x_{f})$ no matter what the initial functional value $f(x_{0})$ is. This is the important property of the classical halting protocol. This property is also very important for the reversible and unitary halting protocol.
If the classical computational program $Q_{cl}$ would be reversible and unitary, then it could be suitably used to build up an efficient quantum search process to solve the quantum search problem in a cyclic-group state space, and such a quantum search process would be much simpler than that one in the previous paper \[24\]. For the quantum search problem in the multiplicative cyclic-group state space $S(C_{p-1})$ \[24\] the periodic function $f(x)$ in the program $Q_{cl}$ may be chosen as the modular exponential function $f_{k}(x)=(g^{M_{k}})^{x}\func{mod}p$ with the order (or period) $x_{T}=m_{k}$ and the integer variable $x=0,1,...,m_{k}-1$ for $%
k=1,2,...,r$. Then the corresponding reversible functional operation $U_{f}$ may be defined as the unitary cyclic-group operation $U_{g^{M_{k}}}:f_{k}(x)%
\rightarrow f_{k}(x+1)$. Here $p$ is a prime and $g$ the generator of the multiplicative cyclic group $C_{p-1},$ and the group operation of a multiplicative cyclic group is the modular multiplication operation. The multiplicative cyclic group $C_{p-1}$ has the order $%
p-1=p_{1}^{a_{1}}p_{2}^{a_{2}}...p_{r}^{a_{r}}$, where $p_{1},$ $p_{2},$ $%
...,$ $p_{r}$ are distinct primes and the exponents $a_{1},$ $a_{2},$ $...,$ $a_{r}>0.$ Here the integer $m_{k}=p_{k}^{a_{k}}$ and $p-1=m_{k}M_{k}$ for $%
k=1,2,...,r$. Any pair of the integers $m_{i}$ and $m_{j}$ are coprime to each other for $1\leq i<j\leq r,$ and for convenience usually set $%
m_{1}<m_{2}<...<m_{r}$. Obviously, every functional value is distinct for the modular exponential function $f_{k}(x)$ which satisfies $f_{k}(x)\geq 1$ for $x=0,1,...,m_{k}-1$. The functional states $\{|f_{k}(x)\rangle \}$ really form the multiplicative-cyclic-group state subspace $%
S(m_{k})=\{|(g^{M_{k}})^{s_{k}}\func{mod}p\rangle ,$ $s_{k}=0,1,...,m_{k}-1%
\} $ of the factor cyclic subgroup $C_{p_{k}^{a_{k}}}$ of the multiplicative cyclic group $C_{p-1}=\{g^{s}\func{mod}p\}=C_{p_{1}^{a_{1}}}\times
C_{p_{2}^{a_{2}}}\times ...\times C_{p_{r}^{a_{r}}},$ whereas the multiplicative-cyclic-group state space $S(C_{p-1})$ with dimension $p-1$ is the direct product of the multiplicative-cyclic-group state subspaces $%
\{S(m_{k})\}$ with dimensions $\{m_{k}\}:$$$S(C_{p-1})=S(m_{1})\bigotimes S(m_{2})\bigotimes ...\bigotimes S(m_{r}).$$If the quantum search problem is solved in the additive-cyclic-group state space $S(Z_{p-1})$ with dimension $p-1=m_{1}m_{2}...m_{r},$ then the periodic function $f(x)$ may be taken as the modular function $f_{k}(x)=x%
\func{mod}m_{k}$ with the period $x_{T}=m_{k}$ and $x=0,1,...,m_{k}-1$ for $%
k=1,2,...,r$. Every modular functional value $f_{k}(x)$ is distinct and $%
f_{k}(x)=0,1,...,m_{k}-1.$ Obviously, the modular functional states $%
\{|f_{k}(x)\rangle \}$ form really the additive-cyclic-group state subspace $%
S(Z_{m_{k}})=\{|s_{k}\func{mod}m_{k}\rangle ,$ $s_{k}=0,1,...,m_{k}-1\}.$ The additive cyclic group $Z_{p-1}=\{0,1,...,p-2\}$ may be decomposed into the direct sum of the factor additive cyclic subgroups $\{Z_{m_{k}}%
\}:Z_{p-1}=Z_{m_{1}}\bigoplus Z_{m_{2}}\bigoplus ...\bigoplus Z_{m_{r}},$ here the group operation of an additive cyclic group is the modular addition operation. The additive-cyclic-group state space $S(Z_{p-1})$ then may be decomposed into the direct product of the factor additive-cyclic-group state subspaces $\{S(Z_{m_{k}})\}$ with dimensions $\{m_{k}\}:$$$S(Z_{p-1})=S(Z_{m_{1}})\bigotimes S(Z_{m_{2}})\bigotimes ...\bigotimes
S(Z_{m_{r}}).$$Though the $2^{n}-$dimensional Hilbert space $S(Z_{2^{n}})$ of an $n-$qubit quantum system may be thought of as an additive-cyclic-group state space under the modular addition operation ($\func{mod}2^{n}$), for the Hilbert space $S(Z_{2^{n}})$ there is not a direct-sum decomposition like $%
S(Z_{p-1}) $ above with $r>1.$ However, the Hilbert space $S(Z_{2^{n}})$ may be decomposed into the direct product of $n$ $2-$dimensional additive-cyclic-group state subspaces $\{S(Z_{2})\}$, $$S(Z_{2^{n}})=S(Z_{2})\bigotimes S(Z_{2})\bigotimes ...\bigotimes S(Z_{2}).$$If now the periodic function $f(x)$ is chosen as the modular function $%
f_{k}(x)=x\func{mod}2$ with $x=0,1$ for $k=1,2,...,n$, then the quantum search problem in the $2^{n}-$dimensional Hilbert space of the $n-$qubit quantum system may also be solved just like that one in the additive-cyclic-group state space $S(Z_{p-1})$ or in the multiplicative-cyclic-group state space $S(C_{p-1})$ \[24\]. The symmetric properties and structures of both the cyclic groups $C_{p-1}$ and $Z_{p-1}$ have been suggested to help the unitary quantum dynamics to solve the quantum search problem in the cyclic-group state spaces $S(C_{p-1})$ and $%
S(Z_{p-1}),$ respectively \[24\].
Obviously, the classical computational program $Q_{cl}$ is not reversible due to the irreversible operations in the program and especially due to the fact that the halting operation within the program is not reversible. However, in order that it can be used to construct a quantum search process to solve the quantum search problem in the cyclic-group state space the whole program $Q_{cl}$ including the halting operation must be made reversible and unitary. Generally, a classical irreversible computational program may be made reversible in the frame of the reversible computational model \[1\] with the help of the reversible mathematical logic gates and especially the reversible operation of a general function mathematically \[1\], the universal quantum gates and especially the conditional quantum gates \[25\], and other unitary operations quantum mechanically \[26\]. One can also construct directly a unitary quantum circuit equivalent to a classical computational program on the universal quantum circuit model \[4\]. It has been shown that there is an equivalent quantum circuit to a given reversible quantum program on the universal quantum Turing machine \[27\]. However, these conventional methods by which an irreversible classical computational program is made reversible and unitary could not be always suitable for transforming the irreversible halting protocol to the reversible and unitary one in the universal quantum computational models. First, there could not be a universal halting protocol in the universal quantum computational models. Second, although there could be a halting protocol in the universal quantum computational models when the input state of a quantum computational process is limited to a single basis state, this halting protocol could not be thought to be reversible in the sense that an information loss occurs in the halting protocol due to the non-unitary quantum measurement. For example, by the quantum measurement one could know what time the halting state $%
|0\rangle $ is changed to the state $|1\rangle $ and hence obtains the information of the instant of time at which the quantum system arrives in the end state of the quantum computational process, implying that the quantum computational process loses the information of the instant of time. However, even if the current quantum measurement could not be non-unitary according to quantum mechanics \[5b\] as the measured base now is a single basis state of the measurement operator and such an information loss could not have a significant effect on some quantum computational processes, this information is not available according as the mathematical logic principles of the quantum search problem and can not be yet allowed to use if the halting protocol is used in solving the quantum search problem based on the unitary quantum dynamics. This is because a quantum computational process obeys the quantum physical laws and is compatible with the used mathematical logic principles. Therefore, the classical program $Q_{cl}$ may be made reversible and unitary according as these conventional methods except the irreversible halting operation within the program. Of course, in the case that any initial state is restricted to be a single basis state the halting operation may also be made reversible by the conventional methods \[11, 54\] for the conventional computational tasks other than the current tasks. In the reversible computational model the reversible halting operation in the program $Q_{cl}$ may be achieved in such a way \[54\] that after the computational process from the initial functional value $f(x_{0})$ to the output functional value $f(x_{f})$ is done the conditional halting operation is executed and then the inverse computational process is performed from the functional value $f(x_{f})$ back to the original functional value $f(x_{0}),$ and the cyclic process consisting of the computational process and inverse computational process repeats incessantly. Evidently, this extra inverse computational process is dependent upon the cyclic index $i$ of the program as a different initial functional value $f(x_{0})$ is changed to the functional value $f(x_{f})$ in a different cycle in the program. One could also use the halting protocol \[11\] to achieve the reversible halting operation for the program. When the program arrives at the output functional state (corresponding to $f(x_{f})$, see below) the conditional halting operation is executed and then an extra unitary evolution process starts, that is, the state of the auxiliary register starts to evolve \[11\]. If now the step number of the program $Q_{cl}$ is fixed (for example the step number may be set to $x_{T})$, then a different initial functional state (corresponding to $f(x_{0})$) will result in a different output state of the auxiliary register. Thus, the total output state of the program is really dependent upon the initial functional state. One therefore concludes that if any of the two reversible halting protocols \[11, 54\] is used in the program, then the program is dependent sensitively upon the initial functional state. Such a reversible program is not suited as a component of the quantum search process based on the unitary quantum dynamics. It is necessary to use the specific method to make the halting operation within the program $Q_{cl}$ reversible and unitary if one wants to use the program to construct a quantum search process. This specific scheme to make the halting protocol reversible and unitary may be stated below. As the first point of the scheme, in order that the halting protocol is reasonable from the viewpoint of the mathematical logic principle any initial state is limited to a single basis state for the halting protocol. The cost for this point is that the quantum parallel principle could become less important. As the second point, the halting protocol should not contain any quantum measurement so as to keep away from any irreversible and non-unitary process and avoid any information loss of the quantum system. Generally, the unitary quantum dynamics avoids using any non-unitary quantum measurement as its quantum operation within quantum computational processes. This is quite different from the early proposals that the quantum measurement may also be used as a quantum operation to build up quantum circuits and algorithms \[13, 28\]. If the halting operation within the classical program $Q_{cl}$ is achieved by the brute-force method, then the end state (i.e., the output state) of the classical program is independent of any initial functional value $f(x_{0}).$ But the brute-force method to stop the program from the outside could cause the whole program irreversible. Therefore, for the third point of the scheme the brute-force method is replaced with the unitary operation conditionally depending upon the halting state to stop the program \[11, 12, 54\]. However, all these three points above in the scheme can not ensure that the end state of the program which could include the computational state, halting-qubit state, and auxiliary state is independent of any initial functional state, that is, all these three points can not lead to the important property of the halting operation that the output state is independent of any initial state. It is of crucial importance to make the end state of the program independent or almost independent of any initial functional state. This is because whether or not the quantum search process based on the reversible and unitary halting protocol is efficient is mainly dependent upon this point. Thus, as the fourth point, the end state of the program is locked by the state-locking pulse field so that it is not dependent sensitively upon any initial functional state. This scheme which consists of the four points above has been used in the previous paper \[24\]. However, this scheme is not intuitive to understand the reversibility and unitarity of the reversible and unitary halting protocol mainly due to that it is hard to understand how the state-locking pulse field is capable of keeping a quantum state almost unchanged in a unitary form for a long time in a quantum computational process.
It was proposed in the previous paper \[24\] that a unitary quantum control unit which consists of a trigger pulse ($P_{t}$), a state-locking pulse field ($P_{SL}$), and a two-state control state subspace is used to simulate efficiently the reversible and unitary halting protocol. This quantum control unit does not use any non-unitary quantum measurement as its component. The state-locking pulse field which is the key component of the unitary quantum control unit is used to keep the desired state almost unchanged in a unitary form for a long time in the quantum computational process under study. Now by using a similar unitary quantum control unit to simulate faithfully and efficiently the halting operation within the classical program $Q_{cl}$ and with the help of the reversible mathematical logic operations \[1\], the universal quantum gates and especially the conditional quantum gates \[25\], and any other unitary quantum operations, a reversible and unitary quantum computational program $Q_{c}$ and its equivalent quantum circuit which correspond to the classical program $Q_{cl}$ can be constructed explicitly in the frames of the reversible computational model \[1\] and the universal quantum circuit model \[4\], respectively, and they may be represented intuitively by $$\text{State-Locking Pulse Field }(P_{SL}):ON \tag{P1}$$$$|n_{h}\rangle =|0\rangle \tag{P2}$$$$|b_{h}\rangle =|0\rangle \tag{P3}$$$$|f_{r}(x)\rangle =|f_{r}(x_{0})\rangle \tag{P4}$$$$\text{For }i=1\text{ to }m_{r} \tag{P5}$$$$\text{If }|f_{r}(x)\rangle =|1\rangle \text{ then }U_{b}^{c}:|b_{h}\rangle
\rightarrow |b_{h}+1\rangle \text{ end if} \tag{P6}$$$$\text{While }|f_{r}(x)\rangle =|1\rangle ,\text{ Do }U_{h}^{c}:|n_{h}\rangle
|f_{r}(x)\rangle =|0\rangle |1\rangle \rightarrow |0\rangle |0\rangle ,$$$$\quad \qquad P_{t}:|n_{h}\rangle |f_{r}(x)\rangle =|0\rangle |0\rangle
\rightarrow |c_{1}\rangle |0\rangle ,\text{ }P_{SL}:|c_{1}\rangle
\rightarrow |c_{2}\rangle \tag{P7}$$$$\text{If }|b_{h}\rangle =|0\rangle \text{ then }U_{f}:|f_{r}(x)\rangle
\rightarrow |f_{r}(x+1)\rangle \text{ end if} \tag{P8}$$$$\text{end for} \tag{P9}$$$$\text{State-Locking Pulse Field }(P_{SL}):OFF \tag{P10}$$Note that the halting bit $n_{h}$ and the function $f(x)$ in the classical program $Q_{cl}$ have already been replaced with the halting state $%
|n_{h}\rangle $ and the functional state $|f(x)\rangle $ in the quantum program $Q_{c},$ respectively. In the quantum program $Q_{c}$ the functional state $|f(x)\rangle $ is set to the modular exponentiation state $%
|f_{r}(x)\rangle $ with the period $x_{T}=m_{r}$ of the multiplicative-cyclic-group state subspace $S(m_{r})=\{|f_{r}(x)\rangle
=|(g^{M_{r}})^{x}\func{mod}p\rangle ;$ $x=0,1,...,m_{r}-1\}$ and the desired functional state $|f_{r}(x_{f})\rangle $ set to the state $|1\rangle .$ Note that the multiplicative-cyclic-group state subspace $S(m_{r})$ does not contain the state $|0\rangle $ because the functional value $f_{r}(x)\geq 1$ for any integer $x=0,1,...,m_{r}-1$. Owing to $f_{r}(x)\neq 0$ here the state $|f_{r}(x)\rangle =|0\rangle $ means that the state in the register of the functional state $|f_{r}(x)\rangle $ takes the state $|0\rangle $ rather than that the functional value $f_{r}(x)=0.$ The unitary functional operation $U_{f}$ is defined by $U_{f}|f_{r}(x)\rangle =|f_{r}(x+1)\rangle $ for $|f_{r}(x)\rangle \in S(m_{r}),$ here $U_{f}$ is really the cyclic-group unitary operation $U_{g^{M_{r}}}$ of the factor cyclic subgroup $%
C_{p_{r}^{a_{r}}}$ of the multiplicative cyclic group $C_{p-1}$. If the state $|f_{r}(x)\rangle =|0\rangle ,$ then $U_{f}|f_{r}(x)\rangle
=U_{f}|0\rangle =|0\rangle $ as the state $|0\rangle $ does not belong to the multiplicative-cyclic-group state subspace $S(m_{r})$. More generally the functional operation $U_{f}$ satisfies $U_{f}|g(x)\rangle =|g(x)\rangle $ if the state $|g(x)\rangle $ is not in the state subspace $S(m_{r})$. Though here the quantum program $Q_{c}$ and its equivalent quantum circuit are designed for the multiplicative-cyclic-group state subspaces $\{S(m_{k}),$ $%
k=1,2,...,r\}$ (the index $r$ may be replaced with $k$ in the quantum program $Q_{c}$), similar quantum programs and circuits can also be constructed for the additive-cyclic-group state subspace $%
S(Z_{m_{k}})=\{|f_{k}(x)\rangle =|x\func{mod}m_{k}\rangle ;$ $%
x=0,1,...,m_{k}-1\}$ for $k=1,2,...,r$ and for a general periodic-function state space. The quantum program $Q_{c}$ contains mainly $m_{r}$ cycles with the cyclic index $i=1$ to $m_{r}$ and the state-locking pulse field ($P_{SL}$).
The quantum program $Q_{c}$ consists of ten statements, which are denoted conveniently as the statement P1, statement P2, ..., statement P10, respectively. In particular, the statement P7 consists of the conditional unitary operation $U_{h}^{c}$, the trigger pulse $P_{t}$, and the state-locking pulse field $P_{SL}.$ The quantum program $Q_{c}$ is more complex than the classical one $Q_{cl}$ mainly due to the statement P7. The statement P7 contains the unitary quantum control unit that simulates efficiently the reversible and unitary halting protocol. The unitary quantum control unit consists of the trigger pulse $P_{t},$ the state-locking pulse field $P_{SL},$ and the control state subspace $S(C)=\{|c_{1}\rangle
,|c_{2}\rangle \}$. Any one of these two states $|c_{1}\rangle $ and $%
|c_{2}\rangle $ of the control state subspace is different from the initial halting state $|n_{h}\rangle =|0\rangle $. Actually the three states, i.e., the halting state $|n_{h}\rangle =|0\rangle $ and these two states $%
|c_{1}\rangle $ and $|c_{2}\rangle ,$ should be orthogonal to each other and belong to the same register which is named the halting register here. These three states are also called the halting-register states or briefly the halting states. It will be seen in next section that the halting register may be generated from the simple physical system of an atomic ion or a neutral atom in the double-well potential field. The physical system of the halting register may also be called the quantum control system whose Hilbert space contains the control state subspace $S(C)$ and probably other relevant states. Both the trigger pulse $P_{t}$ and the state-locking pulse field $%
P_{SL}$ are applied only to the physical system of the halting register. Actually, in the quantum program $Q_{c}$ the state-locking pulse field $%
P_{SL}$ is applied only to the control state subspace $S(C).$ The trigger pulse $P_{t}$ acts on the initial halting state $|n_{h}\rangle =|0\rangle $ to convert conditionally it into the state $|c_{1}\rangle $ of the control state subspace only if the state $|f_{r}(x)\rangle $ in the register of the functional state is the state $|0\rangle .$ Thus, here the unitary transformation $U_{t}$ during the trigger pulse $P_{t}$ could be defined simply by $$U_{t}:|n_{h}\rangle |f_{r}(x)\rangle =|0\rangle |0\rangle \leftrightarrow
|c_{1}\rangle |0\rangle .$$Both the trigger pulse and the state-locking pulse field are very important in the unitary quantum control unit and will be discussed separately in detail later. In order that the halting quantum bit $\{|n_{h}\rangle
,n_{h}=0,1\}$ is separated from other qubits of the computational state subspace in the quantum program an extra quantum bit named the branch-control quantum bit $\{|b_{h}\rangle ,b_{h}=0,1\}$ is used to control directly the functional operation of the function $f_{r}(x)$ in place of the halting quantum bit. Therefore, there is the unitary conditional functional operation $U_{f}^{c}$ for the functional state $|f_{r}(x)\rangle $ defined by $$U_{f}^{c}|b_{h}\rangle |f_{r}(x)\rangle =\left\{
\begin{array}{c}
|0\rangle |f_{r}(x+1)\rangle \text{ if }b_{h}=0 \\
|b_{h}\rangle |f_{r}(x)\rangle \text{ if }b_{h}\neq 0\quad \
\end{array}%
\right.$$This definition shows that the conditional functional operation $U_{f}^{c}$ is applied only to both the functional state $|f_{r}(x)\rangle $ and the branch-control state $|b_{h}\rangle .$ If the branch-control state $%
|b_{h}\rangle $ is the state $|0\rangle $ the functional operation $U_{f}$ changes the functional state $|f_{r}(x)\rangle $ to another functional state $|f_{r}(x+1)\rangle ,$ otherwise the operation $U_{f}$ does not change the functional state. Therefore, the branch-control state $|b_{h}\rangle $ can control conditionally the action of the functional operation $U_{f}$ upon the functional state. When the branch-control state $|b_{h}\rangle $ is changed from the state $|0\rangle $ to the state $|1\rangle $ the functional operation $U_{f}$ is halted to act on the functional state $|f_{r}(x)\rangle
$ even though the conditional functional operation $U_{f}^{c}$ continues to apply to the functional state. In the quantum program the halting quantum bit $\{|n_{h}\rangle \}$ is designed to control the branch-control quantum bit $\{|b_{h}\rangle \}$ indirectly and hence controls ultimately the action of the functional operation $U_{f}$ on the functional state. The conditional unitary transformation $U_{b}^{c}$ in the quantum program could be defined simply by $$U_{b}^{c}|b_{h}\rangle |f_{r}(x)\rangle =\left\{
\begin{array}{c}
|(b_{h}+1)\func{mod}2\rangle |f_{r}(x)\rangle ,\text{ if }|f_{r}(x)\rangle
=|1\rangle \\
|b_{h}\rangle |f_{r}(x)\rangle ,\text{ if }|f_{r}(x)\rangle \neq |1\rangle
\qquad \qquad \quad \
\end{array}%
\right.$$and the conditional unitary transformation $U_{h}^{c}:|n_{h}\rangle
|f_{r}(x)\rangle =|0\rangle |1\rangle $ $\leftrightarrow |0\rangle |0\rangle
$ is defined explicitly by $$U_{h}^{c}|n_{h}\rangle |f_{r}(x)\rangle =\left\{
\begin{array}{c}
|0\rangle |0\rangle ,\text{ if }|f_{r}(x)\rangle =|1\rangle \text{ and }%
|n_{h}\rangle =|0\rangle \\
|0\rangle |1\rangle ,\text{ if }|f_{r}(x)\rangle =|0\rangle \text{ and }%
|n_{h}\rangle =|0\rangle \\
|n_{h}\rangle |f_{r}(x)\rangle ,\text{ otherwise\qquad \qquad \qquad\ \ }%
\end{array}%
\right.$$
The detailed analysis for the quantum program $Q_{c}$ is given below. The state-locking pulse field $P_{SL}$ is first switched on at the beginning of the quantum program (see the statement P1 in the program) and could be switched off (or partly switched off) after the quantum program finished (the statement P10). It is mainly used to manipulate the states $%
|c_{1}\rangle $ and $|c_{2}\rangle $ and lock the state $|c_{2}\rangle $ of the control state subspace $S(C)$. Both the initial halting state $%
|n_{h}\rangle $ (statement P2) and the initial branch-control state $%
|b_{h}\rangle $ (the statement P3) are simply set to the state $|0\rangle .$ The initial functional state $|f_{r}(x_{0})\rangle $ (the statement P4) could be unknown and may be stored in the memory of the quantum computer in advance or is input from outside the quantum program. There are $m_{r}$ possible initial functional states $\{|f_{r}(x_{0})\rangle ,$ $%
x_{0}=0,1,...,m_{r}-1\}$ at most. The quantum program first checks whether or not the initial functional state $|f_{r}(x_{0})\rangle $ is the desired functional state $|f_{r}(x_{f})\rangle =|1\rangle $ (the statement P6). Then there are two possible cases to be considered, that is, either $%
|f_{r}(x_{0})\rangle =|f_{r}(x_{f})\rangle $ or $|f_{r}(x_{0})\rangle \neq
|f_{r}(x_{f})\rangle $. Consider the first case that $|f_{r}(x_{0})\rangle
=|f_{r}(x_{f})\rangle =|1\rangle $. Since the state $|f_{r}(x_{0})\rangle
=|f_{r}(x_{f})\rangle $ the branch-control state $|b_{h}\rangle =|0\rangle $ is first changed to the state $|1\rangle $ by the conditional unitary operation $U_{b}^{c}$ (the statement P6). Then the desired state $%
|f_{r}(x_{f})\rangle $ is changed conditionally to the state $|0\rangle $ (the statement P7) by the conditional unitary operation $U_{h}^{c}$ due to that the halting state $|n_{h}\rangle $ now is the state $|0\rangle ,$ the initial halting state $|n_{h}\rangle =|0\rangle $ then is changed to the state $|c_{1}\rangle $ of the control state subspace by the trigger pulse $%
P_{t}$ and then the state $|c_{1}\rangle $ to another orthogonal state $%
|c_{2}\rangle $ of the control state subspace, since then the state $%
|c_{2}\rangle $ is kept unchanged by the state-locking pulse field $P_{SL}$ (the statement P7). When the quantum program executes the statement P8 the conditional functional operation $U_{f}^{c}$ will not have a net effect on the functional state $|f_{r}(x)\rangle $ according as the definition of the operation $U_{f}^{c}$ because the branch-control state $|b_{h}\rangle $ now is the state $|1\rangle $ and the state $|f_{r}(x)\rangle =|0\rangle ,$ although now the operation $U_{f}^{c}$ is still applied to the whole quantum system of the quantum computer. This shows that the functional operation $%
U_{f}$ acting on the functional state $|f_{r}(x)\rangle $ is really halted after the initial branch-control state $|0\rangle $ is changed to the state $%
|1\rangle $. Now the quantum program finished the first cycle with the index $i=1$. It then returns and executes the statement P5 of the second cycle with the index $i=2$. When the quantum program executes the statement P6 in the second cycle, the current branch-control state $|b_{h}\rangle =|1\rangle
$ keeps unchanged under the action of the operation $U_{b}^{c}$ as the current state $|f_{r}(x)\rangle =|0\rangle .$ Since the current state $%
|n_{h}\rangle |f_{r}(x)\rangle =|c_{2}\rangle |0\rangle ,$ that is, the state $|n_{h}\rangle \neq |0\rangle ,$ the unitary operation $U_{h}^{c}$ does not have a real effect on the quantum system when the quantum program executes the statement P7, and since now the halting-register state $%
|n_{h}\rangle $ is the state $|c_{2}\rangle ,$ that is, the state $%
|n_{h}\rangle $ is neither $|0\rangle $ nor $|c_{1}\rangle ,$ the trigger pulse $P_{t}$ does not yet have a real effect on the quantum system. The key point in the quantum program is that the state $|c_{2}\rangle $ of the control state subspace has been locked by the state-locking pulse field $%
P_{SL}$ since the quantum program executes the statement P7 in the first cycle $(i=1).$ Hence the state $|c_{2}\rangle $ is not yet changed when the statement P7 is executed in the second cycle. Actually, the state $%
|c_{2}\rangle $ may be kept unchanged till the end of the quantum program after the statement P7 was executed in the first cycle. If now the program continues to execute the rest statements and even run till the end of the program, then all these states $|n_{h}\rangle =|c_{2}\rangle ,$ $%
|b_{h}\rangle =|1\rangle ,$ and $|f_{r}(x)\rangle =|0\rangle $ of the whole quantum system of the quantum computer are still kept unchanged due to the fact that the halting-register state $|n_{h}\rangle $ is kept in the state $%
|c_{2}\rangle $ by the state-locking pulse field. Note that the conditional functional operation $U_{f}^{c}$ is applied continuously to the quantum system of the quantum computer even after the branch-control state $%
|b_{h}\rangle =|0\rangle $ is changed to the state $|1\rangle ,$ which leads to that the functional operation $U_{f}$ acting on the functional state $%
|f_{r}(x)\rangle $ is halted. Of course, in this case the conditional functional operation $U_{f}^{c}$ has not a net effect on the functional state $|f_{r}(x)\rangle .$ This process is repeated from the second cycle ($%
i=2$) to the end ($i=m_{r}$) of the quantum program. The analysis above shows that if the functional state $|f_{r}(x)\rangle $ is the desired state $%
|f(x_{f})\rangle $, then when the quantum program executes the statement P6 the branch-control state $|b_{h}\rangle =|0\rangle $ is changed to the state $|1\rangle ,$ and then on the statement P7 the initial halting state $%
|n_{h}\rangle =|0\rangle $ is changed to the state $|c_{1}\rangle $ and then further to the state $|c_{2}\rangle $ which is ultimately kept unchanged by the state-locking pulse field, since then the state $|n_{h}\rangle
|b_{h}\rangle |f_{r}(x)\rangle =$ $|c_{2}\rangle |1\rangle |0\rangle $ of the whole quantum system is kept unchanged till the end of the program. Therefore, the quantum program outputs the final state $|n_{h}\rangle
|b_{h}\rangle |f_{r}(x)\rangle =$ $|c_{2}\rangle |1\rangle |0\rangle $ (the state $|f_{r}(x)\rangle =$ $|0\rangle $ is easily changed to the desired state $|f_{r}(x_{f})\rangle =|1\rangle ).$ Next consider the second case: the state $|f_{r}(x_{0})\rangle \neq |f_{r}(x_{f})\rangle .$ If the initial functional state $|f_{r}(x_{0})\rangle \neq |f_{r}(x_{f})\rangle =|1\rangle
, $ then the initial branch-control state $|b_{h}\rangle =|0\rangle $ is not changed to the state $|1\rangle $ by the unitary operation $U_{b}^{c}$ when the program executes the statement P6. Again due to that the state $%
|f_{r}(x_{0})\rangle \neq |1\rangle $ and $|0\rangle $ the unitary operation $U_{h}^{c}$ does not really act on the state $|n_{h}=0\rangle
|f_{r}(x_{0})\rangle $ and both the trigger pulse $P_{t}$ and the state-locking pulse field $P_{SL}$ do not yet act on the initial halting state $|n_{h}\rangle =|0\rangle $ when the program executes the statement P7. Since the branch-control state $|b_{h}\rangle =|0\rangle $ the functional state $|f_{r}(x_{0})\rangle $ is changed to the state $%
|f_{r}(x_{0}+1)\rangle $ by the conditional functional operation $U_{f}^{c}$ after executing the statement P8. Now the quantum program returns to execute the statement P6 of the second cycle after the cyclic index $i=1$ is changed to $i=2$ on the statement P5. Again the program first checks whether or not the functional state $|f_{r}(x_{0}+1)\rangle $ is the desired state $%
|f_{r}(x_{f})\rangle $. Just like before there are also two possible cases, the first case is $|f_{r}(x_{0}+1)\rangle =|f_{r}(x_{f})\rangle $ and the second $|f_{r}(x_{0}+1)\rangle \neq |f_{r}(x_{f})\rangle $. As shown above, for the first case $|f_{r}(x_{0}+1)\rangle =|f_{r}(x_{f})\rangle $ the program will output the final state $|n_{h}\rangle |b_{h}\rangle
|f_{r}(x)\rangle =$ $|c_{2}\rangle |1\rangle |0\rangle .$ For the second case $|f_{r}(x_{0}+1)\rangle \neq |f_{r}(x_{f})\rangle $ the functional state $|f_{r}(x_{0}+1)\rangle $ will be changed to the state $%
|f_{r}(x_{0}+2)\rangle $ at the end of the second cycle ($i=2$) of the program (the statement P8). This process is repeated till the $k-$th cycle ($%
m_{r}>k\geq 1$) when the functional state $|f_{r}(x_{0}+k)\rangle
=|f_{r}(x_{f})\rangle $ at the end of the $k-$th cycle$.$ Here the index $k$ is unique for $0\leq k<m_{r}$ and $k=0$ corresponds to the earlier case $%
|f_{r}(x_{0})\rangle =|f_{r}(x_{f})\rangle .$ Now for the $(k+1)-$th cycle the initial branch-control state $|b_{h}\rangle =|0\rangle $ is first changed to the state $|1\rangle $ (the statement P6), following the statement P6 the functional state $|f_{r}(x_{0}+k)\rangle
=|f_{r}(x_{f})\rangle $ is changed to the state $|0\rangle ,$ then the initial halting state $|n_{h}\rangle =|0\rangle $ to the state $%
|c_{1}\rangle $ by the trigger pulse $P_{t}$ and further to the state $%
|c_{2}\rangle $ by the state-locking pulse field (the statement P7), and since then the state $|c_{2}\rangle $ is kept unchanged by the state-locking pulse field till the end of the program. Therefore, the quantum program outputs finally the state $|n_{h}\rangle |b_{h}\rangle |f_{r}(x)\rangle =$ $%
|c_{2}\rangle |1\rangle |0\rangle $. This shows that after executing quantum program $Q_{c}$ the initial state $|n_{h}\rangle |b_{h}\rangle
|f_{r}(x)\rangle =$ $|0\rangle |0\rangle |f_{r}(x_{0})\rangle $ is always transferred to the output state $|c_{2}\rangle |1\rangle |0\rangle $ no matter what the initial functional state $|f_{r}(x_{0})\rangle $ is with $%
x_{0}=0,1,...,m_{r}-1.$ Note that the quantum program $Q_{c}$ is reversible and unitary because all these operations of the quantum program are reversible and unitary. One therefore concludes that by the unitary quantum program $Q_{c}$ different initial states $\{|0\rangle |0\rangle
|f_{r}(x_{0})\rangle \}$ are transferred to the same output state $%
|c_{2}\rangle |1\rangle |0\rangle $ and hence the output state of the quantum program is not dependent sensitively upon any initial states. However, the first part of the conclusion is apparently in conflict with the fact that different input states can not be completely transferred to the same output state by a given unitary transformation. Therefore, the first part of the conclusion is expressed exactly as that different initial states $\{|0\rangle |0\rangle |f_{r}(x_{0})\rangle \}$ are transferred to the same output state $|c_{2}\rangle |1\rangle |0\rangle $ in probabilities approaching infinitely 100% in theory by the unitary quantum program $Q_{c}$. In theory there is only one initial state $|0\rangle |0\rangle
|f_{r}(x_{0})\rangle $ that may be completely transferred to the output state $|c_{2}\rangle |1\rangle |0\rangle $ by the unitary quantum program. As shown in next sections, due to the fact that the quantum program $Q_{c}$ is reversible and unitary the output state $|c_{2}\rangle |1\rangle
|0\rangle $ can be really obtained from the initial state $|0\rangle
|0\rangle |f_{r}(x_{0})\rangle $ only in a probability close to 100% rather than in the probability 100% for any initial functional state $%
|f_{r}(x_{0})\rangle $ in a real physical system.
The state-locking pulse field $P_{SL}$ plays a key role in the quantum control process that simulates efficiently the reversible and unitary halting protocol in the quantum program $Q_{c}$. It is the state-locking pulse field that keeps the state $|c_{2}\rangle $ of the control state subspace unchanged till the end of the quantum program after the functional state $|f_{r}(x)\rangle $ is changed to the desired functional state $%
|f_{r}(x_{f})\rangle $, while keeping the state $|c_{2}\rangle $ unchanged for a long time is the key step to achieve the reversible and unitary halting protocol. As pointed out before, the statement P7 of the quantum program which is involved in the state-locking pulse field $P_{SL}$ is mainly used to simulate the reversible and unitary halting protocol. The statement P7 really forms a unitary quantum control process (or unit). This process (or unit) is almost independent of the quantum computational process (or unit) to compute the desired functional state $|f_{r}(x_{f})\rangle $ in the quantum program, but it really controls the quantum computational process (or unit). A quantum program and its quantum circuit may be generally divided into two parts, one part is the quantum computational unit (or process) and another the quantum control unit (or process). As an example, for the quantum program $Q_{c}$ the quantum computational unit is used to compute the desired functional state $|f_{r}(x_{f})\rangle ,$ while the quantum control unit is used to perform the reversible and unitary halting protocol. As pointed out before, the unitary quantum control unit consists of the trigger pulse $P_{t}$, the state-locking pulse field $P_{SL}$, and the control state subspace $S(C)$. Generally, the control state subspace $S(C)$ is different from the computational state subspace such as the multiplicative-cyclic-group state subspace $S(m_{r})$ in the quantum program, but they all belong to the Hilbert space of the whole quantum system of the quantum computer. The simplest control state subspace is a two-state subspace such as $S(C)=\{|c_{1}\rangle ,|c_{2}\rangle \},$ but generally a control state subspace is not restricted to be a two-state subspace in the quantum program. The trigger pulse $P_{t}$ could be used for communication between the control state subspace and the computational state subspace. It generally triggers a time- and space-dependent unitary evolution process in the quantum control system with the control state subspace $S(C)$. The unitary transformation for the trigger pulse may be defined explicitly. For example, in the quantum program $Q_{c}$ the initial halting state $|n_{h}\rangle =|0\rangle $ is converted into the state $%
|c_{1}\rangle $ and vice versa by the trigger pulse $P_{t}$ only if the state $|f(x)\rangle $ is the state $|0\rangle $. Then the unitary transformation for the trigger pulse $P_{t}$ is defined simply by $U_{t},$ as shown before. Here the state $|f_{r}(x)\rangle =|0\rangle $ in the register of the functional state could be thought to be related to the computational state subspace, while the state $|c_{1}\rangle $ is of the control state subspace. A different definition for the unitary transformation during the trigger pulse $P_{t}$ can be seen in next section. The unitary quantum control process that simulates the reversible and unitary halting protocol may be simply described below. Here it is first pointed out that the state-locking pulse field $P_{SL}$ is applied only to the quantum control system with the control state subspace $S(C)$ in the quantum program. The state-locking pulse field is first switched on to apply to the quantum control system of the quantum computer at the beginning of the quantum program (see the statement P1)$,$ but it usually does not take a real action on the quantum control system at the beginning time. However, when the functional state $|f_{r}(x)\rangle $ is changed to the desired state $|f_{r}(x_{f})\rangle $ and then the initial halting state $%
|n_{h}\rangle =|0\rangle $ changed conditionally to the state $|c_{1}\rangle
$, the state-locking pulse field $P_{SL}$ which has been applying to the control state subspace since the beginning of the quantum program starts to take a real action on the states of the control state subspace. The state $%
|c_{1}\rangle $ is first sent to the state $|c_{2}\rangle $ in the control state subspace under the state-locking pulse field. This process usually is a time- and space-dependent unitary evolution process. Then the state $%
|c_{2}\rangle $ is kept unchanged by the state-locking pulse field to the end of the quantum program and circuit. Because now the branch-control state $|b_{h}\rangle $ leaves the initial one and is kept in the state $|1\rangle $ unchanged due to that the state $|c_{2}\rangle $ is kept unchanged by the state-locking pulse field the computational process is halted conditionally and the reversible and unitary halting operation therefore is achieved. According to this picture that the quantum control process simulates the reversible and unitary halting protocol the state-locking pulse field $%
P_{SL} $ is applied continuously to the quantum control system from the beginning to the end of the quantum program$.$ If the quantum system of the quantum computer which includes the quantum control system now is acted on by a unitary operation such as one of the unitary operations $U_{b}^{c},$ $%
U_{h}^{c},$ $U_{f}^{c},$ and the trigger pulse $P_{t}$ of the quantum program, then actually it is acted on by both the state-locking pulse field and the unitary operation simultaneously. Then the state-locking pulse field could be designed in such a way that the effect of the state-locking pulse field on the quantum system can be negligible during the period of the unitary operation applying to the quantum system. For example, the unitary transformation on the quantum system is approximately equal to the single unitary transformation of the functional operation $U_{f}^{c}$ when both the functional operation $U_{f}^{c}$ and the state-locking pulse field $P_{SL}$ are applied to the quantum system simultaneously in the quantum program. This is because in this case the state-locking pulse field has a negligible effect on the quantum system. However, as shown in next section, there may also be another case that the unitary transformation during the trigger pulse $P_{t}$ could be really generated by both the state-locking pulse field and the trigger pulse. Then in this case the contribution of the state-locking pulse field is not negligible. These general properties of an ideal state-locking pulse field could be used to measure the performance of a real state-locking pulse field used in the reversible and unitary halting protocol.
The quantum control process that simulates the reversible and unitary halting protocol could be a single time-dependent unitary evolution process, but generally it may be a time- and space-dependent unitary evolution process of the quantum control system. However, if the quantum control process is restricted to be dependent only upon a single time variable, there could be a large drawback for such a quantum control process with a two-state control state subspace $S(C)$. This can be explained in detail below. Suppose that the state $|c_{1}\rangle $ of the control state subspace is generated completely from the initial halting state $|n_{h}\rangle
=|0\rangle $ by the trigger pulse $P_{t}$ at the instant of time $t_{0i}$ in the $i-$th cycle of the quantum program (see statement P7). The instant of time $t_{0i}$ is special in that the state-locking pulse field really starts to act on the control state subspace $S(C)$ at the instant of time $t_{0i}$ in the quantum program $Q_{c}$. Here suppose that the state-locking pulse field has a negligible effect on the state $|c_{1}\rangle $ during the trigger pulse $P_{t}$. Evidently, a different initial functional state $%
|f_{r}(x_{0})\rangle $ corresponds to a different time $t_{0i}$, and there are $m_{r}$ possible and different times $\{t_{0i},$ $i=1,2,...,m_{r}\}$ at most in the quantum program because there are $m_{r}$ different initial functional states $\{|f_{r}(x_{0})\rangle ,$ $x_{0}=0,1,...,m_{r}-1\}$ of the cyclic-group state subspace $S(m_{r}).$ Suppose that $\Delta T_{i}$ $%
(1\leq i\leq m_{r})$ is the time period of the $i-$th cycle of the quantum program. Obviously, $t_{0i}=t_{0(i-1)}+\Delta T_{i}$ for $i=1,2,...,m_{r},$ where $t_{00}$ may be defined as $t_{00}=t_{01}-\Delta T_{1}$. For convenience, here the time period $\Delta T_{i}$ is set to the same one $%
\Delta T_{c}$ for every cycle $i=1,2,...,m_{r}.$ Suppose that in the quantum program $Q_{c}$ the duration of the trigger pulse $P_{t}$ is $\delta t_{r}$ and the duration is denoted as $\Delta t_{0}$ during which the state $%
|c_{1}\rangle $ is converted completely into the state $|c_{2}\rangle $ in the control state subspace by the state-locking pulse field. In order to show that a quantum control process that is restricted to be dependent only upon a single time variable is not a good one to simulate efficiently the reversible and unitary halting operation there are two possible situations to be investigated in the quantum control process. Consider the first situation. Note that in the $i-$th cycle of the quantum program the initial halting state $|n_{h}\rangle =|0\rangle $ is changed completely to the state $|c_{1}\rangle $ by the trigger pulse $P_{t}$ in the time interval from the time $t_{0i}-\delta t_{r}$ to the time $t_{0i}.$ This means that the functional state $|f_{r}(x)\rangle $ is changed to the desired state $%
|f_{r}(x_{f})\rangle $ at the end of the $(i-1)-$th cycle, and during the period from the time $t_{0i}$ to the time $t_{0i}+\Delta t_{0}$ the state $%
|c_{1}\rangle $ is converted into the state $|c_{2}\rangle $, and from the time $t_{0i}+\Delta t_{0}$ on, the state $|c_{2}\rangle $ is kept unchanged to the end of the quantum program by the state-locking pulse field $P_{SL}$. Evidently, before the instant of time $t_{0(i+1)}-\delta t_{r}$ in the $%
(i+1)-$th cycle the state $|c_{1}\rangle $ must be completely converted into the state $|c_{2}\rangle $ and since then the state $|c_{2}\rangle $ is kept unchanged by the state-locking pulse field, otherwise it is possible that the residual state $|c_{1}\rangle $ could be changed back to the initial halting state $|n_{h}\rangle =|0\rangle $ by the trigger pulse $P_{t}$ during the period from the time $t_{0(i+1)}-\delta t_{r}$ to the time $%
t_{0(i+1)}$ in the $(i+1)-$th cycle. Therefore, for the first situation the quantum control process requires that the quantum control system be completely in the state $|c_{2}\rangle $ in the time interval between $%
t_{0(i+1)}-\delta t_{r}$ and $t_{0(i+1)}$ in the $(i+1)-$th cycle of the quantum program. Now consider the second situation. There is also another possibility that unlike the first situation the functional state $%
|f_{r}(x)\rangle $ is converted into the desired state $|f_{r}(x_{f})\rangle
$ in the $i-$th cycle instead of the $(i-1)-$th cycle, because there are different initial functional states $\{|f_{r}(x_{0})\rangle \}$ in the quantum program$.$ Then during the period from the time $t_{0(i+1)}-\delta
t_{r}$ to the time $t_{0(i+1)}$ in the $(i+1)-$th cycle the initial halting state $|n_{h}\rangle =|0\rangle $ is changed completely to the state $%
|c_{1}\rangle $ by the trigger pulse $P_{t}.$ This shows that for the second situation the quantum control system is in the state $|c_{1}\rangle $ at the time $t_{0(i+1)}$ in the $(i+1)-$th cycle. Then it can be seen from the two possible situations that at the instant of time $t_{0(i+1)}$ in the $(i+1)-$th cycle of the quantum program the quantum control system is either completely in the state $|c_{1}\rangle $ for the second situation or completely in the state $|c_{2}\rangle $ for the first situation. Note that these two states $|c_{1}\rangle $ and $|c_{2}\rangle $ of the control state subspace are orthogonal to one another. Now one considers the $(i+2)-$th cycle of the quantum program. The quantum program requires for any one of the two possible situations that the quantum control system be completely in the state $|c_{2}\rangle $ so as to avoid any real effect of the trigger pulse on the quantum system during the period of the trigger pulse between the time $t_{0(i+2)}-\delta t_{r}$ and the time $t_{0(i+2)}$ in the $(i+2)-$th cycle. Evidently, given a state-locking pulse field and a trigger pulse as well as other unitary operations in the quantum program there is the same time evolution propagator $U(t_{0(i+2)}-\delta t_{r},t_{0(i+1)})$ of the quantum control system during the period from the time $t_{0(i+1)}$ in the $%
(i+1)-$th cycle to the time $t_{0(i+2)}-\delta t_{r}$ in the $(i+2)-$th cycle no matter that the quantum control system is in the state $%
|c_{1}\rangle $ for the second situation or in the state $|c_{2}\rangle $ for the first situation at the instant of time $t_{0(i+1)}.$ Thus, there are two possibilities to be considered. The first one is that the state of the control state subspace is the state $|c_{2}\rangle $ at the instant of time $%
t_{0(i+1)}$ in the $(i+1)-$th cycle, which corresponds to the first situation above. Since during the period between the time $t_{0(i+2)}-\delta
t_{r}$ and the time $t_{0(i+2)}$ in the $(i+2)-$th cycle the quantum control system must be in the state $|c_{2}\rangle $ as required by the quantum program and circuit, one has the unitary transformation for the state $%
|n_{h}\rangle |b_{h}\rangle |f_{r}(x)\rangle $ of the whole quantum system: $$U(t_{0(i+2)}-\delta t_{r},t_{0(i+1)})|c_{2}\rangle |1\rangle |0\rangle
=|c_{2}\rangle |1\rangle |0\rangle$$where the state of the whole quantum system is $|c_{2}\rangle |1\rangle
|0\rangle $ at the time $t_{0(i+1)}$ in the $(i+1)-$th cycle and also $%
|c_{2}\rangle |1\rangle |0\rangle $ at the time $t_{0(i+2)}-\delta t_{r}$ in the $(i+2)-$th cycle. The second one is that the state of the control state subspace is the state $|c_{1}\rangle $ at the instant of time $t_{0(i+1)}$ in the $(i+1)-$th cycle, which corresponds to the second situation above. Then for this situation the unitary state transformation is given by $$U(t_{0(i+2)}-\delta t_{r},t_{0(i+1)})|c_{1}\rangle |1\rangle |0\rangle
=|c_{2}\rangle |1\rangle |0\rangle$$where the state of the whole quantum system is also $|c_{2}\rangle |1\rangle
|0\rangle $ at the time $t_{0(i+2)}-\delta t_{r}$, as required by the quantum program and circuit. One sees that these two orthogonal states $%
|c_{1}\rangle $ and $|c_{2}\rangle $ are changed completely to the same state $|c_{2}\rangle $ in the control state subspace by the same unitary transformation $U(t_{0(i+2)}-\delta t_{0},t_{0(i+1)})$ during the period from the time $t_{0(i+1)}$ to the time $t_{0(i+2)}-\delta t_{r}$. Obviously, this is impossible and there is a conflict between the unitarity of the propagator $U(t_{0(i+2)}-\delta t_{0},t_{0(i+1)})$ and the requirement of the quantum program and circuit that the quantum control system be in the state $|c_{2}\rangle $ at the time $t_{0(i+2)}-\delta t_{r}$ in the $(i+2)-$th cycle, because the requirement leads to non-unitarity of the propagator $%
U(t_{0(i+2)}-\delta t_{0},t_{0(i+1)})$. Therefore, the quantum control process with the two-state control state subspace $S(C)$ could not simulate faithfully and efficiently the reversible and unitary halting protocol and could fail to control the quantum computational process in the quantum program $Q_{c}$ if it is constrained to be a single time-dependent evolution process. This conflict could be related to the square speedup limit of the quantum search algorithm if the quantum program $Q_{c}$ is used to construct the quantum search algorithm. Here it should be pointed out that this conflict could be avoided in a larger control state subspace rather than the two-state control state subspace, but this could lead to that the output state of the quantum program is still dependent sensitively upon initial states so that the quantum program $Q_{c}$ becomes unvalued for building up an efficient quantum search process. Of course, it is usually better to use a simpler control state subspace to control the quantum computational process.
It is well known that a quantum system with a time-independent Hamiltonian satisfies the time-displacement symmetry (or invariance) \[5a\]. The time evolution process of such a quantum system is independent of any initial time but depends upon the time difference between the end time and the initial time of the process. Therefore, one possible scheme to make the output state of the quantum program $Q_{c}$ independent of any initial state could be that the Hamiltonian that governs the quantum control process is restricted to be time-independent. As shown before, the times $\{t_{0i}\}$ could be thought of as the starting times for the state-locking pulse field to really act on the control state subspace in the quantum control process. Actually, the quantum control process starts to work after the trigger pulse is applied at the instant of time $t_{0i}-\delta t_{r},$ and it may be stated simply as that the initial halting state $|n_{h}\rangle =|0\rangle $ is changed completely to the state $|c_{1}\rangle $ by the trigger pulse $%
P_{t}$ in the time interval from the time $t_{0i}-\delta t_{r}$ to the time $%
t_{0i},$ then the state $|c_{1}\rangle $ is changed to the state $%
|c_{2}\rangle $ by the state-locking pulse field in the time interval $%
\Delta t_{0}$ from the time $t_{0i}$ to the time $t_{0i}+\Delta t_{0},$ and from the time $t_{0i}+\Delta t_{0}$ on, the state $|c_{2}\rangle $ is locked by the state-locking pulse field. According to this picture the quantum control process could be expressed conveniently in terms of a sequence of unitary transformations on the state $|n_{h}\rangle |b_{h}\rangle
|f_{r}(x)\rangle $ of the whole quantum system, $$\{P_{SL}(\{\varphi _{k}\},t_{0i},t_{0i}-\delta t_{r}),P_{t}\}|0\rangle
|1\rangle |0\rangle =|c_{1}\rangle |1\rangle |0\rangle ,\qquad \quad \quad
\quad \label{1}$$$$P_{SL}(\{\varphi _{k}\},t,t_{0i})|c_{1}\rangle |1\rangle |0\rangle \quad
\hspace{1in}\hspace{1in}\qquad$$$$=\{\varepsilon (t,t_{0i})|c_{1}\rangle +e^{-i\gamma (t,t_{0i})}\sqrt{%
1-|\varepsilon (t,t_{0i})|^{2}}|c_{2}\rangle \}|1\rangle |0\rangle ,\text{ }%
t_{0i}\leq t. \label{2}$$Here $P_{SL}(\{\varphi _{k}\},t,t_{0i})$ is the unitary propagator of the state-locking pulse field $P_{SL}$ applying separately to the quantum system during the period from the initial time $t_{0i}$ to the time $t$ and $%
\{P_{SL}(\{\varphi _{k}\},t_{0i},t_{0i}-\delta t_{r}),P_{t}\}$ represents the unitary propagator when the trigger pulse $P_{t}$ and the state-locking pulse field $P_{SL}$ are applied to the quantum system simultaneously during the pulse duration $\delta t_{r}$ of the trigger pulse from the time $%
t_{0i}-\delta t_{r}$ to the time $t_{0i}.$ The parameters $\{\varphi _{k}\}$ in the unitary propagator $P_{SL}(\{\varphi _{k}\},t,t_{0})$ are the control parameters of the state-locking pulse field which may be generally dependent on the time variable, the spatial variables, or even the quantum field variables. Here the unitary propagator $\{P_{SL}(\{\varphi
_{k}\},t_{0i},t_{0i}-\delta t_{r}),P_{t}\}$ is really equal to the unitary transformation $U_{t}$ during the trigger pulse $P_{t}$, indicating that the unitary transformation $U_{t}$ is really generated by both the state-locking pulse field and the trigger pulse. As supposed before, the state-locking pulse field is negligible during the trigger pulse in the quantum program $%
Q_{c}$, and hence the unitary transformation $U_{t}$ is really generated approximately by the single trigger pulse. In the unitary transformation (2) $\gamma (t,t_{0i})$ is the phase angle of the amplitude of the state $%
|c_{2}\rangle $ and $\varepsilon (t,t_{0i})$ the residual amplitude of the state $|c_{1}\rangle $ at the time $t$ after the unitary transformation $%
P_{SL}(\{\varphi _{k}\},t,t_{0i})$ acts on the state $|c_{1}\rangle $. As required by the quantum program $Q_{c}$, the state $|c_{1}\rangle $ must be converted completely into the state $|c_{2}\rangle $ in the control state subspace by the unitary transformation $P_{SL}(\{\varphi _{k}\},t,t_{0i})$ within the period $\Delta t_{0}$ from the time $t_{0i}$ to the time $%
t=t_{0i}+\Delta t_{0}.$ Here the time interval $\Delta t_{0}$ is shorter than the cyclic period $\Delta T_{c}$ minus the duration $\delta t_{r}$ of the trigger pulse, that is, $\Delta t_{0}<\Delta T_{c}-\delta t_{r}.$ Evidently, when the time $t\geq t_{0i}+\Delta t_{0}$ the absolute amplitude value $|\varepsilon (t,t_{0i})|$ should approach infinitely zero in theory but can not equal exactly zero for every time $t_{0i}$ for $i=1,2,...,m_{r}$ even for an ideal state-locking pulse field, and generally it should be close to zero for a real state-locking pulse field. One may say that the amplitude $\varepsilon (t,t_{0i})$ equals zero in theory if it approaches infinitely zero. The amplitude value $|\varepsilon (t,t_{0i})|$ measures the performance of a real state-locking pulse field, that is, the smaller the amplitude value $|\varepsilon (t,t_{0i})|$ is for $t\geq t_{0i}+\Delta
t_{0}, $ the better the performance is for a real state-locking pulse field $%
P_{SL}$. If now the quantum control process is governed by a time-independent Hamiltonian during the state-locking pulse field, then the time evolution process of Eq. (2) from the state $|c_{1}\rangle $ to the state $|c_{2}\rangle $ does not depend directly upon any initial time $%
t_{0i} $ or the end time $t$ but it is dependent upon the time difference $%
\Delta t_{i}=t-t_{0i}$, that is, $P_{SL}(\{\varphi
_{k}\},t,t_{0i})=P_{SL}(\{\varphi _{k}\},\Delta t_{i})$. This also means that both the amplitude $\varepsilon (t,t_{0i})$ of the state $|c_{1}\rangle
$ and $e^{-i\gamma (t,t_{0i})}\sqrt{1-|\varepsilon (t,t_{0i})|^{2}}$ of the state $|c_{2}\rangle $ in Eq. (2) are dependent upon the time difference $%
\Delta t_{i},$ that is, $\varepsilon (t,t_{0i})=\varepsilon (\Delta t_{i})$ and $\gamma (t,t_{0i})=\gamma (\Delta t_{i}).$ It was pointed out before that there are $m_{r}$ different times $\{t_{0i},$ $i=1,2,...,m_{r}\}$ at most for the quantum control process in the quantum program. If the state-locking pulse field $P_{SL}$ is designed in such a way that the Hamiltonian that governs the quantum control process of Eq. (2) is time-independent, then the time difference $\Delta t_{i}=t-t_{0i}$ for $%
i=1,2,...,m_{r}$ replaces the time variable $t$ to become a new time variable of the quantum control process of Eq. (2), and consequently it is not necessary to deal with directly the time variable $t$ in the quantum control process of Eq. (2). The unitary transformations (1) and (2) indicate that by the quantum program $Q_{c}$ with the quantum control process of Eq. (2) governed by a time-independent Hamiltonian these $m_{r}$ different initial functional states $\{|f_{r}(x_{0})\rangle \}$ (as well as the initial halting state and branch-control state) can be transferred one-by-one to the $m_{r}$ states on the right-hand side of Eq. (2) whose phases and amplitudes are dependent upon the time differences $\{\Delta
t_{i}\}$ for $i=1,2,...,m_{r}$, respectively. That the Hamiltonian that governs the quantum control process of Eq. (2) under the state-locking pulse field is constrained to be time-independent could be helpful for designing a state-locking pulse field with a good performance to control the quantum computational process in the quantum program $Q_{c}$.
However, the time-displacement symmetry is not sufficient to solve thoroughly the conflict mentioned above and it can not yet figure out completely a state-locking pulse field. This is because although the amplitude $\varepsilon (\Delta t_{i})$ of the state $|c_{1}\rangle $ in Eq. (2) is independent of any single initial time $t_{0i}$, it is still dependent upon the time difference $\Delta t_{i}$. Obviously, a different time difference $\Delta t_{i}$ generally leads to a different residual amplitude $\varepsilon (\Delta t_{i})$ of the state $|c_{1}\rangle $ in Eq. (2). Then there could be still a problem that the amplitude $|\varepsilon
(\Delta t_{i})|$ could fail to be close to zero during the trigger pulse $%
P_{t}$ from the time $t_{0k}-\delta t_{r}$ to time $t_{0k},$ here the time difference $(t_{0k}-\delta t_{r}-t_{0i})>\Delta t_{0}$ for $%
k=i+1,i+2,...,m_{r}.$ As a result, the residual state $|c_{1}\rangle $ with a large amplitude $|\varepsilon (\Delta t_{i})|$ may be changed back to the initial halting state $|n_{h}\rangle =|0\rangle $ by the trigger pulse $%
P_{t} $ again. Obviously, this result is not consistent with the requirement of quantum program that any residual amplitude $|\varepsilon (\Delta t_{i})|$ of the state $|c_{1}\rangle $ in Eq. (2) equal zero in theory when $\Delta
t_{i}\geq \Delta t_{0}$ for $i=1,2,...,m_{r}$. Now suppose that these two states $|c_{1}\rangle $ and $|c_{2}\rangle $ of the control state subspace are degenerative eigenstates of the time-independent Hamiltonian $H$ and the unitary propagator $P_{SL}(\{\varphi _{k}\},t,t_{0i})=\exp (-iH\Delta t_{i})$ ($\hslash =1$)$.$ Then these two eigenstates have the same eigenvalue and the eigen-equations are written as $H|c_{1}\rangle =\lambda |c_{1}\rangle $ and $H|c_{2}\rangle =\lambda |c_{2}\rangle $ with the common eigenvalue $%
\lambda ,$ respectively. Thus, the time evolution process similar to Eq. (2) with the initial superposition state $a|c_{1}\rangle +b|c_{2}\rangle $ of the control state subspace $S(C),$ which is drived by the time-independent Hamiltonian $H$, may be generally expressed by $$P_{SL}(\{\varphi _{k}\},\Delta t_{i})(a|c_{1}\rangle +b|c_{2}\rangle
)|1\rangle |0\rangle =\exp (-i\lambda \Delta t_{i})(a|c_{1}\rangle
+b|c_{2}\rangle )|1\rangle |0\rangle .$$Evidently, only the global phase factor of the state $a|c_{1}\rangle
+b|c_{2}\rangle $ is changed by the unitary propagator $P_{SL}(\{\varphi
_{k}\},\Delta t_{i})$. The global phase factor is dependent upon the time difference $\Delta t_{i},$ but the absolute amplitude of the state does not change as the time difference. Therefore, the time-independent Hamiltonian that drives the time evolution process of Eq. (2) in the two-state control state subspace is very useful for keeping the amplitude of any state of the control state subspace unchanged for a long time. However, such a Hamiltonian could not be suitable for transferring the state $|c_{1}\rangle $ to the state $|c_{2}\rangle $ in the control state subspace, while the state transfer is necessary for a quantum control process such as Eq. (2) to simulate the reversible and unitary halting protocol.
It is shown above that it is not sufficient to build up the state-locking pulse field with a good performance by constraining the Hamiltonian of the quantum control system with the two-state subspace $S(C)$ under the state-locking pulse field to be time-independent. A more suitable Hamiltonian that governs the quantum control process of Eq. (2) may be dependent upon the spatial variables and/or the quantum field variables but independent of the time variable so that the propagator $P_{SL}(\{\varphi
_{k}\},\Delta t)$ is still dependent upon the time difference $\Delta t$. In a quantum computer architecture different quantum bits of the quantum system of the quantum computer must be addressed spatially or distinguished from each other by some properties of the quantum system such as the spectroscopic properties so that they can be manipulated at will. However, such spatial-dependent properties of the quantum system are static and different from a space-dependent evolution process. A quantum computational process generally is considered as a unitary time evolution process of a quantum system \[3\] which may be generally dependent upon both time and space. According as quantum mechanics \[5\], time- and space-dependent evolution processes of a quantum system such as the conventional quantum scattering process, the quantum tunneling process, the quantum collision process, the molecular chemical dissociation process, and so on, obey the Schrödinger equation as well and hence they are also governed by the unitary quantum dynamics in time and space. The force to drive a time- and space-dependent evolution process such as a quantum scattering process usually could be the motional momentum of a particle or an electromagnetic field and so on. The time- and space-dependent unitary evolution processes could also be used to build up quantum computational processes just like the conventional quantum gates \[25\], although space-dependent unitary evolution processes usually are more complicated and difficult to be manipulated at will than those space-independent ones. A large advantage of a time- and space-dependent unitary evolution process over a space-independent one for building up a quantum computational process could be that a time- and space-dependent unitary evolution process may be manipulated separately either in the time dimension or in the space dimensions or in both the time and space dimensions. While a time-dependent and space-independent unitary evolution process could be inadequate as the quantum control process of Eq. (2) for the two-state control system, a time- and space-dependent unitary evolution process could be better to act as the quantum control process. Thus, a time- and space-dependent unitary evolution process could be very useful for some specific purposes in quantum computation, although a quantum computational process usually is simply designed to be a space-independent unitary evolution process of a quantum system and any space-dependent evolution processes of the quantum system are suppressed so that in algorithm the quantum computational process may be constructed as simply as possible. As shown before, if the quantum control process is purely time-dependent in the two-state subspace $S(C),$ there is a conflict between the unitarity of the quantum control process and the performance of the quantum control process that the state $|c_{2}\rangle $ in the control state subspace $S(C)$ is kept unchanged for a long time by the state-locking pulse field. Then it could be better to use a time- and space-dependent unitary evolution process such as a quantum scattering process to realize the quantum control process of Eq. (2), meanwhile the quantum computational process may be set to a single time-dependent unitary evolution process of the quantum system of the quantum computer. This is because both the single time-dependent quantum computational process and the time- and space-dependent quantum control process may be manipulated separately in time and space and hence they become almost independent upon each other. If the Hamiltonian to drive the time- and space-dependent quantum control process of Eq. (2) is space-dependent and time-independent, then the energy of the quantum control system is conservative during the quantum control process of Eq. (2) and hence both these two states $|c_{1}\rangle $ and $%
|c_{2}\rangle $ of the control state subspace have the same energy. Then the quantum control process of Eq. (2) in the quantum control system is directly dependent upon the time interval $\Delta t_{i}=t-t_{0i}$ rather than the initial time $t_{0i}$ or the time variable $t$ separately. Since now these two states $|c_{1}\rangle $ and $|c_{2}\rangle $ are degenerate in energy only their global phase is dependent upon the time difference $\Delta t_{i}$ but their amplitudes are not during the quantum control process of Eq. (2), and hence the state $|c_{2}\rangle $ may be kept unchanged for a long time under the state-locking pulse field. However, here the state transfer from the state $|c_{1}\rangle $ to the state $|c_{2}\rangle $ in the control state subspace $S(C)$ could be achieved by the time- and space-dependent unitary evolution process such as the quantum scattering process if the time-independent and space-dependent Hamiltonian of the quantum control system is chosen suitably. Obviously, the quantum scattering process should be designed suitably according to the properties of an ideal state-locking pulse field.
The detailed analysis in the former sections for the quantum control process shows that a quantum control process that simulates faithfully and efficiently the reversible and unitary halting protocol should contain a time- and space-dependent unitary evolution process such as a quantum scattering process and the control state subspace $S(C)$ should not be restricted to be only the smallest two-state subspace. Generally, the quantum control process to simulate efficiently the reversible and unitary halting protocol in the quantum program $Q_{c}$ may be implemented in a real quantum physical system. A trapped atomic-ion system has been proposed as a real physical system to implement quantum computation \[33\]. Here a simple quantum physical system of an atomic ion or a neutral atom in one-dimensional double-well potential field is proposed as the quantum control system of the quantum program and circuit $Q_{c}$ with the control state subspace $S(C)$ which is larger than the two-state control state subspace. Now the simple atomic physical system will be used to illustrate how the quantum control process simulates really the reversible and unitary halting protocol and how to construct explicitly the state-locking pulse field. In this simple physical system the atomic ion or the neutral atom in the double-well potential field is called the halting-quantum-bit atom or the halting-qubit atom briefly. Hereafter the halting-qubit atom is referred to as the atomic ion or the neutral atom in the double-well potential field unless otherwise stated. In the double-well potential field the left-hand potential well could be approximately a conventional harmonic potential well, while the right-hand potential well could be simply a square potential well. Here also suppose that the right-hand square potential well is sufficiently wide such that the halting-qubit atom motions almost freely in one-dimensional space in the square potential well. The intermediate part between these two potential wells is a square potential barrier which is used to block free transportation of the halting-qubit atom from one potential well to another in the double-well potential field. Here the maximum height of the right potential wall of the left-hand potential well is just equal to the height of the intermediate square potential barrier. The left and right potential walls of the double-well potential field may be infinitely high in theory, but the intermediate square potential barrier is finitely high and wide. Actually the left-hand potential well should be an asymmetric harmonic potential well with an infinitely high left potential wall and a finitely high right potential wall, respectively. If the intermediate square potential barrier is infinitely high and finitely wide then any two states of the halting-qubit atom in the left- and right-hand potential wells respectively are completely orthogonal to one another, and if the square potential barrier is high enough and finitely wide then the two states are also considered to be orthogonal to one another approximately. The time-independent double-well potential field in the atomic physical system could be generated by the external electromagnetic field \[30\]. More generally the double-well potential field could be thought of as an effective potential field of the halting-qubit atom, that is, this potential field could be generated effectively by the interaction between the halting-qubit atom and the external electromagnetic field, the interactions between the halting-qubit atom and those atoms of the computational state subspace in the quantum system of the quantum computer, and those interactions between the halting-qubit atom and its environment. The halting-qubit atom in the double-well potential field could be coupled to those quantum-bit atoms of the computational state subspace either through the Coulomb interactions between charged atomic ions \[31\] or through the atomic dipole-dipole interactions between atomic ions in an atomic-ion physical system \[32b\], while the halting-qubit atom could also be coupled to other atoms by the dipole-dipole interactions of neutral atoms in a neutral atomic physical system \[32a, 32c\]. These interactions may be used to set up two-qubit quantum gates between the halting-qubit atom and those quantum-bit atoms of the computational state subspace in the quantum system. With these two-qubit quantum gates and one-qubit quantum gates one can build up those efficient unitary operations that act on both the halting qubit and those qubits of the computational state subspace such as the unitary operation $%
U_{h}^{c}$ in the quantum program $Q_{c}$ and $V_{h}^{c}$ in the quantum control unit $Q_{h}$ (see below). It is required by the quantum control process that these interactions be available only when the halting-qubit atom is in the left-hand harmonic potential well, while they are negligible when the halting-qubit atom is in the right-hand potential well due to that the halting-qubit atom in the right-hand potential well is much farther from those atoms of the computational state subspace, and both the Coulomb interaction and the dipole-dipole interaction may become very weak as the distance between the interacting atoms become large \[31, 32\]. Therefore, when the halting-qubit atom enters the right-hand potential well from the left-hand one these interactions between the halting-qubit atom and those atoms of the computational state subspace should be decoupled and can be negligible so that the two-qubit quantum gates between the halting-qubit atom in the right-hand potential well and those atoms of the computational state subspace can not be built up effectively. More generally any quantum operations involved in the halting-qubit atom in the left-hand potential well could be really hung up when the halting-qubit atom leaves the left-hand potential well. Therefore, the halting operation could be achieved due to that these quantum operations are hung up when the halting-qubit atom enters into the right-hand potential well from the left-hand one.
Generally, an atom has both the internal electronic states and the center-of-mass motional states. Here a center-of-mass motional state of an atom may be a wave-packet motional state, and for a heavy particle the wave-packet picture in quantum mechanics is close to the classical particle picture \[5a\]. The internal electronic states of an atom are generally quantized bound states, but the center-of-mass motional states may be either the quantized bound states in a potential field or the continuous states in free space \[5a, 5c\]. A quantum bit of an atom generally may be chosen suitably as a pair of the internal electronic ground states of the atom, but sometime the quantized motional states of an atom in a harmonic potential field are also taken as the quantum bits in quantum computation \[30, 33\]. Thus, the halting-qubit atom has the internal electronic states and also the center-of-mass motional states in the double-well potential field. The halting quantum bit generally may be chosen as a pair of the specific internal electronic ground states of the halting-qubit atom. The time-independent double-well potential field generally affects the center-of-mass motional states of the halting-qubit atom \[30\], but it could have a negligible effect on the internal electronic states of the halting-qubit atom so that these internal electronic states could keep unchanged when the halting-qubit atom moves from one potential well to another \[34\]. Actually, the internal electronic states of the halting-qubit atom are determined mainly by the internal interactions of the halting-qubit atom itself, although the complete quantum structure of the halting-qubit atom in the double-well potential field is determined by both the external double-well potential field and the internal interactions of the halting-qubit atom itself. As shown in the previous section, the implementation of the reversible and unitary halting protocol in the atomic physical system is involved in the time- and space-dependent unitary evolution process that the halting-qubit atom moves from one potential well to another in the double-well potential field. Then the center-of-mass motional states and especially the wave-packet motional states of the halting-qubit atom in the double-well potential field will play an important role in implementing the reversible and unitary halting protocol in the atomic physical system.
If now the atomic physical system of the halting-qubit atom in the one-dimensional double-well potential field is considered to be a quantum control system, then one must define explicitly the initial halting state $%
|n_{h}\rangle $ and the states $|c_{1}\rangle $ and $|c_{2}\rangle $ of the control state subspace $S(C)$ in the quantum program $Q_{c}$. First of all, the total wavefunction of the halting-qubit atom in the one-dimensional double-well potential field may be generally written as $$|n_{h}^{\prime },CM,R\rangle =|n_{h}^{\prime }\rangle |CM,R\rangle .$$Here the states $|n_{h}^{\prime }\rangle $ and $|CM,R\rangle $ represent the internal electronic state and the center-of-mass motional state of the halting-qubit atom, respectively. The integer $n_{h}^{\prime }$ and the index $CM$ are the quantum numbers of the internal state $|n_{h}^{\prime
}\rangle $ and the motional state $|CM,R\rangle ,$ respectively, and $R$ is the spatial coordinate of the center of mass of the halting-qubit atom with the motional state $|CM,R\rangle $ in the double-well potential field. The center-of-mass spatial position $R$ is generally time-dependent, i.e., $%
R=R(t)$. Actually, $CM$ is also used to represent the expectation value or eigenvalue of the motional energy (or momentum) of the halting-qubit atom in the double-well potential field particularly when the halting-qubit atom is in a wave-packet motional state. Before the quantum program $Q_{c}$ is performed the halting-qubit atom is located in the left-hand harmonic potential well of the double-well potential field by the ionic or atomic trapping techniques \[30, 35\]. For convenience the left-hand potential well may be prepared temporarily as a conventional harmonic potential well before the quantum program starts to work. This can be achieved easily by setting the height of the right-hand wall of the left-hand potential well to be sufficiently large, since the left-hand potential well can be thought of approximately as a conventional harmonic potential well when the right-hand potential wall is sufficiently high (note that the left-hand potential wall is infinitely high in theory). Thus, before the quantum program starts the internal and motional states of the halting-qubit atom may be really prepared to be the ground internal state and the ground motional state of the conventional harmonic oscillator by the laser cooling techniques \[30, 36\], respectively. Now the global ground state of the halting-qubit atom in the left-hand harmonic potential well may be written as $|0,CM0,R_{0}\rangle
=|0\rangle |CM0,R_{0}\rangle ,$ which is the product state of the ground internal state $|n_{h}^{\prime }\rangle =|0\rangle $ and the ground motional state $|CM0,R_{0}\rangle $ of the atom in the harmonic potential well. Note that the ground motional state $|CM0,R_{0}\rangle $ of the atom in the conventional one-dimensional harmonic potential well is a Gaussian wave-packet motional state \[5a\]. After the total ground state $%
|0,CM0,R_{0}\rangle $ is prepared the left-hand harmonic potential well is suddenly changed back to the original double-well potential field at the starting time of the quantum program. Actually, this process changes merely the sufficiently high right-hand wall of the harmonic potential well to the finitely high one of the left-hand potential well of the double-well potential field. According as quantum mechanics \[5a\] that a wavefunction of a quantum system must be continuous in time, the state $|0,CM0,R_{0}\rangle $ still remains unchanged at the starting time of the quantum program when the harmonic potential field is suddenly changed back to the double-well potential field. Then at the starting time of the quantum program the motional state for the halting-qubit atom in the left-hand potential well of the double-well potential field is just $|CM0,R_{0}\rangle $ and hence still a one-dimensional Gaussian wave-packet motional state. The wave-packet state $|0,CM0,R_{0}\rangle $ could not be exactly the global ground state of the halting-qubit atom in the double-well potential field. However, the energy of the wave-packet state $|0,CM0,R_{0}\rangle $ may be very close to that one of the global ground state of the halting-qubit atom in the double-well potential field if the double-well potential field is designed suitably. Therefore, the initial halting state $|n_{h}\rangle =|0\rangle $ in the quantum program $Q_{c}$ may be set to the wave-packet state $%
|0,CM0,R_{0}\rangle $. Then the halting quantum bit may be chosen as the two ground internal states $\{|n_{h}^{\prime }\rangle ,$ $n_{h}^{\prime }=0,1\}$ of the wave-packet states $\{|n_{h}^{\prime },CM0,R_{0}\rangle \}$. The state $|c_{1}\rangle $ of the control state subspace in the quantum program $%
Q_{c}$ could be taken as the wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})%
\rangle $ with the internal state $|n_{h}^{\prime }\rangle =|1\rangle $ and the wave-packet motional state $|CM1(t_{0i}),R_{1}(t_{0i})\rangle $ of the halting-qubit atom in the left-hand potential well. Here the internal state $%
|n_{h}^{\prime }\rangle =|1\rangle $ could be chosen as another hyperfine ground internal electronic state of the halting-qubit atom different from the hyperfine ground internal electronic state $|n_{h}^{\prime }\rangle
=|0\rangle ,$ and the wave-packet spatial position $R_{1}(t_{0i})$ is within the left-hand potential well. The wave-packet motional state $%
|CM1(t_{0i}),R_{1}(t_{0i})\rangle $ is generated from the wave-packet motional state $|CM0,R_{0}\rangle $ by the trigger pulse $P_{t}$ (see below). The mean motional energy ($CM1$) of the wave-packet motional state $%
|CM1(t_{0i}),R_{1}(t_{0i})\rangle $ is much higher than that one ($CM0$) of the motional state $|CM0,R_{0}\rangle $ and also the height of the intermediate potential barrier in the double-well potential field. Thus, the wave-packet state $|c_{1}\rangle =|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ is an unstable state. When the halting-qubit atom in the left-hand potential well is in the unstable wave-packet state $|c_{1}\rangle $, its motional energy ($CM1$) is much higher than the height of the intermediate potential barrier so that the halting-qubit atom is driven by the high motional momentum ($CM1$) of the atom to pass the intermediate potential barrier to enter into the right-hand potential well. This is a time- and space-dependent quantum scattering process for the halting-qubit atom in the double-well potential field. This quantum scattering process will be taken as the quantum control process of Eq. (2). Note that the quantum scattering process starts at the initial time $t_{0i}$. Now suppose that the halting-qubit atom enters completely into the right-hand potential well at the time $t_{mi}=t_{0i}+\Delta t_{0}$ in the quantum scattering process and at the time $t_{mi}$ the wave-packet state of the halting-qubit atom in the right-hand potential well is denoted as $|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle
$. Here the wave-packet spatial position $R_{2}(t_{mi})$ is within the right-hand potential well. Since the quantum scattering process for the halting-qubit atom from one potential well to another in the double-well potential field does not change the ground internal states of the atom \[34\] both the wave-packet states $|c_{1}\rangle
=|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ and $|c_{2}^{\prime }\rangle
=|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle $ have the same internal state with $%
n_{h}^{\prime }=1$. Now another state $|c_{2}\rangle $ of the control state subspace in the quantum program $Q_{c}$ could be temporarily set to the wave-packet state $|c_{2}^{\prime }\rangle $. Actually, the state $%
|c_{2}\rangle $ in the quantum program $Q_{c}$ will correspond to any wave-packet state of the halting-qubit atom in the right-hand potential well, as can be seen later, for the control state subspace of the atomic physical system is not a two-state subspace. Due to the motional energy conservation during the quantum scattering process the motional energy ($CM1$) of the wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ of the halting-qubit atom in the left-hand potential well is really equal to that one ($CM2$) of the wave-packet state $|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle $ of the atom in the right-hand potential well. Thus, the wave-packet state $%
|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle $ is also an unstable state just like the wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $. The spatial spreads of these two wave-packet motional states $|CM1(t_{0i}),R_{1}(t_{0i})%
\rangle $ and $|CM2(t_{mi}),R_{2}(t_{mi})\rangle $ could be generally different from each other, but these two motional states could be approximately Gaussian in coordinate space \[5a, 5c\] just like the motional state $|CM0,R_{0}\rangle $. Though these two wave-packet motional states could not be exactly orthogonal to one another, they could be almost completely orthogonal to one another if the wave-packet spatial distance $%
|R_{2}(t_{mi})-$ $R_{1}(t_{0i})|$ between their spatial positions $%
R_{1}(t_{0i})$ and $R_{2}(t_{mi})$ is large enough, because the overlapping integral between these two wave-packet motional states decays exponentially as the wave-packet spatial distance increases. Therefore, the width of the intermediate potential barrier of the double-well potential field must be large enough such that any wave-packet motional state of the halting-qubit atom within one potential well is almost completely orthogonal to that one within another potential well in the double-well potential field.
In the atomic physical system the time-independent double-well potential field could be considered as one of the components of the state-locking pulse field $P_{SL}$. More generally, a complete state-locking pulse field in the atomic physical system could consist of three parts below. The first part is the double-well potential field itself. The second is the sequences of the time- and space-dependent electromagnetic pulse fields which are applied only to the right-hand potential well. As can be seen below, these sequences include mainly the unitary decelerating sequence and the unitary accelerating sequence. The unitary decelerating sequence (the unitary accelerating sequence) is used to decelerate (accelerate) the halting-qubit atom in motion in the right-hand potential well. In this part the sequences of the electromagnetic pulse fields are used to manipulate coherently the wave-packet motional state of the halting-qubit atom in the right-hand potential well so that the halting-qubit atom can stay in the right-hand potential well for a long time required by the quantum computational process in the quantum program. The third part is the sequences of the time- and space-dependent electromagnetic pulse fields and includes also the time-dependent potential fields which are applied mainly to the left-hand potential well after the quantum program $Q_{c}$ terminates. In this part the electromagnetic pulse fields associated with the unitary accelerating sequence in the second part are used to drive the halting-qubit atom in the right-hand potential well to go back to the left-hand potential well after the quantum computational process finished, and at the same time transfer each of possible wave-packet states of the halting-qubit atom in the right-hand potential well to some given wave-packet state of the atom in the left-hand potential well so that the final state of the halting-qubit atom in the left-hand potential well is not dependent sensitively upon these possible wave-packet states. Therefore, in both the second and third parts these electromagnetic pulse fields should have a negligible effect on any motional state of the halting-qubit atom if the atom locates within the left-hand potential well during the quantum computational process. In this and next sections it is discussed how the double-well potential field affects the wave-packet state of the halting-qubit atom before the atom leaves the left-hand potential well in the quantum computational process, while in the section 5 it will be discussed how the sequences of the time- and space-dependent electromagnetic pulse fields acts on the halting-qubit atom to keep the atom in the right-hand potential well for a long time and how by the electromagnetic pulse fields and the time-dependent potential fields each of the possible wave-packet states of the halting-qubit atom in the right-hand potential well is changed to some given wave-packet state of the atom in the left-hand potential well after the quantum computational process finished. The time-independent double-well potential field must be designed properly. Before the halting-qubit atom leaves the left-hand potential well during the quantum computational process the atom may have its zero-point oscillatory motion in the left-hand harmonic potential well, and it could also penetrate through the intermediate potential barrier and enter into the right-hand potential well due to the quantum tunneling effect \[5a\] even if the motional energy of the halting-qubit atom in the left-hand potential well is lower than the height of the intermediate potential barrier. The zero-point oscillatory motion is allowed normally in quantum computation \[30\], but the quantum control process that simulates efficiently the reversible and unitary halting protocol could become degraded if the halting-qubit atom enters into the right-hand potential well in a non-negligible probability due to the quantum tunneling effect before the halting operation is performed according as the quantum program. Thus, the double-well potential field should be designed in such a way that the probability for the halting-qubit atom going from the left-hand potential well to the right-hand one due to the quantum tunneling effect should be minimized and can be neglected for the quantum computational process. The height and width of the intermediate potential barrier in the double-well potential field may control the quantum tunneling effect, that is, the higher and wider the intermediate potential barrier, the lower the penetrating probability for the halting-qubit atom. For example, consider a simple physical model that a free particle with motional energy $E_{h}$ and mass $m_{h}$ hits a square potential barrier with height $V_{0}$ and width $%
a $ \[5a\]$.$ The particle will be reflected and/or transmitted by the square potential barrier. If the motional energy $E_{h}$ of the particle is much less than the potential barrier height $V_{0}$ such that $\beta a>>1$ with $%
\beta \hslash =\sqrt{2m_{h}(V_{0}-E_{h})},$ then the transmission coefficient of the particle is approximately proportional to the exponential factor $\exp (-2\beta a)$ \[5a\]$.$ Therefore, the probability for the free particle to penetrate through the potential barrier is also proportional to the exponential factor $\exp (-2\beta a)$. When the potential barrier height $V_{0}$ and width $a$ are large enough, this probability falls off rapidly. Now a bound particle with the same motional energy $E_{h}<<V_{0}$ like the halting-qubit atom in the left-hand potential well is more difficult to penetrate through the same square potential barrier than a free particle. An atom has a much heavier mass $m_{h}$ than an electron. The probability for the halting-qubit atom with the motional energy $E_{h}<<V_{0}$ in the left-hand potential well to penetrate through the intermediate square potential barrier with the height $V_{0}$ and width $a$ decays rapidly in an exponential form as the height $V_{0},$ the width $a$, and the atomic mass $%
m_{h}$ increase. Evidently, if the intermediate potential barrier is high and wide enough such that the ground-state motional energy ($CM0$) of the halting-qubit atom is much lower than the barrier height, then the quantum tunneling effect could have a negligible effect on the initial halting state $|n_{h}\rangle =|0,CM0,R_{0}\rangle $ and also the state $|n_{h}\rangle
=|1,CM0,R_{0}\rangle $ of the halting quantum bit, and hence the wave-packet states $\{|n_{h}^{\prime },CM0,R_{0}\rangle \}$ with $n_{h}^{\prime }=0,1$ of the halting qubit atom in the left-hand potential well may keep almost unchanged during the quantum computational process before the halting-qubit atom leaves the left-hand potential well due to the halting operation in the quantum program.
The coherent stimulated Raman adiabatic passage ($STIRAP$) method has been used to prepare the $^{\prime }$Schrödinger Cat$^{\prime }$ superposition state of a trapped atom \[37\]. Here the coherent $STIRAP$ method \[37–41\] may also be used to transfer selectively the halting state $%
|n_{h}\rangle =|1,CM0,R_{0}\rangle $ of the halting-qubit atom in the left-hand potential well to the unstable state $|c_{1}\rangle
=|1,CM1(t_{0i}),$ $R_{1}(t_{0i})\rangle $ of the control state subspace. Therefore, the state-dependent trigger pulse $P_{t}$ in the quantum program $%
Q_{c}$ could be chosen as the coherent Raman adiabatic laser pulse. The spatial action zone of the coherent Raman adiabatic laser pulse $P_{t}$ is confined within the left-hand potential well. A coherent Raman adiabatic laser pulse $P_{t}$ consists of a pair of the coherent adiabatic laser beams. Here denote these two adiabatic laser beams as $A$ and $B$, respectively. When the coherent Raman adiabatic laser pulse $P_{t}$ is applied to the halting-qubit atom in the left-hand potential well, one of these two adiabatic laser beams (e.g., the beam $A$) first excites selectively the transition of the halting-qubit atom from the wave-packet state $|n_{h}^{\prime },CM0,R_{0}\rangle $ with the internal state $%
|n_{h}^{\prime }\rangle =|1\rangle $ to some specific excited state $%
|n_{e},CM_{e},R_{e}\rangle ,$ meanwhile the halting-qubit atom in the excited state $|n_{e},CM_{e},R_{e}\rangle $ is stimulated by another adiabatic laser beam $B$ to jump to the state $|c_{1}\rangle
=|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $. Here the excited internal state $%
|n_{e}\rangle $ $(n_{e}\neq 0,1)$ of the excited state $|n_{e},CM_{e},R_{e}%
\rangle $ has a higher internal energy than the ground internal states $%
\{|n_{h}^{\prime }\rangle ,$ $n_{h}^{\prime }=0,1\}$ and the halting-qubit atom with the wave-packet motional state $|CM_{e},R_{e}\rangle $ is still within the left-hand potential well. For example, if the halting-qubit atom is chosen as a single $^{9}B_{e}^{+}$ ion, then these two internal states $%
\{|n_{h}^{\prime }\rangle ,$ $n_{h}^{\prime }=0,1\}$ could be taken as the ionic hyperfine ground states $^{2}S_{1/2}$ $(F=2,m_{F}=-2)$ and $%
^{2}S_{1/2} $ $(F=1,m_{F}=-1),$ respectively, while the excited internal state $|n_{e}\rangle $ could be the excited electronic state $^{2}P_{1/2}$ $%
(F=2,m_{F}=-2)$ of the ion \[37a\]. This state-dependent excitation process under the coherent Raman adiabatic laser trigger pulse $P_{t}$ may be simply expressed as $$|1,CM0,R_{0}\rangle \overset{A}{\leftrightarrow }|n_{e},CM_{e},R_{e}\rangle
\overset{B}{\leftrightarrow }|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle .$$In order that only the two desired transitions: $|1,$ $CM0,R_{0}\rangle
\leftrightarrow |n_{e},CM_{e},R_{e}\rangle $ and $|n_{e},CM_{e},R_{e}\rangle
\leftrightarrow |1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ are excited selectively by the coherent Raman adiabatic laser pulse $P_{t}$ the frequencies of both the laser beams $A$ and $B$ must be close to the resonance frequencies of these two desired transitions, respectively, and they are also much far from resonance frequencies of any other transitions including the transition: $|0,CM0,R_{0}\rangle \leftrightarrow
|n_{e},CM_{e},R_{e}\rangle $. Any wave-packet state $|n_{h}^{\prime
},CM,R\rangle $ of the halting-qubit atom with the internal state $%
|n_{h}^{\prime }\rangle \neq |1\rangle $ will not be affected effectively by any one of these two adiabatic laser beams. Thus, the coherent Raman adiabatic laser pulse $P_{t}$ does not act on any wave-packet state with the internal state $|n_{h}^{\prime }\rangle \neq |1\rangle $ such as the initial halting state $|0,CM0,R_{0}\rangle $. On the other hand, in order to suppress irreversible spontaneous emission processes both the laser beams $A$ and $B$ are detuned properly from the excitation state $|n_{e},CM_{e},R_{e}%
\rangle $. It might be better that the wave vector difference of these two laser beams $A$ and $B$ is set to point to the left-hand potential wall of the double-well potential field as the left-hand potential wall may be sufficiently high in practice. This means that the halting-qubit atom moves along $-x$ axis toward the left-hand potential wall under the action of the coherent Raman adiabatic laser pulse $P_{t}$. Obviously, in the state-dependent excitation process the halting state $|1,CM0,R_{0}\rangle $ is excited to the higher motional energy state $|c_{1}\rangle
=|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ by the coherent Raman adiabatic laser pulse $P_{t}$. In particular, the mean motional energy of the motional state $|CM1(t_{0i}),R_{1}(t_{0i})\rangle $ of the halting-qubit atom must be much higher than the height of the intermediate potential barrier so that the unitary quantum scattering process can take place automatically for the halting-qubit atom from the left-hand potential well to the right-hand one.
The quantum control process in the quantum program $Q_{c}$ usually should be modified properly when it is implemented in a real quantum physical system such as the simple atomic physical system mentioned above. For the quantum control system of the atomic physical system the initial halting state $%
|n_{h}\rangle $ and these two states $|c_{1}\rangle $ and $|c_{2}\rangle $ of the control state subspace $S(C)$ in the quantum program $Q_{c}$ are defined as the wave-packet states of the halting-qubit atom in the left-hand potential well: $|n_{h}\rangle =|0,CM0,R_{0}\rangle ,$ $|c_{1}\rangle
=|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle ,$ and $|c_{2}\rangle =|c_{2}^{\prime
}\rangle =|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle ,$ respectively. The state-dependent trigger pulse $P_{t}$ is taken as a suitable coherent Raman adiabatic laser pulse that is applied to within the left-hand potential well in space. The state-locking pulse field in the atomic physical system consists of the double-well potential field and the sequences of the time- and space-dependent electromagnetic pulse fields and the time-dependent potential fields, as mentioned in the previous paragraphs. Now a quantum control process (or unit) $Q_{h}$ of the atomic physical system which replaces the statement P7 of the quantum program $Q_{c}$ is designed to simulate efficiently the reversible and unitary halting protocol. The quantum control process (or unit) $Q_{h}$ is written as $$\text{While }|f_{r}(x)\rangle =|f_{r}(x_{f})\rangle ,\text{ \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \quad }$$$$\quad \text{Do }U_{h}^{c}:|n_{h}\rangle |f_{r}(x_{f})\rangle
=|0,CM0,R_{0}\rangle |1\rangle \rightarrow |0,CM0,R_{0}\rangle |0\rangle ,$$$$\quad \quad V_{h}^{c}:|n_{h}\rangle |f_{r}(x)\rangle =|0,CM0,R_{0}\rangle
|0\rangle \rightarrow |1,CM0,R_{0}\rangle |0\rangle$$$$\text{State-dependent excitation process }(P_{t}):\qquad \ \quad \quad
\qquad \qquad$$$$|n_{h}\rangle =|1,CM0,R_{0}\rangle \rightarrow |c_{1}\rangle
=|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle$$$$\text{Quantum scattering process in time and space }(P_{SL}):\qquad \quad$$$$\qquad \qquad |c_{1}\rangle =|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle \rightarrow
|c_{2}^{\prime }\rangle =|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle$$Here the desired functional state is $|f_{r}(x_{f})\rangle =|1\rangle $ if the quantum program $Q_{c}$ works in the multiplicative-cyclic-group state subspace $S(m_{r})$ and $|f_{r}(x_{f})\rangle =|0\rangle $ if the quantum program $Q_{c}$ works in the additive-cyclic-group state subspace $%
S(Z_{m_{r}})$ and in later case the unitary operation $U_{h}^{c}$ may be omitted from the quantum control unit $Q_{h}$. In the quantum control unit $%
Q_{h}$ the unitary operation $U_{h}^{c}$ is the same as the original one in the quantum program $Q_{c}$: $$U_{h}^{c}:|0,CM0,R_{0}\rangle |b_{h}\rangle |1\rangle \leftrightarrow
|0,CM0,R_{0}\rangle |b_{h}\rangle |0\rangle ,$$and the new conditional unitary operation $V_{h}^{c}$ is defined as $$V_{h}^{c}:|0,CM0,R_{0}\rangle |b_{h}\rangle |0\rangle \leftrightarrow
|1,CM0,R_{0}\rangle |b_{h}\rangle |0\rangle$$with $b_{h}=0,1$, and the conditional unitary operation $U_{tr}^{c}$ during the state-dependent trigger pulse $P_{t}$ is simply defined by $$U_{tr}^{c}:|1,CM0,R_{0}\rangle |b_{h}\rangle |f_{r}(x)\rangle
\leftrightarrow |1,CM1(t_{0i}),R_{1}(t_{0i})\rangle |b_{h}\rangle
|f_{r}(x)\rangle .$$Here the unitary operation $U_{tr}^{c}$ corresponds to the original one $%
U_{t}$ in the quantum program $Q_{c}$. The conditional unitary operations $%
U_{h}^{c}$ and $V_{h}^{c}$ generally are dependent upon both the functional states and the internal states of the halting-qubit atom but may be independent of any motional states of the atom. They may be built up efficiently out of the interactions between the halting-qubit atom in the left-hand potential well and those atoms of the computational state subspace. It follows from the quantum control unit $Q_{h}$ that only when the functional state $|f_{r}(x)\rangle $ becomes the desired state $%
|f_{r}(x_{f})\rangle $ can the unitary operations $U_{h}^{c}$ and $V_{h}^{c}$ take a real action on the quantum system of the quantum computer. One of the important processes in the quantum control unit $Q_{h}$ is the state-dependent excitation process involved in the trigger pulse $P_{t}$ with the pulse duration $\delta t_{r}$. Though the coherent Raman adiabatic laser trigger pulse $P_{t}$ is applied to the halting-qubit atom in the left-hand potential well at the starting time $t_{0i}-\delta t_{r}$ for every cycle of the quantum program $Q_{c}$ with the cyclic index $%
i=1,2,...,m_{r}$, it can only take a real action on those wave-packet states of the atom with the internal state $|n_{h}^{\prime }\rangle =|1\rangle $ such as the halting state $|1,CM0,R_{0}\rangle $. Therefore, only when the initial halting state $|n_{h}\rangle =|0,CM0,R_{0}\rangle $ is changed to the halting state $|n_{h}\rangle =|1,CM0,R_{0}\rangle $ by the conditional unitary operation $V_{h}^{c}$ in the quantum program can the trigger pulse $%
P_{t}$ excite the state $|n_{h}\rangle =|1,CM0,R_{0}\rangle $ with the ground motional energy ($CM0$) to the unstable wave-packet state $%
|c_{1}\rangle =|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ with a much higher motional energy ($CM1$) and a different spatial position $R_{1}(t_{0i})\neq
R_{0}$ in the left-hand potential well. Since the mean motional energy ($CM1$) of the wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ is much higher than the height of the intermediate potential barrier the halting-qubit atom in the left-hand potential well with the wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ can easily pass the intermediate potential barrier to enter into the right-hand potential well. This process is a quantum scattering process in space and time: $%
|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle \rightarrow
|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle ,$ that is, a time- and space-dependent unitary evolution process which is driven by the motional momentum of the halting-qubit atom in the unstable state $|1,CM1(t_{0i}),R_{1}(t_{0i})%
\rangle $. This is a key process for the quantum control process $Q_{h}$ to achieve the reversible and unitary halting operation. The halting operation will take place when the halting-qubit atom leaves the left-hand potential well because the coherent Raman adiabatic laser trigger pulse $P_{t}$ does not have a real action on any wave-packet state of the halting-qubit atom if the atom is not in the left-hand potential well. On the other hand, if the halting-qubit atom enters into the right-hand potential well, then any two-qubit quantum gates become unavailable between the halting-qubit atom and those atoms of the computational state subspace due to that the effective interactions vanish between the halting-qubit atom and those atoms. Then any one of the unitary operations $U_{h}^{c}$ and $V_{h}^{c}$ in the quantum control unit $Q_{h}$ becomes yet unavailable and consequently the halting operation is achieved too.
Now the original quantum control unit (the statement P7) of the quantum program $Q_{c}$ is replaced with the quantum control unit $Q_{h}$ of the atomic physical system. The time evolution process of the atomic physical system of the quantum computer under the quantum program and circuit $Q_{c}$ is investigated in detail below. This investigation will be helpful for understanding more clearly and deeply the general properties of a state-locking pulse field and especially the properties of the unitary transformations related to the state-locking pulse field. First of all, the quantum program and circuit $Q_{c}$ with the quantum control unit $Q_{h}$ may be written in the simple form $$Q_{c}=\{P_{SL}:OFF\}\{U_{f}^{c}P_{SL}(\{\varphi _{i}\},\Delta
t_{0})U_{tr}^{c}V_{h}^{c}U_{h}^{c}U_{b}^{c}\}^{m_{r}}\{P_{SL}:ON\}.$$Here the initial state of the quantum circuit $Q_{c}$ is set to the basis state $|n_{h}\rangle |b_{h}\rangle |f_{r}(x)\rangle =|0,CM0,R_{0}\rangle
|0\rangle |f_{r}(x_{0})\rangle .$ The state-locking pulse field $P_{SL}$ is applied continuously to the quantum system from the beginning to the end of the quantum circuit. During the periods of these unitary operations $%
U_{f}^{c},$ $U_{tr}^{c},$ $U_{h}^{c},$ $V_{h}^{c},$ and $U_{b}^{c}$ the quantum system is really acted upon simultaneously by both the state-locking pulse field and these unitary operations. The unitary transformation $%
P_{SL}(\{\varphi _{i}\},\Delta t_{0})$ represents the quantum scattering process during the period from the time $t_{0i}$ to the time $%
t_{mi}=t_{0i}+\Delta t_{0}.$ Obviously, if a unitary operation is applied only to the computational state subspace it commutes with the unitary propagator $P_{SL}(\{\varphi _{k}\},t,t_{0})$ of the state-locking pulse field as the state-locking pulse field is applied only to the halting-qubit atom in the double-well potential field. Now the unitary functional operation $U_{f}^{c}$ and the unitary operation $U_{b}^{c}$ are applied only to the computational state subspace in the quantum circuit. Thus, the unitary propagator $P_{SL}(\{\varphi _{k}\},t,t_{0})$ always commutes with these conditional unitary operations$:$ $$P_{SL}(\{\varphi _{k}\},t,t_{0})U_{b}^{c}\equiv U_{b}^{c}P_{SL}(\{\varphi
_{k}\},t,t_{0}), \label{3}$$$$P_{SL}(\{\varphi _{k}\},t,t_{0})U_{f}^{c}\equiv U_{f}^{c}P_{SL}(\{\varphi
_{k}\},t,t_{0}). \label{4}$$Though the conditional unitary operations $U_{h}^{c}$ and $V_{h}^{c}$ may be independent of any atomic motional states according as their definitions, these two unitary operations may require that the wave-packet motional state $|CM0,R_{0}\rangle $ of the halting states {$|n_{h}^{\prime
},CM0,R_{0}\rangle ,n_{h}^{\prime }=0,1$} of the halting-qubit atom in the left-hand potential well keep unchanged up to a global phase factor when these two unitary operations are applied to the quantum system in the period from the initial time $t_{0}$ of the quantum circuit to the time $%
t_{0i}-\delta t_{r}$ before the state $|1,CM0,R_{0}\rangle $ is changed to the state $|c_{1}\rangle =|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ by the state-dependent trigger pulse $P_{t}.$ Here the quantum circuit $Q_{c}$ starts to run the first cycle at the initial time $t_{0}$. The requirement may be necessary when both the unitary operations are built up out of the dipole-dipole interactions or the Coulomb interactions which are dependent on the interdistances of atoms. Since the double-well potential field is considered as one of the components of the state-locking pulse field $P_{SL}$ in the atomic physical system, any motional state of the halting-qubit atom in the double-well potential field is affected inevitably by the state-locking pulse field. However, as pointed out before, the intermediate potential barrier is so high and wide that up to a global phase factor the wave-packet motional state $|CM0,R_{0}\rangle $ of the halting states $%
\{|n_{h}^{\prime },CM0,R_{0}\rangle \}$ is almost unchanged in the period from the initial time $t_{0}$ to the time $t_{0i}-\delta t_{r}$ before the state $|1,CM0,R_{0}\rangle $ is changed to the unstable state $|c_{1}\rangle
.$ This property of the state-locking pulse field could be simply expressed by the unitary transformation: $$P_{SL}(\{\varphi _{k}\},t,t_{0})|n_{h}^{\prime },CM0,R_{0}\rangle \qquad
\qquad \qquad \qquad \qquad \qquad$$$$=\exp [-iE_{0}(t-t_{0})/\hslash ]|n_{h}^{\prime },CM0,R_{0}\rangle ,\text{ }%
t_{0}\leq t\leq t_{0i}-\delta t_{r}. \label{5}$$where $n_{h}^{\prime }=0,$ $1$ and $E_{0}$ is the motional energy of the ground motional state $|CM0,R_{0}\rangle .$ Hereafter for convenience the global phase factor such as $\exp [-iE_{0}(t-t_{0})/\hslash ]$ in Eq. (5) is omitted without confusion. Therefore, both the unitary operations $U_{h}^{c}$ and $V_{h}^{c}$ commute approximately with the unitary propagator $%
P_{SL}(\{\varphi _{k}\},t,t_{0})$ ($t_{0}\leq t\leq t_{0i}-\delta t_{r}$) of the state-locking pulse field, $$P_{SL}(\{\varphi _{k}\},t,t_{0})U_{h}^{c}=U_{h}^{c}P_{SL}(\{\varphi
_{k}\},t,t_{0}), \label{6}$$$$P_{SL}(\{\varphi _{k}\},t,t_{0})V_{h}^{c}=V_{h}^{c}P_{SL}(\{\varphi
_{k}\},t,t_{0}). \label{7}$$Moreover, the commutation relations (6) and (7) still hold when the halting-qubit atom enters into the right-hand potential well from the left-hand one, because in this case these two unitary operations $U_{h}^{c}$ and $V_{h}^{c}$ have not any real effect on the halting-qubit atom and are reduced theoretically to the unity operation. Therefore, the commutation relations (6) and (7) hold for the whole quantum circuit. The commutation relations (5), (6), and (7) lead to that there hold the state unitary transformations, $$\{P_{SL}(\{\varphi _{k}\},t,t_{0}),U_{h}^{c}\}|n_{h}^{\prime
},CM0,R_{0}\rangle |b_{h}\rangle |f_{r}(x)\rangle$$$$\hspace{1in}=U_{h}^{c}|n_{h}^{\prime },CM0,R_{0}\rangle |b_{h}\rangle
|f_{r}(x)\rangle ,\text{ }t_{0}\leq t\leq t_{0i}-\delta t_{r}, \label{8}$$$$\{P_{SL}(\{\varphi _{k}\},t,t_{0}),V_{h}^{c}\}|n_{h}^{\prime
},CM0,R_{0}\rangle |b_{h}\rangle |f_{r}(x)\rangle$$$$\hspace{1in}=V_{h}^{c}|n_{h}^{\prime },CM0,R_{0}\rangle |b_{h}\rangle
|f_{r}(x)\rangle ,\text{ }t_{0}\leq t\leq t_{0i}-\delta t_{r}. \label{9}$$Evidently, the unitary propagator $P_{SL}(\{\varphi _{k}\},t,t_{0})$ generally could not commute with the unitary operation $U_{tr}^{c}$ of the trigger pulse $P_{t}$ in the atomic physical system, $$P_{SL}(\{\varphi _{k}\},t,t_{0})U_{tr}^{c}\neq U_{tr}^{c}P_{SL}(\{\varphi
_{k}\},t,t_{0}).$$This is because in the atomic physical system the state-dependent excitation process of the trigger pulse $P_{t}$ from the state $|1,CM0,R_{0}\rangle $ to the unstable state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ is affected inevitably by the left-hand harmonic potential field. Then the coherent Raman adiabatic laser trigger pulse $P_{t}$ must be designed suitably such that when both the trigger pulse $P_{t}$ and the state-locking pulse field $%
P_{SL}$ are applied simultaneously to the halting-qubit atom in the left-hand potential well the unitary propagator $\{P_{SL}(\{\varphi
_{k}\},t_{0i},t_{0i}-\delta t_{r}),P_{t}\}$ satisfies the relation: $$\{P_{SL}(\{\varphi _{k}\},t_{0i},t_{0i}-\delta
t_{r}),P_{t}\}|1,CM0,R_{0}\rangle |b_{h}\rangle |f_{r}(x)\rangle \qquad
\qquad \qquad \quad$$$$\equiv U_{tr}^{c}|1,CM0,R_{0}\rangle |b_{h}\rangle |f_{r}(x)\rangle
=|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle |b_{h}\rangle |f_{r}(x)\rangle .
\label{10}$$The approximate calculation in theory for the unitary propagator $%
U_{tr}^{c}\equiv \{P_{SL}(\{\varphi _{k}\},t_{0i},$ $t_{0i}-\delta
t_{r}),P_{t}\}$ could be carried out on the simple physical model of the time-dependent forced harmonic oscillator \[5c, 37, 50g, 50h, 50i, 51b\].
Suppose again that the six unitary operations $U_{f}^{c},$ $P_{SL}(\{\varphi
_{k}\},\Delta t_{0}),$ $U_{tr}^{c},$ $V_{h}^{c},$ $U_{h}^{c},$ and $U_{b}^{c}
$ in the quantum circuit $Q_{c}$ have the durations $\delta t_{f},$ $\Delta
t_{0},$ $\delta t_{r},$ $\delta t_{h}^{\prime },$ $\delta t_{h},$ and $%
\delta t_{b},$ respectively, and the period of each cycle of the quantum circuit is $\Delta T=\delta t_{b}+\delta t_{h}+\delta t_{h}^{\prime }+\delta
t_{r}+\Delta t_{0}+\delta t_{f}.$ As stated before, the unstable state $%
|c_{1}\rangle =|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ of the control state subspace is generated completely at the instant of time $t_{0i}$ in the $i-$th cycle of the quantum circuit$.$ Then it follows that the functional state $|f_{r}(x)\rangle $ is converted into the desired functional state $%
|f_{r}(x_{f})\rangle $ during the period $\delta t_{f}$ from the time $%
(t_{0i}-\delta t_{r}-\delta t_{h}^{\prime }-\delta t_{h}-\delta
t_{b})-\delta t_{f}$ to the time $(t_{0i}-\delta t_{r}-\delta t_{h}^{\prime
}-\delta t_{h}-\delta t_{b})$ in the $(i-1)-$th cycle. Note that the initial state of the quantum circuit is the wave-packet state $|0,CM0,R_{0}\rangle
|0\rangle |f_{r}(x_{0})\rangle $ and at the initial time $t_{0}$ the halting-qubit atom is in the state $|0,CM0,R_{0}\rangle $ and in the left-hand potential well. Then the desired functional state $%
|f_{r}(x_{f})\rangle $ takes the integer $x_{f}=(x_{0}+i-1)\func{mod}m_{r}$. Before the $i-$th cycle the initial halting state $|n_{h}\rangle
=|0,CM0,R_{0}\rangle $ and the initial branch-control state $|b_{h}\rangle
=|0\rangle $ keep unchanged but only the initial functional state $%
|f_{r}(x_{0})\rangle $ is consecutively changed to other functional state $%
|f_{r}(x_{0}+j)\rangle $ for $j=0,1,2,...,i-1$ in the quantum circuit$,$ where the last functional state $|f_{r}(x_{0}+i-1)\rangle $ is the desired functional state $|f_{r}(x_{f})\rangle $. Obviously, before the $i-$th cycle the time evolution process of the whole quantum system of the quantum computer with the initial state $|0,CM0,R_{0}\rangle |0\rangle
|f_{r}(x_{0})\rangle $ can be expressed as $$\{P_{SL}(\{\varphi _{k}\},t_{j},t_{0}),(U_{T})^{j}\}|0,CM0,R_{0}\rangle
|0\rangle |f_{r}(x_{0})\rangle \qquad \quad \quad$$$$=(U_{T})^{j}|0,CM0,R_{0}\rangle |0\rangle |f_{r}(x_{0})\rangle \ \ \qquad
\qquad \hspace{1in}\qquad \qquad$$$$=|0,CM0,R_{0}\rangle |0\rangle |f_{r}(x_{0}+j)\rangle ,\text{ }%
t_{j}=t_{0}+j\Delta T,\text{ }0\leq j\leq i-1, \label{11}$$where the unitary operator $U_{T}$ is denoted as the unitary operation sequence $U_{f}^{c}P_{SL}(\{\varphi _{k}\},\Delta
t_{0})U_{tr}^{c}V_{h}^{c}U_{h}^{c}U_{b}^{c}$ of the quantum circuit $Q_{c}.$ In the unitary transformation (11) the first equation shows that up to a global phase factor the unitary propagator of the quantum system under both the state-locking pulse field $P_{SL}$ and the unitary operation sequence $%
(U_{T})^{j}$ for $0\leq j\leq i-1$ acting on the initial state is really equal to the single unitary operation sequence $(U_{T})^{j}$ acting on the same initial state before the $i-$th cycle of the quantum circuit. Here the integer $j$ is also used as the cyclic index of the quantum circuit.
For example, there are some cases to be considered for the unitary transformation (11). For the first case $(i)$ the initial functional state $%
|f_{r}(x_{0})\rangle $ happens to be just the desired functional state $%
|f_{r}(x_{f})\rangle .$ In this case the index $i=1$. Then $x_{f}=x_{0}$ and the cyclic index takes $j=0$ in the unitary transformation (11). Therefore, $%
t_{j}=t_{0}$ for $j=0.$ Here $t_{01}=t_{0}+\delta t_{b}+\delta t_{h}+\delta
t_{h}^{\prime }+\delta t_{r}.$ Both the unitary propagator $P_{SL}(\{\varphi
_{k}\},t_{j},t_{0})$ and the unitary operation sequence $(U_{T})^{j}$ become the unity operator in effect as the cyclic index $j=0$ and the unitary transformation (11) is a state identity. For the second case $(ii)$ the initial functional state $|f_{r}(x_{0})\rangle $ could be changed to the desired functional state $|f_{r}(x_{f})\rangle $ at the end of the first cycle of the quantum circuit. In this case the index $i=2$. Then $%
x_{f}=(x_{0}+1)\func{mod}m_{r}$ and the cyclic index takes $j=0$ and $1$ in the unitary transformation (11). Note that the end time of the first cycle is just equal to the beginning time of the second cycle in the quantum circuit. Thus, $t_{0}=t_{0},$ $t_{1}=t_{0}+\Delta T=t_{01}+\Delta
t_{0}+\delta t_{f}=t_{02}-\delta t_{r}-\delta t_{h}^{\prime }-\delta
t_{h}-\delta t_{b}.$ Here $t_{01}=t_{0}+\delta t_{b}+\delta t_{h}+\delta
t_{h}^{\prime }+\delta t_{r}$ and $t_{02}=t_{01}+\Delta T.$ Generally, for the third case $(iii)$ the initial functional state $|f_{r}(x_{0})\rangle $ could be changed to the desired functional state $|f_{r}(x_{f})\rangle $ at the end of the $(i-1)-$th cycle of the quantum circuit. In this case the cyclic index $i>1$. Then $x_{f}=(x_{0}+i-1)\func{mod}m_{r}$ and the cyclic index takes $j=0,1,...,(i-1)$ in the unitary transformation (11). Here the end time of the $(i-1)-$th cycle is just the beginning time of the $i-$th cycle in the quantum circuit. Thus, $t_{0}=t_{0}$ and $t_{j}=t_{0}+j\Delta
T=t_{0j}+\Delta t_{0}+\delta t_{f}=t_{0(j+1)}-\delta t_{r}-\delta
t_{h}^{\prime }-\delta t_{h}-\delta t_{b}$ for $j=1,...,(i-1).$ Here $%
t_{01}=t_{0}+\delta t_{b}+\delta t_{h}+\delta t_{h}^{\prime }+\delta t_{r}$ and $t_{0(j+1)}=t_{01}+j\Delta T$ for $j=1,...,(i-1).$
The $i-$th cycle ($i\geq 1$) of the quantum circuit will be analyzed separately as follows. In the $i-$th cycle of the quantum circuit the starting time is $t_{(i-1)}=t_{0}+(i-1)\Delta T=t_{0i}-\delta t_{r}-\delta
t_{h}^{\prime }-\delta t_{h}-\delta t_{b}$ and the starting state $%
|0,CM0,R_{0}\rangle |0\rangle |f_{r}(x_{f})\rangle $ with $x_{f}=(x_{0}+i-1)%
\func{mod}m_{r}.$ Suppose that now the quantum circuit starts to execute the $i-$th cycle. First, the unitary transformation $U_{b}^{c}$ changes the initial branch-control state $|b_{h}\rangle =|0\rangle $ to the state $%
|1\rangle ,$$$\{P_{SL}(\{\varphi _{k}\},t_{(i-1)}+\delta
t_{b},t_{(i-1)}),U_{b}^{c}\}|0,CM0,R_{0}\rangle |0\rangle
|f_{r}(x_{f})\rangle$$$$\equiv P_{SL}(\{\varphi _{k}\},t_{(i-1)}+\delta
t_{b},t_{(i-1)})U_{b}^{c}|0,CM0,R_{0}\rangle |0\rangle |f_{r}(x_{f})\rangle$$$$=|0,CM0,R_{0}\rangle |1\rangle |f_{r}(x_{f})\rangle \hspace{1.5in}\qquad
\quad \quad . \label{12}$$Here the unitary propagator $\{P_{SL}(\{\varphi _{k}\},t_{(i-1)}+\delta
t_{b},t_{(i-1)}),U_{b}^{c}\}$ is identical to the unitary operation $%
P_{SL}(\{\varphi _{k}\},t_{(i-1)}+\delta t_{b},t_{(i-1)})U_{b}^{c}$ due to the fact that both the unitary transformation $P_{SL}(\{\varphi
_{k}\},t_{(i-1)}+\delta t_{b},t_{(i-1)})$ of the state-locking pulse field and the unitary operation $U_{b}^{c}$ commute with each other, as shown in Eq. (3). The second equation holds due to the state unitary transformation $%
P_{SL}(\{\varphi _{k}\},t_{(i-1)}+\delta t_{b},t_{(i-1)})|0,CM0,R_{0}\rangle
=|0,CM0,R_{0}\rangle ,$ as shown in the unitary transformation (5). Then, the unitary operation $U_{h}^{c}$ converts the desired functional state $%
|f_{r}(x_{f})\rangle =|1\rangle $ into the state $|0\rangle ,$$$\{P_{SL}(\{\varphi _{k}\},t,t_{(i-1)}+\delta
t_{b}),U_{h}^{c}\}|0,CM0,R_{0}\rangle |1\rangle |f_{r}(x_{f})\rangle$$$$=P_{SL}(\{\varphi _{k}\},t,t_{(i-1)}+\delta
t_{b})U_{h}^{c}|0,CM0,R_{0}\rangle |1\rangle |f_{r}(x_{f})\rangle$$$$=|0,CM0,R_{0}\rangle |1\rangle |0\rangle ,\qquad \text{ }t=t_{(i-1)}+\delta
t_{b}+\delta t_{h},\qquad \label{13}$$and the unitary operation $V_{h}^{c}$ further changes the initial halting state $|n_{h}\rangle =|0,CM0,R_{0}\rangle $ to the state $%
|1,CM0,R_{0}\rangle $, $$\{P_{SL}(\{\varphi _{k}\},t,t_{(i-1)}+\delta t_{b}+\delta
t_{h}),V_{h}^{c}\}|0,CM0,R_{0}\rangle |1\rangle |0\rangle$$$$=P_{SL}(\{\varphi _{k}\},t,t_{(i-1)}+\delta t_{b}+\delta
t_{h})V_{h}^{c}|0,CM0,R_{0}\rangle |1\rangle |0\rangle$$$$=|1,CM0,R_{0}\rangle |1\rangle |0\rangle ,\ \ \hspace{1.5in}\qquad \qquad
\label{14}$$where $t=t_{(i-1)}+\delta t_{b}+\delta t_{h}+\delta t_{h}^{\prime
}=t_{0i}-\delta t_{r}.$ These two unitary transformations (13) and (14) may be obtained by the relations (8) and (9), respectively, and here equation (5) has also been used. Now the state-dependent trigger pulse $P_{t}$ starts to act on the halting state $|n_{h}\rangle =|1,CM0,R_{0}\rangle $ at the time $t_{0i}-\delta t_{r}$ due to that the halting state has the internal state with $n_{h}^{\prime }=1$ and the halting-qubit atom now is in the left-hand potential well. With the help of the relation (10) the state-dependent excitation process of the trigger pulse $P_{t}$ from the halting state to the unstable state $|c_{1}\rangle $ of the control state subspace may be expressed as $$\{P_{SL}(\{\varphi _{k}\},t_{0i},t_{0i}-\delta
t_{r}),U_{tr}^{c}\}|1,CM0,R_{0}\rangle |1\rangle |0\rangle$$$$=|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle |1\rangle |0\rangle \label{15}$$where $t_{0i}=t_{(i-1)}+\delta t_{b}+\delta t_{h}+\delta t_{h}^{\prime
}+\delta t_{r}$ and $\{P_{SL}(\{\varphi _{k}\},t_{0i},t_{0i}-\delta
t_{r}),U_{tr}^{c}\}$ of the quantum circuit $Q_{c}$ is just defined as the unitary operation $U_{tr}^{c}$ (see Eq. (10)). This excitation process increases the motional energy of the halting-qubit atom in the left-hand potential well so that the following unitary quantum scattering process for the atom can take place automatically.
From the instant of time $t_{0i}$ on, the quantum circuit $Q_{c}$ really starts to execute simultaneously its own two almost independent processes: the quantum control process and the quantum computational process. The quantum control process could be really thought to start at the time $t_{0i}$ and at the initial state $|c_{1}\rangle =|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle
$ for the physical system of the halting-qubit atom. Here for convenience the state-dependent excitation process of the trigger pulse is not included in the quantum control process. In the quantum control process the halting-qubit atom with the state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ first carries out a unitary quantum scattering process in space and time. During the period $\Delta t_{0}$ of the quantum scattering process the halting-qubit atom is driven by its own motional momentum to leave the left-hand potential well and pass the intermediate potential barrier to enter into the right-hand potential well. This quantum scattering process may be expressed formally by the unitary transformation: $$\{P_{SL}(\{\varphi _{k}\},t_{mi},t_{0i}),P_{SL}(\{\varphi _{k}\},\Delta
t_{0})\}|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle |1\rangle |0\rangle \newline
\newline$$$$\equiv P_{SL}(\{\varphi _{k}\},\Delta
t_{0})|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle |1\rangle |0\rangle \qquad$$$$=|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle |1\rangle |0\rangle ,\text{ }%
t_{mi}=t_{0i}+\Delta t_{0},\ \ \label{16}$$where $\{P_{SL}(\{\varphi _{k}\},t_{mi},t_{0i}),P_{SL}(\{\varphi
_{k}\},\Delta t_{0})\}$ in the quantum circuit $Q_{c}$ is defined as $%
P_{SL}(\{\varphi _{k}\},\Delta t_{0}),$ and the duration $\Delta t_{0}$ must be long enough so that by the quantum scattering process the wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ of the halting-qubit atom in the left-hand potential well can be almost completely transferred to the wave-packet state $|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle $ of the atom in the right-hand potential well. Actually, the duration $\Delta t_{0}$ must ensure that the wave-packet spatial distance $|R_{2}(t_{mi})-R_{0}|$ is large enough such that both the wave-packet state $|1,CM0,R_{0}\rangle $ and $%
|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle $ are almost orthogonal to each other. Note that the distance $|R_{2}(t_{mi})-R_{1}(t_{0i})|$ is longer than $%
|R_{2}(t_{mi})-R_{0}|.$ This means that such a duration $\Delta t_{0}$ ensures that both the wave-packet states $|1,CM1(t_{0i}),R_{1}(t_{0i})%
\rangle $ and $|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle $ are also almost orthogonal to each other. The quantum scattering process (16) is dependent only upon the time difference $\Delta t_{0}$ rather than the instant of time $t_{mi}$ or $t_{0i},$ that is, $P_{SL}(\{\varphi _{k}\},t_{mi},t_{0i})\equiv
P_{SL}(\{\varphi _{k}\},\Delta t_{0})$, and it is also an energy conservative process that the motional energy of the final state $%
|c_{2}^{\prime }\rangle =|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle $ is really equal to that of the initial state $|c_{1}\rangle
=|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle .$ After the quantum scattering process the quantum computational process continues to execute the functional unitary operation $U_{f}^{c}$ as usual, but from the time $t_{mi}$ on, the quantum computational process is really halted in effect and meanwhile the halting-qubit atom also motions continuously in the right-hand potential well under the state-locking pulse field. This process may be expressed in terms of the unitary transformation: $$\{P_{SL}(\{\varphi _{k}\},t_{mi}+\delta
t_{f},t_{mi}),U_{f}^{c}\}|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle |1\rangle
|0\rangle$$$$\equiv P_{SL}(\{\varphi _{k}\},t_{mi}+\delta
t_{f},t_{mi})U_{f}^{c}|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle |1\rangle |0\rangle$$$$=|n_{h}^{\prime }(t_{mi}+\delta t_{f}),CM2(t_{mi}+\delta
t_{f}),R_{2}(t_{mi}+\delta t_{f})\rangle |1\rangle |0\rangle ,\quad
\label{17}$$where $|n_{h}^{\prime }(t_{mi}+\delta t_{f})\rangle $ is the internal state of the halting-qubit atom at the time $t_{mi}+\delta t_{f}.$ The functional operation $U_{f}^{c}$ does not have a real effect on the state $%
|f_{r}(x)\rangle =|0\rangle $ also due to the branch-control state $%
|b_{h}\rangle =|1\rangle .$ Here it must be ensured that the halting-qubit atom is in the right-hand potential well during the period from the time $%
t_{mi}$ to the time $t_{mi}+\delta t_{f}$ so that the wave-packet state $%
|n_{h}^{\prime }(t),CM2(t),R_{2}(t)\rangle $ with $t_{mi}\leq t\leq
t_{mi}+\delta t_{f}$ is almost orthogonal to any one of the three wave-packet states $|n_{h}^{\prime },CM0,R_{0}\rangle $ ($n_{h}^{\prime
}=0,1 $) and $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle .$ Obviously, here the width of the intermediate potential barrier should be large enough so that a wave-packet state of the halting-qubit atom in one potential well is almost orthogonal to any wave-packet state of the atom in another potential well in the double-well potential field.
From the $(i+1)-$th cycle to the end of the quantum circuit the conditional unitary operations $U_{b}^{c},$ $U_{h}^{c},$ $V_{h}^{c},$ $U_{tr}^{c}$, and $%
U_{f}^{c}$ do not really affect the quantum system, although according as the quantum circuit these unitary operations are still applied continuously to the quantum system of the quantum computer, and only the state-locking pulse field takes a real action on the halting-qubit atom in the right-hand potential well. Then the time evolution process of the quantum system from the time $t_{i}=t_{mi}+\delta t_{f}$ to the end of the quantum computational process could be generally written as $$\{P_{SL}(\{\varphi _{k}\},t_{i+j},t_{i}),(U_{T})^{j}\}|n_{h}^{\prime
}(t_{i}),CM2(t_{i}),R_{2}(t_{i})\rangle |1\rangle |0\rangle$$$$=P_{SL}(\{\varphi _{k}\},t_{i+j},t_{i})|n_{h}^{\prime
}(t_{i}),CM2(t_{i}),R_{2}(t_{i})\rangle |1\rangle |0\rangle$$$$=|n_{h}^{\prime }(t_{i+j}),CM2(t_{i+j}),R_{2}(t_{i+j})\rangle |1\rangle
|0\rangle .\text{ } \label{18}$$Here the time $t_{i+j}=t_{i}+j\Delta T$ for $0\leq j\leq m_{r}-i$ and the end time of the computational process is given by $t_{m_{r}}=t_{0}+m_{r}%
\Delta T$. From Eq. (11) and Eq. (18) one sees that before the initial halting state $|0,CM0,R_{0}\rangle $ is changed to the state $%
|1,CM0,R_{0}\rangle $ the unitary propagator $\{P_{SL}(\{\varphi
_{k}\},t_{j},t_{0}),(U_{T})^{j}\}$ acting on the initial state $%
|0,CM0,R_{0}\rangle |0\rangle |f_{r}(x_{0})\rangle $ is really equal to the single unitary operation sequence $(U_{T})^{j}$ acting on the same initial state (see Eq. (11)), but after the halting-qubit atom enters into the right-hand potential well the unitary propagator $\{P_{SL}(\{\varphi
_{k}\},t_{i+j},t_{i}),(U_{T})^{j}\}$ acting on the starting state $%
|n_{h}^{\prime }(t_{i}),CM2(t_{i}),R_{2}(t_{i})\rangle |1\rangle |0\rangle $ is really equal to the single unitary propagator $P_{SL}(\{\varphi
_{k}\},t_{i+j},t_{i})$ of the state-locking pulse field acting on the same state (see Eq. (18)). Here any wave-packet state $|n_{h}^{\prime
}(t),CM2(t),R_{2}(t)\rangle $ for $t_{i}\leq t\leq t_{m_{r}}$ of the halting-qubit atom in the right-hand potential well must be orthogonal or almost orthogonal to any one of the three wave-packet states $|n_{h}^{\prime
},CM0,R_{0}\rangle $ ($n_{h}^{\prime }=0,1$) and $%
|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ of the atom in the left-hand potential well. Therefore, the quantum program and circuit requires that the halting-qubit atom be in the right-hand potential well from the time $t_{mi}$ to the end of the quantum computational process.
Though the halting-qubit atom must be in the right-hand potential well at the end of the computational process no matter what the initial functional state $|f_{r}(x_{0})\rangle $ is with $x_{0}=0,1,...,m_{r}-1$, each possible wave-packet state of the halting-qubit atom $|n_{h}^{\prime
}(t_{m_{r}}),CM2(t_{m_{r}}),R_{2}(t_{m_{r}})\rangle $ with $%
t_{m_{r}}=t_{i}+(m_{r}-i)\Delta T$ for $i=0,1,...,m_{r}-1$ could be different in spatial position in the right-hand potential well at the end of the computational process. This is because a different initial functional state $|f_{r}(x_{0})\rangle $ corresponds to a different wave-packet state $%
|n_{h}^{\prime }(t_{m_{r}}),CM2(t_{m_{r}}),R_{2}(t_{i}+(m_{r}-i)\Delta
T)\rangle $ which is located at a different spatial position $%
R_{2}(t_{i}+(m_{r}-i)\Delta T)$ in the right-hand potential well. Here $%
n_{h}^{\prime }(t_{m_{r}})$ and $CM2(t_{m_{r}})$ with $%
t_{m_{r}}=t_{i}+(m_{r}-i)\Delta T$ each takes a same value for all different index values $i=0,1,...,m_{r}-1,$ respectively$.$ Actually, similar to the situations in the conventional halting protocol \[11, 54\], all these $m_{r}$ possible wave-packet states $\{|n_{h}^{\prime
}(t_{m_{r}}),CM2(t_{m_{r}}),R_{2}(t_{i}+(m_{r}-i)\Delta T)\rangle \}$ for $%
i=0,1,...,m_{r}-1$ should be almost orthogonal to one another. Therefore, the output wave-packet states of the quantum circuit at the end of the computational process are dependent sensitively upon the initial functional states $\{|f_{r}(x_{0})\rangle \}.$ However, as pointed out before, the quantum circuit $Q_{c}$ will not be a suitable component of the quantum search processes based on the unitary quantum dynamics \[24\] if its output state is dependent sensitively upon any initial functional state $%
|f_{r}(x_{0})\rangle .$ Evidently, if each of these $m_{r}$ possible wave-packet states $\{|n_{h}^{\prime
}(t_{m_{r}}),CM2(t_{m_{r}}),R_{2}(t_{i}+(m_{r}-i)\Delta T)\rangle \}$ at the end of the computational process can be further transferred to some desired state in a high probability close to 100% by a given unitary transformation, then the output state of the quantum circuit could be considered to be almost independent of any initial functional state. Of course, it is impossible that the unitary transformation can change all these $m_{r}$ wave-packet states to the same desired state in the probability 100%. It could be better to choose the desired state as the wave-packet state $|n_{h}\rangle =|n_{h}^{\prime },CM0,R_{0}\rangle $ $%
(n_{h}^{\prime }=0$ or $1)$ of the halting-qubit atom in the left-hand potential well as the wave-packet state is stable in the double-well potential field. Thus, it is necessary to manipulate and control coherently the halting-qubit atom by the state-locking pulse field after the halting-qubit atom enters into the right-hand potential well. The coherent manipulation process have two purposes in the quantum control process to simulate efficiently the reversible and unitary halting protocol. The first one is that the halting-qubit atom can stay in the right-hand potential well for a long time till the computational process finished after the halting-qubit atom enters into the right-hand potential well. The second is that after the computational process finished the halting-qubit atom can return the left-hand potential well from the right-hand one and the returning halting-qubit atom in the left-hand potential well is in the wave-packet state $|n_{h}^{\prime },CM0,R_{0}\rangle $ $(n_{h}^{\prime }=0$ or $1)$ with a high probability close to 100% no matter what the wave-packet state $|n_{h}^{\prime
}(t_{m_{r}}),CM2(t_{m_{r}}),R_{2}(t_{i}+(m_{r}-i)\Delta T)\rangle $ is for $%
i=0,1,...,m_{r}-1$. The coherent manipulation of the halting-qubit atom in the right-hand potential well generally starts after the computational process finished, but actually this manipulating process may start at a much earlier time $t_{mi}$ when the halting-qubit atom enters into the right-hand potential well rather than at the end of the computational process.
In this section the issues to discuss are focused on how the state-locking pulse field manipulates coherently the wave-packet states of the halting-qubit atom in the right-hand potential well of the double-well potential field so that the halting-qubit atom can stay in the right-hand potential well for a long time till the computational process finished and how the quantum control process that simulates efficiently the reversible and unitary halting protocol in the atomic physical system makes its output state insensitive to any initial functional state of the quantum circuit $%
Q_{c}$. As shown in the previous section, when the halting-qubit atom leaves the left-hand potential well and enters into the right-hand one at the time $%
t_{mi}=t_{0i}+\Delta t_{0}$ in the $i-$th cycle of the quantum circuit, the conditional unitary operations $U_{h}^{c}$ and $V_{h}^{c}$ and the state-dependent trigger pulse $P_{t}$ of the quantum circuit become unavailable. Thus, from the time $t_{mi}$ on, the quantum computational process really terminates in effect, although it runs continuously to the end of the quantum circuit. Meanwhile, the halting-qubit atom evolves continuously in the right-hand potential well under the state-locking pulse field. Note that the state-locking pulse field consists of the double-well potential field itself and the sequences of the time- and space-dependent electromagnetic field pulses which are applied only to the right-hand potential well during the computational process and then to the double-well potential field after the computational process finished. Obviously, these sequences of the time- and space-dependent electromagnetic field pulses of the state-locking pulse field have a negligible effect on any state of the halting-qubit atom when the atom is in the left-hand potential well during the computational process. Thus, it follows from the unitary transformations (17) and (18) that, from the time $t_{mi}$ on, the time evolution process of the whole quantum system of the quantum computer may be reduced to the simpler quantum control process of the halting-qubit atom in the right-hand potential well under the state-locking pulse field, $$P_{SL}(\{\varphi _{k}\},t,t_{mi})|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle$$$$\hspace{1in}=|n_{h}^{\prime }(t),CM2(t),R_{2}(t)\rangle ,\text{ }t_{mi}\leq
t. \label{19}$$Here the wave-packet state $|n_{h}^{\prime }(t),CM2(t),R_{2}(t)\rangle $ ($%
t_{mi}\leq t$) generally could be expanded in terms of the motional basis states of the halting-qubit atom in the right-hand potential well \[38–43\], $$|n_{h}^{\prime }(t),CM2(t),R_{2}(t)\rangle =\underset{n_{h}^{\prime }}{\sum }%
\underset{CM2}{\sum }a(n_{h}^{\prime },CM2,t)|n_{h}^{\prime }\rangle
|CM2,R_{2}\rangle .$$Now the first purpose of the quantum control process is to design a unitary sequence of the time- and space-dependent electromagnetic pulses and/or the shaped potential fields of the state-locking pulse field $P_{SL}(\{\varphi
_{k}\},t,$ $t_{mi})$ to manipulate the halting-qubit atom so that the atom is able to stay in the right-hand potential well till the end of the computational process. For convenience, suppose that the lowest point of the left-hand harmonic potential well is equal to the bottom of the right-hand square potential well and both are set to zero. As stated before, when the halting-qubit atom is in the unstable wave-packet state $%
|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ the total motional energy of the atom which includes the kinetic and potential energies in the left-hand harmonic potential well but not the atomic internal energy is much higher than the height of the intermediate potential barrier in the double-well potential field. Since the quantum scattering process is an energy-conservative process the total motional energy of the halting-qubit atom remains unchanged when the halting-qubit atom enters into the right-hand potential well from the left-hand one, but it is completely converted into the kinetic energy as the potential energy of the halting-qubit atom is zero in the right-hand square potential well. Suppose that the relativistic effect is negligible for the motional halting-qubit atom in the double-well potential field. Then the motional velocity of the halting-qubit atom is given by $%
v_{h}=\sqrt{2E_{h}/m_{h}}$ at the time $t_{mi}$ after the atom enters into the right-hand potential well, where $E_{h}$ and $m_{h}$ are the total motional energy and mass of the halting-qubit atom, respectively. Therefore, the motional velocity $v_{h}$ of the halting-qubit atom could become very large when the atom enters into the right-hand potential well from the left-hand one. Since the geometric length of the right-hand square potential well is limited it is impossible to keep the halting-qubit atom in the right-hand potential well for a long time required by the computational process if the motional velocity $v_{h}$ is very large. Thus, the motional velocity of the halting-qubit atom must be decelerated greatly by the state-locking pulse field so that the halting-qubit atom does not leave in a short time the right-hand potential well. This decelerating process could be achieved by the unitary sequence of the time- and space-dependent laser light pulses of the state-locking pulse field. The decelerating process of the halting-qubit atom is really very similar to the conventional atomic laser cooling processes \[36\]. The interactions between the halting-qubit atom in motion and the laser light pulse field become important in the decelerating process, while it is known that the dipole force of the laser light pulse field exerting a motional atom plays an important role in the atomic laser cooling processes \[36b\]. There are a variety of the atomic laser cooling methods and techniques which have been discovered and developed in the past decades and used extensively to cool an atomic ensemble to an extremely low temperature \[36\], but most of these atomic laser cooling methods are non-unitary. Thus, not all these atomic laser cooling methods are suitable for building up the unitary decelerating sequence because the decelerating process for the halting-qubit atom must be a unitary process. This is quite different from a conventional atomic laser cooling process in an atomic ensemble. Only when an atomic laser cooling method is unitary or can be made unitary can it be exploited to decelerate the halting-qubit atom in the quantum control process. Thus, only the coherent atomic laser cooling methods could be used to build up the unitary decelerating sequence. Another difference between an atomic laser cooling process and the unitary decelerating process is that the unitary decelerating process is simpler as the current atomic physical system is the pure quantum-state system of an atomic ion or a neutral atom in the double-well potential field, while it is usually more complex to cool an atomic ensemble by an atomic laser cooling technique. The third difference is that it could not be necessary to decelerate the halting-qubit atom to zero velocity, while the target of a conventional laser cooling technique is to cool an atomic ensemble to an extremely low temperature as close to zero degree as possible. Therefore, the unitary decelerating process for the halting-qubit atom is relatively easy to be achieved by a coherent atomic laser cooling technique.
The mechanisms for the laser cooling in an atomic ensemble have been studied extensively and thoroughly in the past years \[36\]. However, in order to investigate the mechanism of the unitary decelerating process for the halting-qubit atom here the basic atomic-laser-cooling mechanism is introduced briefly. A conventional atomic laser cooling method usually consists of both the atomic optical-pumping process (or the atomic optical-absorption process) and the atomic optical-emission process \[36c\]. The optical pumping process is that an atom under cooling is excited from the ground state to the excited state by absorbing photons from the laser light field, while the optical emission process is that the atom in the excited state returns the ground state by emitting photons to the laser light field or environment. The optical pumping process generally may be made unitary easily, but the optical emission process usually is simply chosen as a spontaneous and random process in a conventional laser cooling technique partly due to that the lifetime of the atomic excited state usually is very short. Hence the optical emission process usually is a non-unitary process in an atomic ensemble for a conventional laser cooling technique. Suppose that a free atomic ion or neutral atom with the mass $m$ in the ground state is irradiated by a laser light field and makes a transition from the ground state to the excited state by absorbing a photon from the laser light field. For convenience, here only nonrelativistic limit is considered for the atomic motion. The optical pumping process generally obeys the energy-, momentum-, and angular momentum-conservative laws. Denote that before the transition the atom in the ground state has the internal-state energy $E_{a}$, the kinetic energy $p_{a}^{2}/(2m)$, the momentum $p_{a}=m\mathbf{v}_{a}$, and the angular momentum $J_{a}$, respectively, and the photon that will be absorbed by the atom has the photonic energy $E_{c}=\hslash \omega ,$ the momentum $p_{c}=\hslash \mathbf{%
k}$ ($|p_{c}|=\hslash \omega /c$), and the angular momentum $J_{c}$, respectively. After the transition the photon is absorbed by the atom and hence the photonic energy, momentum, and angular momentum are transferred to the atom. The atom now is in the excited state. Suppose that after the transition the atom in the excited state has the internal-state energy $%
E_{b} $, the kinetic energy $E_{b}=p_{b}^{2}/(2m),$ the momentum $p_{b}=m%
\mathbf{v}_{b},$ and the angular momentum $J_{b},$ respectively. Then the energy conservation before and after the transition shows that there holds the relation: $$E_{a}+p_{a}^{2}/(2m)+\hslash \omega =E_{b}+p_{b}^{2}/(2m) \label{20}$$where $\omega $ is the photonic frequency. The momentum conservation leads to that $$m\mathbf{v}_{a}+\hslash \mathbf{k}=m\mathbf{v}_{b}, \label{21}$$where $\mathbf{k}$ ($|\mathbf{k|=}\omega /c$) is the wave vector of the photon before the transition and $\mathbf{v}_{a}$ and $\mathbf{v}_{b}$ are the motional velocity vectors of the atom before and after the transition, respectively. The angular momentum conservation between the angular momentum $J_{a}$ of the atom and the photonic angular momentum $J_{c}$ before the transition and the angular momentum $J_{b}$ of the atom after the transition will not be discussed in detail here. If the optical pumping process (or the photon absorption process) for the atom is one-dimensional, then the wave vector $\mathbf{k}$ of the photon is either co-direction to the motional velocity $\mathbf{v}_{a}$ of the atom or opposite to the velocity $\mathbf{v}%
_{a}$ before the transition. For the first case that both the wave propagating of the laser light field with the vector $\mathbf{k}$ and the motion of the atom with the velocity $\mathbf{v}_{a}$ are co-direction to each other the motional velocity of the atom after the transition is given by $$v_{b}=v_{a}+\hslash k/m>v_{a}, \label{22}$$hence the motional atom is accelerated by the copropagating laser light field, while for the second case that the laser light field propagates in the opposite direction to the atomic motion the atomic motional velocity after the transition is written as $$v_{b}=v_{a}-\hslash k/m<v_{a}, \label{23}$$hence the motional atom is decelerated by the opposite propagating laser light field. Obviously, here the atom is slowed down by $\hslash k/m$ when the atom absorbs a photon from the opposite propagating laser light field. When the atom is in the excited state it no longer absorbs any photons from the opposite propagating laser light field and hence can not be further slowed down. One of the schemes to decelerate further the atom is that the atom in the excited state first jumps back to the ground state without changing significantly its total motional momentum, and then it absorbs a photon from the opposite propagating laser light field again and hence is decelerated further. Note that the process that the excited-state atom jumps back to the ground state by emitting photons to the laser light field or its environment is just the atomic optical-emission process. Thus, the atomic laser cooling process consists of a number of the atomic optical absorption-emission cycles that the atom in the ground state absorbs a photon to make a transition to the excited state and then jumps back to the ground state from the excited state by emitting photons. The optical emission process is either a spontaneous emission process in a random form or the stimulated emission process in a coherent form. In the spontaneous emission process photons are emitted in a random form by the atoms in the excited state so that the total momentum of the emitting photons is zero and hence the atomic motional momentum does not change significantly after the emission process. Therefore, the atom is slowed down basically by the optical pumping process in every optical absorption-emission cycle if the optical emission process is spontaneous and random. Most of the conventional atomic laser cooling methods and techniques use the spontaneous optical-emission process as their key component to cool an atom ensemble. Every optical absorption-emission cycle can make the atom to be slowed down by $\hslash k/m$ and the atom can be slowed down continuously by a sequence of the optical absorption-emission cycles. However, the spontaneous optical-emission process of the conventional laser cooling methods is not allowed due to its own non-unitarity if these laser cooling methods are used to slow down the halting-qubit atom in the quantum control process. On the other hand, the coherent optical-emission process is different from the spontaneous optical-emission process in that the coherent optical-emission process may be a unitary process. The coherent optical-emission process generally may be stimulated by an external laser light field \[5a\]. The momentum of the emitting photons from the atom in the excited state in the coherent optical-emission process is not zero on average and hence can make a significant contribution to the motional momentum of the atom after the emission process. The coherent atomic optical-emission process still obeys the energy-, momentum-, and angular momentum-conservative laws. Therefore, if the emitting photon travels along the same direction to the atomic motion, then the atom will lose part of its motional momentum after the emission process and hence is slowed down. In the optical-emission process not only the atomic internal energy ($E_{b}-E_{a}$) of the excited state is transferred to the photonic energy but also part of the motional energy of the atom in the excited state is converted into the photonic energy. However, the atom will receive a recoil momentum from the emitting photon and hence is accelerated after the emission process if the emitting photon travels along the opposite direction to the atomic motion. In this atomic optical-emission process the atomic internal energy of the excited state is transferred partly to the photonic energy and partly to the atomic kinetic energy at the same time. In the quantum control process the halting-qubit atom in the right-hand potential well must be first slowed down greatly so that it is able to stay in the right-hand potential well for a long time till the end of the computational process in the quantum circuit $Q_{c}$, and then the atom is sped up in a unitary form after the computational process finished such that the atom can return to the left-hand potential well. The coherent atomic laser cooling methods and techniques therefore provide a possible way to generate both the decelerating and accelerating processes for the halting-qubit atom in the quantum control process.
A conventional laser cooling method based on the optical absorption-emission (spontaneous) cycles usually is realized more easily than a coherent one in an atomic ensemble. In general, the spontaneous optical-emission process from the atomic excited state to the ground states in the atomic ensemble occurs easily in nature as the atomic excited state usually has a much shorter lifetime than those ground states of the atom. This in turn implies that a coherent atomic laser cooling method could be more complex as the non-unitary spontaneous optical emission must be avoided in the coherent laser cooling process. The stimulated Raman adiabatic passage ($STIRAP$) laser cooling method is one of the important atomic laser cooling methods. The $STIRAP$ method has been used extensively to cool an atomic ensemble to an extremely low temperature \[42, 43\], to cool a trapped atomic ion to the ground state for quantum computation \[30, 44\], to manipulate a coherent atomic beam in the atomic interferometry \[38, 39, 40, 41\], and to prepare and manipulate a nonclassical motional state in a trapped-ion physical system \[37, 45\]. In particular, the $STIRAP$ laser cooling method could be used conveniently to cool a multi-level atomic ensemble with many internal states to an extremely low temperature. The coherent $STIRAP$ laser cooling (or decelerating) method could be a better candidate to avoid the non-unitary spontaneous optical emission of the atom from the excited state to the ground states. This is because the cooling (or decelerating) atom does not stay in the excited state at all or could stay in the excited state in a much shorter time than the lifetime of the excited state during the coherent $STIRAP$ laser cooling process if the Raman adiabatic laser pulses are detuned properly from the excited state. Since an adiabatic laser beam usually has a much wider frequency bandwidth than a conventional $CW$ laser beam the $STIRAP$ laser cooling method is able to take the Doppler effect into account conveniently during the atomic laser cooling process. It is well known that a general Raman adiabatic laser pulse consists of a pair of the adiabatic laser beams with the specific characteristic parameters. Generally, the characteristic parameters for the adiabatic laser beams of a Raman adiabatic laser pulse include the carrier frequencies and detunings, the frequency bandwidths, the amplitudes and phases of the adiabatic laser light fields, the laser-beam durations, the propagation directions and polarizations (e.g., $\sigma _{+}$ or $\sigma _{-}$), and the spatial action positions and zones. Suppose now that the states $|g_{1}\rangle ,$ $%
|g_{2}\rangle ,$ and $|n_{e}\rangle $ are three different internal states of the cooling atom and their corresponding wave-packet states of the atom are written as $|g_{1},CM_{1},R_{1}\rangle ,$ $|g_{2},CM_{2},R_{2}\rangle ,$ and $|n_{e},CM_{e},R_{e}\rangle ,$ respectively. These two internal states $%
|g_{1}\rangle $ and $|g_{2}\rangle $ usually may be chosen as a pair of ground internal states or two lowest energy-level internal states of the atom, while the internal state $|n_{e}\rangle $ may be an excited state whose energy level is much higher than those of the internal states $%
|g_{1}\rangle $ and $|g_{2}\rangle .$ In the coherent $STIRAP$ laser cooling method an adiabatic laser beam $A$ may be used to pump the atom from the ground internal state $|g_{1}\rangle $ to the excited state $|n_{e}\rangle $ and at the same time another adiabatic laser beam $B$ is applied to stimulate the atom in the excited state $|n_{e}\rangle $ to jump back to the internal state $|g_{2}\rangle $ \[42, 43, 44\]. The coherent $STIRAP$ process may be formally expressed in terms of the unitary transition process: $$|g_{1},CM_{1},R_{1}\rangle \overset{A}{\leftrightarrow }|n_{e},CM_{e},R_{e}%
\rangle \overset{B}{\leftrightarrow }|g_{2},CM_{2},R_{2}\rangle .$$Obviously, the carrier frequency of the adiabatic laser beam $A$ should be close to the resonance frequency between the ground state $|g_{1}\rangle $ and the excited state $|n_{e}\rangle ,$ while the carrier frequency for the adiabatic laser beam $B$ is close to the resonance frequency between the ground state $|g_{2}\rangle $ and the excited state $|n_{e}\rangle .$ In order to avoid occurring the non-unitary spontaneous optical emission for the atom from the excited state $|n_{e}\rangle $ to the ground states both the adiabatic laser beams $A$ and $B$ are detuned properly from the excited state $|n_{e}\rangle .$ For example, if the unitary state transfer $%
|g_{1}\rangle \leftrightarrow |g_{2}\rangle $ is achieved by the conventional $CW$ laser light irradiation, that is, the internal state $%
|g_{1}\rangle $ is first transferred completely to the excited state $%
|n_{e}\rangle $ and then to $|g_{2}\rangle $ by the $CW$ irradiation method, then the decoherence effect usually affects largely the state transfer since lifetime of the excited state $|n_{e}\rangle $ usually is very short and also much shorter than those of the ground states $|g_{1}\rangle $ and $%
|g_{2}\rangle $, whereas the $STIRAP$ method can avoid such decoherence effect on the state transfer. While the direct transition from the ground state $|g_{1}\rangle $ $(|g_{2}\rangle )$ to another ground state $%
|g_{2}\rangle $ $(|g_{2}\rangle )$ is prohibited under the $CW$ laser light irradiation, the coherent $STIRAP$ method is a better scheme to excite indirectly the transition between these two ground states. Thus, the coherent $STIRAP$ method has some advantages over the conventional $CW$ irradiation method to transfer the ground state $|g_{1}\rangle $ $%
(|g_{2}\rangle )$ to another ground state $|g_{2}\rangle $ $(|g_{2}\rangle )$ in a unitary form. Obviously, the effective spatial bandwidth ($ESB$) for the Raman adiabatic laser pulse must be greater than the spatial displacement ($SD$) of the atom during the Raman adiabatic laser pulse. The spatial displacement $(SD)$ is not more than the pulse duration ($t_{p}$)of the Raman adiabatic laser pulse times the maximum velocity ($v_{M}$) of the atom during the Raman adiabatic laser pulse, that is, $SD<v_{M}\times
t_{p}\leq ESB.$
While the ground internal state $|g_{1}\rangle $ ($|g_{2}\rangle $) is completely transferred to the state $|g_{2}\rangle $ ($|g_{1}\rangle $) by the Raman adiabatic laser pulse, the corresponding motional state $%
|CM_{1},R_{1}\rangle $ ($|CM_{2},R_{2}\rangle $) of the atom is also changed to another motional state $|CM_{2},R_{2}\rangle $ ($|CM_{1},R_{1}\rangle $). During the coherent $STIRAP$ process the atom could be either sped up or slowed down and this is mainly dependent upon the characteristic parameter settings for these two adiabatic laser beams $A$ and $B$ of the Raman adiabatic laser pulse and also the initial atomic motional velocity and direction, as mentioned earlier. An example is given below. Suppose that the atom is in the wave-packet state $|g_{1},CM_{1},R_{1}\rangle $ at the initial time, the propagating directions of these two adiabatic laser beams $%
A$ and $B$ are opposite to each other, and the beam $A$ propagates in the opposite direction to the atomic motion. Then the atom will be slowed down by $\hslash k_{A}/m+\hslash k_{B}/m$ when the wave-packet state $%
|g_{1},CM_{1},R_{1}\rangle $ is transferred to $|g_{2},CM_{2},R_{2}\rangle $ by the Raman adiabatic laser pulse \[43\]. Here suppose that the initial atomic velocity is much greater than $\hslash k_{A}/m+\hslash k_{B}/m.$ This decelerating process could be understood intuitively: $(i)$ when the state $%
|g_{1},CM_{1},R_{1}\rangle $ is induced a transition to the excited state $%
|n_{e},CM_{e},R_{e}\rangle $ by the laser beam $A$ the atom is slowed down by $\hslash k_{A}/m$ because the atom absorbs the photonic momentum $\hslash
k_{A}$ from the laser light field of the beam $A$ which travels along the opposite direction to the atomic motion; $(ii)$ when the atom is stimulated by the laser beam $B$ to jump to the state $|g_{2},CM_{2},R_{2}\rangle $ from the excited state $|n_{e},CM_{e},R_{e}\rangle $ it releases the momentum $\hslash k_{B}$ to the laser light field of the beam $B$ and the atom therefore is slowed down further by $\hslash k_{B}/m$ as the atomic motional direction is the same as the propagating direction of the beam $B$. Evidently, the atom can also be sped up when the atomic wave-packet state $%
|g_{1},CM_{1},R_{1}\rangle $ is transferred to $|g_{2},CM_{2},R_{2}\rangle $ by the Raman adiabatic laser pulse with the proper characteristic parameter settings. Furthermore, the atom may be slowed down or sped up continuously by many Raman adiabatic laser pulses with the proper characteristic parameter settings. For example, suppose that one wants the atom to be decelerated further after the atom is slowed down by $\hslash
k_{A}/m+\hslash k_{B}/m$ by the Raman adiabatic laser pulse $R(A,B)$ with the beams $A$ and $B.$ Then one may apply another Raman adiabatic laser pulse $R(A_{1},B_{1})$ with the beams $A_{1}$ and $B_{1}$ to the state $%
|g_{2},CM_{2},R_{2}\rangle $ to decelerate further the atom$.$ Since both the spatial positions $R_{1}$ and $R_{2}$ are different for these two wave-packet states $|g_{1},CM_{1},R_{1}\rangle $ and $|g_{2},CM_{2},R_{2}%
\rangle $ the applying spatial position $(R_{2})$ of the Raman adiabatic laser pulse $R(A_{1},B_{1})$ is different from that one ($R_{1}$) of the Raman adiabatic laser pulse $R(A,B).$ Here the adiabatic laser beam $A_{1}$ should travel along the opposite direction to the atomic motion, while the beam $B_{1}$ propagates in the opposite direction to the beam $A_{1}$. Then the atom is decelerated by $\hslash k_{A_{1}}/m+\hslash k_{B_{1}}/m$ again after the wave-packet state $|g_{2},CM_{2},R_{2}\rangle $ is transferred to another state $|g_{1},CM_{3},R_{3}\rangle $ by the Raman adiabatic laser pulse $R(A_{1},B_{1}):|g_{2},CM_{2},R_{2}\rangle \overset{A_{1}}{%
\leftrightarrow }|n_{e},CM_{e}^{\prime },R_{e}^{\prime }\rangle \overset{%
B_{1}}{\leftrightarrow }|g_{1},CM_{3},R_{3}\rangle .$ The unitary decelerating (or accelerating) sequence of the state-locking pulse field used to decelerate (or accelerate) the halting-qubit atom in the right-hand potential well is built up out of these coherent Raman adiabatic laser pulses with the proper characteristic parameter settings. Obviously, both the unitary decelerating and accelerating sequences are time- and space-dependent. The conversion efficiency from one internal state ($%
|g_{1}\rangle )$ to another internal state ($|g_{2}\rangle )$ measures the performance of a coherent Raman adiabatic laser pulse. A good-performance Raman adiabatic laser pulse should be able to convert completely the internal state $|g_{1}\rangle $ ($|g_{2}\rangle $) into the state $%
|g_{2}\rangle $ ($|g_{1}\rangle $). It has been shown theoretically \[46, 38\] that in a three-state atomic system a ground internal state $|g_{1}\rangle $ ($|g_{2}\rangle $) can be transferred completely to another ground state $%
|g_{2}\rangle $ ($|g_{1}\rangle $) in a unitary form through the excited state $|n_{e}\rangle $ by a Raman adiabatic laser pulse with the proper characteristic parameter settings.
The halting-qubit atom generally may be chosen as a multi-level atom with many internal states in addition to the two internal states $%
\{|n_{h}^{\prime }\rangle ,$ $n_{h}=0,1\}$ of the halting quantum bit. Now the coherent $STIRAP$ method is used to manipulate the halting-qubit atom after the atom enters into the right-hand potential well. Here in the $%
STIRAP $ method these two internal states $\{|n_{h}^{\prime }\rangle ,$ $%
n_{h}^{\prime }=0,1\}$ could be conveniently set to the internal states $%
|g_{1}\rangle $ and $|g_{2}\rangle ,$ respectively, and the excited state $%
|n_{e}\rangle $ to some specific excited electronic state of the halting-qubit atom. A unitary decelerating sequence $U_{D}(\{\varphi
_{k}\},t_{mi}+T_{D},t_{mi})$ consisting of the Raman adiabatic laser pulses with the proper parameter settings $\{\varphi _{k}\}$ then is constructed to decelerate the halting-qubit atom when the atom enters into the right-hand potential well at the time $t_{mi}$, where $T_{D}$ is the total duration of the unitary decelerating sequence. The total duration $T_{D}$ must be much shorter than the period $\Delta T$ of each cycle of the quantum circuit. Note that there are $m_{r}$ possible different times $\{t_{mi},$ $%
i=1,2,...,m_{r}\}$ for the quantum circuit $Q_{c}$. The halting-qubit atom may enter into the right-hand potential well at any time $t_{mk}$ of these $%
m_{r}$ possible times $\{t_{mi}\}$ for $k=1,2,...,m_{r}.$ In order to decelerate the halting-qubit atom the unitary decelerating sequence must be applied at every time $t_{mi}$ of these $m_{r}$ possible times $\{t_{mi}\}$ in the quantum circuit. As known in the previous sections, the halting-qubit atom completely enters into the right-hand potential well at the time $%
t_{mi}=t_{0i}+\Delta t_{0}$ in the $i-$th cycle of the quantum circuit and at the time $t_{mi}$ the halting-qubit atom is in the wave-packet state $%
|c_{2}^{\prime }\rangle =|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle $. Then the unstable wave-packet state $|c_{2}^{\prime }\rangle $ will be changed to the stable wave-packet state $|c_{2}(t_{mi}+T_{D})\rangle $ of the control state subspace after the unitary decelerating sequence $U_{D}(\{\varphi
_{k}\},t_{mi}+T_{D},t_{mi})$ acts on the halting-qubit atom at the time $%
t_{mi}$ in the right-hand potential well, $$\begin{aligned}
|c_{2}(t_{mi}+T_{D})\rangle &=&U_{D}(\{\varphi
_{k}\},t_{mi}+T_{D},t_{mi})|1,CM2(t_{mi}),R_{2}(t_{mi})\rangle \\
&=&|0,CM2(t_{mi}+T_{D}),R_{2}(t_{mi}+T_{D})\rangle ,\end{aligned}$$meanwhile the initial motional velocity $v_{h}$ of the halting-qubit atom at the time $t_{mi}$ is slowed down to the velocity $v_{0}<<v_{h}$ by the unitary decelerating sequence and the initial internal state $|1\rangle $ is also changed to $|0\rangle $. Here it is important that after the halting-qubit atom is acted on by the unitary decelerating sequence it is no longer acted on by next unitary decelerating sequences. Then the motional velocity $v_{0}$ of the halting-qubit atom must be greater than zero so that the halting-qubit atom itself can leave in the velocity $v_{0}$ the effective spatial action zone of the unitary decelerating sequence before next decelerating sequence starts to apply at the time $t_{m,i+1}=t_{mi}+%
\Delta T$. The spatial displacement of the halting-qubit atom is given by $%
SD=v_{0}\times (\Delta T-T_{D})$ during the period $(\Delta T-T_{D})$ from the time $t_{mi}+T_{D}$ after the atom is acted on by the unitary decelerating sequence to the time $t_{m,i+1}$ before next decelerating sequence is applied. The spatial displacement $SD$ must be large enough to ensure that the entire wave-packet state $|c_{2}(t_{m,i+1})\rangle $ of the halting-qubit atom at the time $t_{m,i+1}$ is outside the effective spatial action zone of the unitary decelerating sequence, and hence it is also much greater than the effective wave-packet spread of the wave-packet state $%
|c_{2}(t_{m,i+1})\rangle $. Now both the atomic motional velocity $v_{0}$ and energy $E_{0}=mv_{0}^{2}/2$ are much less than the initial velocity $%
v_{h}=\sqrt{2E_{h}/m_{h}}$ and energy $E_{h},$ respectively. The wave-packet state $|c_{2}(t_{mi}+T_{D})\rangle $ is stable in the sense that the atomic motional energy $E_{0}$ of the wave-packet state is much lower than the height of the intermediate potential barrier in the double-well potential field. If now the halting-qubit atom goes in the velocity $v_{0}$ through a fixed distance $\Delta R$ in the right-hand potential well it spends the time equal to $\Delta R/v_{0}.$ Because $v_{h}>>v_{0}$ this time interval $%
\Delta R/v_{0}$ is much longer than the time period $\Delta R/v_{h}$ during which the atom passes the same distance $\Delta R$ in the initial velocity $%
v_{h}$. Thus, one may imagine that the wave-packet state of the halting-qubit atom is locked in the right-hand potential well for a long time $(\Delta R/v_{0})$ after the atom is slowed down greatly by the unitary decelerating sequence. As required by the quantum program and circuit, the halting-qubit atom should stay in the right-hand potential well until the end time ($t_{m_{r}}$) of the computational process. Then the time interval $%
\Delta R/v_{0}\geq t_{m_{r}}-(t_{m1}+T_{D}),$ where the time $t_{m1}$ is the earliest one among all these $m_{r}$ possible times $\{t_{mi}\}.$ For convenience, suppose $\delta t_{f}>T_{D}.$ Then at the end time $%
t_{m_{r}}=t_{0}+m_{r}\Delta T$ of the computational process the center-of-mass spatial position $R_{2}(t_{m_{r}})$ of the wave-packet state $%
|c_{2}(t_{m_{r}})\rangle $ of the halting-qubit atom is given by $$R_{2,i}(t_{m_{r}})=R_{2}(t_{mi}+T_{D})+v_{0}\times (t_{m_{r}}-t_{mi}-T_{D}),%
\text{ }1\leq i\leq m_{r}, \label{24}$$when the halting-qubit atom enters into the right-hand potential well at the time $t_{mi}$ in the $i-$th cycle of the quantum circuit. Here the spatial position $R_{2}(t_{m_{r}})$ is denoted as $R_{2,i}(t_{m_{r}})$ so as to show explicitly that the position is dependent of the cyclic index value $%
i=1,2,...,m_{r}$. Though each possible spatial position $R_{2}(t_{mi}+T_{D})$ is the same for the index value $i=1,2,...,m_{r}$ just like $R_{2}(t_{mi}),$ the wave-packet spatial position $R_{2,i}(t_{m_{r}})$ at the end time $%
t_{m_{r}}$ of the computational process is different for a different index value $i$. This is because the halting-qubit atom stays in the right-hand potential well for a longer time and hence passes a longer spatial distance before the end time $t_{m_{r}}$ of the computational process if it enters into the right-hand potential well at an earlier time $t_{mi}$. The maximum and minimum wave-packet spatial positions for the halting-qubit atom at the end time $t_{m_{r}}$ correspond to the halting-qubit atom entering into the right-side potential well in the first and the last cycle of the quantum circuit, respectively. Evidently, these $m_{r}$ possible different wave-packet spatial positions of the halting-qubit atom at the end time $%
t_{m_{r}}$ satisfy the following inequality: $$R_{2,m_{r}}(t_{m_{r}})<...<R_{2,2}(t_{m_{r}})<R_{2,1}(t_{m_{r}}). \label{25}$$Here as usual the $+x$ coordinate direction is defined as from the left-hand potential well toward the right-hand one in the double-well potential field. The inequality (25) is also correct for the case $\delta t_{f}\leq T_{D}$. Of course, in this case the atomic wave-packet state and its spatial position $R_{2,i}(t_{m_{r}}+T_{D}-\delta t_{f})$ at the time $%
t_{m_{r}}+T_{D}-\delta t_{f}$ correspond to the wave-packet state and its spatial position $R_{2,i}(t_{m_{r}})$ at the end time $t_{m_{r}}$ in the case $\delta t_{f}>T_{D},$ respectively. Actually, the time $%
t_{m_{r}}+T_{D}-\delta t_{f}$ is the end time of the total quantum circuit consisting of the quantum circuit $Q_{c}$ and the unitary decelerating sequence for the case $\delta t_{f}\leq T_{D}$, while if $\delta t_{f}>T_{D}$ the end time of the total quantum circuit is really just the end time $%
t_{m_{r}}$ of the single quantum circuit $Q_{c}$. Hereafter only the situation $\delta t_{f}>T_{D}$ is considered. Obviously, the halting-qubit atom at the time $t_{m_{r}}$ is always in the spatial region $%
[R_{2,m_{r}}(t_{m_{r}})-\delta R(t_{m_{r}})/2,$ $R_{2,1}(t_{m_{r}})+\delta
R(t_{m_{r}})/2]$ of the right-hand potential well, that is, any center-of-mass spatial position $R_{2,i}(t_{m_{r}})$ of the wave packet state $|c_{2}(t_{m_{r}})\rangle $ for $i=1,2,...,m_{r}$ satisfies: $%
R_{2,i}(t_{m_{r}})\in \lbrack R_{2,m_{r}}(t_{m_{r}}),$ $R_{2,1}(t_{m_{r}})].$ Here for convenience suppose that the spatial shape of the wave-packet state $|c_{2}(t_{m_{r}})\rangle $ is symmetrical and $\delta R(t_{m_{r}})$ is the effective spatial spread of the wave-packet state at the time $t_{m_{r}}$. Since the spatial distance between any two nearest wave-packet states takes the same value: $\Delta
R_{2,i,i+1}(t_{m_{r}})=R_{2,i}(t_{m_{r}})-R_{2,i+1}(t_{m_{r}})=v_{0}\times
\Delta T$ for $i=1,2,...,m_{r}-1,$ as shown in Eq. (24), these $m_{r}$ possible wave-packet states distribute uniformly in the spatial region $%
[R_{2,m_{r}}(t_{m_{r}})-\delta R(t_{m_{r}})/2,$ $R_{2,1}(t_{m_{r}})+\delta
R(t_{m_{r}})/2]$ of the right-hand potential well. In general, it follows from Eq. (24) that the spatial distance between the spatial positions $%
R_{2,i}(t_{m_{r}})$ and $R_{2,j}(t_{m_{r}})$ $(1\leq i<j\leq m_{r})$ of the halting-qubit atom at the time $t_{m_{r}}$ can be calculated by $$\Delta
R_{2,i,j}(t_{m_{r}})=R_{2,i}(t_{m_{r}})-R_{2,j}(t_{m_{r}})=v_{0}(j-i)\Delta
T. \label{26}$$Here the time difference $(j-i)\Delta T$ is the duration between the halting-qubit atom entering into the right-hand potential well in the $j-$th cycle and the $i-$th cycle ($j>i)$ in the quantum circuit. Obviously, the maximum spatial distance which is the dimensional size of the spatial region $[R_{2,m_{r}}(t_{m_{r}}),$ $R_{2,1}(t_{m_{r}})]$ is given by $\Delta
R_{2,1,m_{r}}(t_{m_{r}})=v_{0}(m_{r}-1)\Delta T.$ From the end time $%
t_{m_{r}}$ on, there are no longer any unitary operation of the quantum circuit and any unitary decelerating sequence applying to the whole quantum system of the quantum computer. However, in order that the output state of the reversible and unitary halting protocol is not dependent sensitively upon any initial functional state in the quantum circuit the wave-packet state $|c_{2}(t_{m_{r}})\rangle =|0,CM2(t_{m_{r}}),R_{2,i}(t_{m_{r}})\rangle
$ of the halting-qubit atom must be changed back to the stable halting state such as the state $|1,CM0,R_{0}\rangle $ in a high probability. Thus, the halting-qubit atom must ultimately return the left-hand potential well from the right-hand one after the computational process finished. Here the control state subspace $S(C)$ in the atomic physical system consists of a series of wave-packet states and is not a two-state subspace. Then the state $|c_{2}\rangle $ of the control state subspace $S(C)$ in the quantum program $Q_{c}$ really corresponds to these wave-packet states of the halting-qubit atom in the right-hand potential well and also the stable halting state $%
|1,CM0,R_{0}\rangle $ finally.
One possible scheme to force the halting-qubit atom in the right-hand potential well to return to the left-hand potential well is to increase the atomic motional energy and invert the motional direction of the atom. Now a unitary accelerating sequence which consists of the Raman adiabatic laser pulses with the proper characteristic parameter settings is constructed to speed up in a unitary form the halting-qubit atom in the right-hand potential well after the computational process and the unitary decelerating sequence finished. The characteristic parameter settings for the Raman adiabatic laser pulses of the unitary accelerating sequence are clearly different from those of the unitary decelerating sequence. A unitary accelerating process could be thought of as the inverse process of a unitary decelerating process except the atomic motional direction and the spatial action zone. There are two purposes for the unitary accelerating process to speed up the halting-qubit atom. The first purpose is simply that after the halting-qubit atom is sped up by the unitary accelerating sequence it hits the right-hand wall of the right-hand potential well to change its motional direction and then returns to the left-hand potential well in a higher velocity so that the halting-qubit atom can pass the intermediate potential barrier to arrive in the left-hand potential well in a shorter time. Here define the arriving time ($T_{i},$ $i=1,2,...,m_{r};$ see below) as the instant of time at which the entire effective wave-packet state of the halting-qubit atom enters into the left-hand potential well from the right-hand one and moreover the center-of-mass position of the wave-packet state is some given spatial position (e.g., $R_{1}(t_{0i})$) within the left-hand potential well. For a heavy atom the wave-packet picture in quantum mechanics is very similar to the classical particle picture \[5a\]. Then from viewpoint of the particle picture it could be better to choose the given spatial position such that the mean motional speed and kinetic energy of the halting-qubit atom is zero at the given spatial position within the left-hand potential well, that is, at the given spatial position the total motional energy of the halting-qubit atom is pure potential energy. Evidently, when the halting-qubit atom arrives in the left-hand potential well there are $m_{r}$ possible different arriving times for the halting-qubit atom with the $m_{r}$ possible wave-packet states $%
\{|0,CM2(t_{m_{r}}),R_{2,i}(t_{m_{r}})\rangle \}$ ($i=1,2,...,m_{r}$) at the end time $t_{m_{r}}$ of the computational process. Each arriving time corresponds one-to-one to a possible wave-packet state ($%
|0,CM2(t_{m_{r}}),R_{2,i}(t_{m_{r}})\rangle $) which locates at a different spatial position ($R_{2,i}(t_{m_{r}})$) in the right-hand potential well. Generally, the first wave-packet state $|0,CM2(t_{m_{r}}),R_{2,1}(t_{m_{r}})%
\rangle $ will arrive in the left-hand potential well at the earliest time, while the last one $|0,CM2(t_{m_{r}}),R_{2,m_{r}}(t_{m_{r}})\rangle $ enters into the left-hand potential well at the latest time. Then the second purpose is particularly important for the quantum control process in that the unitary accelerating sequence is really used to shorten greatly any time differences among the $m_{r}$ possible different arriving times for the halting-qubit atom. As pointed out earlier, all these $m_{r}$ possible wave-packet states $\{|0,CM2(t_{m_{r}}),R_{2,i}(t_{m_{r}})\rangle \}$ of the halting-qubit atom at the end time $t_{m_{r}}$ are within the spatial region $[R_{2,m_{r}}(t_{m_{r}})-\delta R(t_{m_{r}})/2,$ $R_{2,1}(t_{m_{r}})+\delta
R(t_{m_{r}})/2]$ in the right-hand potential well. In order that any one of these $m_{r}$ possible wave-packet states can be changed back to the stable halting state $|n_{h}^{\prime },CM0,R_{0}\rangle $ $(n_{h}^{\prime }=0$ or $%
1)$ in a high probability each of these $m_{r}$ possible wave-packet states may be acted on by the same unitary accelerating sequence such that any time differences among these $m_{r}$ possible arriving times can be shorten greatly. Here the effective width of the spatial action zone of every Raman adiabatic laser pulse of the unitary accelerating sequence must be greater than the dimensional size of the spatial region $[R_{2,m_{r}}(t_{m_{r}})-%
\delta R(t_{m_{r}})/2,$ $R_{2,1}(t_{m_{r}})+\delta R(t_{m_{r}})/2].$ Since the halting-qubit atom moves also a spatial displacement during the Raman adiabatic laser pulse the effective spatial-action-zone width of the Raman adiabatic laser pulse must also take the spatial displacement into account in addition to the dimensional size of the spatial region. In technique it could be better to choose those spatially uniform ultra-broadband adiabatic laser pulses \[47\] as the adiabatic laser beams of the Raman adiabatic laser pulses of the unitary accelerating sequence. Denote such an ultra-broadband unitary accelerating sequence as $U_{A}(\{\varphi _{k}\},t+T_{A},t),$ where $%
T_{A}$ is the total duration of the accelerating sequence. When the ultra-broadband unitary accelerating sequence acts on the halting-qubit atom at the end time $t_{m_{r}}$ of the computational process any one of these $%
m_{r}$ possible wave-packet states $\{|0,CM2(t_{m_{r}}),R_{2,i}(t_{m_{r}})%
\rangle \}$ of the halting-qubit atom is transferred to the corresponding unstable wave-packet state: $$\begin{aligned}
|c_{2,i}^{\prime }(t_{m_{r}}+T_{A})\rangle &=&U_{A}(\{\varphi
_{k}\},t_{m_{r}}+T_{A},t_{m_{r}})|0,CM2(t_{m_{r}}),R_{2,i}(t_{m_{r}})\rangle
\\
&=&|1,CM2(t_{m_{r}}+T_{A}),R_{2,i}(t_{m_{r}}+T_{A})\rangle\end{aligned}$$where the internal state $|0\rangle $ of the halting-qubit atom is changed to the state $|1\rangle $ after the unitary accelerating sequence, meanwhile the halting-qubit atom is sped up from the initial motional velocity $v_{0}$ to a great velocity $v>>v_{0}$. The motional velocity $v$ usually may be greater than the motional velocity $v_{h}=\sqrt{2E_{h}/m_{h}}$, that is, $%
v\geq v_{h}>>v_{0},$ so that the halting-qubit atom has an enough high motional energy to pass the intermediate potential barrier to enter into the left-hand potential well. Note that the motional velocity $v$ of the halting-qubit atom has an upper-bound value $c,$ where $c$ is the light speed in vacuum, and usually $v<<c.$ The important point for the unitary accelerating process is that the center-of-mass spatial distance between any pair of the wave-packet states among these $m_{r}$ possible wave-packet states $\{|0,CM2(t_{m_{r}}),R_{2,i}(t_{m_{r}})\rangle \}$ of the halting-qubit atom is kept unchanged before and after the unitary accelerating process, although the halting-qubit atom is accelerated under the action of the unitary accelerating sequence and its wave-packet spatial position has been changing along the $+x$ direction. Therefore, after the unitary accelerating sequence these $m_{r}$ possible wave-packet spatial positions $\{R_{2,i}(t_{m_{r}}+T_{A})\}$ of the halting-qubit atom still satisfy the inequality (25) and their possible spatial distances remain also unchanged and are still given by Eq. (26), that is, $\Delta
R_{2,i,j}(t_{m_{r}}+T_{A})=v_{0}(j-i)\Delta T$ for $1\leq i<j\leq m_{r}.$ After the action of the accelerating sequence the halting-qubit atom moves in the velocity $v$ along the direction $+x$ toward the right-hand potential wall of the double-well potential field and hits ultimately the potential wall in an elastic form. Then the halting-qubit atom is bounced off the right-hand potential wall and its motional direction therefore is reversed and hence changed to the direction $-x$. Evidently, this elastic bouncing process is unitary \[5a, 48\]. Now the halting-qubit atom moves in the velocity $v$ ($v\geq v_{h}$) along the direction $-x$ toward the left-hand potential well. It first goes across the right-hand potential well, then passes the intermediate potential barrier, and finally enters into the left-hand harmonic potential well. Note that the spatial position $%
R_{2,1}(t_{m_{r}}+T_{A})$ is nearest the right-hand potential wall among all these $m_{r}$ possible wave-packet spatial positions $%
\{R_{2,i}(t_{m_{r}}+T_{A})\}.$ Evidently, the shortest spatial distance between the spatial position $R_{2,1}(t_{m_{r}}+T_{A})$ and the right-hand potential wall must be much greater than half the wave-packet spread: $%
\delta R(t_{m_{r}}+T_{A})/2.$ The halting-qubit atom needs to spend a short time period when the atom moves from the spatial position $%
R_{2,1}(t_{m_{r}}+T_{A})$ to the right-hand potential wall, bounces off the potential wall, and then returns the original spatial position $%
R_{2,1}(t_{m_{r}}+T_{A})$. Denote this short period as the atomic bouncing dead time $t_{d}$. Suppose that the time period is denoted as $t_{a}$ when the halting-qubit atom arrives in the left-hand potential well from the spatial position $R_{2,1}(t_{m_{r}}+T_{A})$ after the atom bounces off the potential wall. Then the longest time period of the quantum control process from the starting time $(t_{0i})$ of the quantum scattering process to the time when the halting-qubit atom arrives in the left-hand potential well is not longer than $(t_{m_{r}}-t_{01})+T_{A}+t_{d}+t_{a}.$ It follows from the inequality (25) that if the halting-qubit atom enters into the right-hand potential well from the left-hand one in the first cycle of the quantum circuit, then it will first return the left-hand potential well from the right-hand one after the unitary decelerating and accelerating sequences, whereas the halting-qubit atom returns the left-hand potential well at the latest time if it enters into the right-hand potential well in the latest cycle of the quantum circuit. Suppose that the halting-qubit atom returns to the left-hand potential well from the right-hand one and arrives at some given spatial position within the left-hand potential well at the arriving time $T_{i}$ ($T_{i}>t_{m_{r}}+T_{A}$) for $i=1,2,...,m_{r}$ if the atom enters into the right-hand potential well from the left-hand one in the $i-$th cycle of the quantum circuit. It follows from the inequality (25) that these $m_{r}$ possible arriving times $\{T_{i},$ $i=1,2,...,m_{r}\}$ satisfy the following inequality: $$T_{1}<T_{2}<...<T_{m_{r}} \label{27}$$and the equation (26) shows that the arriving-time difference $\Delta
T_{j,i}=T_{j}-T_{i}$ for $1\leq i<j\leq m_{r}$ is given by $$\Delta T_{j,i}=\Delta R_{2,i,j}(t_{m_{r}}+T_{A})/v=(j-i)\Delta Tv_{0}/v,
\label{28}$$and the maximum arriving-time difference equals $$\Delta T_{m_{r},1}=(m_{r}-1)\Delta Tv_{0}/v. \label{29}$$It is known that the time difference between the halting-qubit atom entering into the right-hand potential well in the $j-$th cycle and the $i-$th cycle $%
(j>i)$ of the quantum circuit is given by $(j-i)\Delta T.$ But after the halting-qubit atom is acted on by the unitary decelerating and accelerating sequences in the right-hand potential well the corresponding arriving-time difference becomes $\Delta T_{j,i}=(j-i)\Delta Tv_{0}/v.$ Since the motional velocity $v$ is much greater than the velocity $v_{0},$ that is, the time-compressing factor $v_{0}/v<<1,$ the arriving-time difference $\Delta
T_{j,i}$ is much shorter than the original time difference $(j-i)\Delta T,$ indicating that the original time difference is greatly compressed after the time- and space-dependent quantum control process which contains the unitary decelerating and accelerating processes.
During the quantum control process the halting-qubit atom carries out consecutively the quantum scattering process in which the atom goes from the left-hand potential well to the right-hand one, decelerating process, accelerating process, elastic bouncing process, the second decelerating process (see below), and finally the second quantum scattering process in which the atom returns from the right-hand potential well to the left-hand one. But the internal state of the halting-qubit atom could be changed only in the unitary decelerating and accelerating processes among these processes. For simplicity, here the second decelerating process is not considered temporarily. For convenience, for the case that the halting-qubit atom enters into the right-hand potential well in the $i-$th cycle of the quantum circuit and then returns and arrives in the left-hand potential well at the arriving time $T_{i}$ the wave-packet state of the halting-qubit atom in the left-hand potential well at the arriving time $T_{i}$ is denoted as $%
|1,CM1(T_{i}),R_{1,i}(T_{i})\rangle $ for $i=1,2,...,m_{r}$. Here the wave-packet state $|1,CM1(T_{i}),R_{1,i}(T_{i})\rangle $ has the same internal state as the state $%
|1,CM2(t_{m_{r}}+T_{A}),R_{2,i}(t_{m_{r}}+T_{A})\rangle $ of the halting-qubit atom in the right-hand potential well after the unitary accelerating process. Evidently, in an ideal case all these wave-packet states $\{|1,CM1(T_{i}),R_{1,i}(T_{i})\rangle \}$ are really the same$:$$$\begin{aligned}
|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle &=&|1,CM1(T_{2}),R_{1,2}(T_{2})\rangle \\
&=&...=|1,CM1(T_{m_{r}}),R_{1,m_{r}}(T_{m_{r}})\rangle ,\end{aligned}$$although the arriving time $T_{i}$ for the halting-qubit atom is different for a different cycle index value $i=1,2,...,m_{r}$, as can be seen in (27). However, the wave-packet motional state $|CM1(T_{i}),R_{1,i}(T_{i})\rangle $ for $i=1,2,...,m_{r}$ is generally different from the desired motional state $|CM0,R_{0}\rangle $. Since the motional energy of the halting-qubit atom with the wave-packet state $|1,CM1(T_{i}),R_{1,i}(T_{i})\rangle $ is much higher than the height of the intermediate potential barrier in the double-well potential field the wave-packet state $|1,CM1(T_{i}),$ $%
R_{1,i}(T_{i})\rangle $ is unstable and hence it must be transferred to the stable halting state $|1,CM0,R_{0}\rangle $. In general, there is a unitary transformation that transfers completely the wave-packet state $%
|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle $ to the halting state $%
|1,CM0,R_{0}\rangle .$ This unitary transformation is defined as $$U(\{\varphi _{k}\},T_{t}+T_{1},T_{1})|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle
=|1,CM0,R_{0}\rangle . \label{30}$$Here the duration $T_{t}$ of the unitary operation $U(\{\varphi
_{k}\},T_{t}+T_{1},T_{1})$ usually could be much longer than the maximum arriving-time difference $\Delta T_{m_{r},1}=T_{m_{r}}-T_{1}$ (see Eq. (29)) and $\{\varphi _{k}\}$ are the control parameters of the unitary operation. The unitary operation $U(\{\varphi _{k}\},T_{t}+T_{1},T_{1})$ starts to act on the wave-packet state $|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle $ of the halting-qubit atom in the left-hand potential well at the arriving time $%
T_{1}$ and it does not change any internal state of the halting-qubit atom. It could be generated by applying the coherent Raman adiabatic laser pulse and the time-dependent potential field to the left-hand harmonic potential well. The coherent Raman adiabatic laser trigger pulse $P_{t}$ transfers the lower motional-energy state $|1,CM0,R_{0}\rangle $ to the higher motional-energy wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $, while here the unitary operation $U(\{\varphi _{k}\},T_{t}+T_{1},T_{1})$ converts the higher motional-energy wave-packet state $%
|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle $ into the lower motional-energy state $%
|1,CM0,R_{0}\rangle $ in the left-hand potential well. However, the unitary operation $U(\{\varphi _{k}\},T_{t}+T_{1},T_{1})$ is more complex than the coherent Raman adiabatic laser trigger pulse $P_{t}$. There is a difference between these two motional states $|CM1(T_{1}),R_{1,1}(T_{1})\rangle $ and $%
|CM0,R_{0}\rangle .$ The difference is in the atomic motional energy, momentum, spatial position, wave-packet shape (e.g., the effective spread) and phase, and so forth. Thus, in order to convert the wave-packet motional state $|CM1(T_{1}),R_{1,1}(T_{1})\rangle $ into the ground motional state $%
|CM0,R_{0}\rangle $ one could apply the coherent Raman adiabatic laser pulses and also an external time-dependent potential field to the halting-qubit atom in the left-hand potential well. Here the external time-dependent potential field may be used to modulate the left-hand harmonic potential field or even the whole double-well potential field. Both the external time-dependent potential field and the coherent Raman adiabatic laser pulses could also be thought of as the components of the state-locking pulse field.
As shown in the inequality (27), the arriving time $T_{i}$ is different for $%
i=1,2,...,m_{r}$ for the halting-qubit atom returning and arriving in the left-hand potential well after the atom enters into the right-hand potential well in a different cycle ($i-$th cycle) of the quantum circuit. The earliest and latest arriving times are $T_{1}$ and $T_{m_{r}},$ respectively. It is certain that the halting-qubit atom is within the left-hand potential well at the arriving time $T_{i}$ if the atom enters into the right-hand potential well in the $i-$th cycle of the quantum circuit. However, the halting-qubit atom could not be in the left-hand potential well at the arriving time $T_{i}$ if the atom enters into the right-hand potential well in the $j-$th cycle of the quantum circuit with the cyclic index $j\neq i$ rather than in the $i-$th cycle. If the halting-qubit atom enters into the right-hand potential well in the $j-$th cycle ($m_{r}\geq j>1$) rather than in the first cycle of the quantum circuit, then at the arriving time $T_{1}$ the wave-packet state of the halting-qubit atom could not be the state $|1,CM1(T_{1}),R_{1,1}(T_{1})%
\rangle $ or the state $|1,CM1(T_{j}),R_{1,j}(T_{j})\rangle $, but it could be another wave-packet state $|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle $ different from the state $|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle .$ Here the spatial position $R_{1,j}(T_{1})$ for $m_{r}\geq j>1$ is also different from $R_{1,1}(T_{1})$ or $R_{1,j}(T_{j})$ and will not be constrained to be within the left-hand potential well. Then the unitary operation $U(\{\varphi
_{k}\},T_{t}+T_{1},T_{1})$ of Eq. (30) transfers the wave-packet state $%
|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle $ with the cyclic index $m_{r}\geq j>1$ to the state $|1,CM0,R_{0}\rangle $ in a probability less than 100%. The unitary state transformation may be generally written as $$|1,CM0(j),R_{0}(j)\rangle \equiv U(\{\varphi
_{k}\},T_{t}+T_{1},T_{1})|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle$$$$=A_{j}(CM0,R_{0},T_{t},T_{1})|1,CM0,R_{0}\rangle$$$$\quad \quad \quad \quad +\underset{CM}{\sum }%
A_{j}(CM,R_{1,j}(T_{1}),T_{t},T_{1})|1,CM,R\rangle , \label{31}$$where the first term on the right-hand side is the desired state $%
|1,CM0,R_{0}\rangle $ and the second term a superposition state which is orthogonal to the desired state $|1,CM0,R_{0}\rangle $. The absolute amplitude $|A_{j}(CM0,R_{0},T_{t},T_{1})|$ measures the conversion efficiency from the state $|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle $ ($m_{r}\geq
j\geq 1$) to the state $|1,CM0,R_{0}\rangle $ under the action of the unitary operation $U(\{\varphi _{k}\},T_{t}+T_{1},T_{1})$. By comparing Eq. (30) with Eq. (31) one sees that the amplitude $A_{1}(CM0,$ $%
R_{0},T_{t},T_{1})=1$ and $A_{1}(CM,R_{1,1}(T_{1}),T_{t},T_{1})=0$ for any index value $CM.$ A theoretical calculation for the amplitude $%
A_{j}(CM0,R_{0},T_{t},$ $T_{1})$ ($m_{r}\geq j>1$) usually could be more complex.
Denote that $H_{0}$ and $U_{0}(t,t_{0})$ are the Hamiltonian and time evolution propagator of the halting-qubit atom in the time-independent double-well potential field without the Raman adiabatic laser pulses and the time-dependent external potential field, respectively. The propagator $%
U_{0}(t,t_{0})$ does not change any internal state of the halting-qubit atom. Then there is the relation between both the wave-packet states $%
|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle $ and $|1,CM1(T_{j}),$ $%
R_{1,j}(T_{j})\rangle $ for $j=1,2,...,m_{r},$$$U_{0}(T_{j},T_{1})|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle
=|1,CM1(T_{j}),R_{1,j}(T_{j})\rangle . \label{32}$$This is because the halting-qubit atom in the wave-packet state $%
|1,CM1(T_{1}),$ $R_{1,j}(T_{1})\rangle $ at the time $T_{1}$ will arrive in the left-hand potential well at the arriving time $T_{j}\geq T_{1}$ and moreover it is in the wave-packet state $|1,CM1(T_{j}),$ $%
R_{1,j}(T_{j})\rangle $ at the arriving time $T_{j}$. By the equation (32) and the relation $|1,CM1(T_{j}),$ $R_{1,j}(T_{j})\rangle =|1,CM1(T_{1}),$ $%
R_{1,1}(T_{1})\rangle $ the unitary state transformation (31) may be reduced to the form $$U(\{\varphi _{k}\},T_{t}+T_{1},T_{1})|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle
\qquad \qquad \qquad \qquad$$$$=U(\{\varphi
_{k}\},T_{t}+T_{1},T_{1})U_{0}(T_{j},T_{1})^{+}|1,CM1(T_{1}),R_{1,1}(T_{1})%
\rangle . \label{33}$$The wave-packet motional state $|CM1(T_{1}),R_{1,1}(T_{1})\rangle $ generally is not an exact eigenstate of the Hamiltonian $H_{0}$ of the halting-qubit atom in the double-well potential field, although the motional state $|CM0,R_{0}\rangle $ could be approximately an eigenstate of the Hamiltonian $H_{0}$ with the motional energy eigenvalue $E_{0}=\hslash
\omega _{0}/2$ since it is approximately the ground motional state of the halting-qubit atom in the left-hand harmonic potential well. Thus, equation (33) shows that not every state $|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle $ for the cyclic index $j=1,2,....,m_{r}$ can be converted completely into the same state $|1,CM0,R_{0}\rangle $ by the same unitary operation $U(\{\varphi
_{k}\},T_{t}+T_{1},T_{1}).$ It follows from Eqs. (30), (31), and (33) that the amplitude $A_{j}(CM0,R_{0},T_{t},T_{1})$ of the state $%
|1,CM0,R_{0}\rangle $ on the right-hand side of Eq. (31) can be written as $$A_{j}(CM0,R_{0},T_{t},T_{1})=$$$$\langle
1,CM1(T_{1}),R_{1,1}(T_{1})|U_{0}(T_{j},T_{1})^{+}|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle
\label{34}$$This equation may be used to calculate the amplitude $%
A_{j}(CM0,R_{0},T_{t},T_{1})$ if the wave-packet state $%
|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle $ and the unitary operation $%
U_{0}(T_{j},T_{1})$ are explicitly given for $j=1,2,...,m_{r}$. It follows from Eqs. (32) and (34) that the probability $%
|A_{j}(CM0,R_{0},T_{t},T_{1})|^{2}$ is really the project probability of the wave-packet state $|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle $ for $%
j=1,2,...,m_{r} $ to the wave-packet state $|1,CM1(T_{1}),R_{1,1}(T_{1})%
\rangle .$
The quantum control process really starts at the time $t_{0i}$ (here for convenience the excitation process of the trigger pulse $P_{t}$ is not considered) and its initial state is the unstable wave-packet state $%
|1,CM1(t_{0i}),$ $R_{1}(t_{0i})\rangle $ of the halting-qubit atom in the left-hand potential well. For convenience the mean motional velocity and kinetic energy of the halting-qubit atom may be prepared to be zero when the atom is prepared to be in the wave-packet state $|1,CM1(t_{0i}),$ $%
R_{1}(t_{0i})\rangle $ at the time $t_{0i}.$ This can be achieved by the suitable state-dependent coherent Raman adiabatic laser trigger pulse $P_{t}$, as shown in the quantum control unit $Q_{h}$. From the viewpoint of the particle picture the total motional energy $E_{h}$ of the halting-qubit atom in the left-hand harmonic potential well is really pure potential energy at the time $t_{0i}$ due to zero motional velocity of the atom when the atom is in the wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $. Thus, the center-of-mass position $R_{1}(t_{0i})$ of the halting-qubit atom in the left-hand harmonic potential well is really determined uniquely by the total motional energy $E_{h}$. The wave-packet motional state $%
|CM1(t_{0i}),R_{1}(t_{0i})\rangle $ may be considered as a coherent state \[49\] for the halting-qubit atom in the left-hand harmonic potential well as it is generated from the ground motional state $|CM0,R_{0}\rangle $ by the coherent Raman adiabatic laser trigger pulse $P_{t}$, as shown in Ref. \[37\]. The wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ is transferred to the wave-packet state $|1,CM1(T_{i}),R_{1,i}(T_{i})\rangle $ ($%
i=1,2,...,m_{r}$) by the quantum control process that the halting-qubit atom starts at the position $R_{1}(t_{0i})$ and at the time $t_{0i}$ from the left-hand potential well to enter into the right-hand potential well by the quantum scattering process, then it is manipulated by the decelerating process, the accelerating process, and the elastic bouncing process in the right-hand potential well, and finally it returns to the left-hand potential well by another quantum scattering process and arrives in the left-hand potential well at the arriving time $T_{i}$. For convenience, consider the specific case that the halting-qubit atom arrives at the original position $%
R_{1}(t_{0i})$ in the left-hand potential well at the arriving time $T_{i}$ after it goes a cycle along the double-well potential field in the quantum control process. Then in an ideal condition the wave-packet state $%
|1,CM1(T_{i}),R_{1,i}(T_{i})\rangle $ is just the original wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ and hence it is also a coherent state. In order to achieve this point the total motional energy of the halting-qubit atom in the wave-packet state $|1,CM1(T_{i}),R_{1,i}(T_{i})%
\rangle $ must be equal to that one $E_{h}$ of the atom in the wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle .$ Then this requires that the motional velocity $v$ of the halting-qubit atom after the unitary accelerating process be equal to the motional velocity $v_{h}$ of the atom just before the unitary decelerating process. Actually, this means that for the halting-qubit atom the quantum scattering process from the left-hand potential well to the right-hand one is the inverse process of the second quantum scattering process from the right-hand potential well to the left-hand one. On the other hand, both the wave-packet shapes for the motional states $|CM1(t_{0i}),R_{1}(t_{0i})\rangle $ and $%
|CM1(T_{i}),R_{1,i}(T_{i})\rangle $ could be different from each other in practice. The wave-packet spread of the motional state $%
|CM1(T_{i}),R_{1,i}(T_{i})\rangle $ usually is greater than that of the motional state $|CM1(t_{0i}),R_{1}(t_{0i})\rangle $ \[5a, 48, 53\] even if the halting-qubit atom returns the original position $R_{1}(t_{0i})$ at the arriving time $T_{i}$ in the left-hand potential well after it goes a cycle along the double-well potential field. This is because those processes including the quantum scattering processes, the decelerating and accelerating processes, and the elastic bouncing process as well as the freely atomic motional processes in the quantum control process can change the wave-packet shape of motional state of the halting-qubit atom and usually could broaden the wave-packet spread of the motional state. In order to minimize the effect of these processes on the wave-packet spread of the motional state in the quantum control process the unitary accelerating process may be constructed as the inverse process of the unitary decelerating process except the atomic motional direction and spatial position. This means that the motional velocity $v_{h}$ of the halting-qubit atom before the unitary decelerating process equals the velocity $v$ of the atom after the unitary accelerating process. Then in the case $v=v_{h}$ the effect of the decelerating process could cancel that effect of the accelerating process on the wave-packet spread of the motional state and likely the effect of the first quantum scattering process could cancel that of the last quantum scattering process in the quantum control process. As a result, the net effect for both the decelerating and accelerating processes and both the first and the last quantum scattering processes on the wave-packet spread of the motional state could become so small that it can be neglected. Thus, in the quantum control process the effect on the wave-packet spread of the motional state could mainly come from the elastic bouncing process and the freely atomic motional processes. However, this effect could also be small and negligible if the halting-qubit atom has a large mass $m_{h}$ and there is a short period for the quantum control process \[53\]. Therefore, in the case $v=v_{h}$ the wave-packet motional state $|CM1(T_{i}),R_{1,i}(T_{i})\rangle $ ($i=1,2,...,m_{r}$) still could be considered approximately as the original motional state $%
|CM1(t_{0i}),R_{1}(t_{0i})\rangle $ in practice, although the wave-packet spread of a motional state of the halting-qubit atom could change slightly after the quantum control process. Since the motional state $%
|CM1(t_{0i}),R_{1}(t_{0i})\rangle $ ($i=1$) is a coherent state the wave-packet motional state $|CM1(T_{1}),R_{1,1}(T_{1})\rangle $ may be approximately expressed in a coherent-state form \[49, 37, 5c\], $$|CM1(T_{1}),R_{1,1}(T_{1})\rangle =|\alpha \rangle \equiv \exp (-\frac{1}{2}%
|\alpha |^{2})\overset{\infty }{\underset{k=0}{\sum }}\frac{\alpha ^{k}}{%
\sqrt{k!}}|k\rangle \label{35}$$where the relevant global phase factor is omitted and the state $|k\rangle $ is the energy eigenstate of the Hamiltonian of the halting-qubit atom in the left-hand harmonic potential well and $\alpha $ a complex parameter$.$ The absolute value $|\alpha |^{2}$ is the mean motional energy of the halting-qubit atom in the wave-packet motional state $%
|CM1(T_{1}),R_{1,1}(T_{1})\rangle $ in the left-hand harmonic potential well \[5c\]. Now consider other wave-packet motional states $%
\{|CM1(T_{1}),R_{1,j}(T_{1})\rangle \}$ ($m_{r}\geq j>1)$. If the entire effective wave-packet motional state $|CM1(T_{1}),$ $R_{1,j}(T_{1})\rangle $ is within the left-hand harmonic potential well at the time $T_{1},$ it is also a coherent state approximately just like the coherent state $%
|CM1(T_{1}),$ $R_{1,1}(T_{1})\rangle $ or $|CM1(T_{j}),R_{1,j}(T_{j})\rangle
.$ Here these two coherent states $|CM1(T_{1}),$ $R_{1,j}(T_{1})\rangle $ and $|CM1(T_{1}),R_{1,1}(T_{1})\rangle $ are connected by the unitary transformation (32). If now the right-hand potential wall of the left-hand potential well is sufficiently high, then that the time evolution process of the halting-qubit atom in the left-hand potential well is governed by the Hamiltonian of the double-well potential field is really reduced to the simple one that the time evolution process is governed by the Hamiltonian of the single left-hand harmonic potential well. Here the Hamiltonian $H_{0}$ and the propagator $U_{0}(t,t_{0})=\exp [-iH_{0}(t-t_{0})/\hslash ]$ of the double-well potential field are also reduced to those of the single left-hand harmonic potential field, respectively. Since the state $|k\rangle
$ in the coherent state of Eq. (35) is an eigenstate of the Hamiltonian of the single left-hand harmonic potential well, there holds the eigen-equation: $U_{0}(T_{j},T_{1})^{+}|k\rangle =\exp [i(k+1/2)\omega
_{0}(T_{j}-T_{1})]|k\rangle .$ It follows from Eqs. (32) and (35) that the coherent state $|CM1(T_{1}),R_{1,j}(T_{1})\rangle $ for $j=1,2,...,m_{r}$ can be written as $$|CM1(T_{1}),R_{1,j}(T_{1})\rangle =|\alpha \exp [i\omega
_{0}(T_{j}-T_{1})]\rangle \quad$$$$=\exp (-\frac{1}{2}|\alpha |^{2})\overset{\infty }{\underset{k=0}{\sum }}%
\frac{\{\alpha \exp [i\omega _{0}(T_{j}-T_{1})]\}^{k}}{\sqrt{k!}}|k\rangle
\tag{35a}$$where the global phase factor is also omitted and $\omega _{0}=\sqrt{K/m_{h}}
$ is the basic oscillatory frequency of the harmonic oscillator, i.e., the halting-qubit atom in the left-hand harmonic potential well, and $K$ a force constant of the harmonic oscillator.
There are two different situations to be considered below. For the first situation the maximum arriving-time difference $T_{m_{r}}-T_{1}$ for the halting-qubit atom is much shorter than the basic oscillatory period $2\pi
/\omega _{0}$ of the halting-qubit atom in the left-hand harmonic potential well, that is, $T_{m_{r}}-T_{1}<<\pi /\omega _{0}.$ In the first situation there is not the second unitary decelerating process in the quantum control process. From the viewpoint of the particle picture the halting-qubit atom at the position $R_{1}(t_{0i})$ leaving the left-hand potential well needs to spend roughly the time $\pi /\omega _{0}.$ Then this really means that in the case $T_{m_{r}}-T_{1}<<\pi /\omega _{0}$ each of these $m_{r}$ possible wave-packet states $\{|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle \}$ for the halting-qubit atom is able to enter completely into the left-hand potential well at the time $T_{1}$. Once each of these $m_{r}$ possible wave-packet states $\{|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle \}$ of the halting-qubit atom enters completely into the left-hand potential well at the time $T_{1}$, one may suddenly change the double-well potential field to the single left-hand harmonic potential field. Then in this case it is much easier to calculate theoretically the amplitude $A_{j}(CM0,R_{0},T_{t},T_{1})$ in Eq. (31) as the quantum dynamics can be exactly solved in a single harmonic oscillator even when the Hamiltonian of the harmonic oscillator is time-dependent. Now all these $m_{r}$ possible wave-packet states $%
\{|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle \}$ are the coherent states which are given explicitly by Eqs. (35) and (35a). With the help of Eq. (32) and by inserting the coherent states of Eqs. (35) and (35a) into Eq. (34) the absolute amplitude $|A_{j}(CM0,R_{0},T_{t},T_{1})|$ of Eq. (34) is reduced to the simple form $$|A_{j}(CM0,R_{0},T_{t},T_{1})|=|\langle
1,CM1(T_{1}),R_{1,1}(T_{1})|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle |$$$$\qquad \qquad =\exp \{-|\alpha |^{2}(1-\cos [\omega _{0}(T_{j}-T_{1})])\}
\label{36}$$where the relation for a pair of coherent states $|\alpha \rangle $ and $%
|\beta \rangle :\langle \alpha |\beta \rangle =\exp \{-|\alpha
|^{2}/2-|\beta |^{2}/2+\alpha ^{\ast }\beta \}$ \[49, 51b, 52\] is used. When the arriving time $T_{j}=T_{1}$ it follows from Eq. (36) that the absolute amplitude $|A_{1}(CM0,R_{0},T_{t},T_{1})|$ $=1.$ This result is consistent with Eq. (30). However, if the arriving time $T_{j}>T_{1}$ ($m_{r}\geq j>1$)$%
,$ then equation (36) shows that the absolute amplitude $|A_{j}(CM0,R_{0},$ $%
T_{t},T_{1})|$ decays exponentially as the mean motional energy $|\alpha
|^{2}$ of the halting-qubit atom and the term $(1-\cos [\omega
_{0}(T_{j}-T_{1})])$. In order that the absolute amplitude $%
|A_{j}(CM0,R_{0},T_{t},T_{1})|$ is close to unity one should decrease greatly either the mean motional energy $|\alpha |^{2}$ and the basic oscillatory frequency $\omega _{0}$ or the arriving-time differences $%
\{(T_{j}-T_{1})\}$. Since for the current situation the basic oscillatory frequency $\omega _{0}$ is fixed and the mean motional energy $|\alpha |^{2}$ of the halting-qubit atom must be much greater than the height of the intermediate potential barrier one can only decrease greatly the arriving-time differences $\{(T_{j}-T_{1})\}$ to make the absolute amplitude $|A_{j}(CM0,R_{0},T_{t},T_{1})|$ close to unity. Because the arriving-time difference $T_{j}-T_{i}=(j-i)\Delta Tv_{0}/v$ for $1\leq i<j\leq m_{r},$ as shown in Eq. (28), and the atomic motional velocity $v=v_{h}$ is determined by the mean motional energy $|\alpha |^{2}$ in the current situation, one has to slow down greatly the atomic motional velocity $v_{h}$ to a much smaller velocity $v_{0}$ which could be close to zero by the unitary decelerating process, so that the arriving-time differences can be shortened greatly and thus the probability $|A_{j}(CM0,R_{0},T_{t},T_{1})|^{2}$ of Eq. (36) can be close to 100%. Though a larger atomic motional velocity $v$ leads to a smaller arriving-time difference $(T_{j}-T_{1}),$ it also leads to a larger mean motional energy $|\alpha |^{2}\varpropto v^{2},$ and hence the absolute amplitude $|A_{j}(CM0,R_{0},T_{t},T_{1})|$ does not become larger and closer to unity significantly for a larger atomic motional velocity $v=v_{h}$.
Now consider the second situation that the arriving-time differences are not always much shorter than the basic oscillatory period $2\pi /\omega _{0}$ of the halting-qubit atom in the left-hand harmonic potential well, that is, the condition $T_{j}-T_{1}<\pi /\omega _{0}$ may not hold for some index values $j=1,2,...,m_{r}$. In this case not every of these $m_{r}$ possible wave-packet states $\{|1,CM1(T_{1}),$ $R_{1,j}(T_{1})\rangle \}$ for the halting-qubit atom is able to enter completely into the left-hand potential well at the time $T_{1}$. Since in this case some arriving-time differences $%
\{(T_{j}-T_{1})\}$ are larger the corresponding amplitudes $%
\{|A_{j}(CM0,R_{0},$ $T_{t},T_{1})|\}$ of Eq. (36) are not close to unity. One scheme to solve the problem is that the basic oscillatory frequency $%
\omega _{0}$ is switched to a smaller value such that the new basic oscillatory period can be much longer than the maximum arriving-time difference and meanwhile the mean motional energy $|\alpha |^{2}$ of the halting-qubit atom is lowed down accordingly before the halting-qubit atom enters into the left-hand harmonic potential well. In this case the amplitudes $\{|A_{j}(CM0,R_{0},$ $T_{t},T_{1})|\}$ become larger and may be close to unity, as can be seen from Eq. (36). This scheme may be described as follows. Since in the current situation the shortest arriving-time difference $\Delta T_{i+1,i}=\Delta Tv_{0}/v$ ($1\leq i\leq m_{r}-1$) generally is long and it is also much longer than the pulse duration of the Raman adiabatic laser pulse, one may use a unitary decelerating sequence consisting of the Raman adiabatic laser pulses to decelerate the halting-qubit atom so that the atom has a low motional energy and speed before the atom enters into the left-hand potential well from the right-hand one by the second quantum scattering process. Here every arriving-time difference $\Delta T_{j,i}$ for $1\leq i<j\leq m_{r}$ must be kept unchanged before and after the unitary decelerating process. This decelerating process is the second unitary decelerating process of the quantum control process. It is really a space-compressing process for the $m_{r}$ possible wave-packet states of the halting-qubit atom. The unitary decelerating sequence may be applied after the atom bounces off the right-hand potential wall of the double-well potential field and its spatial action zone could be the same as that one of the first unitary decelerating sequence of the quantum control process. After the quantum computational process finished and long before the halting-qubit atom returns the left-hand potential well the original oscillatory frequency $\omega _{0}$ of the halting-qubit atom in the left-hand potential well is switched to a much smaller oscillatory frequency and the height of the intermediate potential barrier is changed to be much lower than the motional energy of the halting-qubit atom such that the atom with a low motional energy can also enter completely into the left-hand potential well even after it is decelerated by the second unitary decelerating sequence. Of course, the intermediate potential barrier may also be cancelled (i.e., switched off) and in the case the height of the right-hand potential wall of the left-hand potential well has to be lowed down correspondingly. These operations can be achieved by applying external potential fields to modulate the left-hand harmonic potential well and the intermediate potential barrier. Denote that $\omega _{c}$ is the basic oscillatory frequency for the halting-qubit atom in the new left-hand harmonic potential well and $|\alpha _{c}|^{2}$ the mean motional energy of the halting-qubit atom after the atom is decelerated by the second unitary decelerating sequence. Then $\omega _{c}<<\omega _{0}$ and $|\alpha
_{c}|^{2}<<$ $|\alpha |^{2}.$ Now the basic oscillatory period $2\pi /\omega
_{c}$ is longer for a smaller basic oscillatory frequency $\omega _{c}$. Since the time-compressing factor $v_{0}/v=v_{0}/v_{h}<<1$ and $2\pi /\omega
<<2\pi /\omega _{c}$ now the condition: $T_{m_{r}}-T_{1}=(m_{r}-1)\Delta
T(v_{0}/v_{h})<<\pi /\omega _{c}$ may be met easily by setting an enough small basic oscillatory frequency $\omega _{c}$. Then in this case each of these $m_{r}$ possible wave-packet states $\{|1,CM1(T_{1}),R_{1,j}(T_{1})%
\rangle \}$ is able to enter completely into the new left-hand potential well at the time $T_{1}$ after the second decelerating process. After each of these $m_{r}$ possible states $\{|1,CM1(T_{1}),R_{1,j}(T_{1})\rangle \}$ enters completely into the new left-hand potential well one may change suddenly the double-well potential field to the single left-hand harmonic potential field by increasing greatly and quickly the height of the right-hand potential wall of the left-hand potential well. Here the spatial position $R_{1,1}^{c}(T_{1})$ of the wave-packet state $%
|1,CM1(T_{1}),R_{1,1}^{c}(T_{1})\rangle $ in the new left-hand potential well is determined by the mean motional energy $|\alpha _{c}|^{2}$ instead of $|\alpha |^{2}.$ Obviously, the wave-packet state $|\alpha _{c}\rangle
=|1,CM1(T_{1}),R_{1,1}^{c}(T_{1})\rangle $ of the new left-hand potential well is fully different from the original state $|\alpha \rangle
=|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle $ and the state $%
|1,CM1(t_{0i}),R_{1}(t_{0i})\rangle $ of the original left-hand potential well, but both the mean motional velocity and kinetic energy are zero for the halting-qubit atom with any one of the three wave-packet states. Evidently, these wave-packet states $\{|1,CM1(T_{1}),R_{1,j}^{c}(T_{1})%
\rangle \}$ are still approximately coherent states of the new left-hand harmonic potential well and the equation (36) holds also for them, where the oscillatory frequency $\omega _{0}$ and the mean motional energy $|\alpha
|^{2}$ are replaced with $\omega _{c}$ and $|\alpha _{c}|^{2},$ respectively. Now the time-compressing factor $v_{0}/v_{h}<<1$ leads to that $\omega _{c}(T_{j}-T_{1})<<1$ and $|\alpha _{c}|^{2}(1-\cos [\omega
_{c}(T_{j}-T_{1})])<<1$ for $j=1,2,...,m_{r}.$ Thus, it follows from Eq. (36) that the project probability of the wave-packet state $%
|1,CM1(T_{1}),R_{1,j}^{c}(T_{1})\rangle $ to the wave-packet state $%
|1,CM1(T_{1}),R_{1,1}^{c}(T_{1})\rangle $ for the halting-qubit atom in the new left-hand potential well may be given by $$|\langle
1,CM1(T_{1}),R_{1,j}^{c}(T_{1})|1,CM1(T_{1}),R_{1,1}^{c}(T_{1})\rangle |^{2}$$$$=\exp \{-2|\alpha _{c}|^{2}(1-\cos [\omega _{c}(T_{j}-T_{1})])\}. \label{37}$$Here one must pay attention to that unlike in Eq. (36) the project probability in Eq. (37) generally is not equal to the probability $%
|A_{j}(CM0,R_{0},T_{t},T_{1})|^{2}$ of the desired state $%
|1,CM0,R_{0}\rangle $ in Eq. (31), because the current left-hand potential well is not the original one.
In the definition (30) of the unitary transformation $U(\{\varphi
_{k}\},T_{t}+T_{1},T_{1})$ the wave-packet state $%
|1,CM1(T_{1}),R_{1,1}(T_{1})\rangle $ now is replaced with the state $%
|1,CM1(T_{1}),R_{1,1}^{c}(T_{1})\rangle $ of the halting-qubit atom in the new left-hand potential well, while the state $|1,CM0,R_{0}\rangle $ is still of the atom in the original left-hand potential well. If now there is a unitary transformation $V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1})$ such that the motional state $|\alpha _{c}\rangle
=|CM1(T_{1}),R_{1,1}^{c}(T_{1})\rangle $ of the new left-hand potential well is completely converted into another coherent state $|\beta \rangle $ of the original left-hand potential well and there is also another unitary transformation $V_{2}(\{\varphi _{k}\},T_{t}+T_{1},\tau +T_{1})$ such that the coherent state $|\beta \rangle $ is further converted completely into the desired motional state $|CM0,R_{0}\rangle $, then it can turn out that the project probability of Eq. (37) is really just the probability $%
|A_{j}(CM0,R_{0},T_{t},T_{1})|^{2}$ of the desired state $%
|1,CM0,R_{0}\rangle $ in Eq. (31), and the probability $%
|A_{j}(CM0,R_{0},T_{t},T_{1})|^{2}$ now can be expanded as the power series of the arriving-time difference $(T_{j}-T_{1})$, up to the second order approximation, $$|A_{j}(CM0,R_{0},T_{t},T_{1})|^{2}=1-|\alpha _{c}|^{2}(\omega
_{c})^{2}(\Delta T)^{2}(j-1)^{2}(v_{0}/v_{h})^{2}+.... \label{38}$$Therefore, the motional state $|CM1(T_{1}),R_{1,j}^{c}(T_{1})\rangle $ of the new left-hand potential well for $j=1,2,...,m_{r}$ is converted into the desired motional state $|CM0,R_{0}\rangle $ in the probability $%
|A_{j}(CM0,R_{0},T_{t},T_{1})|^{2}$ of Eq. (38) by the unitary transformation $U(\{\varphi _{k}\},T_{t}+T_{1},T_{1})$ of Eq. (30) which now is a product of the unitary transformations $V_{1}$ and $V_{2}$. The equation (38) shows that the lower bound of the probability $|A_{j}(CM0,$ $%
R_{0},T_{t},T_{1})|^{2}$ of the desired state $|1,CM0,R_{0}\rangle $ in Eq. (31) is dependent upon the time-compressing factor $v_{0}/v_{h}$ in a quadratic form when the time-compressing factor $v_{0}/v_{h}<<1$. One sees from Eq. (38) that the probability $|A_{j}(CM0,$ $R_{0},T_{t},T_{1})|^{2}$ becomes closer to unity when the initial motional velocity ($v_{h}=v)$ increases. This point is quite different from that one in the first situation (see Eq. (36)) and it may make the scheme to increase the probability $|A_{j}(CM0,$ $R_{0},T_{t},T_{1})|^{2}$ in the second situation better than that one in the first situation.
Obviously, the unitary transformation $U(\{\varphi _{k}\},T_{t}+T_{1},T_{1})$ of Eq. (30) now may be explicitly expressed as $$U(\{\varphi _{k}\},T_{t}+T_{1},T_{1})=V_{2}(\{\varphi
_{k}\},T_{t}+T_{1},\tau +T_{1})V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1}).$$Here the unitary transformations $V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1})$ and $V_{2}(\{\varphi _{k}\},T_{t}+T_{1},\tau +T_{1})$ are respectively defined as $$V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1})|\alpha _{c}\rangle =|\beta \rangle
,$$$$V_{2}(\{\varphi _{k}\},T_{t}+T_{1},\tau +T_{1})|\beta \rangle
=|CM0,R_{0}\rangle . \label{40}$$The unitary transformation $V_{2}(\{\varphi _{k}\},T_{t}+T_{1},\tau +T_{1})$ can be generated simply by the coherent Raman adiabatic laser pulse applying to the original left-hand harmonic potential well \[37\] as both the motional states $|\beta \rangle $ and $|CM0,R_{0}\rangle $ are coherent states of the halting-qubit atom in the original left-hand potential well. Of course, the unitary operation $V_{2}(\{\varphi _{k}\},T_{t}+T_{1},\tau +T_{1})$ is the unity operation if the motional state $|\beta \rangle $ is just $%
|CM0,R_{0}\rangle .$ Now one needs to construct the unitary transformation $%
V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1})$. Firstly, one must pay attention to that both the desired motional state $|CM0,R_{0}\rangle $ and the coherent motional state $|\beta \rangle $ belong to the halting-qubit atom in the original left-hand harmonic potential well whose basic oscillatory frequency is $\omega _{0},$ while the coherent motional state $|\alpha
_{c}\rangle =|CM1(T_{1}),R_{1,1}^{c}(T_{1})\rangle $ is of the atom in the new left-hand harmonic potential well whose basic oscillatory frequency is $%
\omega _{c}.$ Therefore, the unitary transformation $V_{1}(\{\varphi
_{k}\},\tau +T_{1},T_{1})$ is involved in the time-dependent oscillatory-frequency-varying evolution process of the halting-qubit atom in the left-hand potential well and in the process the initial and final oscillatory frequencies are $\omega _{c}$ and $\omega _{0}$, respectively. In order to construct the unitary transformation $V_{1}(\{\varphi
_{k}\},\tau +T_{1},T_{1})$ one may apply the time-dependent potential field to modulate the left-hand harmonic potential well with the initial and final basic oscillatory frequencies $\omega _{c}$ and $\omega _{0}$, respectively, after the halting-qubit atom enters into the new left-hand harmonic potential well. Note that the time-dependent potential field does not change any internal state of the halting-qubit atom in the left-hand potential well. Denote that $H(t)$ and $U(t,t_{0})$ are the Hamiltonian and evolution propagator of the halting-qubit atom in the time-dependent frequency-varying left-hand harmonic potential well, respectively. Then the time evolution propagator $U(t,t_{0})$ of the atomic physical system which can be considered as a conventional harmonic oscillator may be written as $$U(t,t_{0})=T\exp \{-\frac{i}{\hslash }\overset{t}{\underset{t_{0}}{\int }}%
H(t^{\prime })dt^{\prime }\} \label{41}$$where the operator $T$ is the Dyson time-ordering operator and the time-dependent frequency-modulation Hamiltonian $H(t)$ of the harmonic oscillator is simply written as $$H(t)=\frac{1}{2m_{h}}p_{x}^{2}+\frac{1}{2}m_{h}\omega (t)^{2}x^{2}
\label{42}$$where the frequency-modulation function $\omega (t)$ satisfies $\omega
(t_{0})=\omega _{c}$ for $t_{0}=T_{1}$ and $\omega (t)=\omega _{0}$ for $%
t=\tau +T_{1}.$ The unitary transformation $V_{1}(\{\varphi _{k}\},\tau
+T_{1},T_{1})$ is just the propagator $U(\tau +T_{1},T_{1})$. The unitary propagator $U(\tau +T_{1},T_{1})$ may be generally expressed as \[50g, 50h, 50i, 51, 52\] $$\begin{aligned}
U(\tau +T_{1},T_{1}) &=&\exp [\frac{1}{2}z(\tau ,T_{1})a^{2}-\frac{1}{2}%
z(\tau ,T_{1})^{\ast }(a^{+})^{2}] \notag \\
&&\times \exp [-i\phi (\tau ,T_{1})a^{+}a] \label{43}\end{aligned}$$where the operators $a$ and $a^{+}$ are creation and destruction operators of the conventional harmonic oscillator, respectively, the complex parameter $z(\tau ,T_{1})=|z(\tau ,T_{1})|\exp [i\varphi (\tau ,T_{1})],$ and $\phi
=\phi (\tau ,T_{1})$ is the real angular frequency. The time-dependent frequency-modulation function $\omega (t)$ of the Hamiltonian (42) must be designed suitably. The unitary operation $U(\tau +T_{1},T_{1})$ may be generally determined from Eq. (41) and the Hamiltonian of Eq. (42) by the Magnus expansion method \[50a, 50c, 50d, 50e\] or by the Lie algebra approach \[50b, 50c, 50f, 50g, 50h, 50j\]. The most convenient method to calculate approximately the parameters $|z(\tau ,T_{1})|,$ $\varphi (\tau ,T_{1}),$ and $\phi (\tau ,T_{1})$ in Eq. (43) from the Hamiltonian $H(t)$ of Eq. (42) could be the Magnus expansion method \[50a, 50c\] or the average Hamiltonian theory \[50d, 50e, 50j\].
The unitary operation $V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1})=U(\tau
+T_{1},T_{1})$ of Eq. (43) can be determined from the Hamiltonian of Eq. (42) with the time-dependent frequency-modulation function $\omega (t)$, but here the important thing is how to generate a proper time-dependent frequency-modulation function $\omega (t)$ so that the coherent state $%
|\alpha _{c}\rangle =|CM1(T_{1}),R_{1,1}^{c}(T_{1})\rangle $ can be completely transferred to another coherent state $|\beta \rangle $ by the unitary operation $V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1})$ in Eq. (39). Firstly, the second term $\exp [-i\phi (\tau ,T_{1})a^{+}a]$ of the unitary operation $U(\tau +T_{1},T_{1})$ can transfer the coherent state $|\alpha
_{c}\rangle $ to another coherent state \[49, 5c, 51, 52\], $$\exp [-i\phi (\tau ,T_{1})a^{+}a]|\alpha _{c}\rangle =|\alpha _{c}\exp
[-i\phi (\tau ,T_{1})]\rangle . \label{44}$$Note that the first term of the unitary operation $U(\tau +T_{1},T_{1})$ is a double-photon unitary operator $S(z)\equiv \exp [\frac{1}{2}%
(za^{2}-z^{\ast }(a^{+})^{2})]$ of the conventional harmonic oscillator \[51, 52\]. Then, the double-photon unitary operator $S(z(\tau ,T_{1}))$ acting on the coherent state of Eq. (44) will transfer the coherent state to other coherent states. There is a general formula to calculate the transition amplitude between a pair of coherent states $|\alpha \rangle $ and $|\beta
\rangle $ induced by the double-photon unitary operator $S(z)$ \[51\]: $$\begin{aligned}
\langle \alpha _{1}|S(z)|\beta _{1}\rangle &=&C_{r}^{-1/2}\exp \{-\frac{1}{2}%
(|\alpha _{1}|^{2}+|\beta _{1}|^{2})+C_{r}^{-1}\alpha _{1}^{\ast }\beta
_{1}\} \notag \\
&&\times \exp \{\frac{1}{2}S_{r}C_{r}^{-1}[\beta _{1}^{2}\exp (i\varphi
)-(\alpha _{1}^{\ast })^{2}\exp (-i\varphi )]\} \label{45}\end{aligned}$$where the parameters $z=r\exp (i\varphi ),$ $C_{r}=\cosh r,$ and $%
S_{r}=\sinh r.$ Now inserting the unitary operation $V_{1}(\{\varphi
_{k}\},\tau +T_{1},T_{1})=U(\tau +T_{1},T_{1})$ of Eq. (43) into the equation (39), then using the state transformation (44) and the formula (45), and setting $\beta _{1}=\alpha _{c}\exp [-i\phi (\tau ,T_{1})]$ $%
=|\alpha _{c}|\exp (i\gamma _{c})\exp [-i\phi (\tau ,T_{1})],$ $\alpha
_{1}=\beta =|\beta |\exp (i\gamma ),$ $r=|z(\tau ,T_{1})|,$ and $\varphi
=\varphi (\tau ,T_{1}),$ one obtains $$|\langle \beta |V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1})|\alpha _{c}\rangle
|=$$$$C_{r}^{-1/2}\exp \{-\frac{1}{2}(|\alpha _{c}|^{2}+|\beta
|^{2})+C_{r}^{-1}|\alpha _{c}\beta |\cos [-\gamma +\gamma _{c}-\phi (\tau
,T_{1})]\}$$$$\times \exp \{\frac{1}{2}S_{r}C_{r}^{-1}|\alpha _{c}|^{2}\cos [2\gamma
_{c}-2\phi (\tau ,T_{1})+\varphi (\tau ,T_{1})]\}$$$$\times \exp \{-\frac{1}{2}S_{r}C_{r}^{-1}|\beta |^{2}\cos [2\gamma +\varphi
(\tau ,T_{1})]\}. \label{46}$$In order that the coherent state $|\alpha _{c}\rangle
=|CM1(T_{1}),R_{1,1}^{c}(T_{1})\rangle $ can be transferred completely to the coherent state $|\beta \rangle $ by the unitary operation $%
V_{1}(\{\varphi _{k}\},$ $\tau +T_{1},T_{1})$ the absolute amplitude $%
|\langle \beta |V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1})|\alpha _{c}\rangle
|$ must be equal to unity. Then it follows from Eq. (46) that there holds the relation when $|\langle \beta |V_{1}(\{\varphi _{k}\},\tau
+T_{1},T_{1})|\alpha _{c}\rangle |^{2}=1:$ $$\ln C_{r}=-|\alpha _{c}|^{2}(1-S_{r}C_{r}^{-1}\cos [2\gamma _{c}-2\phi (\tau
,T_{1})+\varphi (\tau ,T_{1})])$$$$+2|\alpha _{c}\beta |C_{r}^{-1}\cos [-\gamma +\gamma _{c}-\phi (\tau ,T_{1})]$$$$-|\beta |^{2}(1+S_{r}C_{r}^{-1}\cos [2\gamma +\varphi (\tau ,T_{1})]).
\label{47}$$Equation (47) is used to construct the time-dependent frequency-modulation function $\omega (t)$ in the Hamiltonian $H(t)$ of Eq. (42), because the equation (47) has to be met if there exists the Hamiltonian $H(t)$ of Eq. (42) such that the coherent state $|\alpha _{c}\rangle $ can be completely converted into the coherent state $|\beta \rangle $ by the unitary transformation $U(\tau +T_{1},T_{1}).$ Here the time-dependent frequency-modulation function $\omega (t)$ must satisfy the initial and final conditions: $\omega (T_{1})=\omega _{c}$ and $\omega (\tau
+T_{1})=\omega _{0}.$ The amplitude $|\alpha _{c}|$ and phase $\gamma _{c}$ are given by the coherent state $|\alpha _{c}\rangle $ and hence there are only four independent variables $|\beta |,$ $\gamma ,$ and $r=|z(\tau
,T_{1})|,$ $\phi (\tau ,T_{1})$ in Eq. (47). In particular, when $r=0$ such that $C_{r}=1,$ $\ln C_{r}=0,$ and $S_{r}=0,$ the equation (47) is reduced to the form $$(|\alpha _{c}|-|\beta |)^{2}+2|\alpha _{c}\beta |(1-\cos [-\gamma +\gamma
_{c}-\phi (\tau ,T_{1})])=0.$$This equation has the unique solution: $|\beta |=|\alpha _{c}|$ and $\gamma
=2k\pi +\gamma _{c}-\phi (\tau ,T_{1}),$ where $k$ is an integer and usually set to zero. This result is obvious because the unitary transformation $%
V_{1}(\{\varphi _{k}\},\tau +T_{1},T_{1})$ becomes the unitary operation $%
\exp [-i\phi (\tau ,T_{1})a^{+}a]$ if $r=0$. For a general case $r=|z(\tau
,T_{1})|\neq 0$ the solutions to the equation (47) are more complex. For simplicity, suppose that the phase $\gamma =const.$ The equation (47) is reduced to the form $$a_{1}|\beta |^{2}+b_{1}|\beta |+c_{1}=0 \label{48}$$where the three coefficients $a_{1}$, $b_{1}$, and $c_{1}$ are given by $$a_{1}=1+S_{r}C_{r}^{-1}\cos [2\gamma +\varphi (\tau ,T_{1})],$$$$b_{1}=-2|\alpha _{c}|C_{r}^{-1}\cos [-\gamma +\gamma _{c}-\phi (\tau
,T_{1})],$$$$c_{1}=\ln C_{r}+|\alpha _{c}|^{2}(1-S_{r}C_{r}^{-1}\cos [2\gamma _{c}-2\phi
(\tau ,T_{1})+\varphi (\tau ,T_{1})]).$$Here the parameters $r=|z(\tau ,T_{1})|,$ $\varphi (\tau ,T_{1}),$ and $\phi
(\tau ,T_{1})$ are determined from the Hamiltonian $H(t)$ of Eq. (42). The equation (48) has the solutions when the coefficient $a_{1}\neq 0:$ $$|\beta |=\frac{1}{2a_{1}}\{-b_{1}\pm \sqrt{b_{1}^{2}-4a_{1}c_{1}}\}.$$The equation (48) has a real solution at least only when $%
b_{1}^{2}-4a_{1}c_{1}\geq 0.$ However, the reasonable solutions to the equation (48) are those positive real solutions. Therefore, the time-dependent frequency-modulation function $\omega (t)$ of the Hamiltonian $H(t)$ of Eq. (42) must be constructed such that there exists at least one positive real solution to the equation (48).
After finishing both the quantum computational and quantum control processes the initial state $|0,CM0,R_{0}\rangle |0\rangle |f_{r}(x_{0})\rangle $ with the initial functional state $|f_{r}(x_{0})\rangle $ is transferred by the quantum circuit $Q_{c}$ to the output state $|1,CM0,R_{0}\rangle |1\rangle
|0\rangle $ in the probability $|A_{i}(CM0,R_{0},T_{t},T_{1})|^{2}$ given by Eq. (36) or (38), where the indices $i$ and $x_{0}$ satisfy $%
x_{0}=(x_{f}-i+1)\func{mod}m_{r}$ or $i=(x_{f}-x_{0}+1)\func{mod}m_{r}$ for $%
i=1,2,...,m_{r}.$ Obviously, the halting state $|1,CM0,R_{0}\rangle $ and the branch-control state $|b_{h}\rangle =|1\rangle $ in the output state can also be further converted completely into the halting state $%
|0,CM0,R_{0}\rangle $ and the state $|0\rangle $ by the unitary transformations, respectively. Now the quantum circuit $Q_{c}$ with the initial state $|0,CM0,R_{0}\rangle |0\rangle |f_{r}(x_{0})\rangle $ outputs the desired state $|0,CM0,R_{0}\rangle |0\rangle |0\rangle $ in the probability $|A_{i}(CM0,R_{0},$ $T_{t},T_{1})|^{2}$ given by Eq. (36) or (38). As shown in Eqs. (36) and (38), the probability $%
|A_{i}(CM0,R_{0},T_{t},T_{1})|^{2}$ for $i=(x_{f}-x_{0}+1)\func{mod}%
m_{r}=1,2,...,m_{r}$ is close to 100% when the time-compressing factor $%
v_{0}/v_{h}<<1$ no matter what the initial functional state $%
|f_{r}(x_{0})\rangle $ is in the initial state $|0,CM0,R_{0}\rangle
|0\rangle |f_{r}(x_{0})\rangle $ of the quantum circuit. This result shows that the desired output state (or the state of Eq. (31)) of the quantum circuit $Q_{c}$ is almost independent upon any initial functional state $%
|f_{r}(x_{0})\rangle $ with $x_{0}=0,1,...,m_{r}-1.$ This is just the characteristic feature of the reversible and unitary halting protocol based on the state-locking pulse field. Evidently, such a quantum circuit whose output state is dependent insensitively upon any initial state could be used to build up an efficient quantum search process just like the quantum program and circuit proposed in the previous paper \[24\]. However, the lower bound for the probability $|A_{i}(CM0,R_{0},T_{t},T_{1})|^{2}$ of Eq. (36) or (38) for any index value $i=1,2,...,m_{r}$ must be greater than some threshold value if the quantum circuit $Q_{c}$ is used to solve efficiently the unsorted quantum search problem in the Hilbert space of an $n-$qubit quantum system. Here the Hilbert space should be the quantum search space. Note that the $2^{n}-$dimensional Hilbert state space $S(Z_{2^{n}})$ of an $%
n-$qubit quantum system is really a direct product state space of $n$ two-dimensional additive-cycle-group state subspaces $%
S(Z_{2}):S(Z_{2^{n}})=S(Z_{2})\bigotimes S(Z_{2})\bigotimes ...\bigotimes
S(Z_{2}).$ It can turn out that the lower-bound probability of Eq. (36) or (38) must be greater than $(1-\ln p(n)/n)$, here $p(n)$ is a polynomial of the qubit number $n$, if the minimum transfer probability from the entire initial state $\dbigotimes\limits_{k=1}^{n}\{|0,CM0,R_{0}\rangle |0\rangle
|f_{r}(x_{0})\rangle \}_{k}$ to the final state $\dbigotimes%
\limits_{k=1}^{n}\{|0,CM0,R_{0}\rangle |0\rangle |0\rangle \}_{k}$ is equal to $1/p(n)$ in the $2^{n}-$dimensional quantum search space $S(Z_{2^{n}})$. Thus, if the time-compressing factor $v_{0}/v$ is small enough, that is, $%
v_{0}/v<<1,$ such that the lower-bound probability of $%
\{|A_{i}(CM0,R_{0},T_{t},T_{1})|^{2}\}$ of Eq. (36) or (38) is greater than $%
(1-\ln p(n)/n),$ then the quantum search process based on the quantum circuit $Q_{c}$ is enough powerful to solve efficiently the unsorted quantum search problem in the Hilbert space of the $n-$qubit quantum system. Here the quantum parallel principle may be applied in the way that quantum computation is performed simultaneously in all factor state subspaces of the whole Hilbert space, and in this case the quantum parallel principle is compatible with both the unitary quantum dynamical principle and the mathematical logic principles used in the quantum search process.
The amplitudes $\{A_{i}(CM0,R_{0},T_{t},T_{1})\}$ of the state $%
|1,CM0,R_{0}\rangle $ in Eq. (31) corresponding to different initial states in the quantum circuit $Q_{c}$ are dependent upon the time-compressing factor $v_{0}/v$ and the atomic mean motional energy $|\alpha _{c}|^{2},$ as shown in Eqs. (36) and (38). Of course, each of these amplitudes is also dependent on a global phase factor which may be different for a different index $i$. Obviously, the global phase factor has not a net contribution to the probability $|A_{i}(CM0,R_{0},T_{t},T_{1})|^{2}$. Therefore, the basic principle to make the reversible and unitary halting protocol insensitive to any initial state includes two parts, one is by the time-compressing process to decrease the difference among the dynamical phase factors which determine the probabilities $\{|A_{i}(CM0,R_{0},T_{t},T_{1})|^{2}\}$ of the state $%
|1,CM0,R_{0}\rangle $ in Eq. (31), another is by manipulating the Hamiltonian that governs the quantum control process to decrease the difference among the dynamical phase factors. Take a simple and intuitive physical model to illustrate the principle. Consider the simple physical model: a simple rotation pulse $\exp (-i\omega _{p}I_{x}\Delta t)$ acting upon the state $|0\rangle $ to convert it into the state $|1\rangle :$ $$\exp (-i\omega _{p}I_{x}\Delta t)|0\rangle =\cos \theta |0\rangle -i\sin
\theta |1\rangle .$$Here the pulse field strength $\omega _{p}$ is the control parameter of the Hamiltonian $H_{p}=\omega _{p}I_{x}=\omega _{p}\sigma _{x}/2$ ($\sigma _{x}$ is the Pauli operator) that governs the state transfer process from the state $|0\rangle $ to the state $|1\rangle .$ The Hamiltonian $H_{p}$ is time- and space-independent. Note that the eigenvalue of the Hamiltonian $%
H_{p}$ is $\omega _{p}m_{x}$, here $m_{x}=1/2$ or $-1/2$ is the eigenvalue ($%
\hslash =1$) of the operator $I_{x}$. Then the dynamical phase factor $\exp
[-im_{x}\omega _{p}\Delta t]$ is determined by the dynamical phase angle, i.e., the rotation pulse angle $\theta =\theta (\Delta t)=\omega _{p}\Delta
t/2.$ The dynamical phase angle is proportional to the pulse duration $%
\Delta t$ and the pulse field strength $\omega _{p}.$ The amplitude $\sin
\theta $ of the state $|1\rangle $ and the transfer efficiency ($\sin
^{2}\theta $) from the state $|0\rangle $ to the state $|1\rangle $ induced by the rotation pulse are determined by the dynamical phase angle $\theta $ up to a global phase factor $-i.$ If the rotation pulse angle $\theta
=\omega _{p}\Delta t/2=\pi /2,$ then the state $|0\rangle $ is transferred completely to the state $|1\rangle $ by the rotation pulse and in this case the pulse duration $\Delta t$ is equal to $\Delta t_{p}=\pi /\omega _{p}.$ Now suppose that the pulse duration $\Delta t$ is slightly different from $%
\Delta t_{p},$ for example, $\Delta t=\Delta t_{p}-\delta t$ with $|\delta
t|<<\Delta t_{p}.$ Then there is a difference between the dynamical phase angles $\theta (\Delta t)$ and $\theta (\Delta t_{p})=\pi /2$, which is given by $\Delta \theta (\Delta t)=\theta (\Delta t)-\theta (\Delta
t_{p})=-\omega _{p}\delta t/2.$ Therefore, the transfer efficiency is given by $$\sin ^{2}\theta (\Delta t)=\cos ^{2}[\Delta \theta (\Delta t)]=1-[\omega
_{p}\delta t/2]^{2}+....$$One sees that the transfer efficiency $\sin ^{2}\theta $ is determined by the dynamical-phase-angle difference $\Delta \theta (\Delta t).$ Now consider the case $|\delta t|<\pi /\omega _{p}$ such that $|\Delta \theta
(\Delta t)|<\pi /2.$ Then the smaller the dynamical-phase-angle difference $%
|\Delta \theta (\Delta t)|$, the higher the transfer efficiency $\sin
^{2}\theta $. Note that the dynamical-phase-angle difference $|\Delta \theta
(\Delta t)|$ is proportional to the time difference $\delta t$ and the pulse field strength $\omega _{p}$ which controls the Hamiltonian $H_{p}.$ Thus, the transfer efficiency may be increased and is closer to unity by decreasing either the time difference or the pulse field strength or both. By comparing this formula with Eq. (38) one sees that there is a correspondence $\delta t\leftrightarrow (j-1)\Delta Tv_{0}/v.$ Now suppose that this simple rotation-pulse model is available approximately for the quantum control process of Eq. (2). Then the probability $1-|\varepsilon
(t,t_{0j})|^{2}$ of the state $|c_{2}\rangle $ of the control state subspace in Eq. (2) could be evaluated roughly by setting $\delta t=(j-1)\Delta
Tv_{0}/v$ for $j=1,2,...,m_{r}$ in the transfer efficiency $\sin ^{2}\theta
(\Delta t)$ above$,$$$1-|\varepsilon (t,t_{0j})|^{2}\thickapprox 1-\frac{1}{4}[\omega
_{p}(j-1)\Delta T(v_{0}/v)]^{2}.$$The probability of the state $|c_{2}\rangle $ is dependent on the time-compressing factor $v_{0}/v$ in a quadratic form when $v_{0}/v<<1$. Actually, the current atomic physical model is much more complicated than the simple rotation-pulse model and especially the Hamiltonian that governs the quantum control process is time- and space-dependent in the atomic physical model so that this simple rotation-pulse model is not suited for describing the quantum control process in the atomic physical model, but the basic principle to make the reversible and unitary halting protocol insensitive to the initial states is the same for both the models!
In the paper the reversible and unitary halting protocol and the state-locking pulse field have been investigated in detail and extensively. Though the halting protocol for the universal quantum computational models usually is irreversible and non-unitary, it can be made reversible and unitary in the specific case that any initial state is limited to a single basis state in the halting protocol, and in this case it can also be simulated efficiently and faithfully (in a probability close to 100%) within the universal quantum computational models. From the viewpoint of the conventional halting-operation property that the output state of the halting operation may be completely independent of any initial state the reversible and unitary halting protocol is generally different from the classical irreversible one. In quantum computation the reversible and unitary halting protocol can only achieve such a unitary halting-operation property that the output state of the reversible and unitary halting protocol may be almost independent of any initial state (in a probability approaching to 100%) but it can not be completely independent of any initial state due to the limitation of unitarity, while the classical halting protocol may have completely the conventional halting-operation property in classical computation. The unitary halting-operation property is very important for the reversible and unitary halting protocol because whether or not the quantum search process built up out of the reversible and unitary halting protocol can solve efficiently the quantum search problem is mainly dependent upon this property. The state-locking pulse field plays a key role in constructing such a reversible and unitary halting protocol that has the unitary halting-operation property in quantum computation. It is shown in the paper that a reversible and unitary halting protocol may be simulated efficiently in a real quantum physical system. A simple atomic physical system which consists of an atomic ion or a neutral atom in the double-well potential field therefore is proposed to implement the reversible and unitary halting protocol. The correctness for the atomic physical model to simulate efficiently the reversible and unitary halting protocol is based on these facts that $(i)$ for a heavy atom the atomic wave-packet picture quantum mechanically is very close to the classical particle picture, $(ii)$ the quantum motional behavior of an atom is much like the classical motional behavior of a particle as an atom is generally much heavier than an electron, and $(iii)$ the time evolution processes for the atomic physical system in a variety of atomic motions still obey the Schrödinger equation and hence are governed by the unitary quantum dynamics. Perhaps this simple physical model could not be the best one, but it does provide one with a much intuitive physical picture to understand the mechanism of the reversible and unitary halting protocol and show how the state-locking pulse field works in the quantum control process to simulate efficiently the reversible and unitary halting protocol. From this simple physical model one can see clearly the reason why unitarity in quantum computation and hence the unitary quantum dynamics are so important.
The unitarity of the quantum control process is of crucial importance if the quantum circuit $Q_{c}$ is used to build up an efficient quantum search process based on the unitary quantum dynamics. It follows from the quantum control process that each possible wave-packet state $|1,CM0(j),R_{0}(j)%
\rangle $ of Eq. (31) for $j=1,2,...,m_{r}$ corresponds one-to-one to a unique arriving time $T_{j}$ (see (27)) and a wave-packet spatial position $%
R_{2,j}(t_{m_{r}})$ at the end time $t_{m_{r}}$ of the computational process (see Eq. (24)). On the other hand, it follows from the quantum circuit $%
Q_{c} $ that a different initial functional state $|f_{r}(x_{0})\rangle $ with the index $x_{0}=1,2,...,m_{r}$ corresponds one-to-one to a different wave-packet spatial position $R_{2,j}(t_{m_{r}})$ and also a different arriving time $T_{j}$ for the index $j=(x_{f}-x_{0}+1)\func{mod}m_{r}$. Therefore, an initial functional state $|f_{r}(x_{0})\rangle $ for $%
x_{0}=1,2,...,m_{r}$ corresponds one-to-one to an output wave-packet state $%
|1,CM0(j),R_{0}(j)\rangle $ of Eq. (31) with the index $j=(x_{f}-x_{0}+1)%
\func{mod}m_{r}$. The one-to-one correspondence is ensured by the wave-packet spatial-position order such as the inequality (25): $%
R_{2,m_{r}}(t_{m_{r}})<...<R_{2,2}(t_{m_{r}})<R_{2,1}(t_{m_{r}})$ and also the arriving-time order such as the inequality (27): $%
T_{1}<T_{2}<...<T_{m_{r}}$. The one-to-one correspondence between the initial functional states $\{|f_{r}(x_{0})\rangle \}$ and the output wave-packet states $\{|1,CM0(j),R_{0}(j)\rangle \}$ is not only necessary for the reversible and unitary halting protocol itself, but also it is of crucial importance if the quantum circuit $Q_{c}$ containing the quantum control unit that simulates efficiently the reversible and unitary halting protocol is used to construct the quantum search process. The quantum search process requires that the probabilities of the desired state $%
|1,CM0,R_{0}\rangle $ in the wave-packet states $\{|1,CM0(j),R_{0}(j)\rangle
\}$ (see Eqs. (31), (36), and (38)) become closer and closer to unity as the qubit number increases. This requires further that the time-compressing factor $(v_{0}/v)$ or the atomic motional velocity $v_{0}$ become smaller and smaller, as shown in Eqs. (36) and (38). Note that the atomic motional velocity $v$ is limited by $v<<c$ (the speed of light in vacuum). Therefore, as the qubit number increases the difference among these $m_{r}$ possible arriving times $\{T_{j}\}$ (see Eq. (28)) becomes smaller and smaller and so does the difference among these $m_{r}$ possible wave-packet spatial positions $\{R_{2,j}(t_{m_{r}})\}$ $(1\leq j\leq m_{r})$ (see Eq. (26)). If now the unitarity of the quantum control process is destroyed slightly due to some factors such as imperfect parameter settings, then the wave-packet spatial-position order such as the inequality (25) and/or the arriving-time order such as the inequality (27) could be easily destroyed in the quantum control process when the qubit number is large due to the small differences among these wave-packet spatial positions $\{R_{2,j}(t_{m_{r}})\}$ and among these arriving times $\{T_{j}\}$. Then the one-to-one correspondence between the initial functional states $\{|f_{r}(x_{0})\rangle \}$ and the output states $\{|1,CM0(j),R_{0}(j)\rangle \}$ could be destroyed easily and this will directly result in that the quantum search process based upon the quantum circuit $Q_{c}$ could not work normally.
In particular, the importance of the unitarity of the quantum control process can be seen more clearly from the inverse process of the quantum control process. It is known in the quantum circuit $Q_{c}$ that these $%
m_{r} $ possible initial functional states $\{|f_{r}(x_{0})\rangle \}$ are different from each other and also orthogonal to one another and so are these $m_{r}$ possible initial states $\{|0,CM0,R_{0}\rangle |0\rangle
|f_{r}(x_{0})\rangle \}$ of the quantum circuit. By the quantum circuit $%
Q_{c}$ these $m_{r}$ well-distinguished initial states are converted one-to-one into these $m_{r}$ output states $\{|1,CM0(j),$ $R_{0}(j)\rangle
|1\rangle |0\rangle \}$ for $j=(x_{f}-x_{0}+1)\func{mod}m_{r}=1,2,...,m_{r}$. Now for the $2^{n}-$ dimensional Hilbert state space $S(Z_{2^{n}})$ with a large qubit number $n$ which is taken as the quantum search space it is required by the quantum circuit $Q_{c}$ that all these $m_{r}$ possible probabilities $\{|A_{j}(CM0,R_{0},T_{t},T_{1})|^{2}\}$ of the state $%
|1,CM0,R_{0}\rangle $ in the wave-packet states $\{|1,CM0(j),R_{0}(j)\rangle
\}$ of Eq. (31) be close to unity, that is, $%
|A_{j}(CM0,R_{0},T_{t},T_{1})|^{2}>(1-\ln p(n)/n)$ for $j=1,2,...,m_{r},$ so that the quantum search process based on the quantum circuit can be made efficient. Thus, this means that all these $m_{r}$ wave-packet states $%
\{|1,CM0(j),$ $R_{0}(j)\rangle \}$ are very similar to the same state $%
|1,CM0,R_{0}\rangle $, indicating that all these states $%
\{|1,CM0(j),R_{0}(j)\rangle \}$ are very similar to each other and not yet orthogonal to one another. Thus, all these $m_{r}$ possible output states $%
\{|1,CM0(j),R_{0}(j)\rangle |1\rangle |0\rangle \}$ of the quantum circuit are very close to the same state $|1,CM0,R_{0}\rangle |1\rangle |0\rangle .$ Hence they are very similar to each other and not yet orthogonal to one another. The difference among these output states becomes also smaller and smaller as the qubit number increases. This property for these output states is completely different from that one for their corresponding initial states. When these output states $\{|1,CM0(j),R_{0}(j)\rangle |1\rangle
|0\rangle \}$ are transferred one-to-one back to the initial states $%
\{|0,CM0,R_{0}\rangle |0\rangle |f_{r}(x_{0})\rangle \}$ by the inverse processes of both the quantum control process and the quantum computational process of the quantum circuit the small difference among these output states is amplified to be a large one among those initial states, that is, almost indistinguishable output states are changed one-to-one back to well-distinguished initial states. Then some output states could not be converted correctly into their corresponding initial states by the inverse processes if the unitarity of the inverse processes is slightly destroyed. Therefore, the unitarity of the inverse processes is of crucial importance particularly in the first time stage of the whole inverse process that these output states are converted one-to-one back into those initial states. Here the first time stage of the whole inverse process is really the starting time stage (between $T_{t}+T_{1}$ and $t_{m_{r}}$) of the inverse process of the quantum control process in the quantum circuit.
Evidently, these $m_{r}$ possible initial states $\{|0,CM0,R_{0}\rangle
|0\rangle |f_{r}(x_{0})\rangle \}$ of the quantum circuit have the same global phase factor and are orthogonal to one another. On the other hand, their corresponding $m_{r}$ possible output states $\{|1,CM0(j),R_{0}(j)%
\rangle |1\rangle |0\rangle \}$ are very similar to each other and not yet orthogonal to one another, but the global phase factors for these output states may be very different from each other. This shows that the difference among these discrete and orthogonal initial states is mainly transformed to the global-phase difference among these similar output states by the quantum program or circuit $Q_{c}$ which consists of the time- and space-dependent unitary evolution processes. The conventional unitary operations which are generally space-independent are not generally suitable for constructing such a state unitary transformation. Only the time- and space-dependent unitary evolution processes can provide one with the best way to generate such a state unitary transformation. It is well known that there exists the square speedup limit for the standard quantum search algorithm whose unitary evolution process is space-independent and can be thought of as a single time-dependent evolution process. Then a quantum search process could be able to break through the square speedup limit if it is constructed with the time- and space-dependent unitary evolution processes. The cost to break through the square speedup limit is that the quantum search space is enlarged and the single time-dependent unitary evolution process is extended to a time- and space-dependent unitary evolution process. It is also known \[23\] that an unknown quantum state can be transferred efficiently and completely to a large state subspace from a small subspace in the $2^{n}-$dimensional Hilbert state space of an $n-$qubit spin system whose unitary evolution process is space-independent, while the inverse unitary state-transfer process is relatively hard in the same Hilbert space. Indeed, when the Hilbert state space is added extra space dimensions and hence the unitary evolution processes are extended to be in a multi-dimensional Hilbert state space of time and coordinate spaces, the inverse state-transfer process that an unknown quantum state is transferred to a small subspace from a large one now could become more efficient than before, as shown in the paper. However, there still exists a difference between the unitary state-transfer process and its inverse process in the time and space multi-dimensional Hilbert state space, although the difference may be much smaller than that one in the $2^{n}-$dimensional Hilbert space in which any unitary evolution process is space-independent. The results in the paper show that an unknown quantum state of a large state subspace may be transferred to a given state of a small state subspace in a probability close to 100%, but it is impossible due to the limitation of unitarity that an unknown quantum state is transferred completely (in the probability 100%) into a small state subspace from a large one by any given unitary quantum dynamical process. On the other hand, it has been shown that an unknown quantum state always can be transferred completely (in the probability 100%) into a large state subspace from a small one \[23\]. One therefore obtains an important theorem: *it is impossible to transfer completely (in the probability 100%) every quantum state of a large state subspace into a small state subspace in the Hilbert space of a quantum system by any given unitary dynamical process quantum mechanically*. This theorem could play an important role in understanding deeply the time evolution process of a quantum ensemble from a non-equilibrium state to the equilibrium state from viewpoint of the unitary quantum dynamics. It is well known in statistical physics that such a non-equilibrium evolution process in a quantum ensemble is generally described through the stochastic probability theory. A direct result of the theorem is that there may exist a computational-power difference \[23, 24\] between a time- and space-dependent unitary dynamical process, which transfers an unknown quantum state from one state subspace to another, and its inverse process in a quantum system consisting of many quantum bits. The computational-power difference for the quantum search problem is very large for the discrete $2^{n}-$dimensional Hilbert space of an $n-$qubit quantum system whose unitary evolution processes are space-independent, but it could become much smaller when the quantum system permits the time- and space-dependent unitary evolution processes in the quantum search process. Finally, it can be believed that a quantum system allowing to have time- and space-dependent unitary evolution processes will have a deep effect on the conventional quantum cloning theorem which usually works in a quantum system with single time-dependent unitary evolution processes.
The paper has not answered the question how high it is the mean motional energy of the unstable wave-packet state $|1,CM1(t_{0i}),R_{1}(t_{0i})%
\rangle $ of the halting-qubit atom in the left-hand harmonic potential well so that the halting-qubit atom can overcome the intermediate potential barrier and enter almost completely into the right-hand potential well by the quantum scattering process. One may determine the lower bound of the mean motional energy by solving the quantum scattering problem of the atomic physical system. One needs also to consider the effect of the wave-packet spread on the quantum control process as the wave-packet spread of the motional state of the halting-qubit atom usually broadens during the atomic motional processes in the quantum control process. The effect may be investigated by solving the Schrödinger equation of the atomic physical system for the quantum control process. However, it can be expected that for a heavy halting-qubit atom the wave-packet spread should not have a significant effect on the lower-bound probability of Eq. (36) or (38).
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\sum_{k=1}^{n}I_{kz}$ in the high temperature approximation for a homonuclear $n-$spin ($I=1/2$) system in the thermal equilibrium state in a high magnetic field ($B_{z}$) and at room temperature. The total magnetization $M_{0}$ of the spin system in the thermal equilibrium state is determined by $M_{0}=Tr\{\rho _{eq}\sum_{l=1}^{n}I_{lz}\}=\varepsilon
n2^{n}/4.$ Since all the multiple-quantum coherences in the spin system are created from the total spin magnetization $M_{0}$ by the suitable pulse sequences in the inphase NMR multiple-quantum experiments \[21\] and the factoring multiple-quantum experiment \[20\] the total spectral intensity of the multiple-quantum-transition spectrum is not more than the total spin magnetization $M_{0}$ up to a scale factor. In the factoring multiple-quantum experiment one part of the total spin magnetization $M_{0}$ are converted into the multiple-quantum coherences and the rest into the longitudinal magnetization and spin order components. Thus, the inphase NMR multiple-quantum spectrum for the factoring algorithm consists of those spectral lines of both the multiple-quantum coherences and the longitudinal magnetization and spin order components. Then the total spectral intensity for the NMR multiple-quantum spectrum is kept constant in experiment and equals really the total spin magnetization $M_{0}$ up to a scale factor (also see \[21b\]). Here the spectral line of the longitudinal magnetization and spin order components really overlaps with the zero-order quantum transition spectral line in the NMR multiple-quantum spectrum.
|
---
author:
- Yue Wang
- 'Hai-Hu Wen[^1]'
title: 'Doping dependence of the upper critical field in La$_{2-x}$Sr$_x$CuO$_4$ from specific heat'
---
Introduction
============
Determining the fundamental parameters of the superconducting state in high-$T_c$ cuprates is crucial to understanding the nature of the high-temperature superconductivity. The upper critical field $H_{c2}$ is one such quantity which is directly correlated with the microscopic coherence length $\xi$. Convenient methods for determining $H_{c2}$ mainly come from the resistive transport or magnetization measurements. For conventional low-$T_c$ superconductors, their low or moderate $H_{c2}$ enables one to draw the line of $H_{c2}(T)$ in the field-temperature phase diagram from the critical temperature $T_c$ to $T\rightarrow0$ K with the application of the magnetic field $H$ to suppress the superconductivity. Hence the zero-temperature $H_{c2}(0)$ can be accurately accessed. Moreover, it is shown that the temperature dependence of $H_{c2}(T)$ in these systems can be well described by the Werthamer-Helfand-Hohenberg (WHH) theory [@Werthamer66]. According to this theory, the $H_{c2}(0)$ can be estimated by the slope d$H_{c2}/$d$T$ in the vicinity of $T_c$. Therefore in many cases through investigating the behavior of $H_{c2}(T)$ near $T_c$ one can also satisfactorily obtain the $H_{c2}(0)$ of the sample.
In contrast, for high-$T_c$ cuprates the $H_{c2}(0)$ is inherently huge in parallel with their high $T_c$. In most cases the superconductivity could be removed only with very intense magnetic field $H$, which is not always accessible in experiments [@Ando99]. Thus most determination of $H_{c2}(0)$ in high-$T_c$ cuprates relies on the extrapolation of the high-temperature $H_{c2}(T)$ to $T\rightarrow0$ K based on the WHH theory. However, recent considerable resistive transport measurements indicate that the mapped $H_{c2}(T)$ shows unusual low temperature upward curvature, which is inconsistent with the saturation predicted by the WHH theory [@Vedeneev06]. This observation casts doubt on estimating $H_{c2}(0)$ using such extrapolation procedure.
Presumably the above difficulty is thought to arise from the strong superconducting fluctuations in high-$T_c$ cuprates, especially for the underdoped region [@Bonn06]. The recent Nernst effect and torque magnetometry experiments highlighted this proposal [@Wang06; @Wang05]. It is found that the vortex Nernst signal or diamagnetic magnetization persists well above $T_c$, indicating the survival of superconducting correlations at high $T$ though the coherent superconductivity has disappeared at $T_c$. By tracing the field scale at which the Nernst or diamagnetic signal vanishes, the $H_{c2}$ has been defined which is higher than that determined in resistive transports and importantly does not become zero through the $T_c$.
In the $H$-$T$ phase diagram, the $H_{c2}(T)$ line signifies the transition from the mixed state to the normal state. From the viewpoint of the specific heat (SH), associated with this transition is the increase of the electronic density of states (DOS) with increasing $H$ and eventually the recovery of the normal state DOS at $H=H_{c2}$. Moreover, theory has suggested a robust connection between the DOS in the mixed state and that in the normal state [@Hussey02]. Thus this recognition actually implies that there is a well-defined way to evaluate the $H_{c2}(0)$ from SH [@Gao05]. In this letter we present an analysis of the low-temperature SH in a series of La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) single crystals. The field-induced increase of the electronic SH in the mixed state has been quantitatively determined. Combination with the normal state electronic SH available in the literature, the $H_{c2}(0)$ has been extracted in a wide doping range across the whole superconducting phase diagram. It is found that the $H_{c2}(0)$ becomes larger as one moves from the underdoped region to the optimal doping point, and then falls with increasing doping in the overdoped region, forming a “dome" shape like the $T_c$. We have discussed the implications of this finding and proposed that there is an effective superconducting energy scale responsible for the $H_{c2}(0)$ behavior in the underdoped region.
Experiment
==========
$x$ 0.063 0.069 0.075 0.09 0.11 0.15 0.178 0.19 0.202 0.218 0.22 0.238
---------- ------- ------- ------- ------ ------ ------ ------- ------ ------- ------- ------ -------
$T_c$ 9.0 12.0 15.7 24.6 29.3 36.1 36.0 32.0 30.5 25.0 27.4 20.0
$A$ 0.26 0.28 0.26 0.28 0.32 0.57 0.94 1.2 1.33 1.55 1.8 2.37
$H_{c2}$ 16 17 39 53 75 82 56 45 45 44 36 17
We have performed the low-temperature SH measurement in LSCO throughout the entire superconducting phase diagram [@Wen05; @Wang07; @Wen04]. The samples used for the study were single crystals prepared by the traveling-solvent floating-zone method. The superconducting transition temperatures $T_c$, defined as the onset of the diamagnetic signal in the magnetic susceptibility, are summarized in table \[tab.1\]. The hole doping level of the sample $p$ is simply regarded as the Sr concentration $x$, which ranges from 0.063 to 0.238. The low-temperature SH was carried out on an Oxford Maglab cryogenic system with a thermal relaxation technique [@Wen04; @Liu05]. The data below 12 K were analyzed for magnetic field $H$ up to 12 T with $H$ parallel to the $c$ axis of the sample.
Results and discussion
======================
From the raw data, we have reliably separated the electronic SH $C_{\mathrm{el}}=\gamma T$ from other contributions such as the phonon SH and the possible Schottky anomaly [@Wen04; @Wang07]. In magnetic field $H$, for a superconductor, there is an increase of the electronic SH in the mixed state, denoted as $\gamma(H)T$. To investigate the field dependence of the $\gamma(H)$ of the sample is a useful method to identify the symmetry of the superconducting gap. For a conventional $s$-wave superconductor, in the mixed state the electronic DOS comes mainly from the vortex core regions. Since the vortex number increases linearly as $H$ increases, the $C_{\mathrm{el}}$ and hence the $\gamma(H)$ is proportional to $H$. For a $d$-wave superconductor, however, Volovik first pointed out that the electronic DOS is actually dominated by contributions from the outer regions of the vortex in a magnetic field [@Volovik93]. While the vortex number increases linearly with $H$, the inter-vortex distance is inversely proportional to $\sqrt{H}$. Thus both effects result in the $C_{\mathrm{el}}$ of a $d$-wave superconductor showing a $\sqrt{H}$ dependence. In experiments, the $\sqrt{H}$ behavior of the $C_{\mathrm{el}}$, that is, $\gamma(H)\propto\sqrt{H}$, has been observed in YBCO and LSCO [@Hussey02; @Wen04]. These have been taken as the bulk evidence for the $d$-wave symmetry of the superconducting gap in high-$T_c$ cuprates. Figure \[fig.1\] shows the $H$ dependence of the $\gamma(H)$ for all doping samples. It can be seen the data are well described by $\gamma(H)=A\sqrt{H}$ (shown as the solid curves) with $A$ a doping-dependent constant. The numerical values of the prefactor $A$ are listed in table \[tab.1\]. For completeness, the $A$ for $x=0.19$ reported by Nohara *et al.* is also included [@Nohara00]. The above result suggests that the $d$-wave symmetry of the superconducting gap dominates the whole phase diagram of LSCO, which is consistent with the result of the recent phase-sensitive measurements [@Tsuei04]. Furthermore, we have shown that the field-induced SH is inversely proportional to the nodal gap slope $v_\Delta$ (and the gap maximum $\Delta_0$), that is, $A\propto1/v_\Delta\propto1/\Delta_0$ [@Wen05]. From figure \[fig.1\] and table \[tab.1\] we can see the $A$ essentially decreases with decreasing doping. Thus this suggests that the $\Delta_0$ becomes larger towards underdoping [@Wang07]. Actually it has been shown that in the underdoped region, the $\Delta_0$ quantitatively tracks the pseudogap in the normal state [@Wen05].
As the $H$ keeps rising, the sample would inevitably undergo a phase transition into the normal state from the mixed state. In SH, this means the field-induced DOS increases and eventually the normal state electronic DOS is recovered. In other words, the $\gamma(H)$ rises and saturates to the normal state electronic SH coefficient $\gamma_N$ at $H=H_{c2}$. Specifically, theory shows that for a $d$-wave superconductor the relation between the above two quantities can be written as $$\label{eq.1}
\frac{\gamma(H)}{\gamma_N}=\sqrt{\frac{8}{\pi}}a\sqrt{\frac{H}{H_{c2}}},$$ where $a$ is a constant depending only on the vortex lattice geometry (=0.465 for a triangular vortex lattice) [@Volovik93; @Kubert98].
Combining eq. (\[eq.1\]) with the experimental result $\gamma(H)=A\sqrt{H}$, we get $H_{c2}=8a^2\gamma_N^2/\pi A^2$. This indicates that we may evaluate the $H_{c2}(0)$ from low-temperature SH provided we could also access the $\gamma_N$ of the sample at $T\rightarrow0$ K. Note that due to the large $H_{c2}(0)$ needed to suppress the superconductivity, the $\gamma_N$ at $T\rightarrow0$ K is hardly to be directly measured in SH for high-$T_c$ cuprates. One alternative way is to investigate the superconducting transition of the sample and extrapolate the $\gamma_N$ above $T_c$ to $T=0$ K based on the entropy conservation. In principle, we should do this performance in our own measurements. However, the uncertainty associated with the separation of the electronic SH from the phonon SH at elevated $T$ makes our relaxation technique not well suitable for such a purpose. In this respect the community agrees that the differential calorimetry may give more accurate results [@Fisher07]. Hence we instead look for the existing data in the literature. Actually, Matsuzaki *et al.* have recently estimated the $\gamma_N$ at $T\rightarrow0$ K of LSCO across the entire phase diagram in a systematic differential calorimetry study [@Matsuzaki04]. The doping dependence of the determined $\gamma_N$ is reproduced in the inset of fig. \[fig.2\]. A roughly linear increase of the $\gamma_N$ with increasing doping is established up to $x\simeq0.2$. It is interesting to notice that the electronic DOS at the Fermi level in LSCO revealed by the angle-integrated photoemission spectroscopy (AIPES) follows essentially the same doping evolution [@Ino98].
For underdoped cuprates, we note that the above result implies there exists finite DOS in the ground state when superconductivity is suppressed at $T=0$ K. Linked with the findings from the angle-resolved photoemission spectroscopy (ARPES), it is natural to speculate that this DOS resides on the Fermi arcs near ($\pi/2$, $\pi/2$) nodal points for underdoped cuprates. ARPES has revealed that at $T_c$ the Fermi surface is truncated forming arcs near the nodal regions in the pseudogap phase [@Norman98; @Yoshida03]. It is further found that with increasing doping the length of the arc increases approximately linearly and at high doping level the arcs eventually connect with each other near ($\pi$, 0) forming a large Fermi surface [@Yoshida06]. Therefore, to reconcile with the SH it is not unreasonable to expect that for underdoped cuprates this Fermi arc state persists to $T=0$ K as the ground state if the superconductivity was totally suppressed, which results in the finite DOS as shown in the SH [@Wen07]. Note that this notion has been supported by the independent nuclear magnetic resonance (NMR) study in Bi$_2$Sr$_{2-x}$La$_x$CuO$_{6+\delta}$ [@Zheng05].
With the knowledge of the $A$ and $\gamma_N$, we have derived the $H_{c2}(0)$ for LSCO according to the above expression, and plotted the result in the main panel of fig. \[fig.2\]. The numerical values are also presented in table \[tab.1\]. The determination of $H_{c2}(0)$ in LSCO from specific heat allows us to see its general behavior in a wide doping regime. As shown in fig. \[fig.2\], towards either underdoping or overdoping, the $H_{c2}(0)$ falls from its maximum value at optimal doping concomitant with the $T_c$, forming a similar “dome" shape as the $T_c$.
We have drawn the location of the $H_{c2}(0)$ in the $H$-$p$ diagram for LSCO from SH. To gain more insight into the experimental result, let us compare our $H_{c2}(0)$ with that determined by other methods. Ando *et al.* have reported the $c$ axis (interlayer) resistivity in intense magnetic field $H\parallel c$ for LSCO [@Ando99]. Choosing 90% of the normal-state resistivity as criterion, the $H_{c2}$ was determined at low $T$ and is plotted as the up triangles in fig. \[fig.2\]. We can see it shows good consistency with our result. Note that a similar “dome" shape of the $H_{c2}(0)$ in Bi$_2$Sr$_2$CaCu$_2$O$_{8+y}$ (Bi2212) was also obtained from the high-field interlayer resistive transport [@Krusin-Elbaum04]. In fig. \[fig.2\], the down triangles and diamonds represent the $H_{c2}$ for LSCO mapped in Nernst effect and torque magnetometry measurements, respectively [@Wang06; @Li07]. By extrapolating the high-field Nernst or diamagnetic signal to zero, the scale of $H_{c2}$ has been determined. It is shown that, aside from very underdoped region, the $H_{c2}$ obtained in Nernst effect and in SH agree reasonably with each other.
The quantitative agreement of the $H_{c2}(0)$ inferred from different methods supports the validity of our estimation of the $H_{c2}(0)$ from SH. Now we examine the implications of the experimental findings. From above we have shown $A\propto1/\Delta_0$ and $H_{c2}\propto(\gamma_N/A)^2$, thus we obtain $H_{c2}\propto(\gamma_N\Delta_0)^2$. This indicates that the $H_{c2}(0)$ of the sample is governed by both the superconducting pairing strength and the DOS contributing to the superconducting condensation [@Wen07]. In the overdoped region, as shown in the inset of fig. \[fig.2\], the $\gamma_N$ increases slightly with increasing doping and finally becomes almost a constant. In spite of this, the $H_{c2}(0)$ is found to drop down monotonically as doping increases from the optimal doped point. This means that it is the reduction of the $\Delta_0$ that dominates the behavior of $H_{c2}(0)$ in this region. In other words, the decrease of $H_{c2}(0)$ with overdoping should originate mainly from the reduction of the pairing strength.
Let us turn to the underdoped region, where the situation seems different. The SH measured in underdoped region has revealed that the $\Delta_0$ continues growing in the underdoped region [@Wen05], which is also found by the thermal conductivity [@Takeya02; @Sutherland03]. In analogy with the overdoped side, one may expect the $H_{c2}(0)$ would keep rising with underdoping provided that the pairing strength is still the decisive factor to determine the $H_{c2}(0)$. Clearly this is at odds with the present experimental result which shows that the $H_{c2}(0)$ declines as doping reduces from the optimal doped point. Therefore, contrary to the overdoped region, this indicates that in the underdoped region it is the fall of the $\gamma_N$ that overwhelms the rise of the $\Delta_0$ and leads to the dropping down of the $H_{c2}(0)$ as the doping decreases. In the above we have argued that in the underdoped region the $\gamma_N$ in SH corresponds to the DOS on the Fermi arcs remaining in the pseudogap phase. Thus we can say for underdoped cuprates the $H_{c2}(0)$ represents the field scale necessary to recover the DOS on Fermi arcs from the superconducting state. Since the Fermi arcs shrink, that is, the DOS available to the superconducting condensation reduces towards underdoping, the $H_{c2}(0)$ naturally decreases concomitantly.
The above analysis actually indicates that in the underdoped region the coherent superconductivity below $T_c$ may be triggered by the pairing of the carriers on nodal Fermi arcs. Since this pair process would be associated with the formation of an energy gap near the nodal region, it means that besides the pseudogap or the $\Delta_0$, there is an effective superconducting energy scale, denoted as $\Delta_{\mathrm{eff}}$, for underdoped cuprates. With decreasing doping, though the $\Delta_0$ increases, the $\Delta_{\mathrm{eff}}$ actually decreases since it represents the maximum gap on nodal Fermi arcs while the length of the Fermi arc reduces towards underdoping. For underdoped cuprates, the $T_c$ or $H_{c2}(0)$ is just the temperature or field scale necessary to close this $\Delta_{\mathrm{eff}}$ in a BCS-like fashion, respectively [@Wen98]. As one increases the $T$ from below to $T=T_c$ or applies the field $H=H_{c2}(0)$, the $\Delta_{\mathrm{eff}}$ closes and the carriers on nodal Fermi arcs are depaired, and therefore the coherent superconductivity is destroyed with the appearance of nodal Fermi arcs while the $\Delta_0$ near ($\pi$, 0) could remain unchanged. Note that this suggestion seems to be consistent with the very recent ARPES experiment which observed that a second energy gap opens at $T_c$ and has a BCS-like temperature dependence in underdoped Bi2212 [@Lee07]. In contrast, for overdoped cuprates, however, the situation is much simpler. With $H=H_{c2}(0)$, the superconductivity disappears with the vanishing of the $\Delta_0$ and the recovery of the DOS on the large Fermi surface.
In the very underdoped region, it is worth noting that the $H_{c2}(0)$ determined in Nernst effect and torque magnetometry seems to be larger than that determined in the present SH. This suggests that the $H_{c2}(0)$ probed by both methods may be different in this very region. We believe that it may originate from the strong superconducting fluctuations for underdoped cuprates [@Emery95]. In SH, since the $H_{c2}$ marks the disappearance of the coherent superconductivity and the recovery of the normal state electronic DOS near nodal regions, it vanishes as the $T_c$ is approached from the low temperature. However, due to sensitive to short-lived vortices, the Nernst effect or torque magnetometry may detect the fluctuating superconducting correlations above $T_c$ and thus give a higher $H_{c2}(0)$ which remains a finite value at $T_c$. Note that a recent scanning tunnelling microscopy (STM) experiment in Bi2212 reported a nucleation of pairing gaps in nanoscale regions above T$_c$, which was considered as a microscopic basis for the above fluctuating superconducting response [@Gomes07]. Under this circumstance, we view the $H_{c2}(0)$ defined in our SH as the field scale to destroy the phase coherence of the superconductivity in the very underdoped region and thus have the suggestion that the pairing of the carriers on nodal Fermi arcs is crucial to and occurs simultaneously with the onset of the phase coherence. While at the same region the $H_{c2}(0)$ defined in Nernst effect or torque magnetometry may mark a higher field scale to destroy the phase-disordered condensate. Note that this is in particular indicated by the fact that, as shown in fig. \[fig.2\], the $H_{c2}(0)$ probed in torque magnetometry varies continuously with $x$ down to $x=0.03$ while the coherent superconductivity only appears down to $x\approx0.055$ [@Li07]. On the other hand, it is interesting to note that for underdoped LSCO the $H_{c2}(0)$ probed in SH is actually roughly consistent in value with the field revealed in torque magnetometry which marks the melt of the vortex solid at $T\rightarrow0$ K and falls to zero as $x\rightarrow0.055$ [@Li07]. In the end, We mention that the above speculation is also backed by the observation that in the overdoped region where fluctuations are most weak or absent the $H_{c2}(0)$ determined in different methods agree with each other.
Finally, from $H_{c2}=\Phi_0/(2\pi\xi^2)$, we can obtain the doping dependence of the coherence length $\xi$. Figure \[fig.3\] shows the calculated $\xi$ from the $H_{c2}(0)$ (circles). It is shown that from the optimal doping point the $\xi$ grows towards underdoping or overdoping. Previously the doping dependence of $\xi$, regarded as the size of the vortex core, had been drawn from the systematic magnetization measurements in LSCO thin films [@Wen03], which is also plotted in fig. \[fig.3\] (squares). It can be seen that both experiments show a reasonable consistency, which assures again the reliability of obtaining $H_{c2}$ from low-temperature SH. Moreover, it should be noted that an increase of the $\xi$ with underdoping from the optimal doping point has also been suggested by the fluctuation magneto-conductivity [@Ando02] and the reversible magnetization [@Gao06] measurements in YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO). Furthermore, it could be instructive to compare $\xi$ with the the Pippard coherence length $\xi_p=\alpha\hbar v_F/k_BT_c$, where $v_F$ is the Fermi velocity and $\alpha$ a numerical constant of order unity [@Tinkham96]. As $v_F$ is nearly doping independent [@Zhou03], $\xi_p\propto
1/T_c$ and thus the variation of $\xi_p$ with doping is qualitatively the same as that of $\xi$ since both $T_c$ and $H_{c2}$ form a “dome" shape with doping. In terms of $\Delta_0$, on the other hand, $\xi_p$ can be also expressed as $\xi_p=\hbar
v_F/\beta\Delta_0$ with $\beta$ another numerical constant. Note that there seems to be inconsistency between the above two expressions for $\xi_p$ for underdoped region since $T_c$ and $\Delta_0$ have opposite doping dependence. Actually this seeming contradiction could be naturally resolved by substituting $\Delta_0$ with $\Delta_{\mathrm{eff}}$ in the expression of $\xi_p$, that is, $\xi_p=\hbar v_F/\beta\Delta_{\mathrm{eff}}$ for underdoped cuprates. This further suggests that there may be an effective superconducting energy scale determining the coherent superconductivity in the underdoped region.
Conclusion
==========
In summary, we have analyzed the low-temperature SH in LSCO throughout the whole superconducting dome to evaluate the zero-temperature $H_{c2}$ which is defined as the field scale necessary to remove the coherent superconductivity and recover the normal state electronic DOS. It is found that the doping dependence of $H_{c2}(0)$ essentially follows that of $T_c$. In the underdoped region, the decrease of $H_{c2}(0)$ concomitant with $T_c$ suggests that the coherent pairing of the carriers on nodal Fermi arcs plays important role in establishing the high-temperature superconductivity.
This work was supported by the National Science Foundation of China, the Ministry of Science and Technology of China (973 Projects No. 2006CB601000, No. 2006CB921802), and Chinese Academy of Sciences (Project ITSNEM).
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abstract: 'The multi-level Monte Carlo method proposed by M. Giles (2008) approximates the expectation of some functionals applied to a stochastic process with optimal order of convergence for the mean-square error. In this paper, a modified multi-level Monte Carlo estimator is proposed with significantly reduced computational costs. As the main result, it is proved that the modified estimator reduces the computational costs asymptotically by a factor $(p/\alpha)^2$ if weak approximation methods of orders $\alpha$ and $p$ are applied in case of computational costs growing with same order as variances decay.'
address:
- 'University of Southern Denmark, Department of Mathematics and Computer Science (IMADA), Campusvej 55, 5230 Odense M, Denmark'
- 'Universität zu Lübeck, Institut für Mathematik, Ratzeburger Allee 160, D-23562 Lübeck, Germany'
author:
- Kristian Debrabant
- 'Andreas Rö[ß]{}ler'
title: 'On the Acceleration of the Multi-Level Monte Carlo Method'
---
and
Multi-level Monte Carlo ,Monte Carlo ,variance reduction ,weak approximation ,stochastic differential equation\
MSC 2000: 65C30 ,60H35 ,65C20 ,68U20
Introduction {#Introduction}
============
The multi-level Monte Carlo method proposed in [@Gil08a] approximates the expectation of some functional applied to some stochastic processes like e. g. solutions of stochastic differential equations (SDEs) at a lower computational complexity than classical Monte Carlo simulation, see also [@Gil08b; @Hei01; @Ke05]. Multi-level Monte Carlo approximation is applied in many fields like mathematical finance [@Av09; @Gil09a], for SDEs driven by a Lévy process [@Der11] or by fractional Brownian motion [@KNP11]. The main idea of this article is to reduce the computational costs further by applying the multi-level Monte Carlo method as a variance reduction technique for some higher order weak approximation method. As a result, the computational effort can be significantly reduced while the optimal order of convergence for the root mean-square error is preserved.\
\
The outline of this paper is as follows. We give a brief introduction to the main ideas and results of the multi-level Monte Carlo method in Section \[Section2:MLMC-Simulation-Original\]. Based on these results, in Section \[Sec3:Improved-MLMC-Estimator\] we present as the main result a modified multi-level Monte Carlo algorithm that allows to reduce the computational costs significantly. Depending on the relationship between the orders of variance reduction and of the growth of the costs, there exists a reduction of the computational costs by a factor depending on the weak order of the underlying numerical method. As an example, the modified multi-level Monte Carlo algorithm is applied to the problem of weak approximation for stochastic differential equations driven by Brownian motion in Section \[Sec4:Numerical-Examples-SDEs\].
Multi-level Monte Carlo simulation {#Section2:MLMC-Simulation-Original}
==================================
Let $(\Omega, \mathcal{F}, {\operatorname{P}})$ be a probability space with some filtration $(\mathcal{F}_t)_{t \geq 0}$ and let $X=(X_t)_{t \in I}$ denote a stochastic process on the interval $I=[t_0,T]$ adapted to the filtration. In the following, we are interested in the approximation of ${\operatorname{E}}_{{\operatorname{P}}}(f(X))$ for some functional $f \in \mathbb{F}$ where $\mathbb{F}$ denotes a suitable class of functionals that are of interest. Further, let an equidistant discretization $I_h = \{t_0, t_1,
\ldots, t_N\}$ with $0 \leq t_0 < t_1 < \ldots < t_N =T$ of the time interval $I$ with step size $h$ be given. Then, we consider a probability space $(\tilde{\Omega}, \tilde{\mathcal{F}},
\tilde{{\operatorname{P}}})$ with some filtration $(\tilde{\mathcal{F}}_t)_{t \in
I_h}$ and we denote by $Y=(Y_{t})_{t \in I_h}$ a discrete time approximation of $X$ on the grid $I_h$, adapted to $(\tilde{\mathcal{F}}_t)_{t \in I_h}$. Here, the probability spaces $(\Omega, \mathcal{F}, {\operatorname{P}})$ and $(\tilde{\Omega},
\tilde{\mathcal{F}}, \tilde{{\operatorname{P}}})$ may be but do not have to be equal and we assume that $Y$ approximates $X$ in the weak sense with some order $p >0$, i.e.$$\| {\operatorname{E}}_{\tilde{{\operatorname{P}}}}(f(Y)) - {\operatorname{E}}_{{\operatorname{P}}}(f(X)) \| = {O}(h^p)$$ for all $f \in \mathbb{F}$.\
\
In order to approximate the expectation of $f(X)$ we apply the multi-level Monte Carlo estimator introduced in [@Gil08a]. For some fixed $M \in \mathbb{N}$ with $M \geq 2$ and some $L \in
\mathbb{N}$ we define the step sizes $h_l = \frac{T}{M^{l}}$ and let $Y^l=(Y_t)_{t \in I_{h_l}}$ denote the discrete time approximation process on the grid $I_{h_l}$ based on step size $h_l$ for $l=0,1,
\ldots, L$. Then, the multi-level Monte Carlo estimator is defined by $$\label{MLMC-estimator-Giles}
\hat{Y}_{ML} = \sum_{l=0}^L \hat{Y}^l$$ for some $L \in \mathbb{N}$ using the estimators $\hat{Y}^0 =
\frac{1}{N_0} \sum_{i=1}^{N_0} f({Y^0}^{(i)})$ and $$\hat{Y}^l = \frac{1}{N_l} \sum_{i=1}^{N_l} \left( f({Y^l}^{(i)})
- f({Y^{l-1}}^{(i)}) \right)$$ for $l=1, \ldots, L$. Then, we get $$\begin{split}
{\operatorname{E}}_{\tilde{{\operatorname{P}}}}(\hat{Y}_{ML})
&= {\operatorname{E}}_{\tilde{{\operatorname{P}}}}(f(Y^0)) + \sum_{l=1}^L {\operatorname{E}}_{\tilde{{\operatorname{P}}}}(f(Y^l) - f(Y^{l-1})) \, .
\end{split}$$ Here, we have to point out, that both approximations ${Y^l}^{(i)}$ and ${Y^{l-1}}^{(i)}$ are simulated simultaneously based on the same realisation of the underlying driving random process whereas $({Y^l}^{(i)},{Y^{l-1}}^{(i)})$ and $({Y^l}^{(j)},{Y^{l-1}}^{(j)})$ are independent realisations for $i \neq j$.\
\
Now, there are two sources of errors for the approximation. On the one hand, we have a systematical error due to the discrete time approximation $Y^h$ based on step size $h$ which is given by the bias of the method. On the other hand, there is a statistical error from the estimator for the expectation of $f(Y)$ by the Monte Carlo simulation. Therefore, we consider the root mean-square error $$\label{Sec2:root-mean-square-error}
e(\hat{Y}_{ML}) = \left( {\operatorname{E}}_{\tilde{{\operatorname{P}}}}(| \hat{Y}_{ML}
- {\operatorname{E}}_{{\operatorname{P}}}(f(X)) |^2) \right)^{1/2}$$ of the multi-level Monte Carlo method in the following. In order to rate the performance of an approximation method, we will analyse the root mean-square error of the method compared to the computational costs. Therefore, we denote by $C(Y)$ the computational costs of the approximation method $Y$. In order to determine $C(Y)$, one may use a cost model where e.g. each operation or evaluation of some function is charged with the price of one unit, i.e. one counts the number of needed mathematical operations or function evaluations. Further, each random number that has to be generated to compute $Y$ may also be charged with the price of one unit.\
\
Then, it is well known that the optimal order of convergence in the case of the classical Monte Carlo estimator $\hat{Y}_{MC} =
\frac{1}{N} \sum_{i=1}^N f({Y}^{(i)})$ is given by $e(\hat{Y}_{MC})
= O \left( (1/C(\hat{Y}_{MC}) )^{\frac{p}{2p+1}} \right)$ where $p$ is the weak order of convergence of the approximations $Y$, see Duffie and Glynn [@DuGl95]. Thus, higher order weak approximation methods result in a higher order of convergence with respect to the root mean-square error. Clearly, the best root mean-square order of convergence that can be achieved is at most $1/2$. However, the order bound $1/2$ can not be reached by any weak order $p$ approximation method in the case of the classical Monte Carlo simulation. Therefore, in order to attain the optimal order of convergence for the root mean-square error we apply the multi-level Monte Carlo estimator (\[MLMC-estimator-Giles\]). The following theorem due to Giles [@Gil08a] is presented in a slightly generalized version suitable for our considerations.
\[Main-Theorem-Giles\] For some $L \in \mathbb{N}$, let $Y^l$ denote the approximation process on the grid $I_{h_l}$ with respect to step size $h_l=\frac{T}{M^l}$ for each $l=0,1, \ldots, L$, respectively. Suppose that there exist some constants $\alpha > 0$ and $c_{1,\alpha}, c_{2,0}, c_{2}, c_{2,L} >0$ and $\beta,\beta_L>0$ such that for the bias
1. $|{\operatorname{E}}_{{\operatorname{P}}}(f(X))-{\operatorname{E}}_{\tilde{{\operatorname{P}}}}(f(Y^{L}))| \leq c_{1,\alpha} \, h_L^{\alpha}$
and for the variances
1. ${\operatorname{Var}}_{\tilde{{\operatorname{P}}}}(f(Y^0(T))) \leq c_{2,0} \, h_0^{\beta}$,
2. ${\operatorname{Var}}_{\tilde{{\operatorname{P}}}}(f(Y^l(T))-f(Y^{l-1}(T))) \leq c_{2} \, h_l^{\beta}$ for $l=1, \ldots, L-1$,
3. ${\operatorname{Var}}_{\tilde{{\operatorname{P}}}}(f(Y^L(T))-f(Y^{L-1}(T))) \leq c_{2,L} \, h_L^{\beta_{L}}$.
Further, assume that there exist constants $c_{3,0}, c_3, c_{3,L} >0$ and $\gamma, \gamma_{L} \geq 1$ such that for the computational costs
1. $C(Y^0(T)) \leq c_{3,0} \, T \, h_0^{-\gamma}$,
2. $C(Y^l(T),Y^{l-1}(T)) \leq c_{3} \, T \, h_l^{-\gamma}$ for $l=1, \ldots, L-1$,
3. $C(Y^L(T),Y^{L-1}(T)) \leq c_{3,L} \, T \, h_L^{-\gamma_{L}}$.
Then, for some arbitrarily prescribed error bound $\varepsilon>0$ there exist values $L$ and $N_l$ for $l=0,1, \ldots, L$, such that the root mean-square error of the multi-level Monte Carlo estimator $\hat{Y}_{ML}$ has the bound $$e(\hat{Y}_{ML}) < \varepsilon$$ with computational costs bounded by $$C(\hat{Y}_{ML}) \leq \begin{cases}
c_4 \, \varepsilon^{-2} & \text{ if } \beta>\gamma, \,
\beta_{L} \geq \gamma_{L}, \, \alpha \geq \frac{1}{2} \, \max \{\gamma, \gamma_L\} , \\
c_4 \, \varepsilon^{-2} \left( \log(\varepsilon) \right)^2 &
\text{ if } \beta=\gamma, \, \beta_{L} \geq \gamma_{L}, \, \alpha \geq \frac{1}{2} \, \max \{\gamma, \gamma_L\} , \\
c_4 \, \varepsilon^{-2-\frac{\max \{\gamma-\beta,\gamma_{L}-\beta_{L} \}}{\alpha}} &
\text{ if } \beta<\gamma, \,
\alpha \geq \frac{\max\{\gamma, \gamma_L\}
- \max \{ \gamma-\beta, \gamma_L-\beta_L\} }{2}, \end{cases}$$ for some positive constant $c_4$.
The calculations for the proof follow the lines of the original proof due to Giles [@Gil08a]. Considering the mean square-error $$e(\hat{Y}_{ML}) = \left( |{\operatorname{E}}_{{\operatorname{P}}}(f(X))-{\operatorname{E}}_{\tilde{{\operatorname{P}}}}(f(Y^{L}))|^2
+ {\operatorname{Var}}_{\tilde{{\operatorname{P}}}}(\hat{Y}_{ML}) \right)^{1/2} < \varepsilon$$ we make use of the weight $q \in \,]0,1[\,$ and claim that $$|{\operatorname{E}}_{{\operatorname{P}}}(f(X))-{\operatorname{E}}_{\tilde{{\operatorname{P}}}}(f(Y^{L}))|^2 < q \, \varepsilon^2 \quad \text{ and } \quad
{\operatorname{Var}}_{\tilde{{\operatorname{P}}}}(\hat{Y}_{ML}) < (1-q) \, \varepsilon^2 \, .$$ Then, we can calculate $L$ from the bias and we have to solve the minimization problem $$\min_{N_l:0 \leq l \leq L} C(\hat{Y}_{ML})$$ under the constraint that ${\operatorname{Var}}_{\tilde{{\operatorname{P}}}}(\hat{Y}_{ML}) <
(1-q) \, \varepsilon^2$. As a result of this, we obtain the following values for $L$ and $N_l$:\
\
Let $$L = \left\lceil \frac{\log(q^{-\frac{1}{2}} \, c_{1,\alpha} \, \varepsilon^{-1} \, T^{\alpha})}
{\alpha \, \log(M)} \right\rceil$$ and $N_0 = \left\lceil \frac{1}{1-q} \, \varepsilon^{-2} \, h_0^{\frac{\beta+\gamma}{2}}
\left( \frac{c_{2,0}}{c_{3,0}} \right)^{\frac{1}{2}} \, \kappa \right\rceil$, $$\label{N_l_Def_Allgemein}
N_l = \left\lceil \frac{1}{1-q} \, \varepsilon^{-2} \, h_l^{\frac{\beta+\gamma}{2}}
\left( \frac{c_{2}}{c_{3}} \right)^{\frac{1}{2}} \, \kappa \right\rceil$$ for $l=1, \ldots, L-1$ and $ N_L = \left\lceil \frac{1}{1-q} \, \varepsilon^{-2} \, h_L^{\frac{\beta_{L}+\gamma_{L}}{2}}
\left( \frac{c_{2,L}}{c_{3,L}} \right)^{\frac{1}{2}} \, \kappa \right\rceil $ for some $q \in \, ]0,1[\,$ with:
- In case of $\beta>\gamma$ and $\beta_{L} \geq \gamma_{L}$ or in case of $\beta<\gamma$ and $\gamma_{L}-\beta_{L} \leq \gamma-\beta$: $$\kappa = \left(c_{2,0} c_{3,0} \right)^{\frac{1}{2}} T^{\frac{\beta-\gamma}{2}}
+ \left( c_2 c_3 \right)^{\frac{1}{2}} \frac{(M^{-1} T)^{\frac{\beta-\gamma}{2}}
- h_L^{\frac{\beta-\gamma}{2}}}
{1-M^{\frac{\gamma-\beta}{2}}} + \left( c_{2,L} c_{3,L} \right)^{\frac{1}{2}}
h_L^{\frac{\beta_{L}-\gamma_{L}}{2}} \, .$$
- In case of $\beta=\gamma$ and $\beta_{L} \geq \gamma_{L}$: $$\kappa = \left(c_{2,0} c_{3,0} \right)^{\frac{1}{2}}
+ (L-1) \left( c_{2} c_{3} \right)^{\frac{1}{2}}
+ \left( c_{2,L} c_{3,L} \right)^{\frac{1}{2}} \,
h_L^{\frac{\beta_{L}-\gamma_{L}}{2}} \, .$$
The improved multi-level Monte Carlo estimator {#Sec3:Improved-MLMC-Estimator}
==============================================
The order of convergence of the multi-level Monte Carlo estimator $\hat{Y}_{ML}$ given in (\[MLMC-estimator-Giles\]) is optimal in the given framework. However, the computational costs can be reduced if a modified estimator is applied. As yet, the estimator $\hat{Y}_{ML}$ is based on some weak order $\alpha$ approximations $Y^l$ for $l=0,1, \ldots, L$ on each level. Now, let us apply some cheap low order weak approximation $Y^l$ on levels $l=0,1, \ldots,
L-1$ combined with some probably expansive high order weak approximation $\check{Y}^L$ on the finest level $L$. The idea is, that the approximations $Y^l$ contribute a variance reduction while the approximation $\check{Y}^L$ results in a small bias of the multi-level Monte Carlo estimator, thus reducing the number of levels needed to attain a prescribed accuracy.\
\
Let $Y$ be an order $\alpha$ weak approximation method and let $\check{Y}$ be an order $p$ weak approximation method applied on the finest level. Further, let $L=L_p$ with $$L_p = \left\lceil \frac{\log(q^{-\frac{1}{2}} \, c_{1,p} \, \varepsilon^{-1} \, T^{p})}
{p \, \log(M)} \right\rceil$$ denote the number of levels in order to indicate the dependence on the weak order $p$. Then, we define the modified multi-level Monte Carlo estimator by $$\label{Sec2:modified-MLMC-estimator}
\hat{Y}_{ML(\alpha,p)} = \sum_{l=0}^{L_p} \hat{Y}^l$$ with the estimators $\hat{Y}^l$ for $l=0,1, \ldots, L_p-1$ based on the order $\alpha$ weak approximations $Y^l$ as defined in Section \[Section2:MLMC-Simulation-Original\], however now applying the modified estimator $$\hat{Y}^{L_p} = \frac{1}{N_{L_p}} \sum_{i=1}^{N_{L_p}}
\left( f({\check{Y}^{L_p}})^{(i)} - f({Y^{L_p-1}})^{(i)} \right)$$ which combines the weak order $\alpha$ approximations $Y^{L_p-1}$ with the weak order $p$ approximations $\check{Y}^{L_p}$. Clearly, all conditions of Theorem \[Main-Theorem-Giles\] have to be fulfilled for $Y^L$ replaced by $\check{Y}^L$. Then, in the case of $p > \alpha$, the improved multi-level Monte Carlo estimator $\hat{Y}_{ML(\alpha,p)}$ features significantly reduced computational costs compared to the originally proposed estimator $\hat{Y}_{ML}=\hat{Y}_{ML(\alpha,\alpha)}$.
\[Main-Prop-Improvement\] Let conditions 1)–7) of Theorem \[Main-Theorem-Giles\] be fulfilled and suppose that there exist constants $\hat{c}_{3,0}, \hat{c}_3,
\hat{c}_{3,L_p}, \delta_i > 0$ and $\hat{c}_{3,0}^{(i)}, \hat{c}_3^{(i)},
\hat{c}_{3,L_p}^{(i)} \geq 0$ such that for the computational costs
1. $C(Y^0(T)) = \hat{c}_{3,0} \, T \, h_0^{-\gamma}
+ \sum_{i=1}^{k} \hat{c}_{3,0}^{(i)} \, T \, h_0^{-\gamma+\delta_i}$,
2. $C(Y^l(T),Y^{l-1}(T)) = \hat{c}_{3} \, T \, h_l^{-\gamma}
+ \sum_{i=1}^{k} \hat{c}_{3}^{(i)} \, T \, h_l^{-\gamma+\delta_i}$ for $l=1, \ldots, L_p-1$,
3. $C(\check{Y}^{L_p}(T),Y^{L_p-1}(T)) = \hat{c}_{3,L_p} \, T \, h_{L_p}^{-\gamma_{L_p}}
+ \sum_{i=1}^{k} \hat{c}_{3,L_p}^{(i)} \, T \, h_{L_p}^{-\gamma_{L_p}+\delta_i}$
with some $\gamma, \gamma_{L_p} \geq 1$ such that $\gamma-\delta_i \geq 1$ and $\gamma_{L_p}-\delta_i \geq 1$. Then, the multi-level Monte Carlo estimator $\hat{Y}_{ML(\alpha,p)}$ based on a weak order $\alpha>0$ approximation scheme on levels $0,1, \ldots, L_p-1$ and some weak order $p > \alpha$ approximation scheme on level $L_p$ has reduced computational costs:
i) In case of $\beta>\gamma$ and $\beta-\gamma<\beta_{L_p}-\gamma_{L_p}$, there exists some $\varepsilon_0>0$ such that for all $\varepsilon \in \, ]0,\varepsilon_0]$ holds $$\label{Main-Prop-Improvement-Aussage1}
\frac{C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon)}
{C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)} > 1$$ provided that $\alpha \geq \frac{\gamma}{2}$, $p \geq \frac{1}{2} \max \{\gamma, \gamma_{L_p}\}$ and $p>\tfrac{1}{4} \max \{\beta+\gamma,\beta-\gamma+2\gamma_{L_p} \}$. In case of $\beta>\gamma$ and $\beta-\gamma=\beta_{L_p}-\gamma_{L_p}$ then (\[Main-Prop-Improvement-Aussage1\]) holds if in addition $c_2 c_3 > (1-M^{\frac{\gamma-\beta}{2}})^2 c_{2,L_p} c_{3,L_p}$ and $ \hat{c}_3^2 \frac{c_2}{c_3} > (1-M^{\frac{\gamma-\beta}{2}})^2
\hat{c}_{3,L_p}^2 \frac{c_{2,L_p}}{c_{3,L_p}}$. Further, for $0<\beta-\gamma \leq \beta_{L_p}-\gamma_{L_p}$ it holds $C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)
= O( \varepsilon^{-2} )$ if $\alpha>0$ and $p \geq \tfrac{1}{2} \max \{\gamma,
\gamma_{L_p}\}$.\
ii) In case of $\beta=\gamma$ and $\beta_{L_p} \geq \gamma_{L_p}$ and if $p \geq \frac{1}{2} \max \{\gamma, \gamma_{L_p} \}$, $\alpha \geq \frac{\gamma}{2}$, we get $$\label{Main-Prop-Improvement-Aussage2}
\lim_{\varepsilon \to 0} \frac{C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon)}
{C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)} \geq \left( \frac{p}{\alpha}
\right)^2$$ and $C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)
= O( \varepsilon^{-2} ( \log(\varepsilon) )^2 )$ if $\alpha>0$ and $p \geq \frac{1}{2} \max \{\gamma, \gamma_{L_p}\}$.\
iii) In case of $\beta<\gamma$ and $\gamma-\beta=\gamma_{L_p}-\beta_{L_p}$ we obtain $$\label{Main-Prop-Improvement-Aussage3}
\begin{split}
\lim_{\varepsilon \to 0} \frac{C(\hat{Y}_{ML(p,p)})(\varepsilon)}
{C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)}
\geq & \,
M^{2(\gamma-\beta)} \left( \frac{\hat{c}_3 c_2}{\hat{c}_{3,L_p} c_{2,L_p}} + \frac{\hat{c}_3 (c_2 c_{3,L_p})^{1/2}}{\hat{c}_{3,L_p} (c_{2,L_p} c_3)^{1/2}}
\left( M^{\frac{\gamma-\beta}{2}} -1 \right) \right. \\
& \left. + \left( \frac{c_2 c_3}{c_{2,L_p} c_{3,L_p}} \right)^{1/2}
\left( M^{\frac{\gamma-\beta}{2}} -1 \right) + \left( M^{\frac{\gamma-\beta}{2}} -1 \right)^2 \right)^{-1}
\end{split}$$ if $p > \frac{1}{2} (\max\{\gamma, \gamma_{L_p}\} - \gamma+\beta)$. If the parameter $q \in \,]0,1[\,$ is chosen as $$\label{Main-Prop-Improvement-Aussage4}
q = \frac{\gamma-\beta}{\gamma-\beta+2p}$$ then the computational costs $C(\hat{Y}_{ML(\alpha,p)})$ are asymptotically minimal. In general, if $\beta<\gamma$ or if $\beta_{L_p}<\gamma_{L_p}$ then it holds that $C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)
= O \big( \varepsilon^{-2-\frac{\max \{\gamma-\beta,\gamma_{L_p}-\beta_{L_p} \}}{p}} \big)$ for $p \geq \frac{1}{2} (\max\{\gamma, \gamma_{L_p}\} -
\min \{\gamma-\beta,\gamma_{L_p}-\beta_{L_p}\})$.
[**[Proof.]{}**]{} Assume that $\varepsilon<1$. Let $\delta_0=0$, $\hat{c}_{3,0}^{(0)}=\hat{c}_{3,0}$, $\hat{c}_{3}^{(0)}=\hat{c}_{3}$ and $\hat{c}_{3,L_p}^{(0)}=\hat{c}_{3,L_p}$. Then, the computational costs for $\hat{Y}_{ML(\alpha,p)}$ are $$\begin{split}
C(\hat{Y}_{ML(\alpha,p)}) = & \, \sum_{i=0}^k \hat{c}_{3,0}^{(i)} \, T \, h_0^{-\gamma+\delta_i}
\, N_0 + \sum_{i=0}^k \sum_{l=1}^{L_p-1} \hat{c}_3^{(i)} \, T \, h_l^{-\gamma+\delta_i}
\, N_l \\
& \, + \sum_{i=0}^k \hat{c}_{3,L_p}^{(i)} \, T \, h_{L_p}^{-\gamma_{L_p}+\delta_i}
\, N_{L_p}
\end{split}$$ with $L=L_p = \left\lceil \frac{\log(q^{-\frac{1}{2}} \, c_{1,p} \, \varepsilon^{-1} \, T^p)}
{p \, \log(M)} \right\rceil$ and $N_l$ for $l=0,1, \ldots, L_p$ given in . Without loss of generality, suppose that $\delta_i \neq \delta_j$ for $i \neq j$ and that $\delta_{k}=\tfrac{\gamma-\beta}{2}$ with $\hat{c}_{3,0}^{(k)} = \hat{c}_3^{(k)} = \hat{c}_{3,L_p}^{(k)} = 0$ in the case of $\beta \geq \gamma$. In the following, we make use of the two estimates $$\begin{aligned}
L_{\alpha} &\geq \frac{\log(\varepsilon^{-1})}{\alpha \, \log(M)}
+ \frac{\log(q^{-\frac{1}{2}} \, c_{1,\alpha} \,
T^{\alpha})}{\alpha \, \log(M)} \, ,
\label{L-lower-bound} \\
L_p - 1 &\leq \frac{\log(\varepsilon^{-1})}{p \log(M)}
+ \frac{\log(q^{-\frac{1}{2}} c_{1,p} T^p)}{p \log(M)} \, .
\label{L-1-upper-bound}\end{aligned}$$ Let $\beta \neq \gamma$. Then, we obtain the lower bound [$$\begin{aligned}
C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon)
\geq &\, \frac{T \kappa \, \varepsilon^{-2}}{1-q}
\sum_{i=0}^k \left(
h_0^{\frac{\beta-\gamma}{2}+\delta_i} \hat{c}_{3,0}^{(i)}
\left(\frac{c_{2,0}}{c_{3,0}} \right)^{1/2}
+ \sum_{l=1}^{L_{\alpha}} h_l^{\frac{\beta-\gamma}{2}+\delta_i}
\hat{c}_3^{(i)} \left( \frac{c_{2}}{c_3} \right)^{1/2}
\right) \notag \\
\geq& \, \frac{T}{1-q} \, \varepsilon^{-2} \Bigg[
\sum_{i=0}^k T^{\beta-\gamma+\delta_i} \hat{c}_{3,0}^{(i)} c_{2,0}\notag \\
& + \sum_{i=0}^k T^{\frac{\beta-\gamma}{2}+\delta_i} \hat{c}_{3,0}^{(i)}
\left( \frac{c_{2,0} c_2 c_3}{c_{3,0}} \right)^{1/2}
\frac{ T^{\frac{\beta-\gamma}{2}} - h_{L_{\alpha}}^{\frac{\beta-\gamma}{2}}}{M^{\frac{\beta-\gamma}{2}}-1} \notag \\
& + \sum_{i=0}^{k-1} \hat{c}_{3}^{(i)} \left( \frac{c_2 c_{2,0} c_{3,0}}{c_3} \right)^{1/2} T^{\frac{\beta-\gamma}{2}} \cdot
\frac{T^{\frac{\beta-\gamma}{2}+\delta_i}
- h_{L_{\alpha}}^{\frac{\beta-\gamma}{2}+\delta_i}}{M^{\frac{\beta-\gamma}{2}+\delta_i}-1} \notag \\
& + \hat{c}_3^{(k)} \left( \frac{c_2 c_{2,0} c_{3,0}}{c_3} \right)^{1/2}
T^{\frac{\beta-\gamma}{2}} \left( \frac{\log(\varepsilon^{-1})}{\alpha \log(M)} + \frac{\log(q^{-\frac{1}{2}}
c_{1,\alpha} T^{\alpha})}{\alpha \log(M)} \right) \notag \\
& + \sum_{i=0}^{k-1} \hat{c}_{3}^{(i)} c_2 \frac{T^{\frac{\beta-\gamma}{2}+\delta_i}
- h_{L_{\alpha}}^{\frac{\beta-\gamma}{2}+\delta_i}}{M^{\frac{\beta-\gamma}{2}+\delta_i}-1} \cdot
\frac{T^{\frac{\beta-\gamma}{2}} - h_{L_{\alpha}}^{\frac{\beta-\gamma}{2}}}{M^{\frac{\beta-\gamma}{2}}-1} \notag \\
&+ \hat{c}_3^{(k)} c_2
\frac{T^{\frac{\beta-\gamma}{2}} - h_{L_{\alpha}}^{\frac{\beta-\gamma}{2}}}{M^{\frac{\beta-\gamma}{2}}-1}
\left( \frac{\log(\varepsilon^{-1})}{\alpha \log(M)} + \frac{\log(q^{-\frac{1}{2}}
c_{1,\alpha} T^{\alpha})}{\alpha \log(M)} \right)
\Bigg] \label{Proof-Main-Prop-Lower-Bound-Ieq-alg}
$$ ]{} where $\hat{c}_{3,L_{\alpha}}^{(i)} = \hat{c}_{3}^{(i)}$, $c_{2,L_{\alpha}}=c_2$, $c_{3,L_{\alpha}}=c_3$, $\beta_{L_{\alpha}}=\beta$ and $\gamma_{L_{\alpha}}=\gamma$ for $\hat{Y}_{ML(\alpha,\alpha)}$.\
\
Next, we calculate for the case of $\beta \neq \gamma$ the upper bound [$$\begin{aligned}
C(\hat{Y}_{ML(\alpha,p)})(\varepsilon) \leq &\, \frac{T \kappa \, \varepsilon^{-2}}{1-q}
\sum_{i=0}^k \left(
h_0^{\frac{\beta-\gamma}{2}+\delta_i} \hat{c}_{3,0}^{(i)} \left(\frac{c_{2,0}}{c_{3,0}} \right)^{1/2}
+ \sum_{l=1}^{L_p-1} h_l^{\frac{\beta-\gamma}{2}+\delta_i}
\hat{c}_3^{(i)} \left( \frac{c_{2}}{c_3} \right)^{1/2} \right. \notag \\
& \left. \, + h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}+\delta_i}
\hat{c}_{3,L_p}^{(i)} \left( \frac{c_{2,L_p}}{c_{3,L_p}} \right)^{1/2}
\right) \notag \\
& + T \sum_{i=0}^k \left( \hat{c}_{3,0}^{(i)} h_0^{-\gamma+\delta_i}
+ \hat{c}_3^{(i)} \sum_{l=1}^{L_p-1} h_l^{-\gamma+\delta_i}
+ \hat{c}_{3,L_p}^{(i)} h_{L_p}^{-\gamma_{L_p}+\delta_i} \right) \notag \\
\leq & \, \frac{T}{1-q} \, \varepsilon^{-2} \Bigg[
\sum_{i=0}^k \hat{c}_{3,0}^{(i)} \left( c_{2,0} T^{\beta-\gamma+\delta_i}
+ \left( \frac{c_{2,0} c_2 c_3}{c_{3,0}}
\right)^{1/2} \Lambda_0 T^{\frac{\beta-\gamma}{2}+\delta_i} \right. \notag \\
& + \left. \left( \frac{c_{2,0} c_{2,L_p} c_{3,L_p}}{c_{3,0}}
\right)^{1/2} T^{\frac{\beta-\gamma}{2}+\delta_i} h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}}
\right) \notag \\
& + \sum_{i=0}^{k-1} \hat{c}_3^{(i)} \left( \frac{c_{2,0} c_{3,0} c_2}{c_3} \right)^{1/2}
T^{\frac{\beta-\gamma}{2}} \Lambda_i
+ \sum_{i=0}^{k-1} \hat{c}_3^{(i)} c_2 \Lambda_i \Lambda_0 \notag \\
& + \left( \hat{c}_3^{(k)} c_2 \Lambda_0
+ \hat{c}_3^{(k)} \left( \frac{c_{2} c_{2,L_p} c_{3,L_p}}{c_3} \right)^{1/2}
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}} \right. \notag \\
& + \left. \hat{c}_3^{(k)} \left( \frac{c_{2,0} c_{3,0} c_2}{c_3} \right)^{1/2}
T^{\frac{\beta-\gamma}{2}} \right) \left(
\frac{\log(\varepsilon^{-1})}{p \log(M)}
+ \frac{\log(q^{-\frac{1}{2}} c_{1,p} T^p)}{p \log(M)} \right) \notag \\
& + \sum_{i=0}^{k-1} \hat{c}_3^{(i)} \left( \frac{c_{2} c_{2,L_p} c_{3,L_p}}{c_3} \right)^{1/2}
\Lambda_i h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}} \notag \\
& + \sum_{i=0}^k \hat{c}_{3,L_p}^{(i)}
\left( \frac{c_{2,0} c_{3,0} c_{2,L_p}}{c_{3,L_p}} \right)^{1/2} T^{\frac{\beta-\gamma}{2}}
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}+\delta_i} \notag \\
&+ \sum_{i=0}^k \hat{c}_{3,L_p}^{(i)} \left(
\left( \frac{c_{2} c_{3} c_{2,L_p}}{c_{3,L_p}} \right)^{1/2} \Lambda_0
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}+\delta_i}
+ c_{2,L_p} h_{L_p}^{\beta_{L_p}-\gamma_{L_p}+\delta_i} \right) \Bigg] \notag \\
& + T \sum_{i=0}^k \left( \hat{c}_{3,0}^{(i)} T^{\delta_i-\gamma}
+ \hat{c}_3^{(i)} \frac{(M^{-1} T)^{\delta_i-\gamma} - h_{L_p}^{\delta_i-\gamma}}
{1-M^{\gamma-\delta_i}} + \hat{c}_{3,L_p}^{(i)} h_{L_p}^{\delta_i-\gamma_{L_p}} \right)
\label{Proof-Main-Prop-Upper-Bound-Ieq-alg}
$$ ]{} with $\Lambda_i = \frac{(M^{-1}
T)^{\frac{\beta-\gamma}{2}+\delta_i} -
h_{L_p}^{\frac{\beta-\gamma}{2}+\delta_i}}{1-M^{\frac{\gamma-\beta}{2}-\delta_i}}$ for $i=0,\dots,k-1$.\
\
In case of $\beta>\gamma$ and $\beta_{L_p} > \gamma_{L_p}$, we prove that there exists some $\varepsilon_0>0$ such that for all $\varepsilon \in \, ]0,\varepsilon_0]$ follows $C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon) >
C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)$. From the lower bound (\[Proof-Main-Prop-Lower-Bound-Ieq-alg\]) for $C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon)$ and the upper bound (\[Proof-Main-Prop-Upper-Bound-Ieq-alg\]) for $C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)$ we get the estimate [$$\begin{aligned}
& C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon) - C(\hat{Y}_{ML(\alpha,p)})(\varepsilon) \notag \\
& \geq \frac{T}{1-q} \, \varepsilon^{-2} \left( \sum_{i=0}^{k-1} T^{\frac{\beta-\gamma}{2}+\delta_i} \hat{c}_{3,0}^{(i)}
\left( \frac{c_{2,0} c_2 c_3}{c_{3,0}} \right)^{1/2}
\frac{h_{L_p}^{\frac{\beta-\gamma}{2}} - M^{\frac{\gamma-\beta}{2}}
h_{L_{\alpha}}^{\frac{\beta-\gamma}{2}}}{1-M^{\frac{\gamma-\beta}{2}}} \right. \notag \\
& + \sum_{i=0}^{k-1} \hat{c}_{3}^{(i)} \left( \frac{c_2 c_{2,0} c_{3,0}}{c_3} \right)^{1/2} T^{\frac{\beta-\gamma}{2}} \cdot
\frac{h_{L_p}^{\frac{\beta-\gamma}{2}+\delta_i}
- M^{\frac{\gamma-\beta}{2}-\delta_i} h_{L_{\alpha}}^{\frac{\beta-\gamma}{2}+\delta_i}}{1-M^{\frac{\gamma-\beta}{2}-\delta_i}} \notag \\
& + \sum_{i=0}^{k-1} \hat{c}_{3}^{(i)} c_2 \left(
\frac{(M^{-1} T)^{\frac{\beta-\gamma}{2}+\delta_i} \left( h_{L_p}^{\frac{\beta-\gamma}{2}}
- M^{\frac{\gamma-\beta}{2}} h_{L_{\alpha}}^{\frac{\beta-\gamma}{2}} \right)
}
{(1-M^{\frac{\gamma-\beta}{2}-\delta_i}) (1-M^{\frac{\gamma-\beta}{2}}) } \right. \notag \\
& + \left. \frac{(M^{-1} T)^{\frac{\beta-\gamma}{2}} \left( h_{L_p}^{\frac{\beta-\gamma}{2}+\delta_i}
- M^{\frac{\gamma-\beta}{2}-\delta_i} h_{L_{\alpha}}^{\frac{\beta-\gamma}{2}+\delta_i} \right)
- h_{L_p}^{\beta-\gamma+\delta_i} + M^{\gamma-\beta-\delta_i} h_{L_{\alpha}}^{\beta-\gamma+\delta_i} }
{(1-M^{\frac{\gamma-\beta}{2}-\delta_i}) (1-M^{\frac{\gamma-\beta}{2}}) } \right) \notag \\
& - \sum_{i=0}^{k-1} \hat{c}_{3,0}^{(i)}
\left( \frac{c_{2,0} c_{2,L_p} c_{3,L_p}}{c_{3,0}}
\right)^{1/2} T^{\frac{\beta-\gamma}{2}+\delta_i} h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}}
\notag \\
& - \sum_{i=0}^{k-1} \hat{c}_3^{(i)} \left( \frac{c_{2} c_{2,L_p} c_{3,L_p}}{c_3} \right)^{1/2}
\Lambda_i h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}} \notag \\
& - \sum_{i=0}^{k-1} \hat{c}_{3,L_p}^{(i)}
\left( \frac{c_{2,0} c_{3,0} c_{2,L_p}}{c_{3,L_p}} \right)^{1/2} T^{\frac{\beta-\gamma}{2}}
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}+\delta_i} \notag \\
& \left. - \sum_{i=0}^{k-1} \hat{c}_{3,L_p}^{(i)} \left(
\left( \frac{c_{2} c_{3} c_{2,L_p}}{c_{3,L_p}} \right)^{1/2} \Lambda_0
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}+\delta_i}
+ c_{2,L_p} h_{L_p}^{\beta_{L_p}-\gamma_{L_p}+\delta_i} \right) \right) \notag \\
& - T \sum_{i=0}^{k-1} \left( \hat{c}_{3,0}^{(i)} T^{\delta_i-\gamma}
+ \hat{c}_3^{(i)} \frac{(M^{-1} T)^{\delta_i-\gamma} - h_{L_p}^{\delta_i-\gamma}}
{1-M^{\gamma-\delta_i}} + \hat{c}_{3,L_p}^{(i)} h_{L_p}^{\delta_i-\gamma_{L_p}} \right) .
\label{Proof-Main-Prop-Difference-Ieq-alg}
$$ ]{} In the following, we make us of $M^{-1}
c_{1,\alpha}^{-\frac{1}{\alpha}} q^{\frac{1}{2 \alpha}}
\varepsilon^{\frac{1}{\alpha}} \leq h_{L_{\alpha}} \leq
c_{1,\alpha}^{-\frac{1}{\alpha}} q^{\frac{1}{2 \alpha}}
\varepsilon^{\frac{1}{\alpha}}$ and $M^{-1} c_{1,p}^{-\frac{1}{p}}
q^{\frac{1}{2 p}} \varepsilon^{\frac{1}{p}} \leq h_{L_{p}} \leq
c_{1,p}^{-\frac{1}{p}} q^{\frac{1}{2 p}} \varepsilon^{\frac{1}{p}}$, i.e. we have $h_{L_{p}} \to 0$ and $h_{L_{\alpha}} \to 0$ as $\varepsilon \to 0$.\
\
Multiplying both sides of (\[Proof-Main-Prop-Difference-Ieq-alg\]) with $\frac{1-q}{T} \, \varepsilon^2 \, h_{L_p}^{-\frac{\min
\{\beta-\gamma,\beta_{L_p}-\gamma_{L_p}\}}{2}}$ and taking into account the assumptions $4p>\beta+\gamma$ and $4p>\beta-\gamma+2\gamma_{L_p}$ results in [$$\begin{aligned}
& \frac{1-q}{T} \, \varepsilon^2 \,
h_{L_p}^{-\frac{\min \{\beta-\gamma,\beta_{L_p}-\gamma_{L_p}\}}{2}}
\left( C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon) - C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)
\right) \notag \\
& \geq \Bigg[
\sum_{i=0}^{k-1} T^{\frac{\beta-\gamma}{2}+\delta_i} \hat{c}_{3,0}^{(i)}
\left( \frac{c_{2,0}}{c_{3,0}} \right)^{1/2} \left( \left( c_2 c_3 \right)^{1/2}
\frac{h_{L_p}^{\frac{\beta-\gamma}{2}} }{1-M^{\frac{\gamma-\beta}{2}}}
- \left( c_{2,L_p} c_{3,L_p} \right)^{1/2} h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}}
\right) \notag \\
& + T^{\frac{\beta-\gamma}{2}} (c_{2,0} c_{3,0})^{1/2}
\left( \hat{c}_{3}^{(0)} \left( \frac{c_2}{c_3} \right)^{1/2}
\frac{h_{L_p}^{\frac{\beta-\gamma}{2}}}{1-M^{\frac{\gamma-\beta}{2}}}
- \hat{c}_{3,L_p}^{(0)} \left( \frac{c_{2,L_p}}{c_{3,L_p}} \right)^{1/2} h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}} \right) \notag \\
& + \sum_{i=0}^{k-1} \hat{c}_{3}^{(i)} {c_2}^{1/2}
\frac{(M^{-1} T)^{\frac{\beta-\gamma}{2}+\delta_i} }
{1-M^{\frac{\gamma-\beta}{2}-\delta_i} } \left( (c_2)^{1/2}
\frac{h_{L_p}^{\frac{\beta-\gamma}{2}} }{1-M^{\frac{\gamma-\beta}{2}} }
- \left( \frac{c_{2,L_p} c_{3,L_p}}{c_3} \right)^{1/2}
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}} \right)
\notag \\
& + \frac{(M^{-1} T)^{\frac{\beta-\gamma}{2}} }
{1-M^{\frac{\gamma-\beta}{2}} } {c_2}^{1/2} \left( \hat{c}_{3}^{(0)} {c_2}^{1/2}
\frac{h_{L_p}^{\frac{\beta-\gamma}{2}} }
{1-M^{\frac{\gamma-\beta}{2}} }
- \hat{c}_{3,L_p}^{(0)} \left( \frac{c_{3} c_{2,L_p}}{c_{3,L_p}} \right)^{1/2}
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}} \right) \notag \\
&
+ o \left( h_{L_p}^{\frac{\min \{\beta_{L_p}-\gamma_{L_p}, \beta-\gamma \}}{2}} \right)
\Bigg]
h_{L_p}^{-\frac{\min \{\beta-\gamma,\beta_{L_p}-\gamma_{L_p}\}}{2}}\,. \label{Proof-Main-Prop-Difference2-Ieq-alg}
$$ ]{} As a result of (\[Proof-Main-Prop-Difference2-Ieq-alg\]) follows that in the case of $\beta-\gamma<\beta_{L_p}-\gamma_{L_p}$ there exists some $\varepsilon_0>0$ such that $$\label{Proof-Main-Prop-Result-case-1}
\frac{C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon)}
{C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)} > 1$$ for all $\varepsilon \in \, ]0,\varepsilon_0]$. In the case of $\beta-\gamma=\beta_{L_p}-\gamma_{L_p}$ there exists some $\varepsilon_0>0$ such that (\[Proof-Main-Prop-Result-case-1\]) holds for all $\varepsilon \in \, ]0,\varepsilon_0]$ if $c_2 c_3 >
(1-M^{\frac{\gamma-\beta}{2}})^2 c_{2,L_p} c_{3,L_p}$ and $\left(
\hat{c}_3^{(0)} \right)^2 \frac{c_2}{c_3} >
(1-M^{\frac{\gamma-\beta}{2}})^2 \left( \hat{c}_{3,L_p}^{(0)}
\right)^2 \frac{c_{2,L_p}}{c_{3,L_p}}$. Finally, $C(\hat{Y}_{ML(\alpha,p)})(\varepsilon) = O( \varepsilon^{-2} )$ follows from (\[Proof-Main-Prop-Upper-Bound-Ieq-alg\]).\
\
In case of $\beta<\gamma$ and $\beta<2p$, we have to compare the dominating terms as $\varepsilon \to 0$. Therefore, we get from the lower bound that $$\begin{aligned}
C(\hat{Y}_{ML(p,p)})(\varepsilon) \geq \,
& \frac{q^{\frac{\beta-\gamma}{2p}}}{1-q} \,
\varepsilon^{-2-\frac{\gamma-\beta}{p}} \, T \,
\hat{c}_{3,L_p}^{(0)} \, c_{2,L_p} \,
c_{1,p}^{\frac{\gamma-\beta}{p}}
\, M^{\gamma-\beta} \left( M^{\frac{\beta-\gamma}{2}}-1
\right)^{-2} \notag \\
& + o(\varepsilon^{-2-\frac{\gamma-\beta}{p}}) \label{Proof-Main-Prop-Lower-Bound-Ieq-simp}\end{aligned}$$ and from the upper bound $$\begin{aligned}
C(\hat{Y}_{ML(\alpha,p)})(\varepsilon) \leq \, &
\frac{q^{\frac{\beta-\gamma}{2p}}}{1-q} \,
\varepsilon^{-2-\frac{\gamma-\beta}{p}} \, T \,
c_{1,p}^{\frac{\gamma-\beta}{p}} \left(
\frac{\hat{c}_{3}^{(0)} c_{2}}{\left( 1-M^{\frac{\gamma-\beta}{2}} \right)^{2}}
- \frac{\hat{c}_{3}^{(0)} \left( c_{2} c_{2,L_p} c_{3,L_p} \right)^{1/2}}{c_3^{1/2}
\left( 1-M^{\frac{\gamma-\beta}{2}} \right)} \right. \notag \\
& - \left. \frac{\hat{c}_{3,L_p}^{(0)} \left( c_{2} c_3 c_{2,L_p} \right)^{1/2}}{c_{3,L_p}^{1/2}
\left( 1-M^{\frac{\gamma-\beta}{2}} \right)} + \hat{c}_{3,L_p}^{(0)} c_{2,L_p} \right)
+ o(\varepsilon^{-2-\frac{\gamma-\beta}{p}}) \label{Proof-Main-Prop-Upper-Bound-Ieq-simp}\end{aligned}$$ Making use of these two estimates (\[Proof-Main-Prop-Lower-Bound-Ieq-simp\]) and (\[Proof-Main-Prop-Upper-Bound-Ieq-simp\]), this results in the estimate (\[Main-Prop-Improvement-Aussage3\]) where $\beta_{L_p}<\gamma_{L_p}$ because we require that $\beta_{L_p}-\gamma_{L_p}=\beta-\gamma<0$.\
\
In general, it follows that $C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)
= O \big( \varepsilon^{-2-\frac{\max
\{\gamma-\beta,\gamma_{L_p}-\beta_{L_p} \}}{p}} \big)$ from the upper bound (\[Proof-Main-Prop-Upper-Bound-Ieq-alg\]) for $\beta<\gamma$ and any $\beta_{L_p}>0$, $\gamma_{L_p} \geq 1$. Further, there is an asymptotically optimal choice for the parameter $q \in \,]0,1[\,$ such that the computational costs are asymptotically minimal. Calculating a lower bound for $C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)$ and taking into account the upper bound (\[Proof-Main-Prop-Upper-Bound-Ieq-simp\]), we get $$ C(\hat{Y}_{ML(\alpha,p)})(\varepsilon) =
\frac{1}{1-q} \, \varepsilon^{-2-\frac{\gamma-\beta}{p}} \,
q^{\frac{\beta-\gamma}{2p}} \, C + o(\varepsilon^{-2-\frac{\gamma-\beta}{p}})$$ with some constant $C>0$ independent of $q$ and $\varepsilon$. Now, we have to find some $\hat{q} \in \,]0,1[\,$ such that $$C \varepsilon^{-2-\frac{\gamma-\beta}{p}}
\frac{\hat{q}^{\frac{\beta-\gamma}{2p}}}{1-\hat{q}}
= \min_{q \in \, ]0,1[ \,} C \varepsilon^{-2-\frac{\gamma-\beta}{p}}
\frac{q^{\frac{\beta-\gamma}{2p}}}{1-q}$$ for all $0<\varepsilon<1$. Solving this minimization problem leads to $$\hat{q} = \frac{\gamma-\beta}{\gamma-\beta+2p} .$$ which is asymptotically the optimal choice for $q \in \, ]0,1[\,$ in case of $\beta<\gamma$.\
\
In case of $\beta=\gamma$, we get the following lower bound [$$\begin{aligned}
C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon) \geq &\, \frac{T}{1-q} \, \varepsilon^{-2} \left(
\sum_{i=0}^k \hat{c}_{3,0}^{(i)} c_{2,0} h_0^{\delta_i}
+ \sum_{i=0}^k \hat{c}_{3,0}^{(i)} \left(\frac{c_{2,0} c_2 c_3}{c_{3,0}}\right)^{1/2}
L_{\alpha} h_0^{\delta_i} \right. \notag \\
& + \hat{c}_3^{(0)} \left(\frac{c_2 c_{2,0} c_{3,0}}{c_3} \right)^{1/2}
L_{\alpha} + \sum_{i=1}^k \hat{c}_3^{(i)} \left(\frac{c_2 c_{2,0} c_{3,0}}{c_3} \right)^{1/2}
\frac{T^{\delta_i}-h_{L_{\alpha}}^{\delta_i}}{M^{\delta_i}-1} \notag \\
& + \left. \hat{c}_3^{(0)} c_2 L_{\alpha}^2 + \sum_{i=1}^k \hat{c}_3^{(i)} c_2 L_{\alpha}
\frac{T^{\delta_i}-h_{L_{\alpha}}^{\delta_i}}{M^{\delta_i}-1} \right) \notag \\
\geq & \, \frac{T}{1-q} \, \varepsilon^{-2} \left(
\sum_{i=0}^k \hat{c}_{3,0}^{(i)} c_{2,0} T^{\delta_i}
+ \left( \sum_{i=0}^k \hat{c}_{3,0}^{(i)} \left( \frac{c_{2,0} c_2 c_3}{c_{3,0}} \right)^{1/2}
T^{\delta_i} \right. \right. \notag \\
& + \left. \hat{c}_3^{(0)} \left(\frac{c_2 c_{2,0} c_{3,0}}{c_3} \right)^{1/2}
+ \sum_{i=1}^k \hat{c}_3^{(i)} c_2
\frac{T^{\delta_i}-h_{L_{\alpha}}^{\delta_i}}{M^{\delta_i}-1} \right) \notag \\
& \times \left( \frac{\log(\varepsilon^{-1})}{\alpha \log(M)} + \frac{\log(q^{-\frac{1}{2}}
c_{1,\alpha} T^{\alpha})}{\alpha \log(M)} \right)
+ \hat{c}_3^{(0)} c_2 \left( \frac{\log(\varepsilon^{-1})}{\alpha \log(M)} \right)^2 \notag \\
& + 2 \hat{c}_3^{(0)} c_2 \frac{\log(\varepsilon^{-1}) \log(q^{-\frac{1}{2}}
c_{1,\alpha} T^{\alpha})}{\alpha^2 (\log(M))^2}
+ \hat{c}_3^{(0)} c_2 \left(\frac{\log(q^{-\frac{1}{2}} c_{1,\alpha} T^{\alpha})}{\alpha \log(M)}
\right)^2 \notag \\
& + \left. \sum_{i=1}^k \hat{c}_3^{(i)} \left(\frac{c_2 c_{2,0} c_{3,0}}{c_3} \right)^{1/2}
\frac{T^{\delta_i}-h_{L_{\alpha}}^{\delta_i}}{M^{\delta_i}-1} \right)
\label{Proof-Main-Prop-Lower-Bound-Eq-alg}
$$ ]{} where $\hat{c}_{3,L_{\alpha}}^{(i)}=\hat{c}_{3}^{(i)}$, $c_{2,L_{\alpha}}=c_2$, $c_{3,L_{\alpha}}=c_3$, $\beta_{L_{\alpha}}=\beta$ and $\gamma_{L_{\alpha}}=\gamma$ for $\hat{Y}_{ML(\alpha,\alpha)}$.\
\
Next, we calculate for $\beta=\gamma$ the upper bound [$$\begin{aligned}
C(\hat{Y}_{ML(\alpha,p)}) (\varepsilon)
\leq &\, \frac{T}{1-q} \, \varepsilon^{-2} \Bigg[
\sum_{i=0}^k \hat{c}_{3,0}^{(i)} c_{2,0} T^{\delta_i}
+ \sum_{i=1}^k \hat{c}_3^{(i)} \left( \frac{c_{2,0} c_{3,0} c_2}{c_3} \right)^{1/2}
\Lambda_i\notag \\
& + \sum_{i=0}^k \hat{c}_{3,0}^{(i)} \left( \frac{c_{2,0} c_{2,L_p} c_{3,L_p}}{c_{3,0}} \right)^{1/2}
T^{\delta_i} h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}} \notag \\
& + \left( \hat{c}_3^{(0)} \left( \frac{c_{2,0} c_{3,0} c_2}{c_3} \right)^{1/2}
+ \sum_{i=0}^k \hat{c}_{3,0}^{(i)} \left( \frac{c_{2,0} c_2 c_3}{c_{3,0}} \right)^{1/2}
T^{\delta_i} \right. \notag \\
& + \hat{c}_3^{(0)} \left( \frac{c_2 c_{2,L_p} c_{3,L_p}}{c_3} \right)^{1/2}
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}} \notag \\
& + \left. \sum_{i=0}^k \hat{c}_{3,L_p}^{(i)} \left( \frac{c_2 c_3 c_{2,L_p}}{c_{3,L_p}} \right)^{1/2}
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}+\delta_i}
+ \sum_{i=1}^k \hat{c}_3^{(i)} c_2 \Lambda_i \right) \notag \\
& \times \left( \frac{\log(\varepsilon^{-1})}{p \log(M)}
+ \frac{\log(q^{-\frac{1}{2}} c_{1,p} T^p)}{p \log(M)} \right) \notag \\
& + \hat{c}_3^{(0)} c_2 \left( \frac{\log(\varepsilon^{-1})}{p \log(M)}
+ \frac{\log(q^{-\frac{1}{2}} c_{1,p} T^p)}{p \log(M)} \right)^2 \notag \\
& + \sum_{i=1}^k \hat{c}_3^{(i)} \left( \frac{c_2 c_{2,L_p} c_{3,L_p}}{c_3} \right)^{1/2}
\Lambda_i h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}} \notag \\
& + \sum_{i=0}^k \hat{c}_{3,L_p}^{(i)} \left(
\left( \frac{c_{2,0} c_{3,0} c_{2,L_p}}{c_{3,L_p}} \right)^{1/2}
h_{L_p}^{\frac{\beta_{L_p}-\gamma_{L_p}}{2}+\delta_i} + c_{2,L_p} h_{L_p}^{\beta_{L_p}-\gamma_{L_p}+\delta_i} \right) \Bigg] \notag \\
& + T \sum_{i=0}^k \left( \hat{c}_{3,0}^{(i)} T^{\delta_i-\gamma}
+ \hat{c}_3^{(i)} \frac{(M^{-1} T)^{\delta_i-\gamma} - h_{L_p}^{\delta_i-\gamma}}{1-M^{\gamma-\delta_i}}
+ \hat{c}_{3,L_p}^{(i)} h_{L_p}^{\delta_i-\gamma_{L_p}} \right)
\label{Proof-Main-Prop-Upper-Bound-Eq-alg}
$$ ]{} where we applied the relation (\[L-1-upper-bound\]).\
\
Suppose that $\beta_{L_p} \geq \gamma_{L_p}$ and $\gamma,
\gamma_{L_p} \leq 2p$. Then, we get from the upper bound (\[Proof-Main-Prop-Upper-Bound-Eq-alg\]) that $C(\hat{Y}_{ML(\alpha,p)})(\varepsilon) = O( \varepsilon^{-2} (
\log(\varepsilon) )^2 )$. Further, comparing the lower and the upper bounds (\[Proof-Main-Prop-Lower-Bound-Eq-alg\]) and (\[Proof-Main-Prop-Upper-Bound-Eq-alg\]), we asymptotically obtain that $$\begin{aligned}
\lim_{\varepsilon \to 0} \frac{C(\hat{Y}_{ML(\alpha,\alpha)})(\varepsilon)}
{C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)}
\geq \lim_{\varepsilon \to 0} \frac{\frac{T}{1-q} \varepsilon^{-2} \hat{c}_3^{(0)}
c_2 \left( \frac{\log(\varepsilon^{-1})}{\alpha \, \log(M)} \right)^2
+ o(\varepsilon^{-2} (\log(\varepsilon))^2)}
{\frac{T}{1-q} \varepsilon^{-2} \hat{c}_3^{(0)}
c_2 \left( \frac{\log(\varepsilon^{-1})}{p \, \log(M)} \right)^2
+ o(\varepsilon^{-2} (\log(\varepsilon))^2)}
= \frac{p^2}{\alpha^2}\end{aligned}$$ which proves statement (\[Main-Prop-Improvement-Aussage2\]). $\Box$
Especially, if $c_3 = \hat{c}_3 $ and $c_{3,L_p}=\hat{c}_{3,L_p} $, then it follows in case of $\beta<\gamma$ and $\beta<2p$ that $$\label{Main-Prop-Improvement-Aussage-Bemerkung-1}
\lim_{\varepsilon \to 0} \frac{C(\hat{Y}_{ML(p,p)})(\varepsilon)}
{C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)} \geq M^{\gamma-\beta} \left(
1 - M^{\frac{\beta-\gamma}{2}}
\left(1 - \left( \frac{c_2 c_3}{c_{2,L_p} c_{3,L_p}} \right)^{1/2} \right)
\right)^{-2} \, .$$ Thus, if $c_2 c_3 < c_{2,L_p} c_{3,L_p}$ it follows directly that $$\label{Main-Prop-Improvement-Aussage-Bemerkung-2}
\lim_{\varepsilon \to 0} \frac{C(\hat{Y}_{ML(p,p)})(\varepsilon)}
{C(\hat{Y}_{ML(\alpha,p)})(\varepsilon)} > 1 \, .$$
Numerical examples in case of SDEs {#Sec4:Numerical-Examples-SDEs}
==================================
For illustration of the improvement that can be realized with the proposed modified multi-level Monte Carlo estimator, we consider the problem of weak approximation for stochastic differential equations (SDEs) $$\label{SDE-general}
\mathrm{d} X_t = a(X_t) \, \mathrm{d}t + \sum_{j=1}^m b^j(X_t) \, \mathrm{d}B_t^j$$ with initial value $X_{t_0}=x_0 \in \mathbb{R}^d$ driven by $m$-dimensional Brownian motion.\
\
In the following, we compare for several numerical examples the root mean-square errors versus the computational costs for the multi-level Monte Carlo estimator $\hat{Y}_{ML}$ proposed in [@Gil08b; @Gil09a; @Gil08a] and described in Section \[Section2:MLMC-Simulation-Original\] with the proposed modified multi-level Monte Carlo estimator $\hat{Y}_{ML(\alpha,p)}$ described in Section \[Sec3:Improved-MLMC-Estimator\]. As a measure for the computational costs, we count the number of evaluations of the drift and diffusion functions taking into account the dimension $d$ of the solution process as well as the dimension $m$ of the driving Brownian motion.\
\
In the following, we consider on each level $l=0,1, \ldots, L$ an equidistant discretization $I_{h_l}=\{t_0, \ldots,
t_{\frac{T}{2^l}}\}$ of $[t_0,T]$ with step size $h_l=\frac{T}{2^l}$. Further, we denote by $Y_n=Y_{t_n}$ the approximation at time $t_n$. In case of the multi-level Monte Carlo estimator $\hat{Y}_{ML}$ we apply on each level $l=0,1, \ldots, L$ the Euler-Maruyama scheme on the grid $I_{h_l}$ given by $Y_0=x_0$ and $$Y_{n+1} = Y_n + a(Y_n) \, h_n + \sum_{j=1}^m b^j(Y_n) \, I_{(j),n}$$ where $h_n=h_l$ and $I_{(j),n} = B_{t_{n+1}}^j - B_{t_n}^j$ for $n=0,1, \ldots, \tfrac{T}{2^l}-1$. The Euler-Maruyama scheme converges with order $1/2$ in the mean-square sense and with order $\alpha=1$ in the weak sense to the solution of the considered SDE at time $T$ [@KP99].\
\
On the other hand, for the modified multi-level Monte Carlo estimator $\hat{Y}_{ML(\alpha,p)}$ the Euler-Maruyama scheme is applied on levels $0,1, \ldots, L_p-1$ whereas on level $L_p$ a second order weak stochastic Runge-Kutta (SRK) scheme RI6 proposed in [@Roe09] is applied. The SRK scheme RI6 on level $L_p$ is defined on the grid $I_{h_{L_p}}$ by $\check{Y}_0=x_0$, $$\begin{split}
\check{Y}_{n+1} &= \check{Y}_n + \tfrac{1}{2} \left( a(\check{Y}_n) + a(\Upsilon) \right) \, h_{n}
+ \tfrac{1}{2} \sum_{k=1}^m \left( b^k(\Upsilon_+^{(k)}) - b^k(\Upsilon_-^{(k)}) \right)
\, \tfrac{\hat{I}_{(k,k),n}}{\sqrt{h_{n}}} \\
&+ \sum_{k=1}^m \left( \tfrac{1}{2} b^k(\check{Y}_n) + \tfrac{1}{4} b^k(\Upsilon_+^{(k)})
+ \tfrac{1}{4} b^k(\Upsilon_-^{(k)}) \right) \, I_{(k),n} \\
&+ \tfrac{1}{2} \sum_{k=1}^m \left( b^k(\hat{\Upsilon}_+^{(k)}) - b^k(\hat{\Upsilon}_-^{(k)})
\right) \, \sqrt{h_{n}} \\
&- \sum_{k=1}^m \left(\tfrac{1}{2} b^k(\check{Y}_n) - \tfrac{1}{4} b^k(\hat{\Upsilon}_+^{(k)})
- \tfrac{1}{4} b^k(\hat{\Upsilon}_-^{(k)}) \right) \, I_{(k),n}
\end{split}$$ where $h_n=h_{L_p}$ and $I_{(k),n} = B_{t_{n+1}}^k - B_{t_n}^k$ for $n=0,1, \ldots, \tfrac{T}{2^{L_p}}-1$ with stages $$\begin{split}
\Upsilon &= \check{Y}_n + a(\check{Y}_n) \, h_{n} + \sum_{j=1}^m b^j(\check{Y}_n) \, I_{(j),n} , \\
\Upsilon_{\pm}^{(k)} &= \check{Y}_n + a(\check{Y}_n) \, h_{n} \pm b^k(\check{Y}_n) \, \sqrt{h_{n}} ,
\quad \quad
\hat{\Upsilon}_{\pm}^{(k)} = \check{Y}_n \pm
\sum_{\substack{j=1 \\ j \neq k}}^m b^j(\check{Y}_n) \, \tfrac{\hat{I}_{(k,j),n}}{\sqrt{h_{n}}}
\end{split}$$ where $\hat{I}_{(k,k),n} = \tfrac{1}{2} (I_{(k),n}^2-h_{n})$ and $$\hat{I}_{(k,j)_n} = \begin{cases}
\tfrac{1}{2} (I_{(k),n} I_{(j),n} - \sqrt{h_{n}} \tilde{I}_{(k),n}) & \text{ if } k<j \\
\tfrac{1}{2} (I_{(k),n} I_{(j),n} + \sqrt{h_{n}} \tilde{I}_{(j),n}) & \text{ if } j<k
\end{cases}$$ based on independent random variables $\tilde{I}_{(k),n}$ with ${\operatorname{P}}(\tilde{I}_{(k),n} = \pm \sqrt{h_{n}})=\tfrac{1}{2}$. Thus, we have $\alpha=1$ and $p=2$ for the modified multi-level Monte Carlo estimator $\hat{Y}_{ML(\alpha,p)}$ in the following. Further, for both schemes the variance decays with the same order as the computational costs increase, i. e.$\beta=\beta_{L_p}=\gamma=\gamma_{L_p}=1$. Then, the optimal order of convergence attained by the multi-level Monte Carlo method is $O(\varepsilon^{-2} (\log(\varepsilon))^2)$ due to Theorem \[Main-Theorem-Giles\]. For the presented simulations, we denote by MLMC EM the numerical results for $\hat{Y}_{ML}$ based on the Euler-Maruyama scheme only and by MLMC SRK the results for $\hat{Y}_{ML(\alpha,p)}$ based on the combination of the Euler-Maruyama scheme and the SRK scheme RI6.\
\
As a first example, we consider the scalar linear SDE with $d=m=1$ given by $$\label{Test-SDE-1}
{\mathrm{d}} X_t = r \, X_t \, {\mathrm{d}}t
+ \sigma X_t \, {\mathrm{d}}B_t \, ,
\quad X_0 = 0.1 \, ,$$ using the parameters $r=1.5$ and $\sigma=0.1$. We choose $T=1$ and apply the functionals $f(x)=x$ and $f(x)=x^2$, see Figure \[Bild-test1\]. The presented simulations are calculated using the prescribed error bounds $\varepsilon=4^{-j}$ for $j=0,1,
\ldots, 5$. In Figure \[Bild-test1\] we can see the significantly reduced computational effort for the estimator $\hat{Y}_{ML(1,2)}$ (MLMC SRK) compared to the estimator $\hat{Y}_{ML}$ (MLMC EM) in case of a linear and a nonlinear functional.
![Error vs. computational effort for SDE using $f(x)=x$ (left) and $f(x)=x^2$ (right).[]{data-label="Bild-test1"}](MLMC_SRK_EM_f01_d1m1_m1_20120206.eps "fig:"){width="5.5cm"} ![Error vs. computational effort for SDE using $f(x)=x$ (left) and $f(x)=x^2$ (right).[]{data-label="Bild-test1"}](MLMC_SRK_EM_f01_d1m1_m2_20120206.eps "fig:"){width="5.5cm"}
The second example is a nonlinear scalar SDE with $d=m=1$ given by $$\label{Test-SDE-2}
{\mathrm{d}} X_t = \tfrac{1}{2} X_t +\sqrt{X_t^2+1} \, {\mathrm{d}}t
+ \sqrt{X_t^2+1} \, {\mathrm{d}}B_t \, ,
\quad X_0 = 0 \, .$$ We apply the functional $f(x)= (\log(x + \sqrt{x^2 + 1}))^3 - 6
(\log(x + \sqrt{x^2 + 1}))^2 + 8 \log(x + \sqrt{x^2 + 1})$. Then, the approximated expectation is given by ${\operatorname{E}}(f(X_t)) = t^3-3 t^2 + 2
t$. Here, the results presented in Figure \[Bild-test2\] (left) are calculated for $T=2$ applying the prescribed error bounds $\varepsilon=4^{-j}$ for $j=0,1, \ldots, 6$. Here, the improved estimator $\hat{Y}_{ML(1,2)}$ performs much better than $\hat{Y}_{ML}$ also for nonlinear functionals and a nonlinear SDE.
![Error vs. computational effort for the nonlinear SDE (left) and SDE (right) with non-commutative noise.[]{data-label="Bild-test2"}](MLMC_SRK_EM_f02_d1m1_m1_20120312.eps "fig:"){width="5.5cm"} ![Error vs. computational effort for the nonlinear SDE (left) and SDE (right) with non-commutative noise.[]{data-label="Bild-test2"}](MLMC_SRK_EM_f12_d4m6_m1_20120229.eps "fig:"){width="5.5cm"}
Finally, we consider a nonlinear multi-dimensional SDE with a $d=4$ dimensional solution process driven by an $m=6$ dimensional Brownian motion with non-commutative noise: $$\label{Test-SDE-3}
\begin{split}
{\mathrm{d}} &\begin{pmatrix} X_t^1 \\ X_t^2 \\ X_t^3 \\ X_t^4 \end{pmatrix}
= \begin{pmatrix} \frac{243}{154} X_t^1 - \frac{27}{77} X_t^2 + \frac{23}{154} X_t^3
- \frac{65}{154} X_t^4 \\
\frac{27}{77} X_t^1 - \frac{243}{154} X_t^2 + \frac{65}{154}
X_t^3 - \frac{23}{154} X_t^4 \\
\frac{5}{154} X_t^1 - \frac{61}{154} X_t^2 + \frac{162}{77}
X_t^3 - \frac{36}{77} X_t^4 \\
\frac{61}{154} X_t^1 - \frac{5}{154} X_t^2 + \frac{36}{77}
X_t^3 - \frac{162}{77} X_t^4 \end{pmatrix} \, {\mathrm{d}}t \\
&+ \frac{1}{9} \sqrt{(X_t^2)^2 + (X_t^3)^2 + \frac{2}{23}}
\begin{pmatrix} \tfrac{1}{13} \\ \tfrac{1}{14} \\ \tfrac{1}{13} \\
\tfrac{1}{15} \end{pmatrix} \, {\mathrm{d}} B_t^1
+ \frac{1}{8} \sqrt{(X_t^4)^2 + (X_t^1)^2 + \frac{1}{11}}
\begin{pmatrix} \frac{1}{14} \\ \frac{1}{16} \\ \frac{1}{16} \\
\frac{1}{12} \end{pmatrix} \, {\mathrm{d}} B_t^2 \\
&+ \frac{1}{12} \sqrt{(X_t^1)^2 + (X_t^2)^2 + \frac{1}{9}}
\begin{pmatrix} \frac{1}{6} \\ \frac{1}{5} \\ \frac{1}{5} \\
\frac{1}{6} \end{pmatrix} \, {\mathrm{d}} B_t^3
+ \frac{1}{14} \sqrt{(X_t^3)^2 + (X_t^4)^2 + \frac{3}{29}}
\begin{pmatrix} \frac{1}{8} \\ \frac{1}{9} \\ \frac{1}{8} \\
\frac{1}{9} \end{pmatrix} \, {\mathrm{d}} B_t^4 \\
&+ \frac{1}{10} \sqrt{(X_t^1)^2 + (X_t^3)^2 + \frac{1}{13}}
\begin{pmatrix} \frac{1}{11} \\ \frac{1}{15} \\ \frac{1}{13} \\
\frac{1}{11} \end{pmatrix} \, {\mathrm{d}} B_t^5
+ \frac{1}{11} \sqrt{(X_t^2)^2 + (X_t^4)^2 + \frac{2}{25}}
\begin{pmatrix} \frac{1}{12} \\ \frac{1}{13} \\ \frac{1}{16} \\
\frac{1}{13} \end{pmatrix} \, {\mathrm{d}} B_t^6
\end{split}$$ with initial condition $X_0 = (\tfrac{1}{8}, \tfrac{1}{8}, 1,
\tfrac{1}{8})^T$. Then, the approximated first moment of the solution is given by $E(X_T^i)= X_0^i \, \exp(2T)$ for $i=1,2,3,4$. The simulation results calculated at $T=1$ for the error bounds $\varepsilon=4^{-j}$ for $j=0,1, \ldots, 6$ are presented in Figure \[Bild-test2\] (right). Again, in the multi-dimensional non-commutative noise case the proposed estimator $\hat{Y}_{ML(1,2)}$ needs significantly less computational effort compared to the estimator $\hat{Y}_{ML}$ which reveals the theoretical results in Proposition \[Main-Prop-Improvement\].
Conclusions {#Sec5:Conclusions}
===========
In this paper we proposed a modification of the multi-level Monte Carlo method introduced by M. Giles which combines approximation methods of different orders of weak convergence. This modified multi-level Monte Carlo method attains the same mean square order of convergence like the originally proposed method that is in some sense optimal. However, the newly proposed multi-level Monte Carlo estimator can attain significantly reduced computational costs. As an example, there is a reduction of costs by a factor $(p/\alpha)^2$ for the problem of weak approximation for SDEs driven by Brownian motion in case of $\beta=\gamma$. This has been approved by some numerical examples for the case of $p=2$ and $\alpha=1$ where four times less calculations are needed compared to the standard multi-level Monte Carlo estimator. Here, we want to point out that there also exist higher order weak approximation schemes, e. g.$p=3$ in case of SDEs with additive noise [@De10], that may further improve the benefit of the modified multi-level Monte Carlo estimator. Future research will consider the application of this approach to, e.g., more general SDEs like SDEs driven by Lévy processes [@Der11] or fractional Brownian motion [@KNP11] and to the numerical solution of SPDEs [@SchGit11]. Further, the focus will be on numerical schemes that feature not only high orders of convergence by also minimized constants for the variance estimates.
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|
---
abstract: 'Our high time resolution observations of individual giant pulses in the Crab pulsar show that both the time and frequency signatures of the interpulse are distinctly different from those of the main pulse. Giant main pulses can occasionally be resolved into short-lived, relatively narrow-band nanoshots. We believe these nanoshots are produced by soliton collapse in strong plasma turbulence. Giant interpulses are very different. Their dynamic spectrum contains narrow, microsecond-long emission bands. We have detected these proportionately spaced bands from 4.5 to 10.5 GHz. The bands cannot easily be explained by any current theory of pulsar radio emission; we speculate on possible new models.'
author:
- 'J. A. Eilek'
- 'T. H. Hankins'
title: Radio Emission Physics in the Crab Pulsar
---
Introduction
============
What is the pulsar radio emission mechanism? Does the same mechanism always operate? Three types of models have been proposed to explain the radio emission: coherent charge bunches, plasma masers and strong plasma turbulence ([*e.g.*]{}, Hankins [*et al.*]{} 2003, “HKWE”). Because each model makes different predictions for the time signature of the emission, our group has carried out ultra-high time resolution observations in order to compare the observed time signatures to those predicted by the models.
We have focused on the Crab nebula pulsar, because its occasional, very strong giant pulses are ideal targets for our observations. The dominant features of this star’s mean profile are a main pulse (MP) and an interpulse (IP). Although the relative amplitudes and detailed profiles of these features change with frequency, they can be identified from low radio frequencies ($\ltw 300$ MHz) up to the optical and hard X-ray bands (Moffet & Hankins 1996). Some models suggest that the MP and IP come from low altitudes, above the star’s two magnetic poles. Other models suggest they come from higher altitudes, possibly relativistic caustics (Dyks [*et al.*]{} 2004) which connect to the two poles. In either case, the physical conditions in the emission region should be similar, and one would expect the same radio emission mechanism to be active in the IP and the MP. We were surprised, therefore, to find that the IP and MP have very different properties. It seems likely that they differ in their emission mechanisms, their propagation within the magnetosphere, or both.
Giant main pulses: strong plasma turbulence
===========================================
We initially studied the MP at nanosecond time resolution, because it is usually brighter, and because giant pulses are more common at the rotation phase of the MP (Cordes [*et al.*]{} 2004). We found that most giant main pulses (GMPs) consist of one to several “microbursts”, each lasting a few microseconds at 5 GHz (HKWE). We recently extended our observations to higher frequencies, where 2 GHz of bandwidth is available at Arecibo. We found the temporal structure of GMPs is the same at higher frequencies, although the microburst duration is typically shorter than at 5 GHz. Figure 1 shows a typical example. The dynamic spectrum of the microbursts turns out to be broadband, filling our entire observing bandwidth. An occasional MP, however, contains much shorter, relatively narrow-band, “nanoshots” (HKWE; also Figures 2 and 3). Most of the time the nanoshots overlap, which is consistent with previous modelling of pulsar emission as amplitude-modulated noise; but in sparse GMPs the nanoshots can sometimes be individually resolved.
We used simple scaling arguments, and numerical simulations from Weatherall (1998), to compare the nanoshots to predictions of the three competing theoretical models of the radio emission mechanisms. The time signature of the nanoshots disagrees with predictions of the maser and charge bunching models; but both the time and frequency signatures are consistent with Weatherall’s numerical models of plasma emission by soliton collapse in strong plasma turbulence. His models predict nanoshot durations at frequency $\nu$ to be $\nu \delta t \sim O(10)$; an individual nanoshot is relatively narrow-band, $\delta \nu /\nu \sim O(0.1)$. In HKWE we suggested, based on the time signature of the nanoshots, that strong plasma turbulence is the emission mechanism in GMPs. The time and spectral signatures of the nanoshots in our recent high-frequency work are also consistent with these models. We thus propose that microbursts in giant main pulses are collections of nanoshots, produced by strong plasma turbulence in the emission region.
If our suggestion is correct, it has one important consequence. Plasma flow in the radio emission region should be highly dynamic. The plasma flow will be smooth only if the local charge density is exactly the Goldreich-Julian (GJ) value, so that the rotation-induced electric field, $\mathbf E$, is fully shielded. Because plasma turbulent emission is centered on the comoving plasma frequency ($\nu_p \propto \sqrt{\gamma_b n}$, for number density $n$ and bulk Lorentz factor $\gamma_b$), we can determine the local density in the radio emission region ([*cf.*]{} also Kunzl [*et al.*]{} 1998). We find that low radio frequencies come from densities too low to match the GJ value anywhere in the magnetosphere. Because the emitting plasma feels an unshielded $\mathbf
E$ field, and feeds back on that field as its charge density fluctuates, we expect unsteady plasma flow (and consequently unsteady radio emission).
Giant interpulses: emission bands
=================================
In order to test our hypothesis that strong plasma turbulence governs the emission physics in the Crab pulsar, we went to higher frequencies to get a larger bandwidth and shorter time resolution. In addition to the MP, we observed giant pulses from the IP, because at high frequencies giant pulses are more common at the rotation phase of the IP. When we used the method described in HWKE to observe giant interpulses (GIPs) with a broad bandwidth, from 6-8 or 8-10 GHz, we were astonished to find that GIPs have very different properties from giant main pulses. GIPs differ from GMPs in time signature, polarization, dispersion and spectral properties. In this paper we summarize our new results; we will present more details in a forthcoming paper (Hankins & Eilek 2007).
Emission bands in the interpulse
--------------------------------
The most striking difference between the IP and the MP is found in the dynamic spectrum. A giant IP contains microsecond-long trains of [*emission bands*]{}, as illustrated in Figures 4 and 5. The bands are grouped into regular “sets”; 2 or 3 band sets can usually be identified in a given IP. Individual band sets last a few $\mu$s. In some pulses new band sets turn on partway through the pulse, often coincident with a secondary burst of total intensity. [*Every*]{} giant interpulse we recorded between 4.5 and 10.5 GHz, during 20 observing days from 2004 to 2006, displays these emission bands. However, giant main pulses observed at the same time and processed identically do not show the bands. The bands are, therefore, not due to instrumental or interstellar effects, but are intrinsic to the star.
At first glance the bands appear to be uniformly spaced. However, closer inspection of our data shows that the bands are [*proportionally spaced*]{}. The spacing between two adjacent bands, at $\nu_1$ and $\nu_2$, depends on the mean frequency, as $\Delta \nu/ \nu = 2(\nu_2 -
\nu_1)/( \nu_2 + \nu_1) \simeq 0.06$. Thus, two bands near 6 GHz are spaced by $\sim 360$ MHz; two bands near 10 GHz are spaced by $\sim 600$ MHz. This proportional spacing is robust; a set of emission bands can drift in frequency (usually upwards, as in Figures 3 and 4), but their frequency spacing stays constant. All bands in a particular set appear almost simultaneously, to within $\sim 0.1 \mu$s; they must all come from a region no larger than $\sim 30$ m across.
We suspect the bands extend over at least a $5\!-\!6$ GHz range in a single GIP, but do not occur below $\sim 4$ GHz. While we have not been able to observe more than 2 GHz simultaneously, we have seen no evidence that a given band set cuts off within our observable bandwidth. The characteristics of the bands (proportional spacing, duration, onset relative to total intensity microbursts) are unchanged from 5 to 10 GHz. In addition, the rotation phase of the high-frequency IP is slightly shifted relative to the low-frequency IP (Moffett & Hankins 1996). This phase offset suggests that the bands do not continue to frequencies below $\sim 4$GHz.
Possible causes of the emission bands
-------------------------------------
The dynamic spectrum of the giant interpulses does not match any of the three types of emission models described above. Because each of the models predicts narrow-band emission at the plasma frequency, [*none of them can explain the dynamic spectrum of the IP*]{}. A new approach is required here, which may “push the envelope” of pulsar radio emission models.
While we remain perplexed by the dramatic dynamic spectrum of the interpulse, we are exploring possible models. This exercise is made particularly difficult by the fact that the emission bands are not regularly spaced. Because of this, models that initially seemed attractive must be rejected. As an example, if the emission bands were uniformly spaced they could be the spectral representation of a regular emission pulse train. Many authors have invoked regularly spaced plasma structures (sparks or filaments), whose passage across the line of sight could create such a pulse train. Alternatively, strong plasma waves with a characteristic frequency will also create a regular emission pulse train. The dynamic spectrum of either of these models would contain emission bands at constant spacing; the [*proportional*]{} spacing we observe disproves both of these hypotheses.
We have looked to solar physics for insight. We initially remembered split bands in the dynamic spectra of Type II solar flares, which are thought to be plasma emission from low and high density regions associated with a shock propagating through the solar corona. This does not seem to be helpful for the Crab pulsar emission bands, because the radio-loud plasma would have to contain 10 or 15 different density stratifications, which seems unlikely. However, “zebra bands” seen in Type IV solar flares may be germane. These are parallel, drifting, narrow emission bands seen in the dynamic spectra of Type IV flares. Band sets containing from a few up to $\sim 30$ bands have been reported, with fractional spacing $\Delta \nu / \nu \sim .01
- .03$ ([*e.g.*]{}, Chernov [*et al.*]{} 2005). While zebra bands have not yet been satisfactorily explained, two classes of models have been proposed, invoking either resonant plasma emission or geometrical effects. Can similar models explain the emission bands in the Crab pulsar?
[*Resonant cyclotron emission.*]{} One possibility is plasma emission at the cyclotron resonance, $\omega - k_{\parallel} v_{\parallel} - s
\Omega_o / \gamma = 0$ (where $\gamma$ is the particle Lorentz factor, $\Omega_o = e B / m c$, and $s$ is the integer harmonic number). Kazbegi [*et al.*]{} (1991) proposed that this resonance operates at high altitudes in the pulsar magnetosphere, and generates X mode waves which can escape the plasma directly. Alternatively, “double resonant” cyclotron emission at the plasma resonant frequency has been proposed for solar flares ([*e.g.*]{}, Winglee & Dulk 1986). In solar conditions, this resonance generates O mode waves, which must mode convert in order to escape the plasma. The emission frequency in these models is determined by local conditions where the resonance is satisfied; the band separation is $\Delta \nu \simeq \Omega_o / 2 \pi
\gamma$. Resonant emission models face several challenges before they can be considered successful. The emission must occur at high altitudes, in order to bring the resonant (cyclotron) frequency down to the radio band. Close to the light cylinder, where $B \sim 3 \times 10^5$G, particle energies $\gamma \sim 10^3\!-\!
10^4$ are needed. In addition, such models must be developed with specific calculations which address the fundamental plasma modes as well as their stability, under conditions likely to exist at high altitudes in the pulsar’s magnetosphere. It is not clear how the specific, proportional band spacing can be explained; perhaps a local gradient in the magnetic field must be invoked.
[*Geometrical models.*]{} Alternatively, the striking regularity of the bands calls to mind a special geometry. If some mechanism splits the emission beam coherently, so that it interferes with itself, the bands could be interference fringes. For instance, a downwards beam which reflects off a high density region could return and interfere with its upwards counterpart on the way back up. Simple geometry suggests that fringes occur if the two paths differ in length by only $c / \Delta \nu \ltw 1$m. Another geometrical possibility is that cavities form in the plasma and trap some of the emitted radiation, imposing a discrete frequency structure in the plasma ([*e.g.*]{}, LaBelle [*et al.*]{} 2003 for solar zebra bands). The scales required here are also small; the cavity scale must be some multiple of the wavelength.
Geometrical models also face several obstacles before they can be considered successful. The basic geometry is a challenge: what long-lived plasma structures can lead to the necessary interference or wave trapping? In addition, the proportional band spacing must be explained, perhaps by a variable index of refraction in the interference or trapping region.
Geometrical models also need an underlying broad-band radiation source, with at least 5 GHz bandwidth, in order to produce the emission bands we observe. Because standard pulsar radio emission mechanisms lead to relatively narrow-band radiation, at the local plasma frequency, they seem unlikely to work here. A double layer might be the radiation source; charges accelerated within the layer should radiate broadband, up to $\nu
\sim L / 2 \pi c$, if $L$ is the thickness of the acceleration region within the double layer. Once again this is a small-scale effect; emission at 10 GHz requires $L \sim 1$ cm.
Final thoughts
==============
Our high time resolution observations of giant pulses from the Crab pulsar have raised as many questions as they have answered. The time and frequency signatures of giant main pulses are consistent with predictions of one current model of pulsar radio emission, namely, strong plasma turbulence. However, the time and frequency signatures of giant interpulses are totally different, and do not seem to match the predictions of any current model. This result is especially surprising because magnetospheric models generally ascribe the main pulse and the interpulse to physically similar regions, which simply happen to be on opposite sides of the star. One important clue may be the offset in rotation phase between the high-radio-frequency interpulse, and the interpulse which is seen at low radio frequencies and also in optical and X-ray bands. Does the high-frequency interpulse originate in an unexpected part of the star’s magnetosphere, where different physical conditions produce such different radiation signatures?
0.4cm
We appreciate helpful conversations with Joe Borovsky, Alice Harding, Axel Jessner, Jan Kuijpers, Maxim Lyutikov, and the members of the Socorro pulsar group. This work was partially supported by the National Science Foundation, through grant AST0139641 and through a cooperative agreement with Cornell to operate the Arecibo Observatory.
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|
---
abstract: 'We bound the supnorm of half-integral weight Hecke eigenforms in the Kohnen plus space of level $4$ in the weight aspect, by combining bounds obtained from the Fourier expansion with the amplification method using a Bergman kernel.'
address: 'Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK'
author:
- 'Raphael S. Steiner'
bibliography:
- 'Bibliography.bib'
title: 'Supnorm of Modular Forms of half-integral Weight in the Weight Aspect'
---
This work is based on my master’s thesis, which I completed during November 2013 - April 2014 in Bristol, UK, and further improved upon during my PhD in Bristol, UK. I would like to thank Prof. Kowalski, for enabling me to do my master’s thesis abroad, Dr. Saha, for his useful comments and discussions on the topic, and Prof. Harcos for his helpful remarks.
|
---
abstract: 'As an important component of multimedia analysis tasks, audio classification aims to discriminate between different audio signal types and has received intensive attention due to its wide applications. Generally speaking, the raw signal can be transformed into various representations (such as Short Time Fourier Transform and Mel Frequency Cepstral Coefficients), and information implied in different representations can be complementary. Ensembling the models trained on different representations can greatly boost the classification performance, however, making inference using a large number of models is cumbersome and computationally expensive. In this paper, we propose a novel end-to-end collaborative learning framework for the audio classification task. The framework takes multiple representations as the input to train the models in parallel. The complementary information provided by different representations is shared by knowledge distillation. Consequently, the performance of each model can be significantly promoted without increasing the computational overhead in the inference stage. Extensive experimental results demonstrate that the proposed approach can improve the classification performance and achieve state-of-the-art results on both acoustic scene classification tasks and general audio tagging tasks.'
author:
- 'Liang Gao, Kele Xu, Huaimin Wang, Yuxing Peng [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'reference.bib'
title: 'Multi-Representation Knowledge Distillation For Audio Classification'
---
[Shell : Bare Demo of IEEEtran.cls for IEEE Journals]{}
convolutional neural networks, acoustic classification, knowledge distillation.
Introduction
============
classification task refers to identify a pre-defined label for an audio signal [@virtanen2018computational]. The potential applications of audio classification seem to be evident in several fields, such as multimedia retrieval, security surveillance [@clavel2005events], health care monitoring [@peng2009healthcare] and context-aware services [@ma2003context]. Due to the dramatic increase of the sound recordings, the demand for automatic audio classification is growing rapidly in last decades. Sustainable efforts have been made to address the audio classification problems [@virtanen2018computational; @xu2018large; @dhanalakshmi2009classification; @lee2006automatic; @dhanalakshmi2011classification; @sainath2015learning; @choi2017convolutional].
The launch of the Detection and Classification of Acoustic Scenes and Events (DCASE) [@7100934] challenge promoted the development of audio classification, which was organized by the IEEE Audio and Acoustic Signal Processing (AASP) technical committee. Many audio processing techniques have been proposed during the past years, and the applications of deep learning in the audio classification have witnessed a significant increase, especially the convolutional neural network (CNN). The traditional method, which commonly involve models like Gaussian Mixture Model (GMM) [@dhanalakshmi2011classification], Support Vector Machine (SVM) [@dhanalakshmi2009classification] or Hidden Markov Model (HMM) [@lee2006automatic] are trained using the frame-level features such as Mel-frequency Cepstral Coefficients (MFCC) [@mesaros2018detection].
[@lee2018samplecnn; @lee2017sample-level] used one-dimensional convolution and fully connected layers to learn from the original raw signal; [@piczak2015environmental] combined MFCC and its delta into two-channel data, and used convolutional neural networks for feature extraction and classifier training. [@8085174] extracted three channels of log Mel-spectrograms (static, delta, and delta delta) as the DCNN input for speech emotion recognition. [@sercu2016dense] explored the use of dilated convolutions to use more contextual information to classify audio. [@sainath2015convolutional; @sainath2015learning; @hoshen2015speech] applied the time-domain convolution method to different tasks, such as speech recognition and sound event detection. [@choi2017convolutional] used convolutional recurrent neural network (CRNN) for music labeling tasks.
In most existing audio classification works [@piczak2015environmental; @sercu2016dense; @sainath2015convolutional; @sainath2015learning; @choi2017convolutional; @zhang2019large; @wang2019acoustic], the raw signal was transformed into one single representation (for example, Short Time Fourier Transform (STFT) [@ramalingam2006gaussian] logMel and MFCC), then train the classifiers based on the single representation. The performance of deep neural networks heavily relied on the representation of the audio clip while one single representation may cannot present the information effectively and efficiently. The conversion process from the original audio signal to the advanced representation undergone a variety of transformations and information compression operations, which undoubtedly led to the loss of audio information. Thus, single representation-based deep models are still short of accuracy. Fusion the knowledge obtained using different representations can greatly improve the classification performance [@yin2018learning; @xu2019general], as the single representation may be stuck at poor local minimums during the training phase.
Different representations represent different aspects of signals, joint using the knowledge of different representations could enhance model generalization performance [@xu2018meta]. For example, in image classification, [@8059841] combined the discriminative power of different views to jointly learn the classifiers and transformation matrices. There are many works that improve model performance by ensemble networks [@dietterich2000ensemble; @rokach2010ensemble; @wang2018weakly] which trained on different representations. But it leads to the increase of model complexity, some researchers tried to fusion information of different representations into a single network. In audio classification, [@8490588] combines convolutional neural networks (CNNs) with long short-term memory (LSTM) to exploit the correlative information from multiple views. These methods require careful model design, The universal approach to employ complementary information provided by different representations is still under-explored.
Intuitively, there are two kinds of approaches for the utilization of complementary information from multiple representations. One straightforward way is early-fusion, which concatenates different representations as a whole input to a single network, with each representation being a separate channel. However, this method decreases the network performance in the practical settings. Another method is later-fusion [@yin2018learning; @dietterich2000ensemble], which ensembles the predictions generated by different classifiers, which can empirically boost the classification performance. Nevertheless, the inference of a large number of models is cumbersome and computationally expensive.
Recently, it has been found that knowledge distillation [@hinton2015distilling; @zhang2018deep] can be used to transfer knowledge between different models, which could improve the classification without increasing the computational complexity in the inference phase [@zhang2018deep; @sun2017ensemble]. Inspired by the knowledge distillation, in this paper, a multi-representation based knowledge distillation approach was proposed for audio classification, with the goal to fully utilize complementary information introduced by different representations of the audios. Moreover, our method uses only one model in the inference phase, so the computational cost is independent of the number of models that participated in the distillation framework. Overall, our contributions are three-fold:
- Firstly, a novel collaborative learning framework is proposed for the audio classification task. Unlike most of the traditional approaches which only use a single representation, we leverage multiple representations within the framework. Complementary information embedded in multiple representations is extracted through different neural networks and fused in the end-to-end distillation framework. The fused knowledge is then fed back to each network during the training phase and can effectively improve the performance of different models. Consequently, the performance of each model has been improved by using collaborative distillation in the training stage. Moreover, different network architectures can be easily integrated into the framework.
- Secondly, our method provides a novel ensemble approach without additional inferring cost. Due to the decoupling nature of the knowledge distillation framework, no dependency is enforced between participated models. In other words, each model can be used independently in the inference stage. Those lightweight models trained by distillation become very effective in resource-constrained computing scenarios.
- Thirdly, extensive experiments are conducted on acoustic scene classification (DCASE 2018 Challenge Task 1A) and general audio tagging (DCASE 2018 Challenge Task 2). We find that the learning framework can improve the performance of audio classification and achieve state-of-the-art results on both acoustic scene classification and general audio tagging task. More specifically, with our framework single network obtained the mAP@3 of 93.26% in the acoustic scene classification task, and the accuracy of 72.48% in the acoustic scene classification task.
The paper is organized as follows. We firstly discuss the relationship between our method and prior works in section \[related\], while the details of the proposed approach are presented in section \[method\]. The experimental settings, the analysis of results and conclusions are given in the last three parts.
Related Work {#related}
============
In this section, we discuss the relationship between our work and previous work, which includes two parts: audio classification and knowledge distillation.
Audio classification
--------------------
Audio tagging aims to predict the presence or absence of certain acoustic events in the interested acoustic scene. Some traditional methods for audio classification like SVM, HMM and GMM are applied in industry. In the first edition of DCASE in 2013, SVM [@burges1998tutorial; @phan2016label] and bagging of decision trees [@dietterich2000experimental] were used. Recently, deep neural networks have shown improved performance for the audio classification task. In brief, the main modifications of current deep learning-based audio classification can be divided into four types: jointly using different representations of the audio signal [@kulkarni2009audio; @aytar2016soundnet]; more sophisticated deep learning architectures [@huang2017densely; @he2016deep; @zhu2018environmental]; and the applications of different regularization methods (such as data augmentation) [@poggio1990regularization; @xu2018mixup; @wei2018sample; @feng2018sample].
Among all of the studies, the selection of representation for the audio signal is one of the key factors for classification performance, while only a few attempts have been made in previous studies [@poggio1990regularization]. The audio signal can be transformed into various representations, such as raw wave signal, MFCC [@aucouturier2007bag], i-vector [@dehak2010front] and so on. A suitable representation can effectively improve the generalization ability of the model, MFCC and logMel have been proven to be useful in CNNs. [@7927482] presents a novel two-phase method for audio representation, they take into account both global structure and local structure, the learned representation can effectively represent the structure of audio. [@7592459] argue that an image-like spectrogram cannot well capture the complex texture details of the spectrogram, so that they proposed a multichannel LBP feature to improve the robustness to the audio noise. Combining knowledge of multiple audio representations can obtain more comprehensive features and strengthen the generalization of models. It is found that ensembling [@kuncheva2003measures; @sollich1996learning] the predictions generated by different classifiers can greatly boost the audio classification performance. However, making inferences using a large number of models is cumbersome and computationally expensive. On the other hand, an efficient fusion of different representations within the end-to-end manner also draws lots of attention [@yin2018learning]. Still, the audio classification using multi-representation needs to be thoroughly explored.
Knowledge distillation
----------------------
Knowledge Distillation was firstly proposed in [@bucilua2006model] and re-popularized with the goal of model compression in [@hinton2015distilling]. With the knowledge distillation method, knowledge of pre-trained complex models can be transferred to a small network, which would help to improve the model performance. Except for the traditional supervised learning objective such as the cross-entropy loss which based on the ground truth label, distillation hopes to introduce the extra supervision from the teacher model to the student model. The extra supervision can be in the forms of classification probabilities [@hinton2015distilling; @gao2019multistructure] or feature representation [@kim2018paraphrasing].
[@sun2017ensemble] uses the knowledge distillation method to extract knowledge from an integrated model and compress the knowledge into a single network. In [@zhang2018deep], two models are training at the same time, exchanging their prediction probability with each other to enhance the model performance. For classifiers that use label smoothing, soft labels replaced one-hot hard labels. Label smoothing can reduce the interference of noise data or wrong labels. In online distillation [@anil2018large] studied the co-distillation of multiple examples of neural networks, which using exactly the same settings, and achieved training acceleration. [@batra2017cooperative] proposed cooperative learning, in which jointly trains multiple models in different fields. For example, in the image detection task, one model inputs using the RGB image and the other model inputs using the depth image. The two models exchange the unchanged object attributes in the task so that the same task can be trained with different data inputs. Only a few attempts have been made to leverage knowledge distillation for the audio analysis tasks.
Methodology {#method}
===========
In this section, we present our approach for the audio classification task using knowledge distillation. The overview of the multi-representation distillation framework is illustrated in Figure \[fig1\]. The framework contains multi-branch networks with leveraging multi-representation as the input. The audio signals can be transformed to different representation, which exhibits heterogeneous properties. Each representation presents a different view of the raw audio, and each view has its own individual representation space and dynamics. The training framework consists of multiple branches, the information aggregation unit and the similarity measure units. Each of branches in the framework is a network of full function with feature layers capturing features, fully-connected layers and logits layer for audio classification. The information aggregation unit which aggregates knowledge from multi-branch networks, and networks in the framework learn from the aggregated knowledge by minimizing the loss of similarity unit. The raw signal of audio can be transformed into a variety of representations with complementary knowledge. And each branch trains network with one representation. After the information aggregation unit aggregated knowledge from multi-branch networks, the branches in the framework learning from the aggregated knowledge.
To train networks with the framework, first we transform the audio into different representations. And then we use the strategy of cyclic distillation for network training. Each cycle of the training process is divided into three small phases: the training of single branches, the information fusion and the distillation training.
Assuming that given a training set which contains $N$ samples $X=\{x_1,x_2,..,x_N\}$, the samples come from $M$ categories, and their corresponding labels are $Y=\{y_1,y_2,...y_N\}$. Assuming there are $\Gamma$ branches in the framework.
Data preparation
----------------
In the data preparation phase, multiple representations of the audio data are generated and used as input for different branches. In acoustic classification tasks, converting the original audio data into a suitable feature representation often leads to better performance. In general, the audio signal conversion process includes framed windowing, Fourier transforming, power spectrum calculation, filtering, and discrete cosine transforming, etc.
The continuous sound signal is converted into a high-level representation through time-frequency transformation, which can highlight the frequency domain features of the audio. And the transformed features representations have the advantages of low dimensions which can be represented as two-dimensional images. On the other hand, advanced audio representations are more in line with human ear characteristics. For example, the MFCC could simulate the human ear’s masking effect (the human ears are more sensitive to low-frequency sounds than to high-frequency sounds and more sensitive to high loudness sounds than low loudness sounds). In the transform processing of raw signal to MFCC or logMel, a set of bandpass filters which distributed gradually sparse from low to high frequencies are arranged in the critical frequency bandwidth to filter the input signal. The basic features obtained with the bandpass filters can be deployed as inputs for networks, and the generated representation is more robust and has better recognition performance in the signal with a lower noise ratio. On the other hand, the constant Q transform (CQT) avoids the disadvantage of uniform time-frequency resolution. For low-frequency waves, its bandwidth is very small, but it has higher frequency resolution to decompose similar notes; and for high-frequency waves, the bandwidth is relatively large, so that there is a higher time resolution at high frequencies to track rapidly changing overtones.
![Different representations of audio data. (a)WAVE (b) MFCC (c) CQT and (d) logMel[]{data-label="fig2"}](modal.pdf)
As shown in Figure \[fig2\], the audio signals could be transformed into different representations, such as log-scaled Mel-spectrograms (logMel), constant-Q transform (CQT) and MFCC and so on. Different audio representations have their own advantages, and one single representation cannot present the information effectively and efficiently. Learning to integrate multiple representations can utilize their mutual complement information. The neural networks learned with multiple representations may give better classification performance due to the complementary information contains in different representations.
In the framework with $\Gamma$ branches, we convert the raw signal $D= (X, Y)$ into $\Gamma$ kinds of different representations, which are denoted as $D^i= (X^i, Y^i), i\in{(1,2,...,\Gamma)}$.
Using knowledge distillation methods to combine with networks of different structures. Each branch in the framework can independently select the network structure, assuming that the networks in i-th branch denoted as $f_i$ in the knowledge distillation framework. And $D_i$ is the corresponding training set of the classifier $f_i$.
Single branch training
----------------------
In the phase of single branch training, the learning objective is to fit the ground-truth label. We minimize the cross-entropy loss between the predicted values with target labels, and the formula is as follows: $$\label{equation3}
L_{ce} = -\frac{1}{N}\sum_{i=0}^{N}[y_i\log{f(x_i)}],$$
where $L_{ce}$ is the loss of single branch training phase for the branch network $f$. The cross-entropy represents the distance between the predicted value and the expected value (ground-truth label).
Information fusion
------------------
In order to aggregate knowledge from multiple branches, we averaged the predicted values, using soft labels as information carriers. The One-hot label only indicates the category to which the sample belongs, but ignores the similar relationship between the sample and different categories. Soft labels are the soften softmax probability of the logits layer, which consists of the sample similarity information. For each branch, we calculate the soft labels for all training data and then send them to the information aggregation unit. For the sample $x_i$, the formula to calculate soft label is:
$$\label{equation5}
p_{i}= \frac{\exp(g_i/T)}{\sum_{j=0}^{M}\exp(g_j/T)},$$
where $g_i$ is the logits layer output of a branch network $f$ corresponding to the i-th category. And $T$ is a soften hyper-parameter. The larger the value of $T$, the smoother the soft label distribution.
The information from different branch networks is aggregated in the aggregation unit, which would be adopted as teacher information during the distillation training phase. The aggregated information was calculated from the multiple branches with average ensemble method:
$$\label{equation6}
\overrightarrow{P} = \frac{1}{\Gamma}\sum_{i=1}^{\Gamma}{(P_i)}.$$
In this equation, $P_i$ represents the training set’s soft labels of branch networks $f_i$, and $\Gamma$ represents the number of branches. The averaged soft labels $\overrightarrow{P}$ has the same effect as probability of ensembled network, which is smoother and stronger generalized than the soft labels in single branch network.
Knowledge distillation
----------------------
Knowledge transfer in knowledge distillation is accomplished by reducing the difference of information between teachers and students. Kullback-Leibler (KL) divergence could measure the difference of two distributions. In the distillation training process, the averaged soft labels which had aggregated knowledge from all branches were used as the teacher. For each branch, the similarity between soft labels $P$ and the averaged soft labels $\overrightarrow{P}$ is calculated as follows:
$$\label{equation7}
L_{kl} = -\frac{1}{N}\sum_{i=0}^{N}\sum_{j=0}^M[P_{i}^j\log{\frac{\overrightarrow{P_{i}^j}}{P_{i}^j}}],$$
where $P_{i}$ denote the soft labels for i-th sample of branch networks $f$, and $j$ donate the j-th category.
In previous attempts for knowledge distillation [@zhang2018deep; @lan2018knowledge], it has been found that combining supervision from onehot labels with supervision of teacher information leads to smoother optimization and better-performed network. The distillation loss for each branch of in the distillation training phase is given as follows:
$$\label{equation9}
L_{d} = L_{ce}+L_{kl}.$$
The training process of the framework is summarized in Algorithm \[algorithm1\]. Different from traditional two-stage distillation, we adopted cyclic distillation strategy, a large cycle including three phases: the branch training, knowledge aggregation, and distillation training. Three phases correspond to the information generation, information aggregation, and information feedback, respectively. The cyclic training could help networks better capture and utilize complementary information of multiple representations. After the model converges, any branch classifier in the framework could be applied for inferring according to the data representation or the resources limits. If better classification performance is sought, an ensemble network of multiple branch networks can also be used.
INPUT: Dataset $D=(X,Y)$,learning rate $lr$, the number of training epoch $Q$, the number of branches $\Gamma$, soften temperature parameter $T$, single branch training epoch $b$ and distillation training epoch $d$.\
PREPARING WORKS: Transforming the raw audio signal to multiple representations. Determining the network architecture $f_i$ and input representation $X_i$ for each branch in the framework.
**CYCLIC DISTILLATION:**
Initialization: $q=1$, load pre-train parameters $\Theta$ for branch networks.
Calculate $L_{ce}$ as Eq. (\[equation3\]) Update the network’s parameters: $ \Theta=\Theta+\frac{\partial{L_{ce}}}{\partial{\Theta}}$ Calculate soft label $P_i$ as Eq (\[equation5\])\
Send soft label $P_i$ to the information aggregation unit\
Aggregate knowledge: $\overrightarrow{P}=\frac{1}{\Gamma}\sum_{i=0}^{\Gamma}{P_i}$ Calculate $L_{ce}$ as Eq. (\[equation3\]) Calculate $L_{kl}$ as Eq. (\[equation7\]) The final loss: $L_d=L_{ce}+L_{kl}$ Update the network’s parameters: $ \Theta=\Theta+\frac{\partial{L_d}}{\partial{\Theta}}$ q=q+1\
Update learning rate\
Experiment
==========
Two widely-used datasets are applied to verify the efficacy of our distillation framework, (1) the FSDKaggle2018 audio tagging dataset and (2) the 2018 TUT Urban acoustic scene classification dataset.
Dataset
-------
**FSDKaggle2018 dataset.** The FSDKaggle2018 dataset [@fonseca2018general] was adopted for the general-purpose audio tagging task in 2018 DCASE, which aims to explore efficient models for general-purpose audio tagging problem. The samples in this dataset were annotated by Freesound [@fonseca2017freesound] with 41 labels (from Google’s AudioSet Ontology). The data format is unified to mono audio files of PCM 16 bit, frequency 44.1 KHz. There are about 9.5k samples in the training set, and 1.6K samples manually-verified annotated in the test set. Among the training set, samples are unequally distributed, which consists of about 3.7K manually-verified samples and about 5.8K non-verified samples whose quality estimated to be around 65-70%. The sample clips range from 92 to 300 for different classes in the training set, while the duration of audio files differs from 300ms to 30s.
**TUT Urban acoustic scenes 2018 dataset.** The TUT Urban Acoustic Scenes 2018 development dataset [@mesaros2018multi] (the dataset for DCASE 2018 task 1) for the acoustic scene classification task was used in our experiments, which tries to classify the characterizes of the environment where a recording from. Every sample in the training set is 10 seconds segments, and all of them are divided into 10 categories (acoustic scenes). Each acoustic scene contains 864 segments, in total 8640 segment. In the dataset, 6122 segments are used for training and 2518 segments are used for testing.
Networks
--------
Convolutional neural networks (CNN) have shown their superiority in the audio classification tasks. However, few researchers have explored the application of the knowledge distillation method to the acoustic CNN model. In this paper, two representative CNN networks, VGGNet [@vgg] and ResNet [@he2016deep], were used for experimental verification.
**VGGNet** replaces the large convolution kernel (such as 7$\times$7 in AlexNet [@krizhevsky2012imagenet]) with a 3$\times$3 convolution kernel and improves its performance by deepening the network architecture. Three 3$\times$3 convolutional layers connected in series have the same effect with one 7$\times$7 convolutional layer, that is, the three 3$\times$3 convolutional layers have a receptive field size equivalent to one 7$\times$7 convolutional layer. However, the former has only about half of the latter’s parameters and reducing linear operations which enhances the learning ability of models.
**ResNet** uses the residual connection to solve the problems of information loss and vanishing gradients while training deep networks. The use of shortcut connections in ResNet directly bypasses the input information to the output, which protecting the integrity of the information and simplifying the learning objectives.
In this paper, we use the classic 19-layer VGGNet network VGGNet19 and the 101-layer ResNet network ResNet101 in the distillation framework [@xu2018meta].
Experiment setting
------------------
We use Pytorch to implement the network architecture and the librosa[^2] toolkit package for data processing. In addition, the gRPC[^3] framework is used for information transfer between networks. All the experiments are conducted on NVIDIA GeForce GTX 1080Ti GPU. For the experiment setting, SGD algorithm was adopted with the learning rate initialized as 0.001. The learning rate decays according to Pytorch CosineAnnealingLR function. The mini batch size set as 64 and the number of train epoch is 150. The mixup-data augmentation method [@zhang2018mixup] was adopted in all experiments to avoid overfitting. Setting the single branch training epoch $s=1$ and distillation training epoch $d=1$. In all experiments the models loading the parameters of pre-trained on ImageNet dataset.
It is worthwhile to notice that many different audio representations can be deployed for the experiments. In this paper, log Mel, CQT and MFCC are used, while it is trial to extent our approach to other representations. To produce logMel and MFCC, we follow the setting of [@kelearticle], the mel filter banks as 64, frameshift as 10 ms and frame width as 80 ms. Thus there will be 150 frames in an audio clip. The delta and delta-delta features are calculated using a window size of 9. To determine the relationship between features at different scales, logMel features of different resolution were used. In this paper, the logMel feature determined as “logMel128” where the number of mel filter banks is 128, and for the logMel feature whose number of mel filter banks is 64 determined as “logMel64”. And in the tables without ambiguity, “logMel” defaults to logMel feature whose number of mel filter banks is 64.
Results and analysis
====================
In this part, we present our experimental results of different configurations. In our experiments, the mean average precision (mAP) and accuracy were applied as the main evaluation criterion. All results reported are the audio level scores.
Knowledge distillation using cross multi-resolution representations
-------------------------------------------------------------------
Table \[tab1\] compares the results of networks trained with our distillation framework (masked with \*) using cross resolution representations as inputs and the results of networks training independently. We can observe that from the table: (1) The performance of network distilled using cross resolution representations are better compared to the independently trained networks. (2) The networks with inputs of logMel128 perform better than the networks with inputs of logMel64, which indicates that logMel128 maintains more information about audio signals than logMel64. The reason may be that more mel filter banks are beneficial to preserve the detailed information. (3) Although the complementary information has been used to improve the performance of each branch in the distillation framework, the ensembled network are better than the single branch network in the framework, which demonstrates that there is still complementary information works in the ensemble network. The two groups (network as ResNet and VGGNet) of experiments have the same trend. In addition, although the classification results based on the VGGNet network are not as good as the experimental group based on the ResNet, our method achieves a bigger performance improvement on the experiments based on VGGNet. Knowledge distillation using cross resolution representations can be beneficial compared to the network training with the conventional method, indicating that different resolution representations enable networks to learn useful features to reach sufficient agreement.
**Network** **Input** **Accuracy** **mAP@3**
------------------ ------------- -------------- -----------
\*[**ResNet**]{} logMel64 88.01 91.06
logMel128 88.42 91.75
logMel64\* 89.43 92.65
logMel128\* 90.12 93.16
Ensemble\* **91.02** **93.84**
\*[**VGGNet**]{} logMel64 82.4 87.58
logMel128 83.12 88.45
logMel64\* 89.75 92.66
logMel128\* 89.81 92.98
Ensemble\* **90.5** **93.48**
: The results (%) of distillation using cross resolution representations with network architecture as ResNet on FSDKaggle2018 dataset. In the table, \* indicates the branch network in our knowledge distillation framework, while network not distilled does not indicate with \*. The Ensemble\* is the results of ensemble networks of the branches in the framework. The same in the following tables. \[tab1\]
Knowledge distillation using multiple representations
-----------------------------------------------------
To verify the effectiveness of multiple representations distillation, we use ResNet as the two branches’ network architecture, while MFCC and LogMel are used as inputs, respectively. Table \[tab2\] reports the accuracy and mAP@3 on the FSDKaggle2018 dataset of ResNet trained with independent training method and the results of distillation using multiple representations. And Table \[tab2-2\] reports the results of distillation on multiple representations on the THU Urban Acoustic Scenes 2018 dataset. From the table, we can conclude that our distillation framework can leverage the complementary information in different representations to enhance the performance. It also can be seen that the branch with inputs of MFCC is better improved than the branch with inputs of logMel, more useful knowledge flow from the logMel branch to the MFCC branch during the distillation process. This is noteworthy that the basic performance of the logMel branch is better than the MFCC branch, that means that the branch of low-performance can get more performance gain from the high-performance branch. Another noteworthy phenomenon is that the accuracy and the mAP@3 of the logMel branch improved more after knowledge distillation with CQT than distillation with MFCC. This is in line with our expectations, because the conversion process from original audio to MFCC and logMel is similar, resulting in fewer feature differences between them. The CQT and logMel have more complementary information, which leads to a better effect of knowledge distillation.
**Model** **Input** **Accuracy** **mAP@3**
------------------ ------------ -------------- -----------
\*[**ResNet**]{} logMel 88.01 91.06
MFCC 84.18 88.78
logMel\* 88.31 91.95
MFCC\* 87.19 90.86
Ensemble\* **90.44** **93.28**
\*[**ResNet**]{} logMel 88.01 91.06
CQT 85.81 89.68
logMel\* 89.63 92.72
CQT\* 87.69 91.29
Ensemble\* **91.06** **93.76**
: The results (%) of distillation using multiple representations with ResNet as network architecture on FSDKaggle2018 dataset. \[tab2\]
**Model** **Input** **Accuracy** **mAP@3**
------------------ ------------ -------------- -----------
\*[**VGGNet**]{} logMel 65.29 76.88
MFCC 63.46 75.11
logMel\* 66.12 77.44
MFCC\* 65.8 76.22
Ensemble\* **66.76** **77.7**
\*[**ResNet**]{} logMel 70.33 80.67
MFCC 67.71 78.32
logMel\* 72.43 81.65
MFCC\* 68.59 79.82
Ensemble\* **72.86** **82.25**
: The results (%) of distillation using multiple representations with ResNet on TUT Urban acoustic scenes 2018 dataset. \[tab2-2\]
Knowledge distillation using different network architectures
------------------------------------------------------------
As the different inputs could produce complementary information, the network architectures may also lead to differences in knowledge. Table \[tab3\] and Table \[tab3-2\] compare the results in the distillation framework which distilled with ResNet and VGGNet network and the results of independent training method on TUT Urban acoustic scenes 2018 dataset and FSDKaggle2018 dataset. As expected, the knowledge distillation framework provided sufficient promotion compared to the independently training. Different network structures are also sources of complementary knowledge. Although our framework is useful on both the FSDKaggle2018 dataset (for general-purpose audio tagging task) and the TUT Urban acoustic scenes 2018 dataset (for acoustic scenes classification task), it is apparent that the promotion is greater on the general-purpose audio tagging task. There are a large number of non-verified samples on the FSDKaggle2018 dataset, and in distillation the soft labels re-marking the erroneous data can greatly reduce the impact of the error label. The process of relabeling the target by the knowledge distillation method can obtain similar benefits to semi-supervised learning and reduce the false induction of confusing labels.
[**Input**]{} [**Network**]{} **Accuracy** **mAP@3**
--------------- ----------------- -------------- -----------
\*[logMel]{} VGGNet 65.29 76.88
ResNet 70.33 80.67
VGGNet\* 66.6 77.59
ResNet\* 70.49 80.51
Ensemble\* **70.84** **80.85**
: The results (%) of distillation using different network architectures with inputs of logMel on TUT Urban acoustic scenes 2018 dataset. \[tab3\]
[**Input**]{} [**Network**]{} **Accuracy** **mAP@3**
--------------- ----------------- -------------- -----------
\*[logMel]{} VGGNet 82.4 87.58
ResNet 88.01 91.06
VGGNet\* 86.43 90.58
ResNet\* 89.18 92.61
Ensemble\* **89.88** **93.05**
\*[**CQT**]{} VGGNet 75.13 82.55
ResNet 85.81 89.68
VGGNet\* 86.2 90.42
ResNet\* 87.94 91.31
Ensemble\* **89.93** **93.18**
: The results (%) of distillation using different network architectures with inputs of CQT on FSDKaggle2018 dataset. \[tab3-2\]
--------------- --------- ----------- ----------- ------------- --------------- --------- ----------- ----------- ------------- ----------- -------------
**I1,N1** **ACC** **mAP@3** **ACC\*** **mAP@3\*** **I2,N2** **ACC** **mAP@3** **ACC\*** **mAP@3\*** **ACC\*** **mAP@3\***
logMel,ResNet 88.01 91.06 90.18 93.26 MFCC,VGGNet 81.54 86.88 84.31 88.96 **90.69** **93.27**
MFCC,ResNet 84.18 88.78 87.5 91.13 logMel,VGGNet 82.4 87.58 89.5 92.78 **89.93** **93.18**
logMel,ResNet 88.01 91.06 90.6 93.16 CQT,VGGNet 75.12 82.55 86.5 90.73 **91.31** **94.11**
--------------- --------- ----------- ----------- ------------- --------------- --------- ----------- ----------- ------------- ----------- -------------
--------------- --------- ----------- ----------- ------------- --------------- --------- ----------- ----------- ------------- ----------- -------------
**I1,N1** **ACC** **mAP@3** **ACC\*** **mAP@3\*** **I2,N2** **ACC** **mAP@3** **ACC\*** **mAP@3\*** **ACC\*** **mAP@3\***
logMel,ResNet 70.33 80.67 **72.48** **81.65** MFCC,VGGNet 63.46 75.11 64.29 75.24 71.17 80.98
MFCC,ResNet 67.71 78.32 67.91 77.87 logMel,VGGNet 65.29 76.88 68.19 78.38 **70.29** **80.14**
--------------- --------- ----------- ----------- ------------- --------------- --------- ----------- ----------- ------------- ----------- -------------
Knowledge distillation using multiple representations and different network architectures
-----------------------------------------------------------------------------------------
Further experiments explored the impact of multi-representations and different network architectures on knowledge distillation. Table \[tab5\] and Table \[tab6\] report the results where two branches use completely different inputs and network architectures on the general-purpose audio tagging task and acoustic scenes classification task, respectively. Theoretically, the more different factors are set in the branches, the more complementary information produced. From Table \[tab5\], we can find that the branches distilled using multi-representations and different network architectures are much better than trained independent. And the improvements (both accuracy and mAP@3) of distilled branches are much bigger than distillation just using multiple representations or different architectures. This indicates that the difference between the branches no matter input or network architecture directly determines effect of distillation. In Table \[tab6\], similar trends as Table \[tab5\] can be observed. These conclusions also hold on the TUT Urban acoustic scenes 2018 dataset. In addition, in the first set of experiments in TUT Urban acoustic scenes 2018 dataset, the ensemble collapse phenomenon occurred for the reason of the large performance gap of the branch networks, which indicates that our knowledge distillation method is more stable than the ensemble method.
![The results (%) comparison of different mode distillation on FSDKaggle2018 dataset, we only report the results of the branch with logMel as input and ResNet as the network in distillation framework. DN is an abbreviation for distillation using different network architectures, DR is an abbreviation for distillation using different representations, and D-RN stands for distillation using different representations and different network architectures.[]{data-label="fig3"}](cross.pdf)
The results comparison of distillation in different settings
------------------------------------------------------------
Figure \[fig3\] compare the results in different setting of our distillation framework. We can find that the networks with our knowledge distillation still perform better that the results of baseline (independently trained network). And it is obvious that the results of distillation using multiple representations and different network architectures are better than only distillation using multiple representations or distillation using different network architectures. From the all above results we conclude that (1) The knowledge distillation method always improves the performance of the branch network, and the ensemble method works for the distilled branches. (2) Both multiple representations distillation and different architectures provided supplementary information needed for distillation. (3) The biggest improvement comes from the distillation simultaneous different representations and different architectures, where branch network gets the best performance. This suggests that complementary knowledge from different sources can be superimposed.
![The T-SNE visualization figure without using our method.[]{data-label="fig4"}](without.jpg){width="8cm" height="5cm"}
![The T-SNE visualization figure of using our method.[]{data-label="fig5"}](with.jpg){width="8cm" height="5cm"}
The T-SNE visualization analysis
--------------------------------
To show the effectiveness of our method clearly, we using the t-distributed Stochastic Neighbor (T-SNE) embedding visualization method. The T-SNE method is an efficient manifold learning method that can compress high-dimensional data to a low-dimensional structure. Figures \[fig4\] shows the T-SNE visualization figure of a ResNet network (CQT as input) trained independently and Figures \[fig5\] was the results with our multi-representation distillation method. We used the 1600 manually verified samples on the validation set of the THU 2018 dataset, and got the logits layer outputs of the network, then using the T-SNE method to compress logits outputs into two-dimensional space to visualize the results. It can be found that without knowledge distillation, the points of samples are more scattered and there are many crossovers between samples of different categories, which makes it more difficult to distinguish the category boundaries. In contrast, in the T-SNE visualization figure of the model trained by our knowledge distillation method, the samples are compact and the sample category confusion is reduced. The figures show that our method can effectively enhance the classification performance of audio classification models.
Conclusions
===========
In this paper, we propose a novel collaborative learning framework for the audio classification task. It takes multiple representations as input and trains a classifier separately on each representation. A collaborative distillation framework is employed to share knowledge across different models. Extensive experiments demonstrate that the proposed approach can improve the classification performance and achieve competitive results on both acoustic scenes classification task and general audio tagging task (experiments were conducted on the FSD-Kaggle2018 dataset and the TUT Urban acoustic scenes classification 2018 dataset). Moreover, it is worthwhile to notice that leveraging this approach is capable of promoting the performance of the model, without increasing the computational complexity in the inference phase. A direction of future work is using the multi-representation distillation method to improve the performance on the tasks of the sound event detection. Moreover, we would like to explore of knowledge distillation for multi-modal datasets.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to thank the National Key R&D Program of China (2016YFB1000101).
[^1]: L. Gao, K. Xu, H. Wang and Y. Peng were with the National Key Laboratory of Parallel and Distributed Processing, College of Computer, National University of Defense Technology, Changsha 410073, China e-mail: (Kelele.xu@gmail.com).
[^2]: https://github.com/librosa/librosa
[^3]: https://github.com/grpc/grpc
|
---
address: |
Physik Department\
Technische Universität München\
D-85747 München, Germany\
E-mail: peter.kolb@ph.tum.de
author:
- 'Peter F. Kolb'
title: Hydrodynamic aspects of Relativistic Heavy Ion Collisions at RHIC
---
Overview
========
Among the many predictions[@Eskola01] made for observables of collisions at the Relativistic Heavy Ion Collider (RHIC), hydrodynamic calculations regarding the expected flow characteristics[@KSH00; @TLS01] turned out to be surprisingly accurate already a few months after running the new facility.[@STARv201] In fact, those predictions were so convincing that they were awarded a bottle of wine in the RHIC prediction competition.[@RHICpredictioncontest] That RHIC creates systems which – [*for the dominant part of their lifetimes*]{} – expand according to hydrodynamic principles is of utmost importance: The dynamical evolution is governed by the nuclear equation of state at extreme energies, whose thermodynamic features are thus accessible through collisions in the laboratory.
In the following, I will review indications of the strong transverse expansion that can be deduced from the observable final state of central and non-central Au+Au collisions. Thereby, I will pay particular attention to signals of collectivity at mid-rapidity, i.e. in the plane perpendicular to the beam-axis in the center of mass of the reaction (at vanishing pseudorapidity $\eta$). Although interesting phenomena are to be expected at $\eta \ne 0$, to which I will return later, the midrapidity region is of particular interest for the observation of collectivity: Before the reaction, there is no transverse component of motion. Observing collectivity in the transverse plane at midrapidity must thus be a result of the dynamical evolution of the fireball [*after*]{} the initial impact. In contrast, at rapidities $\eta \ne 0$, it is difficult to disentangle collectivity created by the fireball region and the longitudinal dynamics of the nuclei that was present already before the reaction.
To address the questions of the how and when those collective features in the transverse plane are established, I will turn to dynamical models and specifically introduce the formalism of hydrodynamics to study the temporal evolution of the fireball matter in this framework. This analysis makes the success of the simple formulas of blast wave parameterizations understandable and validates such a simplistic approach as a tool to rapidly characterize and summarize the main thermodynamic information of the final state.
I will then highlight the most important comparisons of experimental data and hydrodynamic calculations and their interpretation, before I give a detailed account on recent and future developments that can be expected in the field. Almost every single observable of heavy ion physics is influenced by the bulk matter, as the bulk provides the background of the evolution through which even the rare probes have to propagate. Now, with hydrodynamics, we have a tool to calculate the dynamics of this evolution.
Indications of strong transverse flow
=====================================
If the microscopic constituents of a flowing medium share a common average flow velocity $v_\perp$, heavier particles in this medium gain larger momenta than the lighter constituents. In the case of the late stages of heavy ion collisions, the flowing medium is the hadronic soup consisting of hadrons of widely varying masses. From experimental evidence we find that the hadronic abundancies reflect a composition of a hadronic gas at a temperature $T_{\rm chem} \sim 175\,{\rm MeV}$.[@BMMRS01] Refinements of the simple [blast wave model]{}[@SR79; @SSH93] assume the knowledge of a certain flow profile at the break up stage of the reaction as well as the shape of this surface of decoupling from which particles supposedly do not interact any further. In this way one achieves a handy formalism,[@BF01] that can easily be used to fit to the vast amount of data currently available from RHIC experiments, in particular to single particles spectra and eventually their anisotropies.[@RL03] Although the temperature parameter of these fits differs widely from 175 MeV [@BF01] to 100 MeV,[@RL03] the common finding of these model calculation is that a vast amount of transverse flow (with an average velocity greater than $c/2$) is achieved on a rather short time scale of about 10 fm$/c$. While the stunningly good description hints that there possibly is some physical truth underlying these parameterizations, these simple descriptions of the final state do not offer any insight [*how*]{} and [*when*]{} such strong transverse flow develops during the course of the evolution. For this, we have to turn to a fully dynamical description of the transverse expansion stage of the reaction.
Hydrodynamic formalism and phenomenology
========================================
There are two main philosophies to model the dynamical evolution of a system. If the system is small and the scattering processes can be treated individually, one can adapt a microscopic viewpoint, treating all the collisions of the constituents individually.[@MG00] In a very dense, strongly interacting medium, however, this approach becomes quickly impractical, particularly if one also were to consider low-momentum transfer processes consistently. Alternatively, in this limit of high density and strong rescattering, one can give up the particles’ individual personality, and characterize the system in terms of density fields and continuity equations. Ideally, if scattering is strong enough to allow for local thermodynamic equilibrium, the equations of energy momentum conservation can be formulated by the thermodynamic equivalent, the conservation of the thermodynamic energy-momentum tensor, $\partial_\mu \left[ (e+p) u^\mu u^\nu - p \, g^{\mu \nu} \right] = 0$. The energy density $e$ and the pressure $p$ are related by the thermodynamic equation of state of nuclear matter. $u^\mu$ is the collective four velocity, whose time evolution we can study within this formalism. Once supplied with appropriate initial conditions, which are often taken from geometric considerations[@KHHET01] or more fundamental models such as the saturation model[@KHHET01] or the color glass condensate,[@HN04] the system’s evolution is fully determined. At this point, the computer takes over to solve for the time evolution of the thermodynamic fields.
Fig. \[Fig1\] shows the temporal evolution of the entropy density on a double logarithmic scale for three different positions in the fireball of a central collision, 0, 3 and 5 fm away from the fireball center.[@Kolb03] The tangents drawn to the curve stress the occurring transition from the initial one-dimensional, longitudinal expansion to a fully 3-dimensional expansion at late stages. The right plot of Fig. \[Fig1\] shows the radial flow profile at different times throughout the evolution. Striking here is the fast transition to a linear profile, which persists throughout time. Still, as matter is continually moved to larger radii, the mean transverse velocity of the medium increases and transverse flow thus continues to increase. The main part of the fireball matter is know to freeze-out over a rather short time frame as the rescattering rate drops sharply as a function of temperature, and therefore even more so as function of time. This fact, together with the rapid development of a flat flow profile is at the heart of the applicability of the simple and the improved blast wave parameterizations.
=5.6cm =5.6cm
In contrast to the overall radial flow which increases throughout the lifetime of the system, anisotropies in the transverse flow profile are generated during the earliest instances of the reaction. Such anisotropies arise from the eccentricity in coordinate space of non-central collisions. Larger pressure gradients in the short direction (the direction of the impact parameter) lead to a larger transport of matter in this direction, thereby reducing the eccentricity and undermining the source of its own origin.[@Sorge97] In the course of this process more particles are transported into the direction of the impact parameter and the mean transverse momentum of the particles is greater, both of which manifests itself experimentally in anisotropies $v_n$ of the transverse momentum spectra $
\frac{dN}{p_T dp_T dy d\varphi}
=
\frac{dN}{2 \pi p_T dp_T dy} \left (1 + \sum_n v_n \cos (n \varphi)\right)\,.
$ It has been shown in many studies, both microscopically[@ZGK99] as well as macroscopically[@KSH00; @Kolb03] that those momentum anisotropies are generated during the first 5 fm/$c$ of the reaction, where most of the matter is still at temperatures exceeding the critical temperature of nuclear matter. Anisotropies in the transverse momentum spectra of hadrons thus originate from the partonic stage of the fireball and signals their collective behavior.[@KSH00]
Application to experimental observations
========================================
As mentioned before, the transverse momentum distribution at moderate $p_T$ are the natural choice of observables to study collective dynamical effects in the medium. We use the transverse momentum spectra of identified particles to determine the parameters of the calculation for RHIC at 200 GeV center of mass energy per nucleon. Those spectra are well reproduced applying an equilibration time of $\tau_{\rm equ} = 0.6 \,{\rm fm}/c$ and a central fireball temperature of 360 MeV.[@KR03] Although the particle ratios reflect chemical equilibrium at a temperature of about 175 MeV,[@BMMRS01] the slopes in particular of heavier baryons require further transverse accelertion down to temperatures of about 100 MeV.[@KH03] To keep the particle ratios fixed to their value at 175 MeV, chemical potentials seem to be dynamically generated throughout the evolution.[@Rapp02; @Teaney02; @HT02]
=5.6cm =5.6cm
Although the good reproduction of the spectra of central collisions is reassuring, it has been achieved to a great deal by adjusting the parameters of the calculation. Important [*predictions*]{} come into play in non-central collisions. Changing centrality, which introduces a breaking of the azimuthal symmetry inherent in central collisions, renders essentially all observables sensitive on the azimuthal angle of observation. This literally opens up an entirely new dimension of observable space. However, for fully dynamical simulations no additional parameters are introduced when stepping out to explore observables in this new dimension. The modified initialization for non-central collisions is stricktly determined by the geometry of the underlying initialization scenario. The only appearing [*parameter*]{} – the impact parameter $b$ – is estimated experimentally by giving the range of centrality for certain collisions.
We already mentioned the great significance of momentum anisotropies as they provide a signal from the first few fm/$c$ of the collision. Microscopic studies[@MG00] in comparison with experimental data show that the fireball constituents must undergo an incredible amount of rescattering among themselves. Hydrodynamic calculations, which exploit the limit of negligible mean free pathlengths, provide an impressive account of a large collection of momentum anisotropies.[@KH03] They give a good overall description of momentum anisotropies up to transverse momenta of 2 GeV (Pions) and even 4 GeV for Baryons. The centrality dependence of the hydrodynamic elliptic anisotropy compares well with experimental data as long as the impact parameter does not get too large. Finally, the mass characteristics, a flatter onset of momentum anisotropies at small transverse momenta,[@HKHRV01] is confirmed by experiment in stunningly good agreement. A recent compilation of momentum anisotropies of heavy particles is reprinted in Fig. \[Fig2\] (right panel).[@STAR02v2KLambda] Still more recently the momentum anisotropies of Cascades and Omegas have become available.[@Castillo04] Again, those anisotropies have been stunningly large, as predicted 3 years ago by hydrodynamic calculations.[@HKHRV01] As the multistrange resonances are believe to not interact strongly in a hadron gas, this is the clearest signal that strangeness anisotropy is generated in the partonic stage of the reaction. It appears that all quark flavors share the same flow anisotropy on the partonic level. This fact could be most prominently (dis-)proved by investigating momentum anisotropies of higher order and their ratios.[@v4] That the momentum anisotropies are as large as observed is a strong indication that the thermalization of the medium is achieved very rapidly.[@thermalization] The fact that hydrodynamics overestimates anisotropies at large transverse momenta and in peripheral collisions is well understood, considering that high $p_T$ particles escape the fireball too rapidly to follow the collective motion of the bulk, and that small systems do not provide enough rescattering in the limited volume. Here, it is also important to remark that while hydrodynamics does not seem to be fully applicable at the lower beam energies of the SPS in peripheral collisions, there might still be a fair chance for its quantitative application in [*central*]{} collisions. This could be in particular true for the early, plasma part of the reaction, whereas the later hadronic stage might drop out of local thermal equilibrium. Only detailed quantitative comparisons of hydrodynamic calculations and the most recent results[@NA49] from SPS experiments can answer this question and address important topics close to thermal equilibrium and just above the transition temperature.
Current trends and future requirements
======================================
Most of the hydrodynamic calculations that are applied to experimental data at present include some great simplifications in order to keep the numerical efforts to a reasonable degree. One of these is the strong bias to investigations of observables at midrapidity. Although the collectivity observed at midrapidity is clearly produced during the evolution of the produced fireball and therefore contains clean probes of the dynamics, a broader view of the full collision volume is highly desirable, in particular in the light of recent data extending the analysis of momentum anisotropies to large rapidities. Fully three-dimensional calculations have been successfully performed in recent years.[@MMNH02] However not too much is gained by applying the [*ideal*]{} hydrodynamic calculations at large forward rapidities as the system is thought reach thermal equilibrium only at later times[@KH04] or not at all[@Hirano01] in regions far away from the center of mass of the reaction.
Nevertheless, there is a lot to be gained from investigations of hydrodynamic calculations at non-zero rapidity. Due to the (net)-baryon content of the initial nuclei, one can expect that the (net) baryon-number density increases with increasing rapidity. As one leaves the ’clean’ midrapidity region with its antibaryon to baryon ratio of about 0.75, one gets closer to the initial nuclei region where the antibaryon content effectively vanishes. Thus, at larger rapidity, a hydrodynamical system evolves according to the equation of state at larger baryon chemical potential than at mid-rapidity. This offers the exciting possibility of tracing signals of the expected tricritical point at finite rapidity. The effect of the tricritical point on the hydrodynamic evolution of RHIC collisions is currently explored by Nonaka.[@NonakaBNL03] It was found that such a point acts as an attractor for the thermodynamic paths of a constant fraction of entropy and baryon-density. Clearly, due to the large fluctuations that appear around the critical point, the ideal hydrodynamic treatment will become invalid, but it will be interesting to explore how the system drops out of equilibrium, and which quantitative experimental measurements will be predicted.
The less efficient energy density production and deposition at large rapidity eventually leads to a breakdown of the assumption of full thermalization and an ideal expansion. Viscous effects will become more and more important in smaller and more dilute systems. It is of fundamental interest to get estimates of the viscosity of the QCD plasma state. Viscous hydrodynamics can deliver valuable information in this direction. The realization of fully dynamical hydrodynamical calculations is however difficult. The viscous terms have to be solved dynamically, increasing the number of differential equations by up to 5 (depending on the symmetries one exploits explicitly). Although difficult to solve,[@Muronga04] first results on the dynamical evolution of central collisions have recently been presented.[@Teaney04] The message of these dynamical studies is that viscous matter sticks together for a longer period, just to blow up quicker in the late stages. The left panel of Fig. \[Fig3\] shows plots of the energy density as a function of the transverse coordinate for a viscous calculation compared to the ideal Euler calculation.[@Teaney04] Although the densities change slower at first, velocities build up faster due to a restructuring of the pressure components, which is also observed in the case of only transverse thermalization and longitudinal free streaming.[@HW02] Finally, in the late stage of the reaction, the left over matter dissipates more rapidly, due to the larger velocities that were achieved. The net result is that the system stays longer in the hot phase of its reaction and disperses faster through the late stages. The late stages, for which viscosity become increasingly important, thus have a shorter lifetime and freeze-out occurs more rapidly. Comforting is the observation that viscosity does not lead to self-enhancing features!
=5.0cm =5.6cm
In this context it is also interesting to cast doubt on the assumption of having particles distributed in momentum space according to the simple ideal distribution laws of Fermions and Bosons. It has been shown[@Teaney03] that a large deviation from these distributions smears out the directed collective motion observed in the momentum anisotropies and is thus not supported by the data (see right panel of Fig. \[Fig3\].)
Summary
=======
We have learned [*a lot*]{} about the collective phenomena exhibited in RHIC data of the first few years. The data clearly show many characteristics of rapid thermalization and an extended stage of hydrodynamic expansion of the fireball into the surrounding vacuum. On a linear timescale, the hydrodynamic stage is the most significant stage of the reaction, thereby influencing many other relevant observables. Hydrodynamics is a clean (meaning well defined) tool, with only a small number of parameters and easily reproducible by anybody having access to a computer. The two most important facts learned from hydrodynamics and the experimental data so far is the need for rapid thermalization (in a time of less than 1 fm/c) where matter obtains a large transverse push due to a hard equation of state ($p \sim e/3$). Many quantitative questions are still unanswered: What is the frame of error around the equilibration time, what different kinds of equations of state and phase transitions would still account for the data, how does one feed back on the other? What is the role of viscosity in the expansion? Why does viscosity of high temperature QCD matter seem to be so low as to allow for an ideal hydrodynamic description of the data?
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank the organizers for inviting me to present this overview. I deeply acknowledge the contributions of my collaborators and friends without whom I would not have received this honor.
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|
---
abstract: 'I introduce an algorithm for estimating parameters from multidimensional data based on forward modelling. It performs an iterative local search to effectively achieve a nonlinear interpolation of a template grid. In contrast to many machine learning approaches it avoids fitting an inverse model and the problems associated with this. The algorithm makes explicit use of the sensitivities of the data to the parameters, with the goal of better treating parameters which only have a weak impact on the data. The forward modelling approach provides uncertainty (full covariance) estimates in the predicted parameters as well as a goodness-of-fit for observations, thus providing a simple means of identifying outliers. I demonstrate the algorithm, , with the estimation of stellar astrophysical parameters (APs) from simulations of the low resolution spectrophotometry to be obtained by Gaia. The AP accuracy is competitive with that obtained by a support vector machine. For zero extinction stars covering a wide range of metallicity, surface gravity and temperature, can estimate to an accuracy of 0.3% at G=15 and to 4% for (lower signal-to-noise ratio) spectra at G=20, the Gaia limiting magnitude (mean absolute errors are quoted). and can be estimated to accuracies of 0.1–0.4dex for stars with G$\leq18.5$, depending on the magnitude and what priors we can place on the APs. If extinction varies a priori over a wide range (0–10mag) – which will be the case with Gaia because it is an all sky survey – then and can still be estimated to 0.3 and 0.5dex respectively at G=15, but much poorer at G=18.5. and can be estimated quite accurately (3–4% and 0.1–0.2mag respectively at G=15), but there is a strong and ubiquitous degeneracy in these parameters which limits our ability to estimate either accurately at faint magnitudes. Using the forward model we can map these degeneracies (in advance), and thus provide a complete probability distribution over solutions. Additional information from the Gaia parallaxes, other surveys or suitable priors should help reduce these degeneracies.'
author:
- |
C.A.L. Bailer-Jones\
Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany[^1]\
date: 'Accepted 2009 November 26. Received 2009 November 26; in original form 2009 October 19'
title: The ILIUM forward modelling algorithm for multivariate parameter estimation and its application to derive stellar parameters from Gaia spectrophotometry
---
\[firstpage\]
surveys – methods: data analysis, statistical – techniques: spectroscopic – stars: fundamental parameters – ISM: extinction
Introduction \[introduction\]
=============================
Inferring parameters from multidimensional data is a common task in astronomy, whether this be inference of cosmological parameters from CMB experiments, photometric-redshifts of galaxies or physical properites from stellar spectra. Important questions about the structure and evolution of stars and stellar populations require knowledge of abundances and ages, which must be obtained spectroscopically via the stellar atmospheric parameters effective temperature (), surface gravity () and iron-peak metallicity ().
Numerous publications have presented methods for estimating such astrophysical parameters (APs) from spectra. The methods can be divided into two broad categories. The first, based on high resolution and high signal-to-noise ratio (SNR) spectra, makes use of specific line indices selected to be sensitive predominantly to the phenomena of interest. Examples include the detection of BHB stars via Calcium and Balmer lines (e.g. Brown [@brown03]) and spectral type classification of M, L and T dwarfs via molecular band indices (e.g. Hawley et al. [@hawley02]). In each case these methods use a specific, identified phenomenon and soare generally limited to a narrow part of the AP space where the values of the other APs are reasonably well known. While simple and useful, these methods do not use all available information nor can they normally be used with low resolution data, because the effects of individual APs cannot then be separated.
The second category of methods is global pattern recognition, which try to use the full set of available data. These are generally used when we want to estimate one or more APs over a wide range of parameter space. In such cases there is rarely a simple relation between a single feature and the physical quantity of interest. (An example is determination of surface gravity, where the relevant lines to use change with temperature.) We must therefore infer which features are relevant to which APs in which parts of the AP space. As this is generally a nonlinear, multidimensional problem, the standard approach is to use a machine learning algorithm to learn the mapping from the data space to the AP space based on labelled template spectra (spectra with known APs). Various models have been used in astronomy for a variety of problems, including (to name just a few): neural networks for stellar parameter estimation (e.g. Re Fiorentin et al. [@rf07]) or photometric redshift estimation (e.g. Firth et al. [@firth03]); support vector machines for quasar classification (e.g. Gao et al. [@gao08], Bailer-Jones et al. [@cbj08]) or galaxy morphology classification (e.g. Huertas-Company et al. [@hc08]); classification trees for identifying cosmic rays in images (e.g. Salzberg et al. [@salzberg95]); linear basis function projection methods (e.g. the method of Recio-Blanco et al. [@recioblanco06] for spectral parameter estimation, which constructs the basis functions from model spectra). More examples of machine learning methods and their use in astronomy can be found in the volume edited by Bailer-Jones [@cbj08b].
Note that the first category of methods, line indices, is really just a special case of the second in which drastic feature selection has taken place to enable use of low (one or two) dimensional models. In both cases we must learn some relationship ${\rm AP} = g'({\rm Data})$ from a model or based on some labelled templates. But this is an [*inverse*]{} relation: more than one set of APs may fit a given set of data (e.g. a low extinction cool star or high extinction hot star could produce the same colours). Despite this non-uniqueness we nonetheless try and fit a unique model. This causes fitting problems which become more severe the lower the quality of the data (the lower the number of independent measures) and the larger the number of APs we want to estimate from it, and could lead to poor AP estimates or biases. In contrast, the forward mapping (or [*generative model*]{}), ${\rm Data} = g({\rm AP})$ is unique, because this is a causal, physical model (e.g. a stellar atmosphere and radiative transfer model of a spectrum). A further issue is that the model must learn the sensitivity of each input to each AP (and how noise affects this). Yet this information we in principle have already from the gradients of the generative model.
A pattern recognition method which tries to overcome some of these problems is $k$-nearest neighbours, which has also been applied to many problems in astronomy (e.g. Katz et al. [@katz98], Ball et al. [@ball08]; plus extensions thereof such as kernel density estimation, e.g. Richards et al. [@richards04], or the method of Shkedy et al. [@shkedy07], which converts the distances into likelihoods and uses priors to create a full probabilistic solution.) In many ways this is the most natural way to solve the problem: we create a grid of labelled templates and find which are closest to our observation (perhaps smoothing over several neighbours). On the assumption that the APs vary smoothly with the data between the grid points, this may provide a good estimate. But for this to be accurate (and not too biased), the grid must be sufficiently dense that multiple grid points lie within the error ellipse of the observation (the covariance of these neighbours then provides a measure of the uncertainty in the estimated APs). If the SNR is high, the grid must therefore be very dense. Moreover, as the number of APs increases, the required grid density grows exponentially with it: For a stellar parametrization problem with 5 APs we might need an average of 100 samples per AP, resulting in $100^{5}=10^{10}$ templates. There is also the issue of what distance metric to use. The covariance-weighted Mahalanobis distance is often used, but it ignores the sensitivities of the inputs. (If some inputs are sometimes dominated by irrelevant cosmic scatter, this will add unmodelled noise to the distance estimate.) This is a problem when we have a mix of APs, some of which have a large and others a small impact on the variance (“strong” and “weak” APs). If we just use the Mahalanobis distance we loose sensitivity to the weak APs.
We could overcome these problems if we did on-the-fly interpolation of the template grid to generate new templates as we need them. Running stellar models is far too time consuming for this, but also unnecessary because the generative model is smooth: We can instead fit a forward model to a low density grid of templates as an approximation to the generative model. As we shall see, possession of a forward model opens up opportunities not available to the inverse methods, such as direct uncertainty estimates and goodness-of-fit assessment of the solution.
In this article I introduce an algorithm for AP estimation based on this forward modelling idea and iterative interpolation. I will demonstrate it using simulations of low resolution spectra to be obtained from the Gaia mission (e.g. Lindegren et al. [@lindegren08]).[^2] Gaia will observe more than $10^9$ stars down to 20$^{th}$ magnitude over the whole Galaxy, stars which a priori span a very wide range in several APs. This includes the line-of-sight extinction parameter, , which must be estimated accurately if we want to derive intrinsic stellar luminosities from the Gaia parallaxes. AP estimation (Bailer-Jones [@cbj05]) is therefore an integral part of the overall Gaia data processing and comprises one of the Coordination Units in the Gaia Data Processing and Analysis Consortium (DPAC) (Mignard & Drimmel [@aoresponse], O’Mullane et al. [@omullane07]).
I will now describe the basic algorithm (section \[algorithm\]). In section \[data\] I then introduce the simulated Gaia spectroscopy to which is applied, with the results and discussion thereof presented in sections \[sect:tefflogg\], \[sect:tefffeh\] and \[sect:2d1d\]. The latter section also reports on a strong and ubiquitous degeneracy between and . I summarize and conclude in section \[conclusions\]. Additional plots, results and discussions can be found in a series of four Gaia technical notes (Bailer-Jones 2009a,b,c,d) available from [http://www.mpia.de/Gaia]{}.
The algorithm {#algorithm}
==============
I outline the algorithm using the terminology of spectra and stellar astrophysical parameters, although it is quite general and applies to any multivariate data. Table \[tab:notation\] summarizes the notation. “Band” refers to a flux measurement in the spectrum. In general it could be a photometric band, a single pixel in the spectrum, or a function of many pixels.
-------------------- ---------------------------------------------------------------------------
$I$ number of bands (pixels in spectrum)
$i$ counter over band, $i=1 \dots I$
$p_i$ photon counts in band $i$ ($\bmath p$ is a spectrum)
$J$ number of APs (astrophysical parameters)
$j$ counter over AP, $j=1 \dots J$
$\phi_j$ AP $j$ ($\bmath \phi$ is a set of APs)
$s_{ij}$ sensitivity of band $i$ to AP $j$, $\frac{\partial p_i}{\partial \phi_j}$
sensitivity matrix, $I \times J$ matrix with elements $s_{ij}$
$f_i(\bmath \phi)$ forward model for band $i$
$n$ iteration; e.g. $\phi(n)$ is the AP at iteration $n$
-------------------- ---------------------------------------------------------------------------
: Notation\[tab:notation\]
Forward modelling
-----------------
I will call the true relationship between the APs and the flux in a band $i$ the [*generative model*]{}, $g_i(\bmath \phi)$. This provides the observed spectrum for a given set of APs and thus encapsulates the underlying stellar model, radiative transfer, interstellar extinction, instrument model etc. (There is a separate function for each band, but for simplicity I will refer to it in the singular.) Generally we don’t have an explicit function for this model, so it remains unknown. All we have for doing AP estimation is a discrete [*template grid*]{} of example spectra with known APs generated by the generative model. We approximate the generative model using a [*forward model*]{}, $f_i(\bmath \phi)$, which is a (nonlinear) parametrized fit to this grid and provides flux estimates at arbitrary APs, i.e. off the grid. (Forward models can be fit independently for each band.) Demanding the forward models to be continuous functions ensures we can also use them to calculate the sensitivities, which by definition are the gradients of the flux with respect to each AP. The forward model fitting is done just once for a given grid and is kept fixed when predicting APs. In other words it is a training procedure.
Core algorithm
--------------
The basic idea of is to use the Newton-Raphson method to find that forward model-predicted spectrum (and associated APs) which best fits the oberved spectrum.
In detail, the algorithm is as follows (Fig. \[fig:principle\]). Consider first a single AP and single band. The measurement is $p(0)$ and we want to estimate its AP. The forward model, $\hat p = f(\phi)$, has been fit and remains fixed. The procedure is as follows ($n$ is the iteration number)
1. Initialize: find nearest grid neighbour to $p(0)$, i.e. the one which minimizes the sum-of-squares residual $\delta \bmath p^T \delta \bmath p$. Call this $[p(1), \phi(1)]$. $\phi(1)$ is the initial AP estimate.
2. Use the forward model to calculate the local sensitivities, $\frac{\partial p}{\partial \phi}$, at the current AP estimate.
3. Calculate the discrepancy (residual) between the predicted flux and the measured flux, $\delta p(n) = p(n) - p(0)$.
4. Estimate the AP offset as $\delta \phi(n) = \left (\frac{\partial \phi}{\partial p} \right)_{\phi(n)} \times \delta p(n) $, i.e. a Taylor expansion truncated to the linear term. (Note that this partial derivative is the reciprocal sensitivity.)
5. Make a step in AP space, $ \phi(n+1) = \phi(n) - \delta \phi(n)$, [*toward*]{} the better estimate. This is the new AP prediction.
6. Use the forward model to predict the corresponding (off-grid) flux, $p(n+1)$
7. Iterate steps ii–vi until convergence is achieved or a stop is imposed.
The algorithm is basically minimizing $|\delta \bmath p|$. At each iteration we obtain an estimate of the APs (step v) and the corresponding spectrum (step vi). Convergence could be defined in several ways, e.g. when changes in the spectrum or the APs (or their rate of change) drop below some threshold. Alternatively we could simply stop after some fixed number of iterations. There is no guarantee of convergence. For example, if the AP steps were sufficiently large to move to a part of the function with a sensitivity of the opposite sign, then the model could diverge or get stuck in a limit cycle. Likewise, if initialized too far from the true solution the algorithm could become stuck in a local minimum far from the true solution. For this, and other reasons, the algorithm in practice has some additional features (discussed in section \[sect:practical\]). Also,
Generalization to multiple APs and bands
----------------------------------------
In general we have several bands and several APs. The flux perturbation due to small changes in the APs is then $$\delta {\bmath p} \, = \, \sens \, \delta{\bmath \phi}
\label{eqn:deltap}$$ where $\sens$ is the $I \times J$ sensitivity matrix with elements $s_{ij} = {\partial p_i}/{\partial \phi_j}$. Note that $I>J$. Multiplying this equation on the left by $(\sens^T \sens)^{-1}\sens^T$ gives $$\delta {\bmath \phi} \, = \, (\sens^T \sens)^{-1}\sens^T \delta{\bmath p}
\label{eqn:deltaap}$$ so the AP update equation (step v in the algorithm) becomes $${\bmath \phi}(n+1) = {\bmath \phi}(n) - (\sens^T \sens)^{-1}\sens^T \delta{\bmath p}(n)
\label{eqn:apupdate}$$ The $I$ forward models are now functions of $J$ variables, and this turns out to be a critical matter.
The forward model {#sect:forward_model}
-----------------
The core algorithm just described can make use of any form for the forward model, on the condition that it provides values of the function and its first derivatives for arbitrary values of the APs.
The most obvious forward model would be a multidimensional, nonlinear regression of the form $\hat{p} = f({\bmath \phi})$, which in principle works for any number of APs. However, I found that it was difficult to get a model which simultaneously fits both , a strong AP, and , a weak AP to sufficient accuracy. [*Strong*]{} here means that it explains much of the variance in the flux data, i.e. is a strong predictor of the flux. [*Weak*]{} is a relative term, indicating that the AP explains much less of the variance. The reason for poor fits over the weak APs is that the model is fit by minimizing a single objective function, namely the error in reproducing the flux. As the weak AP has very little impact on the flux, its influence has little impact on the error, so the model optimization does little to produce a good fit over this AP.
![Schematic diagram of a two-component forward model. Note the stronger variation in the flux in the direction of the strong AP (the contrast is typically much larger for the problems considered in this article).[]{data-label="fig:forward_model_schematic"}](figures/forward_model_schematic){width="40.00000%"}
To overcome this problem I use a two-component forward model to separately fit the strong and weak APs. Consider the case of a single strong AP and a single weak AP (this is generalized later). The strong component is a 1D nonlinear function of the strong AP which is fit by marginalizing over the weak AP. This fits most of the flux variation. Then, at each discrete value of the strong AP in the grid, we fit the residual flux as a function of the weak AP; these are the second components (also 1D). They provide a flux increment dependent on the weak AP, which is added to the flux predicted given the strong AP. A schematic illustration of such a two-component forward model is illustrated in Fig. \[fig:forward\_model\_schematic\].
![Schematic diagram of the grid of AP values, $k = 1 \ldots 20$, $l = 1 \dots 7$. The solid (red) points denote those used to fit the weak component of the forward model for one value of the strong AP.[]{data-label="fig:schematic_grid"}](figures/schematic_grid){width="33.00000%"}
![Schematic illustration of the two components of the forward model fit over all the strong and weak points in the grid in Fig. \[fig:schematic\_grid\] Top: the fit over the strong APs. Bottom: one of the fits over the weak APs (the solid/red points in the top panel at k=14).[]{data-label="fig:two_component_forward_model"}](figures/two_component_forward_model_2){width="33.00000%"}
To be more precise the model is fit as follows. Let $\phi^S$ denote a strong AP, $\phi^W$ a weak AP and $f_i(\phi^S, \phi^W)$ the complete (2D) forward model for band $i$. Let subscripts $k$ and $l$ denote specific values in the grid of the strong and weak APs respectively (see Fig. \[fig:schematic\_grid\]). We model the flux at an arbitrary AP point as $$f_i(\phi^S, \phi^W) \, = \, f^S_i(\phi^S) \, + \, f^W_{i,k}(\phi^W ; \phi^S=\phi^S_k)$$ where both $f^S_i$ and $ f^W_{i,k}$ are 1D functions. $f^S_i(\phi^S)$ is the single strong component (for band $i$). It is a fit to the average value of $\phi^W$ at each $\phi^S$ (the curve in the top panel of Fig. \[fig:two\_component\_forward\_model\]). $f^W_{i,k}$ is the $k^{th}$ weak component, which is a fit with respect to the weak AP with the strong AP fixed at $\phi^S_k$ (e.g. the solid/red points in Fig. \[fig:schematic\_grid\]), i.e. it is fit to the residuals $$\{ \, p(\phi^S = \phi^S_k) - \overline{ p(\phi^S = \phi^S_{k^{'}}) } \, \}$$ as illustrated in in the bottom panel of Fig. \[fig:two\_component\_forward\_model\].
This fitting approach clearly requires us to have a “semi-regular” grid: one which has a range of values of the weak AP for each value of the strong AP. (This requirement is easily fulfilled when using synthetic grids.) The number of weak components in the model is equal to the number of unique values of $\phi^S$, 20 in this schematic case. If we have more than one strong or weak component then we raise the dimensionality of the strong or weak component (see section \[sect:2d1d\]).
Applying the forward model is easy. Given $(\phi^S, \phi^W)$ we
1. evaluate $f^S_i$, the strong component;
2. find the nearest value, $\phi^S_k$, in the grid to $\phi^S$, i.e. identify the closest weak component;
3. evaluate $f^W_{i,k}$, the increment from the weak component;
4. sum the two components, $f^S_i + f^W_{i,k}$, to give the forward model prediction.
Although the weak component we use changes discontinuously as $\phi^S$ varies, the weak component is only specifying an [*increment*]{} to the strong component fit. As both components are smooth in their respective APs the combined function is also smooth along any direction parallel to the AP axes. It is not smooth along arbitrary directions, but this is unimportant because explicit calculations with the forward model (e.g. of the sensitivities) are only carried out parallel to the AP axes.
Practical algorithm {#sect:practical}
-------------------
The practical realities of working with real (noisy) data mean that the basic algorithm should be extended in order to make it more robust. These I now describe together with other implementation aspects used for the experiments described later. For the purposes of this paper the algorithm has been implemented in R[^3]. A Java implementation is in progress, which will be necessary for larger scale applications.
### Standardized variables
The spectral variables are observed photon counts (to within an irrelevant constant factor). The APs are all on logarithmic scales: in magnitudes, and in dex, and . In order to bring each variable to the same level I standardize the flux in each band and each AP (linearly scale each to have zero mean and unit variance). If there were correlations in the spectra these could be removed by “sphereing” (“prewhitening”), a covariate generalization of standardization which gives the data unit diagonal covariance (e.g. Bishop [@bishop06]).
### Forward model functions {#fmfunctions}
The strong and weak components of the forward model are fit using smoothing splines (e.g. Hastie et al. [@hastie01]). Conventional cubic splines have the drawback that one must control their complexity (smoothness) using the number and position of the knots. Smoothing splines circumvent this problem by setting a knot at every point (which would overfit the data) and then applying a smoothing penalty which is controlled by specifying the effective degrees-of-freedom (dof). I set this by trial and error via inspection of the resulting fits. For the 2D problems TG ( and ) and TM ( and ) described in sections \[sect:tefflogg\] and \[sect:tefffeh\], both the strong and the weak model splines are 1D. The strong model uses dof$ = n_{\rm Teff}/2=16.5$ where $n_{\rm Teff}$ is the number of unique points. As the maximum number of points is 10 (for the training data), and because the variation with is quite smooth, I set the dof for these fits to be 4. However, many of the values in the training grid have fewer or points: To avoid overfitting, if $n_{\rm logg}(T_{\rm eff}) \leq 4$ then a linear fit is used. If $n_{\rm
logg}(T_{\rm eff}) = 1$, then no fit is performed and this weak component of the forward model is zero. (Practical aspects of higher dimensional forward models are described in section \[sect:2d1d\].)
The forward model is always fit to noise-free data. It is well known (and the author’s experience) that inverse modelling methods such as ANNs and SVMs perform best when trained on data with a similar noise level as the target data (e.g. Snider et al. [@snider01]).This precipitates the need for multiple models when used on survey data with a range of SNRs, something which is not an issue for .
### Sensitivity estimation
If the forward model is a simple analytic function then it may have analytical first derivatives which can be used to calculate the sensitivities. But in the general case we can use the method of first differences $$\left . \frac{\partial p}{\partial \phi_j} \right |_{\bmath \phi} \, \simeq \, \frac{f(\bmath \phi + \delta \phi_j) - f(\bmath \phi - \delta \phi_j)}{2 \, \delta \phi_j}
\label{eqn:firstdiff}$$ I select $\delta \phi_j$ to be slightly smaller than the maximum precision a priori possible in an AP. As the forward model must be smooth at this resolution, the first difference approximation is sufficiently accurate. For the examples shown later, I choose $\delta \phi_j$ to be 0.05 dex for and , 0.0005 for (0.1% for ) and 0.03mag for .
### Lower limit on sensitivities (singularity avoidance)
Given that the AP updates depend upon the inverse of the sensitivity matrix (equation \[eqn:deltaap\]), it is prudent to prevent the sensitivities being too small in order to avoid $\sens^T \sens$ being singular. For this reason, a lower limit is placed on the absolute value of each sensitivity, $s_{ij}$, of 0.001 (with $p$ and $\phi$ in standardized units). For the examples shown later, this limit rarely had to be applied in practice and had negligible impact on the results. To avoid singularity $\sens^T \sens$ must also have a rank of at least $J$, i.e. to estimate $J$ APs we need at least $J$ independent measures in the spectrum.
### AP update contribution clipping {#sect:apcontclip}
Equation \[eqn:deltaap\] can be written $${\bmath \phi}(n+1) = {\bmath \phi}(n) - {\mathbfss M} \, \delta{\bmath p}(n)
\label{eqn:mupdate}$$ where $${\mathbfss M} = (\sens^T \sens)^{-1}\sens^T
\label{eqn:m}$$ is a $J \times I$ matrix. Equation \[eqn:mupdate\] gives $J$ update equations, one for each AP. The update for AP $j$ can be written as the dot product of two vectors, the $j^{th}$ row of ${\mathbfss M}$, ${\bmath m}_j$ with ${\bmath p}(n)$, i.e.$$\begin{aligned}
\label{eqn:apcont}
\phi_j(n+1) &=& \phi_j(n) - {\bmath m}_j \, \delta{\bmath p}(n) \nonumber \\
&=& \phi_j(n) - \sum_i m_{ij} \delta p_i(n) \nonumber \\
&=& \phi_j(n) - \sum_i u_{ij}(n)\end{aligned}$$ which defines $u_{ij}$. Thus we see that the update to AP $j$ is a sum over $I$ individual updates, which we can view as an update “spectrum”. If we inspect these updates (see Bailer-Jones[@cbj09a]) we see that on occasion some are much larger than the others. This dominance of the update by just one of a few elements is undesirable, because they may be affected by noise ($\delta{\bmath p}(n)$ is a noisy measurement). For this reason, I clip outliers in this spectrum. (It is valid to compare the updates for different bands, because we work with standardized fluxes.) To be robust, I set an upper (lower) limit which is a multiple $c$ of the median of those points above (below) the median. Using the notation $\theta()$ to denote median, the limits are $$\begin{aligned}
u_{\rm upper} &=& \theta(u_i) + c[\theta(u_i > \theta(u_i)) - \theta(u_i)] \nonumber \\
u_{\rm lower} &=& \theta(u_i) + c[\theta(u_i < \theta(u_i)) - \theta(u_i)]\end{aligned}$$ I somewhat arbitrarily set $c=10$ so as to be relatively conservative in clipping.
### Upper limit on AP update step size {#sect:apupdates}
The AP steps at each iteration (equation \[eqn:deltaap\]) could be very large. This is undesirable, because the updates are based on a local [*linear*]{} approximation to the generative model. The code therefore imposes upper limits on the AP updates, corresponding to steps no larger than about 2.0dex in and , 0.04 in (10% in ) and 0.3mag in . A larger step size is permitted for the weaker APs because the initial nearest neighbour offset can be quite incorrect. These limits are imposed more often for noisy data, but still relatively rarely.
### Limit AP extrapolation {#sect:apextrapolate}
We do not expect the forward model to make good predictions beyond the AP extremes of the grid, so I set upper and lower limits on the AP estimates which can provide. These are set as $e$ times the range of each AP, i.e.$$\begin{aligned}
{\rm upper~limit} &=& \max{\phi_j} + e(\max{\phi_j} - \min{\phi_j}) \nonumber \\
{\rm lower~limit} &=& \min{\phi_j} - e(\max{\phi_j} - \min{\phi_j})\end{aligned}$$ I set $e=0.1$.
### Stopping criterion
The algorithm is simply run for a fixed number of iterations (20). We often observe good natural convergence, so a more sophisticated stopping criterion is not applied at this time, although it may be important on high variance data sets (low SNR or more APs).
Performance statistics {#errstat}
----------------------
The model performance is assessed via the AP residuals (estimated minus true, $\delta \phi$) on an evaluation data set. I report three statistics : (1) the root-mean-square (RMS) error, which I abbreviate with ; (2) the mean absolute error, $\overline{|\delta \phi|}$, abbreviated as ; (3) the mean residual, $\overline{\delta \phi}$, a measure of the systematic error, abbreviated as . (Note that as (1) and (2) are statistics with respect to the true values they also include any systematic errors.) I mostly use rather than RMS because the former is more robust. If the residuals had a Gaussian distribution then the RMS would equal the Gaussian $1\sigma$ which is $\sqrt{\pi/2}=1.25$ times larger than . But usually there are outliers which increase the RMS significantly beyond this.
Uncertainty estimates {#sect:errest}
---------------------
If vectors ${\bmath y}$ and ${\bmath x}$ are related by a transformation ${\bmath y} = {\mathbfss A} {\bmath x}$ then a standard result of matrix algebra is that the covariance of ${\bmath y}$ is ${\mathbfss C}_y = {\mathbfss A} {\mathbfss C}_x {\mathbfss A}^T$ where ${\mathbfss C}_x$ is the covariance of ${\bmath x}$. Applying this to equation \[eqn:deltaap\] gives us an expression for the covariance in the APs $${\mathbfss C}_{\phi} = (\sens^T \sens)^{-1}\sens^T {\mathbfss C}_p \sens (\sens^T \sens)^{-1}
\label{eqn:apcov}$$ as a function of the sensitivity (calculated at the estimated APs) and the covariance in the measured photometry, ${\mathbfss C}_p$. (This equation assumes that provides unbiased AP estimates and that the sensitivities have zero covariance. It can also be written ${\mathbfss C}_{\phi} = {\mathbfss M} {\mathbfss C}_p {\mathbfss M}^T$ where [M]{} is the update matrix introduced in equation \[eqn:m\].) ${\mathbfss C}_p$ can be estimated from a photometric error model, and will be diagonal if the photometric errors in the bands are independent. Even in this case ${\mathbfss C}_{\phi}$ is generally non-diagonal: the AP estimates are correlated on account of the sensitivities.
Because we have a forward model we can calculate a goodness-of-fit (GoF) for any estimate of the APs. Here I simply use the reduced- to measure the difference between the observed spectrum and predicted spectrum[^4] $${\rm GoF} = \frac{1}{I-1} \sum_{i=1}^{i=I} \left ( \frac{p_i - {\hat p_i}}{\sigma_{p_i}} \right )^2
\label{eqn:gof_chisq}$$ where ${\hat p_i} = f_i({\bmath \phi_j})$ is the forward model prediction and $\{ \sigma^2_{p_i} \} = {\rm diag}( {\mathbfss C}_p)$ is the expected photometric noise. (Despite the name, a larger value refers to a poorer fit!) As the GoF can be measured without knowing the true APs, it can be used for detection of poor solutions or outliers.
Conventional methods of AP estimation via direct inverse modelling (e.g. with SVMs or ANNs) do not naturally provide uncertainty estimates and must usually resort to time-intensive sampling methods, such as resampling the measured spectrum according to its estimated covariance. They cannot provide a GoF at all because they lack a forward model.
Signal-to-noise weighted AP updates
-----------------------------------
The update equation (\[eqn:deltaap\]) only takes into account the sensitivity of the bands, not their SNR. However, even if a band is very sensitive to an AP in principle, if its measurement is very noisy then it is less useful. We could accommodate this by including a factor proportional to ${\mathbfss C}_p^{-1}$ into equation \[eqn:deltaap\] which would down-weight noisier measurements. Preliminary results using this on the TG problem (see section \[sect:datasets\]) show it actually degrades performance at G=15, but gives some improvement at G=18.5 (Bailer-Jones [@cbj09c]).
Assessing the algorithm {#data}
=======================
Gaia simulations {#gaiasims}
----------------
To illustrate I apply it to estimate stellar APs from simulated Gaia stellar spectra and thereby also make preliminary predictions of the mission performance. Gaia will observe all of its targets with two low-dispersion slitless prism spectrographs, together covering the wavelength range from 350–1050nm. (These are creatively called “BP” for blue photometer and “RP” for red photometer.) The dispersion varies from 3nm/pixel at the blue end to 30nm/pixel at the red end (Brown [@brown06]). The blue and the red spectra are each sampled with 60 pixels, but as the line-spread-function of the spectrograph is much broader, these samples are not independent. After removing low SNR regions of the modelled spectra, I retain 34 pixels in BP covering 338–634nm and 34 pixels in RP covering 667–1035nm. This is a slightly narrower range (and 18 pixels fewer) than the one adopted by Bailer-Jones et al. [@cbj08] for quasar classification with similar spectra.
![The – grid of the data used in the experiments.[]{data-label="fig:apgrid_teff_logg"}](figures/apgrid_teff_logg){width="34.00000%"}
Extensive libraries of BP/RP spectra have been simulated by the Gaia DPAC using the GOG (Gaia Object Generator; Luri et al. [@luri05], Isasi [@isasi09]) instrument model and libraries of input spectra.[^5] Here I use the Basel (Lejeune et al. [@lejeune97]) and Marcs (e.g. Gustafsson et al. [@gustafsson08]) stellar libraries. The former (as used here) includes 17 values from 8000–15000K with non-uniform spacing and the latter 17 values from 4000–8000K in uniform steps of 250K. (Together there are 33 unique values because 8000K is in both.) Together the grids span values from $-0.5$ to $5.0$dex in steps of $0.5$dex, although the grid is incomplete for astrophysical reasons (Fig. \[fig:apgrid\_teff\_logg\]). ranges from $-4.0$dex to $+1.0$dex with 13 discrete values for the cooler stars (with $\leq 8000$K): The spacing is 0.25dex from $+1$ to $-1$, followed by points at $-1.5$, $-2.0$, $-3.0$ and $-4.0$. Not all are present at all – combinations. Each star has been simulated at one of ten values of the interstellar extinction, $\in \{0, 0.1, 0.5, 1, 2, 3, 4, 5, 8,10\}$ with =3.1. (Note that is the extinction parameter defined by Cardelli et al. [@cardelli89]. It is not the extinction in the $V$ band.) The Marcs library additionally shows variation in the alpha element abundances, , representing five values from 0.0–0.4dex in 0.1dex steps. Hence this combined library shows variance in five APs, , , , , the first two of which are “strong” and the latter three “weak”. The total number of spectra is 46310. will be used to estimate the first four APs; will be ignored and contributes cosmic scatter.
![Noise-free Gaia spectra for solar metallicity dwarfs at zero extinction with ={4000, 5000, 6000, 7000, 8500, 10000, 15000}K, increasing monotonically from bottom (red) to top (violet) at long wavelengths. They are composed of two spectra (BP and RP) to the left and right of about 660nm. The ordinate is in units of photoelectrons (to within some constant multiple), not energy flux. \[fig:teff\_spectra\]](figures/teff_spectra){width="47.00000%"}
![Noise-free Gaia spectra at a range of (as in Fig. \[fig:teff\_spectra\]) and (0.0, 0.1, 0.5, 1, 2, 3, 4, 5, 8, 10) ranging from 0.0mag (lowest line at long wavelengths; in red) to 10.0mag (highest line at long wavelengths; in violet). Each temperature block has been offset by 1200 counts for clarity (the zero levels are shown by the dashed lines). =4.0dex, =0.0dex and =0.0dex in all cases. \[fig:teff\_av\_spectra\]](figures/teff_av_spectra){width="47.00000%"}
![Noise-free Gaia spectra at a range of and $(-0.5,0.5,2,3,4,5)$, with the lowest gravity (red) forming the lowest curve at the red end of the spectrum and the highest gravity (violet) the highest. Note that not all gravities are present at all due to the limitations of reality. Each temperature block has been offset by 600 counts for clarity (the zero levels are shown by the dashed lines) =0.0mag, =0.0dex and =0.0dex in all cases. \[fig:teff\_logg\_spectra\]](figures/teff_logg_spectra){width="47.00000%"}
![Noise-free Gaia spectra at a range of and $(-3,-2,-1,0,+0.5)$, with the lowest metallicity (red) forming the lowest curve at the red end of the spectrum and the highest metallicity (violet) the highest. Note that not all metallicities are present at all due to limitations of the simulated libraries. Each temperature block has been offset by 600 counts for clarity (the zero levels are shown by the dashed lines) =0.0mag, =4.0dex and =0.0dex in all cases. \[fig:teff\_feh\_spectra\]](figures/teff_feh_spectra){width="47.00000%"}
GOG simulates the number of photoelectrons (“counts”) in the spectral bands. (While these will be calibrated in physical flux units before being published to the community, the classification work by DPAC is currently done in photoelectron space.) The variance in the spectra due to the four APs of interest is demonstrated in the example spectra plotted in Figs. \[fig:teff\_spectra\], \[fig:teff\_av\_spectra\], \[fig:teff\_logg\_spectra\] and \[fig:teff\_feh\_spectra\]. The first plot shows the variance due to only. The break between the BP and RP instruments around 660nm is clear, as is the highly variable dispersion. The lower counts in RP immediately to the right of the break compared to BP is primarily due to the higher dispersion (fewer photons per band). In the other plots two APs are varied while the other two are held constant. Fig. \[fig:teff\_av\_spectra\] demonstrates the strong impact of both and variations. (As I will discuss in section \[degeneracy\], these two APs are highly degenerate in these data.) The third and fourth figures clearly demonstrate why and are weak parameters: they have little impact on the spectra compared to the variance due to or . In particular, at high temperatures the spectra show essentially no sensitivity to .
![The median signal-to-noise ratio (SNR) (solid line) and 0.1 and 0.9 quartiles (dashed lines) across the set of zero extinction dwarf stars for G=18.5 (red/thicker lines) and G=20.0 (blue/thinner lines). Compared to the G=18.5 curve, the SNR at G=15 is 8–6 times larger between 400 and 660nm and 7–12 times larger between 660 and 1000nm.[]{data-label="fig:snr_dwarfs"}](figures/snr_G185_G200){width="35.00000%"}
The AP estimation accuracy is, of course, a strong function of the SNR in the spectrum. As Gaia has a fixed sky scanning law, the SNR depends on the source magnitude and the number of observations (because the individual observations are combined into a single end-of-mission spectrum). I adopt here 72 observations for all spectra (the mean number of observations per source for a 5-year mission), and report results as a function of just the magnitude. The simulator noise model takes into account the source, background and background-subtraction photon (Poisson) noise as well as the CCD readout noise. At present errors due to the combination of spectra, charge-transfer inefficiency and CCD radiation damage are not explicitly included. There is instead a factor to account for general processing and calibration errors. All of these noise terms are combined into a zero mean Gaussian model for each pixel, the standard deviation of which is a function of the G-band apparent magnitude. (The G-band is the filterless band – defined by the mirror and CCD response – in which the Gaia astrometry is obtained, covering a range similar to BP/RP.) This defines a “sigma spectrum” for each star from which I generate random numbers in order to simulate noisy spectra at G=15, 18.5 and 20. The resulting SNR is a strong function of wavelength and the specific source, and is summarized in Fig. \[fig:snr\_dwarfs\].
Data sets and forward model fitting {#sect:datasets}
-----------------------------------
In the following sections I will apply Gaia to four distinct problems according to the APs we are trying to determine. In each case the forward model is fit to a grid varying in at least those APs, and in some cases the grid also shows variance in another AP (which therefore acts to provide additional “cosmic scatter”). For each case the forward model is fit (trained) using the full, noise-free data set over the complete range of the represented APs. The corresponding noisy data set (at G=15, 18.5 or 20) is split randomly into two equal halves: one half is used for initialization (selecting the nearest neighbour) and is applied to the other half (or a random subset of it where indicated below) on which the performance is evaluated.[^6] (The legitimacy of this procedure for evaluating performance is discussed in appendix \[sect:modelassess\].) In addition to reporting global results (over the full AP ranges present in the training set) I also measure performance on subsets of the evaluation data set, in particular the just for cool stars (the forward model is [*not*]{} refit). The problems and their grids are as follows (the name indicates the APs being determined: Temperature, Gravity, Metallicity and/or Extinction)
- TG: Estimation of $+$ (1 strong and 1 weak AP), for stars with =0 and =0, some 274 stars. I also build a second TG model (TG-allmet) which is trained and evaluated with the full range in the grid ($-4$ to $+1$dex). This contains 4361 stars, of which a quarter are used in the evaluation set.
- TM: Estimation of $+$ (1 strong and 1 weak AP), for stars with =0 and either $\in\!\{4.0, 4.5, 5.0\}$ (TM-dwarfs; 1716 stars), or $\in\!\{1.0, 1.5, 2.0, 2.5, 3.0\}$, (TM-giants; 1882 stars). I also build a third model (TM-allgrav) which is trained and evaluated with the full range in the grid ($-0.5$ to $+5$dex). This contains 4361 stars (it’s the same grid as used for TG-allmet of course), of which a quarter are used in the evaluation set.
- TAG: Estimation of (, )$+$ (2 strong and 1 weak AP), for stars with =0. This has 2740 stars, of which a random selection of 1000 is used for evaluation.
- TAM: Estimation of (, )$+$ (2 strong and 1 weak AP), for dwarfs with $\in\!\{4.0, 4.5, 5.0\}$. This has 17160 stars of which a random selection of 1000 is used for evaluation.
- TGM: Estimation of $+$(, ) (1 strong and 2 weak APs, for stars with =0. This has 4361 stars of which a random selection of 1000 is used for evaluation.
In appendix \[sect:compare\] the results are compared with results from an SVM on some of these problems.
[ll\*[12]{}[r]{}]{} model & evaluation & & & &\
& sample & & & & & & & & & & & &\
TG & F G=15 & & & & $<$ & 0.0010 & 0.0014 & $<$ & 0.065 & 0.093 & & &\
TG & F G=18.5 & & & & $<$ & 0.0057 & 0.0080 & $<$ & 0.35 & 0.51 & & &\
TG & F G=20 & & & & 0.0045 & 0.019 & 0.027 & $<$ & 1.14 & 1.53 & & &\
TG-allmet & F G=15 & & & & $7.4e^{-4}$ & 0.0058 & 0.0078 & $<$ & 0.53 & 0.90 & & &\
TG-allmet & F G=18.5 & & & & $8.2e^{-4}$ & 0.0083 & 0.011 & $<$ & 0.66 & 0.99 & & &\
TG-allmet & F G=20 & & & & 0.0025 & 0.019 & 0.027 & $-$0.14 & 1.19 & 1.59 & & &\
TM-dwarfs & L G=15 & & & & $<$ & 0.0017 & 0.0020 & & & & $<$ & 0.14 & 0.24\
TM-dwarfs & L G=18.5 & & & & $<$ & 0.0024 & 0.0033 & & & & $-$0.037 & 0.26 & 0.42\
TM-dwarfs & L G=20 & & & & $<$ & 0.0070 & 0.0090 & & & & $<$ & 0.82 & 1.14\
TM-giants & L G=15 & & & & $<$ & 0.0028 & 0.0037 & & & & $<$ & 0.22 & 0.34\
TM-giants & L G=18.5 & & & & $3.4e^{-4}$ & 0.0035 & 0.0045 & & & & $<$ & 0.31 & 0.50\
TM-giants & L G=20 & & & & $0.0011$ & 0.0073 & 0.0092 & & & & $-$0.11 & 0.74 & 1.08\
TM-allgrav & L G=18.5 & & & & $<$ & 0.0052 & 0.0070 & & & & $<$ & 0.40 & 0.60\
TM-allgrav & F G=18.5 & & & & 0.0031 & 0.012 & 0.017 & & & & & &\
TAG & F G=15 & $-$0.0067 & 0.072 & 0.15 & 0.0014 & 0.013 & 0.026 & $-$0.052 & 0.29 & 0.57 & & &\
TAG & F G=18.5 & 0.039 & 0.30 & 0.45 & 0.017 & 0.061 & 0.094 & $-$0.22 & 1.10 & 1.53 & & &\
TAM & L G=15 & 0.032 & 0.18 & 0.32 & 0.0022 & 0.018 & 0.029 & & & & $<$ & 0.46 & 0.79\
TAM & L G=18.5 & 0.047 & 0.52 & 0.74 & 0.010 & 0.056 & 0.080 & & & & $-$0.38 & 1.34 & 1.81\
TGM & L G=15 & & & & $8.3e^{-5}$ & 0.0007 & 0.0012 & $-$0.022 & 0.15 & 0.26 & 0.014 & 0.084 & 0.19\
TGM & L G=18.5 & & & & $8.8e^{-4}$ & 0.0043 & 0.0060 & $-$0.11 & 0.84 & 1.15 & $-$0.045 & 0.35 & 0.60\
TGM & F G=18.5 & & & & $2.3e^{-3}$ & 0.0089 & 0.014 & $-$0.06 & 0.59 & 0.86 & & &\
Application to the + problem (TG) {#sect:tefflogg}
=================================
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First the forward model is fit for each band to the – grid shown in Fig. \[fig:apgrid\_teff\_logg\]. As a reminder, each forward model comprises the 1D function over (the strong component) and 33 1D functions in (the weak components), as described in section \[sect:forward\_model\]. All of these are smoothing splines (section \[fmfunctions\]). Figs. \[fig:formod\_logg=4\] and \[fig:formod\_teff=5000\] show the forward model fit for 12 bands at cuts of constant and respectively. The fits are good: they show the degree of smoothness we would expect for these data plus a robust extrapolation. If we compare the flux scales between Figs. \[fig:formod\_logg=4\] and \[fig:formod\_teff=5000\] (these are standardized variables) we see how small the flux variation is as varies over its full range compared to : this is what it means to be a weak AP. Note also the small discontinuity in some of the bands in Fig. \[fig:formod\_logg=4\] at 8000K (=3.903) where the Marcs and Basel libraries join.
![AP evolution for 5 stars in the evaluation data set at G=15 ( left, right) for the TG problem. The true APs are written at the top of each panel pair and plotted as the red horizontal line. GoF is the reduced goodness-of-fit (equation \[eqn:gof\_chisq\]).[]{data-label="fig:test10.iter"}](figures/itup_iterations_002 "fig:"){width="45.00000%"} ![AP evolution for 5 stars in the evaluation data set at G=15 ( left, right) for the TG problem. The true APs are written at the top of each panel pair and plotted as the red horizontal line. GoF is the reduced goodness-of-fit (equation \[eqn:gof\_chisq\]).[]{data-label="fig:test10.iter"}](figures/itup_iterations_032 "fig:"){width="45.00000%"} ![AP evolution for 5 stars in the evaluation data set at G=15 ( left, right) for the TG problem. The true APs are written at the top of each panel pair and plotted as the red horizontal line. GoF is the reduced goodness-of-fit (equation \[eqn:gof\_chisq\]).[]{data-label="fig:test10.iter"}](figures/itup_iterations_067 "fig:"){width="45.00000%"} ![AP evolution for 5 stars in the evaluation data set at G=15 ( left, right) for the TG problem. The true APs are written at the top of each panel pair and plotted as the red horizontal line. GoF is the reduced goodness-of-fit (equation \[eqn:gof\_chisq\]).[]{data-label="fig:test10.iter"}](figures/itup_iterations_079 "fig:"){width="45.00000%"} ![AP evolution for 5 stars in the evaluation data set at G=15 ( left, right) for the TG problem. The true APs are written at the top of each panel pair and plotted as the red horizontal line. GoF is the reduced goodness-of-fit (equation \[eqn:gof\_chisq\]).[]{data-label="fig:test10.iter"}](figures/itup_iterations_124 "fig:"){width="45.00000%"}
Having trained I apply it to the evaluation data set at G=15. Fig. \[fig:test10.iter\] shows five examples of how the AP estimates evolve. The first iteration is the nearest neighbour initialization; the final is the adopted AP estimate. The red (horizontal) lines show the true APs. Looking at many examples we see a range of convergence behaviours. Sometimes convergence is rapid, for other stars it takes longer. Sometimes it is smooth, other times not. It can be quite different for the two APs for a given star and depends also on the specific spectrum (which is noisy). Sometimes the nearest neighbour estimate is the correct one, and may actually iterate away from this and converge on a different value. Convergence (on something) is almost always achieved on this problem, even though there is nothing adaptive in the algorithm. This is an encouraging property. Limit cycles are also seen in a handful of cases, but with negligible amplitudes on this problem (high SNR). For this specific problem, the extrapolation limits on the AP estimations (section \[sect:apextrapolate\]) and the limits on the AP updates at each step (section \[sect:apupdates\]) never had to be enforced by the algorithm.
![AP residuals for the TG problem at G=15[]{data-label="fig:itup3_test10_itup_residuals"}](figures/itup3_test10_itup_residuals){width="54.00000%"}
Fig. \[fig:itup3\_test10\_itup\_residuals\] plots the residuals (estimated minus true APs) on the 137 stars in the evaluation set; the statistics are summarized in line 1 of Table \[sumres\]. We see that the APs can be estimated very accurately (no significant systematic error, ) and very precisely (low scatter, or ). Applying the same model to noisier spectra obviously increases the errors (lines 2 and 3 in the table), but at G=18.5 the mean absolute errors are still an acceptable 1% in and 0.35dex in logg. At G=20 cannot be estimated accurately enough to reliably distinguish dwarfs and giants, although is still okay with an expected error of 260K at 6000K, for example.
This evaluation set is relatively small so the error statistics are subject to variation. Applying to an ensemble of randomly selected evaluation sets (at G=18.5), we see that the error statistics vary by 5–10% (inter-quartile range) of their mean. The reported values are therefore reasonably representative.
The performance of an SVM on these TG problems is given in Table \[svmres\]. For both APs and all three magnitudes is similar to or significantly better than SVM. (As there is some variance in the results from both methods I only consider the performance significantly different if the better one has a at least 25% smaller.) So in this limited variance problem, at least, the forward modelling approach improves performance.
It is hard to fairly compare the performance with nearest neighbours (NN), because on the one hand NN is limited by the density of its template grid, but on the other hand it can report the [*exact*]{} AP values. If we used a full, noise-free template grid and noisy evaluation objects, then provided the noise is low enough, 1-NN will give exact results. If we instead split the template and evaluation sets to have no common objects, then the precision of at least one of the APs is limited by the grid spacing. We could instead average over the $k$ nearest neighbours, but then NN does badly because it averages over a wide range of the weak AP (a shortcoming which party motivates ). To give some comparison, however, I estimate the parameters of all 274 stars in the TG grid of noise-free spectra via leave-one-out cross validation. The errors in both APs are six times larger than the result at G=15. This at least confirms that overcomes the grid resolution limitation of NN.
The above results are for stars with =0dex. I retrained and evaluated on a grid with the full range of metallicities ($-4$ to $1.0$dex; TG-allmet). The errors averaged over this full data set at G=15, reported in line 4 of Table \[sumres\], are 6–8 times larger than the solar metallicity case. This is entirely due to the new metallicity variance which is not accounted for (modelled) by and so is a confusing factor. (A accuracy of 1% at G=15 and 2% at G=18.5 is nonethelss good for stars which a priori show variance over the full range of , and ). Curiously, if we apply the model to G=20 data, then we see that the performance is no worse than when we limited the problem to solar metallicity stars (compare lines 3 and 6 of Table \[sumres\]). This is because the photometric noise dominates over the variance introduced by the (unmodelled) metallicity range.
If the AP residuals had a Gaussian distribution, then =1.25 in Table \[sumres\]. The fact that is always larger (sometimes much larger), indicates that there are outliers, which justifies the use of as a more robust error statistic.
In terms of overall error, an SVM does better than on the TG-allmet problem at all three magnitudes (Table \[svmres\]). It’s not clear why this is so, given that was better on the problem limited to solar metallicity (see section \[sect:compare\]).
Application to the + problem (TM) {#sect:tefffeh}
=================================
I now use to estimate and on the two grids TM-dwarfs and TM-giants defined in section \[sect:datasets\]. In each case we effectively assume we already have a rough estimate. The remaining spread in in each case acts as cosmic scatter. The models are then applied to the corresponding evaluation sets with noise levels at three magnitudes. The summary performance statistics are show in six lines in Table \[sumres\]. As there is essentially no sensitivity to metallicity at above 7000K (Fig. \[fig:teff\_feh\_spectra\]), I only report results on cooler stars, even though the models were fit to the full range.
Looking first at the performance, we see that the precision is twice as good as TG at all magnitudes (e.g. 0.007dex compared to 0.019dex). This just indicates that we can estimate more precisely for cool stars. Within the range 4000–7000K there is no strong dependence of the precision with or : can estimate equally well at all metallicities.
![ residuals for the TM-dwarfs model at G=15 for the full range (top) and for cool stars only ($\leq$7000K; bottom). Note the different scales on the ordinate. The systematic error (due to the lack of sensitivity to for hot star spectra) vanishes in the lower panel.[]{data-label="fig:sys_demo"}](figures/tm_dwarfs_G150_allteff_fehresid "fig:"){width="35.00000%"} ![ residuals for the TM-dwarfs model at G=15 for the full range (top) and for cool stars only ($\leq$7000K; bottom). Note the different scales on the ordinate. The systematic error (due to the lack of sensitivity to for hot star spectra) vanishes in the lower panel.[]{data-label="fig:sys_demo"}](figures/tm_dwarfs_G150_lowteff_fehresid "fig:"){width="35.00000%"}
Turning now to metallicity, we see good performance at G=15 and G=18.5 for dwarfs and giants: random errors of 0.3dex or less and negligible systematics. At G=20 the performance is quite a lot worse (0.7–0.8dex). There is little dependence of precision or accuracy with , as can be seen in the lower panel of Fig. \[fig:sys\_demo\]: Even for the most metal poor stars in the sample at =$-$4.0dex the precision is still 0.5dex. This plot also shows that the AP estimates hardly ever exceed the limits of the training grid, even though they are allowed to (section \[sect:apextrapolate\]), again suggesting natural convergence properties of the algorithm.
We gain some insight into how works if we include hot stars in the evaluation set. There is now a strong systematic error in the estimates, as can seen in the upper panel of Fig. \[fig:sys\_demo\]. The reason is that the sensitivity of all the spectral bands to metallicity is essentially zero in hot stars. In that case there is no contribution from metallicity to the flux updates in the algorithm, so the flux prediction is entirely from the strong component of the forward model. That predicts a flux corresponding to the average value of the metallicity (Fig. \[fig:two\_component\_forward\_model\]) in the training grid, so this is the AP value which reports. This is obviously higher than the lowest metallicities, with the result that overestimates . This is logical and desirable: reports the average value in the training data when the spectrum provides no information. The AP distribution is acting as a prior.
The above results have implicitly assumed the of the star to be known well enough to identify it as a dwarf or giant. We saw from the results in the previous section that itself can do this (even if the metallicity is not known: line 4 of Table \[sumres\]). But what if we relaxed this assumption? To test this, I trained and evaluated a new model, TM-allgrav, on the full range of (see section \[sect:datasets\]). Even at G=18.5 this model can estimate to a precision of 0.4dex, which would be enough to trace the metallicity distribution in the Galaxy and identify very metal-poor stars.
![Predicted AP uncertainties (top) and actual residuals (bottom) for from the TM-dwarf problem at G=20. The area of the plotted circle (not the diameter) is proportional to the size of the uncertainty/residual. In the lower panel, positive residuals are shown in black and negative in red. Points below some size limit are not plotted at all.[]{data-label="fig:resid_cov"}](figures/tm_dwarfs_g200_apcovapgrid "fig:"){width="45.00000%"} ![Predicted AP uncertainties (top) and actual residuals (bottom) for from the TM-dwarf problem at G=20. The area of the plotted circle (not the diameter) is proportional to the size of the uncertainty/residual. In the lower panel, positive residuals are shown in black and negative in red. Points below some size limit are not plotted at all.[]{data-label="fig:resid_cov"}](figures/tm_dwarfs_g200_residapgrid "fig:"){width="45.00000%"}
Section \[sect:errest\] gave a formula for the covariance of the APs estimated by . For this TM problem, ${\mathbfss C}_{\phi}$ is a $2\times2$ matrix with elements $c_{jj'}$. The corresponding uncertainty estimates, $\sqrt{c_{jj}}$, are plotted as a function of the two APs in the upper panel of Fig. \[fig:resid\_cov\]. This can be compared to the actual errors (residuals) in the lower panel. The fact that they broadly agree indicates that the uncertainty estimation is useful. This is important, because for many purposes knowing the uncertainty in an estimate is as important as the estimate itself.
Application to 3-AP problems (TAG & TAM) {#sect:2d1d}
========================================
Extension to higher dimensions
------------------------------
So far the forward model has comprised two 1D components. For more than two APs the forward modelling approach described in section \[sect:forward\_model\] generalizes almost trivially. The principle is to retain the partition of the APs into the two categories “strong” and “weak”. With $N_S$ strong APs, the strong component of the forward model is a single $N_s$-dimensional function, fit at the $n_s$ unique combinations of the strong APs in the training grid. At any one of these points, $k$, there are $n_w(k)$ grid points which vary over the $N_w$ weak APs. The mean flux over these is used to fit the strong component. The weak component at point $k$ is then built by making an $N_w$-dimensional fit to the flux residuals (with respect to the strong component) over the $n_w(k)$ points at $k$. Thus we have $n_s$ independent weak components. The components are combined and applied as in the 1D+1D case. Nothing else in the algorithm is changed.
Here I apply to two different problems both with two strong APs and one weak AP. This is especially important for stellar parametrization (and Gaia) because in practice we have to accommodate variable interstellar extinction, which can be large at low Galactic latitudes and has at least a large an impact as on the spectrum (see Fig. \[fig:teff\_av\_spectra\]).
I fit the 2D strong component of the forward model using a thin plate spline, a type of smoothing spline closely related to kriging (e.g. Hastie et al. [@hastie01]). The number of degrees of freedom is set to $n_s/2=330/2=155$ in both problems. The 1D weak components are again fit using smoothing splines with the dof set as before (section \[fmfunctions\]).
Results for (+)+ (TAG)
----------------------
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In this problem the forward model is fit over the full range of , and for solar metallicity stars (TAG in section \[sect:datasets\]). The 1D cut through the fit in Fig. \[fig:tag\_formod\_logg=4\_teff=10000\] shows that the newly introduced variance is fit accurately (as are the variations in the other APs). The summary performance when applying with this model to G=15 spectra is shown near the bottom of Table \[sumres\]. Compared to the results on the TG data set at the same magnitude, the errors in and are, at 0.013dex (3%) and 0.3dex respectively, considerably worse. This is due to the extra variance introduced by the very wide range of extinction. Yet these errors are still small enough to be scientifically useful, especially when we note that the systematics are quite small. The new AP, extinction, can be estimated very well: a mean absolute error of just 0.07mag. If we limit the analysis (without changing the fitted model) to just low extinction stars ($\leq1.0$mag), then can be estimated only slightly better (0.056mag) but more so (0.008dex). At G=18.5, the errors of course increase (Table \[sumres\]): while and are still manageable, at 1.1dex is not. Fortunately Gaia will provide parallaxes for many stars to improve the estimates (given the and estimates from ).
At G=15 the SVM has comparable performance on this problem, but is somewhat better at G=18.5 (see section \[sect:compare\]).
Results for (+)+ (TAM)
----------------------
We now swap for and train and evaluate on the TAM problem (which is a dwarf sample: the results are broadly similar for a giant sample). The summary performance at G=15 is listed near the bottom of Table \[sumres\]. The performance is degraded significantly compared to the case with no variance (TM-dwarfs), much more so for than for . At 0.46dex the metallicity error is acceptable. Alas at G=18.5 this degrades to an almost useless level with the additional problem of a large systematic.[^7]
If we limit the analysis (at G=15) to just low extinction stars ($\leq1.0$mag), then the precision is essentially unchanged (0.44dex). This implies that it is no more difficult (on average) to determine the metallicity of stars with high extinction than of stars with low extinction. So why is the error larger here than in the zero extinction case reported for the TM problem? It is because we now have a large extinction range [*a priori*]{}, so an uncertainty in determining corresponds to an uncertainty in . As is weak, a relatively small uncertainty in is magnified into a larger one in .
![The trend in the mean absolute errors for and in the TAM problem on the G=18.5 evaluation data set (which spans the full range of , and )[]{data-label="fig:tam_residtrend"}](figures/tam_residtrend){width="50.00000%"}
Note that the extinction errors are larger than those reported with the TAG problem (0.18mag compared to 0.07mag at G=15). This is a consequence of having limited the analysis to cool stars. Indeed, the mean absolute error on the present problem is only 0.1mag if we analyse the hot stars ($>$7000K) rather than the cool ones: can be estimated more accurately for hotter stars. This is not surprising because hot star spectra are simpler (e.g. no metallicity signature) so it should be easier to untangle the effects of temperature and extinction. This is illustrated in Fig. \[fig:tam\_residtrend\] at G=18.5, which plots the dependence of the and residuals with these parameters (averaged over all ). It also shows that is estimated more precisely for cool rather than hot stars, as found also for the TM problem.
![The correlation between the and residuals for the TAM model applied to the G=18.5 evaluation data set[]{data-label="fig:tam_resid_correlation"}](figures/tam_resid_correlation){width="30.00000%"}
Nonetheless, the and errors are quite large at G=18.5: 8–20% for and 0.2–0.6mag in (Fig. \[fig:tam\_residtrend\]). Moreover, their residuals show a strong positive correlation (Fig. \[fig:tam\_resid\_correlation\]): a tendency to overestimate one results in an overestimation of the other. This suggests that these two APs are degenerate in these low resolution BP/RP spectra, something which Fig. \[fig:teff\_av\_spectra\] also suggests. Let us now investigate this further.
Identification of a – degeneracy {#degeneracy}
--------------------------------
Like most estimation algorithms, simply tries to find the best single solution (plus an uncertainty estimate). If we consider a likelihood function of the data given the APs, $P({\rm Data}|{\bmath \phi})$, then the algorithm is trying to find the maximum of this, plus some measure of the width of this peak. But if there is a degeneracy in the APs then the peak position and width are an inadequate summary of the likelihood. A degeneracy can be defined as two or more stars which have spectra differing by an amount consistent with the photometric noise. On observing one of these spectra we would be unable to distinguish between the AP solutions. This applies trivially to stars with very small AP differences (thus the need to quote an uncertainty). What we are interested in here are objects which have significantly different APs (a large fraction of the grid range), such that a simple approximation of the likelihood function (e.g. a Gaussian in the APs with covariance given by equation \[eqn:apcov\]) fails to represent the uncertainties accurately.
![Multiple random initialization of for three different stars (in the three panels). In each panel the starting points are shown as green dots connected to the solutions (after 20 iterations) shown as blue triangles. The true APs are shown as a red cross.[]{data-label="fig:multi_init"}](figures/itup_multi_init_paper){width="40.00000%"}
As is a local search method, it is a priori possible that when initialized at different points it could converge on different (degenerate) solutions. To explore this I ran 25 times for each star with the initialization chosen at random each time, rather than by the nearest neighbour. (I used the TAG model with G=18.5 spectra.) Fig. \[fig:multi\_init\] plots the various solutions in the – plane for three different stars, chosen to illustrate degeneracy.
For the first star (top panel), we see that 22 of 25 initializations over quite a range of and converge on a small region (slightly offset from the true solution possibly due to noise in the spectrum). The three “wrong” solutions occur because the algorithm is initialized too far from a good solution: the searching method gets stuck in a poor local minimum. Inspection of their predicted spectra shows them to be very different from the true spectrum, with GoF values (equation \[eqn:gof\_chisq\]) of more than a few hundred. Such a high GoF would be used in practice to flag and reject such solutions.
Turning to the second star (middle panel) we see two distinct regions of convergence, one of which is close to the true APs. Are the other solutions at around =13000K simply poor (bad convergence)? When we inspect the predicted spectra (not shown), we see this is not the case. The two sets of spectra are indistinguishable with the noise, all having similar and small GoF values (0.01–15).
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This is better illustrated by the third star (bottom panel), where we see a lack of convergence on one or two solutions in favour of a complete ridge of solutions. Fig. \[fig:photpred\] plots the initial and final (predicted) spectra for these 25 runs of . All but one or two of these final spectra agree very closely with the true spectrum, even though they have very different APs. So this is a true degeneracy, and not poor convergence of . This degeneracy is also reflected by a high value for the expected correlation coefficient between and calculated by equation \[eqn:apcov\]. Identification of similar ridges in AP space of near-identical spectra for other stars suggests that the – degeneracy is widespread.
Mapping of the – degeneracy {#degeneracy2}
---------------------------
We can map this degeneracy systematically using just the forward model to generate spectra on a fine grid of and . For each predicted spectrum I adopt as its sigma spectrum that of the star in the original grid which has the closest APs. The expected Mahalanobis distance between a hypothetically measured spectrum, ${\bmath p}$, and any noise-free spectrum, ${\bmath p'}$, in this grid is, with $\delta{\bmath p} = {\bmath p} - {\bmath p}'$, $$\begin{aligned}
\label{eqn:dist}
D^2 \, &=& \, \delta{\bmath p}^T {\mathbfss C}_p^{-1} \delta{\bmath p} \\
&=& \, \sum_{i=1}^{i=I} \left ( \frac{p_i - p'_i}{\sigma_{p_i}} \right )^2\end{aligned}$$ the simplification following because in these simulations there is no inter-pixel noise correlation. A degeneracy arises between two stars when $D^2$ is sufficiently small that it could arise just from photometric noise. Under the null hypothesis ($H_0$) that the differences between the spectra are only due to Gaussian noise and that each pixel is independent, $D^2$ follows a distribution with $I-1=67$ degrees of freedom. I will define two stars as non-degenerate only if the probability of observing their given distance or more under $H_0$ is 1% or less. This corresponds to a critical value of $D^2_{\rm lim}= 96.8$ (or a reduced- value of 1.44). In other words, stars separated by a smaller distance are considered degenerate.[^8]
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For each star in the fine grid I calculate the distance to all of the other stars. The corresponding probabilities (of getting that distance or more) can then be plotted as a [*degeneracy map*]{} in the – plane. Maps for 15 stars are shown in Fig. \[fig:dgenmap\] (similar patterns are observed throughout the grid). We see that the APs are correlated, the high probabilty region extends over a large range of the grid and in some cases is multimodal. In other words, there is a strong degeneracy over the whole plane. What this means in practice is that if we observed a (noisy) spectrum, ${\bmath p}$, at the position of one of the crosses in Fig. \[fig:dgenmap\], then this spectrum is indistinguishable from all the spectra lying within the contours, on account of the noise. Therefore we also cannot distinguish between the corresponding APs.
We can compare the degeneracy maps with the expected covariance for individual objects calculated from equation \[eqn:apcov\] by treating the latter as the covariance in a Gaussian likelihood function in the APs. While the estimated correlations agree qualitatively with what we observe in Fig. \[fig:dgenmap\], this Gaussian likelihood approximation does not accurately reproduce the shape of the degeneracy ridges. Indeed it cannot, because the degeneracy ridges are not symmetric about the true estimate, and cannot be negative. This lack of detailed agreement is not very surprising, however, because equation \[eqn:apcov\] is a valid approximation only when the linear relation in equation \[eqn:deltap\] holds, i.e. when the errors are small.
The implication of this is significant: It is misleading to report a single estimate for and (even if accompanied by a simple covariance estimate). Rather, we must report a whole ridge of solutions. As we can map these degeneracies in advance, the purpose of a classifier such as would be to identify [*a*]{} solution, which we use to identify the corresponding degeneracy ridge (either as a probability grid or a fitted approximation to it). This study shows that a single nearest neighbour initialization of is usually adequate to allow to find a good solution and therefore the correct ridge. The minority of cases where no good solution is found are usually identifiable by the large GoF value. In those cases we may want to re-initialize from some other point or improve the convergence mechanism.
Note that Fig. \[fig:dgenmap\] is for G=18.5. At G=20 the degeneracy regions are somewhat larger. At G=15 the SNR is sufficiently high that there is no longer any degeneracy on a scale of practical relevance: the contours in the degeneracy map are smaller than the grid point separation (0.2mag in and 0.01dex in ). This is consistent with the small errors in these APs recorded in Table \[sumres\].
Including additional or prior information
-----------------------------------------
Additional information could help to reduce the extent of the degeneracy.
The parallax reported by Gaia combined with the apparent G-band magnitude gives an estimate of $+$, where is the absolute magnitude in the $G$ band and is the extinction in the G-band (which we could estimate directly from instead of ). Combined with an HR-diagram (which is just a prior probability density function over and ) this provides additional constraints on and . A method for doing this is outlined in Bailer-Jones [@cbj10].
We could set weak prior probabilities on the APs – the probability of the APs unconditional on the BP/RP spectrum, parallax or G-band magnitude – based on a simple model of the Galaxy. For example, we know that cool stars are much more common than hot stars, so an unreddened cool star is a priori more likely that a highly extinct OB star. Likewise, if the star is observed at high Galactic latitude we could confidently assign a lower prior probability to high extinctions. This latter prior is particularly useful, because we saw in sections \[sect:tefflogg\] and \[sect:tefffeh\] that if the extinction can be fixed then gives much better estimates. Normally, however, we wouldn’t want to bias the Gaia AP estimates with [*detailed, current*]{} knowledge of Galactic structure, given that the primary goal of Gaia is to improve this knowledge.
For very bright stars (G$\ltsim$12; Gaia will saturate at around G=6) we may have AP estimates coming from the high-resolution RVS spectrograph on Gaia measuring the CaII triplet at 860nm (Katz et al. [@katz04]). In principle we can estimate independently of using these data. Note that because the ridges in Fig. \[fig:dgenmap\] have a low inclination, additional information on is more useful than information on .
Finally, estimates of APs from other surveys could be incorporated and combined with the Gaia ones, provided they can be interpreted as probability distributions. Given the opportunity, one should design an ad hoc spectrophotometric system to address the specific goals of the survey. Particularly desirable is a system which maximizes the separability of the APs (a method for doing this was presented by Bailer-Jones [@cbj04]). This opportunity was taken by the Gaia community, which designed a multiband photometric system to address the specific Gaia scientific goals (Jordi et al. [@jordi06]). But this was later replaced with the present (and less desirable) BP/RP spectrophotometry, ultimately for financial reasons.
Further applications (and the TGM problem) {#sect:more}
==========================================
In the previous section the forward model was extended to be 2D in the strong APs and 1D in the weak. In a similar way it can be adapted to model one strong AP (e.g. ) and two weak APs (e.g. and ) simultaneously. This is of practical use in situations where extinction can be assumed to be low. I used this to fit a forward model to the TGM grid (section \[sect:datasets\]) and applied as before. The results are shown in the bottom section of Table \[sumres\]. At G=15, the performance in and is considerably better than obtaind with the corresponding TM problems in which there was unmodelled variation: we now obtain 0.3% in for the full range and 0.1dex in for cool stars. is also better than found with TG-allmet, but not with TG where metallicity was fixed to zero. This as we would expect: at high SNR, modelling the extra variance improves the results in all APs.
This is not the case at G=18.5. Here the performance of the TGM model on all three parameters is similar to that obtained in the TG-allmet or TG-allgrav problems on the appropriate ranges. I suspect that the extra variance introduced by the noise is complicating the problem, so that the modelling the additional AP (rather than marginalizing over it) brings no improvement in performance (but the obvious advantage that we now determine all three APs instead of just two.) It is also worth pointing out that when using the degeneracy mapping method described in section \[degeneracy2\], I find there is a very strong and complex degeneracy between and at G=18.5.
could be further extended to more or other APs or to different problems, provided the partition between strong and weak APs can be retained. If such a clear distinction were not possible then a different approach to forward modelling would be necessary. If we had more than two or three weak or strong APs, then the smoothing splines are unlikely to provide adequate fits, unless we had a lot of data. To overcome this we would need more structured regression models suited for fitting high dimensional sparse data sets (e.g. neural networks).
Summary and Conclusions {#conclusions}
=======================
I have introduced an algorithm for estimating parameters from multi-dimensional data. It uses a forward model of the data to effectively perform a nonlinear interpolation of a template grid via the Newton-Raphson method. It is intended to overcome both the non-uniqueness issue of direct modelling of inverse problems as well as the finite grid density and metric definition issues of $k$ nearest neighbours. makes use of the sensitivity of the data to the astrophysical parameters to find an optimal solution. This is convenient, because in principle it means we don’t need to worry too much about feature selection: low sensitivity features will automatically be down weighted. An important component of the algorithm is the division of the forward model into two parts which independently model the variance of the “strong” APs (such as and ) and the weak APs (such as and ). Good fits could be obtained using low-dimensional (1 or 2) smoothing splines. As it is based on a forward model, naturally provides AP uncertainty estimates (actually full covariances) and a goodness-of-fit, in contrast to most inverse modelling methods.
I applied to the problem of estimating APs from simulations of the low resolution Gaia spectrophotometry, BP/RP. Results are summarized in Table \[sumres\]. When limited to zero extinction stars, can be estimated at G=15 to a (mean absolute) accuracy of better than 1% when the metallicity and surface gravity are entirely unknown (=$-0.5$ to $5.0$dex; =$-4$ to $+1.0$dex). At G=18.5 and G=20.0 (the limiting Gaia magnitude) the average accuracy over the full range of APs is 2% and 4% respectively. At G=15 can be estimated to 0.5dex if is entirely unknown but to 0.15dex if is also modelled (full range of the three APs). Limiting to solar metallicity stars this improves to better than 0.1dex and is still 0.35dex at G=18.5. for cool stars can be estimated to 0.1dex at G=15 and to 0.15–0.35dex at G=18.5, the better result obtained if we know the star is a dwarf. At G=20 and cannot be estimated to any useful accuracy. Of course, for population studies we can average over many stars to reduce the population estimate of metallicity, limited by the systematic errors and any correlations in the data.
If we extend the strong component of the forward model to simultaneously model over a very wide range (0–10mag), then the performance on the weak APs is significantly degraded on account of this extra variance, such that reasonable accuracies can be reached at G=15 but not at G=18.5. itself can be estimated remarkably well, however, 0.07–0.2mag at G=15 and 0.3–0.5mag at G=18.5. The extra variance also affects the determination. However, these statistical errors do not tell the whole story: I have shown that there is a strong and ubiquitous degeneracy between and intrinsic to the BP/RP spectra. Thus in addition to reporting single optimal estimates, the Gaia catalogue will need to provide the corresponding degeneracy map, which can be built in advance using the forward model. The degeneracies could be reduced (and the weak APs also then estimated more accurately) if we can use additional information. Possible sources include the apparent magnitude and parallax measured by Gaia, external data, and/or weak priors from a very simple Galaxy and stellar evolution model.
For some problems outperformed a support vector machine in terms of smaller residuals, but in other cases the SVM was better. This should be assessed in more detail after improvements to the algorithm have been explored. For example, the updating method is very simple – stopping after a fixed number of iterations – whereas a more adaptive approach may help on larger variance problems (e.g. noisier data). We may also want the AP update weighting to take into account the noise (and not just the sensitivity). Now that we have a forward model, further possibilities for modelling open up. is just a method for locating the best APs. If we define a distance metric (such as equation \[eqn:dist\]), then we can define a likelihood function, $P(D|{\bmath \phi})$, which when combined with a suitable prior defines a (non-analytic) posterior over the APs, $P({\bmath \phi} | D)$. We can then use one of many sampling methods, such as Markov Chain Monte Carlo, to sample this as a function of ${\bmath \phi}$, thus yielding a complete Bayesian probabilistic solution which varies smoothly over the APs. This approach to parameter estimation has been used in several areas of astronomy, such as galaxy classification (e.g. Heavens et al. [@heavens00]) and inference of cosmological parameters (e.g. Percival et al. [@percival07]). (If we dispensed with the forward model and just calculated probabilities at points in the original grid, then the principle is similar to that used by Shkedy et al. [@shkedy07].) While offering some advantages, this approach is much slower than , because for the present application it would require of order $10^4$ to $10^5$ samples and hence this many evaluations of the forward model, compared to about $10^2$ for .
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to thank Anthony Brown, Lennart Lindegren and Carola Tiede for useful discussions. This work makes use of Gaia simulated observations, which have been produced thanks to the efforts of many people in the Gaia DPAC. In this respect I would particularly like to thank Anthony Brown, Yago Isasi, Xavier Luri, Paola Sartoretti, Rosanna Sordo and Antonella Vallenari, without whose efforts the predictions of the Gaia performance would not have been possible. The GOG simulations were produced using the MareNostrum supercomputer at the Barcelona Supercomputing Center – Centro Nacional de Supercomputación. I am also grateful to the Marcs team at Uppsala University for producing new stellar spectral simulations for Gaia data processing purposes.
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Assessing model performance {#sect:modelassess}
===========================
The forward model should obviously be fit on the full available grid and using noise-free data. (We need to achieve a minimum grid density in order to reliably assume that the modelled signal vary smoothly between the grid points, yet the grid need be no denser. Some smoothness assumption must be made by any method of estimating continuous parameters.) Yet at first sight one may consider it illegitimate to fit on the full grid and then assess its performance on (noisy) spectra selected from the same grid. (Note that the initializaton set is never included in the evaluation set!) The equivalent approach is generally considered invalid for inverse models, because these are fit by minimizing the AP error on a training set: If they are not properly regularized the model will overfit the training data by learning non-general aspects of these data (such as the noise). But the situation with is different, because the best forward model fit is identified (interactively at present) according to its smoothness and the photometry error, not the AP error. That is, the regularization is also done independently of the AP error. It might anyway seem preferable to start with a grid twice as dense as required, fit the forward model on half the data and evaluate on the other half. But if the smoothness requirement is met, this will anyway give a similar performance.
The underlying issue here – relevant to all machine learning methods – is how to make a fair assessment of performance. Ultimately all assessments are limited by the fact that training and evaluation data are drawn from a common grid. Yet it would be useless to train the classifier on spectra generated by one set of stellar models and evaluate it on spectra generated by another, because this “performance” would reflect the intrinsic differences between the stellar models. This remains a quandary, also because tests based on synthetic spectra ignore the realities of non-Gaussian noise and cosmic scatter of real data. The most reliable assessment of performance would be to first determine APs of a set of high resolution spectra (using whatever method) based on a particular stellar model and define these APs as true. We would then degrade the spectra to the dispersion, line-spread-function and noise of real Gaia data and compare the estimates with the true ones. Such tests are planned but are considerably more arduous than what’s done here.
Comparative performance of a support vector machine {#sect:compare}
===================================================
It is not the goal of this article to make a detailed comparison between the accuracy of and conventional machine learning methods, but a brief comparison is useful. I therefore applied a support vector machine (SVM) (e.g. Cortes & Vapnik [@cortes95], Burges [@burges98]) to some of the problems presented in the article. The SVM is used to directly model the inverse problem by training on noisy data. Although the SVM training has a unique solution for a given set of data, it has three hyperparameters (the length scale parameter $\gamma$ and two regularization parameters $\epsilon$ and $C$) which must be optimized (“tuned”), which is just a higher-level training procedure. I optimize these hyperparameters simply via a brute force search of a regular grid over the hyperparameters, typically with 512 or 1000 models (8 or 10 values of each hyperparameter). To achieve this the data set must be split into three independent parts: the training set, a test set (used to select the best combination of hyperparameters) and the evaluation set (on which the final performance is calculated). (If we just use the same data for testing and evaluation – as is frequently done – then the results are somewhat better; unfairly so, because then the SVM is tuned on the very data set on which it is finally evaluated. I nonetheless did this for the TG problem because of the small amount of data: just 274 stars.) For each of the problems/data sets described in section \[sect:datasets\], the set is randomly split into three, equal-sized disjoint parts to build these subsets. The SVM is tuned on noisy data, separately for each magnitude, and separately for each AP (as an SVM can only model one output).
[ll\*[4]{}[r]{}]{} model & test sample & & & & \
TG & F G=15 & & [**0.0015**]{} & [**0.10**]{} &\
TG & F G=18.5 & & 0.0064 & [**0.47**]{} &\
TG & F G=20 & & 0.016 & 0.90 &\
TG-allmet & F G=15 & & [*0.0017*]{} & [*0.091*]{} &\
TG-allmet & F G=18.5 & & [*0.0060*]{} & [*0.41*]{} &\
TG-allmet & F G=20 & & [*0.013*]{} & [*0.77*]{} &\
TM-dwarfs & L G=15 & & 0.0019 & & [**0.20**]{}\
TM-dwarfs & L G=18.5 & & [**0.0064**]{} & & [**0.37**]{}\
TM-dwarfs & L G=20 & & [**0.013**]{} & & 0.63\
TAG & F G=15 & 0.084 & 0.015 & 0.029 &\
TAG & F G=18.5 & 0.24 & [*0.045*]{} & [*0.62*]{}\
The SVM results are summarized in Table \[svmres\]. This obviously conceals details, such as the existence of systematic errors in some cases. For example, is systematically underestimated for $\geq$10000K and shows a systematic across the whole range in the TAG problem at G=18.5. Comparing with the results in Table \[sumres\], no very clear pattern emerges, with superior to SVM in some problems and vice versa in others. When comparing performance star-by-star on a given problem, we do not see that one algorithm performs systematically better than the other in some parts of the AP space. Rather, one algorithm is just overall better. The fact that is superior on the TG problems whereas SVM is superior on the TG-allmet problems suggests that SVM copes better with problems where there is more unmodelled variance, although the better performance of on the TM-dwarfs problem and TAG at G=15 does not support this suggestion. Possibly it is when the variance is higher overall (lower SNR and/or unmodelled variance) that SVM gives smaller errors, which would be consistent with that method’s approach to dealing with noise. A more detailed comparison is only warranted after further work has been put into optimizing the algorithm, such as the convergence for noisy data or the update clipping procedures.
SVM has the advantage of being much faster to apply once it has been tuned. As the speed of is probably dominated by the nearest neighbour initialization, this could be replaced with an SVM. On the other hand, provides AP uncertainty estimates, goodness-of-fit estimates (for outlier/poor solution detection) and allows one to inspect the relevance of each input in determining the output for every object (equation \[eqn:apcont\]).
[^1]: Email: calj@mpia.de
[^2]: http://www.rssd.esa.int/Gaia
[^3]: [http://www.r-project.org]{}
[^4]: I use $I-1$ degrees of freedom rather than $I$ because all the spectra have a common G magnitude, so the bands are not all strictly independent.
[^5]: For Gaia pundits: I use the CU8 cycle 3 simulations of the nominal (discrete) libraries (Sordo & Vallenari [@sordo08])
[^6]: These target spectra must first be normalized to have the same counts level as those on which was trained. For this I just use the G magnitude to scale the counts. However, as the G-band is not identical to the BP/RP band adopted, this gives rise to a normalization offset between spectra even for a common G magnitude. For example, over the TG grid the integrated BP/RP counts varies by up to 10%, dependent primarily on . A better normalization might be “area normalization”, i.e. dividing the counts in each band by the sum over all bands for that spectrum. My G-band normalization is in principle conservative, as it mimics a small calibration bias.
[^7]: Contrary to expectations, systematic trends can rarely be corrected for. We would have to plot the residual vs. the [*estimated*]{} AP, and if the scatter is larger than the systematic trend then a correction cannot be made.
[^8]: is the distribution followed by a sum of squares of independent unit Gaussian variables, $N(0,1)$, and has a (unnormalized) density function $P'(D^2) = D^{\nu-2} e^{-D^2/2}$ where $\nu$ is the degrees-of-freedom. $\nu\!=\!I\!-\!1$ because one degree-of-freedom is “lost” by the fact that they have a common G magnitude, although this is of no practical significance here. $D^2_{\rm lim}$ is defined by $P(D^2 \geq D^2_{\rm lim} | H_0) = \int_{D_{\rm lim}^2}^{\infty} \! P' dD^2$ = 0.01. Fig. \[fig:dgenmap\] plots this probability (and several others) as a contour.
|
---
abstract: 'AGN feedback is an important ingredient in galaxy evolution, however its treatment in numerical simulations is necessarily approximate, requiring subgrid prescriptions due to the dynamical range involved in the calculations. We present a suite of SPH simulations designed to showcase the importance of the choice of a particular subgrid prescription for AGN feedback. We concentrate on two approaches to treating wide-angle AGN outflows: thermal feedback, where thermal and kinetic energy is injected into the gas surrounding the SMBH particle, and virtual particle feedback, where energy is carried by tracer particles radially away from the AGN. We show that the latter model produces a far more complex structure around the SMBH, which we argue is a more physically correct outcome. We suggest a simple improvement to the thermal feedback model - injecting the energy into a cone, rather than spherically symmetrically - and show that this markedly improves the agreement between the two prescriptions, without requiring any noticeable increase in the computational cost of the simulation.'
author:
- |
Kastytis Zubovas$^{1,\star}$, Martin A. Bourne$^{2,3}$ and Sergei Nayakshin$^{2}$\
$^{1}$Center for Physical Sciences and Technology, Savanoriu 231, Vilnius LT-02300, Lithuania\
$^{2}$Department of Physics & Astronomy, University of Leicester, Leicester, LE1 7RH, UK\
$^{3}$Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK\
$^{\star}$ [E-mail: ]{} [kastytis.zubovas@ftmc.lt]{}
title: A simple way to improve AGN feedback prescription in SPH simulations
---
[quasars: general — accretion, accretion discs — ISM: evolution — methods: numerical]{}
Introduction
============
Feedback from Active Galactic Nuclei (AGN) is a key ingredient in modern galaxy evolution models. It is required in order to explain the sharp drop-off in the galaxy mass function above $M_* \simeq 10^{11}
\msun$, prevent the cooling catastrophe in galaxy clusters and produce the hot gas atmospheres seen around many galaxies. Observations of massive kpc-scale outflows and pc-scale relativistic winds provide further evidence that AGN affect their host galaxies in a significant way.
One outstanding issue in understanding the precise effects of feedback is the range of spatial scales over which it operates. The AGN jets and winds are launched from the accretion disc on scales $l_{\rm min}
< 0.01$ pc, while the observed effects on host galaxies span $10^3$ pc or more. This means that in order to model the AGN feedback precisely, a simulation should span at least 5 orders of magnitude in linear scale, resulting in $10^{15}$ resolution elements for a 3D model, a resolution which is not likely to be reached any time soon. Even resolving the interaction of an AGN wind with the host galaxy material, on scales of several parsecs and larger, is currently only possible in single-galaxy models, rather than cosmological simulations [@Schaye2015MNRAS].
As a result of this shortcoming, some effects of AGN feedback might not be apparent from simulations. In a recent paper [@Bourne2015MNRAS], we showed that mass resolution has a significant influence upon the AGN feedback effects seen in simulations of turbulent gas. In particular, at mass resolutions typical of cosmological simulations, we found AGN feedback to be much more negative, in the sense of efficiency of gas removal, than at higher resolution. In high-resolution simulations, AGN winds expel only diffuse gas, while denser gas forms filaments which fall in toward the central supermassive black hole (SMBH) and can potentially feed it or form stars. These results suggest that AGN feedback can be positive as well as negative, increasing the star formation rate in its host. Our other previous research [@Nayakshin2012MNRASb; @Zubovas2013MNRAS; @Bourne2014MNRAS] provides some support for this conclusion as well. Several other authors have investigated similar positive AGN feedback coming from the interaction of the jet with the galactic ISM [@Silk2005MNRAS; @Gaibler2012MNRAS].
In this paper, we investigate another numerical aspect of modelling AGN feedback: the prescription detailing how the AGN feedback energy is passed to the surrounding gas. We perform high-resolution simulations of AGN feedback acting upon a turbulent gas reservoir, passing the feedback energy in one of two ways: a Monte Carlo radiative transfer scheme, which self-consistently takes into account gas optical depth; and a kernel-weighted thermal and kinetic energy input scheme. We find that the Monte Carlo scheme, while being numerically far more expensive and difficult to scale, provides qualitatively different results than thermal and kinetic energy injection. In particular, the Monte Carlo scheme allows the formation of simultaneous gas inflows and outflows, resulting in prolonged AGN feeding and faster, but less massive, outflows. We also test an adjusted version of both schemes, where the feedback energy is injected, or virtual particles released, biconically, rather than spherically symmetrically. This change makes the results of the two schemes significantly more alike. We suggest that using this improvement to the thermal feedback prescriptions would make the results of large-scale numerical simulations much more realistic.
The structure of the paper is as follows. In Section \[sec:nummodel\], we describe the numerical model and setup used in the simulations. The simulation results are presented in Section \[sec:results\]. We discuss the implications of our work for both numerical modelling of galaxy evolution and interpretation of observations in Section \[sec:discuss\] and conclude in Section \[sec:concl\].
Numerical model {#sec:nummodel}
===============
Basics of the model
-------------------
The code used is GADGET-3, an updated version of the publicly available GADGET-2 [@Springel2005MNRAS]. It is a hybrid N-body/SPH code with individual particle timesteps and adaptive smoothing. We employ the SPHS extension of the basic SPH scheme [@Read2012MNRAS], which improves the modelling of mixing within a multiphase medium, allowing us to better track the interaction between turbulent gas flows. We use a Wendland kernel [@Wendland95; @Dehnen2012MNRAS] with 100 neighbours.
The initial conditions of all models presented in this paper are the same. They consist of a gas sphere with isothermal density profile, extending from $R_{\rm in} = 100$ pc to $R_{\rm out} = 1$ kpc. The gas sphere is resolved with $10^6$ particles, giving a particle mass of $m_{\rm SPH} = 3000 \msun$ and a mass resolution of $m_{\rm res} =
100m_{\rm SPH} = 3\times 10^5 \msun$. The spatial resolution of the model varies depending on gas density, but is generally of order a few pc; we adopt a minimum gravitational and SPH smoothing length of $0.1$ pc. The gas is embedded in a static isothermal background density profile with velocity dispersion $\sigma = 200$ km/s, and is supported against the background gravity by turbulent motion with characteristic velocity $v_{\rm turb} = \sigma$. The gas is initially cold ($c_{\rm s} \ll v_{\rm turb}$; see below for a discussion on how the gas evolves thermally). A SMBH is embedded in the centre of the gas distribution and can swallow gas which comes within the accretion radius $r_{\rm accr} = 10$ pc of the SMBH particle and is energetically bound to it. We set the SMBH particle mass to $2 \times
10^8 \; \msun$, in order for the radius of its sphere of influence to be $r_{\rm infl} \simeq r_{\rm accr}$. This ensures that the gas moves in a purely isothermal background potential only perturbed by its own self-gravity, rather than the gravity of the SMBH. The SMBH is inactive for the first $1$ Myr of the simulation; this allows the gas to develop a turbulent density structure before it is affected by feedback. In order to prevent the formation of spurious high-density filaments close to the inactive SMBH, we extend the accretion radius to $r'_{\rm accr} = R_{\rm in}$ during this time. After $1$ Myr, the AGN switches on for $1$ Myr and begins affecting the gas in two ways.
First of all, we model the effect of the AGN radiation field by employing a @Sazonov2005MNRAS cooling curve, appropriate for optically thin gas subjected to AGN radiation. Below $T=10^4$ K, we extend the cooling curve with the one proposed by @Mashchenko2008Sci down to a temperature floor, which depends on the gas density such that the Jeans mass of the gas never falls below the resolution limit.
Gas particles which fall on this dynamic temperature floor are stochastically converted into sink particles in order to speed up the simulations. This is equivalent to having a density threshold $$\rho_{\rm J} = \left(\frac{\pi k_{\rm B} T}{\mu m_{\rm p} G}\right)^3
m_{\rm res}^{-2} \simeq 6.7 \times 10^{-15} T_4^3 \; {\rm g}
{cm}^{-3},$$ $$n_{\rm J} \simeq 6.2 \times 10^{9} T_4^3 \; {\rm cm}^{-3},$$ where $T_4 \equiv T/10^4$ K. This crude approximation of the star formation process may affect our results slightly, because sink particles are affected only by gravity and cease to interact with the AGN feedback. Since sink particles are created in dense gas, this slightly reduces the fraction of feedback energy received by dense gas and increases the fraction received by diffuse gas. On the other hand, we do not model the effects of self-shielding of dense gas clumps, so the reduction in received energy caused by the formation of sink particles mimics that effect to some extent. In any case, this effect is small, since our simulations never produce more than $10^4$ sink particles.
The other way that the AGN affects the gas is by means of wind feedback, which we describe below.
Physics and implementation of AGN feedback
------------------------------------------
Model ID Feedback model $\frac{L_{\rm AGN}}{1.3\times10^{46} {\rm erg s}^{-1}}$ $r_{\rm bub}$ (kpc) $\frac{E_{\rm gas}}{E_{\rm input}}$ $\dot{M}_{\rm SMBH}$ ($\msun /$ yr$^{-1}$) $\dot{M}_{\rm out}$ ($\msun /$ yr$^{-1}$)
---------- ---------------------------- --------------------------------------------------------- --------------------- ------------------------------------- -------------------------------------------- -------------------------------------------
NoAGN None $0$ $0$ $-$ $130$ $0$
vp-L1 Virtual particle $1$ $0$ $0.01$ $750$ $40$
vp-L2 Virtual particle $2$ $0.2$ $0.01$ $550$ $200$
vp-L5 Virtual particle $5$ $0.5$ $0.3$ $8$ $1300$
tk-L1 Thermal $1$ $0.1?$ $0.01$ $800$ $15$
tk-L2 Thermal $2$ $0.2$ $<0$ $900$ $20$
tk-L5 Thermal $5$ $0.3$ $0.05$ $0$ $180$
vpc-L1 Virtual particle biconical $1$ $0.25$ $0.01$ $450$ $130$
vpc-L2 Virtual particle biconical $2$ $0.35$ $0.04$ $150$ $470$
vpc-L5 Virtual particle biconical $5$ $0.75$ $0.85$ $20$ $1900$
tkc-L1 Thermal biconical $1$ $0.2$ $0.02$ $650$ $90$
tkc-L2 Thermal biconical $2$ $0.35$ $0.035$ $400$ $220$
tkc-L5 Thermal biconical $5$ $0.6$ $0.3$ $170$ $750$
The AGN wind feedback model [@King2003ApJ; @King2010MNRASa] has a number of appealing properties: it is based upon relatively well-understood physical processes; it can explain the $M-\sigma$ relation without requiring free parameters; it can explain the properties of observed fast AGN winds on sub-parsec scales, as well as those of massive AGN outflows on kpc scales [@Zubovas2012ApJ]. Observations show that winds with velocity $v_{\rm w} \sim 0.1 c$ and power $L_{\rm kin} \sim 0.05 L_{\rm AGN}$ are present in a large fraction of AGN , suggesting a wide opening angle [@Nardini2015Sci]. In the wind feedback model, these winds shock against the galactic ISM and drive large-scale outflows [@King2010MNRASa]. The shocked wind passes its energy and momentum to the ISM, accelerating the outflow to $v_{\rm out}
>1000$ km/s [@Zubovas2012ApJ]. However, a large fraction of the outflow energy can leak out through gaps in the non-uniform ISM, leading to a situation where some of the gas is outflowing with large velocities, while dense gas is simultaneously inflowing toward the SMBH [@Nayakshin2014MNRAS; @Zubovas2014MNRASb].
Therefore, it is important to check how well numerical simulations can capture this complex process of simultaneous inflow and outflow on scales from tens to thousands of parsecs. We do this by exploring two subgrid feedback implementations and their improvements.
### Thermal feedback model
The simplest feedback prescription, often used in cosmological models, is the “thermal feedback” subgrid prescription. In this prescription, the feedback energy of the AGN, which we take to be $\epsilon_{\rm f} = 5\%$ of its luminous energy output over the SMBH timestep, is passed as thermal energy to the SMBH particle neighbours. We choose 100 neighbours for the SMBH particle, the same as for gas particle. Testing showed that using fewer neighbours results in feedback becoming extremely inefficient, as gas particles can be accreted by the SMBH even after being heated; however, if the particles were not accreted, it is likely that the feedback would be more efficient as the temperature would be higher and so the cooling time would increase. Meanwhile, using more neighbours dilutes the feedback and results in faster cooling of gas [@Bourne2015MNRAS].
The feedback energy given to each particle is weighted by the SPH interpolation kernel. This means that gas closer to the SMBH receives proportionately more energy than gas further away.
In addition to thermal energy, each SPH particle receives a similarly kernel-weighted fraction of the AGN wind momentum, $L_{\rm AGN} dt/c$, in a direction radially away from the SMBH. While this is not usually done in cosmological models, we implement this aspect of feedback in order to be able to do a more direct comparison with the Monte Carlo radiative transfer method. Tests showed that the effect of momentum feedback is negligible compared with the thermal energy input.
### Biconical thermal feedback model
One issue with the simple thermal feedback prescription is that the feedback energy is given to the nearest particles without taking their spatial distribution into account. This implicitly assumes that the gas distribution is (approximately) spherically symmetric. A real galaxy contains multiphase gas, with many structures degrading the spherical symmetry, even if such symmetry might be expected on large scales. For the AGN, this results in neighbouring gas particles having very different density and correspondingly very different effective cross-sections. In principle, feedback affecting each particle should be attenuated by calculating the optical depth between the SMBH and this particle. However, doing so is a very time-consuming process, unfeasible to implement in cosmological simulations, which may have many AGN at the same time.
Therefore, we propose a simpler method, which we believe captures some of the important physics. The method relies on injecting the thermal and kinetic energy in a cone centered on the SMBH instead of spherically symmetrically. Spherical symmetry is unlikely on the sub-grid scales on which the wind is launched, since the AGN has a preferred plane - that of the accretion disc - and winds are likely to be launched perpendicular to it. So we would expect the wind to be launched with an axisymmetric geometry. This prescription ensures that gas in all direction does not experience identical feedback. Instead, there is a preferred plane for the gas to accrete in, corresponding to the plane of the sub-grid accretion disc.
Numerically, we implement this model by calculating a separate neighbour list for the SMBH particle, looking for neighbours only in a bicone with a specified opening angle, which is set to 45 degrees in the simulations presented in this paper. For simplicity, we choose the axis of the cone to lie in the Z direction. The number of neighbours in the cone is the same as the number of neighbours used for the hydrodynamics calculations, and the feedback energy input is weighted by the value of the SPH interpolation kernel, only using the larger smoothing length corresponding to the neighbours in the cone.
### Virtual particle model
The “virtual particle” Monte Carlo radiative transfer method [@Nayakshin2009MNRAS] relies on packets of feedback energy, which are emitted by the AGN as particles and move in straight lines with a constant velocity, carrying momentum and energy. When the virtual particle moves into an SPH particle’s smoothing kernel, it starts giving up its momentum and energy over several timesteps, distributing it proportionately among all SPH particles which contain the virtual particle in their smoothing kernels. Since the virtual particles are emitted isotropically, the energy injection is also isotropic, independent of the distribution of gas around the SMBH. This ensures that, for example, a dense and small gas clump close to the SMBH does not receive too much of the feedback energy, because the prescription self-consistently takes into account the cross section of the clump. It also means that even gas very far away from the SMBH may receive a direct energy and momentum input, if the intervening space is empty, for example having been cleared by a progressing outflow.
We test two versions of the virtual particle model: one with spherically symmetric emission of tracer particles, and one where the particles are emitted in a cone with a 45-degree opening angle.
Model parameters
----------------
We present the results of 13 models; their identification labels and main parameters are given in Table \[table:param\]. One simulation, NoAGN, is intended as a control and does not have any AGN feedback (the SMBH is still present and acretes gas). Then four sets of three models each investigate the effects of different subgrid feedback prescriptions on feedback from AGN of different luminosities. We choose $L_{\rm AGN}$ to span a range of observed quasar luminosities; furthermore, these luminosities encompass the critical luminosity necessary to drive away gas from the AGN by purely momentum feedback [@King2010MNRASa]. The opening angle of the cone in the thermal conical (tkc) and virtual particle conical (vpc) simulations is $45$ degrees.
Results {#sec:results}
=======
At first, we briefly discuss the properties of the gas distribution just before the AGN switches on. Then we present the morphology of the spherically symmetric feedback models, vp and tk, and finally move on to their conical feedback counterparts, vpc and tkc. Later, we discuss the resulting gas morphology, outflow energetics, inflow and outflow rates and the density structure.
Initial evolution of the gas shell
----------------------------------
During the first 1 Myr of evolution, the gas shell develops a turbulent density structure and cools down to temperatures in the range $10 - 10^4$ K. Figure \[fig:init\_phase\] shows the phase diagram of the gas just before the AGN switches on. Most of the gas has temperatures just below $10^4$ K, where there is a minimum in the cooling rate. Some gas is located on the temperature floor (diagonal line in the bottom right of the distribution), however the probabilistic conversion of gas into sink particles ensures that only $\sim2000$ particles have been converted into stars. Approximately $15\%$, i.e. $4.5\times10^8\msun$, of the initial shell has been accreted by the SMBH particle. This makes the SMBH mass grow to more than three times its initial mass, however this does not affect the subsequent results, since we fix the AGN luminosity, rather than tying it to the SMBH mass. It is important to note that during this phase, the accretion radius is set to $100$ pc, and most of the gas accreted through this radius is unlikely to reach the SMBH during the time period simulated in our models.
Morphology - spherically symmetric feedback
-------------------------------------------
Figure \[fig:vp\_morph\] shows the density maps of the three virtual particle simulations at $t=1.5$ Myr, i.e. $0.5$ Myr after the AGN switches on, showing the XZ plane. The plots use a wedge-slice projection, such that only particles with $|y/r| < 0.25$ are plotted, to ensure that both the innermost regions and the outskirts are well-sampled. The three plots reveal a clear change in the global effect of feedback. In the vp-L1 model, feedback is not strong enough to prevent inflow or even to produce significant bubbles; there is only one small cavity in the negative-X direction. Simulation vp-L2 shows noticeable outflow bubbles, but simultaneously there exists gas very close to the SMBH, in fact accreting upon the SMBH particle. Most of the feedback energy leaks out in the directions of low density, producing these large bubbles, while the remaining momentum push is too weak to shut off accretion of dense gas. Finally, in vp-L5, we see that feedback is powerful enough to blow away almost all the material, except for a couple of very dense filaments; these remain close to the SMBH, but are also pushed away, albeit slowly.
In Figure \[fig:tk\_morph\], we plot density maps of the analogous tk models, again at at $t=1.5$ Myr. There are significant differences from the vp models. Most noticeably, a well-defined almost spherical feedback bubble is present in simulation tk-L5, without any inflow toward the SMBH. On the other hand, the size of the outflow bubble is much smaller in tk-L5 than in vp-L5, since the energy is injected into the gas closest to the SMBH; this gas is dense and therefore cannot be removed as efficiently as the diffuse gas which absorbs most of the input energy in the vp models. It is also interesting to note that the feedback bubble is surrounded by a boundary layer of dense gas, which is not present in the virtual particle simulations. This dense gas cools down very rapidly, so most of the feedback energy is radiated away rather than used to drive bubble expansion.
The major difference between the two models arises because in the thermal feedback simulation, all of the feedback energy is always injected into gas that is closest to the SMBH, typically within the central $<200$ pc, although the precise radius depends on AGN luminosity and the size of the forming bubble. In the virtual particle simulations, on the other hand, feedback evacuates cavities which allow the virtual particles to stream to large distances of order $500-700$ pc in the highest-luminosity simulation. We now present the effects of changing feedback geometry to conical, which ensures that there is always a channel - the midplane of the gas distribution - through which gas can reach the SMBH, and similarly that there is a channel perpendicular to the first through which feedback can be injected at distances comparable to those in the spherical virtual particle model.
Morphology - conical feedback
-----------------------------
Figure \[fig:vpc\_morph\] shows the gas density in the three conical virtual particle models, vpc-L1 (left), vpc-L2 (middle) and vpc-L5 (right). Even the lowest-luminosity simulation produces a modest outflow, with cavities in the top half and third quadrant of the plot, where the gas was least dense at the time the AGN switched on. The outflow is somewhat conical, but variations in gas density prevent it from acquiring a well-defined shape. In the highest-luminosity model, the feedback bubbles are rather large, touching the edges of the initial gas distribution, but gas close to the midplane has not been blown away, allowing filamentary inflow to continue. It is worth noting that the outflow is wider than the opening angle of the cone within which the virtual particles are emitted. This happens because the low-density cavities are over-pressurized and expand both vertically and laterally.
Figure \[fig:tkc\_morph\] shows the analogous snapshots of the tkc simulations tkc-L1 (left), tkc-L2 (middle) and tkc-L5 (right). Broadly speaking, the simulations evolve similarly to the vpc models; there are two cavities forming in opposite directions and expanding, with material falling in along the equatorial plane. Overall, the differences are much smaller than between the tk and vp models presented above. This happens because with the non-spherical injection of feedback, a channel appears along the equatorial plane which allows inflow and outflow to coexist. Therefore, dense gas no longer accumulates at the edge of a well-defined outflow bubble, but is brought to the midplane by the same processes that act in the vpc models. A similar structure of dense inflow along the midplane and diffuse outflow in the polar direction develops, with some irregularities due to the turbulent gas density distribution. In the L1 and L2 simulations, both tkc and vpc models produce similarly-sized bubbles, while the brightest AGN produces a significantly larger bubble in the vpc simulation. This last size difference is exacerbated by the fact that the bubble reaches the edge of the initial gas distribution in the vpc simulation, and its expansion accelerates. At earlier times, the difference in size between tkc-L5 and vpc-L5 bubbles is much less pronounced.
Despite the morphological differences and similarities, we are more interested in the effect that AGN feedback would have upon the evolution of the host galaxy. Therefore, we now consider several integrated parameters of the system and their time evolution.
Energetics
----------
One of the most important effects that AGN feedback has upon its host is the energy injection, which leads to gas heating and expulsion. The heated gas then cools down and forms structures including clumps and filaments, as seen in the morphology plots. In order to better understand the difference between thermal and virtual particle feedback prescriptions, we show the phase diagrams of the four L2 simulations in Figure \[fig:phase\_L2\] and correspondingly for the L5 simulations in Figure \[fig:phase\_L5\]. In these plots, colour represents the density of particles in the particular parts of the plot, with blue being lowest and red being highest. Dashed lines indicate the bremsstrahlung cooling times.
A clear difference can be seen between the tk-L2 and vp-L2 models (Figure \[fig:phase\_L2\], top row). Thermal feedback creates slightly more dense gas, which cools down quickly to the two horizontal branches of the equilibrium temperature curve, between $10^4$ and $10^5$ K. Meanwhile, virtual particle feedback acts upon diffuse gas and heats it up to $10^6$-$10^8$ K. This gas has long cooling times, $t_{\rm c} > 10^5$ yr and therefore stays hot and flows outward. Thus the virtual particle feedback prescription allows the gas to retain a larger fraction of the AGN input energy than the thermal prescription. Such difference is much less pronounced in the conical feedback runs (bottom row), leading to much more similar morphology.
The difference between the tk-L5 and vp-L5 models is even more pronounced (Figure \[fig:phase\_L5\], top row), as the thermal feedback compresses gas close to the SMBH into a very dense shell ($T
\simeq 10^4$ K, $\rho > 10^{-19}$ g cm$^{-3}$). Meanwhile, the virtual particle feedback creates a large population of gas with $T > 10^7$ K, which expands rapidly outward. Once again, the conical feedback simulations (bottom row) are much more similar to each other, with large amounts of hot expanding gas and much less extremely dense material.
These differences in energy retention are clearly visible when the total gas energy is considered. In Figure \[fig:energy\_evolution\], we show the change of total particle energy compared with input energy as a function of time: $$f_{\rm e}\left(t\right) \equiv \frac{E_{\rm
gas}\left(t\right)-E_{\rm gas}\left(1{\rm
Myr}\right)}{0.05L_{\rm AGN} \left(t-1{\rm Myr}\right)};$$ here, the gas energy $E_{\rm gas}$ includes gravitational potential, thermal and kinetic energy. We do not plot the results of the L1 simulations, because they are qualitatively very similar to those of L2, shown in the top panel. The L5 models are in the bottom panel. Virtual particle simulation results are shown in solid lines (black for spherical, blue for conical virtual particle emission), while the thermal feedback simulations are shown in dashed lines (red for spherical, green for conical feedback injection).
In both cases, the conical feedback simulations lose less energy to cooling than spherically symmetric ones. This happens because with conical feedback, dense gas is pushed toward the midplane and the heated gas is mostly diffuse, therefore has a low cooling rate (the presence of hot diffuse gas is also seen in the phase plots above). In the L2 models, this low cooling rate is still high enough to make the gas lose most of the injected energy, leaving only a few percent as the internal energy. However, this is still a factor few better retention than the spherically symmetric model in the virtual particle case; the spherically symmetric thermal feedback model has gas energy decreasing to values even lower than those at $t=1$ Myr, as gas falls deeper into the potential well. It is worth noting that in the tkc-L2 and vpc-L2 simulations, gas retains very similar amounts of input energy. In the L5 simulations, the virtual particle conical feedback model is essentially purely adiabatic, losing only $10-20\%$ of the input energy by $2$ Myr; the thermal conical feedback model loses a much larger fraction of its energy, since it does not create such a large bubble. Even so, the difference between spherical and conical feedback models is significant: energy retention improves by a factor 5 or more in the thermal model.
Even though the high-luminosity thermal feedback simulation retains significantly less energy than its virtual particle counterpart, the similarity in the L2 models is encouraging. Below we show that similar trends exist in other integrated quantities.
Inflow and outflow rates
------------------------
Another particularly important aspect of the co-evolution of SMBHs and their galaxies is the AGN duty cycle, i.e. the frequency and duration of AGN activity episodes. In a realistic system, one might expect AGN feeding and feedback to occur simultaneously, at least up to some critical AGN luminosity. Within the AGN wind feedback model, this happens because most of the AGN wind energy is carried away by low-density gas, allowing dense clouds to fall in toward the SMBH. Although our simulations do not resolve the scales of SMBH feeding, we nevertheless can investigate the ability of various feedback prescriptions to reproduce simultaneous inflows and outflows. For this, we plot the radial profiles of gas inflow and outflow rates in the L2 and L5 simulations in Figures \[fig:massflow\_profiles\_L2\] and \[fig:massflow\_profiles\_L5\], respectively. The plots are made at three times for each simulation, $0.25$ Myr, $0.5$ Myr and $0.75$ Myr after the AGN switches on. The inflow rate is defined as $$\dot{M}_{\rm in} =
m_{\rm SPH} \sum_{\rm i \in (v_{\rm r} < -\frac{\sigma}{2})} \frac{|v_{\rm r,i}|}{\Delta R} = 4 \pi R^2 v_{\rm r,in} \langle\rho_{\rm in}\rangle,$$ where $\Delta R = 0.1$ kpc is the thickness of the radial bin and the sum goes over all particles with radial velocity directed inward and higher in magnitude than half the background velocity dispersion. The second equality shows that the inflow rate can be expressed via the mean inflowing gas density $\langle\rho\rangle$ and does not depend on the choice of $\Delta R$ so long as the bin is thick enough to sample a large number of particles. We neglect particles with small negative velocities because those are dominated by turbulent motions rather than inflow or outflow. The outflow rate is defined in the same way, but the sum is made over all particles with radial velocities greater than half $\sigma$.
The four L2 models all generally have stronger inflows than outflows. This is expected, since the AGN luminosity is not large enough to drive away all the gas by momentum push alone. However, clear differences are visible among the models, with tk-L2 having essentially zero outflow, vp-L2 having outflow rates $10-20$ times lower than inflow, tkc-L2 having outflow rates only $2-3$ times smaller, especially in the region $r = 0.2-0.5$ kpc, and finally vpc-L2 having inflow and outflow rates of comparable magnitude. We see that the qualitative difference present between the tk-L2 and vp-L2 models disappears when feedback is injected conically, even though a notable quantitative difference remains. It is also worth noting that the morphology of all four L2 models appears very similar in the outer regions, since the outflows are never fast enough to extend beyond $\sim0.5$ kpc in 1 Myr.
The L5 models display much greater differences among themselves. The tk-L5 simulation produces very little outflow and a strong inflow, rather similar to its lower-luminosity counterpart. However, the inflow proceeds only to $r \simeq 0.1$ kpc and stalls there. In the other three simulations, outflows dominate, initially in the inner regions (black lines), but later throughout the simulation volume. The two conical feedback simulations have qualitatively very similar radial profiles of both inflow and outflow, echoing the similarity in their morphologies (Figures \[fig:vpc\_morph\] and \[fig:tkc\_morph\]). It is important to note that even though the outflow is very powerful, the inflow rate in the central $\sim
0.15$ kpc is similar to the outflow rate, showing the the SMBH may be fed even while producing such a massive outflow.
We also plot the time evolution of the rate of gas particle accretion by the SMBH particle (Figure \[fig:mdot\_bh\]) and the total rate of gas outflow, approximated as $$\dot{M}_{\rm out} \simeq
m_{\rm SPH} \sum_{\rm i \in (v_{\rm r} > \frac{\sigma}{2})} \frac{v_{\rm r,i}}{R_{\rm i}}$$ (Figure \[fig:mdot\_out\]). The line colours are as in Figure \[fig:energy\_evolution\]. In all models, there is no accretion for the first $0.2$ Myr of AGN activity, because the gas is falling toward the significantly reduced accretion radius of the SMBH particle.
Here, the two spherically symmetric models are more similar than when the energy retention is considered. However, there is an important qualitative difference. In the tk-L5 model, thermal feedback is able to shut off accretion entirely, by keeping the gas at the edge of a well-defined bubble (see Figure \[fig:tk\_morph\], right panel). On the other hand, the virtual particle model, while producing very large outflow bubbles, also allows some gas to fall on to the SMBH particle. The accretion rate is comparatively small, hardly exceeding $20 \msun$ yr$^{-1}$, but the possibility of accretion is significant. In the L2 models, the situation is reversed: thermal feedback suppresses accretion for $\sim 0.4$ Myr, but then the outflow bubble collapses and accretion rate rises rapidly to $>1000
\msun$ yr$^{-1}$. Meanwhile, the vp-L2 models has a steadily rising accretion rate, without such sudden changes.
Both high-luminosity conical feedback models, unsurprisingly, allow far more accretion on to the SMBH. Simulation vpc-L5 is more efficient at pushing gas away than the tkc-L5 simulation, therefore the accretion rate is significantly lower, only rising to a peak of $\sim
45 \msun$ yr$^{-1}$. The thermal feedback model, on the other hand, allows accretion at rates approaching $200 \msun$ yr$^{-1}$ at $t =
1.25-1.6$ Myr.
In the low-luminosity models, conical feedback actually decreases the gas inflow rate by a factor few. This happens because in these models, outflow bubbles form and prevent accretion in some directions, channeling accreting material toward the midplane. The tkc-L2 simulation has a higher accretion rate at first, since it is initially more efficient in compressing the gas toward the midplane, but from $t
= 1.7$ Myr onward, the two accretion rates become very similar.
The difference among the gas outflow rates (Figure \[fig:mdot\_out\]) is far more striking. The spherical thermal feedback model is incapable of driving a significant outflow in either low- or high-luminosity models, as most of the AGN input energy is radiated away, and the gas is kept in a dense, but small and hardly expanding, shell. The spherical virtual particle simulation produces an outflow with at least several times higher mass flow rate, and in the higher luminosity case, the outflow rate approaches that of the conical simulations by $t = 2$ Myr. This happens because of the complex morphology present in the virtual particle simulation, with outflowing gas being diffuse, and hence reaching very high velocities.
The conical feedback models produce significant outflows, essentially because the dense gas is channeled away and diffuse gas can be accelerated to high velocities. The mass outflow rates approach $2000
\msun$ yr$^{-1}$ in the L5 models. It is interesting that the collimation of feedback into a cone actually increases the outflow rate in the virtual particle simulations, most likely because conical feedback produces more spatially distinct regions of inflow and outflow, so there is less mixing between inflowing and outflowing gas, which would lead to lower outflow rates. The thermal feedback model produces a significantly weaker outflow than the virtual particle model in the conical case as well, but the ratio between outflow rates is much smaller than for the spherical feedback injection simulations.
Density structure
-----------------
In Figure \[fig:rho\_dist\], we plot information regarding the evolution of gas density distribution during the AGN activity episode. More precisely, we plot the values of the 10th, 25th, 75th and 90th percentile of the gas density distribution, in order to evaluate how the different phases of gas (diffuse, medium and dense) behave during the AGN activity episode. The line colours and styles are the same as in previous time evolution figures.
We see that the diffuse gas evolves almost identically for the whole AGN activity episode in the L2 models and for the first $\sim 0.5$ Myr in the L5 models, mostly because most of this gas is far away from the SMBH and is not affected by the AGN outflow at first. None of the L2 models produces a bubble extending to the outskirts of the gas sphere, so there is little evolution of the diffuse gas in this model throughout the time period considered; the same is true for the tk-L5 model. In the other three models, diffuse gas starts to become even more diffuse as the outflow bubble moves past it and disperses the gas in a larger volume.
The dense gas becomes progressively more dense, but with certain differences among the simulations. The virtual particle models compress the dense gas by $\sim 5-10$ times in the period between $\sim 1.05 - 1.3$ Myr, corresponding to the formation of a complex inflow-outflow structure with many dense clumps and filaments. The tkc-L5 model shows a similar behaviour for the densest gas, although the compression takes a slightly longer time. Once this structure is in place, however, density does not increase further, as the densest material either turns into sink particles or is accreted on to the SMBH. The tk-L5 model, on the other hand, shows essentially unrestricted density growth, because here, the densest gas is also the warmest, because it receives all the AGN energy input, and therefore cannot fragment. The tk-L2 model behaves in the same fashion initially, but later the dense gas collapses on to the SMBH and is accreted, resulting in a drop in the densest gas density values.
Discussion {#sec:discuss}
==========
Summary of results
------------------
Our suite of 13 simulations allowed us to explore the effect that the subgrid AGN feedback prescription has upon the evolution of a turbulent gas sphere. Our main results are as follows:
- Spherically symmetric thermal AGN feedback is unable to reproduce the complex structures found in virtual particle models. The reason is that the thermal feedback prescription injects energy only into the surrounding gas, creating a well-defined bubble around the SMBH.
- The virtual particle prescription allows for simultaneous gas inflow and outflow, as well as formation of dense gas filaments and clumps, which are not found in the thermal feedback case.
- Replacing both prescriptions with biconical versions results in a marked improvement in their agreement. In particular, the morphologies of the two models then become very similar, with clear outflow bubbles coexisting with inflow along the equatorial plane.
- The conical feedback models are qualitatively similar to the corresponding spherical virtual particle simulations, but quantitatively very different from both spherical models: the outflow bubbles produced by conical feedback are generally larger, the outflow rates higher and much more of the AGN input energy is retained by the gas.
- The fraction of AGN input energy retained in the gas is very low in the spherically symmetric thermal feedback simulation, even at high AGN luminosity, but increases to $\sim 20-30\%$ in the conical high-luminosity one. In the virtual particle models, the energy retention fraction increases to an ever higher value of $70-80\%$.
- Spherically symmetric thermal feedback either does not allow any accretion on to the SMBH, or allows most of the gas to fall in. It also produces only a weak outflow. Virtual particle and conical models produce both significant inflow and outflow at the same time.
Although the set up of our simulations is strongly idealised, the results have several implications for the understanding of AGN activity and its effects upon the host galaxy. We discuss those implications below.
Major issues with thermal feedback models
-----------------------------------------
As mentioned in the Introduction and Model sections, the commonly-used thermal-kinetic AGN feedback models suffer from a number of drawbacks. The most important drawback is the assumed spherical symmetry, i.e. energy injection into gas close to the SMBH. If it happens that there is gas only to one side of the SMBH, all the energy will be injected into this gas, rather than spread around symmetrically. This creates a paradoxical situation where assumed spherical symmetry results in energy injection which is often very far from symmetric.
The problem with this energy injection in generating outflows lies in the fact that once any parcel of gas escapes far from the SMBH, it is no longer affected by feedback. Therefore, as far as most gas particles in the simulation are concerned, feedback is very intermittent, and this leads to much weaker outflows than would be possible if feedback acted upon a gas parcel for a prolonged period of time. A small amount of gas does experience this prolonged feedback, however, it gets squashed between AGN feedback on one side and inflowing gas on the other. Gas density therefore increases, cooling becomes more efficient and progressively more of the feedback energy is lost to radiation rather than used to drive the outflow. Therefore, the resulting AGN outflow is much weaker than it would otherwise and more energy is required to remove the gas from the host galaxy than it would be if these multi-phase effects were treated properly.
These issues do not necessarily impact the ability of cosmological simulations to reproduce the mass functions and growth histories of present-day galaxies and SMBHs, even though the latter requires appropriate tuning of feedback parameters [@Schaye2015MNRAS; @Crain2015MNRAS]. AGN accretion and feedback self-regulate in these simulations, so that a less efficient AGN feedback allows more accretion which in turn increases the outflow rate, having a negligible effect upon the long-term evolution of the total stellar mass in the galaxy [@Sijacki2007MNRAS; @Schaye2010MNRAS; @Schaye2015MNRAS]. However, as such models, as well as models of isolated galaxies, become more detailed, they start reproducing other, smaller-scale, processes and structures, such as the phase structure of the gas, the spatially-resolved history of star formation, the duration of AGN episodes, and so on. All of these, and many other, properties of the galaxy are affected by the details of how AGN feedback interacts with the ISM, and therefore understanding the limitations imposed by the numerical feedback scheme is of paramount importance when interpreting the results of these models.
Ideally, one would like to minimize the limitations of the numerical method. Our proposed scheme of injecting feedback energy into a cone alleviates the two biggest issues of thermal feedback prescriptions described above. While it does not really resolve the interaction of the AGN outflow with the multiphase gas, it creates a possibility for the gas to remain in two phases (cold dense inflowing and hot diffuse outflowing) while interacting with the outflow. Dense gas is quickly channeled to the bottom of the effective potential well, i.e. to the regions where feedback doesn’t counteract gravity. Once there, dense gas can form filaments which feed the SMBH or fragment into self-gravitating clumps. Meanwhile, diffuse gas gets heated and pushed by the feedback energy injection and forms a massive outflow which cools slowly due to low gas density. This results in an adiabatic outflow, consistent with the analytical predictions of the wind feedback model.
Possible improvements to cosmological simulations
-------------------------------------------------
Given that a conical thermal feedback prescription is capable of resolving a lot of the complexity of simultaneous inflows and outflows with little extra computational cost, we suggest that it may be interesting to explore its effects in cosmological simulations, which self-consistently track the growth and energy release from AGN over long timescales. This would allow one to constrain the AGN growth scenarios, the $M-\sigma$ relation, the effects of AGN outflows upon galaxies and clusters, the triggering of star formation, as well as the history of SMBH growth, with better precision than is currently available [@Schaye2015MNRAS]. One particularly interesting consequence might be that the presence of simultaneous inflows and outflows around SMBH change the qualitative behaviour of AGN self-regulation (see Section \[sec:selfreg\] below).
A few important aspects have to be considered when implementing this scheme into cosmological simulations. The first is numerical resolution, which is typically much worse in cosmological simulations than it is in the simulations presented here. In a previous paper [@Bourne2015MNRAS], we investigated the importance of resolution upon the efficiency of feedback and found that low-resolution simulations produce feedback that is much more negative. This happens in part because those simulations are unable to resolve the complex structures developing in the turbulent ISM. With a conical feedback prescription, it should be easier to resolve those structures, since the cavities and filaments are larger (compare the right panels in Figures \[fig:vp\_morph\] and \[fig:vpc\_morph\]). However, we may still expect a weaker overall effect of AGN feedback due to the fact that the gas density contrast is not resolved as well, leading to overcooling of the hot gas in the bubbles.
Another aspect is defining the appropriate direction of the cone’s axis. We suggest that this axis should coincide with the axis of rotation of the SMBH particle. Rotation can be tracked by tracking the angular momentum magnitude and direction of the infalling gas, and the plane of the accretion disc should be perpendicular to this axis of rotation. Since the AGN outflow originates in the disc, it is reasonable to assume that it goes in the polar direction [@Feldmeier1999ApJ]. The opening angle of the wind cone is a free parameter, which might be constrained by more detailed models, or investigated empirically. However, such an investigation is beyond the scope of this paper.
Finally, cosmological simulations contain galaxies which have varied structures, including discs, large clumps and cavities created by previous outflows. The interaction of a new outflow with these structures may be much more complex than in our simulations presented above. For example, one directed conical outflow might intercept a dense clump and lose a large fraction of its energy, while another may expand into a pre-existing cavity, having little effect upon the mass budget in its host galaxy. A more thorough investigation of feedback prescriptions in a more realistic galaxy model should be done before implementing them into a cosmological simulation. We intend to perform such tests in a future publication.
Self-regulation of AGN activity {#sec:selfreg}
-------------------------------
Our simulations use a fixed AGN luminosity rather than keeping it tied to the central particle accretion rate. Given that the central accretion rates in different simulations can differ by orders of magnitude (see Figure \[fig:mdot\_bh\]), this may seem as a significant drawback of the numerical setup. However, we feel that simulations addressing just the feedback aspect of AGN activity are still very useful since the models with feedback determined by BH accretion rate employ a number of assumptions about this rate. Given the non-linear accretion-feedback connection, it would be hard to arrive at definite conclusions about the feedback efficiency in these more complex and less constrained simulations. In addition, the actual expected evolution of the system is somewhat more complicated than the accretion rate of the central particle would suggest. Infalling gas contains some angular momentum, and therefore forms an accretion disc around the SMBH, rather than falling in directly. Given that in the simulations the accretion rate is measured at a radius of 10 pc, most of this accreting gas will take far longer than $1$ Myr to reach the SMBH, by which time the activity episode in our test run is over. Additionally, some of this gas will fragment and form a nuclear star cluster rather than feeding the SMBH [@Nayakshin2006MNRASb], further reducing the impact upon the AGN luminosity. Therefore, our simulations may be seen as an investigation of how a large-scale gas distribution is affected by AGN activity caused by accreting a cloud of predetermined mass, from $M_{\rm accr} = 2.2\times 10^6 \msun$ (L1 simulations) to $M_{\rm accr} = 1.1\times 10^7 \msun$ (L5 simulations).
In a more realistic simulation, long-term self-regulation of AGN activity becomes important. Here, the temporal evolution of simulations using the thermal-kinetic and virtual particle feedback prescriptions might be very different from one another. When the AGN luminosity increases, simulations using the spherically-symmetric thermal-kinetic AGN feedback prescription abruptly change from a situation where no outflow is occurring to one where the outflow completely shuts off AGN accretion. This latter situation may reverse after some time, as the material piles up at the edge of the outflow bubble, and its weight may overcome the force produced by AGN feedback, leading to collapse and resuming of accretion. However, while the bubble holds, there is no accretion. Conversely, while there is accretion, no outflow bubble can exist. This “all-or-nothing” situation places tight constraints on the regulation of AGN accretion rate. When the AGN is fed at a large enough rate, an outflow forms and shuts off accretion. As the accretion rate drops to zero (with perhaps some delay imposed in the simulation to mimic the draining of the sub-resolution accretion disc), AGN luminosity also decreases and feedback stops, allowing accretion to resume. We may therefore expect AGN activity to keep switching on and off, with the duration of the “on” phase comparable to the viscous timescale of the accretion disc ($t_{\rm on} \sim 10^5 - 10^6$ yr) and the “off” phase lasting for as long as it takes for gas to fall back to the SMBH ($t_{\rm off} >
10^6$ yr). These timescales are comparable to some estimates of the lifetimes of AGN phase [@Schawinski2015arXiv], but this is only coincidental, since the processes described here happen due to numerical, rather than physical, reasons.
The virtual particle prescription, as well as both conical prescriptions explored in this paper, allow simultaneous inflow and outflow to coexist. At some point, the larger gas reservoir gets depleted and accretion switches off, leading to a longer period of inactivity than in the case of spherical thermal feedback. This simulation therefore does not manifest an artificial short-timescale “flickering” and can be used to investigate the physical reasons of the switching on and off of AGN activity.
On the other hand, some integrated parameters, such as the total mass accreted by the SMBH or expelled from the galaxy, may not differ very much between thermal-kinetic and virtual particle feedback simulations. The reason for this is simply that on long timescales SMBH accretion tends to self-regulate so that the total energy release into the surrounding medium is just enough to balance gravity [@Booth2009MNRAS]. A thermal-kinetic feedback prescription would lead to faster accretion followed by a stronger burst of AGN activity, which leads to a larger feedback bubble and shutting off of accretion for a longer time. The virtual particle prescription would result in a more uniform accretion and outflow rate, with neither undergoing such strong changes as in the thermal-kinetic feedback model. As a result, eventually the results of the two simulations might look more similar to each other than might seem from our simulation results. Still, on short timescales, the morphology and temporal variability of inflow and outflow rates would differ significantly among the simulations.
The possibility of having prolonged periods of AGN activity coincident with SMBH growth via accretion also has an implication for the growth of the very first SMBHs. These black holes are known to have masses exceeding $10^9 \msun$ by $z = 6$, when the Universe was $<10^9$ yr old [e.g. @Wu2015Natur]. If these SMBHs grew from stellar-mass progenitors, their growth rate must have been close to the Eddington limit for most of their lives [@King2006MNRAS]. The possibility of having rapid gas inflows during the AGN phase suggests that this might be easier than previously thought and eliminates one of the potential drawbacks of this growth mechanism in explaining the large observed masses. We note, however, that it is possible to grow black holes to the required masses at high-z even with the typical thermal-kinetic AGN feedback prescription [@DiMatteo2012ApJ], at least for seed SMBH masses of $10^5 \msun$. Therefore a change in our understanding of accretion and feedback is not necessary to explain the high-z quasars.
Implications for positive AGN feedback
--------------------------------------
Thermal feedback simulations, both spherically symmetric and conical, eliminate some of the structure present in turbulent gas. Meanwhile, virtual particle simulations, and to some extent the conical thermal feedback simulations, enhance some of the structures, by enveloping dense gas in hot diffuse bubbles and compressing it more than self-gravity alone would. This leads to formation of more clumps and more efficient star formation within them [@Zubovas2014MNRASc], resulting in a higher rate of fragmentation and star formation. In other words, AGN activity has a positive feedback upon star formation in the host galaxy.
This effect has been explored in previous numerical works. @Nayakshin2012MNRAS showed that the outflowing material can fragment as it cools down, leading to potentially rapid star formation and stars ejected from host galaxies. @Gaibler2012MNRAS explored the interaction of an AGN jet with a clumpy ISM and found that the jet compresses dense gas, prevents adiabatic re-expansion of the clumps, and enhances the SFR of the whole galaxy. There is also tentative observational support that galaxies with more luminous AGN have higher star formation rates and efficiencies [e.g., @Wei2010ApJ; @Wang2010ApJb], although the causal connection is not certain.
We chose not to consider in detail the fragmentation of gas in our simulations, as that is not the main point of the present study. Nevertheless, we find qualitatively similar effects, but, importantly, they are much more significant in the virtual particle feedback models, where the multiphase ISM is affected by feedback based on its density. In a more realistic simulation, the effects of AGN outflow, and of the numerical prescription used to track its interaction with the ISM, might significantly affect the fragmentation rate and spatial locations in the galaxy. We suggest that using a more detailed AGN feedback prescription, such as the conical thermal feedback one, would allow investigating this positive feedback in a far more realistic way.
How realistic is the virtual particle simulation?
-------------------------------------------------
One implicit assumption we made so far when analysing the results is that the virtual particle feedback prescription is a good representation of reality. The method certainly has several advantages over the thermal feedback model. It is truly isotropic, i.e. feedback energy is emitted in all directions from the AGN and interacts with the ISM in all directions independently of the shape of the cavity surrounding the AGN. In addition, the model, by construction, accounts for the different optical depths of gas in different directions, leading to situations where a dense clump can shield the gas behind itself from being blown away.
On the other hand, the model suffers from some shortcomings as well. The interaction cross-section between virtual particles and SPH particles is a free parameter of the model. This means that the depth to which virtual particles penetrate the gas can vary significantly between simulations. As a result, dense gas clumps can sometimes be obliterated unphysically rapidly. Conversely, in other cases, gas might be compressed too strongly where a gentler push would be more realistic and represent the many filaments forming along the contact discontinuity between the AGN wind and the shocked ISM.
Another drawback is that the virtual particles are assumed to always propagate radially out from the AGN. This is unrealistic in cases of uneven density, where a steep density gradient may be present in a direction not parallel to the direction of propagation of the virtual particle. In this case, a realistic interaction would have the virtual particle flying off at an angle to its initial direction, something which the present model does not allow. This drawback may have many subtle effects in how the gas morphology is affected by AGN activity.
Given these drawbacks, we stress that the results of the virtual particle simulations should not be considered as perfect representations of reality. However, given that they reproduce a far more complex morphology of the turbulent ISM and properties of outflows in agreement with observations, we believe that their results represent real galaxies better than those of the spherical thermal feedback models.
Conclusion {#sec:concl}
==========
In this paper, we presented a number of simulations of AGN feedback affecting a turbulent gas sphere. The simulations are designed to show the impact of different subgrid recipes of AGN feedback upon the evolution of the whole system. We show that the commonly-used spherically-symmetric thermal AGN feedback prescription is unable to reproduce the variety of complex structures observed in more detailed models, fails to reproduce the observed rapid massive outflows and in general has a much weaker effect on the host galaxy than predicted by analytical calculations of feedback models.
On the other hand, a relatively simple improvement to the prescription - injecting energy into a cone rather than a sphere - alleviates most of these issues. The outflows become rapid, massive and close to adiabatic, dense structures and inflowing filaments coexist with outflowing gas, and gas fragmentation rates are significantly increased. In particular, even extremely bright AGN are unable to shut off gas accretion completely, allowing the SMBH to continue growing. We suggest that this improvement could be used in cosmological simulations in order to better model the effects that AGN have on their host galaxies and clusters.
Acknowledgments {#acknowledgments .unnumbered}
===============
KZ is funded by the Research Council of Lithuania grant no. MIP-062/2013. MAB and SN acknowledge an STFC grant. MAB is funded by a STFC research studentship. We thank Justin Read for the use of SPHS. This research used the DiRAC Complexity system, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment is funded by BIS National E-Infrastructure capital grant ST/K000373/1 and STFC DiRAC Operations grant ST/K0003259/1. DiRAC is part of the UK National E-Infrastructure.
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---
abstract: 'A scarcity of known chemical kinetic parameters leads to the use of many reaction rate estimates, which are not always sufficiently accurate, in the construction of detailed kinetic models. To reduce the reliance on these estimates and improve the accuracy of predictive kinetic models, we have developed a high-throughput, fully automated, reaction rate calculation method, AutoTST. The algorithm integrates automated saddle-point geometry search methods and a canonical transition state theory kinetics calculator. The automatically calculated reaction rates compare favorably to existing estimated rates. Comparison against high level theoretical calculations show the new automated method performs better than rate estimates when the estimate is made by a poor analogy. The method will improve by accounting for internal rotor contributions and by improving methods to determine molecular symmetry.'
author:
- 'Pierre L. Bhoorasingh'
- 'Belinda L. Slakman'
- Fariba Seyedzadeh Khanshan
- 'Jason Y. Cain'
- 'Richard H. West'
bibliography:
- 'kinComp.bib'
title: 'Automated transition state theory calculations for high-throughput kinetics'
---
Introduction {#introduction .unnumbered}
============
Detailed chemical kinetic modeling of complex systems has been aided by software that automatically generates reaction mechanisms[@VandeVijver:2015ba; @Blurock:2013et]. One example of such software, Reaction Mechanism Generator (RMG)[@Gao:2016dk], has been applied to systems such as the pyrolysis and combustion of isobutanol[@Merchant:2013kz], the fast pyrolysis of bio-oil[@SeyedzadehKhanshan:2016ga], and the auto-oxidation of a biofuel surrogate[@BenAmara:2013kl]. Mechanism generators can help ensure that important reaction pathways are not missed, but as a result they require thousands or even millions of thermodynamic and kinetic parameters to complete the model construction. These parameters are preferentially sourced from experimental measurements or accurate theoretical calculations, but most of the required parameters are unknown, leading to frequent use of less accurate estimates [@Broadbelt:2005ez].
Parameter estimation methods provide thermodynamic and kinetic values in a computationally efficient manner[@Yu:2004cd]. Estimation methods are typically based on Benson’s group additivity method for thermochemistry[@Benson:1969gq], in which group values are first determined from molecules with known thermodynamics, then used to estimate the thermodynamics of other molecules. Benson’s group contributions have been used to make adequate thermochemistry predictions for a variety of systems, including hydrocarbons[@Sumathi:2002jv; @Sebbar:2003gk] and silicon hydrides[@Wong:2004gl; @Adamczyk:2011ex]. Despite these successes, group contribution methods have been difficult to extend to some cases, such as predicting thermodynamics for polycyclic species, where the ring strain causes the molecule to be poorly described by the sum of its parts. The RMG software addresses this deficiency in the group additive approach by performing semi-empirical or quantum mechanical calculations of thermodynamic parameters for polycyclic species[@Magoon:2012hg].
For estimating reaction kinetics, the Evans-Polanyi relationship is a simple approach in which the change in enthalpy is used to predict the kinetics of the specific reaction[@Evans:1936ti]. Alternative approaches extend group contribution methods to predict kinetic parameters[@Sumathi:2002ii; @Saeys:2004ko; @Adamczyk:2009es; @West:2011ur; @Vandeputte:2012hu]. Group estimation methods can be automated efficiently, making them useful for mechanism generators where specific reaction rates are often unavailable[@Yu:2004cd]. Group-based predictions can be further improved using a hierarchy of reaction rate rules for increasingly specific reacting functional groups[@Curran:1998bx; @Curran:2002kl; @Carstensen:2009hl; @Villano:2013kr]. Unfortunately, appropriate rate rules are rarely available when studying new systems. In these situations more general (less specific) rules are used, and the accuracy of the estimates suffers; sometimes the estimates resulting from these generic or inappropriate rules can be wrong by several orders of magnitude.
Continuing advances in computing power have increased the feasibility of replacing these estimates with more accurate theoretical calculations using transition state theory (TST) and quantum chemistry methods. However, such calculations currently require a great deal of human input, mostly in locating the transition state, preventing their use in a high-throughput manner. Automating transition state calculations in order to calculate unknown kinetic parameters has been identified by the US Department Of Energy as a basic research need for clean and efficient combustion of 21st century transportation fuels[@McIlroy:2006js], and the Combustion Energy Frontier Research Center as an “important grand challenge”[@Law:2010ut].
Many parts of this challenge have been at least partially automated. TST calculations require the three-dimensional structures of the reactants, products, and transition state involved in each reaction. Reactant and product structures can already be found using the automated software integrated in RMG to calculate species thermochemistry[@Magoon:2012hg], using a combination of distance geometry, force fields, and semi-empirical quantum chemical calculations, to propose, optimize, and compare many conformer geometries for each compound. The artificial force induced reaction (AFIR) method[@Maeda:2009kg; @Maeda:2014jq], the KinBot software[@Zador:un], and other methods[@Zimmerman:2013bb; @Zimmerman:2014ko; @Zimmerman:2015dp; @Rooks:2014kz; @Bhoorasingh:2015dza] can use computational chemistry to automatically locate the necessary transition state geometries. Kinetic programs such as CanTherm[@CanTherm:2016], VariFlex[@klippensteinvariflex], MultiWell[@baker2012multiwell], and POLYRATE[@1992CoPhC..71..235L] have been developed to calculate reaction kinetics if provided the quantum chemistry outputs. Integrating geometry search software with kinetic calculators is a promising route to enable high-throughput kinetics calculations.
The present article describes automated algorithms to locate reactants, products, and transition states based on distance geometry[@Magoon:2012hg; @Bhoorasingh:2015dza] and their integration with the CanTherm[@CanTherm:2016] code, and Reaction Mechanism Generator (RMG) software[@Gao:2016dk] to calculate reaction rate expressions. The integrated algorithm (Fig. \[fig:overview\]) is referred to as the Automated Transition State Theory (AutoTST) calculator. Full details of the algorithms are provided in the Methods section below. In this work we demonstrate the method on three families of reactions, chosen to cover the three main classes of molecularity: intramolecular H transfer or migration which is unimolecular isomerization (1 reactant, 1 product), radical addition to a multiple bond which is addition (2 reactants, 1 product), and H abstraction which is bimolecular (2 reactants, 2 products). The reverse of radical addition to a multiple bond is -scission, which is unimolecular decomposition (1 reactant, 2 products). Although there are many more reaction families, these three demonstrate the generality of the approach and together comprise about half of the reactions in typical combustion models (eg. 48% for the butanol model described below). We posit ring opening (and closing) will be algorithmically quite like intramolecular H transfer in terms of determining the 3D geometry of the transition state, although the electronic structure calculations will require additional considerations to find accurate barrier heights.
![\[fig:overview\] Overview of the AutoTST algorithm.](overview-figure){width="11cm"}
Many important reactions in high temperature gas phase systems such as combustion are pressure-dependent, either chemically activated or in the fall-off regime. The RMG software contains procedures to automatically perform master equation calculations for these unimolecular reaction networks to calculate phenomenological $k(T,P)$ from high-pressure-limit information[@Allen:2012jc]. The AutoTST tool presented here complements this approach by providing these high-pressure-limit reaction rates (except for barrierless reactions that will require a variational TST treatment), as well as direct calculations of barrier heights and vibrational frequencies of the transition states. Without these, the microcanonical rate coefficients $k(E)$ must be estimated via inverse Laplace transforms of canonical $k(T)$ rate expression estimates[@Allen:2012jc].
Methods {#methods .unnumbered}
-------
In brief, the transition state geometries were predicted using a modified version of our previously published algorithm[@Bhoorasingh:2015dza] that combines group-additive prediction of interatomic distances with distance-geometry methods, then optimized, and confirmed to correspond to the correct reaction using an intrinsic reaction coordinate (IRC) calculation. Symmetry numbers were determined via point group using the SYMMETRY software[@Patchkovskii:2003oj] and canonical TST calculations were performed using CanTherm[@CanTherm:2016]. The entire process is automated in Python using modules and classes from RMG-Py[@Gao:2016dk]. Full details of the algorithm and methods are described below.
### Automated geometry searches {#automated-geometry-searches .unnumbered}
Reactant and product structures were located using the automated algorithm developed in RMG and described by @Magoon:2012hg[@Magoon:2012hg]. Transition state structures were located using a group contribution method that predicts transition state reaction center distances using training data of known transition states. A brief explanation of the algorithm follows, with more information in ref. .
The algorithm begins with generating upper and lower bounds of the distances between reactant atoms, known as a bounds matrix, which comes from a distance geometry approach in the open-source cheminformatics program RDKit [@RDKitOpensourcec:nApUqBGo]. The bounds matrix is then edited for the transition state, whose reacting atom distances are estimated via the group contribution method. For hydrogen abstraction, these atom distances are those between the abstracted hydrogen, the atom bonded to the abstracted hydrogen, and the radical abstracting the hydrogen.
Molecular structure groups defining the reacting functional groups are organized into a hierarchical tree for each reaction family. The tree was arranged based on chemical intuition of which molecular structure features are most important, i.e. the number of radical electrons on a reacting atom is more important than its bonding configuration. A training set for these reaction center distances was created from previously optimized transition state geometries. The values for each group are calculated by finding a best fit to the training set, using linear least squares regression. When estimating unknown distances for a transition state, the tree is traversed to the most specific group matching the molecular structure of the reacting groups. The distance values for this group are added to base values stored in the top level of the tree, to provide all the reaction center distances needed for the reaction.
Once an estimate for the reaction center distances is made, the atoms are embedded in 3D space to create several conformers that satisfy the overall bounds matrix. These geometries are optimized, with the lowest energy conformer chosen as the transition state estimate, and can then be further optimized at a chosen level of theory. In this work, we use the M06-2X[@Zhao:2006dj; @Zhao:2008kw] functional with a MG3S[@BenjaminJLynch:2003cy] basis set in Gaussian 09[@Gaussian:we]. The transition state is validated using a intrinsic reaction coordinate (IRC) calculation at the same level of theory; the atomic coordinates from either end of the IRC pathway are converted into molecular graphs using a connect-the-dots algorithm[@Bhoorasingh:2015dza] and compared to the original reactants and products to confirm that the saddle point corresponds to the desired reaction.
![\[fig:mod\_ts\_algo\] Automated transition state search algorithm. The steps with bold borders are deviations from the original algorithm described in ref. .](modified_algorithm){width="3.33in"}
The group contribution method for predicting transition state geometries as described in ref. has been modified from its original formulation to improve its performance; the updated algorithm is shown in Figure \[fig:mod\_ts\_algo\]. The distance geometry algorithm in RDKit[@RDKitOpensourcec:nApUqBGo] requires upper and lower limits for the distances between every atom. The difference between the upper and lower limits for the reaction center distances were previously set to 0.05Å, but this was decreased to 0.025Å due to increased confidence in the reaction center predictions. Conformers were constructed in 3D to satisfy the distance limits for every atom pair.
The optimization protocol was also modified, with the transition state geometry prediction algorithm no longer using a universal force field optimization, instead adopting a protocol similar to that used in the AARON code [@Rooks:2014kz]. The geometry estimate undergoes a constrained optimization to an energy minimum with the reaction center distances frozen, followed by a transition state (saddle point) search with all distances frozen except the reaction center. The resulting geometry is then used for a Berny transition state optimization[@JCC:JCC540030212].
The transition state training data used in this study were optimized and validated at M06-2X with a MG3S basis set, so that predictions were made for the same electronic structure method used in this study. M06-2X/MG3S provides sufficiently accurate kinetic parameters at a reasonable computational cost, and is widely available in computational chemistry packages[@Frisch:2009wv; @Valiev:2010bb; @Neese:2012ki]. Our algorithm, previously demonstrated for hydrogen abstraction reactions, was modified and extended to be applied to intramolecular hydrogen migration and radical addition to multiple bond (-scission in reverse) reaction families.
### Kinetic calculations {#kinetic-calculations .unnumbered}
The CanTherm software package was used to determine kinetic parameters using classical transition state theory[@CanTherm:2016]. Symmetry numbers for the rate calculations were determined via point group using the SYMMETRY software[@Patchkovskii:2003oj]. SYMMETRY takes as input the optimized 3-dimensional geometry and a tolerance to allow for small deviations, and calculates the point group. The point group is used to determine the symmetry number[@Irikura:1998ur], and a chirality contribution is added for point groups that lack a superimposable mirror image. Product geometries and energies were also found for these calculations so the Eckart model could be applied to determine tunneling corrections[@Eckart:1930kz]. For the automated calculations the Rigid Rotor Harmonic Oscillator (RRHO) approximation is used for all reactants, products, and transition states. Figure \[fig:kin\_algo\] provides an overview of the automated kinetic calculation method.
![\[fig:kin\_algo\] The automated kinetic calculations involve an automated transition state search (Figure \[fig:mod\_ts\_algo\]), automated search for reactant and product geometries[@Magoon:2012hg], and automatically calculating kinetics using CanTherm[@CanTherm:2016].](kinetics_algorithm){height="4in"}
### Comparison of Automated TST calculations and Rate Rules {#comparison-of-automated-tst-calculations-and-rate-rules .unnumbered}
The AutoTST calculations were compared to rates of hydrogen abstraction, intramolecular hydrogen migration, and radical addition to multiple bond (and -scission) reactions from the LLNL butanol combustion model[@Sarathy:2012fj], and to estimates of these rates generated by the automated rate rule implementation in RMG. Many of the rates in the hand-curated LLNL model were estimated by applying rate rules, while others were calculated or determined by analogy. In RMG kinetics estimation, rates are determined by traversing hierarchical trees of rules based on molecular structure[@Gao:2016dk; @RMGpy:documentation]. If this results in an exact match for the specific molecular structure of the reactant(s), with that combination of nodes in the trees containing data, then this rule is used. However, if the node exactly matching the reacting molecular structures does not contain data, the estimator will “fall up” to more general nodes, i.e. less precisely defined descriptions of the reactant(s).
These more general nodes will be at some distance from the specific reactants’ nodes as measured by the number of levels in the tree. Because there are several trees (e.g. one for each reactant), the Euclidean distance is calculated. This “Nodal distance” can therefore be interpreted as the inappropriateness of the rate estimate used for each reaction, with an exact match corresponding to a Nodal distance of 0.
Kinetics were compared at 1000K, since the rate rules were determined for a combustion model in that temperature range. For pressure-dependent rates in the LLNL model, the high pressure limits were used.
### Comparison to benchmark calculations {#comparison-to-benchmark-calculations .unnumbered}
As will be seen in the Results section, in some cases there are large differences between rate rule predictions and AutoTST calculations. For two such cases from each reaction family, more thorough theoretical calculations were performed. The reactions were selected if there was at least a hundredfold discrepancy between the rate calculated by AutoTST and both the LLNL and RMG rates at 1000 K. These benchmark calculations were compared to both the rate rule predictions and the AutoTST rates.
The geometries for the benchmark calculations were determined using the same DFT functional and basis set as the automatically calculated rates, but the benchmark calculations used an [*ultrafine*]{} grid. For the benchmark calculations, the 1-D hindered rotor approximation[@Pfaendtner:2007kv] was applied to torsional modes, instead of the RRHO approximation. AutoTST did not always find the lowest energy conformer, so when the hindered rotor scans revealed a lower-energy conformer this was re-optimized and adopted for the benchmark calculations. Barrier heights were also recalculated using single point coupled-cluster calculations (see the ‘Computational chemistry’ section for details). Finally, symmetry numbers were manually checked, and corrected if the AutoTST approach was in error.
These improvements allowed comparison between AutoTST and the benchmark calculations to identify the sources of error in the AutoTST calculations. To distinguish between the sources of error, each of the four improvements on the AutoTST method was systematically removed. This analysis requires five manual quantum chemistry calculations in total for each reaction chosen, plus the automated calculation, so was only performed on six reactions not all 781 of the reactions successfully calculated with AutoTST.
### Computational chemistry {#computational-chemistry .unnumbered}
Geometry optimization and path analysis calculations used the M06-2X DFT functional[@Zhao:2006dj; @Zhao:2008kw] with the MG3S basis set[@BenjaminJLynch:2003cy] (equivalent to 6-311+G(2df,2p) for systems containing C, H, and O)[@Clark:1983gu; @Frisch:1984ep] in the Gaussian 09 quantum chemistry package[@Gaussian:we]. For benchmark calculations, electronic energies were computed using the CCSD(T)-F12/RI method with the cc-VTZ-F12[@Peterson:2008eo] and cc-VTZ-F12-CABS[@Yousaf:2008jw] basis sets in ORCA[@Neese:2012ki].
Results {#results .unnumbered}
=======
Kinetics calculated with the integrated AutoTST algorithm were compared to two sets of estimates: the first were from a butanol combustion model[@Sarathy:2012fj] from the Lawrence Livermore National Laboratory (LLNL) which used a combination of literature data, estimates, analogies, calculations, and rate rules; and the second set were rate rule predictions generated by RMG.
The butanol combustion model contained 855 hydrogen abstraction, 78 intramolecular hydrogen migration, and 184 radical addition to multiple bond reactions (including some reactions in the reverse -scission direction). For each reaction family, AutoTST calculated kinetics for approximately 70% of the reactions (Table \[table:ModelRxns\]). The percentage successfully calculated was consistent across all reaction families, so the AutoTST success rate is so far independent of the reaction type. Failures are usually due to the automated transition state search finding a saddle point that corresponds to a conformational change or a transition other than the intended reaction, and have been previously discussed[@Bhoorasingh:2015dza].
Overall, the AutoTST kinetics corresponded with with rate rule predictions from both sources (Figure \[fig:KinComp\_all\_compare\]), with most rate rules being within an order of magnitude (factor of ten) of each other. Figure \[fig:comp\_RMG\] displays the RMG rate rule predictions vs. AutoTST, with the color of each point corresponding to its nodal distance, a measure of inappopriateness of the rate rule used by RMG (see Methods section). The data and scripts used to produce Figure \[fig:comp\_RMG\] are freely available[@Bhoorasingh2017data].
For both hydrogen abstraction and radical addition to multiple bond reactions, the AutoTST rates that matched best with the RMG rates, generally, were for rates with low nodal distances. Where significant “banding” is seen on the plots, RMG is using the same reaction rate for several reactions; these points are mainly in orange indicating a higher nodal distance. In the intramolecular hydrogen migration reactions, the nodal distances never exceed 1 and less scatter is seen than for the other reaction families. Using the nodal distance to estimate errors in the rate rule estimates could guide when it is appropriate to use AutoTST calculations in the context of automated mechanism generation using RMG.
AutoTST agreement with LLNL rates is similar to its agreement with RMG rate rules, as indicated by Figure \[fig:comp\_Sarathy\]. Banding in the AutoTST vs. LLNL plots for hydrogen abstraction and radical addition to multiple bond, where the same rate is being used for 10 or more reactions in some cases, illustrates that this published model also applies rate rules and analogies with varying degrees of applicability. Both sets of comparisons indicate that AutoTST agrees with rate rules when the rules are applied appropriately, but provides a better method of calculation in the absence of good rate estimates.
The Gaussian 09 log files and CanTherm input and output files for all 781 reactions and 282 corresponding species are available online in a FigShare repository[@Bhoorasingh2016data].
[@lccc]{}
------------------------------------------------------------------------
Reaction Family & Number of & Kinetics successfully & Percentage\
& Reactions & calculated & calculated\
------------------------------------------------------------------------
Hydrogen abstraction & 855 & 598 & 70%\
Intramolecular hydrogen migration & 78 & 52 & 67%\
Radical addition to multiple bond & 184 & 131 & 71%\
------------------------------------------------------------------------
Total & 1117 & 781 & 70%\
![Rate rule estimates from RMG (a) and LLNL model rates (b) plotted against automated algorithm TST calculations evaluated at 1000 K. From ref. .[]{data-label="fig:KinComp_all_compare"}](RateRulesvsAutoTST.pdf){width="\textwidth"}
[0.94]{} ![Rate rule estimates from RMG (a) and LLNL model rates (b) plotted against automated algorithm TST calculations evaluated at 1000 K. From ref. .[]{data-label="fig:KinComp_all_compare"}](SarathyvsAutoTST.pdf "fig:"){width="\textwidth"}
Benchmark calculations
======================
Despite the overall agreement, a number of reactions had significant discrepancy between AutoTST rates and the rate rule predictions. Six of the reactions with significant discrepancies, two from each reaction family (Table \[table:BenchRxns\]), were selected for benchmark calculations as described above, to determine the accuracy of the three prediction methods and the sources of error. For reaction 5, the rate from the LLNL model was provided in the reverse (-scission) direction, so the rate shown (Figure \[fig:RAddMult\_comp1\]) was calculated using the provided rate and thermodynamics from the model.
[@c l c]{}
------------------------------------------------------------------------
Label & Family & Reaction\
------------------------------------------------------------------------
R1 & H abstraction &\
R2 & H abstraction &\
R3 & Intramolecular H migration &\
R4 & Intramolecular H migration &\
R5 & Radical addition &\
R6 & Radical addition &\
[3in]{} ![Comparison of kinetic estimates for hydrogen abstraction reactions R1 (a) and R2 (b). []{data-label="fig:HAbs_Comp"}](R1 "fig:"){width="\textwidth"}
[3in]{} ![Comparison of kinetic estimates for hydrogen abstraction reactions R1 (a) and R2 (b). []{data-label="fig:HAbs_Comp"}](R2 "fig:"){width="\textwidth"}
[3in]{} ![Comparison of kinetic estimates for intramolecular hydrogen migration reactions R3 (a) and R4 (b).[]{data-label="fig:iHA_Comp"}](R3 "fig:"){width="\textwidth"}
[3in]{} ![Comparison of kinetic estimates for intramolecular hydrogen migration reactions R3 (a) and R4 (b).[]{data-label="fig:iHA_Comp"}](R4 "fig:"){width="\textwidth"}
[3in]{} ![Comparison of kinetic estimates for radical addition to multiple bond reactions R5 (a) and R6 (b).[]{data-label="fig:RAddMult_Comp"}](R5 "fig:"){width="\textwidth"}
[3in]{} ![Comparison of kinetic estimates for radical addition to multiple bond reactions R5 (a) and R6 (b).[]{data-label="fig:RAddMult_Comp"}](R6 "fig:"){width="\textwidth"}
For one case (reaction 1, Figure \[fig:habs\_comp1\]), both rate rule estimates outperformed the AutoTST calculation, in that they were closer to the benchmark rate. The rate rule used in the LLNL model was developed for [@Carstensen:2007ks], and the RMG rate rule was developed for [@Walker:1989fv], both of which are quite similar to R1 (). The accuracy of the rate rule predictions should be expected since the rules were developed for reactions resembling R1, and in such cases AutoTST should not be used since the rate rules could provide a good rate prediction at a far lower computational cost.
AutoTST outperformed the rate rule predictions for all the other reactions (Figures \[fig:habs\_comp2\], \[fig:iHA\_Comp\], and \[fig:RAddMult\_Comp\]). In one such example (reaction 3, Figure \[fig:iha\_comp1\]), the RMG rate prediction was made from a generalized rate rule (high nodal distance), so the kinetic data used to make the prediction was very unlike the reaction, leading to the large discrepancy in the reaction rate. The value used in the LLNL model was not an estimate but a theoretically calculated value for the specific reaction [@Lee:2003kt], which agrees favorably with the benchmark calculation.
Overall, the comparison of the 6 reactions with large discrepancies between rate rule predictions and AutoTST show that the automated method performs well for all tested reaction families. This is particularly true when considering the performance of the kinetics across a wide temperature range, where the kinetics calculated with AutoTST agree with the high accuracy calculations, but the rate rules become less accurate outside the temperature range for which they were developed. Discrepancies between the AutoTST calculations and the benchmark calculations presented an opportunity to identify sources of error in the AutoTST method: using the DFT energies instead of CCSD(T)-F12/RI, neglecting hindered rotors, using an automatically determined symmetry number that may be incorrect, and using randomly guessed conformers which may not be the lowest energy choices. To isolate the effects of each error, each correction was individually removed from the benchmark calculations and replaced with the equivalent used for the automated calculations.
Figure \[fig:k1000\_error\] shows the magnitude of the difference in the rate coefficient at 1000 K due to each source of error. Table \[table:Activation\_Energy\] displays the difference in activation energy due to each source of error and the benchmark calculation, and Table \[table:A\_Factor\] shows the changes to the pre-exponential “$A$ factor” due to the same effects.
![\[fig:k1000\_error\] Sources of error in each automated algorithm compared to its respective benchmark calculation. $\Delta \log(k(1000\text{ K})) = \log_{10}\left((k \text{ with a correction omitted})/(\text{benchmark } k\text{ with all corrections})\right)$. Omitting all the corrections simultaneously gives the “Fully Automated Algorithm” result.](error_comparison){width="\textwidth"}
[@|c|c|cccc|c|]{}
------------------------------------------------------------------------
Reaction & Benchmark & Inaccurate & Hindered & Incorrect & Incorrect & Overall\
& E$_a$ & Energy & Rotors & Symmetry & Conformer & Discrepancy\
------------------------------------------------------------------------
R1 & 111.98 & –5.46 & –0.34 & 0.00 & –0.91 & –6.01\
R2 & 119.38 & –9.92 & –20.70 & 0.00 & +4.57 & –24.53\
R3 & 88.54 & +6.16 & +0.28 & 0.00 & –11.05 & –4.06\
R4 & 85.19 & +7.03 & +1.08 & 0.00 & –12.52 & –3.35\
R5 & 97.24 & –7.73 & 0.00 & 0.00 & –2.52 & –7.68\
R6 & 84.89 & –6.74 & –8.35 & 0.00 & +0.01 & +1.91\
[@|c|c|cccc|c|]{}
------------------------------------------------------------------------
Reaction & Benchmark &Inaccurate & Hindered & Incorrect & Incorrect & Overall\
& log$_{10}$A & Energy & Rotors & Symmetry & Conformer & Discrepancy\
------------------------------------------------------------------------
R1 & 13.96 & +0.020 & –0.772 & –1.079 & 0.000 & –2.337\
R2 & 13.27 & +0.010 & –2.659 & 0.000 & -0.016 & –1.854\
R3 & 12.00 & –0.001 & +0.416 & 0.000 & +0.004 & +0.620\
R4 & 11.64 & –0.002 & +1.253 & 0.000 & +0.004 & +1.369\
R5 & 12.83 & 0.000 & 0.000 & –1.079 & 0.000 & –1.169\
R6 & 11.53 & 0.000 & –0.483 & 0.000 & 0.000 & +0.141\
The major source of error for AutoTST calculations was using the RRHO approximation for treatment of internal rotations, although not for reactions that contained few rotors (e.g. R5). Symmetry was also a major source of error when the automated method determined symmetry numbers incorrectly. This was not consistent for all tested reactions as the automated method correctly determined symmetry for some cases. As expected, the activation energy is unaffected by correcting the symmetry number but is changed somewhat by using coupled-cluster calculations instead of DFT for barrier heights. Correcting the DFT energy had little effect on the rate calculations in the combustion temperature range, but at lower temperatures the DFT energy led to rates that were approximately an order of magnitude different from the benchmark calculations. AutoTST was not always successful in finding the lowest energy conformer for all structures, which contributed to errors of varying degrees. The intramolecular hydrogen migration reactions were most affected by these, where a single wrong conformer would contribute significantly to an error in the barrier height.
Discussion {#discussion .unnumbered}
==========
While all sources of error need to be addressed, automating the treatment of hindered internal rotors and providing a more robust algorithm for determining symmetry should be targeted first. Automating hindered rotor calculations will also help to identify the lowest energy conformer, thus correcting errors due to the incorrect conformer selection. The additional computational cost of more accurate electronic structure calculations to improve barrier heights would have to be balanced against available computational resources, as using the DFT energy was not a major source of discrepancy.
Conclusion {#conclusion .unnumbered}
==========
In summary, the AutoTST method is extensible, and has now been applied to hydrogen abstraction, intramolecular hydrogen migration or transfer, and radical addition to multiple bond reactions. The successful extension of AutoTST motivates further work to include other reaction families with a reaction barrier. Since the method is shown to work for all three main classes (isomerization, decomposition, bimolecular), extending to additional reaction families is mostly a case of training new group values to predict the interatomic distances, as described in ref. .
Despite current sources of error, good kinetic estimates can be calculated using the automated algorithm, and the protocol should be used for mechanism generation alongside other rate estimation methods. The algorithm can also be used as a stand-alone tool to obtain specific reaction rates or transition state geometries. Further, the tool can be used semi-manually to set up a transition state geometry calculation with good interatomic distances, and the geometry can be optimized further and used for kinetics and other calculations at the user’s desired level of theory.
AutoTST outperforms rule-based kinetic estimation methods when specific rate rules are unavailable for a reaction. The current estimation methods should not be abandoned, however, as they can still provide good kinetic predictions when used appropriately (e.g. R1), and in a computationally efficient manner. A new module in RMG allows the user to obtain an estimate for the uncertainty of a reaction rate. When rates from estimation methods are unavailable, or the rates prove too uncertain to be used confidently in model generation, AutoTST now provides an alternate method to determine kinetics with little human input. The fully automated process represents an important step towards high-throughput calculation of many thousands of accurate reaction rates, which can now be feasibly used in the construction of detailed kinetic models. Another potential use of this tool is for generating large quantities of training data as input to other prediction algorithms.
The authors thank C. Franklin Goldsmith (Brown University), Jorge Aguilera Iparraguirre (Harvard University, now at Kyulux), and the developers of RMG at both Northeastern University and the Massachusetts Institute of Technology for helpful discussions. Acknowledgment is made to Northeastern University Department of Chemical Engineering and the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research.
Supporting Information Available {#supporting-information-available .unnumbered}
================================
The following files are available free of charge.
- The Gaussian 09 log files and CanTherm input and output files for all 781 reactions and 282 corresponding species are available in the figshare repository[@Bhoorasingh2016data] at\
[https://doi.org/10.6084/m9.figshare.4234160]{}
- The results in CSV format and Python scripts required to generate Figure \[fig:KinComp\_all\_compare\] are available in the figshare repository [@Bhoorasingh2017data] at\
[https://doi.org/10.6084/m9.figshare.5244640]{}
|
---
abstract: 'Watkins (1969) first introduced the generalized Petersen graphs (GPGs) by modifying Petersen graph. Zhou and Feng (2012) modified GPGs and introduced the double generalized Petersen graphs (DGPGs). Kutnar and Petecki (2016) proved that DGPGs are Hamiltonian in special cases and conjectured that all DGPGs are Hamiltonian. In this paper, we construct Hamilton cycles in all DGPGs.'
address: 'Department of Informatics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan'
author:
- Yutaro Sakamoto
title: Hamilton Cycles in Double Generalized Petersen Graphs
---
Hamilton cycle ,Double generalized Petersen graph
Introduction
============
In [@Wat] Watkins (1969) first introduced the GPGs to discover Tait coloring of the graphs and Castagna and Prins (1972) proved Watkins’ conjecture about GPGs in [@Cas]. After they introduced GPGs, some properties of GPGs have been studied. For instance, Alspach (1983) determined which GPGs have a Hamilton cycle in [@Als]. Fu, Yang and Jiang (2009) studied the domination number of GPGs in [@Fu].
Now we define double generalized Petersen graphs DP$(n, t)$ (DGPG for short) as follows.
Let $n$ and $t$ be integers that satisfy $n \geq 3$ and $2 \leq 2t < n$. The double generalized Petersen graph DP$(n, t)$ is an undirected simple graph with vertex set $V$ and edge set $E$, where $$\begin{aligned}
V &= \{x_i, u_i, v_i, y_i \mid i \in \mathbb{Z}_n \}, \\
E &= \{x_ix_{i+1}, y_iy_{i+1}, x_iu_i, y_iv_i, u_iv_{i+t}, v_iu_{i+t} \mid i \in \mathbb{Z}_n \}
\end{aligned}$$
Note that $\mathbb{Z}_n$ denotes a set of integers $\mathbb{Z}/n\mathbb{Z}$ throughout this paper.
![DP$(7, 3)$[]{data-label="fig:dp73"}](./dp73-trim.eps){height="0.4\textheight"}
Zhou and Feng (2012) first introduced the double generalized Petersen graphs by modifying the generalized Petersen graphs in [@Jin1]. In [@Jin2], Zhou and Feng (2014) determined all non-Cayley vertex-transitive graphs and all vertex-transitive graphs among DGPGs. From their result, Kutnar and Petecki (2016) gave the complete classification of automorphism groups of DGPGs in [@Kutnar]. They also proved that DP$(n, t)$ is Hamiltonian if $n$ is even or $n$ is odd and the greatest common divisor of $n$ and $t$ equals to $1$ in [@Kutnar]. In addition, a computer-assisted search verified that DP$(n,t)$ have Hamilton cycles for all $n \leq 31$ and they conjectured that all DP$(n, t)$ are Hamiltonian. This paper gives the following theorem.
\[thm\] All DP$(n, t)$ are Hamiltonian.
Preliminaries
=============
As mentioned in the previous section, $\mathbb{Z}_n$ denotes a set of integers $\mathbb{Z}/n\mathbb{Z}$. A sequence of vertices $w_0 w_1 w_2 \dots w_n$ denotes a path in a graph. A path whose end points are the same vertex is called a cycle. $V(G)$ denotes the vertex set of a graph $G$. Let $G$ be an arbitrary subgraph of DP$(n, t)$. We define functions $V_x, V_y, V_u, V_v$ as follows. $$\begin{aligned}
V_x(G) &= V(G) \cap \{x_i \mid i \in \mathbb{Z}_n\}\\
V_y(G) &= V(G) \cap \{y_i \mid i \in \mathbb{Z}_n\}\\
V_u(G) &= V(G) \cap \{u_i \mid i \in \mathbb{Z}_n\}\\
V_v(G) &= V(G) \cap \{v_i \mid i \in \mathbb{Z}_n\}\\
\end{aligned}$$
The construction of Hamilton cycles in DP$(n, t)$
=================================================
We assume that $n$ is even. In this case, Kutnar and Petecki showed that all DP$(n,t)$ are Hamiltonian in [@Kutnar]. Observe that there exist paths $X_i$ for all $i \in \mathbb{Z}_{n/2}$. $$\begin{aligned}
X_i : u_{2i} x_{2i} x_{2i+1} u_{2i+1} v_{2i+1-t} y_{2i+1-t} y_{2i+2-t} v_{2i+2-t} u_{2(i+1)}
\end{aligned}$$ Joining all of the paths gives a Hamilton cycle in DP$(n,t)$.
We assume that $n$ is odd. Let $2k + 1$ be the greatest common divisor of $n$ and $t$. In order to construct a Hamilton cycle in DP$(n, t)$, we define paths $P_i, Q_i, R_i, S_i$ for all $i \in \mathbb{Z}_{2k+1}$. $$\begin{aligned}
P_i &\colon u_{a_i+t} x_{a_i+t} x_{a_i+t+1} x_{a_i+t+2} \dots x_{a_{i+2}+t-1} u_{a_{i+2}+t-1} \\
Q_i &\colon v_{a_i} y_{a_i} y_{a_i+1} y_{a_i+2} \dots y_{a_{i+2}-1} v_{a_{i+2}-1} \\
R_i &\colon u_{a_{i+1}+t-1} v_{a_{i+1}+2t-1} u_{a_{i+1}+3t-1} \cdots v_{a_i} \\
S_i &\colon v_{a_{i+1}-1} u_{a_{i+1}-t-1} v_{a_{i+1}-2t-1} \cdots u_{a_i+t}
\end{aligned}$$ where $a_0, a_1, a_2, \dots ,a_{2k} \in \mathbb{Z}_{2k+1}$ satisfy the following conditions $$\begin{aligned}
&\forall i \in \mathbb{Z}_{2k+1}, a_i \equiv i \ (\text{mod}\ 2k+1) \\
&0 \leq a_0 < a_2 < a_4 < \dots < a_{2k} < a_1 < a_3 < a_5 < \dots < a_{2k-1} < n
\end{aligned}$$ For instance, if $a_0=0, a_2=2, a_4=4, \dots, a_{2k}=2k, a_1=2k+2, a_3=2k+4, a_5=2k+6, \dots ,a_{2k-1}=4k$, the above conditions are met. Joining the paths in the following way gives a Hamilton cycle in DP$(n,t)$. $$\begin{aligned}
((S_0 - P_0) - (R_1 - Q_1) - (S_2 - P_2) - (R_3 - Q_3) - \cdots \\
\cdots - (R_{2k-1} - Q_{2k-1}) - (S_{2k} - P_{2k})) - \\
- ((R_0 - Q_0) - (S_1 - P_1) - (R_2 - Q_2) - (S_3 - P_3) - \cdots \\
\cdots - (S_{2k-1} - P_{2k-1}) - (R_{2k} - Q_{2k}))
\end{aligned}$$
An example of a Hamilton cycle in DP$(n, t)$ is shown in Figure \[fig:cycle\].
![A Hamilton cycle in DP$(n, t)$ $(2k + 1 = 5)$[]{data-label="fig:cycle"}](./cycle3-trim.eps){height="0.44\textheight"}
Proof of Theorem \[thm\]
========================
In this section, we prove that the cycle described in the previous section contains all vertices of DP$(n,t)$ exactly once for all odd integers $n \geq 3$. Let $G$ be DP$(n, t)$ and $2k+1$ be the greatest common divisor of $n$ and $t$.
Firstly, we prove that paths $Q_0, Q_1, \dots, Q_{2k}$ contain all of $V_y(G)$. $$\begin{aligned}
\bigcup_{i=0}^{2k} V_y(Q_i)
= &\left(\bigcup_{i=0}^{k} V_y(Q_{2i}) \right) \cup \left(\bigcup_{i=0}^{k-1} V_y(Q_{2i+1}) \right) \\
= &( \{y_{a_0}, y_{a_0+1}, \dots ,y_{a_2-1}\} \cup\\
&\ \{y_{a_2}, y_{a_2+1}, \dots ,y_{a_4-1}\} \cup\\
&\ \{y_{a_4}, y_{a_4+1}, \dots ,y_{a_6-1}\} \cup\\
&\ \ \vdots\\
&\ \{y_{a_{2k}}, y_{a_{2k}+1}, \dots ,y_{a_1-1}\}) \cup \\
&( \{y_{a_1}, y_{a_1+1}, \dots ,y_{a_3-1}\} \cup\\
&\ \{y_{a_3}, y_{a_3+1}, \dots ,y_{a_5-1}\} \cup\\
&\ \{y_{a_5}, y_{a_5+1}, \dots ,y_{a_7-1}\} \cup\\
&\ \ \vdots\\
&\ \{y_{a_{2k-1}}, y_{a_{2k-1}+1}, \dots ,y_{a_0-1}\})\\
= &\{y_m \mid m \in \mathbb{Z}_n\}\\
= &V_y(G)
\end{aligned}$$
According to the above equation and the definitions of $P_i$ and $Q_i$, we can prove that paths $P_0, P_1, \dots, P_{2k}$ contain all of $V_x(G)$.
Secondly, we will prove that paths $R_0,R_1,\dots,R_{2k},S_0,S_1,\dots,S_{2k}$ contain all of $V_u(G) \cup V_v(G)$. We define cycles $C_i$ in DP$(n, t)$ for all $i \in \mathbb{Z}_{2k+1}$. $$\begin{aligned}
C_i \colon u_i v_{i+t} u_{i+2t} v_{i+3t} \cdots u_{i+(p-1)t} v_i u_{i+t} v_{i+2t} u_{i+3t} \cdots v_{i+(p-1)t} u_i
\end{aligned}$$ Note that odd integers $p$ and $q$ satisfy $n = p(2k + 1)$ and $t = q(2k + 1)$. For all $i \in \mathbb{Z}_{2k+1}$, $C_i$ consists of paths $D_i \colon u_i v_{i+t} u_{i+2t} v_{i+3t} \cdots u_{i+(p-1)t}$ and $E_i \colon v_i u_{i+t} v_{i+2t} u_{i+3t} \cdots v_{i+(p-1)t}$. Since $p$ is odd, the last vertex of $D_i$ is not $v_{i+(p-1)t}$ but $u_{i+(p-1)t}$. By symmetry, the last vertex of $E_i$ is $v_{i+(p-1)t}$. In addition, $u_{i+(p-1)t}$ and $v_{i+(p-1)t}$ are respectively adjacent to $v_i$ and $u_i$ since $pt = pq(2k + 1)$ is a multiple of $n$. Observe that $u_i, u_{i+t}, u_{i+2t}, u_{i+3t}, \dots ,u_{i+(p-1)t}$ contain no two same vertices since $pt$ is the least common multiple of $n$ and $t$. Hence $v_i, v_{i+t}, v_{i+2t}, v_{i+3t}, \dots ,v_{i+(p-1)t}$ also contain no two same vertices.
We show that cycles $C_0, C_1, \dots, C_{2k}$ contain all of $V_u(G) \cup V_v(G)$. $$\begin{aligned}
\bigcup_{i=0}^{2k} V_u(C_i)
&= \bigcup_{i=0}^{2k} \{u_{i+jt} \mid 0 \leq j < p\} \\
&= \bigcup_{i=0}^{2k} \{u_{i+jq(2k+1)} \mid 0 \leq j < p\} \\
&= \bigcup_{i=0}^{2k} \{u_{i+j(2k+1)} \mid 0 \leq j < p\} \\
&= \bigcup_{i=0}^{2k} \{u_m \mid m \in \mathbb{Z}_n, m \equiv i\ (\text{mod}\ 2k+1)\} \\
&= \{u_m \mid m \in \mathbb{Z}_n\}
\end{aligned}$$ By symmetry, we have $$\begin{aligned}
\left(\bigcup_{i=0}^{2k} V_u(C_i)\right) \cup \left(\bigcup_{i=0}^{2k} V_v(C_i)\right)
&= \{u_m \mid m \in \mathbb{Z}_n\} \cup \{v_m \mid m \in \mathbb{Z}_n\} \\
&= V_u(G) \cup V_v(G)
\end{aligned}$$
Observe that both $R_i$ and $S_i$ are subgraphs of $C_i$. For all $i \in \mathbb{Z}_{2k+1}$, $R_i$ and $Q_i$ share no vertex and contain all vertices in $C_i$ since the first vertex of $R_i$ and the last vertex of $R_i$ are respectively adjacent to the first vertex of $S_i$ and the last vertex of $S_i$. Therefore, paths $R_0,R_1,\dots,R_{2k},S_0,S_1,\dots,S_{2k}$ contain all of $V_u(G) \cup V_v(G)$. This completes the proof of Theorem \[thm\]. [$\blacksquare$]{}
[1]{} url \#1[`#1`]{}urlprefixhref \#1\#2[\#2]{} \#1[\#1]{}
M. E. Watkins, A theorem on [T]{}ait colorings with an application to the generalized [P]{}etersen graphs, J. Combin. Theory 6 (1969) 152–164.
F. Castagna, G. Pins, Every generalized [P]{}etersen graph has a [T]{}ait coloring, Pacific J. Math 40 (1972) 53–58.
B. Alspach, The classification of [H]{}amiltonian generalized [P]{}etersen graphs, J. Combin. Theory Ser. B 34 (1983) 293–312.
Y. Y. Xueliang Fu, B. Jiang, On the domination number of generalized [P]{}etersen graphs ${P}(n,2)$, Discrete Math 309 (2009) 2445–2451.
J.-X. Zhou, Y.-Q. Feng, Cubic vertex-transitive non-[C]{}ayley graphs of order $8p$, Electron. J. Combin 19 (2012) [\#]{}P53.
J.-X. Zhou, Y.-Q. Feng, Cubic bi-[C]{}ayley graphs over abelian groups, European J. Combin 36 (2014) 679–693.
K. Kutnar, P. Petecki, On automorphisms and structural properties of double generalized [P]{}etersen graphs, Discrete Math 339 (2016) 2861–2870.
|
---
abstract: |
#### Background: {#background .unnumbered}
In recent years automated data analysis techniques have drawn great attention and are used in almost every field of research including biomedical. Artificial Neural Networks (ANNs) are one of the Computer- Aided- Diagnosis tools which are used extensively by advances in computer hardware technology. The application of these techniques for disease diagnosis has made great progress and is widely used by physicians. An Electrocardiogram carries vital information about heart activity and physicians use this signal for cardiac disease diagnosis which was the great motivation towards our study.
#### Methods: {#methods .unnumbered}
In this study we are using Probabilistic Neural Networks (PNN) as an automatic technique for ECG signal analysis along with a Genetic Algorithm (GA). As every real signal recorded by the equipment can have different artifacts, we need to do some preprocessing steps before feeding it to the ANN. Wavelet transform is used for extracting the morphological parameters and median filter for data reduction of the ECG signal. The subset of morphological parameters are chosen and optimized using GA. We had two approaches in our investigation, the first one uses the whole signal with 289 normalized and de-noised data points as input to the ANN. In the second approach after applying all the preprocessing steps the signal is reduced to 29 data points and also their important parameters extracted to form the ANN input with 35 data points.
#### Results: {#results .unnumbered}
The outcome of the two approaches for 8 types of arrhythmia shows that the second approach is superior than the first one with an average accuracy of %99.42 .
#### Conclusions: {#conclusions .unnumbered}
We have studied 8 types of arrhythmia with high detection accuracy. In the literature, previous attempts are made on 6 types of arrhythmias. The results of the PNN shows that its performance for reduced input signal along with the morphological parameter has the best performance. Also it was noticed that the proper selection of training and testing data sets are of great importance and all the beats of each arrhythmia should not be selected from a single file in the database.
address: |
(1)Department of Physics, Zanjan University, Zanjan 313, Iran\
(2)Zanjan University of Medical Sciences, Zanjan, Iran\
(3)Department of Computer Engineering, Islamic Azad Univeristy, Zanjan Branch, Zanjan, Iran\
(4)Institute for Advance Studies in Basic Sciences, Zanjan, Iran
author:
- 'M. Bazarghan$^{1}$'
- 'Y. Jaberi $^2$'
- 'R. Amandi $^3$'
- 'M. Abedi $^4$'
bibliography:
- 'ECG\_article.bib'
title: Automatic ECG Beat Arrhythmia Detection
---
\[1995/12/01\]
Background {#background-1 .unnumbered}
==========
The electrocardiogram (ECG) inherently carries important information on functionality of the heart. This signal provides a physician with crucial information on a patient’s heart function and can be used for the prognosis and diagnosis of heart disease. It is one of the most common signals used in diagnosis because of its non-invasive nature and the valuable information it contains. Its analysis can be used to judge the pathophysiological condition of the heart. Several systems have been developed for ECG recording and analysis. Early ECG systems were just recording the signal by printing it. New systems use computer technology to provide automated diagnosis. The latter is a large research field and many methods and approaches have been proposed and implemented for the detection of ischemia, arrhythmia detection and classification, and diagnosis of chronic myocardial diseases. Those methods usually include processing of the signal and removing noise and artifacts, extracting certain key features related to diseases, and analyzing the features to make the final decision. The analysis is usually done by using signal processing, artificial neural networks, and fuzzy logic concepts along with the clinical symptoms provided by medical experts. The performance of those systems is evaluated using standard databases.\
The normal activity of the heart starts at a region in the upper wall of the right atrium known as the Sino Atrial (SA) node or the heart’s pacemaker, and through both atria forwards to the Atrioventricular Node (AV) and enters the ventricle from the AV node. There are different types of heart rhythm disorders or arrhythmias. In cardiac arrhythmias, the electrical activity usually starts at a location other than the sinus node or the propagation and its speed is abnormal. Tachycardia and ventricular fibrillation are examples of dangerous arrhythmias. Atrial premature beats are other examples of cardiac arrhythmia.\
The mechanical and electrical activity of the heart is periodic. The electrical activity of the heart at each cycle generates recordable potential differences of, 300 to 1000 ms during which the voltage as well as the direction changes rapidly. These electrical vectors can be recorded by fixing the electrodes on the patients’ skin. The ECG record on the skin in one direction will not give enough information because generated electrical vectors by the heart are changing and moving in three dimensional space. That is why ECG has 12 vectors recording signals in 12 dimensional space where 6 of them are in the horizontal plane and the other 6 in the frontal plane; they are all called ECG leads. These leads record signal by placing electrodes on the patient’s upper and lower limbs and chest. A physician analyses the ECG by having prior knowledge on the space angle of each lead and looking at the plotted signal of the leads.\
A normal ECG signal includes a set of waves and each one represents the activation or inactivation of certain anatomical parts of the heart. These waves are called P, QRS, and T waves which are shown in Figure \[nsr\]. The P wave is the result of activation of the atria and the QRS wave by activation of the ventricles. As the activation of QRS wave is a complex process, so the QRS wave will also be complex. The ventricle inactivation produces the T wave. A wave corresponding to atrial inactivation occurs, but is ordinarily buried in the QRS complex, and is not identifiable in the ECG. Arrhythmia detection and some heart disease diagnosis is done by ECG analysis and is usually done by the heart specialist.\
![The sample normal beat showing different morphological parameters.[]{data-label="nsr"}](nsr.eps){width="8cm"}
Over the last few decades with the entering of computers into our daily lives, they’ve come to play more important roles in medical science. One of the main areas that computers entered in medicine is disease diagnosis. The Computer Aided Diagnosis (CAD) has made great progress in the last two decades [@Yang; @Kunio; @Steward; @Sutton]. Computers have drawn the attention of physicians and engineers in the areas like Lung Cancer [@Lung], lung disease [@Lung1; @Lung2; @Lung3], breast cancer [@Breast] Colonography and polyp [@Polyp1; @Polyp2]. The first attempts to use computers in medicine for automatic diagnosis started in the 70s, and later in the 90s with further studies led to CAD. The computer processing results used for disease diagnosis are nowadays not used in place of physicians but beside them; meaning, the physician gives the final diagnosis with the help of physician assistant tools. One of the important tools in the CAD is Artificial Neural Networks. The ANN is a system loosely modeled based on the human brain and has wide applications in almost every field of science including medical science. ANNs with the ability to recognize patterns and processing large amount of data can help physicians in disease diagnosis [@IEEEmed]. Nowadays, ANNs are widely used in the diagnosis of, Ovarian cancer [@Ovarian], Prostate Cancer [@Prostate], Pancreatic disease [@Pancreatic], EEG abnormality detection [@EEG], ECG signal [@ECG]. The number of published papers on applications of ANNs in medicine has increasingly risen in the recent years [@Brause; @Annrev1; @Annrev2].\
The ANN uses non-linear mathematical models to solve problems. As humans use prior information to solve new problems, ANNs also use solved examples to construct neuron systems to recognize new patterns [@eftekhari]. In recent years with the advances in computer processing capabilities, ANNs have drawn great attention in different areas such as; space science, transportation, gas and petroleum, and medicine.\
The analysis described here uses ANNs for ECG signal analysis and cardiac arrhythmia diagnosis. The methods section describes the data and the abnormalities present in them. It gives the preprocessing steps and explains the parameter extraction of the data and give the classification methods. The last sections give the results and conclusions.
Methods {#methods-1 .unnumbered}
=======
Data {#data .unnumbered}
----
There are different databanks of cardiac signals. In this work we have used the MIT-BIH database [@MIT-BIH]. This databank consists of 48 signalsrecords of 30 minutes from 48 different patients. The sampling frequency of this data is 360Hz and the sampling time rate is 2.7ms. Approximately 60 percent of these records are for patients with arrhythmia. 25 of 48 records are less common but clinically significant arrhythmias. These data are recorded by Holter [@Gao]. In this work we use the data recorded by lead 2. Each 30 minute long record is a continuous waveform and comes with annotations for each heartbeat. These annotations represent the type of arrhythmia. Since we were interested in only certain types of arrhythmias in the database, atfor each of the 48 records, we extracted interesting individual beats with their associated annotations. The list of arrhythmias which we have investigated in this work with their annotations are given in Table1.\
A total of 2800 beats were extracted with 350 beats belonging to each category viz. Normal, Paced, Premature, Escape, Fusion of paced and normal, Fusion of ventricular and normal, Right Bundle Branch Block and Left Bundle Branch Block. Six different datasets were formed from the complete data. The sample plot of each eight arrhythmias under our consideration is given in Figure \[arryt\].
![Sample signals for eight arrhythmias under consideration.[]{data-label="arryt"}](arryt.eps){width="16cm"}
Pre-processing {#pre-processing .unnumbered}
--------------
The analysis of these data requires some preprocessing steps. The performance of the algorithms applied to the data is very much dependent on these preprocessing steps which are explained below.
### Beat separation
The first step in the preprocessing of signals in this application is to extract the beats for classification of the normal and abnormal beats. For extracting the beats we need to first find the peak of the ‘R’ wave then take 200ms before and 600ms after this peak value as a complete beat. According to practitioners the information for arrhythmia detection lies in this 800ms period, and other parts of the signal do not carry significant information. This 800ms signal usually consists of the entire P, QRS, and T waves.
### Normalization
Normalization is one of the important preprocessing steps, and an ANN with normalized input vectors gives better results and converges faster. Here we have used the Min-Max normalization method. In this method a linear mapping is done, in which, the maximum value among vector elements is mapped to 1 and the minimum is mapped to 0. In other words, the components of normalized vector ($v'_i$) can be found as: $$V'_i=\frac{V_i-min}{max-min}$$ where, $V_i$s are the components of the input vector. And min and max are the minimum and maximum value of the components of the vector respectively[@Dmandkd].
### Data reduction
If the sampling time steps are small, then the size of each heart beat data will be large. The larger the data size, the larger the memory required and the longer the time for processing it. So, it is desired to reduce the size of the data as much as possible. For which, the available filtering methods can be used, and we are using median filter. The median filter is actually replacing the number of consecutive data points with their median value. We are using each 10 consecutive data points of signal and replacing them with their median value to reduce the data size and is shown in Figure \[median\]. In this way our input vectors with 289 data points reduce to vectors with only 29 data points. This leads to not only data size reduction but also reduces the signal’s noise, because, by applying this filtering method the instant noise present in the signal vanishes. Furthermore, data size reduction will also reduce the complexity of the ANNs and hence its confusion, which can lead to faster convergence of the ANNs.
![Median algorithm used for data reduction and 10 consecutive points are replaced by a single point with their mean value.[]{data-label="median"}](median.eps){width="16cm"}
### De-noising
ECG signals are recorded using surface Electromyography(EMG). Voltage sensors are placed on the skin in the vicinity of the heart. The signal measurement and the initial signal processing with the instrument creates some noise. The presence of noise in ECG signals may cause problems in identifying the arrhythmia.\
The five main sources of noise are, power line interference, EMG noise, electrode contact noise, motion artifacts and instrumentation noise. Usually filters are used to remove unwanted noise artifacts.\
The power line interference primarily consists of two mechanisms: capacitive and inductive coupling. Capacitive coupling is due to an undesired capacitor formed between two elements of a circuit in which the energy transfer occurs. Inductive coupling on the other hand is caused by mutual inductance between two conductors of the elements.\
Electrode contact noise is caused by changes in the position of the electrodes on the skin and also the medium between the heart and the electrodes. These changes affect the impedance, meanwhile the impedance in the circuits is assumed to be constant to measure the voltages. If the electrode-skin impedance is large enough, the relative impedance change cannot shift the baseline of the ECG signal, but in practice it is not.\
The position with respect to the heart should be constant but movements of the heart and also the electrode on the skin cause motion artifacts. As the movements are slow these artifacts cause peaks in the Fourier power spectrum in low frequencies compared to the frequency of the heart rate.\
EMG noise is due to the contraction of other muscles besides the heart. This contraction generates depolarization and repolarization waves near the electrodes that is picked up and summed into the main waves of the heart signal.\
The presence of noise in all electrical elements causes the Instrumentation noise. The electrode probes, cables and amplifier are the primary elements for ECG. These three elements and any other element like Analog-to-Digital converter and Signal processor contain white Gaussian noises themselves.\
These noise in the ECG signal are to be removed for the automatic disease diagnosis. In most of the signal processing cases, de-noising is one of the important and required steps for proper signal processing. Here we used Wavelet transform for de-noising[@Waveletdenoise] and the first order Coiflet from Matlab toolbox is used for this transformation which is explained more in detail in the next section.
ECG parameters {#ecg-parameters .unnumbered}
--------------
In addition to the recorded signal and using it as input to the ANN, the important extracted features from the signal can also be concatenated. These features, which are shown in Figure \[nsr\], play a key role in disease diagnosis. Each single bit can be characterized with the following parameters; QRS duration, QRS amplitude, RR interval, PR segment, PR interval, P amplitude, ST segment, ST interval, QT interval, T amplitude. In fact the type of abnormality and cardiac arrhythmia directly influences these parameters and the duration or amplitude of each wave will be affected correspondingly.
### Feature set selection using Genetic Algorithm
The main requirement for diagnosis systems is to achieve high prediction accuracy. Furthermore, a classification learning algorithm is expected to have short training and prediction times. In real classification problems one needs to choose a subset of features from a much larger set that represents the knowledge to be used in the classification. This is because of the fact that the performance of classifier and the cost of classification are sensitive to the selection of the features used in the construction of the classifier. By reducing the set of features, the time required for learning the classification knowledge and the time needed for classification reduces [@Featurega]. Further, by the extraction of relevant features and therefore the elimination of the irrelevant ones, the accuracy of the classifier can be increased. Exhaustive evaluation of possible feature subsets is usually infeasible and not recommended in practice because it requires large amounts of computational effort. This is where the Genetic Algorithms (GAs) can be very handy and offer an attractive approach to find near-optimal solutions to such optimization problems. GAs are adaptive heuristic search used to solve optimization problems guided by the principles of evolution and natural genetics [@Clever]. In GAs, the parameters of the search space (in our example, 10 ECG morphological parameters) are encoded in the form of strings, called chromosomes. A collection of such strings is called a population. In the case of feature selection problem, each chromosome represents a subset of features selected. The number of these features defines the size of a chromosome. Each element of the chromosome string can take the value 1 or 0, where 1 indicates that the corresponding feature is selected, and 0 otherwise. The goal of the search, in our case, is to find a chromosome that represents a set of features that lead to highest accuracy. If the result of the search is the same for multiple feature subsets and all have the best accuracy, the one with the smallest cardinality is the desired one.\
The result of the GA search gives the highest accuracy for 6 and 9 feature subsets. But, the smallest chromosome string is selected as the desired feature subset. The selected feature subset is given in Table2 and clearly shows that the features related to the T wave is ignored. The medical reason behind this fact is that, in the diagnosis of arrhythmias under our consideration the T wave has no impact and usually is not used by practitioners. The parameters which are important for the diagnosis of arrhythmias under investigation are; depolarization of atrium and ventricle, and the relation between these two depolarizations. In other words, as repolarization has no role in heart rhythm detection so the T wave which is the product of repolarization will also lose its importance for our diagnosis.
### Working principle
GA begins with a set of solution and along with the ECG beat is given to the ANN and the output of the ANN is compared with the desired output and hence the accuracy is calculated. If the accuracy is below the accepted level, the feedback loop is given to the GA block, to select the new population and the process repeats until the output error is minimized and the highest accuracy is achieved. Figure \[GAfinal\]-a shows the error plot of the feature subset selection process and Figure \[GAfinal\]-b shows a block diagram of the system.
![(a) the fitness and (b) block diagram of the system[]{data-label="GAfinal"}](GAfinal.eps){width="16cm"}
### Parameter extraction
The ECG feature extraction plays a very important role in disease diagnosis. The amplitudes and interval values of P-QRS-T waves determine the functionality of the heart of every human and the majority of the clinically important information in the ECG is in the intervals and amplitudes of these waves. An accurate diagnosis of arrhythmia will be possible by the precise detection of P, QRS and T waves. If the beat is similar to a normal beat it is easy to detect the waves by eye, but it could be rather challenging in cases of an abnormal shape of the beat. The waves are studied by some parameters of amplitude and time. Some techniques have been proposed for automatic detection of these parameters [@Pantompkins; @Sahoo]. In this work we use the wavelet transform technique. The wavelet transform is a transformation between time and frequency domains. In contrast to some transformation like Fourier transform, wavelet could be localized both in time and frequency[@Waveletdenoise]. The main equation of the transformation is:
$$T(a,b)=\frac{1}{\sqrt{a}} \int _{-\infty} ^{ \infty} x(t)\psi^* ( \frac{t-b}{a})dt$$
$\psi$ is called the ‘mother wavelet’ and has to satisfy some conditions[@Addison]. Parameters *a* and *b* are the width of the signal and the time localization center respectively. In 1989 R. Coifman introduced a new class of wavelets which are called coiflet [@Dau]. In this class of wavelet the mother wavelet must have zero momentum. Another parameter in coiflet transform is order of coiflet. The order of coiflet is chosen in a way that the transformed data best fits the original data. The numerical procedures are done in Matlab using wavelet toolbox and the coiflet transform of order 1 is used [@Coiflet1]. Fig \[QRSestimation\] shows a transformed beat.
![QRS detection by coiflet transform of order 1.[]{data-label="QRSestimation"}](qrsestimation.eps){width="16cm"}
Using this transformation, Q, R, and S are determined; and knowing the approximate distance of P from these points, P is determined. Using this procedure the values in Table2 are obtained which are known as morphological parameters. These parameters and the ECG beat will serve as the final input to the neural network.\
Using wavelet transform, the small frequencies present in the signal are removed. So wavelet transform could be considered as a method to remove noise from the ECG signal as well.
Neural networks and classification {#neural-networks-and-classification .unnumbered}
----------------------------------
We have used Probabilistic Neural Networks (PNN) among the ANN algorithms for classification of input data. This algorithm learns the prediction of Probability Density Function (PDF). It’s architecture consist of four layers, shown in Figure \[PNN\]. The first layer is the input layer and fully connected to the next layer. Input vector $X = (x_1, x_2, ..., x_n ) \in \Re^n$ is applied to *n* neurons of input layer. The second layer in which the input vectors are stored, is called the pattern layer. In the second layer including n\*K neurons, the distance between the input vector and each one of K training examples is calculated; The third layer is the summation layer, which has K elements. Each element in this layer combines via processing elements in the pattern layer through the following estimator: $$S_k (X) = \frac{1}{(2 \pi \sigma^2)^{n/2}}\frac{1}{N_k}\sum_{i=1}^{N_k}\exp \left( -
\frac{\parallel X - X_{k,i} \parallel^2}{2 \sigma^2} \right).$$ where $\sigma$, known as the spread or smoothing parameter is the deviation of the Gaussian function and $ X_{k,i} \in \Re ^n $ is the center of the kernel. Finally, the output layer selects the neuron in the summation layer with the maximum output[@Spetch]. $$C(X)=\rm arg \ max\it(_{\substack{\\ \\ \hskip -0.8 cm 1 \le k \le
K}} S_k).$$
![Structure of the Probabilistic Neural Network.[]{data-label="PNN"}](PNN.eps){width="16cm"}
Results and Discussion {#results-and-discussion .unnumbered}
======================
We have developed an automated system that works with the Probabilistic Neural Network algorithm and gives a promising result; which we evaluated based on Sensitivity, Specificity, and Accurcy which are defined as the following.
$$Sensitivity=\frac{TP}{TP+FN}$$ $$Specificity=\frac{TN}{TN+FP}$$ $$Accuracy=\frac{TN+TP}{TN+TP+FN+FP}$$
One can find the definitions for TP,TN,FP and FN in Table4. Depending on the input type we had two approaches to deal with.
First approach {#first-approach .unnumbered}
--------------
The simplest way is to give the whole information of a beat (the voltage-time), that contains 289 data points, as an input to the ANN. Before giving it to the network, we applied normalization and de-noising on the data and the result is given in Table3-1. As it is shown in the table we used different ratios of beats for training and testing the PNN. For example in feature set I, 250 beats out of 350 are selected for the training set and the rest kept as the testing set. Similarly, the ratio of 300 to 50 for feature set II, ratio of 150 to 200 for feature set III and finally the ratio is 200 to 150 for feature set IV.
Second approach {#second-approach .unnumbered}
---------------
In this scheme all the preprocessing steps are taken and the final reduced ECG beat contains 29 data points. We have also extracted the morphological parameters of each beat and then the optimized feature subset is chosen. The addition of the ECG beat and the selected morphological parameters will form our input vector to the neural network with 35 data points. The result of this method is given in Table3-2. The number of train and test sets for Table3-2 is the same as the previous section.
As it appears in Table3-1 and Table3-2 the results obtained by the second approach has a higher accuracy.\
In this study we are working on 8 types of arrhythmia whereas previous attempts are made on 6 types. By increasing the number of arrhythmias under consideration the risk of misclassification can increase. But, the highest classification accuracy achieved by the second approach is %99.42 which we claim is the highest accuracy for 8 types of arrhythmia ever made.
Conclusions {#conclusions-1 .unnumbered}
===========
In this paper ANN is applied for the classification of ECG beat arrhythmias. The results show a high rate of accuracy in 8 classes of arrhythmias using probabilistic neural networks(PNN). The selection of ECG beats used in training set plays an important role. This set has to contain combination of different arrhythmias and all their possible variations from different patients to achieve good results. Two types of input vectors to the ANN are considered. First the whole ECG signal is taken as input; in which case the results are good. In the second input type the reduced data along with the important parameters of ECG beat is considered as input; this way the ANN has better performance because of the reduced overhead. The time and size of the network is also reduced and the accuracy acheived this way is higher than the first input data type. Wavelet transform of type coilflet is used to extract the parameters from ECG waves. It is the most efficient compared to other methods. The set of parameters fed to the network is also very important. To choose the optimum set of parameters among all parameters Genetic Algorithm is used. GA selects the best parameters without affecting the performance of ANN. The set of parameters chosen by GA is in good agreement with the set suggested by physicians to detect arrhythmia type.\
Similar methods could be used to study the anatomical disease and also classification of EEG signals. We are planing to sudy the brain related disease in future, using MRI images and the EEG signal.
Acknowledgements {#acknowledgements .unnumbered}
================
This study was financially supported by Iran National Science Foundation(Project No 90000947); and also the authors would like to thank Prof. Y. Sobouti and Dr. A. Biglari for initiating this project and establishing the collaboration with Zanjan University of Medical Sciences.
Figures {#figures .unnumbered}
=======
Figure 1 - Sample Normal Beat {#figure-1---sample-normal-beat .unnumbered}
-----------------------------
The sample normal beat showing different morphological parameters.
Figure 2 - Sample signals for eight arrhythmias {#figure-2---sample-signals-for-eight-arrhythmias .unnumbered}
-----------------------------------------------
Sample signals for eight arrhythmias under consideration.
Figure 3 - Filtered signal with median algorithm {#figure-3---filtered-signal-with-median-algorithm .unnumbered}
------------------------------------------------
Median algorithm used for data reduction and 10 consecutive points are replaced by a single point with their mean value.
Figure 4 - Important morphological parameter selection {#figure-4---important-morphological-parameter-selection .unnumbered}
-------------------------------------------------------
\(a) The fitness and (b) block diagram of the system
Figure 5 - The QRS estimation {#figure-5---the-qrs-estimation .unnumbered}
-----------------------------
The result of Wavelet transform (Coiflet1) on sample ECG signal for the QRS estimation
Figure 6 - The PNN architecture {#figure-6---the-pnn-architecture .unnumbered}
-------------------------------
The architecture of the Probabilistic Neural Network
Tables {#tables .unnumbered}
======
Table 1 - Types of arrhythmia {#table-1---types-of-arrhythmia .unnumbered}
-----------------------------
The eight types of arrhythmia
\[Table1\]
symbol arrythmia
-------- ------------------------------------------------
N Normal beat
/ Paced beat
A Premature beat (atrial,aberrated atrial,nodal)
E Escape beat(ventricular,atrial,nodal)
f Fusion of paced and normal beat
F Fusion of ventricular and normal beat
L Left bundle branch block beat
R Right bundle branch block beat
Table 2 - ECG Parameters {#table-2---ecg-parameters .unnumbered}
------------------------
The ECG optimized morphological parameters using Genetic Algorithm.
\[Table2\]
Feature Duration
--------------- --------------
P wave 60-80(ms)
PR-segment 50-120(ms)
PR- interval 120-200(ms)
QRS duration 80-120(ms)
QRS amplitude 0.25-1(v)
RR-interval 400-1200(ms)
Table 3-1 - The result of first approach {#table-3-1---the-result-of-first-approach .unnumbered}
----------------------------------------
The results of first approach for different ratios of training and testing data sets as input to the PNN.\
Table 3-2 - The result of second approach {#table-3-2---the-result-of-second-approach .unnumbered}
-----------------------------------------
The results of second approach for different ratios of training and testing data sets as input to the PNN.
----------- --------- --------- -------- --------- --------- -------- --------- --------- -------- --------- --------- --------
ECG
beat type Sens(%) Spec(%) Acc(%) Sens(%) Spec(%) Acc(%) Sens(%) Spec(%) Acc(%) Sens(%) Spec(%) Acc(%)
N 99.00 100.0 99.72 100.00 99.41 99.48 100.00 99.70 99.74 100.00 99.57 99.63
R 80.00 94.35 95.08 100.00 99.70 99.74 94.00 98.99 98.36 100.00 98.98 99.11
L 88.00 100.0 98.39 100.00 100.00 100.00 99.00 100.00 99.87 99.20 99.69 99.63
F 96.00 98.91 98.52 96.00 99.70 99.23 99.00 99.85 99.74 88.00 98.64 97.28
f 96.00 99.84 99.32 100.00 98.82 98.87 100.00 99.85 99.87 98.40 99.22 99.11
E 93.00 99.22 98.39 90.00 98.84 97.72 95.00 100.00 99.36 84.40 99.46 97.68
A 63.00 98.10 93.63 88.00 100.00 98.47 96.00 99.13 98.73 90.40 99.28 98.14
/ 100.0 100.0 100.0 100.0 99.70 99.74 99.00 99.85 99.74 99.60 98.98 99.06
Average 91.97 98.80 97.88 96.75 99.52 99.16 97.75 99.67 99.42 95.00 99.22 98.71
----------- --------- --------- -------- --------- --------- -------- --------- --------- -------- --------- --------- --------
Table 4 - Definitions of TP, TN, FP, FN {#table-4---definitions-of-tp-tn-fp-fn .unnumbered}
---------------------------------------
Definition of TP, TN, FP, FN
\[Table4\]
---------------------------- ------------------- ----------------------- --------------------
Sample is A type. Sample is not A type.
Outcome of network weather True True Poseitive(TP) False Positive(FP)
the sample is A type. False False Negative(FN) True Negative(TN)
---------------------------- ------------------- ----------------------- --------------------
|
---
abstract: 'We study a generalization of the order-preserving pattern matching recently introduced by Kubica et al. (Inf. Process. Let., 2013) and Kim et al. (submitted to Theor. Comp. Sci.), where instead of looking for an exact copy of the pattern, we only require that the relative order between the elements is the same. In our variant, we additionally allow up to $k$ mismatches between the pattern of length $m$ and the text of length $n$, and the goal is to construct an efficient algorithm for small values of $k$. Our solution detects an order-preserving occurrence with up to $k$ mismatches in ${\mathcal{O}}(n(\log\log m+k\log\log k))$ time.'
author:
- 'Paweł Gawrychowski and Przemysław Uznański[^1]'
bibliography:
- 'biblio.bib'
title: |
Order-preserving pattern matching\
with $k$ mismatches
---
Introduction
============
*Order-preserving pattern matching*, recently introduced in [@Kim] and [@KubicaOrder], and further considered in [@CrochemoreOrder], is a variant of the well-known pattern matching problem, where instead of looking for a fragment of the text which is identical to the given pattern, we are interested in locating a fragment which is order-isomorphic with the pattern. Two sequences over integer alphabet are *order-isomorphic* if the relative order between any two elements at the same positions in both sequences is the same. Similar problems have been extensively studied in a slightly different setting, where instead of a fragment, we are interested in a (not necessarily contiguous) subsequence. For instance, pattern avoidance in permutations was of much interest.
For the order-preserving pattern matching, both [@Kim] and [@KubicaOrder] present an ${\mathcal{O}}(n+m\log m)$ time algorithm, where $n$ is the length of the text, and $m$ is the length of the pattern. Actually, the solution given by [@KubicaOrder] works in ${\mathcal{O}}(n+\text{sort}(m))$ time, where $\text{sort}(m)$ is the time required to sort a sequence of $m$ numbers. Furthermore, efficient algorithms for the version with multiple patterns are known [@CrochemoreOrder]. Also, a generalization of suffix trees in the order-preserving setting was recently considered [@CrochemoreOrder], and the question of constructing a forward automaton allowing efficient pattern matching and developing an average-case optimal pattern matching algorithm was studied [@Vialette].
Given that the complexity of the exact order-preserving pattern matching seems to be already settled, a natural direction is to consider its approximate version. Such direction was successfully investigated for the related case of *parametrized pattern matching* in [@MosheParam], where an ${\mathcal{O}}(nk^{1.5}+mk\log m)$ time algorithm was given for parametrized matching with $k$ mismatches.
We consider a relaxation of order-preserving pattern matching, which we call *order-preserving pattern matching with $k$ mismatches*. Instead of requiring that the fragment we seek is order-isomorphic with the pattern, we are allowed to first remove $k$ elements at the corresponding positions from the fragment and the pattern, and then check if the remaining two sequences are order-isomorphic. In such setting, it is relatively straightforward to achieve running time of ${\mathcal{O}}(nm\log\log m)$, where $n$ is the length of the text, and $m$ is the length of the pattern. Such complexity might be unacceptable for long patterns, though, and we aim to achieve complexity of the form ${\mathcal{O}}(n f(k))$. In other words, we would like our running time to be close to linear if the bound on the number of mismatches is very small. We construct a deterministic algorithm with ${\mathcal{O}}(n(\log\log m+k\log\log k))$ running time. At a very high level, our solution is similar to the one given in [@MosheParam]. We show how to filter the possible starting positions so that a position is either eliminated in ${\mathcal{O}}(f(k))$ time, or the structure of the fragment starting there is simple, and we can verify the occurrence in ${\mathcal{O}}(f(k))$ time. The details are quite different in our setting, though.
A different variant of approximate order-preserving pattern matching could be that we allow to remove $k$ elements from the fragment, and $k$ elements from the pattern, but don’t require that they are at the same positions. Then we get order-preserving pattern matching with $k$ errors. Unfortunately, such modification seems difficult to solve in polynomial time: even if the only allowed operation is removing $k$ elements from the fragment, the problem becomes NP-complete [@Bose].
Overview of the algorithm
=========================
Given a text $(t_{1},\ldots,t_{n})$ and a pattern $(p_{1},\ldots,p_{m})$, we want to locate an order-preserving occurrence with at most $k$ mismatches of the pattern in the text. Such occurrence is a fragment $(t_{i},\ldots,t_{i+m-1})$ with the property that if we ignore the elements at some up to corresponding $k$ positions in the fragment and the pattern, the relative order of the remaining elements is the same in both of them.
The above definition of the problem is not very convenient to work with, hence we start with characterising $k$-isomorphic sequences using the language of subsequences in Lemma \[lemma:subseq\]. This will be useful in some of the further proofs and also gives us a polynomial time solution for the problem, which simply considers every possible $i$ separately. To improve on this naive solution, we need a way of quickly eliminating some of these starting positions. For this we define the signature $S(a_{1},\ldots,a_{m})$ of a sequence $(a_{1},\ldots,a_{m})$, and show in Lemma \[lemma:implication\] that the Hamming distance between the signatures of two $k$-isomorphic sequences cannot be too large. Hence such distance between $S(t_{i},\ldots,t_{i+m-1})$ and $S(p_{1},\ldots,p_{m})$ can be used to filter some starting positions where a match cannot happen.
In order to make the filtering efficient, we need to maintain $S(t_{i},\ldots,t_{i+m-1})$ as we increase $i$, i.e., move a window of length $m$ through the text. For this we first provide in Lemma \[lemma:prune\_structure\] a data structure which, for a fixed word, allows us to maintain a word of a similar length under changing the letters, so that we can quickly generate the first $k$ mismatches between subwords of the current and the fixed word. The structure is based on representing the current word as a concatenation of subwords of the fixed word. Then we observe that increasing $i$ by one changes the current signature only slightly, which allows us to apply the aforementioned structure to maintain $S(t_{i},\ldots,t_{i+m-1})$ as shown in Lemma \[lemma:prune\]. Therefore we can efficiently eliminate all starting positions for which the Hamming distance between the signatures is too large.
For all the remaining starting positions, we reduce the problem to computing the heaviest increasing subsequence, which is a weighted version of the well-known longest increasing subsequence, in Lemma \[lemma:reduction\]. The time taken by the reduction depends on the Hamming distance, which is small as otherwise the position would be eliminated in the previous step. Finally, such weighted version of the longest increasing subsequence can be solved efficiently as shown in Lemma \[lemma:heaviest\]. Altogether these results give an algorithm for order-preserving pattern matching with $k$ with the cost of processing a single $i$ depending mostly on $k$.
An implicit assumption in this solution is that there are no repeated elements in neither the text nor the pattern. In the last part of the paper we remove this assumption while keeping the same time complexity. At a high level the algorithm remains the same, but a few carefully chosen modifications are necessary. First we further generalize the heaviest increasing subsequence into heaviest chain in a plane, and in Lemma \[lemma:heaviest chain\] how to solve this version efficiently. Then we modify the definition of a signature, and prove in Lemma \[lemma:reduction2\] that after such change checking if two sequences are order-isomorphic can be reduced to computing the heaviest chain.
Preliminaries
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We consider strings over an integer alphabet, or in other words sequences of integers. Two such sequences are *order-isomorphic* (or simply *isomorphic*), denoted by $(a_{1},\ldots,a_{m})\sim (b_{1},\ldots,b_{m})$, when $a_{i} \leq a_{j}$ iff $b_{i} \leq b_{j}$ for all $i,j$. We will also use the usual equality of strings. Whenever we are talking about sequences, we are interested in the relative order between their elements, and whenever we are talking about strings consisting of characters, the equality of elements is of interest to us. For two strings $s$ and $t$, their *Hamming distance* ${\mathrm{H}}(s,t)$ is simply the number of positions where the corresponding characters differ.
Given a text $(t_{1},\ldots,t_{n})$ and a pattern $(p_{1},\ldots,p_{m})$, the *order-preserving pattern matching* problem is to find $i$ such that $(t_{i},\ldots,t_{i+m-1}) \sim (p_{1},\ldots,p_{m})$. We consider its approximate version, i.e., order-preserving pattern matching with $k$ mismatches.
We define two sequences *order-isomorphic with $k$ mismatches*, denoted by $(a_{1},\ldots,a_{m}){\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_{1},\ldots,b_{m})$, when we can select (up to) $k$ indices $1\leq i_{1} < \ldots < i_{k}\leq m$, and remove the corresponding elements from both sequences so that the resulting two new sequences are isomorphic, i.e., $a_{j} \leq a_{j'}$ iff $b_{j} \leq b_{j'}$ for any $j,j' \notin\{i_{1},\ldots,i_{k}\}$. In *order-preserving pattern matching with $k$ mismatches* we want $i$ such that $(t_{i},\ldots,t_{i+m-1}) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(p_{1},\ldots,p_{m})$, see Fig. \[fig:occurrence\].
![$[1,4,2,5,11]$ occurs in $[1,10,6,4,8,5,7,9,3]$ (at position 4) with $1$ mismatch.[]{data-label="fig:occurrence"}](fig_matching.pdf){width="0.8\linewidth"}
Our solution works in the word RAM model, where $n$ integers can be sorted in ${\mathcal{O}}(n\log\log n)$ time [@Han], and we can implement dynamic dictionaries using van Emde Boas trees. In the restricted comparison model, where we can only compare the integers, all $\log\log$ in our complexities increase to $\log$.
In Section \[section:algorithm\], we assume that integers in any sequence are all distinct. Such assumption was already made in one of the papers introducing the problem [@Kim], with a justification that we can always perturb the input to ensure this (or, more precisely, we can consider pairs consisting of a number and its position). In some cases this can change the answer, though[^2]. Nevertheless, using a more complicated argument, shown in Section \[section:generalization\], we can generalize our solution to allow the numbers to repeat. Another simplifying assumption that we make in designing our algorithm is that $n\leq 2m$. We can do so using a standard trick of cutting the text into overlapping fragments of length $2m$ and running the algorithm on each such fragment separately, which preserves all possible occurrences.
The algorithm {#section:algorithm}
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First we translate $k$-isomorphism into the language of subsequences.
\[lemma:subseq\] $(a_{1},\ldots,a_{m}) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_{1},\ldots,b_{m})$ iff there exist $i_1,\ldots,i_{m-k}$ such that $a_{i_1} < \ldots < a_{i_{m-k}}$ and $b_{i_1} < \ldots < b_{i_{m-k}}$.
If the sequences are $k$-isomorphic, according to the definition there exists a set of $k$ indices $j_1,j_2,\ldots,j_k$ such that $\forall_{j,j' \not\in \{j_1,\ldots,j_k\}} a_j < a_{j'} \text{ iff } b_j < b_{j'}$. We set $\{i_1,\ldots,i_{m-k}\} = \{1,\ldots,m\} \setminus \{j_1,\ldots,j_k\}$ and order its elements so that $a_{i_1} < \ldots < a_{i_{m-k}}$. Then clearly $b_{i_1} < \ldots < b_{i_{m-k}}$ as well.
From the existence of $i_1,\ldots,i_{m-k}$ such that $a_{i_1} < \ldots < a_{i_{m-k}}$ and $b_{i_1} < \ldots < b_{i_{m-k}}$ we deduce that if we choose $\{j_1,\ldots,j_k\} = \{1,\ldots,m\} \setminus \{i_1,\ldots,i_{m-k}\}$, we have that $\forall_{j,j' \not\in \{j_1,\ldots,j_k\}} a_j < a_{j'} \text{ iff } b_j < b_{j'}$.
The above lemma implies an inductive interpretation of $k$-isomorphism useful in further proofs and a fast method for testing $k$-isomorphism.
\[prop:inductive\] If $(a_{1},\ldots,a_{m}) {\overset{k+1}{\sim}} (b_{1},\ldots,b_{m})$ then there exists $(a'_{1},\ldots,a'_{m})$ such that $(a_{1},\ldots,a_{m}) {\overset{1}{\sim}} (a'_{1},\ldots,a'_{m})$ and $(a'_{1},\ldots,a'_{m}) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_{1},\ldots,b_{m})$.
\[lemma:checking\] $(a_{1},\ldots,a_{m}) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_{1},\ldots,b_{m})$ can be checked in time ${\mathcal{O}}(m \log\log m)$.
Let $\pi$ be the sorting permutation of $(a_{1},\ldots,a_{m})$. Such permutation can be found in time ${\mathcal{O}}(m \log\log m)$. Let $(b'_1,\ldots, b'_m)$ be a sequence defined by setting $b'_i := b_{\pi(i)}$. Then, by Lemma \[lemma:subseq\], $(a_1,\ldots,a_m) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_1,\ldots,b_m)$ iff there exists an increasing subsequence of $b'$ of length $m-k$. Existence of such a subsequence can be checked in time ${\mathcal{O}}(m \log\log m)$ using a van Emde Boas tree [@Hunt].
By applying the above lemma to each of the possible occurrences separately, we can already solve order-preserving pattern matching with $k$ mismatches in time ${\mathcal{O}}(n m \log\log m)$. However, our goal is to develop a faster ${\mathcal{O}}(nf(k))$ time algorithm. For this we cannot afford to verify every possible position using Lemma \[lemma:checking\], and we need a closer look into the structure of the problem.
The first step is to define the [*signature*]{} of a sequence $(a_{1},\ldots,a_{m})$. Let ${\mathrm{pred}}(i)$ be the position where the predecessor of $a_i$ among $\{a_1,\ldots,a_m\}$ occurs in the sequence (or $0$, if $a_{i}$ is the smallest element). Then the signature $S(a_{1},\ldots,a_{m})$ is a new sequence $(1-{\mathrm{pred}}(1),\ldots,m-{\mathrm{pred}}(m))$ (a simpler version, where the new sequence is $({\mathrm{pred}}(1),\ldots,{\mathrm{pred}}(m))$, was already used to solve the exact version). The signature clearly can be computed in time ${\mathcal{O}}(m\log\log m)$ by sorting. While looking at the signatures is not enough to determine if two sequences are $k$-isomorphic, in some cases it is enough to detect that they are not, as formalized below.
\[lemma:implication\] If $(a_{1},\ldots,a_{m}) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_{1},\ldots,b_{m})$, then the Hamming distance between $S(a_{1},\ldots,a_{m})$ and $S(b_{1},\ldots,b_{m})$ is at most $3k$.
We apply induction on the number of mismatches $k$.
For $k=0$, $(a_{1},\ldots,a_{m}) \sim (b_{1},\ldots,b_{m})$ iff $S(a_{1},\ldots,a_{m}) = S(b_{1},\ldots,b_{m})$, so the Hamming distance is clearly zero.
Now we proceed to the inductive step. If $(a_{1},\ldots,a_{m}) {\overset{k+1}{\sim}} (b_{1},\ldots,b_{m})$, then due to Proposition \[prop:inductive\], there exists $(a'_{1},\ldots,a'_{m})$, such that $(a'_1,\ldots,a'_m) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_{1},\ldots,b_{m})$ and $(a_1,\ldots,a_m) {\overset{1}{\sim}} (a'_{1},\ldots,a'_{m})$. Second constraint is equivalent (by application of Lemma \[lemma:subseq\]) to existence of such $i$, that $(a_1,\ldots,a_{i-1},a_{i+1},\ldots,a_m) \sim (a'_{1},\ldots,a'_{i-1},a'_{i+1},\ldots,a'_{m}).$
We want to upperbound the Hamming distance between $S(a_1,\ldots,a_m)$ and $S(a'_1,\ldots,a'_m)$. Let $j,j'$ be indices such that $a_j$ is the direct predecessor of $a_i$ and $a_{j'}$ is the direct successor of $a_i$, both taken from the set $\{a_1,\ldots,a_m\}$. Similarly, let $\ell,\ell'$ be such indices, that $a'_\ell$ is the direct predecessor, and $a'_{\ell'}$ is the direct successor of $a'_i$, both taken from the set $\{a'_1,\ldots,a'_m\}$. That is, $$\ldots < a_j < a_i < a_{j'} < \ldots$$ is the sorted version of $(a_1,\ldots,a_m),$ and $$\ldots < a'_{\ell} < a'_i < a'_{\ell'} < \ldots$$ is the sorted version of $(a'_1,\ldots,a'_m)$. The signatures $S(a_1,\ldots,a_m)$ and $S(a'_1,\ldots,a'_m)$ differ on at most 3 positions: $j'$, $\ell'$, and $i$. Thus ${\mathrm{H}}( S(a_1,\ldots,a_m), S(b_1,\ldots,b_m) )$ can be upperbounded by $${\mathrm{H}}( S(a_1,\ldots,a_m), S(a'_1,\ldots,a'_m) ) + {\mathrm{H}}( S(a'_1,\ldots,a'_m), S(b_1,\ldots,b_m) ) \le 3k + 3,$$ which ends the inductive step.
Consider the following two sequences and their signatures: $$\begin{aligned}
S(11,\mathbf{4},12,1,9,3,\mathbf{10},7,2,5,13,0,6,8) &=& (\mathbf{6},\ \ \mathbf{4},-2,\ \ \mathbf{8},9,3,\mathbf{-2},5,-5,\mathbf{-8},\mathbf{-8},0,-3,-6)\\
S(10,\mathbf{1},11,2,9,4,\mathbf{12},7,3,5,13,0,6,8) &=&(\mathbf{4},\mathbf{10},-2,\mathbf{-2},9,3,\mathbf{-4},5,-5,\mathbf{-4},\mathbf{-4},0,-3,-6).\end{aligned}$$ One can see easily that the sequences are $2$-isomorphic and the Hamming distance between their signatures is $6$.
Our algorithm iterates through $i=1,2,3,\ldots$ maintaining the signature of the current $(t_{i},\ldots,t_{i+m-1})$. Hence the second step is that we develop in the next two lemmas a data structure, which allows us to store $S(t_{i},\ldots,t_{i+m-1})$, update it efficiently after increasing $i$ by one, and compute its Hamming distance to $S(p_{1},\ldots,p_{m})$.
\[lemma:prune\_structure\] Given a string $S^P[1..m]$, we can maintain a string $S^T[1..2m]$ and perform the following operations:
1. replacing any character $S^T[x]$ in amortized time ${\mathcal{O}}(\log \log m)$,
2. generating the first $3k$ mismatches between $S^T[i..(i+m-1)]$ and $S^P[1..m]$ in amortized time ${\mathcal{O}}(k+\log \log m)$.
The structure is initialized in time ${\mathcal{O}}(m \log\log m)$.
We represent the current $S^T[1..2m]$ as a concatenation of a number of fragments. Each fragment is a subword of $S^P$ (possibly single letter) or a special character \$ not occurring in $S^P$. The starting positions of the fragments are kept in a van Emde Boas tree, and additionally each fragment knows its successor and predecessor. In order to bound the amortized complexity of each operation, we maintain an invariant that every element of the tree has $2$ credits available, with one credit being worth ${\mathcal{O}}(\log\log m)$ time. We assume that given any two substrings of $S^P$, we can compute their longest common prefix in ${\mathcal{O}}(1)$ time. This is possible after ${\mathcal{O}}(m)$ preprocessing [@LCA; @SuffixArray].
We initialize the structure by partitioning $S^T$ into $2m$ single characters. The cost of initialization, including allocating the credits, is ${\mathcal{O}}(m \log\log m)$. Replacing $S^T[x]$ with a new character $c$ starts with locating the fragment $w$ containing the position $i$ using the tree. If $w$ is a single character, we replace it with the new one. If $w$ is a longer subword $w[i..j]$ of $S^P$, and we need to replace its $\ell$-th character, we first split $w$ into three fragments $w[i..(i+\ell-1)]$, $w[i+\ell]$, $w[(i+\ell+1)..j]$. In both cases we spend ${\mathcal{O}}(\log \log m)$ time, including the cost of inserting the new elements and allocating their credits.
![Updating the representation. Black boxes represent mismatches, gray areas are full fragments between mismatches. Fragments are either left untouched (on the left), or compressed into a single new one (on the right).[]{data-label="fig:decomposition"}](fig_decomposition.pdf){width="0.8\linewidth"}
Generating the mismatches begins with locating the fragment corresponding to the position $i$. Then we scan the representation from left to right starting from there. Locating the fragment takes ${\mathcal{O}}(\log \log m)$ time, but traversing can be done in ${\mathcal{O}}(1)$ time per each step, as we can use the information about the successor of each fragment. We will match $S^P$ with the representation of $S^T$ while scanning. This is done using constant time longest common prefix queries. Each such query allows us to either detect a mismatch, or move to the next fragment. Whenever we find a mismatch, if the part of the text between the previous mismatch (or the beginning of the window) and the current mismatch contains at least 3 full fragments, we replace them with a single fragment, which is the corresponding subword of $S^P$. If there are less than $3$ full fragments, we keep the current representation intact, see Fig. \[fig:decomposition\]. We stop the scanning after reaching $(3k+1)$-th mismatch, or after the whole window was processed, whichever comes first.
To bound the amortized cost of processing a single mismatch, let $p$ be the number of full fragments between the current mismatch and the previous one (or the beginning of the window). If $p\ge 3$, we concatenate all $p$ fragments by simply erasing every full fragment except the first one. We also need to traverse (and perform longest common prefix queries on) $p+2$ fragments. However, we remove $p-1$ elements from the tree, and hence can use all their $2(p-1)$ credits to pay for the processing. Thus, the amortized cost is ${\mathcal{O}}((p-1)\log \log m + (p+2) - 2(p-1)\log \log m) = {\mathcal{O}}(1)$. Therefore we need ${\mathcal{O}}(k+\log \log m)$ time in total to generate all the mismatches.
\[lemma:prune\] Given a pattern $(p_{1},\ldots,p_{m})$ and a text $(t_{1},\ldots,t_{2m})$, we can maintain an implicit representation of the current signature $S(t_{i},\ldots,t_{i+m-1})$ and perform the following operations:
1. increasing $i$ by one in amortized time ${\mathcal{O}}(\log\log m)$,
2. generating the first $3k$ mismatches between $S(p_{1},\ldots,p_{m})$ and $S(t_{i},\ldots,t_{i+m-1})$ in time ${\mathcal{O}}(k+\log\log m)$.
The structure is initialized in time ${\mathcal{O}}(m \log\log m)$.
First we construct $S(p_{1},\ldots,p_{m})$ in time ${\mathcal{O}}(m\log\log m)$ by sorting. Whenever we increase $i$ by one, just a few characters of $S(t_{i},\ldots,t_{i+m-1})=(s_{1},\ldots,s_{m})$ need to be modified. The new signature can be created by first removing the first character $s_{1}$, appending a new character $s_{m+1}$, and then modifying the characters corresponding to the successors of $t_{i}$ and $t_{i+m}$. By maintaining all $t_{i},\ldots,t_{i+m-1}$ in a van Emde Boas tree (we can rename the elements so that $t_{i} \in \{1,\ldots,2m\}$ by sorting) we can calculate both $s_{m+1}$ and the characters which needs to be modified in ${\mathcal{O}}(\log\log m)$ time. Current $S(t_{i},\ldots,t_{i+m-1})$ is stored using Lemma \[lemma:prune\_structure\]. We initialize $S^T[1..2m]$ to be $S(t_{1},\ldots,t_{m})$ concatenated with $m$ copies of, say, $0$. After increasing $i$ by one, we replace $S^T[i]$, $S^T[i+m]$ and possibly two more characters in ${\mathcal{O}}(\log\log m)$ time. Generating the mismatches is straightforward using Lemma \[lemma:prune\_structure\].
Now our algorithm first uses Lemma \[lemma:prune\] to quickly eliminate the starting positions $i$ such that the Hamming distance between the corresponding signatures is large. For the remaining starting positions, we reduce checking if $(t_{i},\ldots,t_{i+m-1}) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(p_{1},\ldots,p_{m})$ to a weighted version of the well-known longest increasing subsequence problem on at most $3(k+1)$ elements. In the weighted variant, which we call *heaviest increasing subsequence*, the input is a sequence $(a_{1},\ldots,a_{\ell})$ and weight $w_{i}$ of each element $a_{i}$, and we look for an increasing subsequence with the largest total weight, i.e., for $1\leq i_{1} < \ldots < i_{s} \leq \ell$ such that $a_{i_{1}} < \ldots < a_{i_{s}}$ and $\sum_{j}w_{i_{j}}$ is maximized.
\[lemma:reduction\] Assuming random access to $(a_{1},\ldots,a_{m})$, the sorting permutation $\pi_{b}$ of $(b_{1},\ldots,b_{m})$, and the rank of every $b_{i}$ in $\{b_{1},\ldots,b_{m}\}$, and given $\ell$ positions where $S(a_{1},\ldots,a_{m})$ and $S(b_{1},\ldots,b_{m})$ differ, we can reduce in ${\mathcal{O}}(\ell\log\log\ell)$ time checking if $(a_{1},\ldots,a_{m}) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_{1},\ldots,b_{m})$ to computing the heaviest increasing subsequence on at most $\ell+1$ elements.
![Partition into maximal paths. The heaviest increasing subsequence is marked.[]{data-label="fig:chains"}](fig_chains.pdf){width="0.6\linewidth"}
Let $d_{1},\ldots,d_{\ell}$ be the positions where $S(a_{1},\ldots,a_{m})$ and $S(b_{1},\ldots,b_{m})$ differ. From the definition of a signature, for any other position $i$ the predecessors of $a_{i}$ and $b_{i}$ in their respective sequences are at the same position $j$, which we denote by $j\rightarrow i$. For any given $i$, $j\rightarrow i$ for at most one $j$. Similarly, for any given $j$, $j\rightarrow i$ for at most one $i$, because the only such $i$ corresponds to the successor of, say, $a_{j}$ in its sequence. Consider a partition of the set of all positions into maximal *paths* of the form $j_{1} \rightarrow \ldots \rightarrow j_{k}$ (see Fig. \[fig:chains\]). Such partition is clearly unique, and furthermore the first element of every path is one of the positions where the signatures differ (except one possible path starting with the position corresponding to the smallest element). Hence there are at most $\ell+1$ paths, and we denote by $I_{j}$ the path starting with $d_{j}$. If the smallest element occurs at the same position in both sequences, we additionally denote this position by $d_{0}$, and call the path starting there $I_{0}$ (we will assume that this is always the case, which can be ensured by appending $-\infty$ to both sequences).
Recall that our goal is to check if $(a_{1},\ldots,a_{m}) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_{1},\ldots,b_{m})$. For this we need to check if there exist $i_1,\ldots,i_{m-k}$ such that $a_{i_1} < \ldots < a_{i_{m-k}}$ and $b_{i_1} < \ldots < b_{i_{m-k}}$. Alternatively, we could compute the largest $s$ for which there exist a solution $i_1,\ldots,i_{s}$ such that $a_{i_1} < \ldots < a_{i_{s}}$ and $b_{i_1} < \ldots < b_{i_{s}}$. We claim that one can assume that for each path $I$ either none of its elements are among $i_1,\ldots,i_{s}$, or all of its elements are there. We prove this in two steps.
1. If $i_{k}\in I$ and $i_{k}\rightarrow j$, then without losing the generality $i_{k+1}=j$. Assume otherwise, so $i_{k+1}\neq j$ or $k=s$. Recall that it means that $a_{j}$ is the successor of $a_{i_{k}}$ and $b_{j}$ is the successor of $b_{i_{k}}$. Hence $a_{i_{k}} < a_{j}$ and $b_{i_{k}} < b_{j}$. If $k=s$ we can extend the current solution by appending $j$. Otherwise $a_{j} < a_{i_{k+1}}$ and $b_{j} < b_{i_{k+1}}$, so we can extend the solution by inserting $j$ between $i_{k}$ and $i_{k+1}$.
2. If $i_{k}\in I$ and $j\rightarrow i_{k}$, then without losing the generality $i_{k-1}=j$. Assume otherwise, so $i_{k-1}\neq j$ or $k=1$. Similarly as in the previous case, $a_{j}$ is the predecessor of $a_{i_{k}}$ and $b_{j}$ is the predecessor of $b_{i_{k}}$. Hence $a_{j} < a_{i_{k}}$ and $b_{j} < b_{i_{k}}$. If $k=1$ we can extend the current solution by prepending $j$. Otherwise $a_{i_{k+1}} < a_{j}$ and $b_{i_{k+1}} < b_{j}$, so we can insert $j$ between $i_{k-1}$ and $i_{k}$.
Now let the weight of a path $I$ be its length $|I|$. From the above reasoning we know that the optimal solution contains either no elements from a path, or all of its elements. Hence if we know which paths contain the elements used in the optimal solution, we can compute $s$ as the sum of the weights of these paths. Additionally, if we take such optimal solution, and remove all but the first element from every path, we get a valid solution. Hence $s$ can be computed by choosing some solution restricted only to $d_{0},\ldots,d_{\ell}$, and then summing up weights of the corresponding paths. It follows that computing the optimal solution can be done, similarly as in the proof of Lemma \[lemma:checking\], by finding an increasing subsequence. We define a new weighted sequence $(a'_{0},\ldots,a'_{\ell})$ by setting $a'_{j}=b_{\pi_{b}(d_{j})}$ and choosing the weight of $a'_{j}$ to be $|I_{j}|$. Then an increasing subsequence of $(a'_{0},\ldots,a'_{\ell})$ corresponds to a valid solution restricted to $d_{0},\ldots,d_{\ell}$, and moreover the weight of the heaviest such subsequence is exactly $s$. In other words, we can reduce our question to computing the heaviest increasing subsequence.
Finally, we need to analyze the complexity of our reduction. Assuming random access to both $(a_{1},\ldots,a_{m})$ and $\pi_{b}$, we can construct $(a'_{0},\ldots,a'_{\ell})$ in time ${\mathcal{O}}(\ell)$. Computing the weight of every $a'_{j}$ is more complicated. We need to find every $|I_{j}|$ without explicitly constructing the paths. For every $d_{j}$ we can retrieve the rank $r_{j}$ of its corresponding element in $\{b_{1},\ldots,b_{m}\}$. Then $I_{j}$ contains $d_{j}$ and all $i$ such that the predecessor of $b_{i}$ among $\{b_{d_{0}},\ldots,b_{d_{\ell}}\}$ is $b_{d_{j}}$. Hence $|I_{j}|$ can be computed by counting such $i$. This can be done by locating the successor $b_{d_{j'}}$ of $b_{d_{j}}$ in $\{b_{d_{0}},\ldots,b_{d_{\ell}}\}$ and returning $r_{d_{j'}} - r_{d_{j}}-1$ (if the successor does not exist, $m-r_{d_{j}}$). To find all these successors, we only need to sort $\{b_{d_{0}},\ldots,b_{d_{\ell}}\}$, which can, again, be done in time ${\mathcal{O}}(\ell\log\log\ell)$.
\[lemma:heaviest\] Given a sequence of $\ell$ weighted elements, we can compute its heaviest increasing subsequence in time ${\mathcal{O}}(\ell\log\log\ell)$.
Let the sequence be $(a_{1},\ldots,a_{\ell})$, and denote the weight of $a_{i}$ by $w_{i}$. We will describe how to compute the weight of the heaviest increasing subsequence, reconstructing the subsequence itself will be straightforward. At a high level, for each $i$ we want to compute the weight $r_{i}$ of the heaviest increasing subsequence ending at $a_{i}$. Observe that $r_{i}=w_{i} + \max\{ r_{j} : j<i \text{ and } a_{j} < a_{i}\}$, where we assume that $a_{0}=-\infty$ and $r_{0}=0$. We process $i=1,\ldots,\ell$, so we need a dynamic structure where we could store all already computed results $r_{j}$ so that we can select the appropriate one efficiently. To simplify the implementation of this structure, we rename the elements in the sequence so that $a_{i} \in \{1,\ldots,\ell\}$. This can be done in ${\mathcal{O}}(\ell\log\log\ell)$ time by sorting. Then the dynamic structure needs to store $n$ values $v_{1},\ldots,v_{n}$, all initialized to $-\infty$ in the beginning, and implement two operations:
1. increase any $v_{k}$,
2. given $k$, return the maximum among $v_{1},\ldots,v_{k}$.
Then to compute $r_{i}$ we first find the maximum among $v_{1},\ldots,v_{a_{i}-1}$, and afterwards update $v_{a_{i}}$ to be $r_{i}$.
Now we describe how the structure is implemented. First observe that if $v_{i} > v_{j}$ and $i<j$, we will never return the current $v_{j}$ as the maximum, hence we don’t need to store it. In other words, we only need to store $v_{i_{1}}, \ldots, v_{i_{t}}$ such that $i_{1}=1$ and each $i_{j+1}$ is the smallest position on the right of $i_{j}$ such that $v_{i_{j+1}} > v_{i_{j}}$. We store all $i_{j}$ in a van Emde Boas tree, where each element knows its successor and predecessor. Then to find the maximum among $v_{1},\ldots,v_{k}$ we perform a predecessor query in the tree to locate the largest $i_{j}\leq k$, and return the corresponding $v_{i_{j}}$. To increase $v_{k}$, we must consider two cases. First, it might happen that $k=i_{j}$. In such case we update $v_{i_{j}}$ and then look at the successor $i_{j+1}$ of $i_{j}$ in the tree. If $v_{i_{j+1}} \leq v_{i_{j}}$, we remove $i_{j+1}$ and repeat, and otherwise stop. The other case is that $k$ is not in the tree, then we update $v_{k}$ and locate the largest $i_{j}\leq k$. Then if $v_{i_{j}}>v_{k}$, we don’t have to modify the tree. Otherwise we need to insert $k$ into the tree, and then repeatedly remove its successors as long as they correspond to elements which are smaller or equal to $v_{k}$, as in the previous case. Finding the maximum clearly requires ${\mathcal{O}}(\log\log\ell)$ time, as it requires just one predecessor query. Increasing any value requires inserting at most one new element, and removing zero or more already existing elements, hence its amortized complexity is ${\mathcal{O}}(\log\log\ell)$. We execute $\ell$ operations in total, so the running time is as claimed.
\[theorem:algo\] Order-preserving pattern matching with $k$ mismatches can be solved in time ${\mathcal{O}}(n(\log\log m+k\log\log k))$, where $n$ is the length of the text and $m$ is the length of the pattern.
First we focus on the special case when $n\leq 2m$. We iterate over all possible starting positions $i$ in the text while maintaining the signature $S(t_i,\ldots,t_{i+m-1})$ of the current fragment using Lemma \[lemma:prune\]. If the Hamming distance between $S(t_i,\ldots,t_{i+m-1})$ and $S(p_1,\ldots,p_m)$ exceeds $3k$, which can be detected in time ${\mathcal{O}}(k+\log\log m)$, by Lemma \[lemma:implication\] the current $i$ cannot correspond to a match, and we continue. Otherwise we generate at most $3k$ mismatches, and apply Lemma \[lemma:reduction\] to reduce in time ${\mathcal{O}}(k\log\log k)$ checking if $(t_i,\ldots,t_{i+m-1}){\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(p_1,\ldots,p_m)$ to computing the heaviest increasing subsequence on at most $3(k+1)$ elements. This, by Lemma \[lemma:heaviest\], can be done in time ${\mathcal{O}}(k\log\log k)$, too. We get that the total complexity for a single $i$ is ${\mathcal{O}}(\log\log m+k\log\log k)$. We spend ${\mathcal{O}}(m\log\log m)$ to initialize the structure from Lemma \[lemma:prune\], so the total time is ${\mathcal{O}}(m(\log\log m+k\log\log k))$.
Finally, by cutting the input into overlapping fragments of length $2m$ and using the above method on each of them, a text of length $n$ can be processed in time ${\mathcal{O}}(\lceil\frac{n}{m}\rceil m(\log\log m+k\log\log k))={\mathcal{O}}(n(\log\log m+k\log\log k))$.
Allowing repeated elements {#section:generalization}
==========================
In this section we show how to generalize the algorithm as to remove the restriction that the text (and the pattern) has no repeated elements. A simple fix could be that instead of comparing numbers, we compare pairs consisting of the number and its position. This might create new occurrences, though. We will carefully modify all ingredients of the solution described in the previous section to deal with possible equalities. The time complexity will stay the same.
We start with Lemma \[lemma:subseq\]. Now the condition becomes that for all $j=1,2,\ldots,m-k-1$, either $a_{i_j} < a_{i_{j+1}}$ and $b_{i_j} < b_{i_{j+1}}$, or $a_{i_j} = a_{i_{j+1}}$ and $b_{i_j} = b_{i_{j+1}}$. Then checking whether two sequences are $k$-isomorphic seems more complicated, but in fact it is not so. We can reduce the question to a generalized heaviest increasing subsequence problem, which we call the *heaviest chain in a plane*. In this generalization we are given a collection of $\ell$ weighted points in a plane, and the goal is to find the chain with the largest total weight, where a chain is a set $S$ of points such that for any $(x,y),(x',y')\in S$ either $x<x'$ and $y<y'$, or $x>x'$ and $y>y'$, or $x=x'$ and $y=y'$. This can be solved in ${\mathcal{O}}(\ell\log\log\ell)$ time similarly as in Lemma \[lemma:heaviest\].
\[lemma:heaviest chain\] Given $\ell$ weighted points in a plane, we can compute their heaviest chain in time ${\mathcal{O}}(\ell\log\log\ell)$.
Given $\ell$ weighted points in a plane, we reduce the problem of finding the heaviest chain to computing the heaviest increasing subsequence. First, we make sure that the points are unique by collapsing all duplicates into single points with the weight equal to the sum of the weights of the collapsed points. Let the $i$-th point be $p_i=(x_i,y_i)$ and define $x'_i = (x_i, y_i)$ and $y'_i = (y_i, x_i)$. Even though all $x'_i$ and $y'_i$ are pairs of numbers instead of just numbers, we can still consider the question of computing the heaviest chain for the new set of points $p'_i=(x'_i,y'_i)$ if we compare the pairs using the standard lexicographical ordering. Observe that $p'_i$ and $p'_j$ can be in the same chain in the original instance iff $p_i$ and $p_j$ can be in the same chain in the new instance, because $x'_i \leq x'_j$ and $y'_i \leq y'_j$ iff $x_i \leq x_j$ and $y_i \leq y_j$. However, in the new instance we additionally have the property that $x'_i=x'_j$ iff $p_i=p_j$, and symmetrically $y'_i=y'_j$ iff $p_i=p_j$. Since we made sure that the points in the original instance are unique, $x'_i=x'_j$ iff $i=j$, and $y'_i=y'_j$ iff $i=j$. Hence we can reorder the points in the new instance so that their first coordinates are strictly increasing, and then if we look at the sequence of their second coordinates, its increasing subsequence corresponds to a chain (technically, we also need to normalize the second coordinates by sorting, so that they are numbers instead of pairs of numbers). Hence we can find the heaviest chain in ${\mathcal{O}}(\ell\log\log\ell)$ time using Lemma \[lemma:heaviest\].
Now to check if two sequences are $k$-isomorphic, for each $i$ we either create a new point $(a_i,b_i)$ with weight $1$, or if such point already exists, we increase its weight by $1$. Then we check if the weight of the heaviest chain is at least $n-k$.
We have to modify the definition of the signature. Recall that $S(a_1,\ldots,a_m)$ was defined as a new sequences $(1-{\mathrm{pred}}(1),\ldots,m-{\mathrm{pred}}(m))$. Now the predecessor of $a_i$ might be not unique, hence the sequence will consist of pairs from $\{<,=\} \times \mathcal{Z}$. If a given $a_i$ is the rightmost occurrence of the corresponding number, we output the pair $(<,i-{\mathrm{pred}}(i))$, where ${\mathrm{pred}}(i)$ is the position of the predecessor of $(a_i,i)$ in $\{(a_1,1),\ldots,(a_m,m)\}$ (if there is no such predecessor, $0$), where the pairs are compared using the standard lexicographic ordering. If $a_i$ is not the rightmost occurrence of the corresponding number, we output $(=, i-{\mathrm{pred}}(i))$. For such modified definition Lemma \[lemma:implication\] still holds. It also holds that $S(t_i,\ldots,t_{i+m-1})$ differs from $S(t_{i+1},\ldots,t_{i+m})$ on a constant number of positions, hence the time bounds from Lemma \[lemma:prune\] remain the same.
Now the only remaining part is to show how to generalize Lemma \[lemma:reduction\]. First of all, given that the numbers can repeat, it is not clear what the rank of $b_i$ in $\{b_1,\ldots,b_m\}$ exactly is. We define it as the number of all elements smaller than $b_i$, and also define *the equal-rank* of $b_i$ to be the number of equal elements on its left.
{width="0.5\linewidth"} {width="0.4\linewidth"}
\[lemma:reduction2\] Assuming random access to $(a_{1},\ldots,a_{m})$, the sorting permutation $\pi_{b}$ of $((b_{1},1)\ldots,(b_{m}, m))$, and the rank, the equal-rank, and total number of repetitions for every $b_i$, and given $\ell$ positions where $S(a_{1},\ldots,a_{m})$ and $S(b_{1},\ldots,b_{m})$ differ, we can reduce in ${\mathcal{O}}(\ell\log\log\ell)$ time checking if $(a_{1},\ldots,a_{m}) {\overset{k}{\rule{0pt}{.4ex}\smash{\sim}}}(b_{1},\ldots,b_{m})$ to computing the heaviest chain on at most $3(\ell+1)$ elements.
For any position $i$ where the signatures are the same, either both $a_i$ and $b_i$ are not the rightmost occurrence of the corresponding number, and their next occurrences are at the same position $j$ in both sequences, denoted $j\stackrel{=}{\rightarrow} i$, or both $a_i$ and $b_i$ are the rightmost occurrence of the corresponding number, and the leftmost occurrences of their predecessors are at the same position $j$ in both sequences, denoted $j\stackrel{<}{\rightarrow} i$. In both cases, we denote the situation by $j\rightarrow i$, and consider the unique partition of the set of all positions into maximal paths. Now we would like to say that for each such path $i$, either none of its elements belong to the optimal solution, or all of them are there, where the solution is a collection of indices $i_1,\ldots,i_{m-k}$ such that for all $j=1,2,\ldots,m-k+1$ either $a_{i_j}=a_{i_{j+1}}$ and $b_{i_j}=b_{i_{j+1}}$, or $a_{i_j}<a_{i_{j+1}}$ and $b_{i_j}<b_{i_{j+1}}$. Unfortunately, this is not true: one path might end at some $i$, and the other might start at some $j$, such that $a_i=a_j$, yet $b_i\neq b_j$. Then we cannot have these two whole paths in the solution, but it might pay off to have a prefix of the former, or a suffix of the latter, see Fig. \[fig:eq\_paths\]. Our fix is to additionally split every path into three parts. The parts correspond to the maximal prefix $I_{pref}$ of the form $i_1\stackrel{=}{\rightarrow}i_2\stackrel{=}{\rightarrow}\ldots$, the middle part $I_{middle}$, and the maximal suffix $I_{suf}$ of the form $\ldots\stackrel{=}{\rightarrow}i_{k-1}\stackrel{=}{\rightarrow}i_k$ The splitting can be performed efficiently using the ranks, the equal-ranks, and the total number of repetitions. Then we create an instance of the heaviest chain problem by collapsing each path into a single weighted point (with the same coordinates as the first point on the path), and additionally replacing identical points with one (and summing up their weights).
Now we need to prove that solving this instance gives us an optimal solution to the original question. Clearly, if $j\stackrel{=}{\rightarrow} i$ then the optimal solution takes both $i$ and $j$ or none of them, hence merging identical points preserves the optimal solution. We must show that for each chain $I$ decomposed into $I_{pref}\cup I_{middle}\cup I_{suf}$ the optimal solution contains either all points from $I_{middle}$ or none of them. Let $I_{pref} = \ldots \stackrel{=}{\rightarrow}i'$, $I_{middle}=i \rightarrow \ldots \rightarrow j$, and $I_{suf} = j'\stackrel{=}{\rightarrow}\ldots$. We know that $a_{i'}<a_{i}$ and $b_{i'}<b_{i}$, and also $a_{j}<a_{j'}$ and $b_{j}<b_{j'}$. Hence for all $k\in I_{middle}$, all positions $k'$ such that $a_k=a_{k'}$ belong to $I_{middle}$, and similarly all positions $k'$ such that $b_k=b_{k'}$ belong to $I_{middle}$. Furthermore, these two sets of positions are the same. It follows that after collapsing identical points we get that for every $k\in I_{middle}$, there are no $k'$ (inside or outside $I_{middle}$) such that $a_k=a_{k'}$ or $b_k=b_{k'}$. A straightforward modification of the two step proof used in Lemma \[lemma:reduction\] can be used to show that if one element from $I_{middle}$ belongs to the optimal solution, then all its elements are there.
Conclusions
===========
Recall that the complexity of our solution is ${\mathcal{O}}(n(\log\log m+k\log\log k))$. Given that it is straightforward to prove a lower bound of $\Omega(n+m\log m)$ in the comparison model, and that for $k=0$ one can achieve ${\mathcal{O}}(n+\text{sort}(m))$ time [@KubicaOrder], a natural question is whether achieving ${\mathcal{O}}(nf(k))+{\mathcal{O}}(m{\text{\hspace{1mm}polylog}}(m))$ time is possible. Finally, even though the version with $k$ errors seems hard (see the introduction), there might be an ${\mathcal{O}}(nf(k))$ time algorithm, with $f(k)$ being an exponential function.
[^1]: This work was started while the second author was a PhD student at Inria Bordeaux Sud-Ouest, France.
[^2]: More precisely, it might make two non-isomorphic sequences isomorphic, but not the other way around.
|
---
abstract: 'We investigate the stability of theories in which Lorentz invariance is spontaneously broken by fixed-norm vector “” fields. Models with generic kinetic terms are plagued either by ghosts or by tachyons, and are therefore physically unacceptable. There are precisely three kinetic terms that are not manifestly unstable: a sigma model $(\partial_\mu A_\nu)^2$, the Maxwell Lagrangian $F_{\mu\nu}F^{\mu\nu}$, and a scalar Lagrangian $(\partial_\mu A^\mu)^2$. The timelike sigma-model case is well-defined and stable when the vector norm is fixed by a constraint; however, when it is determined by minimizing a potential there is necessarily a tachyonic ghost, and therefore an instability. In the Maxwell and scalar cases, the Hamiltonian is unbounded below, but at the level of perturbation theory there are fewer degrees of freedom and the models are stable. However, in these two theories there are obstacles to smooth evolution for certain choices of initial data.'
author:
- 'Sean M. Carroll, Timothy R. Dulaney, Moira I. Gresham, and Heywood Tam'
bibliography:
- 'lorentz-bib.bib'
title: 'Instabilities in the [Æ]{}ther'
---
Introduction
============
The idea of spontaneous violation of Lorentz invariance through tensor fields with non-vanishing expectation values has garnered substantial attention in recent years [@Will:1972zz; @Gasperini:1987nq; @Kostelecky:1989jw; @Colladay:1998fq; @Jacobson:2000xp; @Eling:2003rd; @Carroll:2004ai; @Jacobson:2004ts; @Lim:2004js; @Eling:2004dk; @Dulaney:2008ph; @Jimenez:2008sq]. Hypothetical interactions between Standard Model fields and Lorentz-violating (LV) tensor fields are tightly constrained by a wide variety of experimental probes, in some cases leading to limits at or above the Planck scale [@Colladay:1998fq; @Kostelecky:2000mm; @Carroll:2004ai; @Elliott:2005va; @Mattingly:2005re; @Will:2005va; @Jacobson:2008aj].
If these constraints are to be taken seriously, it is necessary to have a sensible theory of the dynamics of the LV tensor fields themselves, at least at the level of low-energy effective field theory. The most straightforward way to construct such a theory is to follow the successful paradigm of scalar field theories with spontaneous symmetry breaking, by introducing a tensor potential that is minimized at some non-zero expectation value, in addition to a kinetic term for the fields. (Alternatively, it can be a derivative of the field that obtains an expectation value, as in ghost condensation models [@ArkaniHamed:2003uy; @ArkaniHamed:2005gu; @Cheng:2006us].) As an additional simplification, we may consider models in which the nonzero expectation value is enforced by a Lagrange multiplier constraint, rather than by dynamically minimizing a potential; this removes the “longitudinal” mode of the tensor from consideration, and may be thought of as a limit of the potential as the mass near the minimum is taken to infinity. In that case, there will be a vacuum manifold of zero-energy tensor configurations, specified by the constraint.
All such models must confront the tricky question of stability. Ultimately, stability problems stem from the basic fact that the metric has an indefinite signature in a Lorentzian spacetime. Unlike in the case of scalar fields, for tensors it is necessary to use the spacetime metric to define both the kinetic and potential terms for the fields. A generic choice of potential would have field directions in which the energy is unbounded from below, leading to tachyons, while a generic choice of kinetic term would have modes with negative kinetic energies, leading to ghosts. Both phenomena represent instabilities; if the theory has tachyons, small perturbations grow exponentially in time at the linearized level, while if the theory has ghosts, nonlinear interactions create an unlimited number of positive- and negative-energy excitations [@Carroll:2003st]. There is no simple argument that these unwanted features are necessarily present in any model of LV tensor fields, but the question clearly warrants careful study.
In this paper we revisit the question of the stability of theories of dynamical Lorentz violation, and argue that most such theories are unstable. In particular, we examine in detail the case of a vector field $A_\mu$ with a nonvanishing expectation value, known as the “[æ]{}ther” model or a “bumblebee” model. For generic choices of kinetic term, it is straightforward to show that the Hamiltonian of such a model is unbounded from below, and there exist solutions with bounded initial data that grow exponentially in time.
There are three specific choices of kinetic term for which the analysis is more subtle. These are the sigma-model kinetic term, \_K = - \_A\_\^A\^, \[sigmamodel\] which amounts to a set of four scalar fields defined on a target space with a Minkowski metric; the Maxwell kinetic term, \_K = -F\_F\^, \[maxwell\] where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ is familiar from electromagnetism; and what we call the “scalar” kinetic term, \_K = (\_A\^)\^2, \[scalar\] featuring a single scalar degree of freedom. Our findings may be summarized as follows:
- The sigma-model Lagrangian with the vector field constrained by a Lagrange multiplier to take on a timelike expectation value is the only theory for which the Hamiltonian is bounded from below in every frame, ensuring stability. In a companion paper, we examine the cosmological behavior and observational constraints on this model [@Carroll:2009en]. If the vector field is spacelike, the Hamiltonian is unbounded and the model is unstable. However, if the constraint in the sigma-model theory is replaced by a smooth potential, allowing the length-changing mode to become a propagating degree of freedom, that mode is necessarily ghostlike (negative kinetic energy) and tachyonic (correct sign mass term), and the Hamiltonian is unbounded below, even in the timelike case. It is therefore unclear whether models of this form can arise in any full theory.
- In the Maxwell case, the Hamiltonian is unbounded below; however, a perturbative analysis does not reveal any explicit instabilities in the form of tachyons or ghosts. The timelike mode of the vector acts as a Lagrange multiplier, and there are fewer propagating degrees of freedom at the linear level (a “spin-1” mode propagates, but not a “spin-0” mode). Nevertheless, singularities can arise in evolution from generic initial data: for a spacelike vector, for example, the field evolves to a configuration in which the fixed-norm constraint cannot be satisfied (or perhaps just to a point where the effective field theory breaks down). In the timelike case, a certain subset of initial data is well-behaved, but, provided the vector field couples only to conserved currents, the theory reduces precisely to conventional electromagnetism, with no observable violations of Lorentz invariance. It is unclear whether there exists a subset of initial data that leads to observable violations of Lorentz invariance while avoiding problems in smooth time evolution.
- The scalar case is superficially similar to the Maxwell case, in that the Hamiltonian is unbounded below, but a perturbative analysis does not reveal any instabilities. Again, there are fewer degrees of freedom at the linear level; in this case, the spin-1 mode does not propagate. There is a scalar degree of freedom, but it does not correspond to a propagating mode at the level of perturbation theory (the dispersion relation is conventional, but the energy vanishes to quadratic order in the perturbations). For the timelike field, obstacles arise in the time evolution that are similar to those of a spacelike vector in the Maxwell case; for a spacelike field with a scalar action, the behavior is less clear.
- For any other choice of kinetic term, theories are always unstable.
Interestingly, these three choices of dynamics are precisely those for which there is a unique propagation speed for all dynamical modes; this is the same condition required to ensure that the Generalized Second Law is respected by a Lorentz-violating theory [@Dubovsky:2006vk; @Eling:2007qd].
One reason why our findings concerning stability seem more restrictive than those of some previous analyses is that we insist on perturbative stability in all Lorentz frames, which is necessary in theories where the form of the Hamiltonian is frame-dependent. In a Lorentz-invariant field theory, it suffices to pick a Lorentz frame and examine the behavior of small fluctuations; if they grow exponentially, the model is unstable, while if they oscillate, the model is stable. In Lorentz-violating theories, in contrast, such an analysis might miss an instability in one frame that is manifest at the linear level in some other frame [@Kostelecky:2001xz; @Mattingly:2005re; @Adams:2006sv]. This can be traced to the fact that a perturbation that is “small” in one frame (the value of the perturbation is bounded everywhere along some initial spacelike slice), but grows exponentially with time as measured in that frame, will appear “large” (unbounded on every spacelike slice) in some other frame.
As an explicit example, consider a model of a timelike vector with a background configuration $\bar{A}_\mu = (m, 0, 0, 0)$, and perturbations $\delta a^\mu = \epsilon^\mu e^{-i\omega t} e^{i\vec k \cdot \vec x}$, where $\epsilon^\mu$ is some constant polarization vector. In this frame, we will see that the dispersion relation takes the form \^2 = v\^2 k\^2. Clearly, the frequency $\omega$ will be real for every real wave vector $\vec k$, and such modes simply oscillate rather than growing in time. It is tempting to conclude that models of this form are perturbatively stable for any value of $v$. However, we will see below that when $v > 1$, there exist other frames (boosted with respect to the original) in which $\vec k$ can be real but $\omega$ is necessarily complex, indicating an instability. These correspond to wave vectors for which, evaluated in the original frame, both $\omega$ and $\vec k$ are complex. Modes with complex spatial wave vectors are not considered to be “perturbations,” since the fields blow up at spatial infinity. However, in the presence of Lorentz violation, a complex spatial wave vector in one frame may correspond to a real spatial wave vector in a boosted frame. We will show that instabilities can arise from initial data defined on a constant-time hypersurface (in a boosted frame) constructed solely from modes with real spatial wave vectors. Such modes are bounded at spatial infinity (in that frame), and could be superimposed to form wave packets with compact support. Since the notion of stability is not frame dependent, the existence of at least one such frame indicates that the theory is unstable, even if there is no linear instability in the rest frame.
Several prior investigations have considered the question of stability in theories with LV vector fields. Lim [@Lim:2004js] calculated the Hamiltonian for small perturbations around a constant timelike vector field in the rest frame, and derived restrictions on the coefficients of the kinetic terms. Bluhm et al. [@Bluhm:2008yt] also examined the timelike case with a Lagrange multiplier constraint, and showed that the Maxwell kinetic term led to stable dynamics on a certain branch of the solution space if the vector was coupled to a conserved current. It was also found, in [@Bluhm:2008yt], that most LV vector field theories have Hamiltonians that are unbounded below. Boundedness of the Hamiltonian was also considered in [@Chkareuli:2006yf]. In the context of effective field theory, Gripaios [@Gripaios:2004ms] analyzed small fluctuations of LV vector fields about a flat background. Dulaney, Gresham and Wise [@Dulaney:2008ph] showed that only the Maxwell choice was stable to small perturbations in the spacelike case assuming the energy of the linearized modes was non-zero.[^1] Elliot, Moore, and Stoica [@Elliott:2005va] showed that the sigma-model kinetic term is stable in the presence of a constraint, but not with a potential.
In the next section, we define notation and fully specify the models we are considering. We then turn to an analysis of the Hamiltonians for such models, and show that they are always unbounded below unless the kinetic term takes on the sigma-model form and the vector field is timelike. This result does not by itself indicate an instability, as there may not be any dynamical degree of freedom that actually evolves along the unstable direction. Therefore, in the following section we look carefully at linear stability around constant configurations, and isolate modes that grow exponentially with time. In the section after that we show that the models that are not already unstable at the linear level end up having ghosts, with the exception of the Maxwell and scalar cases. We then examine some features of those two theories in particular.
Models
======
We will consider a dynamical vector field $A_\mu$ propagating in Minkowski spacetime with signature $(-+++)$. The action takes the form S\_A = d\^4x ([L]{}\_K + [L]{}\_V), where ${\cal L}_K$ is the kinetic Lagrange density and ${\cal L}_V$ is (minus) the potential. A general kinetic term that is quadratic in derivatives of the field can be written[^2] $$\label{aLag}
{\cal{L}}_K = -\b_1(\partial_\m A_\n)(\partial^\m A^\n) - \beta_2 (\partial_\m A^\m)^2 %\nonumber \\
- \b_3 (\partial_\m A_\n)(\partial^\n A^\m)
-{\beta_4} {A^\m A^\n \over m^2} (\partial_\m A_\rho)(\partial_\n A^\rho)\,.$$ In flat spacetime, setting the fields to constant values at infinity, we can integrate by parts to write an equivalent Lagrange density as $$\label{aLag2}
{\cal{L}}_K = -{1\over 2}\b_1 F_{\m\n}F^{\m\n} %&
-\bnew(\partial_\m A^\m)^2 - {\beta_4 } {A^\m A^\n \over m^2} (\partial_\m A_\rho)(\partial_\n A^\rho)\,,$$ where $F_{\m\n} = \partial_\m A_\n - \partial_\n A_\m$ and we have defined = \_1 + \_2 +\_3. In terms of these variables, the models specified above with no linear instabilities or negative-energy ghosts are:
- Sigma model: $\b_1 = \bnew$,
- Maxwell: $\bnew = 0$, and
- Scalar: $\b_1 = 0$,
in all cases with $\b_4 = 0$.
The vector field will obtain a nonvanishing vacuum expectation value from the potential. For most of the paper we will take the potential to be a Lagrange multipler constraint that strictly fixes the norm of the vector: \_V = (A\^ A\_ m\^2), \[constraintl\] where $\l$ is a Lagrange multiplier whose variation enforces the constraint A\^ A\_ = m\^2. \[constraint\] If the upper sign is chosen, the vector will be timelike, and it will be spacelike for the lower sign. Later we will examine how things change when the constraint is replaced by a smooth potential of the form ${\cal L}_V = - V(A_\mu) \propto \xi(A_\mu A^\mu \pm m^2)^2$. It will turn out that the theory defined with a smooth potential is only stable in the limit as $\xi \rightarrow \infty$. In any case, unless we specify otherwise, we assume that the norm of the vector is determined by the constraint (\[constraint\]).
We are left with an action \[vf action\] S\_A = d\^4x . The Euler-Lagrange equation obtained by varying with respect to $A_\mu$ is \[ele1\] \_1\_F\^ + \^\_A\^ + \_4 G\^= -A\^, where we have defined \[Gnu\] G\^= . Since the fixed-norm condition (\[constraint\]) is a constraint, we can consistently plug it back into the equations of motion. Multiplying (\[ele1\]) by $A_\nu$ and using the constraint, we can solve for the Lagrange multiplier, = (\_1 \_F\^ + \^\_A\^+ \_4 G\^)A\_. Inserting this back into (\[ele1\]), we can write the equation of motion as a system of three independent equations: $$\begin{aligned}
Q_\r \equiv \left(\eta_{\r \n} \pm {A_\r A_\n \over m^2}\right)
\left(\b_1\partial_\m F^{\m \n} + \bnew \partial^\n \partial_\m A^\m
+ \b_4 G^\n\right) = 0.
\label{eoms}\end{aligned}$$ The tensor $\eta_{\rho\nu} \pm m^{-2}A_\rho A_\nu$ acts to take what would be the equation of motion in the absence of the constraint, and project it into the hyperplane orthogonal to $A_\mu$. There are only three independent equations because $A^\r Q_\r$ vanishes identically, given the fixed norm constraint.
Validity of effective field theory {#Validity of effective field theory}
----------------------------------
As in this paper we will restrict our attention to classical field theory, it is important to check that any purported instabilities are found in a regime where a low-energy effective field theory should be valid. The low-energy degrees of freedom in our models are Goldstone bosons resulting from the breaking of Lorentz invariance. The effective Lagrangian will consist of an infinite series of terms of progressively higher order in derivatives of the fields, suppressed by appropriate powers of some ultraviolet mass scale $M$. If we were dealing with the theory of a scalar field $\Phi$, the low-energy effective theory would be valid when the canonical kinetic term $(\partial \Phi)^2$ was large compared to a higher-derivative term such as (\^2 )\^2 . For fluctuations with wavevector $k^\mu = (\omega, \vec k)$, we have $\partial\Phi \sim k \Phi$, and the lowest-order terms accurately describe the dynamics whenever $|\vec k| < M$. A fluctuation that has a low momentum in one frame can, of course, have a high momentum in some other frame, but the converse is also true; the set of perturbations that can be safely considered “low-energy” looks the same in any frame.
With a Lorentz-violating vector field, the situation is altered. In addition to higher-derivative terms of the form $M^{-2}(\partial^2 A)^2$, the possibility of extra factors of the vector expectation value leads us to consider terms such as \_4 = A\^6 (\^2 A)\^2 . The number of such higher dimension operators in the effective field theory is greatly reduced because $A_\m A^\m = -m^2$ and, therefore, $A_\m \partial_\n A^\m =0$. It can be shown that an independent operator with $n$ derivatives includes at most $2 n$ vector fields, so that the term highlighted here has the largest number of $A$’s with four derivatives. We expect that the ultraviolet cutoff $M$ is of order the vector norm, $M\approx m$. Hence, when we consider a background timelike vector field in its rest frame, |A\_= (m, 0, 0, 0), the ${\cal L}_4$ term reduces to $m^{-2}(\partial^2 A)^2$, and the effective field theory is valid for modes with $k< m$, just as in the scalar case.
But now consider a highly boosted frame, with |A\_= (m, m, 0, 0). At large $\h$, individual components of $A$ will scale as $e^{|\h|}$, and the higher-derivative term schematically becomes \_4 \~ e\^[6||]{} (\^2 A)\^2. For modes with spatial wave vector $k=|\vec k|$ (as measured in this boosted frame), we are therefore comparing $m^{-2}e^{6|\h|}k^4$ with the canonical term $k^2$. The lowest-order terms therefore only dominate for wave vectors with k < e\^[-3||]{}m. In the presence of Lorentz violation, therefore, the realm of validity of the effective field theory may be considerably diminished in highly boosted frames. We will be careful in what follows to restrict our conclusions to those that can be reached by only considering perturbations that are accurately described by the two-derivative terms. The instabilities we uncover are infrared phenomena, which cannot be cured by changing the behavior of the theory in the ultraviolet. We have been careful to include all of the lowest order terms in the effective field theory expansion—the terms in .
Boundedness of the Hamiltonian
==============================
We would like to establish whether there are any values of the parameters $\b_1$, $\bnew$ and $\b_4$ for which the model described above is physically reasonable. In practice, we take this to mean that there exist background configurations that are stable under small perturbations. It seems hard to justify taking an unstable background as a starting point for phenomenological investigations of experimental constraints, as we would expect the field to evolve on microscopic timescales away from its starting point.
“Stability” of a background solution $X_0$ to a set of classical equations of motion means that, for any small neighborhood $U_0$ of $X_0$ in the phase space, there is another neighborhood $U_1$ of $X_0$ such that the time evolution of any point in $U_0$ remains in $U_1$ for all times. More informally, small perturbations oscillate around the original background, rather than growing with time. A standard way of demonstrating stability is to show that the Hamiltonian is a local minimum at the background under consideration. Since the Hamiltonian is conserved under time evolution, the allowed evolution of a small perturbation will be bounded to a small neighborhood of that minimum, ensuring stability. Note that the converse does not necessarily hold; the presence of other conserved quantities can be enough to ensure stability even if the Hamiltonian is not bounded from below.
One might worry about invoking the Hamiltonian in a theory where Lorentz invariance has been spontaneously violated. Indeed, as we shall see, the form of the Hamiltonian for small perturbations will depend on the Lorentz frame in which they are expressed. To search for possible linear instabilities, it is necessary to consider the behavior of small perturbations in every Lorentz frame.
The Hamiltonian density, derived from the action (\[vf action\]) via a Legendre transformation, is $$\begin{aligned}
\label{Hamiltonian Density}
{\cal{H}} &= {\partial \lag_A \over \partial(\partial_0 A_\m)} \partial_0 A_\m - \lag_A \\
&= {\b_1 \over 2} F_{ij}^2 + \b_1 (\partial_0 A_i)^2 -\b_1 (\partial_i A_0)^2
+ \bnew(\partial_i A_i)^2 - \bnew (\partial_0 A_0)^2 \nonumber \\
&\qquad + \beta_4 {A^j A^k \over m^2} (\partial_j A_\rho)(\partial_k A^\rho)
- \beta_4 {A^0 A^0 \over m^2} (\partial_0 A_\rho)(\partial_0 A^\rho) ,\end{aligned}$$ where Latin indices $i, j$ run over $\{1, 2, 3\}$. The total Hamiltonian corresponding to this density is $$\begin{aligned}
H &= \int d^3 x \ham \nonumber \\
&= \int d^3 x \big( \b_1(\partial_\m A_i \partial_\m A_i - \partial_\m A_0 \partial_\m A_0)
+(\b_1- \bnew)[(\pd_0 A_0)^2 - (\pd_i A_i)^2 ] \nonumber \\
&\qquad+ \beta_4 {A_j A_k \over m^2} (\partial_j A_\rho)(\partial_k A^\rho)
- \beta_4 {A_0 A_0 \over m^2} (\partial_0 A_\rho)(\partial_0 A^\rho)\big)\,.
\label{hamiltonian}\end{aligned}$$ We have integrated by parts and assumed that $\partial_i A_j$ vanishes at spatial infinity; repeated lowered indices are summed (without any factors of the metric). Note that this Hamiltonian is identical to that of a theory with a smooth (positive semi-definite) potential instead of a Lagrange multiplier term, evaluated at field configurations for which the potential is minimized. Therefore, if the Hamiltonian is unbounded when the fixed-norm constraint is enforced by a Lagrange multiplier, it will also be unbounded in the case of a smooth potential.
There are only three dynamical degrees of freedom, so we may reparameterize $A_\m$ such that the fixed-norm constraint is automatically enforced and the allowed three-dimensional subspace is manifest. We define a boost variable $\phi$ and angular variables $\theta$ and $\psi$, so that we can write $$\begin{aligned}
A_0 &\equiv m \cosh \phi \\
A_i &\equiv m \sinh \phi f_i(\theta, \psi) \end{aligned}$$ in the timelike case with $A_\m A^\m = - m^2$, and $$\begin{aligned}
A_0 &\equiv m \sinh \phi \\
A_i &\equiv m \cosh \phi f_i(\theta, \psi) \end{aligned}$$ in the spacelike case with $A_\m A^\m = + m^2$. In these expressions, $$\begin{aligned}
f_1 &\equiv \cos \theta \cos \psi \\
f_2 &\equiv \cos \theta \sin \psi \\
f_3 &\equiv \sin \theta\,,\end{aligned}$$ so that $f_if_i = 1$. In terms of this parameterization, the Hamiltonian density for a timelike field becomes $$\begin{aligned}
\label{tham}
{\ham^{(t)} \over m^2} &= \b_1\sinh^2\phi \partial_\m f_i \pd_\m f_i +\b_1\partial_\m \phi \partial_\m \phi
+(\b_1 - \bnew)\left[ (\pd_0 \phi)^2 \sinh^2\phi - (\cosh \phi f_i \pd_i \phi + \sinh \phi \pd_i f_i)^2 \right] \nonumber \\
&\qquad +\beta_4 \sinh^2\phi \left[ (f_i \partial_i \phi)^2 + \sinh^2\phi (f_i \partial_i f_l)(f_j \partial_j f_l)\right] - \beta_4 \cosh^2\phi \left[ (\partial_0 \phi)^2 + \sinh^2\phi (\partial_0 f_i)^2 \right],\end{aligned}$$ while for the spacelike case we have $$\begin{aligned}
\label{sham}
{\ham^{(s)} \over m^2} &= \b_1 \cosh^2\phi \partial_\m f_i \pd_\m f_i
-\b_1 \partial_\m \phi \partial_\m \phi + (\b_1- \bnew) \left[ (\pd_0 \phi)^2 \cosh^2\phi
- (\sinh \phi f_i \pd_i \phi + \cosh \phi \pd_i f_i)^2 \right] \nonumber \\
&\qquad -\beta_4 \cosh^2\phi \left[ (f_i \partial_i \phi)^2 - \cosh^2\phi (f_i \partial_i f_l)(f_j \partial_j f_l)\right] +\beta_4 \sinh^2\phi \left[ (\partial_0 \phi)^2 - \cosh^2\phi (\partial_0 f_i)^2 \right].\end{aligned}$$
Expressed in terms of the variables $\phi, \theta, \psi$, the Hamiltonian is a function of initial data that automatically respects the fixed-norm constraint. We assume that the derivatives $\partial_\m A_\n (t_0, \vec{x})$ vanish at spatial infinity.
Timelike vector field
---------------------
We can now determine which values of the parameters $\{ \b_1, \bnew, \b_4\}$ lead to Hamiltonians that are bounded below, starting with the case of a timelike field. We can examine the various possible cases in turn.
- [**Case One: $\b_1=\bnew$ and $\beta_4 = 0$.**]{}
This is the sigma-model kinetic term (\[sigmamodel\]). In this case the Hamiltonian density simplifies to \^[(t)]{} = m\^2 \_1(\^2\_f\_i \_f\_i +\_\_) . It is manifestly non-negative when $\beta_1 >0$, and non-positive when $\beta_1 < 0$. The sigma-model choice $\b_1=\bnew >0$ therefore results in a theory that is stable. (See also §6.2 of [@Eling:2004dk].)
- [**Case Two: $\b_1 < 0$ and $\beta_4 = 0$.**]{}
In this case, consider configurations with $(\pd_0 f_i) \neq 0$, $(\pd_i f_j) = 0$, $\pd_\m \phi = 0$, $\sinh^2 \phi \gg 1$. Then we have $${\cal{H}}^{(t)} \sim m^2 \b_1 \sinh^2\phi (\partial_0 f_i)^2.$$ For $\b_1<0$, the Hamiltonian can be arbitrarily negative for any value of $\bnew$.
- [**Case Three: $\b_1 \geq 0$, $\bnew < \b_1$, and $\beta_4 = 0$.**]{}
We consider configurations with $\pd_\m f_i = 0$, $f_i \pd_i \phi \neq 0 $, $\pd_0 \phi = 0$, $\cosh^2 \phi \gg 1 $, which gives $${\cal{H}}^{(t)} \sim m^2 (\bnew-\b_1) \cosh^2\phi (f_i \partial_i \phi)^2.$$ Again, this can be arbitrarily negative.
- [**Case Four: $\b_1 \geq 0$, $\bnew > \b_1$, and $\beta_4 = 0$.**]{}
Now we consider configurations with $\pd_\m f_i = 0$, $f_i \pd_i \phi = 0$, $\pd_0 \phi \neq 0$, $\sinh^2 \phi \gg 1 $. Then, $${\cal{H}}^{(t)} \sim m^2 (\b_1-\bnew) \sinh^2\phi (\partial_0 \phi)^2,$$ which can be arbitrarily negative.
- [**Case Five: $\beta_4 \neq 0$.**]{}
Now we consider configurations with $\pd_\m f_i \neq 0$, $\partial_\m \phi =0$ and $\sinh^2 \phi \gg 1 $. Then, $${\cal{H}}^{(t)} \sim m^2 \beta_4 \left[ \sinh^4\phi (f_i \partial_i f_l)(f_k \partial_k f_l) - \sinh^2\phi \cosh^2\phi (\partial_0 f_i)^2\right]\,,$$ which can be arbitrarily negative for any non-zero $\beta_4$ and for any values of $\b_1$ and $\bnew$.
For any case other than the sigma-model choice $\b_1=\bnew$, it is therefore straightforward to find configurations with arbitrarily negative values of the Hamiltonian.
![Hamiltonian density (vertical axis) when $\b_1 = 1$, $\bnew = 1.1$, and $\theta = \psi = \partial_y \phi = \partial_z \phi = 0$ as a function of $\partial_t \phi$ (axis pointing into page) and $\partial_x \phi$ (axis pointing out of page) for various $\phi$ ranging from zero to $\phi_{crit} = \tanh^{-1} \sqrt{\b_1/\bnew}$, the value of $\phi$ for which the Hamiltonian is flat at $\partial_x \phi = 0$, and beyond. Notice that the Hamiltonian density turns over and becomes negative in the $\partial_t \phi$ direction when $\phi > \phi_{crit}$.[]{data-label="hamiltonian plots"}](hamplotcropped.pdf){width="60.00000%"}
Nevertheless, a perturbative analysis of the Hamiltonian would not necessarily discover that it was unbounded. The reason for this is shown in Fig. \[hamiltonian plots\], which shows the Hamiltonian density for the theory with $\b_1 = 1$, $\bnew = 1.1$, in a restricted subspace where $\partial_y\phi = \partial_z\phi = 0$ and $\theta = \phi = 0$, leaving only $\phi$, $\partial_t\phi$, and $\partial_x\phi$ as independent variables. We have plotted $\ham$ as a function of $\partial_t\phi$ and $\partial_x\phi$ for four different values of $\phi$. When $\phi$ is sufficiently small, so that the vector is close to being purely timelike, the point $\partial_t\phi = \partial_x\phi = 0 $ is a local minimum. Consequently, perturbations about constant configurations with small $\phi$ would appear stable. But for large values of $\phi$, the unboundedness of the Hamiltonian becomes apparent. This phenomenon will arise again when we consider the evolution of small perturbations in the next section. At the end of this section, we will explain why such regions of large $\phi$ are still in the regime of validity of the effective field theory expansion.
Spacelike vector field
----------------------
We now perform an equivalent analysis for an field with a spacelike expectation value. In this case all of the possibilities lead to Hamiltonians that are unbounded below, and the case $\b_1=\bnew > 0$ is not picked out.
- [**Case One: $\b_1 < 0$ and $\beta_4 = 0$.**]{}
Taking $(\pd_\m \phi) = 0$, $\pd_j f_i = 0$, $\pd_0 f_ i \neq 0$, we find $${\cal{H}}^{(s)} \sim m^2 \b_1\cosh^2\phi (\partial_0 f_i)^2.$$
- [**Case Two: $\b_1 > 0$, $\bnew \leq \b_1$, and $\beta_4 = 0$.**]{}
Now we consider $\pd_\m f_i = 0$, $\pd_i \phi \neq 0 $, $\pd_0 \phi = 0$, giving $${\cal{H}}^{(s)} \sim m^2 \left[ - \b_1 \pd_i \phi \pd_i \phi + (\bnew-\b_1) \sinh^2\phi (f_i \partial_i \phi)^2\right].$$
- [**Case Three: $\b_1 \geq 0$, $\bnew > \b_1$, and $\beta_4 = 0$.**]{}
In this case we examine $(\pd_0 \phi) \neq 0$, $\pd_\m f_i = 0$, $\pd_i \phi = 0$, which leads to $${\cal{H}}^{(s)} \sim m^2 (\b_1-\bnew) \cosh^2\phi (\partial_0 \phi)^2.$$
- [**Case Four: $\beta_4 \neq 0$.**]{}
Now we consider configurations with $\pd_\m f_i \neq 0$, $\partial_\m \phi =0$ and $\sinh^2 \phi \gg 1 $. Then, $${\cal{H}}^{(s)} \sim m^2 \beta_4 \left( \cosh^4\phi (f_i \partial_i f_l)(f_k \partial_k f_l) - \cosh^2\phi \sinh^2\phi (\partial_0 f_i)^2\right).$$
In every case, it is clear that we can find initial data for a spacelike vector field that makes the Hamiltonian as negative as we please, for all possible $\b_1$, $\beta_4$ and $\bnew$.
Smooth Potential
----------------
The usual interpretation of a Lagrange multiplier constraint is that it is the low-energy limit of smooth potentials when the massive degrees of freedom associated with excitations away from the minimum cannot be excited. We now investigate whether these degrees of freedom can destabilize the theory. Consider the most general, dimension four, positive semi-definite smooth potential that has a minimum when the vector field takes a timelike vacuum expectation value, $$V = {\xi \over 4} (A_\m A^\m + m^2)^2,$$ where $\xi$ is a positive dimensionless parameter. The precise form of the potential should not affect the results as long as the potential is non-negative and has the global minimum at $A_\m A^\m = -m^2$.
We have seen that the Hamiltonian is unbounded from below unless the kinetic term takes the sigma-model form, $(\partial_\m A_\n)(\partial^\m A^\n)$. Thus we take the Lagrangian to be $${\cal{L}} = -{1\over 2}(\partial_\m A_\n)(\partial^\m A^\n) - {\xi \over 4} (A_\m A^\m + m^2)^2.$$
Consider some fixed timelike vacuum $\bar A_\mu$ satisfying $\bar{A}_\m \bar{A}^\m = -m^2$. We may decompose the field into a scaling of the norm, represented by a scalar $\Phi$, and an orthogonal displacement, represented by vector $B_\mu$ satisfying $\bar{A}_\mu B^\mu = 0$. We thus have A\_= |[A]{}\_- [|[A]{}\_m]{} + B\_, where $$B_\m = \left(\eta_{\m\n}+{\bar{A}_\m\bar{A}_\n \over m^2}\right) A^\n~~\text{and}~~~ \Phi = {\bar{A}_\m A^\m \over m}+m.$$ With this parameterization, the Lagrangian is $${\cal{L}} = {1 \over 2} (\partial_\m \Phi)( \partial^\m \Phi) - {1 \over 2} (\partial_\m B_\n)(\partial^\m B^\n) -{\xi \over 4} (2m\Phi + B_\m B^\m - \Phi^2)^2.$$ The field $\Phi$ automatically has a wrong sign kinetic term, and, at the linear level, propagates with a dispersion relation of the form $$\omega_\Phi^2 = \vec{k}^2 - 2\xi m^2.$$ We see that in the case of a smooth potential, there exists a ghostlike mode (wrong-sign kinetic term) that is also tachyonic with spacelike wave vector and a group velocity that generically exceeds the speed of light. It is easy to see that sufficiently long-wavelength perturbations will exhibit exponential growth. The existence of a ghost when the norm of the vector field is not strictly fixed was shown in [@Elliott:2005va].
In the limit as $\xi$ goes to infinity, the equations of motion enforce a fixed-norm constraint and the ghostlike and tachyonic degree of freedom freezes. The theory is equivalent to one of a Lagrange multiplier if the limit is taken appropriately.
Discussion
----------
To summarize, we have found that the action in leads to a Hamiltonian that is globally bounded from below only in the case of a timelike sigma-model Lagrangian, corresponding to $\b_1 = \bnew > 0$ and $\beta_4 = 0$. Furthermore, we have verified (as was shown in [@Elliott:2005va]) that if the Lagrange multiplier term is replaced by a smooth, positive semi-definite potential, then a tachyonic ghost propagates and the theory is destabilized.
If the Hamiltonian is bounded below, the theory is stable, but the converse is not necessarily true. The sigma-model theory is the only one for which this criterion suffices to guarantee stability. In the next section, we will examine the linear stability of these models by considering the growth of perturbations. Although some models are stable at the linear level, we will see in the following section that most of these have negative-energy ghosts, and are therefore unstable once interactions are included. The only exceptions, both ghost-free and linearly stable, are the Maxwell and scalar models.
We showed in the previous section that, unless $\bnew - \beta_1$ and $\beta_4$ are exactly zero, the Hamiltonian is unbounded from below. However, the effective field theory breaks down before arbitrarily negative values of the Hamiltonian can be reached; when $\bnew \neq
\b_1$ and/or $\b_4 \neq 0$, in regions of phase space in which ${\cal H} < 0$ (schematically), \~- m\^2 e\^[4 ||]{} ()\^2 {, , }. The effective field theory breaks down when kinetic terms with four derivatives (the terms of next highest order in the effective field theory expansion) are on the order of terms with two derivatives, or, in the angle parameterization, when m\^2 e\^[4 ||]{} ()\^2 \~e\^[8 ||]{} ()\^4. In other words, the effective field theory is only valid when e\^[2 ||]{} || < m. In principle, terms in the effective action with four or more derivatives could add positive contributions to the Hamiltonian to make it bounded from below. However, our analysis shows that the Hamiltonian (in models other than the timelike sigma model with fixed norm) is necessarily concave down around the set of configurations with constant fields. If higher-derivative terms intervene to stabilize the Hamiltonian, the true vacuum would not have $H=0$. Theories could also be deemed stable if there are additional symmetries that lead to conserved currents (other than energy-momentum density) or to a reduced number of physical degrees of freedom.
Regardless of the presence of terms beyond leading order in the effective field theory expansion, due to the presence of the ghost-like and tachyonic mode (found in the previous section), there is an unavoidable problem with perturbations when the field moves in a smooth, positive semi-definite potential. This exponential instability will be present regardless of higher order terms in the effective field theory expansion because it occurs for very long-wavelength modes (at least around constant-field backgrounds).
Linear instabilities
====================
We have found that the Hamiltonian of a generic model is unbounded below. In this section, we investigate whether there exist actual physical instabilities at the linear level—[*i.e.*]{}, whether small perturbations grow exponentially with time. It will be necessary to consider the behavior of small fluctuations in every Lorentz frame,[^3] not only in the rest frame [@Kostelecky:2001xz; @Mattingly:2005re; @Adams:2006sv]. We find a range of parameters $\beta_i$ for which the theories are tachyon-free; these correspond (unsurprisingly) to dispersion relations for which the phase velocity satisfies $0 \leq v^2 \leq 1$. In §\[ghosts\] we consider the existence of ghosts.
Timelike vector field
---------------------
Suppose Lorentz invariance is spontaneously broken so that there is a preferred rest frame, and imagine that perturbations of some field in that frame have the following dispersion relation: $$\label{disprelation}
v^{-2} \o^2 = \kv \cdot \kv.$$ This can be written in frame-invariant notation as $$\label{Tdispersion}
(v^{-2} - 1) (t^\m k_\m)^2 = k_\m k^\m,$$ where $t^\m$ is a timelike Lorentz vector that characterizes the 4-velocity of the preferred rest frame. So, in the rest frame, $t^\m =
\{1,0,0,0\}$. Indeed, in the Appendix, we find dispersion relations for the modes of exactly the form in with $t^\m = \bar{A}^\m / m$ and $$v^2 = {\b_1 \over \b_1 - \b_4}$$ and $$v^2 = {\b_* \over \b_1 - \b_4}.$$
Now consider the dispersion relation for perturbations of the field in another (“primed”) frame. Let’s solve for $k_0' =
\o'$, the frequency of perturbations in the new frame. Expanded out, the dispersion relation reads $$\o'^2(1 + (v^{-2}-1)(t'^0)^2) + 2 \o' (v^{-2} - 1)t'^0 t'^i k_i' %\\
- \kv' \cdot \kv' + (v^{-2} - 1) (t'^i k_i')^2 = 0$$ where $i \in \{ 1,2,3 \}$. The solution for $\o'$ is: $$\label{omegaTdispersion}
\o' = {-(v^{-2} - 1)t'^0 t'^i k'_i \pm \sqrt{D_{(t)}} \over 1 +(v^{-2}-1)(t'^0)^2}\,,$$ where $${D_{(t)}} = \kv' \cdot \kv' + (v^{-2}-1)\left((t'^0)^2 \kv' \cdot \kv' - (t'^i k'_i)^2 \right).$$ In general, $t'^0 = \cosh \h$ and $t'^i = \sinh \h \, \hat{n}^i$, where $\hat{n}_i \hat{n}^i
= 1$ and $\h = \cosh^{-1}\g$ is a boost parameter. We therefore have $${D_{(t)}} = \kv' \cdot \kv' \left\{1 + (v^{-2}-1)\left[\cosh^2\h -
\sinh^2\h \, (\hat{n}\cdot \hat{k}')^2 \right] \right\},$$ where $\hat{k}' = \vec{k}'/|\vec{k}'|$. Thus ${D_{(t)}} $ is clearly greater than zero if $v \leq 1$. However, if $v > 1$ then ${D_{(t)}} $ can be negative for very large boosts if $\kv'$ is not parallel to the boost direction.
The sign of the discriminant ${D_{(t)}} $ determines whether the frequency $\o'$ is real- or complex-valued. We have shown that when the phase velocity $v$ of some field excitation is greater than the speed of light in a preferred rest frame, then there is a (highly boosted) frame in which the excitation looks unstable—that is, the frequency of the field excitation can be imaginary. More specifically, plane waves traveling along the boost direction with boost parameter $\gamma = \cosh \h$ have a growing amplitude if $\gamma^2 > 1/(1-v^{-2}) > 0$.
In Appendix \[Ap:A\], we find dispersion relations of the form in for the various massless excitations about a constant timelike background ($t^\m = \bA^\m /m$). Requiring stability and thus $0 \leq v^2 \leq 1$ leads to the inequalities, $$\label{spin1timelike}
0 \le {\beta_1 \over \beta_1 - \beta_4} \le 1$$ and $$\label{spin0timelike}
0 \le {\bnew \over \beta_1 - \beta_4} \le 1 \, .$$ Models satisfying these relations are stable with respect to linear perturbations in any Lorentz frame.
Spacelike vector field {#s superluminal}
----------------------
We show in Appendix \[Ap:A\] that fluctuations about a spacelike, fixed-norm, vector field background have dispersion relations of the form $$\label{Sdispersion}
(v^2 - 1) (s^\m k_\m)^2 = - k_\m k^\m,$$ with $s^\m = \bar{A}^\m / m$ and $$v^2 = { \b_1 + \b_4 \over \b_1}$$ and $$v^2 = { \b_1 + \b_4 \over \b_*}.$$ In frames where $s^\m = \{0, \hat{s}\}$, $v$ is the phase velocity in the $\hat{s}$ direction.
Consider solving for $k'_0 = \omega'$ in an arbitrary (“primed”) frame. The solution is as in , but with $v^{-2} \rightarrow
2 - v^{2}$ and $t'^\m \rightarrow s'^\m$. Thus, $$\label{omegaSdispersion}
\o' = {(v^{2} - 1)s'^0 s'^i k'_i \pm \sqrt{D_{(s)}} \over 1 +(1-
v^{2})(s'^0)^2}\,,$$ where $$D_{(s)} = \kv' \cdot \kv' - (v^{2}-1)\left[(s'^0)^2 \kv' \cdot \kv' - (s'^i k'_i)^2 \right].$$ In general, $s'^0 = \sinh \h$ and $s'^i = \cosh \h \, \hat{n}^i$ where $\hat{n}_i \hat{n}^i
= 1$ and $\h = \cosh^{-1}\g$ is a boost parameter. So, $$D_{(s)} = \kv' \cdot \kv' \left\{ 1 - (v^{2}-1)\left[\sinh^2\h -
\cosh^2\h \, (\hat{n}\cdot \hat{k'})^2 \right] \right\}.$$ which can be rewritten, $$D_{(s)} = \kv' \cdot \kv'\left\{ v^{2} + (1-v^{2})\cosh^2\h\left[1-
(\hat{n}\cdot \hat{k'})^2 \right]\right\}.$$ It is clear that $D_{(s)}$ is non-negative for all values of $\eta$ if and only if $ 0 \leq v^2 \leq 1$. The theory will be unstable unless $0 \leq v^2 \leq 1$.
The dispersion relations of the form for the massless excitations about the spacelike background are given in Appendix \[Ap:A\]. The requirement that $ 0 \leq v^2 \leq 1$ implies $$\label{spin1spacelike}
0 \le {\beta_1 + \beta_4 \over \beta_1} \le 1$$ and $$\label{spin0spacelike}
0 \le {\beta_1 + \beta_4 \over \bnew} \le 1 \, .$$ Models of spacelike fields will only be stable with respect to linear perturbations if these relations are statisfied.
The requirements or do not apply in the Maxwell case (when $\b_*=0=\b_4$), and those of or do not apply in the scalar case (when $\b_1 = 0 = \b_4$), since the corresponding degrees of freedom in each case do not propagate.
Stability is not frame-dependent
--------------------------------
The excitations about a constant background are massless (*i.e.* the frequency is proportional to the magnitude of the spatial wave vector), but they generally do not propagate along the light cone. In fact, when $v >1$, the wave vector is timelike even though the cone along which excitations propagate is strictly outside the light cone. We have shown that such excitations blow up in some frame. The exponential instability occurs for observers in boosted frames. In these frames, portions of constant-time hypersurfaces are actually inside the cone along which excitations propagate.
Why do we see the instability in only *some* frames when performing a linear stability analysis? Consider boosting the wave four-vectors of such excitations with complex-valued frequencies and real-valued spatial wave vectors back to the rest frame. Then, in the rest frame, both the frequency and the spatial wave vector will have non-zero imaginary parts. Such solutions with complex-valued $\vec{k}$ require initial data that grow at spatial infinity and are therefore not really “perturbations” of the background. But even though the field defines a rest frame, there is no restriction against considering small perturbations defined on a constant-time hypersurface in any frame. Well-behaved initial data can be decomposed into modes with real spatial wave vectors; if any such modes lead to runaway growth, the theory is unstable.
Negative Energy Modes {#ghosts}
=====================
We found above that manifest perturbative stability in all frames requires $0\le v^2 \le 1$. In the Appendix, we show that there are two kinds of propagating modes, except when $\b_*= \b_4 = 0$ or when $\b_1 = \b_4 = 0$. Based on the dispersion relations for these modes, the $0\le v^2 \le 1$ stability requirements translated into the inequalities for $\b_*, \b_1$, and $\b_4$ in - for timelike and - for spacelike [æ]{}ther. We shall henceforth assume that these inequalities hold and, therefore, that $\o$ and $\vec{k}$ for each mode are real in every frame. We will now show that, even when these requirements are satisfied and the theories are linearly stable, there will be negative-energy ghosts that imply instabilities at the nonlinear level (except for the sigma model, Maxwell, and scalar cases).
For timelike vector fields, with respect to the [æ]{}ther rest frame, the various modes correspond to two spin-1 degrees of freedom and one spin-0 degree of freedom. Based on their similarity in form to the timelike [æ]{}ther rest frame modes, we will label these modes once and for all as “spin-1” or “spin-0,” even though these classifications are only technically correct for timelike fields in the [æ]{}ther rest frame.
The solutions to the first order equations of motion for perturbations $\d A_\m$ about an arbitrary, constant, background $\bA_\m$ satisfying $\bA^\m \bA_\m \pm m^2 = 0$ are (see Appendix \[Ap:A\]): $$\d A_\m = \int d^4 k \, q_\m(k) e^{i k_\m x^\m}, \qquad q_\m(k) = q_\m^*(-k)$$ where either, $$\label{mode1}
q_\m(k) = i \a^\n k^\r {\bA^\s\over m} \e_{\m \n \r
\s}~~\text{and}~~\b_1 k_\m k^\m + \b_4{ ( \bA_\m k^\m)^2 \over m^2}= 0~~\text{and}~~\a^\n \bA_\n = 0 \qquad \text{(spin-1)}$$ where $\a^\n$ are real-valued constants or $$\label{mode2}
q_\m = i \a \left(\h_{\m \n} \pm {\bA_{\m} \bA_{\n}
\over m^2}\right)k^\n \qquad %\\
\text{and} \qquad \left( \bnew \h_{\m \n} + \left(\b_4 \pm (\bnew -
\b_1) \right) {\bA_{\m} \bA_{\n}\over m^2}\right)k^\m k^\n = 0 \qquad \text{(spin-0)}$$ where $\a$ is a real-valued constant.
Note that when $\b_1 = \b_4 = 0$, corresponding to the scalar form of , the spin-1 dispersion relation is satisfied trivially, because the spin-1 mode does not propagate in this case. Similarly, when $\b_* = \b_4 = 0$, the kinetic term takes on the Maxwell form in and the spin-0 dispersion relation becomes $\bA_\m k^\m = 0$; the spin-0 mode does not propagate in that case.
The Hamiltonian for either of these modes is $$\label{k hamiltonian}
H = \int d^3 k \, \left\{ \left[ \b_1 (\o^2+\kv\cdot \kv) +\b_4(-(\ba^0 \o)^2 + (\ba^i k_i)^2) \right] q^\m q_\m^{*} + (\b_1 - \bnew) (\o^2 q_0^*q_0 + k_i q_i^* k_j q_j ) \right\}\,,$$ where $k_0 = \o = \o(\kv)$ is given by the solution to a dispersion relation and where $\ba^\m \equiv \bA^\m / m$. One can show that, as long as $\b_1$ and $\b_4$ satisfy the conditions or that guarantee real frequencies $\o$ in all frames, we will have q\^\*\_q\^0 for all timelike and spacelike vector perturbations. We will now proceed to evaluate the Hamiltonian for each mode in different theories.
Spin-1 energies
---------------
In this section we consider nonvanishing $\b_4$, and show that the spin-1 mode can carry negative energy even when the conditions for linear stability are satisfied.
#### Timelike vector field. {#timelike-vector-field. .unnumbered}
Without loss of generality, set \_= m (, ). where $\hat{n} \cdot \hat{n} = 1$. The energy of the spin-1 mode in the timelike case is given by $$H = \int d^3k (\vec{k}\cdot \vec{k}) q^*_\m q^\m \left[{2 X \mp \beta_4 \sinh(2\h) (\hat{n}\cdot\hat{k})\sqrt{X} \over \beta_1 -\beta_4 \cosh^2\h } \right],$$ where $$X = \beta_1\left\{\beta_1 + \beta_4\left[(\hat{n}\cdot \hat{k})^2 \sinh^2\h - \cosh^2\h
\right]\right\}.$$ Looking specifically at modes for which $\hat{n} \cdot \hat{k} = +1$, we find $$H = \int d^3k (\vec{k}\cdot \vec{k})q^*_\m q^\m \left[{2\beta_1(\beta_1 - \beta_4) \mp \beta_4 \sinh(2\h) \sqrt{\beta_1(\beta_1 - \beta_4)} \over \beta_1 -\beta_4 \cosh^2\h } \right]\,.$$ The energy of such a spin-1 perturbation can be negative when $|\beta_4 \sinh(2\h)| > 2\sqrt{\beta_1(\beta_1 - \beta_4)}$. Thus it is possible to have negative energy perturbations whenever $\beta_4 \neq 0$. Perturbations with wave numbers perpendicular to the boost direction have positive semi-definite energies.
#### Spacelike vector field. {#spacelike-vector-field. .unnumbered}
Without loss of generality, for the spacelike case we set \_= m (, ), where $\hat{n} \cdot \hat{n} = 1$. The energy of the spin-1 mode in this case is given by $$H = \int d^3k (\vec{k}\cdot \vec{k}) q^*_\m q^\m
\left[{2 X \mp \beta_4 \sinh(2\h) (\hat{n}\cdot\hat{k})\sqrt{X} \over \beta_1 -\beta_4 \sinh^2\h } \right],$$ where $$X = \beta_1\left\{\beta_1 + \beta_4\left[(\hat{n}\cdot \hat{k})^2 \cosh^2\h - \sinh^2\h\right]\right\}.$$ Looking at modes for which $\hat{n} \cdot \hat{k} = +1$, we find $$H = \int d^3k (\vec{k}\cdot \vec{k})q^*_\m q^\m \left[{2\beta_1(\beta_1 + \beta_4) \mp \beta_4 \sinh(2\h) \sqrt{\beta_1(\beta_1 + \beta_4)} \over \beta_1 - \beta_4 \sinh^2\h } \right]\,.$$ Thus, the energy of perturbations can be negative when $|\beta_4 \sinh(2\h)| > 2\sqrt{\beta_1(\beta_1 + \beta_4)}$. Thus it is possible to have negative energy perturbations whenever $\beta_4 \neq 0$. Perturbations with wave numbers perpendicular to the boost direction have positive semi-definite energies. In either the timelike or spacelike case, models with $\b_4\neq 0$ feature spin-1 modes that can be ghostlike.
We note that the effective field theory is valid when $k < e^{- 3 |\h|} m$, as detailed in §\[Validity of effective field theory\]. But even if $\h$ is very large, the effective field theory is still valid for very long wavelength perturbations, and therefore such long wavelength modes with negative energies lead to genuine instabilities.
Spin-0 energies
---------------
We now assume the inequalities required for linear stability, or , and also that $\b_4 = 0$. We showed above that, otherwise, there are growing modes in some frame or there are propagating spin-1 modes that have negative energy in some frame. When $\bnew \neq 0$, the energy of the spin-0 mode in is given by $$\label{mode2 energy}
H = 2 \b_1 \a^2 \int d^3 k \, (\ba_\r k^\r)^2\left( \o^2(\kv)\left[\pm 1 - (1 - \b_1/\bnew) \ba_0^2 \right] + \o(\kv)\, \ba_0 (1 - \b_1/\bnew) \ba_i k_i \right)$$ for $\bA_\m \bA^\m \pm m^2 = 0$ and $\ba_\m \equiv \bA_\m / m$.
#### Timelike vector field. {#timelike-vector-field.-1 .unnumbered}
We will now show that the quadratic order Hamiltonian can be negative when the background is timelike and the kinetic term does not take one of the special forms (sigma model, Maxwell, or scalar). Without loss of generality we set $\ba_0 = \cosh \h $ and $\ba_i = \sinh \h \, \hat{n}_i$, where $\hat{n}\cdot \hat{n} = 1$. Then plugging the freqency $\o(\vec{k})$, as defined by the spin-0 dispersion relation, into the Hamiltonian gives $$\label{T spin-0 energy}
H = \b_1 \a^2 \int d^3 k \, (\ba_\r k^\r)^2 \left[{
2 X \pm (1- \beta_1/\bnew)\sinh 2\h (\hat{n}\cdot \hat{k})\sqrt{X}
\over
1 + (\b_1/\bnew - 1)\cosh^2\h
}\right],$$ where $$X = {1+ (\b_1/\bnew -1) [\cosh^2\h - (\hat{n}\cdot\hat{k})^2\sinh^2\h ]}.$$ If $\hat{n}\cdot\hat{k} \neq 0$, the energy can be negative. In particular, if $\hat{n}\cdot\hat{k} = 1$ we have $$H = \b_1 \a^2 \int d^3 k \, (\ba_\r k^\r)^2 \left[{
2 \b_1/\bnew \pm (1- \beta_1/\bnew)\sinh 2\h \sqrt{\b_1 / \bnew}
\over
1+ (\b_1/\bnew - 1)\cosh^2\h
}\right].$$ Given that $ \b_1 / \bnew -1 \geq 0$, $H$ can be negative when $| \sinh 2 \h | > 2 \sqrt{\b_1 / \bnew} / (\b_1 / \bnew - 1)$.
We have thus shown that, for timelike backgrounds, there are modes that in some frame have negative energies and/or growing amplitudes as long as $\b_1 \neq \bnew$, $\b_1 \neq 0$, and $\bnew\neq 0$. Therefore, the only possibly stable theories of timelike fields are the special cases mentioned earlier: the sigma-model ($\b_1 = \bnew$), Maxwell ($\bnew = 0$), and scalar ($\b_1=0$) kinetic terms.
#### Spacelike vector field. {#spacelike-vector-field.-1 .unnumbered}
For the spacelike case, without loss of generality we set $\ba_0 = \sinh \h $ and $\ba_i = \cosh \h \, \hat{n}_i$, where $\hat{n}\cdot \hat{n} = 1$. Once again, plugging the frequency $\o(k)$ into the Hamiltonian gives $$\label{S spin-0 energy}
H = \b_1 \a^2 \int d^3 k \, (\ba_\r k^\r)^2 \left[{
- 2 X \pm (1 - \beta_1/\beta_*)\sinh 2\h (\hat{n}\cdot \hat{k})\sqrt{X}
\over
1 + (1- \b_1 / \bnew)\sinh^2\h
}\right] ,$$ where $$X = {1 + (1- \b_1 / \bnew ) \left[\sinh^2\h - (\hat{n}\cdot\hat{k})^2\cosh^2\h \right]}.$$ Upon inspection, one can see that there are values of $\hat{n}\cdot \hat{k}$ and $\h$ that make $H$ negative, except when $\bnew = 0$ (Maxwell) or $\b_1 = 0$ (scalar). Again, the Hamiltonian density is less than zero for modes with wavelengths sufficiently long ($k < e^{-3 |\h|} m$), so the effective theory is valid.
Maxwell and Scalar Theories
===========================
We have shown that the only version of the theory for which the Hamiltonian is bounded below is the timelike sigma-model theory ${\cal L}_K = -(1/2)(\partial_\mu A_\nu)
(\partial^\mu A^\nu)$, corresponding to the choices $\b_1=\bnew$, $\b_4=0$, with the fixed-norm condition imposed by a Lagrange multiplier constraint. (Here and below, we rescale the field to canonically normalize the kinetic terms.) However, when we looked for explicit instabilities in the form of tachyons or ghosts in the last two sections, we found two other models for which such pathologies are absent: the Maxwell Lagrangian \_K = -F\_F\^ , \[maxwell2\] corresponding to $\bnew = 0 = \b_4$, and the scalar Lagrangian \_K = (\_A\^)\^2 , corresponding to $\b_1 = 0 = \b_4$. In both of these cases, we found that the Hamiltonian is unbounded below,[^4] but a configuration with a small positive energy does not appear to run away into an unbounded region of phase space characterized by large negative and positive balancing contributions to the total energy.
These two models are also distinguished in another way: there are fewer than three propagating degrees of freedom at first order in perturbations in the Maxwell and scalar Lagrangian cases, while there are three in all others. This is closely tied to the absence of perturbative instabilities; the ultimate cause of those instabilities can be traced to the difficulty in making all of the degrees of freedom simultaneously well-behaved. The drop in number of degrees of freedom stems from the fact that $A_0$ lacks time derivatives in the Maxwell Lagrangian and that the $A_i$ lack time derivatives in the scalar Lagrangian. In other words, some of the vector components are themselves Lagrange multipliers in these special cases.
Only two perturbative degrees of freedom—the spin-1 modes—propagate in the Maxwell case (cf. - when $\b_* = 0 = \b_4$). The “mode” in is a gauge degree of freedom; at first order in perturbations the Lagrangian has a gauge-like symmetry under $\d A_\m \rightarrow \d A_\m + \partial_\m \phi(x)$ where $\bA^\m \partial_\m \phi = 0$. As expected of a gauge degree of freedom, the spin-0 mode has zero energy and does not propagate. Meanwhile, the spin-1 perturbations propagate as well-behaved plane waves and have positive energy. We note that the Dirac method for counting degrees of freedom in constrained dynamical systems implies that there are *three* degrees of freedom [@Bluhm:2008yt].[^5] The additional degree of freedom, not apparent at the linear level, could conceivably cause an instability; this mode does not propagate because it is gauge-like at the linear level, but there is no gauge symmetry in the full theory.
In the scalar case, there are no propagating spin-1 degrees of freedom. The spin-0 degree of freedom has a nontrivial dispersion relation but no energy density (cf. -, , and when $\b_1 = 0 = \b_4$) at leading order in the perturbations. Essentially, the fixed-norm constraint is incompatible with what would be a single propagating scalar mode in this model; the theory is still dynamical, but perturbation theory fails to capture its dynamical content.
Each of these models displays some idiosyncratic features, which we now consider in turn.
Maxwell action
--------------
The equation of motion for the Maxwell Lagrangian with a fixed-norm constraint is \_F\^ = -2A\^. Setting $A_\mu A^\mu = \mp m^2$, the Lagrange multiplier is given by = A\_\_F\^. For timelike [æ]{}ther fields, the sign of $\l$ is preserved along timelike trajectories since, when the kinetic term takes the special Maxwell form, there is a conserved current (in addition to energy-momentum density) due to the Bianchi identity[^6]: $$\label{eq:conservedcurrent}
0 = \partial_\n (\partial_\m F^{\m \n}) = -2 \partial_\n (\lambda A^\n).$$ In particular, the condition that $\lambda = 0$ is conserved along timelike $A^\n$ [@Jacobson:2000xp; @Bluhm:2008yt]. In the presence of interactions this will continue to be true only if the coupling to external sources takes the form of an interaction with a conserved current, $A_\mu J^\mu$ with $\partial_\mu J^\mu=0$.
If we take the timelike Maxwell theory coupled to a conserved current and restrict to initial data satisfying $\lambda = 0$ at every point in space, the theory reduces precisely to Maxwell electrodynamics—not only in the equation of motion, but also in the energy-momentum tensor. We can therefore be confident that this theory, restricted to this subset of initial data, is perfectly well-behaved, simply because it is identical to conventional electromagnetism in a nonlinear gauge [@Nambu:1968qk; @Chkareuli:2006yf; @Bluhm:2007bd].
In the case of a spacelike vector expectation value, there is an explicit obstruction to finding smooth time evolution for generic initial data. In this case, the constraint equations are $$- A_0^2 + A_i A_i = m^2 \qquad \text{and} \qquad \pd_i \pd^i A_0 - \pd_0 \pd_i A^i = -2\l A_0.$$ Suppose spatially homogeneous initial conditions for the $A_i$ are given. Without loss of generality, we can align axes such that $$A_\mu(t_0) = (A_0(t_0),0,0,A_3(t_0)),$$ where $-A^2_0 + A^2_3 = m^2$. If $A_i A_i \neq m^2$, the equations of motion are $$\label{badexample}
\pd_\mu {F^{\mu}}_\nu = 0.$$ The $\nu = 3$ equation reads $$\pd_\mu {F^\mu}_3 = -\frac{\pd^2 A_3}{\pd t^2} = 0,$$ whose solutions are given by $$A_3(t) = A_3(t_0) + C (t-t_0),$$ where $C$ is determined by initial conditions. $A_0$ is determined by the fixed-norm constraint $A_0 = \pm \sqrt{A_3^2 - m^2}$. If $C \neq 0$, $A_0$ will eventually evolve to zero. Beyond this point, $A_3$ keeps decreasing, and the fixed-norm condition requires that $A_0$ be imaginary, which is unacceptable since $A_\mu$ is a real-valued vector field. Note that this never happens in the timelike case, as there always exists some real $A_0$ that satisfies the constraint for any value of $A_3$. The problem is that $A_3$ evolves into the ball $A^2_i < m^2$, which is catastrophic for the spacelike, but not the timelike, case. An analogous problem arises even when the Lagrange multiplier constraint is replaced by a smooth potential.
It is possible that this obstruction to a well-defined evolution will be regulated by terms of higher order in the effective field theory. Using the fixed-norm constraint and solving for $A_0$, the derivative is $${\partial_\m A_0} = \frac{A_i}{\sqrt{A_j A_j - m^2}} {\partial_\m A_i }.$$ As $A_j A_j$ approaches $m^2$, with finite derivatives of the spatial components, the derivative of the $A_0$ component becomes unbounded. If higher-order terms in the effective action have time derivatives of the component $A_0$, these terms could become relevant to the vector field’s dynamical evolution, indicating that we have left the realm of validity of the low-energy effective field theory we are considering.
We are left with the question of how to interpret the timelike Maxwell theory with intial data for which $\l \neq 0$. If we restrict our attention to initial data for which $\l < 0$ everywhere, then the evolution of the $A_i$ would be determined and the Hamiltonian would be positive. We have $$\begin{aligned}
H &= {1 \over 2} \int d^3 x \,\left( {1 \over 2}F^2_{ij} + (\partial_0 A_i)^2 - (\partial_i A_0)^2 \right)\\
&= {1 \over 2} \int d^3 x \,\left( {1 \over 2}F^2_{ij} + F_{0i}F_{0i} - 2(\partial_i A_0)F_{i0} \right)\\
&= {1 \over 2} \int d^3 x \,\left( {1 \over 2}F^2_{ij} + F_{0i}F_{0i} + 2A_0 \partial_i F_{i0} \right)\\
&= {1 \over 2} \int d^3 x \,\left( {1 \over 2}F^2_{ij} + F_{0i}F_{0i} - 4 \l A_0^2 \right), \label{ham with lambda}\end{aligned}$$ which is manifestly positive when $ \l < 0 $. However, it is not clear why we should be restricted to this form of initial data, nor whether even this restriction is enough to ensure stability beyond perturbation theory.
The status of this model in both the spacelike and timelike cases remains unclear. However, there are indications of further problems. For the spacelike case, Peloso *et. al.* find a linear instability for perturbations with wave numbers on the order of the Hubble parameter in an exponentially expanding cosmology [@Himmetoglu:2008zp; @Himmetoglu:2008hx]. For the timelike case, Seifert found a gravitational instability in the presence of a spherically symmetric source [@Seifert:2007fr].
Scalar action
-------------
The equation of motion for the scalar Lagrangian with a fixed-norm constraint is $$\partial^\n \partial_\m A^\m = 2 \lambda A^\n.$$ Using the fixed-norm constraint ($A_\m A^\m = \mp m^2$), we can solve for the Lagrange multiplier field, $$\lambda = \mp {1 \over 2 m^2} A_\n \partial^\n \partial_\m A^\m.$$ In contrast with the Maxwell theory, in the scalar theory it is the timelike case for which we can demonstrate obstacles to smooth evolution, while the spacelike case is less clear. (The Hamiltonian is bounded below, but there are no perturbative instabilities or known obstacles to smooth evolution.)
When the vector field is timelike, we have four constraint equations in the scalar case, $$A_0^2 - A_iA_i = m^2 \qquad \text{and} \qquad \partial_i(\partial_\m A^\m) = 2\lambda A_i.$$ Suppose we give homogeneous initial conditions such that $A_0(t_0) > m$. Align axes such that, $$A_\m(t_0) = \left(A_0(t_0),0,0,A_3(t_0) \right),$$ where $A_3(t_0)^2 = A_0(t_0)^2 - m^2$. Note that, since $A_3(t_0) \neq 0$, we have that $\lambda = 0$ from the $\n = 3$ equation of motion. The $\n = 0$ equation of motion therefore gives, $${d^2 A_0 \over d t^2} = 0.$$ We see that the timelike component of the vector field has the time-evolution, $$A_0(t) = A_0(t_0) + C (t-t_0).$$
For generic homogeneous initial conditions, $C \neq 0$. In this case, $A_0$ will not have a smooth time evolution since $A_0$ will saturate the fixed-norm constraint, and beyond this point $A_0$ will continue to decrease in magnitude. To satisfy the fixed-norm constraint, the spatial components of the vector field $A_i$ would need to be imaginary, which is unacceptable since $A_\m$ is a real-valued vector field. This problem never occurs for the spacelike case since there always exist real values of $A_i$ that satisfy the constraint for any $A_0$.
Again, it is possible that this obstruction to a well-defined evolution will be regulated by terms of higher order in the effective field theory. The time derivative of $A_3$ is $${\partial_\m A_3} = \frac{A_0}{\sqrt{A_0 A_0 - m^2}} {\partial_\m A_0 }.$$ As $A_0 A_0$ approaches $m^2$, with finite derivatives of $A_0$, the derivative of the spatial component $A_3$ becomes unbounded. If higher-order terms in the effective action have time derivatives of the components $A_i$, these terms could become relevant to the vector field’s dynamical evolution, indicating that we have left the realm of validity of the low-energy effective field theory we are considering.
Whether or not a theory with a scalar kinetic term and fixed expectation value is viable remains uncertain.
Conclusions
===========
In this paper, we addressed the issue of stability in theories in which Lorentz invariance is spontaneously broken by a dynamical fixed-norm vector field with an action $$S = \int d^4 x \, \left( -{1\over 2}\b_1 F_{\m\n}F^{\m\n} %&
-\bnew(\partial_\m A^\m)^2 -\beta_4 {A^\m A^\n \over m^2} (\partial_\m A_\rho)(\partial_\n A^\rho) + \lambda(A^{\mu} A_{\mu} \pm m^2) \right)\,,$$ where $\lambda$ is a Lagrange multiplier that strictly enforces the fixed-norm constraint. In the spirit of effective field theory, we limited our attention to only kinetic terms that are quadratic in derivatives, and took care to ensure that our discussion applies to regimes in which an effective field theory expansion is valid.
We examined the boundedness of the Hamiltonian of the theory and showed that, for generic choices of kinetic term, the Hamiltonian is unbounded from below. Thus for a generic kinetic term, we have shown that a constant fixed-norm background is not the true vacuum of the theory. The only exception is the timelike sigma-model Lagrangian ($\b_1 = \bnew$, $\beta_4 = 0$ and $A^{\mu} A_{\mu} = -m^2$), in which case the Hamiltonian is positive-definite, ensuring stability. However, if the vector field instead acquires its vacuum expectation value by minimizing a smooth potential, we demonstrated (as was done previously in [@Elliott:2005va]) that the theory is plagued by the existence of a tachyonic ghost, and the Hamiltonian is unbounded from below. The timelike fixed-norm sigma-model theory nevertheless serves as a viable starting point for phenomenological investigations of Lorentz invariance; we explore some of this phenomenology in a separate paper [@Carroll:2009en].
We next examined the dispersion relations and energies of first-order perturbations about constant background configurations. We showed that, in addition to the sigma-model case, there are only two other choices of kinetic term for which perturbations have non-negative energies and do not grow exponentially in any frame: the Maxwell ($\bnew = \b_4 = 0$) and scalar ($\b_1 = \b_4 = 0$) Lagrangians. In either case, the theory has fewer than three propagating degrees of freedom at the linear level, as some of the vector components in the action lack time derivatives and act as additional Lagrange multipliers. A subset of the phase space for the Maxwell theory with a timelike field is well-defined and stable, but is identical to ordinary electromagnetism. For the Maxwell theory with a spacelike field, or the scalar theory with a timelike field, we can find explicit obstructions to smooth time evolution. It remains unclear whether the timelike Maxwell theory or the spacelike scalar theory can exhibit true violation of Lorentz invariance while remaining well-behaved.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are very grateful to Ted Jacobson, Alan Kostelecky, and Mark Wise for helpful comments. This research was supported in part by the U.S. Department of Energy and by the Gordon and Betty Moore Foundation.
Solutions to the linearized equations of motion {#Ap:A}
===============================================
We start by finding the solution to the equations of motion, linearized about a timelike, fixed-norm background, $A_\m$. Then, showing less details, we find the solutions to the equations of motion linearized about a spacelike background. Finally, we put the solutions in both cases into the compact form of -. Our results agree with the solutions for Goldstone modes found in [@Gripaios:2004ms].
The equations of motion for a timelike (+) or spacelike ($-$) vector field are , $$Q_\m \equiv \left(\eta_{\m \n} \pm {A_\m A_\n \over m^2}\right)\left(\b_1 \pd_\r \pd^\r A^\n + (\bnew-\b_1)\pd^\n \pd_\r A^\r + {\beta_4 } G^\n \right) = 0,$$ where $G^\n$ is defined in and $A^\m Q_\m = 0$ identically.
#### Timelike background. {#timelike-background. .unnumbered}
Consider perturbations about an arbitrary, constant (in space and time) timelike background $A_\m = \bar{A}_\m$ that satisfies the constraint: $\bar{A}_\m \bar{A}^\m = - m^2$. Define perturbations by $A_\m = \bA_\m + \d A_\m$. Then, to first order in these perturbations, $\bA^\m Q_\m = 0$ identically, and $\eta^{\m \n} \bA_\m \d A_\n = 0$ by the constraint. We can define a basis set of four Lorentz 4-vectors $n^\alpha$, with components $$\label{T vielbeins}
n^0_\m = \bA_\m / m \ , \qquad n^i_\m\ ;\qquad \; i \in \{ 1, 2, 3\}\,,$$ such that $$\label{T vielbein ortho}
\eta^{\m \n} n^\a_\m n^\b_\n = \eta^{\a \b}.$$
The independent perturbations are $\d a^\a \equiv \eta^{\m \n} n^\a_\m \d A_\n$ for $\a = 1, 2, 3$. ($\d a^0$ is zero at first order in perturbations due to the constraint.) It is then clear that there are three independent equations of motion at first order in pertubations (assuming the constraint) for the three independent perturbations, $$\d Q^i \equiv n^i_\n \left(\b_1 \pd_\r \pd^\r \d A^\n + (\bnew-\b_1)\pd^\n \pd_\r \d A^\r + \beta_4 n^0_\m n^0_\rho \partial^\m \partial^\rho \delta A^\n \right) = 0,$$ where $i \in \{ 1, 2, 3\}$. We look for plane wave solutions for the $\d A$: $$\label{T fourier}
\d A_\m = \int d^4 k \, q_\m (k) e^{i k_\n x^\n}.$$ Since $\eta^{\m \n} n^0_\m \d A_\n = 0$, at first order, $$\label{T q decomp}
q_\m = c_j n^j_\m \qquad \text{where} \qquad j \in \{1,2,3\}.$$ The equations of motion become the algebraic equations: $$\begin{aligned}
0 &= \left( \b_1 k_\r k^\r n^i_\n n^{j \n} + (\bnew - \b_1) n^i_\n k^\n n^j_\m k^\m + \beta_4 n^0_\m n^0_\rho k^\m k^\rho n^i_\n n^{j \n} \right) c_j\\
&= \left( \b_1 k_\r k^\r \d^{i j} + (\bnew - \b_1) n^i_\n k^\n n^j_\m k^\m + \beta_4 n^0_\m n^0_\rho k^\m k^\rho \d^{i j} \right) c_j\\
&\equiv M^{i j} c_j.\end{aligned}$$
The three independent solutions to these equations are given by setting an eigenvalue of the matrix $M$ to zero and setting $c_i$ to the corresponding eigenvector. Setting an eigenvalue of $M$ equal to zero gives a dispersion relation, $$\b_1 k_\r k^\r + \beta_4 (n^0_\m k^\m)^2 = 0,$$ with two linearly independent eigenvectors, $$\label{eq:speedoflightTmodes}
(e_2)_i = \e_{2 i j}n^j_\m k^\m \qquad ; \qquad (e_3)_i = \e_{3 i j}n^j_\m k^\m.$$ The second eigenvalue of $M$ gives the dispersion relation, $$\begin{aligned}
\bnew k_\r k^\r + (\bnew - \b_1+\beta_4)(n^0_\m k^\m)^2= 0,\label{weirdTdispersion}\end{aligned}$$ with corresponding eigenvector, $$\label{eq:weirdTmodes}
c_i = n^i_\m k^\m.$$
#### Spacelike background. {#spacelike-background. .unnumbered}
The first order linearized equations of motion about a spacelike background are: $$\d Q^a \equiv n^a_\n \left(\b_1 \pd_\r \pd^\r \d A^\n + (\bnew-\b_1)\pd^\n \pd_\r \d A^\r + \beta_4 n^3_\m n^3_\rho \partial^\m \partial^\rho \delta A^\n\right) = 0$$ where $a \in \{ 0, 1, 2\}$ and where, similarly to the timelike case, we have defined the set of four Lorentz 4-vectors, $n^\a_\m$, to be $$\label{S vielbeins}
n^3_\m = \bA_\m / m \qquad \text{and} \qquad n^a_\m ; \; a \in \{ 0,1, 2\}$$ such that $$\label{S vielbein ortho}
\eta^{\m \n} n^\a_\m n^\b_\n = \eta^{\a \b}.$$ The independent perturbations are $\d a^\a \equiv \eta^{\m \n}
n^\a_\m \d A_\n$ for $\a = 0,1,2$. ($\d a^3$ is zero at first order in perturbations due to the constraint.)
Again we look for plane wave solutions of the form in . But now, since $\eta^{\m \n} n^3_\m \d A_\n = 0$, at first order, $$\label{S q decomp}
q_\m = c_a n^a_\m \qquad \text{where} \qquad a \in \{0,1,2\}.$$ The equations of motion become the algebraic equations: $$\begin{aligned}
&= \left( \b_1 k_\r k^\r n^a_\n n^{b \n} + (\bnew - \b_1) n^a_\n k^\n n^b_\m k^\m + \beta_4 n^3_\m n^3_\rho k^\m k^\rho n^a_\n n^{b \n}\right) c_b\\
&= \left( \b_1 k_\r k^\r \h^{a b} + (\bnew - \b_1) n^a_\n k^\n n^b_\m k^\m+ \beta_4 n^3_\m n^3_\rho k^\m k^\rho \h^{ab} \right) c_b\\
&\equiv M^{a b} c_b. \qquad a,b\in\{0,1,2\}\end{aligned}$$ Two independent solutions correspond to the dispersion relation ($a \in \{0,1,2\}$) $$\b_1 k_\r k^\r + \beta_4 (n^3_\m k^\m)^2 = 0 \,,$$ with corresponding eigenmodes $$\label{eq:speedoflightSmodes}
(e_1)_a = \e_{a 1 b 3}n^b_\m k^\m \qquad ; \qquad (e_2)_a = \e_{a b
2 3}n^b_\m k^\m.$$ The third solution corresponds to the dispersion relation $$\label{weirdSdispersion}
\bnew k_\r k^\r - (\bnew - \beta_1-\beta_4)(n^3_\m k^\m)^2= 0\,,$$ with corresponding eigenmode $$\label{eq:weirdTmodes}
c_a = \eta_{a b} n^b_\m k^\m.$$
#### General expression. {#general-expression. .unnumbered}
We can express the solutions in the timelike and spacelike cases in a compact form by using the orthonormality of the $n^\a_\m$, , along with , , and the fact that,[^7] $$\label{epsilonrelation}
\e_{\a \b \r \s}n^\a_\m n^\b_\n = \e_{\m \n \a \b}n^\a_\r n^\b_\s.$$ Then plugging and into yields the solutions, $$\label{apA:deltaA}
\d A_\m = \int d^4 k \, q_\m(k) e^{i k_\n x^\n}$$ where either, $$\label{apA:spin-1}
q_\m(k) = i \a^\n k^\r {\bA^\s\over m} \e_{\m \n \r
\s}~~\text{and}~~\b_1 k_\r k^\r + \beta_4 \left(\bar{A}_\m k^\m \over m\right)^2 = 0~~\text{and}~~\a^\n \bA_\n = 0,$$ where $\a^\n$ are real-valued constants or, $$\label{scalar mode}
q_\m = i \a \left(\h_{\m \n} \pm {\bA_{\m} \bA_{\n}
\over m^2}\right)k^\n \qquad %\\
\text{and} \qquad \bnew k_\r k^\r \pm (\bnew-\beta_1 \pm \beta_4) \left(\bar{A}_\m k^\m \over m\right)^2 = 0,$$ where $\a$ is a real-valued constant. The reality of the $\a$’s follows from the condition, $q_\m(k) = q_\m^*(-k)$, that holds if and only if $\d A_\m$ in is real. In , the “$+$” sign corresponds to the timelike background and the “$-$” sign to a spacelike background.
[^1]: This effectively eliminates the scalar case.
[^2]: In terms of the coefficients, $c_i$, defined in [@Jacobson:2004ts] and used in many other publications on [æ]{}ther theories, \_i = [c\_i 16 G m\^2]{} where $G$ is the gravitational constant.
[^3]: The theory of perturbations about a constant background is equivalent to a theory with explicit Lorentz violation because the first order Lagrange density includes the term, $\lambda \bA^\m \d A_\m$, where $\bA^\m$ is effectively some constant coefficient.
[^4]: Boundedness of the Hamiltonian was considered in [@Clayton:2001vy].
[^5]: For a discussion of constrained dynamical systems see [@Henneaux:1992ir].
[^6]: If $\l > 0$ initially, then it must pass through $\l =0$ to reach $\l < 0$—but $\l = 0$ is conserved along timelike trajectories, so $\l$ can at best stop at $\l = 0$.
[^7]: This follows from the invariance of the Levi-Civita tensor, $$\e_{\a \b \g \d}n^\a_\m n^\b_\n n^\g_\r n^\d_\s = \e_{\m \n \r \s}$$ plus orthonormality, .
|
---
abstract: |
We study the spectral shift function (SSF) $\xi(\lambda)$ and the resonances of the operator $H_V := \big( \sigma \cdot
(-i\nabla - \textbf{A}) \big)^{2} + V$ in $L^2(\mathbb{R}^3)$ near the origin. Here $\sigma := (\sigma_1,\sigma_2,\sigma_3)$ are the $2 \times 2$ Pauli matrices and $V$ is a hermitian potential decaying exponentially in the direction of the magnetic field $\textbf{B} := \text{curl} \hspace{0.6mm} \textbf{A}$. We give a representation of the derivative of the SSF as a sum of the imaginary part of a holomorphic function and a harmonic measure related to the resonances of $H_V$. This representation warrant the Breit-Wigner approximation moreover we deduce information about the singularities of the SSF at the origin and a local trace formula.
address: 'Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago de Chile'
author:
- 'Diomba <span style="font-variant:small-caps;">Sambou</span>'
title: Spectral Shift Function and Resonances near the low ground state for Pauli and Schrödinger operators
---
**AMS 2010 Mathematics Subject Classification:** 35P25, 35J10, 47F05, 81Q10.
**Keywords:** Spectral shift function, Pauli operator, Schrödinger operator, Resonances, Breit-Wigner approximation, Trace formula.
Introduction and motivations {#s1}
============================
Unperturbed operator
--------------------
Consider the three-dimensional Pauli operator acting in $L^{2}({{\mathbb{R}}}^3) := L^{2}({{\mathbb{R}}}^3,
\mathbb{C}^{2})$ and describing a quantum non-relativistic spin-$\frac{1}{2}$ particle subject to a magnetic field $\textbf{B} :
{{\mathbb{R}}}^3 \longrightarrow {{\mathbb{R}}}^3$ pointing at the $x_3$ direction: $$\label{eq1,1}
\textbf{B}(\textbf{x}) = \big( 0,0,b(\textbf{x})
\big), \quad \textbf{x} := (x_{\perp},x_3)
\in {{\mathbb{R}}}^{3}, \quad x_{\perp} := (x_1,x_2)
\in {{\mathbb{R}}}^{2}.$$ Then ${x_\perp}= (x_1,x_2) \in \mathbb{R}^{2}$ are the variables on the plane perpendicular to the magnetic field. Let $\textbf{A} =
(a_{1},a_{2},a_{3}) : \mathbb{R}^{3}
\rightarrow \mathbb{R}^{3}$ denote the magnetic potential generating the magnetic field, namely $\textbf{B}(\textbf{x}) :=
\text{curl} \hspace{0.6mm} \textbf{A}
(\textbf{x})$. Since $\text{div}
\hspace{0.6mm} \textbf{B} = 0$ then $b$ is independent of $x_3$. Hence there is no loss of generality in assuming that $a_{j}$, $j = 1$, $2$ are independent of $x_3$ and $a_{3} = 0$: $$\label{eq1,01}
\textbf{A}(\textbf{x})
= \big( a_1({x_\perp}),a_2({x_\perp}),0 \big),
\quad b(\textbf{x}) = b({x_\perp}) =
\partial_1 a_2({x_\perp}) - \partial_2 a_1({x_\perp}).$$ Let $\sigma_{j}$, $j \in \lbrace 1, 2, 3
\rbrace$ be the $2 \times 2$ Pauli matrices given by $$\sigma_{1} := \begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}, \hspace{0.5cm}
\sigma_{2} := \begin{pmatrix}
0 & -i \\
i & 0
\end{pmatrix}, \hspace{0.5cm}
\sigma_{3} := \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}.$$ The free self-adjoint Pauli operator is initially defined on $C_{0}^{\infty}({{\mathbb{R}}}^3,\mathbb{C}^{2})$ (then closed in $L^{2}({{\mathbb{R}}}^3)$) by $$\label{eq1,2}
H_0 := \big( \sigma \cdot (-i\nabla - \textbf{A})
\big)^{2}, \qquad \sigma := (\sigma_1,\sigma_2,\sigma_3).$$ A trivial computation shows that $$\label{eq1,3}
H_0 = \begin{pmatrix}
(-i\nabla - \textbf{A})^{2} - b & 0 \\
0 & (-i\nabla - \textbf{A})^{2} + b
\end{pmatrix}.$$
We will assume (abusing the terminology) that $b : {{\mathbb{R}}}^{2} \rightarrow {{\mathbb{R}}}$ is an admissible magnetic field. This means that there exists a positive constant $b_{0}$ satisfying $b({x_\perp}) = b_{0} +
\tilde{b}({x_\perp})$, $\tilde{b}$ being a function such that the Poisson equation $$\label{eq1,4}
\Delta \tilde{\varphi} = \tilde{b}$$ admits a solution $\tilde{\varphi} \in C^{2}({{\mathbb{R}}}^{2})$ verifying $\sup_{{x_\perp}\in {{\mathbb{R}}}^{2}} \vert D^{\alpha}
\tilde{\varphi}({x_\perp}) \vert < \infty$, $\alpha \in \mathbb{Z}_{+}^{2}$, $\vert \alpha \vert
\leq 2$, (we refer to [@rage Section 2.1] for more details and examples on admissible magnetic fields). Introduce $\varphi_{0}({x_\perp}) = b_{0} \vert {x_\perp}\vert^{2}/4$ and $\varphi := \varphi_{0} + \tilde{\varphi}$ so that we have $\Delta \varphi = b$. Define originally on $C_{0}^{\infty}({{\mathbb{R}}}^{2},{{\mathbb{C}}})$ the operators $$\label{eq1,5}
a = a(b) := -2i \textup{e}^{-\varphi}
\frac{\partial}{\partial \bar{z}} \textup{e}^{\varphi}
\quad \text{and} \quad a^{\ast} = a^{\ast}(b) := -2i
\textup{e}^{\varphi} \frac{\partial}{\partial z}
\textup{e}^{-\varphi}$$ with $z := x_{1} + i x_{2}$, $\bar{z} := x_{1} -
i x_{2}$ and introduce the operators $$\label{eq1,6}
H_{1}(b) = a^{\ast} a \quad \text{and} \quad \quad
H_{2}(b) = a a^{\ast}.$$ The spectral properties of $H_{j} = H_{j}(b)$, $j = 1$, $2$ are well know from [@rage Proposition 1.1]: $$\label{eq1,7}
\begin{cases}
\sigma (H_{1}) \subseteq \lbrace 0 \rbrace \cup
[\zeta,+\infty) \hspace{0.1cm} \text{whith
$0$ an eigenvalue of infinite multiplicity}, \\
\sigma (H_{2}) \subseteq [\zeta,+\infty), \quad \dim
\hspace{0.5mm} \textup{Ker} \hspace{0.5mm} H_{2} = 0,
\end{cases}$$ where $$\label{eq1,8}
\zeta := 2 b_{0} e^{-2 \hspace{0.5mm} \textup{osc}
\hspace{0.5mm} \tilde{\varphi}}, \quad
\textup{osc} \hspace{0.5mm} \tilde{\varphi}:=
\sup_{{x_\perp}\in \mathbb{R}^{2}} \tilde{\varphi} ({x_\perp})
- \inf_{{x_\perp}\in {{\mathbb{R}}}^{2}} \tilde{\varphi} ({x_\perp}).$$ The orthogonal projection onto $\text{Ker} \hspace{0.5mm} H_{1}(b)$ will be denoted by $p = p(b)$. From [@hal Theorem 2.3] we know that it admits a continuous integral kernel $\mathcal{P}_{b}({x_\perp},{x_\perp}')$, ${x_\perp}$, ${x_\perp}' \in {{\mathbb{R}}}^{2}$. Furthermore by [@rage Lemma 2.3] $$\label{eq1,81}
\frac{b_0}{2\pi} e^{-2\textup{osc} \hspace{0.5mm}
\tilde{\varphi}} \leq \mathcal{P}_{b}({x_\perp},{x_\perp}) \leq
\frac{b_0}{2\pi} e^{2\textup{osc} \hspace{0.5mm}
\tilde{\varphi}}, \qquad {x_\perp}\in {{\mathbb{R}}}^2.$$
Under the above considerations by taking $a_1 = -\partial_2 \varphi$ and $a_2 = \partial_1 \varphi$ the operator $H_0$ can be written in $L^{2}({{\mathbb{R}}}^{3})
= L^{2}({{\mathbb{R}}}^{2}) \otimes L^{2}({{\mathbb{R}}})$ as $$\label{eq1,9}
\small{H_0 = \begin{pmatrix}
H_{1}(b) \otimes 1 + 1 \otimes
\left( -\frac{d^{2}}{dx_3^{2}} \right) & 0 \\
0 & H_{2}(b) \otimes 1 + 1 \otimes
\left( -\frac{d^{2}}{dx_3^{2}} \right)
\end{pmatrix} =: \begin{pmatrix}
\mathcal{H}_{1}(b) & 0 \\
0 & \mathcal{H}_{2}(b)
\end{pmatrix}}.$$ The spectrum of $-\frac{d^{2}}{dx_3^{2}}$ originally defined on $C_{0}^{\infty}({{\mathbb{R}}},{{\mathbb{C}}})$ coincides with $[0,+\infty)$ and is absolutely continuous. Then and imply that $$\label{eq1,10}
\sigma (H_0) = \sigma_{\textup{\textbf{ac}}} (H_0) =
[0,+\infty),$$ (see also [@rage Corollary 2.2]).
Perturbed operator and the spectral shift function
--------------------------------------------------
On the domain of ${{H_0}}$ we introduce the perturbed operator $$\label{eq1,11}
{{H_V}}:= {{H_0}}+ V,$$ where $V$ is identified with the multiplication operator by the matrix-valued function $$\label{eq1,12}
V(\textbf{x}) := \begin{pmatrix}
v_{11}(\textbf{x}) & v_{12}(\textbf{x}) \\
v_{21}(\textbf{x}) & v_{22}(\textbf{x})
\end{pmatrix} \in \mathfrak{B}_h({{\mathbb{C}}}^2), \quad \textbf{x}
\in {{\mathbb{R}}}^3,$$ $\mathfrak{B}_h({{\mathbb{C}}}^2)$ being the set of $2 \times 2$ hermitian matrices. Throughout this paper we require an exponential decay along the direction of the magnetic field for the electric potential $V$ in the following sense: $$\label{eq1,13}
\begin{cases}
0 \not\equiv V \in C^0 ({{\mathbb{R}}}^3), \quad
\vert v_{\ell k}(\textbf{x}) \vert \leq \text{Const.}
\hspace{0.5mm} \langle {x_\perp}\rangle^{-{m_\perp}}
\hspace{0.5mm} e^{-\gamma \langle
x_3 \rangle}, \quad 1 \leq \ell,k \leq 2 \\
\textup{with ${m_\perp}> 2$, $\gamma > 0$ constant
and $\langle y \rangle := \sqrt{1 + \vert y \vert^2}$
for $y \in {{\mathbb{R}}}^d$.}
\end{cases}$$
Introduce some notations. Let $\mathscr{H}$ be a separable Hilbert space and ${{\mathcal{S}_\infty}}(\mathscr{H})$ be the set of compact linear operators on $\mathscr{H}$. Denote by $s_k(T)$ the $k$-th singular value of $T \in {{\mathcal{S}_\infty}}(\mathscr{H})$. The Schatten-von Neumann class ideals ${{\mathcal{S}_q}}(\mathscr{H})$, $q \in [1,+\infty)$ are defined by $$\label{eq1,14}
{{\mathcal{S}_q}}(\mathscr{H}) := \Big\lbrace T \in {{\mathcal{S}_\infty}}(\mathscr{H})
: \Vert T \Vert^q_{{\mathcal{S}_q}}:= \sum_k s_k(T)^q < +\infty
\Big\rbrace.$$ For $\lceil q \rceil := \min \big\lbrace n \in \mathbb{N}
: n \geq q \big\rbrace$ and $T \in {{\mathcal{S}_q}}(\mathscr{H})$ the regularized determinant $\textup{det}_{\lceil q \rceil}
(I - T)$ is defined by $$\label{eq1,15}
\small{\textup{det}_{\lceil q \rceil} (I - T)
:= \prod_{\mu \hspace*{0.1cm} \in \hspace*{0.1cm} \sigma (T)}
\left[ (1 - \mu) \exp \left( \sum_{k=1}^{\lceil q \rceil-1}
\frac{\mu^{k}}{k} \right) \right]}.$$ The case $q = 1$ corresponds to the trace class operators while the case $q = 2$ coincides with the Hilbert-Schmidt operators.
Now let $\mathcal{H}_0$ and $\mathcal{H}$ be two self-adjoint operators in $\mathscr{H}$ such that $$\label{eq1,16}
V := \mathcal{H} - \mathcal{H}_0 \in {{\mathcal{S}_1}}(\mathscr{H}).$$ There exists an important object in the theory of scattering associated to the pair of operators $(\mathcal{H},\mathcal{H}_0)$ called the *spectral shift function* (SSF) $\xi(\lambda)$. The concept of SSF was first formally introduced by Lifshits [@lif]. The mathematical theory of the SSF was developed by Krein [@kre1]. For trace class perturbations the SSF is related to the determinant perturbation by the Krein’s formula (see for instance [@kre1], [@kre2]) $$\label{eq1,17}
\xi(\lambda) = \frac{1}{\pi}
\lim_{\varepsilon \longrightarrow 0^+}
\text{Arg} \det
\big( I + V(\mathcal{H}_0 - \lambda - i\varepsilon)^{-1}
\big), \quad \text{a.e.} \hspace*{0.15cm} \lambda \in {{\mathbb{R}}},$$ the branch of the argument being fixed by the condition $$\text{Arg} \det
\big( I + V(\mathcal{H}_0 - z)^{-1} \big)
\longrightarrow 0, \quad {\textup{Im}}(z) \longrightarrow + \infty.$$ Actually on the basis of the invariance principle (see for instance [@BiYa]) the SSF is well defined once there exists $\ell > 0$ such that $$\label{eq1,18}
(\mathcal{H} - i)^{-\ell} - (\mathcal{H}_0 - i)^{-\ell}
\in {{\mathcal{S}_1}}(\mathscr{H}).$$ It’s the function whose derivative is given by the following distribution: $$\label{eq1,19}
\xi' : f \longmapsto - \text{Tr} \big( f(\mathcal{H})
- f(\mathcal{H}_0) \big), \quad
f \in C_0^\infty({{\mathbb{R}}}).$$ Following the Birman-Krein theory (see [@BiKr]) the SSF coincides with the scattering phase $s(\lambda) = -\frac{1}{2\pi} \text{Arg} \det
S(\lambda)$ where $S(\lambda)$ is the scattering matrix. More precisely by the Birman-Krein formula (see [@BiKr]) the SSF is related to $S(\lambda)$ by $\det S(\lambda) = e^{-2i\pi S(\lambda)}$ for almost every $\lambda \in \sigma_{ac}({{H_0}})$. The above interpretation of the SSF as the scattering phase stimulates its investigation in quantum-mechanical problems. We refer to the review [@BiYa] and the book [@Yaf] for a large detailed bibliography about the SSF.
In our case assumption on $V$ implies that there exists $\mathscr{V} \in
\mathscr{L}(\mathscr{H})$ such that $$\label{eq1,191}
\vert V \vert^\frac{1}{2} (\textbf{x})
= \mathscr{V} \left( \langle {x_\perp}\rangle
^{-\frac{m_\perp}{2}} \otimes
e^{-\frac{\gamma}{2}
\langle t \rangle} \right), \quad
\textbf{x} = ({x_\perp},t) \in {{\mathbb{R}}}^3, \quad m_\perp > 2.$$ The standard criterion [@sim Theorem 4.1] implies that $$\label{eq1,20}
\langle {x_\perp}\rangle
^{-\frac{m_\perp}{2}} \otimes
e^{-\frac{\gamma}{2}
\langle t \rangle} (-\Delta + 1)^{-1} \in
{{\mathcal{S}_2}}\big( L^2({{\mathbb{R}}}^3,{{\mathbb{C}}}) \big).$$ Then this together with the diamagnetic inequality (see [@avr Theorem 2.3]-) and the boundedness of the magnetic field $b$ imply that $$\label{eq1,21}
\vert V \vert^\frac{1}{2} ({{H_0}}- i)^{-1} \in
{{\mathcal{S}_2}}\big( L^2({{\mathbb{R}}}^3) \big).$$ Therefore exploiting the resolvent identity we obtain $$\label{eq1,22}
({{H_V}}- i)^{-1} - ({{H_0}}- i)^{-1}
\in {{\mathcal{S}_1}}\big( L^2({{\mathbb{R}}}^3) \big).$$ Namely holds with $\ell = 1$ with respect to the operators ${{H_V}}$, ${{H_0}}$ and the Hilbert space $\mathscr{H} = L^2({{\mathbb{R}}}^3)$. So the distribution $$\label{eq1,23}
\xi' : f \longmapsto - \text{Tr} \big( f({{H_V}})
- f({{H_0}}) \big), \quad
f \in C_0^\infty({{\mathbb{R}}})$$ is well defined. For our purpose it is more convenient to introduce the regularized spectral shift function (see for instance [@kop] or [@bou]) $$\label{eq1,24}
\xi_2(\lambda) = \frac{1}{\pi}
\lim_{\varepsilon \longrightarrow 0^+}
\text{Arg} \hspace*{0.1cm} {\det}_2
\big( I + V({{H_0}}- \lambda - i\varepsilon)^{-1} \big)$$ whose derivative is given by the distribution $$\label{eq1,25}
\xi_2' : f \longmapsto - \text{Tr} \left( f({{H_V}})
- f({{H_0}}) - \frac{d}{d\varepsilon}
f({{H_0}}+ \varepsilon V)_{\vert \varepsilon = 0} \right),
\quad f \in C_0^\infty({{\mathbb{R}}}).$$ From the relation between $\xi'$ and $\xi_2'$ given by Lemma \[l5,1\], we will deduce the properties of the SSF. In the present paper the main result concerns a representation of the derivative of the SSF near the low ground state of the operator ${{H_0}}$ corresponding to the origin as a sum of a harmonic measure (related to the resonances of the operator ${{H_V}}$ near zero) and the imaginary part of a holomorphic function. Such representation justifies the Breit-Wigner approximation (see Theorem \[t2,1\]) and implies a trace formula (see Theorem \[t2,2\]) as in [@pet], [@bru], [@DiZe], [@bon]. We derive also from our main result an asymptotic expansion of the SSF near the origin (see Theorem \[t2,3\]). Similar results are obtained in [@bon] for the SSF near the Landau levels as well in [@kho]. On the other hand the singularities of the SSF associated to the pair $({{H_V}},{{H_0}})$ is also studied in [@rage] with polynomial decay on the electric potential $V$. In Remark \[r2,2\], we compare our results to those of [@rage]. The case of the Dirac Hamiltonian with admissible magnetic fields is considered in [@tda] where the singularities of the SSF near $\pm m$ are investigated. Results obtained there are closely related to those from [@rage].
The paper is organized as follows. In Section \[s2\] we formulate our main results. Sections \[s3\]-\[s4\] are devoted to the study of the resonances of ${{H_V}}$ near the origin. In the first one we define the resonances and in the second one we establish upper bounds on their number near the origin. Sections \[s5\]-\[s7\] are respectively devoted to the proofs of the main results. Section \[sa\] is a brief appendix on finite meromorphic operator-valued functions.
Statement of the main results {#s2}
=============================
First introduce some notations and terminology.
Denote by $\vert V \vert$ the multiplication operator by the matrix-valued function $$\label{eq2,1}
\sqrt{V^\ast V}(\textbf{x}) = \sqrt{V^2}(\textbf{x})
=: \big\lbrace \vert V
\vert_{\ell k}(\textbf{x}) \big\rbrace, \quad 1
\leq \ell,k \leq 2$$ and by $J := sign(V)$ the matrix sign of $V$ which satisfies $V = J \vert V \vert$. We will say that $V$ is of definite sign if the multiplication operator $V(\textbf{x})$ by the matrix-valued function $V(\textbf{x})$ satisfies $$\pm V(\textbf{x}) \geq 0$$ for any $\textbf{x} \in {{\mathbb{R}}}^3$. It is easy to check that in this case we have respectively $V
= J \vert V \vert = \pm \vert V \vert$. Then without loss of generality we will say that $V$ is of definite sign $J = \pm$.
Let $\textbf{W}$ be the multiplication operator by the function $\textbf{W} : {{\mathbb{R}}}^2 \longrightarrow
{{\mathbb{R}}}$ defined by $$\label{eq2,2}
\displaystyle \textbf{W}({x_\perp}) :=
\int_{{\mathbb{R}}}\vert V \vert_{11} ({x_\perp},x_3)dx_3.$$ Hypothesis on $V$ implies that $$\label{eq2,3}
0 \leq \textbf{W} ({x_\perp}) \leq \text{Const.}'
\hspace{0.5mm} \langle {x_\perp}\rangle^{-{m_\perp}},
\quad {m_\perp}> 2, \quad {x_\perp}\in {{\mathbb{R}}}^2,$$ where $\text{Const.}' = \text{Const.} \hspace{0.5mm}
\int_{{\mathbb{R}}}e^{-\gamma \langle x_3 \rangle} dx_3$. Then by [@rage Lemma 2.3] the positive self-adjoint Toeplitz operator $p \textbf{W} p$ is of trace class, $p = p(b)$ being the orthogonal projection onto $\text{Ker}
\hspace{0.5mm} H_{1}(b)$ defined by .
Introduce $e_\pm$ the multiplication operators by the functions $e^{\pm\frac{\gamma}{2} \langle \cdot
\rangle}$ respectively and let $c : L^{2}({{\mathbb{R}}})
\longrightarrow {{\mathbb{C}}}$ be the operator given by $$\label{eqc1}
c(u) := \langle u,
e^{-\frac{\gamma}{2} \langle \cdot \rangle}
\rangle$$ while $c^{\ast} : {{\mathbb{C}}}\longrightarrow
L^{2}({{\mathbb{R}}})$ satisfies $c^{\ast}(\lambda) =
\lambda e^{-\frac{\gamma}{2} \langle \cdot
\rangle}$. Define the operator $K : L^{2}({{\mathbb{R}}}^{3})
\longrightarrow L^{2}({{\mathbb{R}}}^{2})$ by $$\label{eq2,4}
K := \frac{1}{\sqrt{2}} (p \otimes c)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
e_+ \vert V \vert^{\frac{1}{2}}.$$ To be more explicit we have $$(K \psi)(\textbf{x}) = \frac{1}{\sqrt{2}}
\int_{{{\mathbb{R}}}^{3}}
{\mathcal P}_{b}({x_\perp},{x_\perp}^\prime)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
\vert V \vert^{\frac{1}{2}} ({x_\perp}^\prime,x_3^\prime)
\psi ({x_\perp}^\prime,x_3^\prime)
d{x_\perp}^\prime dx_3^\prime,$$ where ${\mathcal P}_{b}(\cdot,\cdot)$ is the integral kernel of the orthogonal projection $p$. Obviously the adjoint operator $K^{\ast} : L^{2}({{\mathbb{R}}}^{2}) \longrightarrow
L^{2}({{\mathbb{R}}}^{3})$ verifies $$(K^{\ast}\varphi)({x_\perp},x_3) =
\frac{1}{\sqrt{2}} \vert V \vert^{\frac{1}{2}}
({x_\perp},x_3)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
(p \varphi)({x_\perp}).$$ Then $$\label{eq2,5}
K K^{\ast} = \begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
\frac{p \textbf{\textup{W}} p}{2}$$ so that it is a self-adjoint positif compact operator.
Now let us introduce technical important conditions. Define the constant $$\label{eq2,51}
N_{\gamma,\zeta} := \min \left( \frac{\gamma}{2},
\sqrt{\zeta} \right),$$ where $\gamma$ and $\zeta$ are respectively defined by and . Let $\mathscr{W}_\pm \Subset \Omega_\pm$ be open relatively compact subsets of $\pm ]0,N_{\gamma,\zeta}^2[ e^{\pm i]-2\theta_0,2\varepsilon_0[}$ such that $0 < \min(\theta_0,\varepsilon_0)$ and $\max(\theta_0,\varepsilon_0) < \frac{\pi}{2}$. Let $r > 0$ be a small parameter and assume that $\mathscr{W}_\pm$ and $\Omega_\pm$ are simply connected sets independent of $r$. We also assume that the intersections between $\pm ]0,N_{\gamma,\zeta}^2[$ and $\mathscr{W}_\pm$, $\Omega_\pm$ are intervals. Hence we set $I_\pm := \mathscr{W}_\pm \cap \pm ]0,N_{\gamma,\zeta}^2[$.
In the case where the potential $V$ is of definite sign $J = sign(V)$ the representation of the SSF near zero can be specified. This required firstly that for $k \in {{\mathbb{C}}}$ small enough the operator $I + \frac{iJ}{k} K^\ast K$ be invertible. That is for $\text{Arg} \hspace*{0.1cm} k \neq -J\frac{\pi}{2}$. Secondly that the condition $$\label{eq2,6}
-J\frac{\pi}{2} \notin \left( \frac{\pi}{2} \right)_\mp
\pm [-\theta_0,\varepsilon_0]$$ be satisfied with respect to the subscript “$\pm$” in $\mathscr{W}_\pm \Subset \Omega_\pm$, $I_\pm := \mathscr{W}_\pm \cap \pm ]0,N_{\gamma,\zeta}^2[$, where $\left( \frac{\pi}{2} \right)_- = 0$ and $\left( \frac{\pi}{2} \right)_+ = \frac{\pi}{2}$.
\[r2,1\] $-$
**(i)** Under our considerations on $\theta_0$ and $\varepsilon_0$ above condition is satisfied in the case “$+$” for $J = \pm$. Namely $$\mp \frac{\pi}{2} \notin
[-\theta_0,\varepsilon_0], \quad J = \pm.$$
**(ii)** In the case “$-$” condition is satisfied for $J = +$. Namely $$- \frac{\pi}{2} \notin
\left[ \frac{\pi}{2} - \varepsilon_0,
\frac{\pi}{2} + \theta_0 \right].$$
From now on the set of the resonances near zero of ${{H_V}}$ (see Definition \[d3,1\]) will be denoted by $\text{Res}({{H_V}})$. Our first main result goes as follows:
\[t2,1\] $\textup{(Breit-Wigner approximation)}$
Assume that assumption holds. Let $\mathscr{W}_\pm \Subset \Omega_\pm$ be open relatively compact subsets of $\pm ]0,N_{\gamma,\zeta}^2[ e^{\pm i]-2\theta_0,2
\varepsilon_0[}$ as above. Choose moreover $0 < s_1 < \sqrt{\textup{dist} \big( \Omega_\pm,0
\big)}$. Then there exists $r_0 > 0$ and holomorphic functions $g_\pm$ in $\Omega_\pm$ satisfying for any $\mu \in rI_\pm$ and $r < r_0$ $$\label{eq2,7}
\xi'(\mu) = \frac{1}{r \pi} {\textup{Im}}\hspace{0.5mm} g'_\pm
\left( \frac{\mu}{r},r \right) +
\sum_{\substack{w \in \textup{Res}({{H_V}}) \cap
r \Omega_\pm \\ {\textup{Im}}(w) \neq 0}}
\frac{{\textup{Im}}(w)}{\pi \vert \mu - w \vert^2}
- \sum_{w \in \textup{Res}({{H_V}}) \cap
r I_\pm} \delta (\mu - w),$$ where the functions $g_\pm(z,r)$ satisfy the bound $$\label{eq2,8}
\begin{split}
g_\pm(z,r) & = \mathcal{O} \left[ \textup{Tr}
\hspace{0.4mm} {{\bf 1}}_{(s_1\sqrt{r},\infty)}
\big( p \textbf{\textup{W}} p \big)
\vert \ln r \vert +
\Tilde{n}_1 \left( \frac{1}{2} s_1\sqrt{r} \right) +
\Tilde{n}_2 \left( \frac{1}{2} s_1\sqrt{r} \right)
\right] \\
& = \mathcal{O} \left( \vert \ln r \vert
r^{-1/m_\perp} \right),
\end{split}$$ uniformly with respect to $0 < r < r_0$ and $z \in
\Omega_\pm$, with $\Tilde{n}_q (\cdot)$, $q = 1$, $2$ defined by .
Furthermore for potentials of definite sign $J = sign(V)$ we have for $\lambda \in rI_\pm$ $$\label{eq2,9}
\frac{1}{r} {\textup{Im}}\hspace{0.5mm} g'_\pm
\left( \frac{\lambda}{r},r \right) =
\frac{1}{r} {\textup{Im}}\hspace{0.5mm} \Tilde{g}'_\pm
\left( \frac{\lambda}{r},r \right) +
{\textup{Im}}\hspace{0.5mm} \Tilde{g}'_{1,\pm}(\lambda)
+ {{\bf 1}}_{(0,N_{\gamma,\zeta}^2)}(\lambda)
J \phi'(\lambda),$$ where the function $\phi$ is defined by $$\label{eq2,10}
\phi(\lambda) := \textup{Tr}
\hspace{0.4mm} \left( \arctan
\frac{K^\ast K}{\sqrt{\lambda}} \right)
= \textup{Tr} \hspace{0.4mm} \left( \arctan
\frac{p\textbf{\textup{W}}p}{2\sqrt{\lambda}}
\right),$$ the functions $z \mapsto \Tilde{g}_\pm (z,r)$ being holomorphic in $\Omega_\pm$ and satisfying $$\label{eq2,11}
\Tilde{g}_\pm (z,r) = \mathcal{O}
\big( \vert \ln r \vert \big),$$ uniformly with respect to $0 < r < r_0$ and $z \in \Omega_\pm$. The functions $z \mapsto
\Tilde{g}_{1,\pm}(z)$ are holomorphic in $\pm ]0,N_{\gamma,\zeta}^2[
e^{\pm i]-2\theta_0,2\varepsilon_0[}$ and there exists a positive constant $C_{\theta_0}$ depending on $\theta_0$ such that $$\label{eq2,12}
\vert \Tilde{g}_{1,\pm}(z) \vert \leq
C_{\theta_0} \sigma_2 \left( \sqrt{\vert z \vert}
\right)^{\frac{1}{2}}$$ for $z \in \pm ]0,N_{\gamma,\zeta}^2[
e^{\pm i]-2\theta_0,2\varepsilon_0[}$, where the quantity $\sigma_2(\cdot)$ is defined by .
As first consequence of the above theorem we have the following result describing the asymptotic behaviour of the SSF on the right of the low ground state.
\[t2,2\] $\textup{(Singularity at the low ground state)}$
Assume that $V$ satisfies assumption with definite sign $J =
sign(V)$. Then $$\label{eq2,13}
\xi(\lambda) = \frac{J}{\pi} \phi(\lambda) +
\mathcal{O} \left( \phi(\lambda)^{\frac{1}{2}}
\right) + \mathcal{O} \bigl( \vert \ln \lambda
\vert^2 \bigr)$$ as $\lambda \searrow 0$, the function $\phi(\lambda)$ being defined by .
\[r2,2\] $-$
**(i)** Since for $\lambda > 0$ $$\xi(-\lambda) = - \# \big\lbrace
\text{discrete eigenvalues of ${{H_V}}$
lying in $(-\infty,-\lambda)$} \big\rbrace$$ then for $V \geq 0$ we have $\xi(-\lambda) = 0$.
**(ii)** In [@rage] the singularities of the SSF near the origin are studied. If $\textbf{\textup{W}}$ satisfies assumptions **(A1)**, **(A2)** or **(A3)** implying respectively , or then it is proved in [@rage] that $$\label{eq2,131}
\xi(\lambda) = \frac{J}{\pi} \phi(\lambda)
\big( 1 + o(1) \big), \quad \lambda
\searrow 0.$$ Thus provides a remainder estimate of when $\textbf{\textup{W}}$ satisfies assumption **(A1)**. However for $V \leq 0$ it is proved in [@rage] that $$\xi(-\lambda) = -\textup{Tr}
\hspace{0.4mm} {{\bf 1}}_{(2\sqrt{\lambda},
\infty)} \big( p \textbf{\textup{W}} p
\big) \big( 1 + o(1) \big), \quad
\lambda \searrow 0.$$
As second consequence of Theorem \[t2,1\] we have the following
\[t2,3\]
Let the domains $\mathscr{W}_\pm \Subset \Omega_\pm$ be as in Theorem \[eq2,1\]. Assume that $f_\pm$ is holomorphic in a neighbourhood of $\Omega_\pm$ and let $\psi_\pm \in C_0^{\infty} \big( \Omega_\pm \cap
{{\mathbb{R}}}\big)$ satisfy $\psi_\pm(\lambda) =
1$ near $\Omega_\pm \cap {{\mathbb{R}}}$. Then under the assumptions of Theorem \[eq2,1\] $$\label{eq2,14}
\textup{Tr} \hspace{0.4mm} \left[ (\psi_\pm f_\pm)
\left( \frac{{{H_V}}}{r} \right) - (\psi_\pm f_\pm)
\left( \frac{{{H_0}}}{r} \right) \right] =
\sum_{w \in \textup{Res}({{H_V}}) \cap
r \mathscr{W}_\pm} f_\pm \left( \frac{w}{r} \right)
+ E_{f_\pm,\psi_\pm}(r)$$ with $$\label{eq2,15}
\vert E_{f_\pm,\psi_\pm}(r) \vert \leq M (\psi_\pm)
\sup \big\lbrace \vert f_\pm(z) \vert : z \in
\Omega_\pm \setminus \mathscr{W}_\pm : {\textup{Im}}(z) \leq 0
\big\rbrace \times N(r),$$ where $$\label{eq2,16}
\begin{split}
N(r) & = \textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(s_1\sqrt{r},\infty)}
\big( p \textbf{\textup{W}} p \big)
\vert \ln r \vert +
\Tilde{n}_1 \left( \frac{1}{2} s_1\sqrt{r} \right) +
\Tilde{n}_2 \left( \frac{1}{2} s_1\sqrt{r} \right)
\\
& = \mathcal{O} \left( \vert \ln r \vert
r^{-1/m_\perp} \right).
\end{split}$$
Our results remain true if instead the operator ${{H_V}}$ defined by we consider in $L^2({{\mathbb{R}}}^3,{{\mathbb{C}}})$ the perturbed Schrödinger operator $$(-i\nabla - \textbf{A})^{2} - b + V$$ on $Dom \big( (-i\nabla - \textbf{A})^{2} -
b \big)$ with $V(\textbf{\textup{x}}) =
\mathcal{O} \left( \langle {x_\perp}\rangle^{-{m_\perp}}
\hspace{0.5mm} e^{-\gamma \langle x_3 \rangle}
\right)$ for any $\textbf{\textup{x}} \in {{\mathbb{R}}}^3$, $m_\perp > 2$, $\gamma > 0$ as in . Here $\textbf{\textup{W}}$ is just given by $\displaystyle
\textbf{\textup{W}}({x_\perp}) = \int_{{\mathbb{R}}}\vert V
({x_\perp},x_3) \vert dx_3$ for any ${x_\perp}\in {{\mathbb{R}}}^2$ and in identities - the matrix $\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}$ is removed.
**Acknowledgements**. The author is partially supported by the Chilean Program *Núcleo Milenio de Física Matemática RC$120002$*. The author wishes to express his gratitude to V. Bruneau for suggesting the study of this problem.
Definition of the resonances {#s3}
============================
The potential $V$ is assumed to satisfy . We recall also that $p = p(b)$ is the orthogonal projection onto $\text{Ker} \hspace{0.5mm}
H_{1}$ with $H_1 = H_1(b)$ defined by .
Set $P := p \otimes 1$, $Q := I - P$. Introduce the orthogonal projections in $L^2({{\mathbb{R}}}^3)$ $$\label{eq3,1}
\textup{P} := \begin{pmatrix}
P & 0 \\
0 & 0
\end{pmatrix}, \hspace{1cm}
\textup{Q} := \textup{I} - \textup{P} =
\begin{pmatrix}
Q & 0 \\
0 & I
\end{pmatrix}.$$ For $z \in {{\mathbb{C}}}\setminus [0,+\infty)$ and imply that $$\label{eq3,2}
({{H_0}}- z)^{-1} \textup{P} =
\begin{pmatrix}
p \otimes \mathscr{R}(z) & 0 \\
0 & 0
\end{pmatrix},$$ where $\mathscr{R}(z) := \left( -\frac{d^2}{dt^2}
- z \right)^{-1}$ acts in $L^{2}({{\mathbb{R}}})$. Thus $$\label{eq3,3}
\big( {{H_0}}- z \big)^{-1} =
\big( p \otimes \mathscr{R}(z) \big)
\begin{pmatrix}
1 & 0\\
0 & 0
\end{pmatrix} + \big( {{H_0}}-
z \big)^{-1} \textup{Q}.$$ The one-dimensional resolvent $\mathscr{R}(z)$ introduced above admits the integral kernel $$\label{eq3,4}
\mathscr{N}_{z}(t,t') =
\frac{i \textup{e}^{i \sqrt{z} \vert t -
t' \vert}}{2 \sqrt{z}}$$ if the branch ${\textup{Im}}(\sqrt{z})$ is chosen such that ${\textup{Im}}(\sqrt{z}) > 0$. In the sequel we set $$\label{eq3,5}
\mathbb{C}^{+} := \big\lbrace z \in {{\mathbb{C}}}:
{\textup{Im}}(z) > 0 \big\rbrace \quad \text{and} \quad
{{\mathbb{C}}}_{1/2}^+ := \big\lbrace k \in
{{\mathbb{C}}}: k^{2} \in {{\mathbb{C}}}^+ \big\rbrace.$$ With respect to the variable $k$ we define the pointed disk $$\label{eq3,6}
D(0,\epsilon)^\ast := \big\lbrace k \in
\mathbb{C} : 0 < \vert k \vert < \epsilon
\big\rbrace$$ with $$\label{eq3,7}
\epsilon < N_{\gamma,\zeta}$$ the constant defined by .
In order to define the resonances near zero first we extend holomorophically $(H_{0} - k^2)^{-1} \textup{P}$ near $k = 0$.
\[p3,1\] Let $\gamma > 0$ be constant and set $z(k) := k^2$.
$\textup{\textbf{(i)}}$ The operator valued-function $$k \longmapsto \left( \big( {{H_0}}- z(k) \big)
^{-1} \textup{P} : e^{-\frac{\gamma}{2}
\langle t \rangle} L^{2}({{\mathbb{R}}}^3) \longrightarrow
e^{\frac{\gamma}{2} \langle t \rangle}
L^{2}({{\mathbb{R}}}^{3}) \right)$$ admits a holomorphic extension from $\mathbb{C}_{1/2}^+ \cap D(0,\epsilon)^\ast$ to $D(0,\epsilon)^\ast$.
$\textup{\textbf{(ii)}}$ For $v_\perp({x_\perp}) :=
\langle {x_\perp}\rangle^{-\alpha}$ with $\alpha > 1$ the operator valued-function $$T_{v_\perp} : k \longmapsto v_\perp({x_\perp})
e^{-\frac{\gamma}{2} \langle t \rangle}
\big( {{H_0}}- z(k) \big)^{-1} \textup{P}
e^{-\frac{\gamma}{2} \langle t \rangle}$$ has a holomorphic extension to $D(0,\epsilon)^{\ast}$ with values in the Hilbert-Schmidt class ${{\mathcal{S}_2}}\left( L^{2}(\mathbb{R}^{3}) \right)$.
**(i)** Introduce $$L(k) = \big{[} p \otimes \mathcal{R}(k^{2})
\big{]} \begin{pmatrix}
1 & 0\\
0 & 0
\end{pmatrix}$$ acting from $e^{-\frac{\gamma}{2}
\langle t \rangle} L^{2}(\mathbb{R}^{3})$ to $e^{\frac{\gamma}{2} \langle t \rangle}
L^{2}(\mathbb{R}^{3})$. The operator $\mathscr{N}(k) := e^{-\frac{\gamma}{2}
\langle t \rangle} \mathscr{R}(k^{2})
e^{-\frac{\gamma}{2} \langle t \rangle}$ admits the integral kernel $$\label{eq3,8}
e^{-\frac{\gamma}{2} \langle t \rangle}
\frac{i \textup{e}^{i k \vert t - t' \vert}}
{2 k} \textup{e}^{-\frac{\gamma}{2} \langle
t' \rangle}.$$ It is easy to check that the integral kernel belongs to $L^{2}({{\mathbb{R}}})$ once ${\textup{Im}}(k)> -\frac{\gamma}{2}$, $k \in {{\mathbb{C}}}^\ast$. Then for $\epsilon < \frac{\gamma}{2}$ we can extend holomorphically $k \longmapsto L(k) \in
\mathscr{L} \left( e^{-\frac{\gamma}{2}
\langle t \rangle} L^{2}({{\mathbb{R}}}^{3}),
e^{\frac{\gamma}{2} \langle t \rangle}
L^{2}({{\mathbb{R}}}^{3}) \right)$ from ${{\mathbb{C}}}_{1/2}^{+}
\cap D(0,\epsilon)^{\ast}$ to $D(0,\epsilon)^\ast$. This together with imply that $k \longmapsto
\big( {{H_0}}- z(k) \big)^{-1} \textup{P}
\in \mathscr{L} \left( e^{-\frac{\gamma}{2}
\langle t \rangle} L^{2}({{\mathbb{R}}}^{3}),
e^{\frac{\gamma}{2} \langle t \rangle}
L^{2}({{\mathbb{R}}}^{3}) \right)$ admits a holomorphic extension to $D(0,\epsilon)^{\ast}$.
**(ii)** Thanks to $$\label{eq3,9}
T_{v_\perp}(k) = \left[ v_\perp p \otimes
\mathscr{N}(k) \right] \begin{pmatrix}
1 & 0\\
0 & 0
\end{pmatrix}.$$ The operator $\mathscr{N}(k) \in
{{\mathcal{S}_2}}\big( L^{2}({{\mathbb{R}}}) \big)$ following the proof of assertion **(i)** for ${\textup{Im}}(k) > -\frac{\gamma}{2}$, $k \in {{\mathbb{C}}}^\ast$. Since $v_\perp^{2} \in L^1({{\mathbb{R}}}^2)$ then by [@rage Lemma 2.3] $p v_\perp^{2} p$ is a trace class operator in $ L^{2}({{\mathbb{R}}}^2)$. That is $v_\perp p v_\perp \in {{\mathcal{S}_1}}\big( L^{2}({{\mathbb{R}}}^2) \big)$. This together with imply that $v_\perp p \in {{\mathcal{S}_2}}\big( L^{2}({{\mathbb{R}}}^2) \big)$ with $$\label{eq3,10}
\Vert v_\perp p \Vert_{{\mathcal{S}_2}}^2 = \textup{Tr}
\hspace{0.4mm} (v_\perp p v_\perp) =
\int_{{{\mathbb{R}}}^2} v_\perp^2({x_\perp})
\mathcal{P}_b ({x_\perp},{x_\perp}) d{x_\perp}\leq \frac{b_0}{2\pi} e^{2\textup{osc}
\hspace{0.5mm} \tilde{\varphi}}
\int_{{{\mathbb{R}}}^2} v_\perp^2({x_\perp}) d{x_\perp}.$$ Thus $k \mapsto T_{v_\perp}(k)$ has a holomorphic extension as above from ${{\mathbb{C}}}_{1/2}^{+} \cap D(0,
\epsilon)^{\ast}$ to $D(0,\epsilon)^{\ast}$ with values in ${{\mathcal{S}_2}}\left( L^{2}({{\mathbb{R}}}^{3}) \right)$. The proof is complete.
Now let us extend holomorphically the operator $({{H_0}}- z)^{-1} \textup{Q}$ from the upper half-plane to the lower half-plane except a semi-axis.
\[p3,2\] Let $\gamma$ be as in Proposition \[p3,1\] and $\zeta$ be defined by .
The operator valued-function $$z \longmapsto \left( ({{H_0}}- z)^{-1}
\textup{Q} : e^{-\frac{\gamma}{2} \langle t
\rangle} L^{2}({{\mathbb{R}}}^{3}) \longrightarrow
e^{\frac{\gamma}{2} \langle t \rangle}
L^{2}({{\mathbb{R}}}^{3}) \right)$$ admits a holomorphic extension from ${{\mathbb{C}}}^+$ to ${{\mathbb{C}}}\setminus [\zeta,\infty)$.
For $v_\perp({x_\perp})
:= \langle {x_\perp}\rangle^{-\alpha}$ with $\alpha > 1$ the operator valued-function $$L_{v_\perp} : z \longmapsto v_\perp({x_\perp})
e^{-\frac{\gamma}{2} \langle t \rangle}
({{H_0}}- z)^{-1} \textup{Q}
e^{-\frac{\gamma}{2} \langle t \rangle}$$ has a holomorphic extension to ${{\mathbb{C}}}\setminus [\zeta,\infty)$ with values in the Hilbert-Schmidt class ${{\mathcal{S}_2}}\left( L^{2}(\mathbb{R}^{3}) \right)$.
**(i)** Consider $z \in {{\mathbb{C}}}^{+}$. Thanks to and we have $$\label{eq3,11}
({{H_0}}- z)^{-1} \textup{Q}
= \left( \begin{smallmatrix}
\big( \mathcal{H}_1(b) - z \big)^{-1}
Q & 0 \\
0 & \big( \mathcal{H}_2(b) - z
\big)^{-1}
\end{smallmatrix} \right) =
\big( \mathcal{H}_1(b) - z \big)^{-1} Q
\oplus \big( \mathcal{H}_2(b) - z
\big)^{-1}.$$ Since ${{\mathbb{C}}}\setminus [\zeta,\infty)$ is contained in the resolvent set of $\mathcal{H}_1(b)$ acting on $Q Dom \big( \mathcal{H}_1(b) \big)$ and $\mathcal{H}_2(b)$ acting on $Dom \big( \mathcal{H}_2(b) \big)$ then ${{\mathbb{C}}}\setminus [\zeta,\infty) \ni z
\longmapsto \big( \mathcal{H}_1(b) - z
\big)^{-1} Q \oplus \big(
\mathcal{H}_2(b) - z \big)^{-1}$ is well defined and holomorphic. So ${{\mathbb{C}}}^+ \ni z
\mapsto e^{-\frac{\gamma}{2} \langle t
\rangle} ({{H_0}}- z)^{-1} \textup{Q}
e^{-\frac{\gamma}{2} \langle t \rangle}$ admits a holomorphic extension to ${{\mathbb{C}}}\setminus [\zeta,\infty)$.
**(ii)** According to $$\label{eq3,110}
L_{v_\perp}(z) = v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} \left( \big(
\mathcal{H}_1(b) - z \big)^{-1} Q \oplus
\big( \mathcal{H}_2(b) - z \big)^{-1} \right)
e^{-\frac{\gamma}{2} \langle t \rangle}.$$ We have $$\label{eq3,12}
\left\Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} \big( \mathcal{H}_1(b) -
z \big)^{-1} Q \right\Vert_{{\mathcal{S}_2}}^2
\leq \left\Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} \big( \mathcal{H}_1(b) + 1
\big)^{-1} \right\Vert_{{\mathcal{S}_2}}^2
\left\Vert \big( \mathcal{H}_1(b) + 1 \big)
\big( \mathcal{H}_1(b) - z \big)^{-1} Q
\right\Vert^2.$$ By the Spectral mapping theorem $$\label{eq3,13}
\left\Vert \big( \mathcal{H}_1(b) + 1 \big)
\big( \mathcal{H}_1(b) - z \big)^{-1} Q
\right\Vert^q \leq
\textup{sup}_{s \in [\zeta,+\infty)}^q
\left\vert \frac{s + 1}{s - z} \right\vert.$$ With the help of the resolvent identity, the boundedness of the magnetic field $b$ and the diamagnetic inequality (see [@avr Theorem 2.3]-) we obtain $$\label{eq3,14}
\begin{split}
\left\Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} \big( \mathcal{H}_1(b)
+ 1 \big)^{-1} \right\Vert_{{\mathcal{S}_2}}^2
& \leq \left\Vert I + \big( \mathcal{H}_1(b)
+ 1 \big)^{-1}b \right\Vert^2
\left\Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle}
\big( (-i\nabla - \textbf{A})^{2} + 1
\big)^{-1} \right\Vert_{{\mathcal{S}_2}}^2 \\
& \leq C \left\Vert v_\perp
e^{-\frac{\gamma}{2} \langle t \rangle}
(-\Delta + 1)^{-1} \right\Vert_{{\mathcal{S}_2}}^2.
\end{split}$$ By the standard criterion [@sim Theorem 4.1] $$\label{eq3,15}
\left\Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} (-\Delta + 1 \vert)^{-1}
\right\Vert_{{\mathcal{S}_2}}^2 \leq C
\Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} \Vert_{L^2}^2
\left\Vert \Bigl( \vert \cdot \vert^{2}
+ 1 \Bigr)^{-1} \right\Vert_{L^2}^2.$$ Putting together , , and we get $$\label{eq3,16}
\left\Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} \big( \mathcal{H}_1(b) -
z \big)^{-1} Q \right\Vert_{{\mathcal{S}_2}}^2
\leq C \Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} \Vert_{L^{2}}^{2}
\textup{sup}_{s \in [\zeta,+\infty)}^2
\left\vert \frac{s + 1}{s - z} \right\vert.$$ By similar arguments we can prove that $$\label{eq3,17}
\left\Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} \big( \mathcal{H}_2(b) -
z \big)^{-1} \right\Vert_{{\mathcal{S}_2}}^2
\leq C \Vert v_\perp e^{-\frac{\gamma}{2}
\langle t \rangle} \Vert_{L^{2}}^{2}
\textup{sup}_{s \in [\zeta,+\infty)}^2
\left\vert \frac{s + 1}{s - z} \right\vert.$$ Since the multiplication operator by the function $e^{-\frac{\gamma}{2}
\langle t \rangle}$ is bounded then , and imply that $L_{v_\perp}(z)$ belongs to ${{\mathcal{S}_2}}\left( L^{2}({{\mathbb{R}}}^{3})
\right)$ and has a holomorphic extension from ${{\mathbb{C}}}^{+}$ to ${{\mathbb{C}}}\setminus [\zeta,
\infty)$. This completes the proof.
For $V$ satisfying assumption , holds. Then this together with , Propositions \[p3,1\]-\[p3,2\] yield to the following
\[l3,1\] Let $D(0,\epsilon)^{\ast}$ be the pointed disk defined by . Assume that $V$ satisfies and set $z(k) := k^2$. Then the operator valued-function $$\mathbb{C}_{1/2}^{+} \cap D(0,\epsilon)
^{\ast} \ni k \longmapsto \mathcal{T}_{V}
\big( z(k) \big) := J \vert V \vert^{1/2}
\big( {{H_0}}- z(k) \big)^{-1} \vert V
\vert^{1/2},$$ where $J := sign(V)$ has a holomorphic extension to $D(0,\epsilon)^{\ast}$ with values in ${{\mathcal{S}_2}}\left( L^{2}({{\mathbb{R}}}^{3})
\right)$. We will denote again this extension by $\mathcal{T}_{V} \big( z(k)
\big)$. Furthermore the operator $\partial_z \mathcal{T}_{V} \big( z(k)
\big) \in {{\mathcal{S}_1}}\left( L^{2}({{\mathbb{R}}}^{3})
\right)$ is holomorphic on $D(0,\epsilon)^{\ast}$.
Now using the identity $$({{H_V}}- z)^{-1} \big( 1 + V({{H_0}}-
z)^{-1} \big) = ({{H_0}}- z)^{-1}$$ derived from the resolvent equation we obtain $$\begin{aligned}
e^{-\frac{\gamma}{2} \langle t \rangle}
\big({{H_V}}- z \big)^{-1}
e^{-\frac{\gamma}{2} \langle t \rangle}
& = e^{-\frac{\gamma}{2} \langle t
\rangle} ({{H_0}}- z)^{-1}
e^{-\frac{\gamma}{2} \langle t \rangle}
\\
& \times \left( 1 + e^{\frac{\gamma}{2}
\langle t \rangle} V({{H_0}}- z)^{-1}
e^{-\frac{\gamma}{2} \langle t \rangle}
\right)^{-1}.\end{aligned}$$ As in assumption on $V$ implies the existence of $\mathscr{M} \in \mathscr{L} \big(
L^{2}({{\mathbb{R}}}^3) \big)$ such that $$\label{eq3,19}
\vert V \vert ({x_\perp},t)
= \mathscr{M} \left( \langle {x_\perp}\rangle
^{-m_\perp} \otimes e^{-\gamma \langle t
\rangle} \right), \quad ({x_\perp},t) \in {{\mathbb{R}}}^3,
\quad m_\perp > 2.$$ Then similarly to Lemma \[l3,1\] it can be proved that $k \longmapsto
e^{\frac{\gamma}{2} \langle t \rangle}
V({{H_0}}- z)^{-1} e^{-\frac{\gamma}{2}
\langle t \rangle}$ is holomorphic with values in ${{\mathcal{S}_\infty}}\left( L^{2}({{\mathbb{R}}}^{3})
\right)$. Thus by the analytic Fredholm theorem the operator valued-function $$k \longmapsto \left( 1 +
e^{\frac{\gamma}{2} \langle t \rangle}
V({{H_0}}- z)^{-1} e^{-\frac{\gamma}{2}
\langle t \rangle} \right)^{-1}$$ admits a meromorphic extension from $\mathbb{C}_{1/2}^{+} \cap
D(0,\epsilon)^{\ast}$ to $D(0,\epsilon)^{\ast}$. Hence we have the following
\[p3,3\] Under the assumptions and the notations of Lemma \[l3,1\] the operator valued-function $$k \longmapsto \left( \big( {{H_V}}- z(k)
\big)^{-1} : \textup{e}^{-\frac{\gamma}
{2} \langle x_{3} \rangle} L^{2}
(\mathbb{R}^{3}) \longrightarrow
\textup{e}^{\frac{\gamma}{2} \langle
x_{3} \rangle} L^{2}(\mathbb{R}^{3})
\right)$$ admits a meromorphic extension from $\mathbb{C}_{1/2}^{+} \cap
D(0,\epsilon)^{\ast}$ to $D(0,\epsilon)^{\ast}$. This extension will be denoted by $R \big( z(k) \big)$.
We can now define the resonances of ${{H_V}}$ near zero. In the following definition the index of a finite-meromorphic operator valued-function appearing in is recalled in the Appendix.
\[d3,1\] We define the resonances of $H$ near zero as the poles of the meromorphic extension $R(z)$ of $\left( {{H_V}}- z \right)^{-1}$ in $\mathscr{L} \left(
\textup{e}^{-\frac{\gamma}{2}
\langle x_{3} \rangle} L^{2}({{\mathbb{R}}}^3),
\textup{e}^{\delta \langle x_{3}
\rangle} L^{2}({{\mathbb{R}}}^3) \right)$. The multiplicity of a resonance $z_0
:= z(k_0)$ is defined by $$\label{eq3,20}
\textup{mult}(z_0) :=
\textup{Ind}_{\mathcal{C}}
\hspace{0.5mm} \Big( I +
\mathcal{T}_{V} \big( z(\cdot) \big)
\Big),$$ $\mathcal{C}$ being a small contour positively oriented containing $k_0$ as the unique point $k \in D(0,\epsilon)^\ast$ satisfying $z(k)$ is a resonance of ${{H_V}}$, and $\mathcal{T}_{V} \big(
z(\cdot) \big)$ being defined by Lemma \[l3,1\].
Results on the resonances {#s4}
=========================
We establish preliminary results on the resonances we need for the proofs of our main results.
A characterisation of the resonances
------------------------------------
\[p4,1\] Let $\mathcal{T}_{V} (\cdot)$ be defined by Lemma \[l3,1\]. Then the following assertions are equivalent:
$\textup{\textbf{(i)}}$ $z _0 :=
z(k_{0})$ is a resonance of ${{H_V}}$ near zero,
$\textup{\textbf{(ii)}}$ $-1$ is an eigenvalue of $\mathcal{T}_{V} \big(
z(k_{0}) \big)$,
$\textup{\textbf{(iii)}}$ $\textup{det}_2 \big( I +
\mathcal{T}_{V} \big( z(k_{0}) \big)
\big) = 0$.\
Moreover the multiplicity of $z_0$ as zero of $\textup{det}_2 \big( I +
\mathcal{T}_{V} (\cdot) \big)$ coincides with its multiplicity as resonance of ${{H_V}}$.
The equivalence **(i)** $\Leftrightarrow$ **(ii)** follows immediately from $$\label{eq3,21}
\left( I + J \vert V \vert^{1/2}
({{H_0}}- z)^{-1} \vert V
\vert^{1/2} \right) \left( I - J
\vert V \vert^{1/2} ({{H_V}}- z)^{-1}
\vert V \vert^{1/2} \right) = I.$$
The equivalence **(ii)** $\Leftrightarrow$ **(iii)** is a direct consequence of the definition of $\textup{det}_2
\bigl( I + \mathcal{T}_{V} \big(
z(k_{0}) \big) \bigr)$ given by with $q = 2$.
Otherwise since by Lemma \[l3,1\] $\mathcal{T}_{V} (\cdot)$ is holomorphic on $D(0,\epsilon)^{\ast}$ then so is $\textup{det}_2 \bigl( I +
\mathcal{T}_{V} (\cdot)
\big)$ on $D(0,\epsilon)^\ast$. Let $\textup{m}(z_0)$ be the multiplicity of $z_0$ as zero of $\textup{det}_2 \big( I +
\mathcal{T}_{V} (\cdot)
\big)$. If $\mathcal{C}'$ is a small contour positively oriented containing $z_0$ as the unique resonance of ${{H_V}}$ near zero then $$\label{eq3,22}
\textup{m}(z_0) = ind_{\mathcal{C}'}
\Big( \textup{det}_2 \bigl( I +
\mathcal{T}_{V} (\cdot)
\bigr) \Big),$$ where the RHS of is the index defined by of the holomorphic function $\textup{det}_2 \bigl( I +
\mathcal{T}_{V} (\cdot)
\bigr)$ with respect to the contour $\mathcal{C}'$. Now the equality on the multiplicities claimed in the proposition is an immediate consequence of the equality $$ind_{\mathcal{C}'}
\Big( \textup{det}_2 \bigl( I +
\mathcal{T}_{V} (\cdot)
\bigr) \Big) = Ind_{\mathcal{C}}
\hspace{0.5mm} \Big( I +
\mathcal{T}_{V} \big( z(\cdot)
\big) \Big),$$ see for instance [@bo (2.6)]. This concludes the proof.
Decomposition of the weighted resolvent
---------------------------------------
We split the weighted resolvent $\mathcal{T}_{V} \big( z(k) \big) :=
J \vert V \vert^{\frac{1}{2}}
\big( {{H_0}}- z(k) \big)^{-1} \vert V
\vert^{\frac{1}{2}}$ into a singular part near $k = 0$ and a holomorphic part on the open disk $D(0,\epsilon)
:= D(0,\epsilon)^\ast \cup \lbrace 0
\rbrace$ with values in ${{\mathcal{S}_2}}\big( L^{2}({{\mathbb{R}}}^3) \big)$.
According to for $k \in D(0,\epsilon)^\ast$ $$\label{eq4,1}
\mathcal{T}_{V} \big( z(k) \big) =
J \vert V \vert^{\frac{1}{2}} p
\otimes \mathscr{R} \big( z(k) \big)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix} \vert V
\vert^{\frac{1}{2}} + J \vert V
\vert^{\frac{1}{2}} \big( {{H_0}}-
z(k) \big)^{-1} \textup{Q} \vert V
\vert^{\frac{1}{2}}.$$ Recall that $e_\pm$ are the multiplications operators by the functions $e^{\pm\frac{\gamma}{2}
\langle \cdot \rangle}$ respectively. We have $$\label{eq4,2}
J \vert V \vert^{\frac{1}{2}}
p \otimes \mathscr{R} \big( z(k)
\big)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix} \vert V
\vert^{\frac{1}{2}}
= J \vert V \vert^{\frac{1}{2}}
e_+ p \otimes e_- \mathscr{R}
\big( z(k) \big) e_-
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix} e_+ \vert V
\vert^{\frac{1}{2}}.$$ Thanks to the integral kernel of $e_- \mathscr{R} \big( z(k) \big)
e_-$ is given by $$\label{eq4,3}
e^{-\frac{\gamma}{2} \langle t
\rangle} \frac{i \textup{e}^{i k
\vert t - t' \vert}}{2 k}
e^{-\frac{\gamma}{2} \langle t'
\rangle}$$ for $k \in D(0,\epsilon)^\ast$. Then $e_- \mathscr{R}
\big( z(k) \big) e_-$ can be decompose as $$\label{eq4,4}
e_- \mathscr{R} \big( z(k) \big) e_-
= \frac{1}{k}a + b(k),$$ where $a : L^{2}(\mathbb{R})
\longrightarrow L^{2}(\mathbb{R})$ is the rank-one operator defined by $$\label{eq4,5}
a(u) := \frac{i}{2} \big\langle u,
e^{-\frac{\gamma}{2} \langle \cdot
\rangle} \big\rangle
e^{-\frac{\gamma}{2} \langle \cdot
\rangle}$$ and $b(k)$ is the operator with integral kernel given by $$\label{eq4,6}
e^{-\frac{\gamma}{2} \langle t
\rangle} i \frac{ \textup{e}^{i k
\vert t - t' \vert} - 1}{2 k}
e^{-\frac{\gamma}{2} \langle t
\rangle}.$$ It is easy to remark that $-2ia =
c^\ast c$ where $c$ is the operator defined by . This together with yield for $k \in D(0,\epsilon)^\ast$ to $$\label{eq4,7}
p \otimes e_- \mathscr{R} \big( z(k)
\big) e_- = \pm \frac{i}{2k} p
\otimes c^\ast c + p \otimes s(k),$$ where $s(k)$ is the operator acting from $e^{-\frac{\gamma}{2} \langle t
\rangle} L^{2}({{\mathbb{R}}})$ to $e^{\frac{\gamma}{2} \langle t
\rangle} L^{2}({{\mathbb{R}}})$ having the integral kernel $$\label{eq4,8}
\frac{ 1 - \textup{e}^{i k
\vert t - t' \vert}}{2 i k}.$$ By combining with we get for $k \in
D(0,\epsilon)^\ast$ $$\label{eq4,9}
\begin{split}
& J
\vert V \vert^{\frac{1}{2}}
p \otimes \mathscr{R} \big( z(k)
\big)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix} \vert V
\vert^{\frac{1}{2}} \\
& = \frac{iJ}{2k} \vert V
\vert^{\frac{1}{2}}
e_+ (p \otimes c^\ast c)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix} e_+ \vert V
\vert^{\frac{1}{2}}
+ J \vert V \vert^{\frac{1}{2}}
e_+ p \otimes s(k)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix} e_+
\vert V \vert^{\frac{1}{2}}.
\end{split}$$ That is $$\label{eq4,10}
J \vert V \vert^{\frac{1}{2}}
p \otimes \mathscr{R} \big( z(k) \big)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix} \vert V
\vert^{\frac{1}{2}} =
\frac{iJ}{k} K^\ast K + J
\vert V \vert^{\frac{1}{2}} e_+ p
\otimes s(k)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
e_+ \vert V \vert^{\frac{1}{2}},$$ where $K$ is the operator defined by . We have then proved the following
\[p4,2\] Let $V$ satisfy assumptions -. For $k \in D(0,\epsilon)^\ast$ $$\label{eq4,11}
\mathcal{T}_{V} \big( z(k) \big) =
\frac{iJ}{k} \mathscr{B}
+ \mathscr{A}(k), \quad \mathscr{B}
:= K^\ast K,$$ the operator $\mathscr{A}(k) \in
{{\mathcal{S}_2}}\big( L^{2}({{\mathbb{R}}}^3) \big)$ being given by $$\label{eq4,111}
\mathscr{A}(k) :=
J \vert V \vert^{\frac{1}{2}} e_+ p
\otimes s(k)
\begin{pmatrix}
1 & 0 \\
0 & 0
\end{pmatrix}
e_+ \vert V \vert^{\frac{1}{2}}
+ J \vert V \vert^{\frac{1}{2}}
\big( {{H_0}}- z(k) \big)^{-1}
\textup{Q} \vert V \vert^{\frac{1}{2}}$$ and holomorphic on the open disk $D(0,\epsilon)$ with $s(k)$ defined by .
$-$
For any $r > 0$ we have $$\label{eq4,12}
\textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(r,\infty)}
\left( K^\ast K \right)
= \textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(r,\infty)}
\left( K K^\ast \right)
= \textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(r,\infty)}
\big( p \textbf{\textup{W}} p
\big)$$ following .
Note that the asymptotic expansion of the quantity $\textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(r,\infty)}
\big( p U p \big)$ is well known once the function $0 \leq U \in L^\infty ({{\mathbb{R}}}^2)$ decays like a power, exponentially or is compactly supported:
**(A1)** If $U \in C^{1}
\big( \mathbb{R}^{2} \big)$ satisfies $U({x_\perp}) = u_{0}\big({x_\perp}/ \vert {x_\perp}\vert\big) \vert {x_\perp}\vert^{-m}
( 1 + o(1) \big)$, $\vert {x_\perp}\vert
\rightarrow \infty$, $0 \not\equiv
u_{0} \in C^0 \big( \mathbb{S}^{1},
{{\mathbb{R}}}_+ \big)$, $\vert \nabla U({x_\perp})
\vert \leq C_{1} \langle {x_\perp}\rangle^{-m-1}$ with $m$, $C_{1} > 0$ constant and if there exists an integrated density of states for the operator $H_{1}(b)$ then $$\label{eq02,2}
\textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(r,\infty)} \big( p U p \big)
= C_{m} r^{-2/m} \big( 1 + o(1) \big),
\hspace{0.2cm} r \searrow 0,$$ where $C_{m} := \frac{b_0}{4\pi}
\int_{\mathbb{S}^{1}} u_{0}(t)^{2/m}
dt$, (see [@rage Lemma 3.3]).
**(A2)** If $U$ satisfies $\ln U({x_\perp}) = -\mu \vert
{x_\perp}\vert^{2\beta} \big( 1 + o(1)
\big)$, $\vert {x_\perp}\vert \rightarrow
\infty$ with $\beta$, $\mu > 0$ constant then $$\label{eq02,3}
\textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(r,\infty)} \big( p U p \big)
= \varphi_{\beta}(r) \big( 1 + o(1)
\big), \hspace{0.2cm} r \searrow 0,$$ where for $0 < r < \textup{e}^{-1}$ $$\varphi_{\beta}(r) :=
\begin{cases}
\frac{1}{2} b_0 \mu^{-1/\beta} \vert
\ln r \vert^{1/\beta} & \text{if }
0 < \beta < 1,\\
\frac{1}{\ln(1 + 2\mu/b_0)} \vert
\ln r \vert & \text{if } \beta = 1,\\
\frac{\beta}{\beta - 1} \big{(} \ln
\vert \ln r \vert \big{)}^{-1} \vert
\ln r \vert & \text{if } \beta > 1,
\end{cases}$$ (see [@rage Lemma 3.4]).
**(A3)** If $U$ is compactly supported and if there exists $C > 0$ constant such that on an open non-empty subset of $\mathbb{R}^{2}$ $U \geq C$ then $$\label{eq02,4}
\textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(r,\infty)} \big( p U p \big)
= \varphi_{\infty}(r) \big( 1 + o(1)
\big), \hspace{0.2cm} r \searrow 0,$$ where $\varphi_{\infty}(r) := \big( \ln
\vert \ln r \vert \big{)}^{-1}
\vert \ln r \vert, \hspace{0.2cm}
0 < r < \textup{e}^{-1}$, (see [@rage Lemma 3.5]).
By an evident adaptation of [@bon Proof of Corollary 1] we obtain the following corollary summarizing useful properties of the operator $\mathscr{B}$ defined by . Therefore we omit the proof.
\[c4,1\] Let $V$ satisfy assumptions -. Then $\mathscr{B} \in {{\mathcal{S}_1}}\big( L^2({{\mathbb{R}}}^3)
\big)$ and satisfies for $r > 0$ small enough $$\textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(r,\infty)} (\mathscr{B})
= \mathcal{O} \big( r^{-2/m_\perp}
\big).$$ For $j \in
{{\mathbb{N}}}^\ast$ the operator-valued functions $$\label{eq4,13}
{{\mathbb{C}}}\setminus \big( \mp i[0,+\infty[
\big) \ni k \mapsto \mathfrak{B}(k) =
\mathfrak{B}_j^\pm(k) :=
\frac{i\mathscr{B}}{k} \left( I \pm
\frac{i\mathscr{B}}{k} \right)^{-j}
\in {{\mathcal{S}_1}}\big( L^2({{\mathbb{R}}}^3) \big)$$ are holomorphic and $$\label{eq4,14}
\Vert \mathfrak{B}(k) \Vert_{{\mathcal{S}_p}}^p
\leq f(\theta)^{pj} \sigma_p \big(
\vert k \vert \big), \qquad p = 1,
\hspace*{0.1cm} 2,$$ where $\theta = \textup{Arg}
\hspace{1mm} k$, $f(\theta) = \big(
1 - (\sin \theta)_-
\big)^{-\frac{1}{2}}$ with $s_- :=
\max(-s,0)$ for $s \in {{\mathbb{R}}}$ and $$\label{eq4,15}
\sigma_p (r) :=
\left\Vert \frac{\mathscr{B}}{r}
\left( I + \frac{\mathscr{B}^2}{r^2}
\right)^{-1/2} \right\Vert_{{\mathcal{S}_p}}^p =
\mathcal{O} \big( r^{-2/m_\perp}
\big), \quad r > 0.$$
Further for any $r > 0$ and $p
\geq 1$ $$\label{eq4,16}
2^{-p/2} \Tilde{n}_p (r) \leq
\sigma_p (r) \leq \Tilde{n}_p (r)
+ \textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(r,\infty)} (\mathscr{B})$$ with $$\label{eq4,17}
\Tilde{n}_p (r) :=
\left\Vert \frac{\mathscr{B}}{r}
{{\bf 1}}_{[0,r]}(\mathscr{B})
\right\Vert_{{\mathcal{S}_p}}^p.$$
Moreover if the function $\textbf{\textup{W}}$ defined by satisfies assumption **(A1)** with $m > 2$ then for $p = 1, 2$ there exists constants $C_{m,p}$ and $\Tilde{C}_{m,p}$ such that $$\label{eq4,18}
\begin{cases}
\sigma_p (r) = C_{m,p} r^{-2/m}
\big( 1 + o(1) \big), \\
\Tilde{n}_p (r) = \Tilde{C}_{m,p}
r^{-2/m}
\big( 1 + o(1) \big),
\end{cases}
\quad r \searrow 0.$$ Finally if $\textbf{\textup{W}}$ satisfies Assumptions **(A2)** then $$\label{eq4,19}
\sigma_p (r) = \varphi_\beta (r)
\big( 1 + o(1) \big), \quad
\Tilde{n}_p (r) = o
\big( \varphi_\beta (r) \big),
\quad r \searrow 0,$$ where the functions $\varphi_\beta (r)$, $\beta
\in (0,\infty]$ are defined by or .
Upper bounds on the number of resonances {#ss4,1}
----------------------------------------
The next result concerns an upper bound on the number of resonances near zero outside a vicinity of $\big\lbrace
z(k) : k \in -iJ[0,+\infty) \big\rbrace$ for potentials $V$ of definite sign $J = \pm$.
\[t4,1\] Assume that $V$ satisfying assumptions - is of definite sign $J$. Let $\mathcal{C}_{\delta}(J)$ be the sector defined by $$\mathcal{C}_\delta(J) := \big\lbrace
k \in {{\mathbb{C}}}: - \delta J {\textup{Im}}(k) \leq
\vert {\textup{Re}}(k) \vert \big\rbrace.$$ Then for any $\delta > 0$ there exists $r_0 > 0$ such that for any $0 < r <r_0 $ $$\label{eq4,20}
\displaystyle \sum_{\substack{z(k)
\hspace{0.5mm} \in \hspace{0.5mm}
\textup{Res}({{H_V}}) \\ k
\hspace{0.5mm} \in \hspace{0.5mm}
\lbrace r < \vert k \vert < 2r
\rbrace \cap \mathcal{C}_\delta(J)}}
\textup{mult} \big( z(k) \big) =
\mathcal{O} \big( \vert \ln r \vert
\big).$$
Thanks to Proposition \[p4,2\] for $k \in D(0,\epsilon)^\ast$ $$\label{eq4,21}
\mathcal{T}_{V} \big( z(k) \big) =
\frac{iJ}{k}
\mathscr{B} + \mathscr{A}(k),$$ where $\mathscr{B}$ is a self-adjoint positive operator which does not depend on $k$ while $\mathscr{A}(k)
\in {{\mathcal{S}_2}}\big( L^{2}({{\mathbb{R}}}^3) \big)$ is holomorphic near $k = 0$. Since $
I + \frac{iJ}{k} \mathscr{B} =
\frac{iJ}{k} (\mathscr{B} - iJk)
$ then $I + \frac{iJ}{k}\mathscr{B}$ is invertible for $iJk \notin \sigma (\mathscr{B})$ and satisfies $$\label{eq4,22}
\small{\left\Vert \left( I +
\frac{iJ}{k} \mathscr{B} \right)^{-1}
\right\Vert \leq \frac{\vert k \vert}
{\sqrt{\big( J{\textup{Im}}(k) \big)_+^2
+ \vert {\textup{Re}}(k) \vert^2}}},
\qquad r_+ := \max (r,0).$$ Further it is easy to check that for $k \in \mathcal{C}_\delta(J)$ we have uniformly with respect to $\vert
k \vert < r_0 \leq \epsilon$ $$\label{eq4,23}
\small{\left\Vert \left( I +
\frac{iJ}{k} \mathscr{B}
\right)^{-1} \right\Vert \leq
\sqrt{1 + \delta^{-2}}}.$$ Then using we can write $$\label{eq4,24}
\small{I + \mathcal{T}_{V}
\big( z(k) \big) = \big( I + A(k)
\big) \left( I + \frac{iJ}{k}
\mathscr{B} \right)},$$ where $A(k)$ is given by $$\label{eq4,25}
\small{A(k) := \mathscr{A} (k)
\left( I + \frac{iJ}{k} \mathscr{B}
\right)^{-1}} \in {{\mathcal{S}_2}}\big( L^{2}
({{\mathbb{R}}}^3) \big).$$ Otherwise a simple computation allows to obtain $$\mathcal{T}_{V} \big( z(k) \big)
- A(k) = \big( I + A(k) \big)
\frac{iJ}{k} \mathscr{B} \in
\mathcal{S}_1 \big( L^{2}({{\mathbb{R}}}^3)
\big)$$ since $\mathscr{B} \in
\mathcal{S}_1 \big( L^{2}({{\mathbb{R}}}^3)
\big)$ by Corollary \[c4,1\]. Then if we approximate $A(k)$ by a finite rank-operator in and use the formula $\textup{det}_2 (I + T) =
\textup{det} (I + T)
e^{-\textup{Tr}(T)}$ for $T \in
\mathcal{S}_1$ we obtain $$\label{eq4,26}
\small{\textup{det}_2 \big( I +
\mathcal{T}_{V} \big( z(k) \big)
\big) = \textup{det} \left( I +
\frac{iJ}{k} \mathscr{B} \right)
\times \textup{det}_2 \big( I +
A(k) \big) e^{-\textup{Tr} \left(
\mathcal{T}_{V} \big( z(k) \big) -
A(k) \right)}}.$$ Then for $\vert k \vert < r_0$ such that $k \in
\mathcal{C}_\delta(J)$ the zeros of $\textup{det}_2 \big( I +
\mathcal{T}_{V} \big( z(k) \big)
\big)$ are those of $\textup{det}_2 \big( I + A(k)
\big)$ with the same multiplicities thanks to Proposition \[p4,1\] and Property applied to .
Estimate and the fact that $\mathscr{A}(k)$ is holomorphic near $k = 0$ with values in ${{\mathcal{S}_2}}\big( L^{2}({{\mathbb{R}}}^3)
\big)$ imply that the Hilbert-Schmidt norm of $A(k)$ is uniformly bounded with respect to $\vert k \vert < r_0$ small enough and $k \in
\mathcal{C}_\delta(J)$. So we obtain uniformly with respect to $k$ $$\label{eq4,27}
\small{\textup{det}_2 \big( I
+ A(k) \big) = \mathcal{O}
\left( e^{\mathcal{O} \big(
\Vert A(k)
\Vert_{\mathcal{S}_2}^2 \big)}
\right) = \mathcal{O}(1).}$$
In what follows below we prove a corresponding lower bound of . Identity implies that $$\label{eq4,28}
\small{ \big( I + A(k)
\big)^{-1} = \left( I +
\frac{iJ}{k} \mathscr{B} \right)
\Big( I + \mathcal{T}_{V} \big(
z(k) \big) \Big)^{-1}}.$$ With the help of we get for ${\textup{Im}}(k^2) > \varsigma
> 0$ $$\label{eq4,281}
\begin{split}
\left\Vert \Big( I +
\mathcal{T}_{V} \big( z(k) \big)
\Big)^{-1} \right\Vert & =
\mathcal{O} \Big( 1 + \left\Vert
\vert V \vert^{1/2} \big( {{H_V}}-
z(k) \big)^{-1} \vert V
\vert^{1/2} \right\Vert \Big) \\
& = \mathcal{O} \Big( 1 +
\big\vert {\textup{Im}}(k^2) \big\vert^{-1}
\Big) = \mathcal{O} \left(
\varsigma^{-1} \right).
\end{split}$$ This together with yield to $$\label{eq4,29}
\small{\left\Vert \big( I + A(k)
\big)^{-1} \right\Vert
= \mathcal{O} \big( s^{-1} \big)
\mathcal{O} \big( \varsigma^{-1}
\big)},$$ uniformly with respect to $(k,s)$ such that $0 < s < \vert k \vert
< r_0$ and ${\textup{Im}}(k^2) > \varsigma
> 0$. Let $(\mu_j)_j$ be the sequence of eigenvalues of $A(k)$. We have $$\label{eq4,30}
\begin{aligned}
\small{\left\vert \Big(
\textup{det}_2 (I + A(k))
\Big)^{-1} \right\vert}
& \small{= \left\vert
\textup{det} \big( (I +
A(k))^{-1} e^{A(k)} \big)
\right\vert} \\
& \small{\leq \prod_{\vert \mu_j
\vert \leq \frac{1}{2}} \left\vert
\frac{e^{\mu_j}}{1 + \mu_j}
\right\vert \times \prod_{\vert
\mu_j \vert > \frac{1}{2}}
\frac{e^{\vert \mu_j \vert}}
{\vert 1 + \mu_j \vert}.}
\end{aligned}$$ Using the uniform bound $\Vert A(k)
\Vert_{\mathcal{S}_2} =
\mathcal{O}(1)$ with respect to $\vert k \vert < r_0$ small enough and $k \in \mathcal{C}_\delta(J)$ we can prove that the first product is uniformly bounded. On the other hand thanks to we have uniformly with respect to $(k,s)$, $0 < s < \vert k \vert
< r_0$ and ${\textup{Im}}(k^2) > \varsigma
> 0$ $$\label{eq4,31}
\small{\vert 1 + \mu_j \vert^{-1}
= \mathcal{O} \big( s^{-1} \big)
\mathcal{O} \big( \varsigma^{-1}
\big)}.$$ Therefore using the fact that the second product has a finite number of terms $\mu_j$ we deduce from that $$\label{eq4,32}
\small{\left\vert \textup{det}_2
\big( I + A(k) \big) \right\vert
\geq C e^{-C \big( \vert \ln
\varsigma \vert + \vert \ln s
\vert \big)},}$$ for some $C > 0$ constant. To conclude the proof we need the following Jensen type lemma (see for instance [@bon Lemma 6]):
\[la,1\] Let $\Delta$ be a simply connected sub-domain of $\mathbb{C}$ and let $g$ be a holomorphic function in $\Delta$ with continuous extension to $\overline{\Delta}$. Assume there exists $\lambda_{0} \in \Delta$ such that $g(\lambda_{0}) \neq 0$ and $g(\lambda) \neq 0$ for $\lambda\in
\partial \Delta$ the boundary of $\Delta$. Let $\lambda_{1},
\lambda_{2}, \ldots, \lambda_{N} \in
\Delta$ be the zeros of $g$ repeated according to their multiplicity. Then for any domain $\Delta' \Subset \Delta$ there exists $C' > 0$ such that $N(\Delta',g)$ the number of zeros $\lambda_{j}$ of $g$ contained in $\Delta'$ satisfies $$\label{eqa,5}
N(\Delta',g) \leq C' \left(
\int_{\partial \Delta}
\textup{ln} \vert g(\lambda) \vert
d\lambda - \textup{ln} \vert
g(\lambda_{0}) \vert \right).$$
Consider the domain $\Delta :=
\big\lbrace k \in
D(0,\epsilon)^\ast : r < \vert k
\vert < 2r \big\rbrace \cap
\mathcal{C}_\delta(J)$ with some ${\textup{Im}}(k_0^2) > \varsigma > 0$, $k_0 \in \Delta$. Then Theorem \[t4,1\] follows immediately by applying the Jensen Lemma \[la,1\] to the function $D(\cdot) := \textup{det}_2
\big( I + A(\cdot) \big)$ on $\Delta$ together with Proposition \[p4,1\], estimates -. The proof is complete.
For general perturbations $V$ without sign restriction we have the following result:
\[t4,2\] [[@diom Theorem 2.1]]{}
Let $V$ satisfy assumptions -. Then there exists $0 < r_{0} <
\epsilon$ small enough such that for any $0 < r < r_{0}$ $$\label{eq4,321}
\displaystyle \sum_{\substack{z(k)
\hspace{0.5mm} \in \hspace{0.5mm}
\textup{Res}({{H_V}}) \\ k
\hspace{0.5mm} \in \hspace{0.5mm}
\lbrace r < \vert k \vert < 2r
\rbrace}} \textup{mult} \big( z(k)
\big) = \mathcal{O} \Big( \textup{Tr}
\hspace{0.4mm} {{\bf 1}}_{(r,\infty)}
\big( p \textbf{\textup{W}}
p \big) \vert \ln r \vert \Big).$$
Proof of Theorem $\ref{t2,1}$: Breit-Wigner approximation {#s5}
=========================================================
We recall that $N_{\gamma,\zeta}$ is the constant defined by .
Preliminary results
-------------------
\[l5,1\] Let $V$ satisfy assumptions - and $\mathcal{T}_V(\cdot)$ be the operator defined by Lemma . Then on $]-N_{\gamma,\zeta}^2,
N_{\gamma,\zeta}^2[ \setminus
\lbrace 0 \rbrace$ $$\label{eq5,1}
\xi' = \xi_2' + \frac{1}{\pi} {\textup{Im}}\textup{Tr} \hspace{0.4mm} \big(
\partial_z \mathcal{T}_V(\cdot)
\big).$$
To get thanks to and it suffices to prove that for any function $f \in C_0^\infty \left(
]-N_{\gamma,\zeta}^2,
N_{\gamma,\zeta}^2[ \setminus
\lbrace 0 \rbrace \right)$ $$\label{eq5,2}
\textup{Tr} \hspace{0.4mm} \left(
\frac{d}{d\varepsilon}
f({{H_0}}+ \varepsilon V)_{\vert
\varepsilon = 0} \right) = -
\frac{1}{\pi} \int_{{\mathbb{R}}}f(\lambda) {\textup{Im}}\textup{Tr}
\hspace{0.4mm} \big( \partial_z
\mathcal{T}_V(\lambda) \big)
d\lambda.$$ Recall that by the Helffer-Sjöstrand formula (see for instance [@dima]) for an analytic extension $\Tilde{f} \in C_0^\infty
({{\mathbb{R}}}^2)$ of $f$ ($i.e.$ $\Tilde{f}_{\vert{{\mathbb{R}}}} = f$ and $\Bar{\partial}_z \Tilde{f}
(z) = \mathcal{O} \big( \vert {\textup{Im}}(z)
\vert^\infty \big)$) we have $$\label{eq5,3}
f({{H_0}}+ \varepsilon V) = -
\frac{1}{\pi} \int_{{\mathbb{C}}}\Bar{\partial}_z \Tilde{f}
(z) (z - {{H_0}}- \varepsilon V)^{-1}
L(dz),$$ $L(dz)$ being the Lebesgue measure on ${{\mathbb{C}}}$. Quantity is differentiable with respect to $\varepsilon$ and it is easy to check that $$\label{eq5,4}
\frac{d}{d\varepsilon}
f({{H_0}}+ \varepsilon V)_{\vert
\varepsilon = 0} = - \frac{1}{\pi}
\int_{{\mathbb{C}}}\Bar{\partial}_z \Tilde{f}
(z) (z - {{H_0}})^{-1}V(z - {{H_0}})^{-1}
L(dz).$$ Exploiting the diamagnetic inequality and the boundedness of the magnetic field $b$ it can be checked that for $\pm {\textup{Im}}(z) > 0$ the operator $(z - {{H_0}})^{-1}V(z - {{H_0}})^{-1}$ is of trace class. For ${\textup{Im}}(z) > 0$ by the cyclicity of the trace we have $$\label{eq5,5}
\textup{Tr} \hspace{0.4mm} \Big(
(z - {{H_0}})^{-1}V(z - {{H_0}})^{-1}
\Big) =
\textup{Tr} \hspace{0.4mm} \Big(
J \vert V \vert^{\frac{1}{2}}
(z - {{H_0}})^{-2} \vert V
\vert^{\frac{1}{2}}
\Big) =
\textup{Tr} \hspace{0.4mm} \Big(
\partial_z \mathcal{T}_V(z) \Big)$$ and for ${\textup{Im}}(z) < 0$ $$\label{eq5,6}
\textup{Tr} \hspace{0.4mm} \Big(
(z - {{H_0}})^{-1}V(z - {{H_0}})^{-1}
\Big) = - \overline{
\textup{Tr} \hspace{0.4mm} \Big(
\partial_z \mathcal{T}_V(\bar{z})
\Big)}.$$ Therefore the operator $\frac{d}{d\varepsilon}
f({{H_0}}+ \varepsilon V)_{\vert
\varepsilon = 0}$ is of trace class and using we get $$\label{eq5,7}
\begin{split}
\textup{Tr} \hspace{0.4mm} \left(
\frac{d}{d\varepsilon}
f({{H_0}}+ \varepsilon V)_{\vert
\varepsilon = 0} \right) = -
\frac{1}{\pi} \int_{{\textup{Im}}(z) > 0} &
\Bar{\partial}_z \Tilde{f}
(z) \textup{Tr}
\hspace{0.4mm} \big( \partial_z
\mathcal{T}_V(z) \big) L(dz) \\
& + \frac{1}{\pi} \int_{{\textup{Im}}(z) < 0}
\Bar{\partial}_z \Tilde{f}
(z) \overline{ \textup{Tr}
\hspace{0.4mm} \big( \partial_z
\mathcal{T}_V(\bar{z}) \big)} L(dz).
\end{split}$$ Now follows immediately from using the Green formula.
For further use we recall complex analysis results due to J. Sjöstrand summarized in the following
\[p5,0\] [[@sj], [@sj1]]{}
Let $\Omega \subset {{\mathbb{C}}}$ be a simply connected domain satisfying $\Omega \cap {{\mathbb{C}}}^+ \neq \emptyset$. Let $z \mapsto F(z,h)$, $0 < h < h_0$ be a family of holomorphic functions in $\Omega$ having at most a finite number $N(h) \in {{\mathbb{N}}}^\ast$ of zeros in $\Omega$. Suppose that $$F(z,h) = \mathcal{O}(1)
e^{\mathcal{O}(1)N(h)}, \quad z \in
\Omega,$$ and that there exists constants $C$, $\varsigma > 0$ with $\Omega_\varsigma
:= \big\lbrace z \in {{\mathbb{C}}}: {\textup{Im}}(z) >
\varsigma \big\rbrace \neq \emptyset$ such that $$\vert F(z,h) \vert \geq e^{-CN(h)},
\quad z \in \Omega_\varsigma.$$ Then for any $\Tilde{\Omega} \Subset
\Omega$ there exists $g(\cdot,h)$ holomorphic in $\Omega$ such that $$F(z,h) = \prod_{j=1}^{N(h)} (z - z_j)
e^{g(z,h)}, \quad \frac{d}{dz} g(z,h)
= \mathcal{O} \big( N(h) \big), \quad
z \in \Tilde{\Omega},$$ where the $z_j$ are the zeros of $F(z,h)$ in $\Omega$.
In the next proposition the domains $\mathscr{W}_\pm \Subset \Omega_\pm$ and the intervals $I_\pm$ are introduced in Section \[s2\] just after .
\[p5,1\] Assume that $V$ satisfies assumptions -. Let $\mathscr{W}_\pm \Subset
\Omega_\pm$ and $I_\pm$ be as above. Then there exists $r_0 > 0$ and holomorphic functions $g_\pm$ in $\Omega_\pm$ satisfying for any $\mu \in rI_\pm$ $$\label{eq5,8}
\begin{split}
\xi_2'(\mu) = \frac{1}{\pi r} {\textup{Im}}\hspace{0.5mm} g'_\pm \left(
\frac{\mu}{r},r \right) + &
\sum_{\substack{w \in \textup{Res}
({{H_V}}) \cap r \Omega_\pm \\ {\textup{Im}}(w)
\neq 0}} \frac{{\textup{Im}}(w)}{\pi \vert
\mu - w \vert^2} \\
& - \sum_{w \in \textup{Res}({{H_V}})
\cap r I_\pm} \delta (\mu - w) -
\frac{1}{\pi} {\textup{Im}}\textup{Tr}
\hspace{0.4mm} \big( \partial_z
\mathcal{T}_V(\mu) \big),
\end{split}$$ where the functions $g_\pm(\cdot,r)$ satisfy $$\label{eq5,9}
\begin{split}
g_\pm(z,r) & = \mathcal{O} \left[
\textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(s_1\sqrt{r},\infty)}
\big( p \textbf{\textup{W}} p \big)
\vert \ln r \vert + \Tilde{n}_1
\left( \frac{1}{2} s_1\sqrt{r}
\right) + \Tilde{n}_2 \left(
\frac{1}{2} s_1\sqrt{r} \right)
\right] \\
& = \mathcal{O} \left( \vert
\ln r \vert r^{-1/m_\perp} \right),
\end{split}$$ uniformly with respect to $0 < r
< r_0$ and $z \in \Omega_\pm$ with $\Tilde{n}_q$, $q = 1$, $2$ defined by .
The first step consists to reduce the study of the zeros of the $2$-regularized perturbation determinant to that of a suitable holomorphic function in $\Omega_\pm$ satisfying the assumptions of Proposition \[p5,0\].
By Proposition \[p4,2\] for $0 < s
< \vert k \vert \leq s_{0} < \epsilon$ $$\mathcal{T}_{V} \big( z(k) \big) =
\frac{iJ}{k} \mathscr{B} + \mathscr{A}
(k).$$ The operator-valued function $k \mapsto
\mathscr{A}(k)$ is analytic near zero with values in ${{\mathcal{S}_2}}\big( L^2({{\mathbb{R}}}^3)
\big)$. Then for $s_{0}$ small enough there exists $\mathscr{A}_{0}$ a finite-rank operator independent of $k$ and $\tilde{\mathscr{A}}(k)$ analytic near zero satisfying $\Vert \tilde{\mathscr{A}}(k) \Vert <
\frac{1}{4}$, $\vert
k \vert < s_{0}$ such that $$\mathscr{A}(k) = \mathscr{A}_{0} +
\tilde{\mathscr{A}}(k).$$ Consider the decomposition $$\label{eq5,10}
\mathscr{B} = \mathscr{B}
\mathbf{1}_{[0,\frac{1}{2}s]}
(\mathscr{B}) + \mathscr{B}
\mathbf{1}_{]\frac{1}{2}s,\infty[}
(\mathscr{B}).$$ Obviously $\left\Vert (iJ/k)
\mathscr{\mathscr{B}} \mathbf{1}_{[0,
\frac{1}{2}s]} (\mathscr{B}) +
\tilde{\mathscr{A}}(k)
\right\Vert < \frac{3}{4}$ for $0 < s
< \vert k \vert < s_{0}$. Then $$\label{eq5,11}
I + \mathcal{T}_{V} \big( z(k) \big)
= \big( I + \mathscr{K}(k,s) \big)
\left( I + \frac{iJ}{k} \mathscr{B}
\mathbf{1}_{[0,\frac{1} {2}s]}
(\mathscr{B}) + \tilde{\mathscr{A}}
(k) \right),$$ where $K(k,s)$ is given by $$\label{eq5,111}
\mathscr{K}(k,s) := \left( \frac{iJ}{k}
\mathscr{B} \mathbf{1}_{]\frac{1}{2}s,
\infty[} (\mathscr{\mathscr{B}}) +
\mathscr{A}_{0} \right) \left( I +
\frac{iJ}{k} \mathscr{B}
\mathbf{1}_{[0,\frac{1}{2}s]}
(\mathscr{B}) + \tilde{\mathscr{A}}(k)
\right)^{-1}.$$ Its rank is of order $$O \left( \textup{Tr} \hspace{0.4mm}
{{\bf 1}}_{(\frac{1}{2}s,\infty)}
(\mathscr{B}) + 1 \right) =
\mathcal{O} \left( \textup{Tr}
\hspace{0.4mm} {{\bf 1}}_{(s,\infty)}
\big( p \textbf{\textup{W}} p \big)
+ 1 \right)$$ according to (\[eq4,12\]) and moreover its norm is bounded by $\mathcal{O} \left( s^{-1} \right)$ for $0 < s < \vert k \vert < s_{0}$. Since $\Vert (iJ/k) \mathscr{B}
\mathbf{1}_{[0,\frac{1}{2}s]}
(\mathscr{B}) + \tilde{\mathscr{A}}(k)
\Vert < 1$ for $0 < s < \vert k \vert
< s_{0}$ then $$\textup{det} \left( \left( I + \frac{iJ}{k}
\mathscr{B} \mathbf{1}_{[0,\frac{1}
{2}s]} (\mathscr{B}) +
\tilde{\mathscr{A}}(k) \right)
e^{-T_V \big( z(k) \big)} \right) \neq 0.$$ Therefore the zeros of $\textup{det}_2 \big( I +
\mathcal{T}_{V} \big( z(k) \big)
\big)$ are those of $$\label{eq5,12}
\mathscr{D}(k,s) := \textup{det}
\big( I + \mathscr{K}(k,s) \big)$$ with the same multiplicities thanks to Proposition \[p4,1\] and Property applied to . The above properties of $\mathscr{K}(k,s)$ imply that $$\label{eq5,13}
\begin{aligned}
\mathscr{D}(k,s) & = \prod_{j=1}^
{\mathcal{O} \big( \textup{Tr}
\hspace{0.4mm} {{\bf 1}}_{(s,\infty)}
(p \textbf{\textup{W}} p) + 1 \big)}
\big{(} 1 + \lambda_{j}(k,s) \big{)}\\
& = \mathcal{O}(1) \hspace{0.5mm}
\textup{exp} \hspace{0.5mm}
\Big( \mathcal{O} \big( \textup{Tr}
\hspace{0.4mm} {{\bf 1}}_{(s,\infty)}
\big( p \textbf{\textup{W}} p
\big) + 1 \big) \vert \ln s \vert
\Big)
\end{aligned}$$ for $0 < s < \vert k \vert < s_{0}$, where the $\lambda_{j}(k,s)$ are the eigenvalues of $\mathscr{K} :=
\mathscr{K}(k,s)$ satisfying $\vert \lambda_{j}(k,s) \vert =
\mathcal{O} \left( s^{-1} \right)$. If ${\textup{Im}}(k^2) > \varsigma > 0$ with $0
< s < \vert k \vert < s_{0}$ then $$\mathscr{D}(k,s)^{-1} = \det \big( I
+ \mathscr{K} \big)^{-1} = \det
\big( I - \mathscr{K} ( I +
\mathscr{K})^{-1} \big).$$ Thus with the help of we can show similarly to that $$\label{eq5,14}
\small{\vert \mathscr{D}(k,s) \vert
\geq C \hspace{0.5mm} \textup{exp}
\hspace{0.5mm} \Big( - C \big(
\textup{Tr} \hspace{0.4mm} {{\bf 1}}_{
(s,\infty)} \big( p \textbf{\textup{W}}
p \big) + 1 \big) \big( \vert \textup{ln}
\hspace{0.5mm} \varsigma \vert + \vert
\textup{ln} \hspace{0.5mm} s \vert \big)
\Big)}.$$
Now for $\mathscr{D}(k,s)$ defined by fix $0 < s_1 <
\sqrt{\textup{dist} \big( \Omega_\pm,0
\big)}$ and consider the functions $$F_\pm : z \in \Omega_\pm \mapsto
\mathscr{D} \left( \sqrt{r}\sqrt{z},
\sqrt{r}s_1 \right)$$ where $$\label{eq5,15}
\displaystyle \sqrt{z} =
\left\{ \begin{array}{ccc} \sqrt{\rho}
e^{i\frac{\theta}{2}} & \hbox{ if }
& z = \rho e^{i\theta} \in \Omega_+, \\
i\sqrt{\rho} e^{-i\frac{\theta}{2}}
& \hbox{ if } & z = -\rho e^{-i\theta}
\in \Omega_-. \end{array} \right.$$ The functions $F_\pm$ are holomorphic in $\Omega_\pm$ and according to Proposition \[p4,1\] $\Tilde{\omega}$ is a zero of $F_\pm$ if and only if $\omega = \Tilde{\omega}r$ is a resonance of ${{H_V}}$. Then by Proposition \[p5,0\] applied to $F = F_+$ and $F(z) = \overline{F_-(-\bar{z})}$ with $h = r$, $N(r) = \textup{Tr}
\hspace{0.4mm} {{\bf 1}}_{(s_1\sqrt{r},
\infty)} \big( p \textbf{\textup{W}}
p \big) \vert \ln r \vert$ there exists holomorphic functions $g_{0,\pm}$ in $\Omega_\pm$ satisfying for any $z \in \Omega_\pm$ $$\label{eq5,16}
\mathscr{D}_\pm \left( \sqrt{r}\sqrt{z},
\sqrt{r}s_1 \right) = \prod_{w \in
\textup{Res}({{H_V}}) \cap r \Omega_\pm}
\left( \frac{zr - \omega}{r} \right)
e^{g_{0,\pm}(z,r)}$$ with $$\label{eq5,17}
\frac{d}{dz} g_{0,\pm}(z,r)
= \mathcal{O} \left( \textup{Tr}
\hspace{0.4mm} {{\bf 1}}_{(s_1\sqrt{r},
\infty)} \big( p \textbf{\textup{W}}
p \big) \vert \ln r \vert \right),$$ uniformly with respect to $z \in
\mathscr{W}_\pm$.
From above - we know that for $z = z \big( \sqrt{r}
k \big)$, $0 < s_1 < \vert k \vert
< s_0$ $$\label{eq5,18}
\begin{split}
\textup{det}_2 & \big( I +
\mathcal{T}_{V} (z) \big) = \\
& \mathscr{D} \left(
\sqrt{r}k, \sqrt{r}s_1
\right) \textup{det} \left(
\left( I + \frac{iJ}{\sqrt{r}k}
\mathscr{B} \mathbf{1}_{[0,\frac{1}
{2}s_1\sqrt{r}]} (\mathscr{B}) +
\tilde{\mathscr{A}}(\sqrt{r}k)
\right) e^{-T_V (z)} \right).
\end{split}$$ By setting $$\mathfrak{A}(k)
:= \frac{iJ}{\sqrt{r}k}
\mathscr{B} \mathbf{1}_{[0,\frac{1}
{2}s_1\sqrt{r}]} (\mathscr{B}) +
\tilde{\mathscr{A}}(\sqrt{r}k)$$ we deduce from that $\mathcal{T}_{V}(z) - \mathfrak{A}(k)$ is a finite-rank operator thanks to the properties of the operator $\mathscr{K} \big( \sqrt{r}k,
\sqrt{r}s_1 \big)$ given by . Then as in we can prove that $$\label{eq5,19}
\begin{split}
\textup{det} \Big(
\Big( I + \frac{iJ}{\sqrt{r}k}
\mathscr{B} \mathbf{1}_{[0,\frac{1}
{2}s_1\sqrt{r}]} & (\mathscr{B}) +
\tilde{\mathscr{A}}(\sqrt{r}k)
\Big) e^{-T_V (z)} \Big) \\
& = \textup{det}_2 \big( I +
\mathfrak{A}(k) \big)
e^{-\textup{Tr} \big( T_V (z) -
\mathfrak{A}(k) \big)}
\end{split}$$ with $\textup{det}_2 \big( I +
\mathfrak{A}(k) \big) \neq 0$ since $\Vert \mathfrak{A}(k) \Vert < 1$ for $0 < s_1 < \vert k \vert
< s_0$. The holomorphicity of $\tilde{\mathscr{A}}(k)$ with values in ${{\mathcal{S}_2}}\big( L^2({{\mathbb{R}}}^3) \big)$ combined with of Corollary \[c4,1\] imply that $$\label{eq5,20}
\Vert \mathfrak{A}(k) \Vert_2^2
= \mathcal{O} \left( \Tilde{n}_2
\left( \frac{1}{2} \sqrt{r}s_1
\right) \right).$$ Then we have $$\label{eq5,21}
\textup{det}_2 \big( I +
\mathfrak{A}(k) \big)
= \mathcal{O}(1) e^{\mathcal{O}(1)
\Tilde{n}_2 \left( \frac{1}{2}
\sqrt{r}s_1 \right)}.$$ On the other hand it can be also checked that $$\label{eq5,22}
\textup{det}_2 \big( I +
\mathfrak{A}(k) \big)^{-1} =
\textup{det}_2 \left( I -
\mathfrak{A}(k) \big( I +
\mathfrak{A}(k) \big)^{-1} \right)
= \mathcal{O}(1) e^{\mathcal{O}(1)
\Tilde{n}_2 \left( \frac{1}{2}
\sqrt{r}s_1 \right)}.$$ Then Proposition \[p5,0\] implies that there exists $g_1(\cdot,r)$ holomorphic in $\Omega_\pm$ such that $$\label{eq5,23}
\textup{det}_2 \big( I +
\mathfrak{A}(k) \big)
= e^{g_1(z,r)}$$ with $$\label{eq5,24}
\frac{d}{dz} g_{1}(z,r)
= \mathcal{O} \left( \Tilde{n}_2
\left( \frac{1}{2} \sqrt{r}s_1
\right) \right),$$ uniformly with respect to $z
\in \mathscr{W}_\pm$. Therefore according to definition of $\xi_2$ and by combining , , with we get for $\mu = z \big( \sqrt{r}k \big)
= rk^2 \in r(\Omega_\pm \cap {{\mathbb{R}}})$ $$\label{eq5,25}
\begin{split}
\xi_2'(\mu) & = \frac{1}{\pi r} {\textup{Im}}\hspace{0.5mm} \partial_\lambda
(g_{0,\pm} + g_1) \left(
\frac{\mu}{r},r \right) +
\sum_{\substack{w \in \textup{Res}
({{H_V}}) \cap r \Omega_\pm \\ {\textup{Im}}(w)
\neq 0}} \frac{{\textup{Im}}(w)}{\pi \vert
\mu - w \vert^2} - \sum_{w \in
\textup{Res}({{H_V}}) \cap r I_\pm}
\delta (\mu - w) \\
& + \frac{1}{\pi} {\textup{Im}}\textup{Tr}
\hspace{0.4mm} \left( \frac{1}{2k}
\partial_k \left( \frac{iJ}{k}
\mathscr{B} \mathbf{1}_{[0,\frac{1}
{2}s_1\sqrt{r}]} (\mathscr{B}) +
\tilde{\mathscr{A}}(k) \right) -
\partial_z \mathcal{T}_V(\mu + i0)
\right)
\end{split}$$ with $$\label{eq5,27}
\displaystyle k =
\left\{ \begin{array}{ccc} \sqrt{\mu}
& \hbox{ if } & \mu > 0, \\
i\sqrt{-\mu} & \hbox{ if } & \mu < 0.
\end{array} \right.$$ By of Corollary \[c4,1\] $$\textup{Tr}
\hspace{0.4mm} \left( \frac{1}{2k}
\partial_k \left( \frac{iJ}{k}
\mathscr{B} \mathbf{1}_{[0,\frac{1}
{2}s_1\sqrt{r}]} (\mathscr{B})
\right) \right) = -
\frac{iJs_1\sqrt{r}}{4k^3}
\Tilde{n}_1 \left( \frac{1}{2}
\sqrt{r}s_1 \right).$$ Thanks to Lemma \[l3,1\] $\partial_z \mathcal{T}_V(z)$ is of trace class. Then since $\mathscr{B}
\in {{\mathcal{S}_1}}\big( L^2({{\mathbb{R}}}^3) \big)$ the operator $$\label{eq5,271}
\partial_k \tilde{\mathscr{A}}(k) =
\partial_k \mathscr{A}(k) =
\partial_k \Big(
\mathcal{T}_V \big( z(k) \big) -
\frac{iJ}{k} \mathscr{B} \Big)$$ is of trace class. Moreover the definition of $\mathscr{A}(k)$ implies that $$\label{eq5,272}
\textup{Tr} \hspace{0.4mm} \left(
\frac{1}{2k} \partial_k
\mathscr{A}(k) \right) =
\textup{Tr} \hspace{0.4mm}
\left( J \vert V
\vert^{\frac{1}{2}} \big( {{H_0}}- k^2 \big)^{-2} \textup{Q} \vert
V \vert^{\frac{1}{2}} \right) =
\textup{Tr} \hspace{0.4mm}
\left( J \vert V
\vert^{\frac{1}{2}} \big( {{H_0}}- \mu \big)^{-2} \textup{Q} \vert
V \vert^{\frac{1}{2}} \right).$$ By setting $g_\pm = g_{0,\pm} +
g_1 + g_2$ with $$g_2(z) = \frac{iJs_1}{2\sqrt{z}}
\Tilde{n}_1 \left( \frac{1}{2}
\sqrt{r}s_1 \right),$$ where $\sqrt{z}$ is defined on $\Omega_\pm$ by we get the desired conclusion.
The representation of the SSF near zero can be specified if the potential $V$ is of definite sign $J = sign(V)$. According to Remark \[r2,1\] in the next proposition the case $"-"$ is with respect the definite sign $J = +$.
\[p5,2\] Assume the assumptions of Theorem \[t2,1\] with $V$ of definite sign $J = sign(V)$. Then for $\lambda \in rI_\pm$ holds with $$\label{eq5,28}
\frac{1}{r} {\textup{Im}}\hspace{0.5mm} g'_\pm
\left( \frac{\lambda}{r},r \right) =
\frac{1}{r} {\textup{Im}}\hspace{0.5mm} \Tilde{g}'_\pm
\left( \frac{\lambda}{r},r \right) +
{\textup{Im}}\hspace{0.5mm} \Tilde{g}'_{1,\pm}(\lambda)
+ {{\bf 1}}_{(0,N_{\gamma,\zeta}^2)}(\lambda)
J \phi'(\lambda),$$ where the function $\phi$ is defined by $$\label{eq5,29}
\phi(\lambda) := \textup{Tr}
\hspace{0.4mm} \left( \arctan
\frac{K^\ast K}{\sqrt{\lambda}} \right)
= \textup{Tr} \hspace{0.4mm} \left( \arctan
\frac{p\textbf{\textup{W}}p}{2\sqrt{\lambda}}
\right),$$ the functions $z \mapsto \Tilde{g}_\pm (z,r)$ being holomorphic in $\Omega_\pm$ and satisfying $$\label{eq5,30}
\Tilde{g}_\pm (z,r) = \mathcal{O}
\big( \vert \ln r \vert \big),$$ uniformly with respect to $0 < r < r_0$ and $z \in \Omega_\pm$. The functions $z \mapsto
\Tilde{g}_{1,\pm}(z)$ are holomorphic in $\pm ]0,N_{\gamma,\zeta}^2[
e^{\pm i]-2\theta_0,2\varepsilon_0[}$ and there exists a positive constant $C_{\theta_0}$ depending on $\theta_0$ such that $$\label{eq5,31}
\vert \Tilde{g}_{1,\pm}(z) \vert \leq
C_{\theta_0} \sigma_2 \left( \sqrt{\vert
z \vert} \right)^{\frac{1}{2}}$$ for $z \in \pm ]0,N_{\gamma,\zeta}^2[
e^{\pm i]-2\theta_0,2\varepsilon_0[}$, where the quantity $\sigma_2(\cdot)$ is defined by .
We use notations of Subsection \[ss4,1\]. Hence for $z = z \big(
\sqrt{r} k \big)$, $0 < s_1 < \vert
k \vert < s_0$ and $k \in
\mathcal{C}_\delta(J)$ implies that $$\label{eq5,32}
\textup{det}_2 \big( I +
\mathcal{T}_{V}(z) \big) =
\textup{det} \left( I +
\frac{iJ}{\sqrt{r}k} \mathscr{B}
\right) \times \textup{det}_2 \big(
I + A(\sqrt{r}k) \big)
e^{-\textup{Tr} \left(
\mathcal{T}_{V}(z) - A(\sqrt{r}k)
\right)},$$ where $A(\sqrt{r}k)$ is given by with $k$ replaced by $\sqrt{r}k$. Then as in the previous proof by applying Proposition \[p5,0\] to $\textup{det}_2 \big(
I + A(\sqrt{r}\sqrt{\cdot}) \big)$ in $\Omega_\pm$ taking into account and we get $$\label{eq5,33}
\textup{det}_2 \big( I +
A(\sqrt{r}\sqrt{z}) \big) =
\prod_{w \in \textup{Res}({{H_V}})
\cap r \Omega_\pm} \left(
\frac{zr - \omega}{r} \right)
e^{\Tilde{g}_\pm(z,r)},$$ where $\Tilde{g}_\pm$ is holomorphic in $\Omega_\pm$ such that $$\label{eq5,34}
\frac{d}{dz} \Tilde{g}_\pm(z,r)
= \mathcal{O} \left( \vert
\ln r \vert \right),$$ uniformly with respect to $z \in
\mathscr{W}_\pm$. Then according to definition of $\xi_2$ and by combining - we get for $\mu = z \big( \sqrt{r}k \big)
= rk^2 \in r(\Omega_\pm \cap {{\mathbb{R}}})$ $$\label{eq5,35}
\begin{split}
\xi_2'(\mu) & = \frac{1}{\pi r} {\textup{Im}}\hspace{0.5mm} \partial_\lambda
\Tilde{g}_\pm \left(
\frac{\mu}{r},r \right) +
\sum_{\substack{w \in \textup{Res}
({{H_V}}) \cap r \Omega_\pm \\ {\textup{Im}}(w)
\neq 0}} \frac{{\textup{Im}}(w)}{\pi \vert
\mu - w \vert^2} - \sum_{w \in
\textup{Res}({{H_V}}) \cap r I_\pm}
\delta (\mu - w) \\
& + \frac{1}{2k\pi} {\textup{Im}}\textup{Tr}
\hspace{0.4mm} \left( \left( I +
\frac{iJ}{k} \mathscr{B} \right)^{-1}
\partial_k \left( \frac{iJ}{k}
\mathscr{B} \right) \right) -
\frac{1}{\pi} {\textup{Im}}\textup{Tr}
\hspace{0.4mm} \left( \partial_z
\mathcal{T}_V(\mu + i0) -
\frac{1}{2k} \partial_k A(k) \right),
\end{split}$$ where $k$ is defined by .
By Lemma \[l3,1\] $\partial_z
\mathcal{T}_V(z)$ is of trace class. Then as in accordingly to definition of $A(k)$ $$\label{eq5,36}
\partial_k A(k) = \partial_k
\mathscr{A}(k) - \partial_k \left(
\mathscr{A}(k) \frac{iJ}{k} \mathscr{B}
\left( I + \frac{iJ}{k} \mathscr{B}
\right)^{-1} \right)$$ is of trace class. For the first term of the RHS of equality holds. For the second term we have $$\label{eq5,37}
{\textup{Im}}\frac{1}{2k} \textup{Tr}
\hspace{0.4mm} \partial_k \left(
\mathscr{A}(k) \frac{iJ}{k} \mathscr{B}
\left( I + \frac{iJ}{k} \mathscr{B}
\right)^{-1} \right) = {\textup{Im}}\frac{1}{2k}
\partial_k \big( \Tilde{g}_{1,\pm}(k^2)
\big),$$ where $\Tilde{g}_{1,\pm}$ is the holomorphic function given by $$\label{eq5,38}
\Tilde{g}_{1,\pm}(z) := \textup{Tr}
\hspace{0.4mm} \left( \mathscr{A}
(\sqrt{z}) \frac{iJ}{\sqrt{z}}
\mathscr{B} \left( I +
\frac{iJ}{\sqrt{z}} \mathscr{B}
\right)^{-1} \right)$$ satisfying bound by Corollary \[c4,1\].
For the fourth term of the RHS of we have $$\label{eq5,39}
\begin{split}
\frac{1}{2k} {\textup{Im}}\textup{Tr}
\hspace{0.4mm} & \left( \left( I +
\frac{iJ}{k} \mathscr{B} \right)^{-1}
\partial_k \left( \frac{iJ}{k}
\mathscr{B} \right) \right) \\
& = -\frac{1}{2k^2} {\textup{Im}}\textup{Tr}
\hspace{0.4mm} \left( \frac{iJ}{k}
\mathscr{B} \left( I + \frac{iJ}{k}
\mathscr{B} \right)^{-1} \right) \\
& \displaystyle =
\left\{ \begin{array}{ccc} 0
& \hbox{ if } & Jk \in i{{\mathbb{R}}}^+, \\
-\frac{1}{2k^2} \textup{Tr}
\hspace{0.4mm} \left( \frac{J}{k}
\mathscr{B} \left( I +
\frac{\mathscr{B}^2}{k^2}
\right)^{-1} \right) = J \Phi'(k^2)
& \hbox{ if } & k \in {{\mathbb{R}}}.
\end{array} \right.
\end{split}$$ Then Proposition \[p5,2\] follows.
Back to the proof of Theorem \[t2,1\]
-------------------------------------
It follows immediately by combining Lemma \[l5,1\] with Propositions \[p5,1\]-\[p5,2\].
Proof of Theorem $\ref{t2,2}$: Singularity at the low ground state {#s6}
==================================================================
We begin by applying Theorem \[t2,1\] on intervals of the form $r_n[1,2]$, $r_n = 2^n\lambda$ with $\lambda > 0$ small enough. Hence for $\Omega_+$ a complex neighbourhood of $[1,2]$ and $\mu \in r_n[1,2]$ we have $$\label{eq5,40}
\begin{split}
\xi'(\mu) = \frac{1}{r_n \pi} {\textup{Im}}\hspace{0.5mm} \Tilde{g}'_\pm \left(
\frac{\mu}{r_n},r_n \right) & +
\sum_{\substack{w \in \textup{Res}({{H_V}})
\cap r_n \Omega_+ \\ {\textup{Im}}(w) \neq 0}}
\frac{{\textup{Im}}(w)}{\pi \vert \mu - w
\vert^2} \\
& - \sum_{w \in \textup{Res}
({{H_V}}) \cap r_n [1,2]} \delta (\mu - w)
+ \frac{1}{\pi} \left( J \phi' + {\textup{Im}}\hspace{0.5mm} \Tilde{g}'_{1,\pm}
\right)(\mu).
\end{split}$$ By Theorem \[t4,1\] there exists at most $\mathcal{O} \big( \vert \ln r_n
\vert \big)$ resonances in $r_n\Omega_+$. Then by integrating on $r_n[1,2]$ we obtain $$\label{eq5,41}
\xi(r_{n+1}) - \xi(r_n) =
\frac{1}{\pi} \big[ {\textup{Im}}\hspace{0.5mm} \Tilde{g}_\pm
(\cdot,r_n) \big]_1^2 + \mathcal{O}
\big( \vert \ln r_n \vert \big)
+ \frac{1}{\pi} \big[ J \phi + {\textup{Im}}\hspace{0.5mm} \Tilde{g}_{1,\pm}
\big]_{r_n}^{r_{n+1}}.$$ Now choose $N \in {{\mathbb{N}}}$ such that $\frac{N_{\gamma,\zeta}^2}{4} \leq
\lambda 2^{N+1} \leq
\frac{N_{\gamma,\zeta}^2}{2}$. Then taking the sum in and exploiting the fact that in $\frac{N_{\gamma,\zeta}^2}{2}
\big[ \frac{1}{2},1 \big]$ the functions $\xi$, $\Phi$, $\Tilde{g}_{1,\pm}$ are uniformly bounded together with $\Tilde{g}_\pm
(\cdot,r_n) = \mathcal{O} \big(
\vert \ln r_n \vert \big)$ we get $$\label{eq5,42}
\xi(\lambda) =
\frac{J}{\pi} \Phi (\lambda)
+ \frac{1}{\pi} {\textup{Im}}\hspace{0.5mm}
\Tilde{g}_{1,\pm}(\lambda) +
\sum_{n=0}^N \mathcal{O} \big( \vert
\ln 2^n \lambda \vert \big) +
\mathcal{O}(1).$$ Since $N = \mathcal{O} \big( \vert
\ln \lambda \vert \big)$ and $\Tilde{g}_{1,\pm}$ satisfies then implies that for $\lambda$ small enough $$\label{eq5,43}
\left\vert \xi(\lambda) -
\frac{J}{\pi} \Phi (\lambda)
\right\vert \leq C \vert
\ln \lambda \vert^2 + C \sigma_2
\left( \sqrt{\lambda}
\right)^{\frac{1}{2}}$$ for some $C > 0$ constant. For a Hilbert-Schmidt operator $L$ on $\mathscr{H}$ we have $\Vert L \Vert_{{\mathcal{S}_2}}^2 = \textup{Tr}
\hspace{0.4mm} (LL^\ast)$. This together with the elementary inequality $$\frac{u^2}{1 + u^2} \leq \arctan u
, \quad u \geq 0$$ imply that $\sigma_2 \left(
\sqrt{\lambda} \right) \leq \Phi
(\lambda)$, which completes the proof.
Proof of Theorem $\ref{t2,3}$: Local trace formula {#s7}
==================================================
For simplicity of notation we ignore in the proof the dependence on the subscript $\pm$. Let $\Tilde{\psi} \in C_0^{\infty} \big(
\Omega \big)$ be an almost analytic extension of $\psi$ such that $\Tilde{\psi} = 1$ on $\mathcal{W}$ and $$\textup{supp} \hspace*{0.1cm}
\Bar{\partial}_z \Tilde{\psi} \subset
\Omega \setminus \mathcal{W}.$$ By Applying and Theorem \[t2,1\] we get $$\label{eq5,44}
\begin{split}
\textup{Tr} \hspace{0.4mm} & \left[
(\psi f) \left( \frac{{{H_V}}}{r} \right)
- (\psi f) \left( \frac{{{H_0}}}{r}
\right) \right] = - \left\langle \xi'
(\lambda),(\psi f) \left(
\frac{\lambda}{r} \right) \right\rangle
\\
& = \sum_{w \in \textup{Res}({{H_V}}) \cap
r \textup{supp} \hspace{0.05cm} \psi}
(\psi f) \left( \frac{w}{r} \right)
- \frac{1}{\pi} \int (\psi f) \left(
\frac{\lambda}{r} \right) {\textup{Im}}\hspace{0.5mm}
g' \left( \frac{\lambda}{r},r \right)
\frac{d\lambda}{r} \\
& + \sum_{\substack{w \in \textup{Res}
({{H_V}}) \cap r \textup{supp}
\hspace{0.05cm} \psi \\ {\textup{Im}}(w)
\neq 0}} \frac{1}{2\pi i} \int (\psi f)
\left( \frac{\lambda}{r} \right) \left(
\frac{1}{\lambda - \overline{w}} -
\frac{1}{\lambda - w}\right) d\lambda.
\end{split}$$ Using the Green formula and on $\textup{supp}
\hspace{0.05cm} \Tilde{\psi}$ we can estimate the integral involving $g'$. On the other hand for $w \in {{\mathbb{C}}}_- :=
\big\lbrace z \in {{\mathbb{C}}}: {\textup{Im}}(z) < 0
\big\rbrace$ by applying the Green formula we get $$- \frac{1}{\pi} \int_{{{\mathbb{C}}}_-}
\Bar{\partial}_z \Tilde{\psi}(z)
\frac{1}{z - w} L(dz) + \Tilde{\psi}(w)
= - \frac{1}{2\pi i} \int_{{\mathbb{R}}}\Tilde{\psi}(\lambda) \frac{1}{\lambda
- w} d\lambda$$ and $$- \frac{1}{\pi} \int_{{{\mathbb{C}}}_-}
\Bar{\partial}_z \Tilde{\psi}(z)
\frac{1}{z - \overline{w}} L(dz)
= - \frac{1}{2\pi i} \int_{{\mathbb{R}}}\Tilde{\psi}(\lambda) \frac{1}{\lambda
- \overline{w}} d\lambda.$$ Since $f$ is holomorphic then with the help of the above formulas and using the fact that $\Tilde{\psi}
= \psi$ on ${{\mathbb{R}}}$ the third term of the RHS of is equal to $$\begin{split}
\sum_{\substack{w \in \textup{Res}
({{H_V}}), {\textup{Im}}(w) \neq 0}} &
(\Tilde{\psi} f) \left( \frac{w}{r}
\right) \\
& + \sum_{\substack{w \in \textup{Res}
({{H_V}}) \cap r \textup{supp}
\hspace{0.05cm} \Tilde{\psi} \\ {\textup{Im}}(w) \neq 0}} \frac{1}{\pi r}
\int_{{{\mathbb{C}}}_-} (\Bar{\partial}_z
\Tilde{\psi}) \left( \frac{z}{r}
\right) f \left( \frac{z}{r} \right)
\left( \frac{1}{z - \overline{w}}
- \frac{1}{z - w} \right) L(dz).
\end{split}$$ Now by using Theorem \[t4,2\] in $\Omega$ and the elementary inequality [@pet (5.3)] $$\int_{\Omega_1}
\frac{1}{\vert z - w \vert} L(dz)
\leq 2 \sqrt{2\pi\text{vol}(\Omega)}$$ we get the result.
Appendix {#sa}
========
We recall in this subsection the notion of index (with respect to a positively oriented contour) of a holomorphic function and a finite meromorphic operator-valued function, see for instance [@bo Definition 2.1].
If a function $f$ is holomorphic in a neighbourhood of a contour $\gamma$ its index with respect to $\gamma$ is defined by $$\label{eqa,1}
ind_{\gamma} \hspace{0.5mm} f
:= \frac{1}{2i\pi} \int_{\gamma} \frac{f'(z)}{f(z)} dz.$$ Let us point out that if $f$ is holomorphic in a domain $\Omega$ with $\partial \Omega = \gamma$ then thanks to the residues theorem $\textup{ind}_{\gamma} \hspace{0.5mm} f$ coincides with the number of zeros of $f$ in $\Omega$ taking into account their multiplicity.
Let $D \subseteq \mathbb{C}$ be a connected domain, $Z \subset D$ be a pure point and closed subset and $A : \overline{D} \backslash Z \longrightarrow
\textup{GL}(E)$ a be finite meromorphic operator-valued function which is Fredholm at each point of $Z$. The index of $A$ with respect to the contour $\partial \Omega$ is defined by $$\label{eqa,2}
\small{Ind_{\partial \Omega} \hspace{0.5mm} A :=
\frac{1}{2i\pi} \textup{Tr} \int_{\partial \Omega} A'(z)A(z)^{-1}
dz = \frac{1}{2i\pi} \textup{Tr} \int_{\partial \Omega} A(z)^{-1}
A'(z) dz}.$$ The following properties are well known: $$\label{eqa,3}
Ind_{\partial \Omega} \hspace{0.5mm} A_{1} A_{2} =
Ind_{\partial \Omega} \hspace{0.5mm} A_{1} +
Ind_{\partial \Omega} \hspace{0.5mm} A_{2};$$ for $K(z)$ a trace class operator $$\label{eqa,4}
Ind_{\partial \Omega} \hspace{0.5mm} (I+K)=
ind_{\partial \Omega} \hspace{0.5mm} \det \hspace{0.5mm} (I + K).$$ We refer for instance [@goh Chap. 4] for more details.
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|
---
abstract: 'Gowers norms have been studied extensively both in the direct sense, starting with a function and understanding the associated norm, and in the inverse sense, starting with the norm and deducing properties of the function. Instead of focusing on the norms themselves, we study associated dual norms and dual functions. Combining this study with a variant of the Szemerédi Regularity Lemma, we give a decomposition theorem for dual functions, linking the dual norms to classical norms and indicating that the dual norm is easier to understand than the norm itself. Using the dual functions, we introduce higher order algebras that are analogs of the classical Fourier algebra, which in turn can be used to further characterize the dual functions.'
address:
- |
Laboratoire d’analyse et de mathématiques appliquées, Université Paris-Est Marne-la-Vallée & CNRS UMR 8050\
5 Bd. Descartes, Champs sur Marne\
77454 Marne la Vallée Cedex 2, France
- |
Department of Mathematics, Northwestern University\
2033 Sheridan Road\
Evanston, IL 60208-2730, USA
author:
- Bernard Host
- Bryna Kra
title: A point of view on Gowers uniformity norms
---
[^1]
Introduction
============
In his seminal work on Szemerédi’s Theorem, Gowers [@gowers] introduced uniformity norms $U(d)$ for each integer $d\geq 1$, now referred to as Gowers norms or Gowers uniformity norms, that have played an important role in the developments in additive combinatorics over the past ten years. In particular, Green and Tao [@GT] used Gowers norms as a tool in their proof that the primes contain arbitrarily long arithmetic progressions in the primes; shortly thereafter, they made a conjecture [@GT3], the Inverse Conjecture for the Gowers norms, on the algebraic structures underlying these norms. Related seminorms were introduced by the authors [@HK1] in the setting of ergodic theory, and the ergodic structure theorem provided a source of motivation in the formulation of the Inverse Conjecture. For each integer $d\geq 1$ and $\delta > 0$, Green and Tao introduce a class ${{\mathcal F}}(d,\delta)$ of “$(d-1)$-step nilsequences of bounded complexity,” which we do not define here, and the proof of the Inverse Conjecture was given:
\[th:inverse\] For each integer $d\geq 1$ and $\delta > 0$, there exists a constant $C = C(d, \delta)> 0 $ such that for every function $f$ on ${{{\mathbb Z}}}/N{{{\mathbb Z}}}$ with $|f|\leq 1$ and ${\lVert f\rVert}_{U(d)}\geq \delta$, there exists $g\in{{\mathcal F}}(d, \delta)$ with $\langle g;f\rangle \geq C$.
See also Szegedy’s approach to the Inverse Conjecture, outlined in the announcement [@CS] for the article [@S].
We are motivated by the work of Gowers in [@gowers1]. Several ideas come out of this work, in particular the motivation that algebra norms are easier to study. The Gowers norms $U(d)$ are classically defined in ${{{\mathbb Z}}}/N{{{\mathbb Z}}}$, but we choose to work in a general compact abelian group. For most of the results presented here, we take care to distinguish between the group ${{{\mathbb Z}}}/N{{{\mathbb Z}}}$ and the interval $[1,\ldots, N]$, of the natural numbers ${{{\mathbb N}}}$, whereas for applications in additive combinatorics, the results may be more directly proved without this separation. This is a conscious choice that allows us to separate what about Gowers norms is particular to the combinatorics of ${{{\mathbb Z}}}/N{{{\mathbb Z}}}$ and what is more general. Our point of view is that of harmonic analysis, rather than combinatorial.
More generally, the Gowers norms can be defined on a nilmanifold. This is particularly important in the ergodic setting where analogous seminorms were defined by the authors in [@HK1] in an arbitrary measure space; these seminorms are exactly norms when the space is a nilmanifold. While we restrict ourselves to abelian groups in this article, most of the results can be carried out in the more general setting of a nilmanifold without significant changes.
Instead of focusing on the Gowers norms themselves, we study the associated dual norms that fit within this framework and the associated dual functions. Moreover, in the statement of the inverse theorem, and more generally in uses of the Gowers norms, one typically assumes that the functions are bounded by $1$. >From the duality point of view, instead we study functions in the dual space itself, we can consider functions that are within a small $L^{1}$ error from functions in this space. This allows us to restrict ourselves to dual functions of functions in a certain $L^{p}$ class (Theorem \[th:k\]). Moreover, we rephrase the Inverse Theorem in terms of dual functions (see Section \[subsec:def\_dual\_functions\] for precise meanings of the term) in certain $L^p$ classes, and in this form the Gowers norms do not appear explicitly (Section \[subsec:approx\]). This reformulates the Inverse Theorem more in a classical analysis context.
The dual functions allow us to introduce algebras of functions on the compact abelian group $Z$. For $d=2$, this corresponds to the classical Fourier algebra. Finding an interpretation for the higher order uniformity norms is hard and no analogs of Fourier analysis and simple formulas, such as Parseval, exist. For $d> 2$, the higher order Fourier algebra are analogs of the classical case of the Fourier algebra. These algebras allow us to further describe the dual functions. Starting with a dual function of level $d$, we find that it lies in the Fourier algebra of order $d$, giving us information on its dual norm $U(d)^{*}$, and by an approximation result, we understand further the original function.
We obtain a result on compactness (Theorem \[th:main\]) of dual functions, by applying a variation of the classical Szemerédi Regularity Lemma.
Gowers norms: definition and elementary bounds
===============================================
Notation
--------
Throughout, we assume that $Z$ is a compact abelian group and let $\mu$ denote Haar measure on $Z$. If $Z$ is finite, then $\mu$ is the uniform measure; the classical case to keep in mind is when $Z = {{{\mathbb Z}}}_{N}={{{\mathbb Z}}}/N{{{\mathbb Z}}}$ and the measure of each element is $1/N$.
All functions are implicitly assumed to be real valued. When $Z$ is infinite, we also implicitly assume that all functions and sets are measurable. For $1\leq p\leq\infty$, ${\lVert \cdot\rVert}_p$ denotes the $L^p(\mu)$ norm; if there is a need to specific the measure, write ${\lVert \cdot\rVert}_{L^{p}(\mu)}$ or ${\lVert \cdot\rVert}_{L^{p}(Z)}$ when we wish to emphasize the space.
We fix an integer $d\geq 1$ throughout and the dependence on $d$ is implicit in all statements.
We have various spaces of various dimensions: $1$, $d$, $2^{d}$. Ordinary letters $t$ are reserved for spaces of one dimension, vector notation ${{\vec{t}}}$ for dimension $d$, and bold face characters ${{\mathbf t}}$ for dimension $2^{d}$.
If $f$ is a function on $Z$ and $t\in Z$, we write $f_t$ for the function on $Z$ defined by $$f_t(x)=f(x+t),$$ where $x\in Z$. If $f$ is a $\mu$-integrable function on $Z$, we write $${{{\mathbb E}}}_{x\in Z}f(x)=\int f(x)\,d\mu(x)\ .$$ We use similar notation for multiple integrals. If $f$ and $g$ are functions on $Z$, we write $$\langle f ;g\rangle ={{{\mathbb E}}}_{x\in Z}f(x)g(x),$$ assuming that the integral on the right hand side is defined.
If $d$ is a positive integer, we set $$V_d=\{0,1\}^d.$$ Elements of $V_d$ are written as ${{\vec{\epsilon}}}=\epsilon_1\epsilon_2\cdots\epsilon_d$, without commas or parentheses. Writing $\vec 0 = 00\cdots 0\in V_d$, we set $${\widetilde}V_d=V_d\setminus\{{{\vec 0}}\}.$$
For ${{\mathbf x}}\in Z^{2^d}$, we write ${{\mathbf x}}= (x_{{\vec{\epsilon}}}\colon {{\vec{\epsilon}}}\in
V_d)$.
For ${{\vec{\epsilon}}}\in V_d$ and ${{\vec{t}}}=(t_1,t_2,\dots,t_d)\in Z^d$ we write $${{\vec{\epsilon}}}\cdot {{\vec{t}}}=\epsilon_1t_1+\epsilon_2t_2+\cdots+\epsilon_dt_d\ .$$
The uniformity norms and the dual functions: definitions {#subsec:def_dual_functions}
--------------------------------------------------------
The *uniformity norms*, or *Gowers norms*, ${\lVert f\rVert}_{U(d)}$, $d\geq 2$, of a function $f\in L^\infty(\mu)$ are defined inductively by $${\lVert f\rVert}_{U(1)}=\bigl|E_xf(x)|$$ and for $d\geq 2$, $${\lVert f\rVert}_{U(d)}=\Bigl(E_t{\lVert f.f_t\rVert}_{U(d-1)}^{2^{d-1}}\Bigr)^{1/2^{d}}.$$ Note that ${\lVert \cdot\rVert}_{U(1)}$ is not actually a norm. (See [@gowers] for more on these norms and [@HK1] for a related seminorm in ergodic theory.) If there is ambiguity as to the underlying group $Z$, we write ${\lVert \cdot\rVert}_{U(Z,d)}$.
These norms can also be defined by closed formulas: $$\label{eq:def-norm}
{\lVert f\rVert}_{U(d)}^{2^d}={{{\mathbb E}}}_{x\in Z,\; {{\vec{t}}}\in Z^d}
\prod_{{{\vec{\epsilon}}}\in V_d}f(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}}).$$
We can rewrite this formula. Let $Z_d$ be the subset of $Z^{2^d}$ defined by $$\label{eq:Zd}
Z_d=\bigl\{ (x+{{\vec{\epsilon}}}\cdot {{\vec{t}}}\colon{{\vec{\epsilon}}}\in V_d)
\colon x\in Z,\ {{\vec{t}}}\in{{{\mathbb Z}}}^d\bigr\}.$$ This set can be viewed as the “set of cubes of dimension $d$” (see, for example, [@gowers] or [@HK1]). It is easy to check that $Z_d$ is a closed subgroup of $Z^{2^d}$. Let $\mu_d$ denote its Haar measure. Then $Z_d$ is the image of $Z^{d+1}=Z\times Z^d$ under the map $(x,{{\vec{t}}})\mapsto
(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}}\colon{{\vec{\epsilon}}}\in V_d)$. Furthermore, $\mu_d$ is the image of $\mu\times\mu\times\ldots\times\mu$ (taken $d+1$ times) under the same map. If $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in V_d$, are functions in $L^\infty(\mu)$, then $${{{\mathbb E}}}_{x\in Z,\; {{\vec{t}}}\in Z^d}\prod_{{{\vec{\epsilon}}}\in V_d}
f_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot{{\vec{t}}})
=\int_{Z_d}\prod_{{{\vec{\epsilon}}}\in V_d}
f_{{\vec{\epsilon}}}(x_{{\vec{\epsilon}}})\,d\mu_d({{\mathbf x}}).$$ In particular, for $f\in L^\infty(\mu)$, $$\label{eq:def-norm2}
{\lVert f\rVert}_{U(d)}^{2^d}=\int_{Z_d}
\prod_{{{\vec{\epsilon}}}\in V_d}
f(x_{{\vec{\epsilon}}})\,d\mu_d({{\mathbf x}}).$$ Associating the coordinates of the set $V_d$ with the coordinates of the Euclidean cube, we have that the measure $\mu_d$ is invariant under permutations that are associated to the isometries of the Euclidean cube. These permutations act transitively on $V_d$.
For $d=2$, by Parseval’s identity we have that $$\label{eq:U2Parseval}
{\lVert f\rVert}_{U(2)}={\lVert {\widehat}f\rVert}_{\ell^4({\widehat}Z)},$$ where ${\widehat}Z$ is the dual group of $Z$ and ${\widehat}f$ is the Fourier transform of $f$. For $d\geq 3$, no analogous simple formula is known and the interpretation of the Gowers uniformity norms is more difficult. A deeper understanding of the higher order norms is, in part, motivation for the current work.
We make use of the “Cauchy-Schwarz-Gowers Inequality” (CSG) used in the proof of the subadditivity of Gowers norms:
\[prop:CSG\] Let $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in V_d$, be $2^d$ functions belonging to $L^\infty(\mu)$. Then $$\begin{gathered}
\tag{CSG}
\Bigl|{{{\mathbb E}}}_{x\in Z,\;{{\vec{t}}}\in Z^d}f_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}})\Bigr|\\
=
\Bigl|\int_{Z_d} \prod_{{{\vec{\epsilon}}}\in V_d}
f_{{\vec{\epsilon}}}(x_{{\vec{\epsilon}}})\,d\mu_d({{\mathbf x}})\Bigr|
\leq \prod_{{{\vec{\epsilon}}}\in\{0,1\}^d}{\lVert f_{{\vec{\epsilon}}}\rVert}_{U(d)}.\end{gathered}$$
Applying the Cauchy-Schwarz-Gowers Inequality with half of the functions equal to $f$ and the other half equal to the constant $1$, we deduce that $${\lVert f\rVert}_{U(d+1)}\geq{\lVert f\rVert}_{U(d)}$$ for every $f\in L^{\infty}(Z)$.
For $f\in L^\infty(\mu)$, define the *dual function* ${{\mathcal D}}_d f$ on $Z$ by $${{\mathcal D}}_df(x)={{{\mathbb E}}}_{{{\vec{t}}}\in Z^d}\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_d}
f(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}}).$$
It follows from the definition that $$\label{eq:norme_Dd}
{\lVert f\rVert}_{U(d)}^{2^d}=\langle{{\mathcal D}}_d f;f\rangle.$$
More generally, we define:
If $f_{{{\vec{\epsilon}}}}\in L^{\infty}$ for $\epsilon\in{\widetilde}V_{d}$, we denote $${{\mathcal D}}_{d}(f_{{{\vec{\epsilon}}}}\colon{{\vec{\epsilon}}}\in {\widetilde}V_{d})(x) =
{{{\mathbb E}}}_{{{\vec{t}}}\in Z^d}\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_d}
f_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}}).$$ We call such a function the [*cubic convolution product of the functions $f_{{{\vec{\epsilon}}}}$*]{}.
There is a formal similarity between the cubic convolution product and the classic convolution product; for example, $${{\mathcal D}}_{2}(f_{01}, f_{10}, f_{11})(x) =
{{{\mathbb E}}}_{t_{1}t_{2}\in Z} f_{01}(x+t_1)f_{10}(x+t_2)f_{11}(x+t_1+t_2).$$
Elementary bounds
-----------------
For ${{\vec{\epsilon}}}\in V_d$ and $\alpha\in\{0,1\}$, we write ${{\vec{\epsilon}}}\alpha=\epsilon_1\dots\epsilon_d\alpha
\in V_{d+1}$, maintaining the convention that such elements are written without commas or parentheses. Thus $$V_{d+1} = \{{{\vec{\epsilon}}}0\colon{{\vec{\epsilon}}}\in V_{d}\}\cup
\{{{\vec{\epsilon}}}1\colon{{\vec{\epsilon}}}\in V_{d}\}.$$
The image of $Z_{d+1}$ under each of the two natural projections on $Z^{2^d}$ is $Z_d$, and the image of the measure $\mu_{d+1}$ under these projections is $\mu_d$.
\[lem:Ddf\] Let $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in{\widetilde}V_d$, be $2^d-1$ functions in $L^\infty(\mu)$. Then for all $x\in Z$, $$\label{eq:boud_DProduct}
\Bigl|
{{\mathcal D}}_{d}(f_{{{\vec{\epsilon}}}}\colon{{\vec{\epsilon}}}\in {\widetilde}V_{d})(x)\Bigr|
\leq
\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_d}
{\lVert f_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}.$$ In particular, for every $f\in L^\infty(\mu)$, $$\label{eq:bouns_Dd}
{\lVert {{\mathcal D}}_d f\rVert}_\infty\leq{\lVert f\rVert}_{2^{d-1}}^{2^d-1}.$$
Without loss, we can assume that all functions are nonnegative. We proceed by induction on $d\geq 2$.
For nonnegative $f_{01}, f_{10}$ and $f_{11}\in L^\infty(\mu)$, $$\begin{aligned}
{{\mathcal D}}_{2}(f_{01}, f_{10},f_{11})(x) & = {{{\mathbb E}}}_{t_1\in Z} f_{01}(x+t_1){{{\mathbb E}}}_{t_2\in Z}
f_{10}(x+t_2)f_{11}(x+t_1+t_2) \\
& \leq {{{\mathbb E}}}_{t_1\in Z} f_{01}(x+t_1) {\lVert f_{10}\rVert}_{L^2(\mu)}
{\lVert f_{11}\rVert}_{L^2(\mu)}\\
& \leq {\lVert f_{01}\rVert}_{L^2(\mu)}
{\lVert f_{10}\rVert}_{L^2(\mu)}
{\lVert f_{11}\rVert}_{L^2(\mu)}.\end{aligned}$$ This proves the case $d=2$. Assume that the result holds for some $d\geq 2$. Let $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in{\widetilde}V_{d+1}$, be nonnegative and belong to $L^{2^d(\mu)}$. Then $$\begin{gathered}
{{\mathcal D}}_{d+1}(f_{{{\vec{\epsilon}}}}\colon{{\vec{\epsilon}}}\in{\widetilde}V_{d+1})(x)\\
= {{{\mathbb E}}}_{{{\vec{s}}}\in {\widetilde}Z^d}
\Bigl(
\prod_{{{\vec{\eta}}}\in {\widetilde}V_d} f_{{{\vec{\eta}}}0}(x+{{\vec{\eta}}}\cdot {{\vec{s}}})
{{{\mathbb E}}}_{u\in Z}\prod_{{{\vec{\theta}}}\in V_d} f_{{{\vec{\theta}}}1}
(x+{{\vec{\theta}}}\cdot {{\vec{s}}}+u) \Bigr).\end{gathered}$$ But, for every ${{\vec{s}}}\in Z^d$ and every $x\in Z$, by the Hölder Inequality, $${{{\mathbb E}}}_{u\in Z}\prod_{{{\vec{\theta}}}\in V_d} f_{{{\vec{\theta}}}1}
(x+{{\vec{\theta}}}\cdot {{\vec{s}}}+u) \leq\prod_{{{\vec{\theta}}}\in V_d}
{\lVert f_{{{\vec{\theta}}}1}\rVert}_{2^d}.$$ On the other hand, by the induction hypothesis, for every $x\in Z$, $${{{\mathbb E}}}_{{{\vec{s}}}\in {\widetilde}Z^d}
\prod_{{{\vec{\eta}}}\in {\widetilde}V_d}f_{{{\vec{\eta}}}0}(x+{{\vec{\eta}}}\cdot {{\vec{s}}})
\leq\prod_{{{\vec{\eta}}}\in {\widetilde}V_d}{\lVert f_{{{\vec{\eta}}}0}\rVert}_{2^{d-1}}
\leq\prod_{{{\vec{\eta}}}\in {\widetilde}V_d}{\lVert f_{{{\vec{\eta}}}0}\rVert}_{2^{d}}$$ and holds for $d+1$.
\[cor:boud\_2d\] Let $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in V_d$, be $2^d$ functions belonging to $L^\infty(\mu)$. Then $$\label{eq:boud_2d}
\Bigl|{{{\mathbb E}}}_{x\in Z, {{\vec{t}}}\in Z^d}\prod_{{{\vec{\epsilon}}}\in V_d}
f_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}})\Bigr|
\leq
\prod_{{{\vec{\epsilon}}}\in V_d}{\lVert f_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}.$$ In particular, for $f\in L^\infty(\mu)$, $$\label{eq:boud_Norm}
{\lVert f\rVert}_{U(d)}\leq{\lVert f\rVert}_{2^{d-1}}.$$
By the corollary, the definition of the Gowers norm $U(d)$ can be extended by continuity to the space $L^{2^{d-1}}(\mu)$, and if $f\in L^{2^{d-1}}(\mu)$, then the integrals defining ${\lVert f\rVert}_{U(d)}$ in Equation exist and holds. Using similar reasoning, if $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in V_d$, are $2^d$ functions belonging to $L^{2^{d-1}}(\mu)$, then the integral on the left hand side of exists, Inequality CSG remains valid, and holds. If we have $2^{d-1}$ functions in $L^{2^{d-1}}(\mu)$, then Inequality remains valid. Similarly, the definitions and results extend to ${{\mathcal D}}_{d}f$ and to cubic convolution products for functions belonging to $L^{2^{d-1}}(\mu)$.
The bounds given here (such as ) can be improved and made sharp. In particular, one can show that $${\lVert f\rVert}_{U(d)}\leq {\lVert f\rVert}_{2^{d}/(d+1)}$$ and $${\lVert {{\mathcal D}}f\rVert}_{\infty}\leq {\lVert f\rVert}_{(2^{d}-1)/d}^{2^{d}-1}.$$ We omit the proofs, as they are not used in the sequel.
When $Z$ is infinite, we define the *uniform space of level $d$* to be the completion of $L^\infty(\mu)$ under the norm $U(d)$. As $d$ increases, the corresponding uniform spaces shrink. A difficulty is that the uniform space may contain more than just functions. For example, if $Z={{{\mathbb T}}}:={{{\mathbb R}}}/{{{\mathbb Z}}}$, the uniform space of level $2$ consists of the distributions $T$ on ${{{\mathbb T}}}$ whose Fourier transform ${\widehat}T$ satisfies $\sum_{n\in{{{\mathbb Z}}}}|{\widehat}T(n) |^4 <+\infty$.
\[cor:alpha\] Let $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in V_d$, be $2^d$ functions on $Z$ and let ${{\vec{\alpha}}}\in V_d$. Assume that $f_{{\vec{\alpha}}}\in L^1(\mu)$ and $f_{{\vec{\epsilon}}}\in L^{2^{d-1}}(\mu)$ for ${{\vec{\epsilon}}}\neq{{\vec{\alpha}}}$. Then $$\Bigl|
{{{\mathbb E}}}_{x\in Z,\;{{\vec{t}}}\in Z^d}
\prod_{{{\vec{\epsilon}}}\in V_d}
f_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}})\Bigr|\leq
{\lVert f_\alpha\rVert}_1
\prod_{\substack{{{\vec{\epsilon}}}\in V_d \\ {{\vec{\epsilon}}}\neq{{\vec{\alpha}}}}}
{\lVert f_{{\vec{\epsilon}}}\rVert}_{L^{2^{d-1}}(\mu)}.$$
The left hand side is equal to $$\Bigl|
\int_{Z_d} f_{{\vec{\alpha}}}(x_{{\vec{\alpha}}})
\prod_{\substack{{{\vec{\epsilon}}}\in V_d \\ {{\vec{\epsilon}}}\neq{{\vec{\alpha}}}}}
f_{{\vec{\epsilon}}}(x_{{\vec{\epsilon}}})\, d\mu_d({{\mathbf x}})\Bigr |$$ Using the symmetries of the measure $\mu_d$, we can reduce to the case that ${{\vec{\alpha}}}={{\vec 0}}$, and then the result follows immediately from Lemma \[lem:Ddf\].
We note for later use:
\[prop:continuous\] For every $f\in L^{2^{d-1}}(\mu)$, ${{\mathcal D}}_df(x)$ is a continuous function on $Z$.
More generally, if $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in{\widetilde}V_d$ are $2^d-1$ functions belonging to $L^{2^{d-1}}(\mu)$, then the cubic convolution product ${{\mathcal D}}_{d}(f_{{{\vec{\epsilon}}}}\colon{{\vec{\epsilon}}}\in {\widetilde}V_{d})(x)$ is a continuous function on $Z$.
By density and , it suffices to prove the result when $f_{{\vec{\epsilon}}}\in L^\infty(\mu)$ for every ${{\vec{\epsilon}}}\in{\widetilde}V_d$. Furthermore, we can assume that $|f_{{\vec{\epsilon}}}| \leq 1$ for every ${{\vec{\epsilon}}}\in{\widetilde}V_d$. Let $g$ be the function on $Z$ defined in the statement. For $x,y\in Z$, we have that $$|g(x)- g(y) | \leq \sum_{{{\vec{\epsilon}}}\in{\widetilde}V_d}{\lVert f_{{{\vec{\epsilon}}},x} -
f_{{{\vec{\epsilon}}},y}\rVert}_1$$ and the result follows.
Duality
=======
Anti-uniform spaces
-------------------
Consider the space $L^{2^{d-1}}(\mu)$ endowed with the norm $U(d)$. By , the dual of this normed space can be viewed as a subspace of $L^{2^{d-1}/(2^{d-1}-1)}(\mu)$, with the duality given by the pairing $\langle\cdot ;\cdot\rangle$. Following Green and Tao [@GT], we define
The *anti-uniform space of level $d$* is defined to be the dual space of $L^{2^{d-1}}(\mu)$ endowed with the norm $U(d)$. Functions belonging to this space are called *anti-uniform functions of level $d$*. The norm on the anti-uniform space given by duality is called the *anti-uniform norm of level $d$* and is denoted by ${\lVert \cdot\rVert}_{U(d)}^*$.
Obviously, when $Z$ is finite, then every function on $Z$ is an anti-uniform function. It follows from the definitions that $${\lVert f\rVert}_{U(d+1)}^{*}\leq {\lVert f\rVert}_{U(d)}^{*}$$ for every $f\in L^{\infty}(Z)$, and thus as $d$ increases, the corresponding anti-uniform spaces increase.
More explicitly, a function $g\in L^{2^{d-1}/(2^{d-1}-1}(\mu)$ is an anti-uniform function of level $d$ if $$\sup\bigl\{|\langle g ; f\rangle|\colon
f\in L^{2^{d-1}}(\mu),\
{\lVert f\rVert}_{U(d)}\leq 1\bigr\} <+\infty$$ and in this case, ${\lVert g\rVert}_{U(d)}^*$ is defined to be equal to this supremum. Again, in case of ambiguity about the underlying space $Z$, we write ${\lVert \cdot\rVert}_{U(Z,d)}^{*}$. We conclude:
\[cor:anti\] For every anti-uniform function $g$ of level $d$, ${\lVert g\rVert}_{U(d)}^*\geq {\lVert g\rVert}_{2^{d-1}/(2^{d-1}-1)}$.
For $d=2$, the anti-uniform space consists in functions $g\in
L^2(\mu)$ with ${\lVert {\widehat}g\rVert}_{\ell^{4/3}({\widehat}Z)}$ finite, and for these functions, $$\label{eq:U2dual}
{\lVert g\rVert}_{U(2)}^*={\lVert {\widehat}g\rVert}_{\ell^{4/3}({\widehat}Z)}.$$ >From this example, we see that there is no bound for the converse direction of Corollary \[cor:anti\].
The dual spaces allow us to give an equivalent reformulation of the Inverse Theorem in terms of dual norms: For each integer $d\geq 1$ and each $\delta > 0$, there exists a family of “$(d-1)$-step nilsequences of bounded complexity,” which we do not define here, such that its convex hull ${{\mathcal F}}'(d,\delta)$ satisfies
\[th:dualinverse\] For each integer $d\geq 1$ and each $\delta > 0$, every function $g$ on ${{{\mathbb Z}}}_{N}$ with ${\lVert g\rVert}_{U(d)}^{*}\leq 1$ can be written as $g = h+\psi$ with $h\in{{\mathcal F}}'(d, \delta)$ and ${\lVert \psi\rVert}_{1}\leq
\delta$.
In this statement, there is no hypothesis on ${\lVert g\rVert}_\infty$, and the function $g$ is not assumed to be bounded.
We show that this statement is equivalent to the Inverse Theorem. First assume the Inverse Theorem and let ${{\mathcal F}}= {{\mathcal F}}(d,\delta)$ be the class of nilsequences and $C=C(d,\delta)$ be as in the formulation of the Inverse Theorem. Let $$K ={\widetilde}{{{\mathcal F}}} + B_{L^{1}(\mu)}(C),$$ where ${\widetilde}{{\mathcal F}}$ denotes the convex hull of ${{\mathcal F}}$ and the second term is the ball in $L^{1}(\mu)$ of radius $C$. Let $g$ be a function with $g\leq C$ on $K$. In particular, $|g|\leq 1$ and $g\leq C$ on ${{\mathcal F}}$. By the Inverse Theorem, we have that ${\lVert g\rVert}_{U(d)}< \delta$. By the Hahn-Banach Theorem, $K \supset B_{U(d)^{*}} (C/\delta)$. Thus $$B_{U(d)^{*}} (1)\subset (\delta /C) {\widetilde}{{\mathcal F}}+
B_{L^{1}(\mu)}(\delta).$$ Taking ${{\mathcal F}}'(d,\delta)$ to be $(\delta/C){\widetilde}{{\mathcal F}}$, we have the statement.
Conversely, we assume the Dual Form and prove the Inverse Theorem. Say that ${{\mathcal F}}' = {{\mathcal F}}'(d,\delta/2)$ is the convex hull of ${{\mathcal F}}_{0}={{\mathcal F}}_{0}(d,\delta)$. Assume that $f$ satisfies $|f|\leq 1$ and ${\lVert f\rVert}_{U(d)}\geq\delta$. Then there exists $g$ with ${\lVert g\rVert}_{U(d)}^{*}\leq 1$ and $\langle g;f\rangle \geq
\delta$. By the dual version, there exists $h\in{{\mathcal F}}'$ and $\psi$ with ${\lVert \psi\rVert}_{1}< \delta/2$ such that $g = h+\psi$. Since $$\delta\leq \langle g;f\rangle = \langle h+\psi;f\rangle =
\langle h;f\rangle + \langle \psi;f\rangle$$ and $\langle \psi;f\rangle\leq\delta/2$, we have that $\langle h;f\rangle\geq \delta/2$. Since $h\in{{\mathcal F}}'$, there exists $h'\in{{\mathcal F}}_{0}$ with $\langle h';f\rangle> \delta/2$ and we have the statement.
Dual functions and anti-uniform spaces
--------------------------------------
\[lem:norm\_Dprod\] Let $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in{\widetilde}V_d$, belong to $L^{2^{d-1}}(\mu)$. Then $${\lVert {{\mathcal D}}_{d}(f_{{{\vec{\epsilon}}}}\colon{{\vec{\epsilon}}}\in {\widetilde}V_{d})\rVert}_{U(d)}^*\leq\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_d}{\lVert f_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}.$$
For every $h\in L^{2^{d-1}}(\mu)$, we have that $$\begin{aligned}
\bigl|\langle h; g\rangle\bigr| = &
\Bigl|{{{\mathbb E}}}_{x\in Z,\;{{\vec{t}}}\in Z^d} h(x+{{\vec 0}}\cdot {{\vec{t}}})
\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}f_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}})\Bigr|\\
\leq & {\lVert h\rVert}_{U(d)}\cdot\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}
{\lVert f_{{\vec{\epsilon}}}\rVert}
\leq{\lVert h\rVert}_{U(d)}\cdot\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}
{\lVert f_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}\end{aligned}$$ by the Cauchy-Schwarz-Gowers Inequality and Inequality .
In particular, for $f\in L^{2^{d-1}}(\mu)$, we have that $ {\lVert {{\mathcal D}}_df\rVert}_{U(d)}^*\leq{\lVert f\rVert}_{U(d)}^{2^d-1}$. On the other hand, by , $${\lVert f\rVert}_{U(d)}^{2^d}= \langle{{\mathcal D}}_d f;f\rangle\leq
{\lVert {{\mathcal D}}_d f\rVert}_{U(d)}^*\cdot{\lVert f\rVert}_{U(d)}$$ and thus ${\lVert {{\mathcal D}}_d f\rVert}_{U(d)}^*\geq{\lVert f\rVert}_{U(d)}^{2^d-1}$. We conclude:
\[prop:norm\_Df\] For every $f\in L^{2^{d-1}}(\mu)$, ${\lVert {{\mathcal D}}_d f\rVert}_{U(d)}^*={\lVert f\rVert}_{U(d)}^{2^d-1}$.
While the following proposition is not used in the sequel, it gives a helpful description of the anti-uniform space:
\[prop:convex\_hull\] The unit ball of the anti-uniform space of level $d$ is the closed convex hull in $L^{2^{d-1}/(2^{d-1}-1)}(\mu)$ of the set $$\bigl\{ {{\mathcal D}}_d f\colon f\in L^{2^{d-1}}(\mu),\
{\lVert f\rVert}_{U(d)}\leq 1\bigr\}.$$
The proof is a simple application of duality.
Let $B\subset L^{2^{d-1}/(2^{d-1}-1)}(\mu)$ be the unit ball of the anti-uniform norm ${\lVert \cdot\rVert}_{U(d)}^*$. Let $K$ be the convex hull of the set in the statement and let $\overline K$ be its closure in $L^{2^{d-1}/(2^{d-1}-1)}(\mu)$.
By Proposition \[prop:norm\_Df\], for every $f$ with ${\lVert f\rVert}_{U(d)}\leq 1$, we have ${{\mathcal D}}_d f\in B$. Since $B$ is convex, $K\subset B$. Furthermore, $B$ is contained in the unit ball of $L^{2^{d-1}/(2^{d-1}-1)}(\mu)$ and is a weak\* compact subset of this space. Therefore, $B$ is closed in $L^{2^{d-1}/(2^{d-1}-1)}(\mu)$ and $\overline K\subset B$.
We check that $\overline K\supset B$. If this does not hold, there exists $g\in L^{2^{d-1}/(2^{d-1}-1)}(\mu)$ satisfying ${\lVert g\rVert}_{U(d)}^*\leq 1$ and $g\notin\overline K$. By the Hahn-Banach Theorem, there exists $f\in L^{2^{d-1}}(\mu)$ with $\langle f ; h\rangle\leq 1$ for every $h\in K$ and $\langle f;g\rangle> 1$. This last property implies that ${\lVert f\rVert}_{U(d)}>1$. Taking $\phi={\lVert f\rVert}_{U(d)}{^{-1}}\cdot f$, we have that ${\lVert \phi\rVert}_{U(d)}=1$ and ${{\mathcal D}}_d\phi\in K$. Thus by the first property of $f$, $\langle {{\mathcal D}}_d\phi;f \rangle\leq 1$. But $$\langle {{\mathcal D}}_d\phi;f\rangle={\lVert f\rVert}_{U(d)}^{-2^d+1}
\langle{{\mathcal D}}_d f;f\rangle= {\lVert f\rVert}_{U(d)}$$ and we have a contradiction.
It can be shown that when $Z$ is finite, the set appearing in Proposition \[prop:convex\_hull\] is already closed and convex:
Assume $Z$ is finite. Then the set $$\bigl\{ {{\mathcal D}}_d f\colon {\lVert f\rVert}_{U(d)}\leq 1\bigr\}$$ is the unit ball of the anti-uniform norm.
We omit the proof of this result, as the proof (for finite $Z$) is similar to that of Theorem \[th:k\] below, which seems more useful. For the general case, the analogous statement is not as clear because the “uniform space” does not consist only of functions.
Approximation results for anti-uniform functions {#subsec:approx}
------------------------------------------------
\[th:k\] Assume $d\geq 1$ is an integer. For every anti-uniform function $g$ with ${\lVert g\rVert}_{U(d)}^*=1$, integer $k\geq d-1$, and $\delta>0$, the function $g$ can be written as $$g={{\mathcal D}}_d f+h,$$ where $$\begin{gathered}
{\lVert f\rVert}_{2^k}\leq 1/\delta;\\
{\lVert h\rVert}_{2^k/(2^k-1)}\leq\delta;\\
{\lVert f\rVert}_{U(d)}\leq 1.\end{gathered}$$
As in the Dual Form of the Inverse Theorem, there is no hypothesis on ${\lVert g\rVert}_\infty$ and we do not assume that the function $g$ is bounded.
Fix $k\geq d-1$ and $\delta>0$. For $f\in L^{2^k}(\mu)$, define $$\label{eq:def_nnorm}
{\lvert\!|\!| f|\!|\!\rvert}=
\begin{cases}
\bigl(
{\lVert f\rVert}_{U(d)}^{2^k}+\delta^{2^k}{\lVert f\rVert}_{2^k}^{2^k}
\bigr)^{1/2^k}
& \text{if }k\geq d;\\
\bigl(
{\lVert f\rVert}_{U(d)}^{2^d}+\delta^{2^d}{\lVert f\rVert}_{2^{d-1}}^{2^d}
\bigr)^{1/2^d}
& \text{if }k=d-1.
\end{cases}$$
Since ${\lVert f\rVert}_{U(d)}\leq{\lVert f\rVert}_{2^{d-1}}\leq{\lVert f\rVert}_{2^k}$ for every $f\in L^{2^k}(\mu)$, ${\lvert\!|\!| f|\!|\!\rvert}$ is well defined on $L^{2^k}(\mu)$ and ${\lvert\!|\!| \cdot|\!|\!\rvert}$ is a norm on this space, equivalent to the norm ${\lVert \cdot\rVert}_{2^k}$.
Let ${\lvert\!|\!| \cdot|\!|\!\rvert} ^*$ be the dual norm of ${\lvert\!|\!| \cdot|\!|\!\rvert}$: for $g\in L^{2^k/(2^k-1)}(\mu)$, $${\lvert\!|\!| g|\!|\!\rvert}^*=
\sup\Bigl\{
\bigl|\langle f; g\rangle\bigr|\colon
f\in L^{2^k}(\mu),\ {\lvert\!|\!| f|\!|\!\rvert}\leq 1\Bigr\}.$$
This dual norm is equivalent to the norm ${\lVert \cdot\rVert}_{2^k/(2^k-1)}$. Since ${\lvert\!|\!| f|\!|\!\rvert}\geq{\lVert f\rVert}_{U(d)}$ for every $f\in L^{2^k}(\mu)$, we have that $${\lvert\!|\!| g|\!|\!\rvert}^*\leq{\lVert g\rVert}_{U(d)}^*\text{ for every }g.$$
Fix an anti-uniform function $g$ with ${\lVert g\rVert}_{U(d)}^*\leq 1$. Since ${\lvert\!|\!| f|\!|\!\rvert}\geq{\lVert f\rVert}_{U(d)}$ for every $f$, we have that $$c:= {\lvert\!|\!| g|\!|\!\rvert}^*\leq{\lVert g\rVert}_{U(d)}^*\leq 1 .$$ Set $g'=c{^{-1}}g$, and so ${\lvert\!|\!| g'|\!|\!\rvert}^*=1$.
Since the norm ${\lvert\!|\!| \cdot|\!|\!\rvert}$ is equivalent to the norm ${\lVert \cdot\rVert}_{2^k}$ and the Banach space $(L^{2^k}(\mu),{\lVert \cdot\rVert}_{2^k})$ is reflexive, the Banach space $(L^{2^k}(\mu),{\lvert\!|\!| \cdot|\!|\!\rvert})$ is also reflexive. This means that $(L^{2^k}(\mu),{\lvert\!|\!| \cdot|\!|\!\rvert})$ is the dual of the Banach space $(L^{2^k/(2^k-1)}(\mu),{\lvert\!|\!| \cdot|\!|\!\rvert}^*)$. Therefore, there exists $f'\in L^{2^k}(\mu)$ with $${\lvert\!|\!| f'|\!|\!\rvert}=1\text{ and }\langle g';f'\rangle = 1.$$ By definition of ${\lvert\!|\!| f'|\!|\!\rvert}$, $$\label{eq:borne}
{\lVert f'\rVert}_{U(d)}\leq 1\text{ and }{\lVert f'\rVert}_{2^k}\leq 1/\delta.$$
Assume first that $k\geq d$. (We explain the modifications needed for the case $k=d-1$ after.)
By , , and the symmetries of the measure $\mu_d$, for every $\phi\in L^{2^k}(\mu)$ and every $t\in{{{\mathbb R}}}$, $$\label{eq:fth_Ud}
{\lVert f'+t\phi\rVert}_{U(d)}^{2^d}={\lVert f'\rVert}_{U(d)}^{2^d}+2^dt\langle{{\mathcal D}}_d
f';\phi\rangle+o(t),$$ where by $o(t)$ we mean any function such that $o(t)/t\to 0$ as $t\to 0$.
Raising this to the power $2^{k-d}$, we have that $${\lVert f'+t\phi\rVert}_{U(d)}^{2^k}=
{\lVert f'\rVert}_{U(d)}^{2^k}+2^kt{\lVert f'\rVert}_{U(d)}^{2^k-2^d}
\langle{{\mathcal D}}_d f';\phi\rangle+o(t).$$ On the other hand, $${\lVert f'+t\phi\rVert}_{2^k}^{2^k}=
{\lVert f'\rVert}_{2^k}^{2^k}+2^kt\langle f'^{2^k-1};\phi\rangle+o(t).$$ Combining these expressions and using the definition of ${\lvert\!|\!| f'+t\phi|\!|\!\rvert}$ and of ${\lvert\!|\!| f'|\!|\!\rvert}$, we have that $$\begin{aligned}
{\lvert\!|\!| f'+t\phi|\!|\!\rvert}^{2^k}= & {\lVert f'+t\phi\rVert}_{U(d)}^{2^k}+
\delta^{2^k}{\lVert f'+t\phi\rVert}_{2^k}^{2^k}\\
= & {\lvert\!|\!| f'|\!|\!\rvert}^{2^k}
+
2^kt{\lVert f'\rVert}_{U(d)}^{2^k-2^d}
\langle{{\mathcal D}}_d f';\phi\rangle
+
\delta^{2^k}2^kt\langle f'^{2^k-1};\phi\rangle+o(t)\\
= & 1+ 2^kt{\lVert f'\rVert}_{U(d)}^{2^k-2^d}
\langle{{\mathcal D}}_d f';\phi\rangle
+
\delta^{2^k}2^kt\langle f'^{2^k-1};\phi\rangle+o(t).\end{aligned}$$ Raising this to the power $1/2^k$, we have that $${\lvert\!|\!| f'+t\phi|\!|\!\rvert}=
1+t{\lVert f'\rVert}_{U(d)}^{2^k-2^d}
\langle{{\mathcal D}}_d f';\phi\rangle
+
\delta^{2^k}t\langle f'^{2^k-1};\phi\rangle+o(t).$$ Since for every $\phi\in L^{2^k}(\mu)$ and every $t\in{{{\mathbb R}}}$ we have $$1+t\langle g';\phi\rangle= \langle g';f'+t\phi\rangle
\leq {\lvert\!|\!| f'+t\phi|\!|\!\rvert},$$ it follows that $$1+t\langle g';\phi\rangle\leq
1+t{\lVert f'\rVert}_{U(d)}^{2^k-2^d}
\langle{{\mathcal D}}_d f';\phi\rangle
+
\delta^{2^k}t\langle f'^{2^k-1};\phi\rangle+o(t)\ .$$ Since this holds for every $t$, we have $$\langle g';\phi\rangle =
{\lVert f'\rVert}_{U(d)}^{2^k-2^d}
\langle{{\mathcal D}}_d f';\phi\rangle
+
\delta^{2^k}\langle f'^{2^k-1};\phi\rangle.$$ Since this holds for every $\phi$, we conclude that $$g'={\lVert f'\rVert}_{U(d)}^{2^k-2^d}{{\mathcal D}}_d f'+
\delta^{2^k}f'^{2^k-1}.$$ Thus $$g= c{\lVert f'\rVert}_{U(d)}^{2^k-2^d}{{\mathcal D}}_d f'+ c\delta^{2^k}f'^{2^k-1}.$$ Set $$f=\bigl(c{\lVert f'\rVert}_{U(d)}^{2^k-2^d}\bigr)^{1/(2^d-1)} f'
\text{ and }
h=c\delta^{2^k}f'^{2^k-1}.$$ Then $$g={{\mathcal D}}_d f+h$$ and by , $$\begin{gathered}
{\lVert f\rVert}_{U(d)}\leq 1\ ;\ {\lVert f\rVert}_{2^k}\leq 1/\delta \\
{\lVert h\rVert}_{2^k/(2^k-1)}=
c\delta^{2^k}{\lVert f'\rVert}_{2^k}^{2^k-1}\leq\delta.\end{gathered}$$
For the case $k=d-1$, for every $\phi\in L^{2^k}(\mu)$ and every $t\in{{{\mathbb R}}}$, we have and $${\lVert f'+t\phi\rVert}_{2^{d-1}}^{2^d}=
{\lVert f'\rVert}_{2^{d-1}}^{2^d}+2^dt {\lVert f'\rVert}_{2^{d-1}}^{2^{d-1}} \langle
f'^{2^{d-1}-1};\phi\rangle+o(t).$$ Thus $${\lvert\!|\!| f'+t\phi|\!|\!\rvert}=1+t\langle{{\mathcal D}}_d f';\phi\rangle+
\delta^{2^d}
{\lVert f'\rVert}_{2^{d-1}}^{2^{d-1}} \langle
f'^{2^{d-1}-1};\phi\rangle+o(t).$$ As above, we deduce that $$g'={{\mathcal D}}_df'+\delta^{2^d}{\lVert f'\rVert}_{2^{d-1}}^{2^{d-1}}
f'^{2^{d-1}-1}.$$ Taking $$f=c^{1/(2^d-1)}f'\text{ and } h=c \delta^{2^d}{\lVert f'\rVert}_{2^{d-1}}^{2^{d-1}}
f'^{2^{d-1}-1},$$ we have the statement.
When $Z$ is finite, we can say more:
\[th:borne\] Assume that $Z$ is finite. Given a function $g$ with ${\lVert g\rVert}_{U(d)}^*=1$ and $\delta>0$, the function $g$ can be written as $$g={{\mathcal D}}_d f+h,$$ where $$\begin{gathered}
{\lVert f\rVert}_\infty\leq 1/\delta;\\
{\lVert h\rVert}_1\leq\delta;\\
{\lVert f\rVert}_{U(d)}\leq 1.\end{gathered}$$
By Theorem \[th:k\], for every $k\geq d-1$ we can write $$g={{\mathcal D}}_d f_k+h_k,$$ where $${\lVert f_k\rVert}_{2^k}\leq 1/\delta;\\
{\lVert h_k\rVert}_{2^k/(2^k-1)}\leq\delta;\\
{\lVert f_k\rVert}_{U(d)}\leq 1.$$ Let $N=|Z|$. Since ${\lVert f_k\rVert}_{2^k}\leq 1/\delta$, we have that ${\lVert f_k\rVert}_\infty\leq N/\delta$. In the same way, ${\lVert h_k\rVert}_\infty\leq N\delta$. By passing to a subsequence, since the functions are uniformly bounded we can therefore assume that $f_k\to f$ and that $h_k\to
h$ pointwise as $k\to+\infty$. Thus ${{\mathcal D}}_df_k\to{{\mathcal D}}_d f$ pointwise and so $$g={{\mathcal D}}_d f+h.$$ Since ${\lVert f_k\rVert}_{U(d)}\to{\lVert f\rVert}_{U(d)}$, it follows that ${\lVert f\rVert}_{U(d)}\leq 1$. For every $k\geq d-1$, we have that ${\lVert h_k\rVert}_1\leq
{\lVert h_k\rVert}_{2^k/(2^k-1)}\leq\delta$. Since ${\lVert h_k\rVert}_1\to{\lVert h\rVert}_1$, it follows that ${\lVert h\rVert}_1\leq \delta$. For $\ell\geq k\geq d-1$, $${\lVert f_\ell\rVert}_{2^k}\leq{\lVert f_\ell\rVert}_{2^\ell}
\leq1/\delta.$$ Taking the limit as $\ell\to+\infty$, we have that ${\lVert f\rVert}_{2^k}\leq 1/\delta$ for every $k\geq d-1$ and so ${\lVert f\rVert}_\infty\leq 1/\delta$.
Does Theorem \[th:borne\] also hold when $Z$ is infinite?
We conjecture that the answer is positive, but the proof given does not carry through to this case.
Applications
------------
Theorems \[th:k\] and \[th:borne\] give insight into the $U(d)$ norm, connecting it to the classical $L^p$ norms. For example, we have:
Let $\phi$ be a function with ${\lVert \phi\rVert} \leq 1$ and ${\lVert \phi\rVert}_{U(d)}=\theta>0$. Then for every $p\geq 2^{d-1}$, there exists a function $f$ such that ${\lVert f\rVert}_p\leq 1$ and $\langle {{\mathcal D}}_{d}f;\phi\rangle > (\theta/2)^{2^d}$.
If $Z$ is finite, there exists a function $f$ with ${\lVert f\rVert}_\infty\leq 1$ and $\langle {{\mathcal D}}_{d}f;\phi\rangle > (\theta/2)^{2^d}$.
It suffices to prove the result when $p= 2^k$ for some integer $k\geq
d-1$. There exists $g$ with ${\lVert g\rVert}_{U(d)}^{*}=1$ and $\langle g;\phi\rangle = \theta$. Taking $\delta = \theta/2$ in Theorem \[th:k\], we have the first statement. For the second statement, apply Theorem \[th:borne\].
Theorem \[th:borne\] leads to an equivalent reformulation of the Inverse Theorem, without any explicit reference to the Gowers norms. For all $d\geq 1$ and $\delta > 0$, there exists a family of “$(d-1)$-step nilsequences of bounded complexity” whose convex hull ${{\mathcal F}}''(d, \delta)$ satisfies:
\[th:reformulated\] For every $\delta > 0$, every function $\phi$ on ${{{\mathbb Z}}}_{N}$ with ${\lVert \phi\rVert}_\infty\leq 1$, the function ${{\mathcal D}}_d\phi$ can be written as ${{\mathcal D}}_d\phi = g +h$ with $g\in{{\mathcal F}}''(d,\delta)$ and ${\lVert h\rVert}_{1}\leq\delta$.
We show that the statement is equivalent to the Dual Form of the Inverse Theorem. First assume the Dual Form. Given $\phi$ with ${\lVert \phi\rVert}_\infty\leq 1$, we have that ${\lVert \phi\rVert}_{U(d)}\leq 1$ and thus ${\lVert {{\mathcal D}}_d\phi\rVert}_{U(d)}^{*}\leq 1$. By the Dual Form, ${{\mathcal D}}_{d}\phi = h+\psi$, where $h\in{{\mathcal F}}'(d,\delta)$ and ${\lVert \psi\rVert}_{1}\leq\delta$, which is exactly the Reformulated Version.
Conversely, assume the Reformulated Version. Let $g\in B_{U(d)^{*}}(1)$. Then by Theorem \[th:k\], $g = {{\mathcal D}}_{d}h +\psi$, where ${\lVert h\rVert}_\infty\leq 2/\delta$ and ${\lVert \psi\rVert}_{1}\leq \delta/2$. Define ${{\mathcal F}}' = {{\mathcal F}}'(d,\delta)$ to be equal to $(2/\delta)^{2^d-1} {{\mathcal F}}''(d,\eta)$, where $\eta$ is a positive constant to be defined later and ${{\mathcal F}}''(d,\eta)$ is as in the Reformulated Version. By the Reformulated Version, ${{\mathcal D}}_{d}h = f+\psi$, with $$f\in{{\mathcal F}}'\text{ and }
{\lVert \psi\rVert}_1\leq (2/\delta)^{2^d-1}\eta\ .$$ Then $g = f+\phi+\psi$ with $f\in{{\mathcal F}}'$ and ${\lVert \phi+\psi\rVert}_{1}\leq \delta/2+(2/\delta)^{2^d-1}\eta$. Taking $\eta=(\delta/2)^{2^d}$, we have the result.
Anti-uniformity norms and embeddings {#sec:embedding}
------------------------------------
This section is a conjectural, and somewhat optimistic, exploration of the possible uses of the theory of anti-uniform norms we have developed. The main interest is not the sketches of proofs included, but rather the questions posed and the directions that we conjecture may be approached using these methods.
If $G$ is a $(d-1)$-step nilpotent Lie group and $\Gamma$ is a discrete, cocompact subgroup of $G$, the compact manifold $X = G/\Gamma$ is [*$(d-1)$-step nilmanifold*]{}. The natural action of $G$ on $X$ by left translations is written as $(g,x)\mapsto g.x$ for $g\in G$ and $x\in X$.
We recall the following “direct” result (a converse to the Inverse Theorem), proved along the lines of arguments in [@HK1]:
\[prop:12\_6\] Let $X=G/\Gamma$ be a $(d-1)$-step nilmanifold, $x\in X$, $g\in G$, $F$ be a continuous function on $X$, and $N\geq 2$ be an integer. Let $f$ be a function on ${{{\mathbb Z}}}_N$ with $|f|\leq 1$. Assume that for some $\eta>0$, $$\bigl|{{{\mathbb E}}}_{0\leq n<N} f(n)F(g^n\cdot x)\bigr|\geq\eta.$$ Then there exists a constant $c = c(X,F, \eta) > 0$ such that $${\lVert f\rVert}_{U(d)}\geq c.$$
The key point is that the constant $c$ depends only on $X$, $F$, and $\eta$, and not on $f$, $N$, $g$ or $x$.
In [@GT2], the average is taken over the interval $[-N/2,N/2]$ instead of $[0,N)$, but the proof of Proposition \[prop:12\_6\] is the same for the modified choice of interval.
A similar result is given in Appendix G of [@GTZ1], and proved using simpler methods, but there the conclusion is about the norm ${\lVert f\rVert}_{U_d({{{\mathbb Z}}}_{N'})}$, where $N'$ is sufficiently large with respect to $N$.
By duality, Proposition \[prop:12\_6\] can be rewritten as
\[prop:12\_6bis\] Let $X=G/\Gamma, x,g,F$ be as in Proposition \[prop:12\_6\]. Let $N\geq 2$ be an integer and let $h$ denote the function $n\mapsto F(g^n\cdot x)$ restricted to $[0,N)$ and considered as a function on ${{{\mathbb Z}}}_N$. Then for every $\eta>0$, we can write $$h=\phi+\psi$$ where $\phi$ and $\psi$ are functions on $Z_N$ with ${\lVert \phi\rVert}_{U(d)}^*\leq c(X,F,\eta)$ and ${\lVert \psi\rVert}_1\leq\eta$.
Proposition \[prop:12\_6\] does not imply that ${\lVert h\rVert}_{U(d)}^*$ is bounded independent of $N$, and using , one can easily construct a counterexample for $d=2$ and $X={{{\mathbb T}}}$. On the other hand, for $d=2$ we do have that ${\lVert h\rVert}_{U(d)}^*$ is bounded independent of $N$ when the function $F$ is sufficiently smooth. Recalling that the Fourier series of a continuously differentiable function on ${{{\mathbb T}}}$ is absolutely convergent and directly computing using Fourier coefficients, we have:
Let $F$ be a continuously differentiable function on ${{{\mathbb T}}}$ and let $\alpha\in{{{\mathbb T}}}$. Let $N\geq 2$ be an integer and let $h$ denote the restriction of the function $n\mapsto F(\alpha^n)$ to $[0,N)$, considered as a function on ${{{\mathbb Z}}}_N$. Then $${\lVert h\rVert}_{U(2)}^*\leq c{\lVert {\widehat}F\rVert}_{\ell^1({{{\mathbb Z}}})},$$ where $c$ is a universal constant.
A similar result holds for functions on ${{{\mathbb T}}}^k$.
It is natural to ask whether a similar result holds for $d>2$. For the remained of this section, we assume that every nilmanifold $X$ is endowed with a smooth Riemannian metric. For $k\geq 1$, we let ${{\mathcal C}}^k(X)$ denote the space of $k$-times continuously differentiable functions on $X$, endowed with the usual norm ${\lVert \cdot\rVert}_{{{\mathcal C}}^k(X)}$. We ask if the dual norm is bounded independent of $N$:
\[qu:smooth\_dual\] Let $X=G/\Gamma$ be a $(d-1)$-step nilmanifold. Does there exist an integer $k\geq 1$ and a positive constant $c$ such that for all choices of a function $F\in{{\mathcal C}}^k(X)$, $g\in G$, $x\in X$ and integer $N\geq 2$, writing $h$ for the restriction to $[0,N)$ of the function $n\mapsto F(g^n\cdot x)$, considered as a function on ${{{\mathbb Z}}}_N$, we have $${\lVert h\rVert}_{U(d)}^*\leq c{\lVert F\rVert}_{{{\mathcal C}}^k(X)}?$$
If $g\in G$ and $x\in X$ are such that $g^N\cdot x=x$, we say that the map $n\mapsto g^n\cdot x$ is an *embedding* of ${{{\mathbb Z}}}_N$ in $X$.
\[prop:embed-ZN\] The answer to Question \[qu:smooth\_dual\] is positive under the additional hypothesis that $n\mapsto g^n\cdot x$ is an embedding of ${{{\mathbb Z}}}_N$ in $X$, that is, that $g^N\cdot x=x$.
The proof of this proposition is similar to that of Proposition 5.6 in [@HK3] and so we omit it.
More generally, we can phrase these results and the resulting question for groups other than ${{{\mathbb Z}}}_N$. We restrict ourselves to the case of ${{{\mathbb T}}}$, as the extension to ${{{\mathbb T}}}^k$ is clear. By the same argument used for Proposition \[prop:12\_6\], we have:
\[prop:12\_6\_T\] Let $X=G/\Gamma$ be a $(d-1)$-step nilmanifold, $x\in X$, $u$ be an element in the Lie algebra of $G$, and $F$ be a continuous function on $X$. Let $f$ be a function on ${{{\mathbb T}}}$ with $|f|\leq 1$. Assume that for some $\eta>0$ we have $$\Bigl|\int f(t)F\bigl(\exp(tu)\cdot x\bigr)\,dt\Bigr|\geq\eta,$$ where we identify ${{{\mathbb T}}}$ with $[0,1)$ in this integral. Then there exists a constant $c = c(X,F, \eta) > 0$ such that $${\lVert f\rVert}_{U(d)}\geq c.$$
By duality, Proposition \[prop:12\_6\_T\] can be rewritten as
\[prop:12\_6\_Tbis\] Let $X=G/\Gamma, x,u,F$, and $c=c(X,F,\eta)$ be as in Proposition \[prop:12\_6\_T\]. Let $h$ denote the restriction of the function $t\mapsto F\bigl(\exp(tu)\cdot x)$ to $[0,1)$, considered as a function on ${{{\mathbb T}}}$. Then for every $\eta>0$, we can write $$h=\phi+\psi,$$ where $\phi$ and $\psi$ are functions on ${{{\mathbb T}}}$ with ${\lVert \phi\rVert}_{U(d)}^*\leq c$ and ${\lVert \psi\rVert}_1\leq\eta$.
We can ask the analog of Question \[qu:smooth\_dual\] for the group ${{{\mathbb T}}}$:
\[qu:smooth\_dual\_T\] Let $X=G/\Gamma$ be a $(d-1)$-step nilmanifold. Does there exist an integer $k\geq 1$ and a positive constant $c$ such that for all choices of a function $F\in{{\mathcal C}}^k(X)$, $u$ in the Lie algebra of $G$, and $x\in X$, writing $h$ for the restriction of the function $t\mapsto F\bigl(\exp(tu)\cdot x\bigr)$ to $[0,1)$, considered a function on ${{{\mathbb T}}}$, we have $${\lVert h\rVert}_{U(d)}^*\leq c{\lVert F\rVert}_{{{\mathcal C}}^k(X)}?$$
Analogous to Proposition \[prop:embed-ZN\], the answer to this question is positive under the additional hypothesis that $t\mapsto\exp(tu)\cdot x$ is an *embedding* of ${{{\mathbb T}}}$ in $X$, meaning that $\exp(u)\cdot x=x$.
Multiplicative structure
========================
Higher order Fourier Algebras
-----------------------------
In light of Theorem \[th:k\], the family of functions $g$ on $Z$ of the form $g={{\mathcal D}}_d f$ for $f\in L^{2^k} (\mu)$ for some $k\geq d-1$ is relevant, and more generally, cubic convolution products for functions $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in{\widetilde}V_d$, belonging to $L^{2^k}(\mu)$ for some $k\geq d-1$. We only consider the case $k=d-1$, as it gives rise to interesting algebras.
For an integer $d\geq 1$, define $A(d)$ to be the space of functions $g$ on $Z$ that can be written as $$\label{eq:defAd}
g(x)=\sum_{j=1}^\infty {{\mathcal D}}_{d}(f_{j,{{\vec{\epsilon}}}}\colon{{\vec{\epsilon}}}\in{\widetilde}V_{d})$$ where all the functions $f_{j,{{\vec{\epsilon}}}}$ belong to $L^{2^{d-1}}(\mu)$ and $$\label{eq:defAd2}
\sum_{j=1}^\infty\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_{d}}
{\lVert f_{j,{{\vec{\epsilon}}}}\rVert}_{2^{d-1}}<+\infty.$$
For $g\in A(d)$, we define $$\label{eq:defNormAd}
{\lVert g\rVert}_{A(d)}=\inf
\sum_{j=1}^\infty\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_{d}}
{\lVert f_{j,{{\vec{\epsilon}}}}\rVert}_{2^{d-1}},$$ where the infimum is taken over all families of functions $f_{j,{{\vec{\epsilon}}}}$ in $L^{2^{d-1}}(\mu)$ satisfying and .
We call $A(d)$ the *Fourier algebra of order $d$*; we show in this section that it is a Banach algebra.
It follows from the definitions that $A(1)$ consists of the constant functions with the norm ${\lVert \cdot\rVert}_{A(1)}$ being absolute value. Clearly, if $Z$ is finite and $d\geq 2$, then every function on $Z$ belongs to $A(d)$ and we can replace each series by a finite sum in the definitions.
It is easy to check that $A(d)$ is a vector space of functions.
Furthermore, by and Lemma \[prop:continuous\], condition implies that the series in converges under the uniform norm and that every function in $A(d)$ is a continuous function on $Z$. Moreover, by , ${\lVert g\rVert}_\infty\leq{\lVert g\rVert}_{A(d)}$ and ${\lVert \cdot\rVert}_{A(d)}$ is a norm on $A(d)$.
For every $g\in A(d)$, we have that $g$ belongs to that anti-uniform space of level $d$ and that $${\lVert g\rVert}_{U(d)}^*\leq{\lVert g\rVert}_{A(d)}.$$
If $(g_n)_{n\in{{{\mathbb N}}}}$ is a sequence in $A(d)$ with $g=\sum_{n=1}^\infty{\lVert g_n\rVert}_{A(d)}<+\infty$, then the series $\sum_{n=1}^\infty g_n$ converges under the uniform norm, the sum $g$ of this series belongs to $A(d)$, and the series converges to $g$ in $A(d)$. This shows that the space $A(d)$ endowed with the norm ${\lVert \cdot\rVert}_{A(d)}$ is a Banach space.
Let ${{\mathcal C}}(Z)$ denote the space of continuous functions on $Z$. We summarize:
\[prop:Ad\_continuous\] $A(d)$ is a linear subspace of ${{\mathcal C}}(Z)$ and of the anti-uniform space of level $d$. For every $g\in A(d)$, we have that ${\lVert g\rVert}_\infty\leq{\lVert g\rVert}_{A(d)}$. The space $A(d)$ endowed with the norm ${\lVert \cdot\rVert}_{A(d)}$ is a Banach space.
Tao’s uniform almost periodicity norms
--------------------------------------
In [@T], Tao introduced a sequence of norms, the uniform almost periodicity norms, that also play a dual role to the Gowers uniformity norms:
For $f\colon Z\to {{{\mathbb C}}}$, define ${\lVert f\rVert}_{\operatorname{U\!A\!P}^0(Z)}$ to be equal to $|c|$ if $f$ is equal to the constant $c$, and to be infinite otherwise. For $d\geq 1$, define ${\lVert f\rVert}_{\operatorname{U\!A\!P}^{d+1}(Z)}$ to be the infimum of all constants $M> 0$ such that for all $n\in{{{\mathbb Z}}}$, $$T^nf = M{{{\mathbb E}}}_{h\in H}(c_{n,h}g_h),$$ for some finite nonempty set $H$, collection of functions $(g_h)_{h\in H}$ from $Z$ to ${{{\mathbb C}}}$ satisfying ${\lVert g_h\rVert}_{L^\infty(Z)}\leq 1$, collection of functions $(c_{n,h})_{n\in Z, h\in H}$ from $Z$ to ${{{\mathbb C}}}$ satisfying ${\lVert c_{n,h}\rVert}_{\operatorname{U\!A\!P}^d(Z)}\leq 1$, and a random variable $h$ taking values in $H$.
When the underlying group is clear, we omit it from the notation and write ${\lVert f\rVert}_{\operatorname{U\!A\!P}^d(Z)} = {\lVert f\rVert}_{\operatorname{U\!A\!P}^d}$.
The definition given in [@T] implicitly assumes that $Z$ is finite; to extend to the case that $Z$ is infinite, take $H$ to be an arbitrary probability space and view the functions $g_h$ and $c_{n,h}$ as random variables.
Tao shows that this defines finite norms $\operatorname{U\!A\!P}^d$ for $d\geq 1$ and that the uniformly almost periodic functions of order $d$ (meaning functions for which the $\operatorname{U\!A\!P}^d$ norm is bounded) form a Banach algebra: $${\lVert fg\rVert}_{\operatorname{U\!A\!P}^d} \leq {\lVert f\rVert}_{\operatorname{U\!A\!P}^d} {\lVert g\rVert}_{\operatorname{U\!A\!P}^d}.$$
The $\operatorname{U\!A\!P}^{d-1}$ and $A(d)$ norms are related: both are algebra norms and they satisfy similar properties, such as $${\lVert f\rVert}_{\operatorname{U\!A\!P}^{d-1}} \geq {\lVert f\rVert}_{U(d)}^*$$ and $${\lVert f\rVert}_{A(d)} \geq {\lVert f\rVert}_{U(d)}^*.$$ For $d=2$, the two norms are in fact the same (an exercise in [@TV] due to Green and Section \[sec:d=2\] below). However, in general we do not know if they are equal:
For a function $f\colon Z\to{{{\mathbb C}}}$, is $${\lVert f\rVert}_{A(d)} = {\lVert f\rVert}_{\operatorname{U\!A\!P}^{d-1}}$$ for all $d\geq 2$?
In particular, while the $\operatorname{U\!A\!P}$ norms satisfy $${\lVert f\rVert}_{\operatorname{U\!A\!P}(d-1)} \geq {\lVert f\rVert}_{\operatorname{U\!A\!P}(d)}$$ for all $d\geq 2$, we do not know if the same inequality holds for the norms $A(d)$.
The case $d=2$ {#sec:d=2}
--------------
We give a further description for $d=2$, relating these notions to the classical objects in Fourier analysis.
We have that ${\widetilde}V_2=\{01,10,11\}$. Every function $g$ defined as a cubic convolution product of $f_{{{\vec{\epsilon}}}}$, ${{\vec{\epsilon}}}\in{\widetilde}V_{2}$, satisfies $$\begin{aligned}
\label{eq:2_3}
\sum_{\xi\in{\widehat}Z}|{\widehat}{g}(\xi)|^{2/3} = &
\sum_{\xi\in{\widehat}Z}\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_{2}}|{\widehat}{f_{{{\vec{\epsilon}}}}}(\xi)|^{2/3}\\
\nonumber
\leq & \prod_{{{\vec{\epsilon}}}\in{\widetilde}V_{2}}\bigl(\sum_{\xi\in{\widehat}Z}|{\widehat}{f_{{{\vec{\epsilon}}}}}(\xi)
|^{2}\bigr)^{1/3} =
\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_{2}}{\lVert f_{{{\vec{\epsilon}}}}\rVert}_{L^{2}(\mu)}^{2/3}.\end{aligned}$$ Thus $$\sum_{\xi\in{\widehat}Z}|{\widehat}g(\xi)|
\leq \prod_{{{\vec{\epsilon}}}\in{\widetilde}V_2}{\lVert f_{{\vec{\epsilon}}}\rVert}_{L^2(\mu)}.$$ It follows that for $g\in A(d)$, we have that $$\sum_{\xi\in{\widehat}Z}|{\widehat}g(\xi)|\leq {\lVert g\rVert}_{A(2)}.$$
On the other hand, let $g$ be a continuous function on $Z$ with $\sum_{\xi\in{\widehat}Z}|{\widehat}g(\xi)|$ $<+\infty$. This function can be written as (in this example, we make an exception to our convention that all functions are real-valued) $$g(x)=\sum_{\xi\in{\widehat}Z}{\widehat}g(\xi)\,\xi(x)=
\sum_{\xi\in{\widehat}Z}{\widehat}g(\xi){{{\mathbb E}}}_{t_1,t_2\in Z}\xi(x+t_1)
\xi(x+t_2)\overline\xi(x+t_1+t_2).$$ It follows that $g\in A(d)$ and ${\lVert g\rVert}_{A(2)}\leq
\sum_{\xi\in{\widehat}Z}|{\widehat}g(\xi)|$.
We summarize these calculations:
The space $A(2)$ coincides with the Fourier algebra $A(Z)$ of $Z$: $$A(Z):=\bigl\{g\in{{\mathcal C}}(Z)\colon \sum_{\xi\in{\widehat}Z}|{\widehat}g(\xi)|<+\infty\bigr\}$$ and, for $g\in A(Z)$, ${\lVert g\rVert}_{A(2)}={\lVert g\rVert}_{A(Z)}$, which is equal by definition to the sum of this series.
$A(d)$ is an algebra of functions
---------------------------------
\[th:Ad\_algebra\] The Banach space $A(d)$ is invariant under pointwise multiplication and ${\lVert \cdot\rVert}_{A(d)}$ is an algebra norm, meaning that for all $g,g'\in A(d)$, $$\label{eq:algebra}
\quad\ {\lVert gg'\rVert}_{A(d)}\leq{\lVert g\rVert}_{A(d)}\,{\lVert g'\rVert}_{A(d)}.$$
Assume that $$g(x)= {{\mathcal D}}_{d}(f_{{{\vec{\epsilon}}}}\colon{{\vec{\epsilon}}}\in {\widetilde}V_{d})(x) \ \text{ and } \
g'(x)={{\mathcal D}}_{d}(f'_{{{\vec{\epsilon}}}}\colon{{\vec{\epsilon}}}\in {\widetilde}V_{d})(x),$$ where $f_{{\vec{\epsilon}}}$ and $f'_{{\vec{\epsilon}}}\in L^{2^{d-1}}(\mu)$ for every ${{\vec{\epsilon}}}\in V_d$. Once we show that $gg'\in A(d)$ and $$\label{eq:product}
{\lVert gg'\rVert}_{A(d)}\leq\prod_{{{\vec{\epsilon}}}\in V_d}{\lVert f_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}
{\lVert f'_{{\vec{\epsilon}}}\rVert}_{2^{d-1}},$$ the statement of the theorem follows from the definitions of the space $A(d)$ and its norm.
We have $$g(x)g'(x)=
{{{\mathbb E}}}_{{{\vec{s}}}\in{{{\mathbb Z}}}^d}\Bigl({{{\mathbb E}}}_{{{\vec{t}}}\in Z^d}
\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}
f_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}})\;
f'_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{s}}})
\Bigr)$$
Writing ${{\vec{u}}}={{\vec{s}}}-{{\vec{t}}}$, we have that $$\begin{gathered}
g(x)g'(x)={{{\mathbb E}}}_{{{\vec{u}}}\in{{{\mathbb Z}}}^d}\Bigl({{{\mathbb E}}}_{{{\vec{t}}}\in Z^d}
\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}
f_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot
{{\vec{t}}})\;f'_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{u}}}+{{\vec{\epsilon}}}\cdot {{\vec{t}}})
\Bigr)\\
={{{\mathbb E}}}_{{{\vec{u}}}\in{{{\mathbb Z}}}^d}\Bigl({{{\mathbb E}}}_{{{\vec{t}}}\in Z^d}\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}
\bigl(f_{{\vec{\epsilon}}}\;.\, f'_{{{\vec{\epsilon}}},{{\vec{\epsilon}}}\cdot {{\vec{u}}}})(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}})
\Bigr)
={{{\mathbb E}}}_{{{\vec{u}}}\in{{{\mathbb Z}}}^d} g^{({{\vec{u}}})}(x),\end{gathered}$$ where $$g^{({{\vec{u}}})}(x):= {{{\mathbb E}}}_{{{\vec{t}}}\in Z^d}\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}
\bigl(f_{{\vec{\epsilon}}}\; .\, f'_{{{\vec{\epsilon}}},{{\vec{\epsilon}}}\cdot {{\vec{u}}}})(x+{{\vec{\epsilon}}}\cdot {{\vec{t}}}).$$
Then $$\begin{gathered}
{{{\mathbb E}}}_{{{\vec{u}}}\in Z^d}\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}
{\lVert f_{{{\vec{\epsilon}}}}.f'_{{{\vec{\epsilon}}}, {{\vec{\epsilon}}}\cdot {{\vec{u}}}}\rVert}_{2^{d-1}}\\
={{{\mathbb E}}}_{u_1,\dots,u_{d-1}\in Z}
\Bigl(
\prod_{\substack{{{\vec{\epsilon}}}\in{\widetilde}V_d \\ \epsilon_d=0}}
{\lVert f_{{\vec{\epsilon}}}\; .\,f'_{{{\vec{\epsilon}}},\epsilon_1u_1+\dots+\epsilon_{d-1}u_{d-1}}\rVert}_{2^{d-1}}
\qquad\strut\\
\prod_{\substack{{{\vec{\epsilon}}}\in{\widetilde}V_d \\ \epsilon_d=1}}
\Bigl(
{{{\mathbb E}}}_{u_d\in Z}{\lVert f_{{\vec{\epsilon}}}\;.\,
f'_{{{\vec{\epsilon}}},\epsilon_1u_1+\dots+\epsilon_{d-1}u_{d-1}+u_d}\rVert}
_{2^{d-1}}^{2^{d-1}}
\Bigr)^{1/2^{d-1}}\Bigr).\end{gathered}$$
But, for all $u_1,u_2,\dots,u_{d-1}\in Z$ and every ${{\vec{\epsilon}}}\in{\widetilde}V_d$ with $\epsilon_d=1$, $$\begin{gathered}
{{{\mathbb E}}}_{u_d\in Z}
{\lVert f_{{\vec{\epsilon}}}\;.\,
f'_{{{\vec{\epsilon}}},\epsilon_1u_1+\dots+\epsilon_{d-1}u_{d-1}+u_d}\rVert}
_{2^{d-1}}^{2^{d-1}}\\
= {{{\mathbb E}}}_{v\in Z}{\lVert f_{{\vec{\epsilon}}}\;.\,f'_{{{\vec{\epsilon}}},v} \rVert}_{2^{d-1}}^{2^{d-1}}
={\lVert f_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}^{2^{d-1}}\;
{\lVert f'_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}^{2^{d-1}}.\end{gathered}$$ On the other hand, $$\begin{aligned}
{{{\mathbb E}}}_{u_1,\dots,u_{d-1}\in Z} &
\prod_{\substack{{{\vec{\epsilon}}}\in{\widetilde}V_d \\ \epsilon_d=0}}
{\lVert f_{{\vec{\epsilon}}}\;.\,
f'_{{{\vec{\epsilon}}},\epsilon_1u_1+\dots+\epsilon_{d-1}u_{d-1}}\rVert}_{2^{d-1}}
\\
\leq \prod_{\substack{{{\vec{\epsilon}}}\in{\widetilde}V_d \\ \epsilon_d=0}} &
\Bigl(
{{{\mathbb E}}}_{u_1,\dots,u_{d-1}\in Z}
{\lVert f_{{\vec{\epsilon}}}\;.\,f'_{{{\vec{\epsilon}}},\epsilon_1u_1+\dots+\epsilon_{d-1}u_{d-1}
}\rVert}_{2^{d-1}}^{2^{d-1}}
\Bigr)^{1/2^{d-1}}.\end{aligned}$$
But, for ${{\vec{\epsilon}}}\in{\widetilde}V_d$ with $\epsilon_d=0$, we have that $\epsilon_1,\dots,\epsilon_{d-1}$ are not all equal to $0$ and $$\begin{aligned}
{{{\mathbb E}}}_{u_1,\dots,u_{d-1}\in Z}
{\lVert f_{{\vec{\epsilon}}}\;.\,f'_{{{\vec{\epsilon}}},\epsilon_1u_1+\dots+\epsilon_{d-1}u_{d-1}
}\rVert}_{2^{d-1}}^{2^{d-1}}
= & {{{\mathbb E}}}_{w\in Z}{\lVert f_{{\vec{\epsilon}}}\;.\,f'_{{{\vec{\epsilon}}},w}\rVert}_{2^{d-1}}^{2^{d-1}}\\
= & {\lVert f_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}^{2^{d-1}}\;
{\lVert f'_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}^{2^{d-1}}.\end{aligned}$$ Combining these relations, we obtain that $${{{\mathbb E}}}_{{{\vec{u}}}\in Z^d} \prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}
{\lVert f\;.\,f'_{{{\vec{\epsilon}}}\cdot {{\vec{u}}}}\rVert} _{2^{d-1}}\leq
\prod_{{{\vec{\epsilon}}}\in V_d}{\lVert f_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}\;
{\lVert f'_{{{\vec{\epsilon}}}}\rVert}_{2^{d-1}}.$$ Therefore, for $\mu\times\dots\times\mu$-almost every ${{\vec{u}}}\in Z^d$ and for every ${{\vec{\epsilon}}}\in{\widetilde}V_d$, the function $f_{{\vec{\epsilon}}}.f'_{{{\vec{\epsilon}}},{{\vec{\epsilon}}}\cdot {{\vec{u}}}}$ belongs to $L^{2^{d-1}}(\mu)$. It follows that for $\mu\times\dots\times\mu$-almost every ${{\vec{u}}}\in Z^d$, the function $g^{({{\vec{u}}})}$ belongs to $A(d)$ and that $${{{\mathbb E}}}_{{{\vec{u}}}\in Z^d}{\lVert g^{({{\vec{u}}})}\rVert}_{A(d)}\leq
\prod_{{{\vec{\epsilon}}}\in V_d}{\lVert f_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}\;
{\lVert f'_{{\vec{\epsilon}}}\rVert}_{2^{d-1}}.$$ Since $gg'(x)={{{\mathbb E}}}_{{{\vec{u}}}\in Z^d}g^{({{\vec{u}}})}(x)$, Inequality follows.
Decomposable functions on $Z_d$
-------------------------------
Recall that $Z_{d}$ is the subset of $Z^{2^d}$ defined in and the elements ${{\mathbf x}}\in Z_{d}$ are written as ${{\mathbf x}}= (x_{{\vec{\epsilon}}}\colon {{\vec{\epsilon}}}\in
V_d)$.
The space $D(d)$ of *decomposable functions* consists in functions $F$ on $Z_d$ that can be written as $$\label{eq:def_decomp}
F({{\mathbf x}})= \sum_{j=1}^\infty\prod_{{{\vec{\epsilon}}}\in V_d}f_{j,{{\vec{\epsilon}}}}
(x_{{\vec{\epsilon}}}),$$ where all the functions $f_{j,{{\vec{\epsilon}}}}$ belong to $L^{2^d}(\mu)$ and $$\label{eq:def_decomp2}
\sum_{j=1}^\infty\prod_{{{\vec{\epsilon}}}\in V_d}
{\lVert f_{j,{{\vec{\epsilon}}}}\rVert}_{L^{2^d}(\mu)}<+\infty.$$
For $F\in D(d)$, define $${\lVert F\rVert}_{D(d)}=\inf\sum_{j=1}^\infty\prod_{{{\vec{\epsilon}}}\in V_d}
{\lVert f_{j,{{\vec{\epsilon}}}}\rVert}_{2^d},$$ where the infimum is taken over all families of functions $f_{j,{{\vec{\epsilon}}}}$ in $L^{2^d}(\mu)$ satisfying and .
By the remark following , a function $F\in D(d)$ belongs to $L^2(\mu_d)$ and $${\lVert F\rVert}_{L^2(\mu_d)}\leq
{\lVert F\rVert}_{D(d)}.$$
Clearly, if $Z$ is finite, then every function on $Z_d$ belongs to $D(d)$ and in the definition, we can replace the series by a finite sum.
We summarize the properties of the space $D(d)$:
$D(d)$ is a linear subspace of $L^2(\mu_d)$ and for $F\in D(d)$, we have that ${\lVert F\rVert}_{L^2(\mu_d)}\leq{\lVert F\rVert}_{D(d)}$. The space $D(d)$ endowed with the norm ${\lVert \cdot\rVert}_{D(d)}$ is a Banach space.
Diagonal translations
---------------------
For $t\in Z$, we write $t^\Delta=(t,t,\dots,t)\in Z_d$. The map ${{\mathbf x}}\mapsto {{\mathbf x}}+t^\Delta$ is called the *diagonal translation by $t$*.
Let ${{\mathcal I}}(d)$ denote the subspace of $L^2(\mu_d)$ consisting of functions invariant under all diagonal translations. The orthogonal projection $\pi$ on ${{\mathcal I}}(d)$ is given by $$\pi F({{\mathbf x}})={{{\mathbb E}}}_{t\in Z}F({{\mathbf x}}+t^\Delta).$$
\[prop:proj\_pi\] If $F$ belongs to $D(d)$, then $\pi F$ belongs to $D(d)$ and ${\lVert \pi F\rVert}_{D(d)}\leq{\lVert F\rVert}_{D(d)}$. Furthermore, $\pi F$ is a continuous function on $Z_d$ satisfying ${\lVert \pi F\rVert}_\infty\leq {\lVert F\rVert}_{D(d)}$.
In particular, functions $F$ belonging to $D(d)\cap{{\mathcal I}}(d)$ are continuous on $Z_d$ and satisfy ${\lVert F\rVert}_\infty\leq {\lVert F\rVert}_{D(d)}$.
Assume that $f$ is given by where the functions $f_{j,{{\vec{\epsilon}}}}$ belong to $L^{2^d}(\mu)$ and is satisfied. Then $$\begin{aligned}
\pi F({{\mathbf x}})
&={{{\mathbb E}}}_{t\in Z}\sum_{j=1}^\infty \prod_{{{\vec{\epsilon}}}\in V_d}
f_{j,{{\vec{\epsilon}}},t}(x_{{\vec{\epsilon}}})\\
&={{{\mathbb E}}}_{t\in Z}\sum_{j=1}^\infty \prod_{{{\vec{\epsilon}}}\in V_d}
f_{j,{{\vec{\epsilon}}},x_{{\vec{\epsilon}}}}(t).\end{aligned}$$ The first equality gives the first part of the proposition and the second implies the second part.
\[th:decomposable\] For $F\in D(d)$ and $G\in D(d)\cap{{\mathcal I}}(d)$, we have that $FG$ belongs to $D(d)$ and that ${\lVert FG\rVert}_{D(d)}\leq{\lVert F\rVert}_{D(d)}{\lVert G\rVert}_{D(d)}$.
In particular, $ D(d)\cap{{\mathcal I}}(d)$, endowed with pointwise multiplication and the norm ${\lVert \cdot\rVert}_{B(d)}$, is a Banach algebra.
Since $\pi G=G$ when $G\in D(d)\cap {{\mathcal I}}(d)$, it suffices to show that for all $F,G\in D(d)$, we have $F.\pi(G)\in D(d)$ and $$\label{eq:product1}
{\lVert F.\pi G\rVert}_{D(d)}\leq {\lVert F\rVert}_{D(d)}{\lVert G\rVert}_{D(d)}.$$
First consider the case that $F$ and $G$ are product function: $$F({{\mathbf x}})=\prod_{{{\vec{\epsilon}}}\in V_d}f_{{\vec{\epsilon}}}(x_{{\vec{\epsilon}}}),\
G({{\mathbf x}})=\prod_{{{\vec{\epsilon}}}\in V_d}g_{{\vec{\epsilon}}}(x_{{\vec{\epsilon}}}),$$ where $f_{{\vec{\epsilon}}}$ and $g_{{\vec{\epsilon}}}\in L^{2^d}(\mu)$ for every ${{\vec{\epsilon}}}\in V_{d}$. Then $$(F.\pi G)({{\mathbf x}})={{{\mathbb E}}}_{t\in Z}\prod_{{{\vec{\epsilon}}}\in V_d}
(f_{{\vec{\epsilon}}}.g_{{{\vec{\epsilon}}},t})(x_{{\vec{\epsilon}}})=E_{t\in Z} H^{(t)}({{\mathbf x}})\ ,$$ where $$H^{(t)}({{\mathbf x}})\prod_{{{\vec{\epsilon}}}\in V_d}
(f_{{\vec{\epsilon}}}.g_{{{\vec{\epsilon}}},t})(x_{{\vec{\epsilon}}})\ .$$ Furthermore, $${{{\mathbb E}}}_{t\in Z}\prod_{{{\vec{\epsilon}}}\in V_d}
{\lVert f_{{\vec{\epsilon}}}.g_{{{\vec{\epsilon}}},t}\rVert}_{2^d}
\leq \prod_{{{\vec{\epsilon}}}\in V_d}\Bigl(
{{{\mathbb E}}}_{t\in Z}{\lVert f_{{\vec{\epsilon}}}.g_{{{\vec{\epsilon}}},t}\rVert}_{2^d}^{2^d}
\Bigr)^{1/2^d}
=\prod_{{{\vec{\epsilon}}}\in V_d}{\lVert f_{{\vec{\epsilon}}}\rVert}_{2^d}{\lVert g_{{\vec{\epsilon}}}\rVert}_{2^d}\ .$$ Thus for $\mu$-almost every $t\in Z$, we have that $f_{{\vec{\epsilon}}}.g_{{{\vec{\epsilon}}},t}$ belongs to $L^{2^d}$ for every ${{\vec{\epsilon}}}$ and the function $H^{(t)}$ belongs to $B(d)$. Finally, $${\lVert F.\pi G\rVert}_{B(d)}\leq E_{t\in Z}{\lVert H^{(t)}\rVert}_{B(d)}
\leq E_{t\in Z}\prod_{{{\vec{\epsilon}}}\in V_d}
{\lVert f_{{\vec{\epsilon}}}\rVert}_{2^d}{\lVert g_{{\vec{\epsilon}}}\rVert}_{2^d}$$ and the statement of the theorem follows from the definitions of the space $D(d)$ and its norm.
A result of finite approximation
================================
A decomposition theorem
-----------------------
For a probability space $(X,\mu)$, we assume throughout that it belongs to one of the two following classes:
- $\mu$ is nonatomic. We refer to this case as the *infinite case*.
- $X$ is finite and $\mu$ is the uniform probability measure on $X$. We refer to this case as the *finite case*.
This is not a restrictive assumption: Haar measure on a compact abelian group always falls into one of these two categories.
As usual, all subsets or partitions of $X$ are implicitly assumed to be measurable.
\[def:almost-uniform\] Let $m\geq 2$ be an integer and let $(X_1,\dots,X_m)$ a partition of the probability space $(X,\mu)$. This partition is *almost uniform* if:
- in the infinite case, $\mu(X_i)=1/m$ for every $i$.
- In the finite case, $|X_i|=\lfloor |X|/m\rfloor$ or $\lceil
|X|/m \rceil$ for every $i$.
The main result of this paper is:
\[th:main\] Let $d\geq 1$ be an integer and let $\delta>0$. There exists an integer $M=M(d,\delta)\geq 2$ and a constant $C=C(d,\delta)>0$ such that the following holds: if $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in{\widetilde}V_{d+1}$, are $2^{d+1}-1$ functions belonging to $L^{2^d}(\mu)$ with ${\lVert f_{{\vec{\epsilon}}}\rVert}_{L^{2^d}(\mu)}\leq 1$ and $$\phi(x)=
{{\mathcal D}}_{d+1}(f_{{{\vec{\epsilon}}}}\colon{{\vec{\epsilon}}}\in {\widetilde}V_{d+1})(x),$$ then for every $\delta>0$ there exist an almost uniform partition $(X_1,\dots,X_m)$ of $Z$ with $m\leq M$ sets, a nonnegative function $\rho$ on $Z$, and for $1\leq i\leq m$ and every $t\in Z$, a function $\phi_{i}^{(t)}$ on $Z$ such that
1. ${\displaystyle}{\lVert \rho\rVert}_{L^2(\mu)}\leq\delta$;
2. ${\lVert \phi_{i}^{(t)}\rVert}_\infty\leq 1$ and ${\displaystyle}{\lVert \phi_{i}^{(t)}\rVert}_{A(d)}\leq C$ for every $i$ and every $t$;
3. $$\label{eq:main}
\Bigl| \phi(x+t)-\sum_{i=1}^m1_{X_i}(x)\phi_{i}^{(t)}(x)
\Bigr|\leq\rho(x)
\text{ for all }x,t\in Z.$$
Combining this theorem with an approximation result, this gives insight into properties of the dual norm.
In fact we show a bit more: each function $\phi_{i}^{(t)}$ is the sum of a bounded number of functions that are cubic convolution products of functions with $L^{2^{d-1}}(\mu)$ norm bounded by $1$.
The function $\phi$ in the statement of Theorem \[th:main\] satisfies $|\phi|\leq 1$ and thus $0\leq \rho \leq 2$.
Furthermore, the function $\phi$ belongs to $A(d+1)$, with ${\lVert \phi\rVert}_{A(d+1)}\leq 1$. But Theorem \[th:main\] can not be extended to all functions belonging to $A(d+1)$, even for $d=1$.
Theorem \[th:main\] holds for $d=1$, keeping in mind that $A(1)$ consists of constant functions and that ${\lVert \cdot\rVert}_{A(1)}$ is the absolute value. In this case, the results can be proven directly and we sketch this approach. In Section \[sec:d=2\], we showed that the Fourier coefficients of the function $\phi$ satisfy $$\sum_{\xi\in{\widehat}Z}|{\widehat}{\phi}(\xi)|^{2/3}\leq 1.$$ Let $\psi$ be the trigonometric polynomial obtained by removing the Fourier coefficients in $\phi$ that are less than $\delta^{3}$. The error term satisfies ${\lVert \phi-\psi\rVert}_{\infty}\leq\delta$ and so the function $\rho$ in the theorem can be taken to be the constant $\delta$. There are at most $1/\delta^{2}$ characters so that $\xi$ such that ${\widehat}{\psi}(\xi)\neq 0$. Taking a finite partition such that each of these characters is essentially constant on each set in the partition, we have that for every $t$ the function $\phi_{t}$ is essentially constant on each piece of the partition.
Before turning to the proof, we need some definitions, notation, and further results. Throughout the remainder of this section, we assume that an integer $d\geq 1$ is fixed, and the dependence of all constants on $d$ is implicit in all statements. For notational convenience, we study functions belonging to $A(d+1)$ instead of $A(d)$.
Regularity Lemma
----------------
\[def:partition\] Fix an integer $D\geq 2$. Let $(X,\mu)$ be a probability space of one of the two types considered in Definition \[def:almost-uniform\].
Let $\nu$ be a measure on $Z^D$ such that each of its projections on $Z$ is equal to $\mu$.
Let ${{\mathcal P}}$ be a partition of $Z$. An atom of the product partition ${{\mathcal P}}\times\ldots\times {{\mathcal P}}$ ($D$ times) of $Z^D$ is called a [*rectangle*]{} of ${{\mathcal P}}$.
A [*${{\mathcal P}}$-function*]{} on $Z^D$ is a function $f$ that is constant on each rectangle of ${{\mathcal P}}$.
For a function $F$ on $Z^D$, we define $F_{{\mathcal P}}$ to be the ${{\mathcal P}}$-function obtained by averaging over each rectangle with respect to the measure $\nu$: for every $x\in Z^D$, if $R$ is the rectangle containing $x$, then $$F_{{\mathcal P}}(x) =
\begin{cases}
{\displaystyle}\frac{1}{\nu(R)}\int F\,d\nu & \text{ if }\nu(R)\neq 0;\\
0 & \text{ if }\nu(R)=0.
\end{cases}$$
An $m$-step function is a ${{\mathcal P}}$-function for some partition ${{\mathcal P}}$ into at most $m$ sets.
As with $d$, we assume that the integer $D$ is fixed throughout and omit the explicit dependencies of the statements and constants on $D$.
We make use of the following version of the Regularity Lemma, a modification of the analytic version of Szemerédi’s Regularity Lemma in [@Lovasz]:
\[th:regularity\] For every $D$ and $\delta > 0$, there exists $M=M(D,\delta)$ such that if $(X,\mu)$ and $\nu$ are as in Definition \[def:partition\], then for every function $F$ on $Z^D$ with $|F|\leq 1$, there is an almost uniform partition ${{\mathcal P}}$ of $Z$ into $m\leq M$ sets such that for every $m$-step function $U$ on $Z^D$ with $|U|\leq 1$, $$\Bigl|\int U(F-F_{{\mathcal P}})\,d\nu \Bigr|\leq\delta \ .$$
We defer the proof to Appendix \[appendix:regularity\]. In the remainder of this section, we carry out the proof of Theorem \[th:main\].
An approximation result for decomposable functions
--------------------------------------------------
We return to our usual definitions and notation. We fix $d\geq 1$ and apply the Regularity Lemma to the probability space $(Z,\mu)$, $D=2^d$ and the probability measure $\mu_d$ on $Z^{2^d}$.
In this section, we show an approximation result that allows to go from weak to strong approximations:
\[th:RL2\] Let $F$ be a function on $Z_d$ belonging to $D(d)$ with ${\lVert F\rVert}_{D(d)}\leq 1$ and ${\lVert F\rVert}_\infty\leq 1$. Let $\theta >0$ and ${{\mathcal P}}$ be the partition of $Z$ associated to $F$ and $\theta$ by the Regularity Lemma (Theorem \[th:regularity\]). Then there exist constants $C=C(d)>0$ and $c=c(d)>0$ such that $${\lVert F-F_{{\mathcal P}}\rVert}_2\leq (C\theta^c+\theta)^{1/2}\ .$$
We first prove a result that allows us to pass from sets to functions:
\[lem:prod\] Assume that $F$ is a function on $Z_d$ with ${\lVert F\rVert}_\infty\leq
1$. Let $\theta >0$ and let ${{\mathcal P}}$ be the partition of $Z$ associated to $F$ and $\theta$ by the Regularity Lemma (Theorem \[th:regularity\]). If $f_{{\vec{\epsilon}}}$, ${{\vec{\epsilon}}}\in V_d$, are functions on $Z$ satisfying ${\lVert f_{{\vec{\epsilon}}}\rVert}_{2^d}\leq 1$ for every ${{\vec{\epsilon}}}$, then $$\label{eq:prod}
\Bigl|{{{\mathbb E}}}_{{{\mathbf x}}\in Z_d}(F-F_{{\mathcal P}})({{\mathbf x}})
\prod_{{{\vec{\epsilon}}}\in V_d}f_{{\vec{\epsilon}}}(x_{{\vec{\epsilon}}})\Bigr|
\leq C\theta^c,$$ where $c=c(d)$ and $C=C(d)$ are positive constants.
In other words, writing ${\lVert \cdot\rVert}_{D(d)}^*$ for the dual norm of the norm ${\lVert \cdot\rVert}_{D(d)}$, we have that $${\lVert F-F_{{\mathcal P}}\rVert}_{D(d)}^*\leq C\theta^c.$$
By construction, ${{\mathcal P}}$ is an almost uniform partition of $Z$ into $m<M(\eta)$ pieces and the function $F=F_{{\mathcal P}}$ satisfies $$\label{eq:regul}
|{{{\mathbb E}}}_{Z_d}U(F-F_{{\mathcal P}})|\leq\eta$$ for every $m$-step function $U$ on $Z_d$ with $|U|\leq 1$. We show .
By possibly changing the constant $C$, we can further assume that the functions $f_{{\vec{\epsilon}}}$ are all non-negative. Let $\eta>0$ be a parameter, with its value to be determined. For ${{\vec{\epsilon}}}\in\{0,1\}^d$, set $$f'_{{\vec{\epsilon}}}(x)=\min\bigl(f_{{\vec{\epsilon}}}(x),\eta\bigr)
\text{ and }
f''_{{\vec{\epsilon}}}(x)=f_{{\vec{\epsilon}}}-f'_{{\vec{\epsilon}}}(x).$$ Thus the average of can be written as a sum of $2^d$ averages, which we deal with separately.
**a)** We first show that $$\label{eq:maj1}
\Bigl|{{{\mathbb E}}}_{{{\mathbf x}}\in Z_d}(F-F_{{\mathcal P}})({{\mathbf x}})
\prod_{{{\vec{\epsilon}}}\in V_d}f'_{{\vec{\epsilon}}}(x_{{\vec{\epsilon}}})\Bigr|\leq
\eta^{2^d}\theta.$$
For $u\in{{{\mathbb R}}}_+$, write $$A({{\vec{\epsilon}}},u)=\{x\in Z\colon f_{{\vec{\epsilon}}}(x)\leq u\}.$$ For each ${{\vec{\epsilon}}}\in \{0,1\}^d$, we have that $$f'_{{\vec{\epsilon}}}(x)=\int_0^\eta {{\boldsymbol 1}}_{A ({{\vec{\epsilon}}},u)}(x)\,du$$ and so the average of the left hand side of is the integral over ${{\mathbf u}}=(u_{{\vec{\epsilon}}}\colon{{\vec{\epsilon}}}\in V_d)\in
[0,\eta]^{2^d}$ of $${{{\mathbb E}}}_{{{\mathbf x}}\in Z_d}(F-F_{{\mathcal P}})({{\mathbf x}})\prod_{{{\vec{\epsilon}}}\in V_d}{{\boldsymbol 1}}_{A
({{\vec{\epsilon}}},u_{{\vec{\epsilon}}})}(x_{{\vec{\epsilon}}}).$$ By , for each ${{\mathbf u}}\in
[0,\eta]^{2^d}$, the absolute value of this average is bounded by $\theta$. Integrating, we have the bound .
**b)** Assume now that for each ${{\vec{\epsilon}}}\in \{0,1\}^d$, the function $g_{{\vec{\epsilon}}}$ is equal either to $f'_{{\vec{\epsilon}}}$ or to $f''_{{\vec{\epsilon}}}$, and that there exists ${{\vec{\alpha}}}\in \{0,1\}^d$ with $g_{{\vec{\alpha}}}=f''_{{\vec{\epsilon}}}$. We show that $$\Bigl|{{{\mathbb E}}}_{{{\mathbf x}}\in Z_d}(F-F_{{\mathcal P}})({{\mathbf x}})
\prod_{{{\vec{\epsilon}}}\in V_d}g_{{\vec{\epsilon}}}(x_\epsilon)\Bigr|
\leq 2\eta^{-2^d+1}.$$ Since $|F-F_{{\mathcal P}}|\leq 2$ and the functions $g_{{\vec{\epsilon}}}$ are nonnegative, it suffices to show that $${{{\mathbb E}}}_{{{\mathbf x}}\in Z_d}
\prod_{{{\vec{\epsilon}}}\in \{0,1\}^d}g_{{\vec{\epsilon}}}(x_{{\vec{\epsilon}}})\leq
\eta^{-2^d+1}.$$ By Corollary \[cor:alpha\], the left hand side is bounded by $$\begin{gathered}
\prod_{\substack{{{\vec{\epsilon}}}\in V_d\\ {{\vec{\epsilon}}}\neq{{\vec{\alpha}}}}}
{\lVert g_{{\vec{\epsilon}}}\rVert}_{L^{2^{d-1}}(\mu)}\cdot{\lVert g_{{\vec{\alpha}}}\rVert}_{L^1(\mu)}
\leq {\lVert g_{{\vec{\alpha}}}\rVert}_{L^1(\mu)}
=\int 1_{f_{{\vec{\alpha}}}>\eta}(x) f_{{\vec{\alpha}}}(x)\\
\leq
{\lVert f_{{\vec{\alpha}}}\rVert}_{2^d}
\mu\{x\in Z\colon f_{{\vec{\alpha}}}(x)\geq\eta\}^{(2^d-1)/2^d}
\leq \eta^{-2^d+1}\end{gathered}$$ and we have the statement.
**c)** The left hand side of is thus bounded by $$\eta^{2^d}\theta+2(2^d-1)\eta^{-2^d+1}.$$ Taking $\eta=\theta^{-1/(2^{d+1}-1)}$, we have the bound .
We now use this to prove the proposition:
Since $F$ belongs to $D(d)$ with ${\lVert F\rVert}_{D(d)}\leq 1$, it follows from the definition of this norm and from Lemma \[lem:prod\] that $|{{{\mathbb E}}}_{{{\mathbf x}}\in Z_d}(F-F_{{\mathcal P}})({{\mathbf x}})F({{\mathbf x}})|\leq C\theta^c$.
On the other hand, $F_{{\mathcal P}}$ is an $m$-step function and by the property of the partition ${{\mathcal P}}$ given by Theorem \[th:regularity\], we have that $|{{{\mathbb E}}}_{{{\mathbf x}}\in Z_d}(F-F_{{\mathcal P}})({{\mathbf x}})F_{{\mathcal P}}({{\mathbf x}})|\leq\theta$. Finally, ${{{\mathbb E}}}_{{{\mathbf x}}\in Z_d}\bigl((F-F_{{\mathcal P}})({{\mathbf x}})^2\bigr)
\leq C\theta^c+\theta$.
Proof of Theorem \[th:main\]
----------------------------
We use the notation and hypotheses from the statement of Theorem \[th:main\].
a) A decomposition {#a-a-decomposition .unnumbered}
------------------
Define $\operatorname{P}\,\,\colon L^1(\mu_d)\to L^1(\mu)$ to be the operator of conditional expectation. The most convenient definition of this operator is by duality: for $h\in L^\infty(\mu)$ and $H\in L^1(\mu_d)$, $$\int_Z h(x)\, \operatorname{P}H(x)\,d\mu(x)=
\int_{Z_d} h(x_{{\vec 0}}) H({{\mathbf x}})\,d\mu_d({{\mathbf x}}).$$ Recall that ${\lVert \operatorname{P}H\rVert}_{L^{1}(\mu_{d}}\leq{\lVert H\rVert}_{L^1(\mu_d)}$.
By definition, when $$H(x)=\prod_{{{\vec{\epsilon}}}\in V_d}f_{{\vec{\epsilon}}}(x_{{\vec{\epsilon}}}),$$ where the functions $f_{{\vec{\epsilon}}}$ belong to $L^{2^{d-1}(\mu)}$, then $$\label{eq:PH_Product}
\operatorname{P}H(x)={{{\mathbb E}}}_{{{\vec{t}}}\in Z_d}\prod_{{{\vec{\epsilon}}}\in V_d}
f_{{\vec{\epsilon}}}(x+{{\vec{\epsilon}}}\cdot{{\vec{t}}}).$$
For ${{\mathbf x}}\in Z_d$, define $$\begin{gathered}
G({{\mathbf x}})=\bigotimes_{{{\vec{\epsilon}}}\in{\widetilde}V_d} f_{{{\vec{\epsilon}}}0}({{\mathbf x}})=
\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_d}f_{{{\vec{\epsilon}}}0}(x_{{\vec{\epsilon}}})\\
F({{\mathbf x}})
=\Bigl(\pi\bigotimes_{{{\vec{\epsilon}}}\in V_d} f_{{{\vec{\epsilon}}}1}\Bigr)
({{\mathbf x}})
={{{\mathbb E}}}_{u\in Z}\prod_{{{\vec{\epsilon}}}\in V_d}
f_{{{\vec{\epsilon}}}1}(x_{{\vec{\epsilon}}}+u).\end{gathered}$$
For $x\in Z$, we have $$\begin{aligned}
\phi(x)= & {{{\mathbb E}}}_{{{\vec{s}}}\in Z_d}\prod_{{{\vec{\epsilon}}}\in{\widetilde}V_d}
\Bigl(
f_{{{\vec{\epsilon}}}0}(x+{{\vec{\epsilon}}}\cdot {{\vec{s}}})\,{{{\mathbb E}}}_{u\in Z}
\prod_{{{\vec{\epsilon}}}\in V_d}f_{{{\vec{\epsilon}}}1}(x+{{\vec{\epsilon}}}\cdot {{\vec{s}}}+u)
\Bigr)\\
= & \operatorname{P}(G\cdot F).\end{aligned}$$
Recall that for $t\in Z$, $\phi_t$ is the function on $Z$ defined by $\phi_t(x)=\phi(x+t)$.
For $t\in Z$ and ${{\mathbf x}}\in Z_d$, define $$G_{t^\Delta}({{\mathbf x}})=G(x+t^\Delta)=
\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_d}f_{{{\vec{\epsilon}}}0}(x_{{\vec{\epsilon}}}+t).$$ Since the function $F$ is invariant under diagonal translations, for $x,t\in Z$ we have that $$\phi_t(x)=\operatorname{P}(G_{t^\Delta}\cdot F)(x).$$ By Proposition \[prop:proj\_pi\], the function $F$ belongs to $D(d)$ and ${\lVert F\rVert}_{D(d)}\leq 1$. Thus ${\lVert F\rVert}_\infty\leq 1$.
Let $\delta>0$. Let $c$ and $C$ be as in Proposition \[th:RL2\] and let $\theta>0$ be such that $(C\theta^c+\theta)^{1/2}<\delta$. Let ${{\mathcal P}}$ and $F_{{\mathcal P}}$ be associated to $F$ and $\theta$ as in the Regularity Lemma. Let ${{\mathcal P}}=(A_1,\dots, A_m)$.
For $x,t\in Z$, we have that $$\phi_t(x)=\operatorname{P}(G_{t^\Delta}\cdot (F-F_{{\mathcal P}}))+
\operatorname{P}(G_{t^\Delta}\cdot F_{{\mathcal P}})$$ and we study the two parts of this sum separately.
b) Bounding the rest {#b-bounding-the-rest .unnumbered}
--------------------
Define $$\rho(x)= \bigl(\operatorname{P}(F-F_{{\mathcal P}})^2\bigr)^{1/2}.$$ We have that $${\lVert \rho\rVert}_{2}= {\lVert \operatorname{P}(F-F_{{{\mathcal P}}})^{2}\rVert}_{L^{2}(\mu_{d})}^{1/2}
\leq{\lVert (F-F_{{{\mathcal P}}})^{2}\rVert}_{L^{1}(\mu_{d})}^{1/2} =
{\lVert F-F_{{\mathcal P}}\rVert}_{L^{2}(\mu_{d})}\leq\delta,$$ where the last inequality follows from Proposition \[th:RL2\].
Moreover, $$\bigl| \operatorname{P}(G_{t^\Delta}\cdot (F-F_{{\mathcal P}}))\bigr|\leq
\bigl(\operatorname{P}(G_{t^\Delta}^2)\bigr)^{1/2}\cdot
\bigl(\operatorname{P}(F-F_{{\mathcal P}})^2\bigr)^{1/2}\leq \rho(x)$$ by and Lemma \[lem:Ddf\].
c) The main term {#c-the-main-term .unnumbered}
----------------
We write elements of $\{1,\dots,m\}^{2^d}$ as $${{\mathbf j}}=(j_{{\vec{\epsilon}}}\colon{{\vec{\epsilon}}}\in V_d).$$ For ${{\mathbf j}}=(j_{{\vec{\epsilon}}}\colon {{\vec{\epsilon}}}\in V_d)\in
\{1,\dots,m\}^{2^d}$, write $$R_{{{\mathbf j}}}=\prod_{{{\vec{\epsilon}}}\in V_d} A_{j_{{\vec{\epsilon}}}}.$$ The function $F_{{\mathcal P}}$ is equal to a constant on each rectangle $R_{{\mathbf j}}$. Let $c_{{\mathbf j}}$ be this constant. We have that $|c_{{{\mathbf j}}}|\leq 1$.
For $1\leq i\leq m$ and $t,x\in Z$, define $$\phi_i^{(t)}(x):={{{\mathbb E}}}_{{{\vec{s}}}\in Z^d}
\sum_{\substack{ {{\mathbf j}}\in\{1,\dots,m\}^{2^d} \\
j_{{\vec 0}}= i}}
c_{{\mathbf j}}\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_d}
1_{A_{j_{{\vec{\epsilon}}}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{s}}}).
\phi_{{{\vec{\epsilon}}}0}(x+{{\vec{\epsilon}}}\cdot {{\vec{s}}}).$$ Since distinct rectangles are disjoint, it follows that $$\Bigl|\sum_{\substack{ {{\mathbf j}}\in\{1,\dots,m\}^{2^d} \\
j_{{\vec 0}}= i}}
c_{{\mathbf j}}\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_d}
1_{A_{j_{{\vec{\epsilon}}}}}(x+{{\vec{\epsilon}}}\cdot {{\vec{s}}}).
\phi_{{{\vec{\epsilon}}}0}(x+{{\vec{\epsilon}}}\cdot {{\vec{s}}})\Bigr|
\leq
\prod_{{{\vec{\epsilon}}}\in {\widetilde}V_d}
|\phi_{{{\vec{\epsilon}}}0}(x+{{\vec{\epsilon}}}\cdot{{\vec{s}}})|.$$ Thus $$|\phi_i^{(t)}(x)|\leq 1.$$ On the other hand, the function $\phi_i^{(t)}$ is the sum of $m^{2^d-1}$ functions belonging to $A(d)$ with norm $\leq 1$ and thus $${\lVert \phi_i^{(t)}\rVert}_{A(d)}\leq C=M^{2^d-1}.$$
We claim that $$\label{eq:FP_phiit}
P(G_{t^\Delta}\cdot{{\mathcal F}}_{{\mathcal P}})=
\sum_{i=1}^m 1_{A_i}(x)\phi_i^{(t)}(x).$$ Via the definitions, we have that $$(G_{t^\Delta}\cdot{{\mathcal F}}_{{\mathcal P}})({{\mathbf x}})
=\sum_{{{\mathbf j}}\in\{1,\dots,m\}^{2^d}}
c_{{\mathbf j}}\prod_{{{\vec{\epsilon}}}\in {{\vec{t}}}V_d}f_{{{\vec{\epsilon}}}0}(x_{{\vec{\epsilon}}})
\prod_{{{\vec{\epsilon}}}\in V_d}1_{A_{j_{{\vec{\epsilon}}}}}(x_{{\vec{\epsilon}}}).$$ Grouping together all terms of the sum with $j_{{\vec 0}}=i$ and using , we obtain . This completes the proof of Theorem \[th:main\].
Further directions
==================
We have carried this study of Gowers norms and associated dual norms in the setting of compact abelian groups. This leads to a natural question: what is the analog of the Inverse Theorem for groups other than ${{{\mathbb Z}}}_{N}$? What would be the generalization for other finite groups or for infinite groups such as the torus, or perhaps even for totally disconnected (compact abelian) groups?
In Section \[sec:embedding\], we give examples of functions with small dual norm, obtained by embedding in a nilmanifold. One can ask if this process is general: does one obtain all functions with small dual norm, up to a small error in $L^{1}$ in this way? In particular, for ${{{\mathbb Z}}}_{N}$ this would mean that in the Inverse Theorem we can replace the family ${{\mathcal F}}(d,\delta)$ by a family of nilsequences with “bounded complexity” that are periodic, with period $N$, meaning that they all come from embeddings of ${{{\mathbb Z}}}_N$ in a nilmanifold.
By the computations in Section \[sec:d=2\], we see a difference between $A(2)$ and the dual functions: the cubic convolution product $f$ of functions belonging to $L^{2}(\mu)$ satisfies $\sum|{\widehat}{f}|^{2/3}< \infty$, while $A(2)$ is the family of functions $f$ such that $\sum|{\widehat}{f}(\xi)|< +\infty$. It is natural to ask what analogous distinctions are for $d> 2$.
Proof of the regularity lemma {#appendix:regularity}
=============================
We make use of the following version of the Regularity Lemma in a Hilbert space introduced in [@Lovasz]:
\[lemma:lovasz\] Let $K_1, K_2, \ldots$ be arbitrary nonempty subsets of a Hilbert space ${{\mathcal H}}$. Then for every $\varepsilon > 0$ and $f\in{{\mathcal H}}$, there exists $k\leq \lceil 1/\varepsilon^2\rceil $ and $f_i\in K_i$, $i=1,
\ldots, k$ and $\gamma_1, \ldots, \gamma_k\in {{{\mathbb R}}}$ such that for every $g\in K_{k+1}$, $$|\langle g, f-(\gamma_1f_1+\ldots + \gamma_kf_k)\rangle| \leq
\varepsilon\cdot{\lvert\!|\!| g|\!|\!\rvert} \cdot {\lvert\!|\!| f|\!|\!\rvert} \ .$$
For the proof of Theorem \[th:regularity\], we follow the proof of the strong form of the Regularity Lemma in [@Lovasz].
We only consider the infinite case only, as the proof in the finite case is similar.
Choose a sequence of integers $(1)< s(2)< \ldots$ such that $$(s(1)s(2)\ldots s(i))^{2} < s(i+1)$$ for each $i\in{{{\mathbb N}}}$ and such that $D/\varepsilon<s(1)$.
Let ${{\mathcal Q}}$ be a partition of $Z$ into at most $s(i)$ sets and let $K_i$ consist of ${{\mathcal Q}}$-functions.
By Lemma \[lemma:lovasz\], there exists $k\leq \lceil 1/\varepsilon ^2\rceil $ and there exists an $s(1)\ldots s(k)$-step function $F^*$ such that $$\label{eq:bound1}
\Bigl|\int U(F-F^*)\, d\nu\Bigr|\leq \varepsilon$$ for any $s(k+1)$-step function $U$. Choose $m$ with $D/\epsilon
<m<s(k+1)$ and refine the partition defining $F^*$ into a partition ${{\mathcal S}}=\{S_1,\dots,S_m\}$ into $m$ sets. Then $F^*$ is a ${{\mathcal S}}$-function and the bound remains valid for every $m$-step function $U$.
Partition each set $S_i$ into subsets of measure $1/m^2$ and a remainder set of measure smaller than $1/m^2$. Take the union of all these remainder sets and partition this union into sets of measure $1/m^2$. Thus we obtain a partition ${{\mathcal P}}=\{A_1,\dots,A_{m^2}\}$ of $Z$ into $m^2$ sets of equal measure.
At least $m^2-m$ of these $m^2$ sets are *good*, meaning that the set is included in some set of the partition ${{\mathcal S}}$. Let $G$ denote the union of these good sets and call it the *good part* of $Z$. We have that $$\nu\bigl(Z^D\setminus G^D\bigr)\leq D/m\leq\varepsilon\ .$$
We claim that if $U$ is an $m$-step function with $|U|\leq 1$, then $$\Bigl\vert \int U(F-F_{{\mathcal P}})\,d\nu
\Bigr\vert \leq 4\varepsilon \ .$$ To show this, set $U'={{\boldsymbol 1}}_{G}\cdot U$. Then $$\Bigl\vert \int(U-U')(F-F_{{\mathcal P}})\,d\nu\Bigr|
\leq 2\int|U-U'|\,d\nu \leq 2\varepsilon\ .$$ Moreover, $U'$ is an $m$-step function with $|U'|\leq 1$ and by hypothesis, $$\Bigl| \int U'(F-F^*)\,d\nu\Bigr|\leq\varepsilon$$ and we are reduced to showing that $$\Bigl\vert\int U'(F^*-F_{{\mathcal P}})\,d\nu
\Bigr\vert \leq \varepsilon \ .$$
Instead, assume that $$\int U'(F^*-F_{{\mathcal P}})\,d\nu
> \varepsilon$$ and we derive a contradiction (the opposite bound is proved in the same way).
Define a new function $U''$ on $Z^D$. Set $U''=0=U'$ outside $G^D$. Let $R$ be a product of good sets. The functions $F^*$ and $F_{{\mathcal P}}$ are constant on $R$ and thus the function $F^*-F_{{\mathcal P}}$ is constant on $R$. Define $U''$ on $R$ to be equal to $1$ if this constant is positive and to be $-1$ if this constant is negative. Then $U''(F^*-F_{{\mathcal P}})\geq U'(F-F_{{\mathcal P}})$ on $R$ and so $$\int U''(F^*-F_{{\mathcal P}})\,d\nu
\geq \int U'(F^*-F_{{\mathcal P}})\,d\nu> \varepsilon \ .$$ On the other hand, $U''$ is a ${{\mathcal P}}$-function and so by definition of $F_{{\mathcal P}}$, $\int U''(F-F_{{\mathcal P}})\,d\nu=0$ and $$\int U''(F^*-F)\,d\nu
> \varepsilon \ .$$ But $U''$ is an $m$-step function with $|U''|\leq 1$ and this integral is $<\varepsilon$ by , leading to a contradiction.
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[^1]: The first author was partially supported by the Institut Universitaire de France and the second author by NSF grant 0900873.
|
---
abstract: 'We investigate the multiquantum vortex states in type-II superconductor both in “clean” and “dirty” regimes defined by impurity scattering rate. Within quasiclassical approach we calculate self-consistently the order parameter distributions and electronic local density of states (LDOS) profiles. In the clean case we find the low temperature vortex core anomaly predicted analytically in G.E. Volovik, JETP Lett. [**58**]{}, 455 (1993) and obtain the patterns of LDOS distributions. In dirty regime the multiquantum vortices feature a peculiar plateau in the zero-energy LDOS profile which can be considered as an experimental hallmark of multiquantum vortex formation in mesoscopic superconductors.'
author:
- 'M. A. Silaev'
title: 'Self-consistent electronic structure of multiquantum vortices in superconductors at $T\ll T_c$'
---
Introduction
============
Modern technology development provides a unique possibility to study superconducting states at the nanoscale. Recently there has been much experimental effort focused on the investigation of exotic vortex states in mesoscopic superconducting samples of the size of several coherence lengths [@Mesovortices; @Geim]. Magnetic field can penetrate the sample in the form of a poligonlike vortex molecule or individual vortices can merge forming multiquantum giant vortex state with a winding number larger than unity [@ExoticVortexStates]. The latter possibility is of particular interest and the search of giant vortices in mesoscopic superconductors was performed by means of various experimental techniques including transport measurements [@Rod10; @Rod11], Bitter decoration [@Rod12], magnetometry [@Rod13], and scanning Hall probe experiments [@Rod14]. Currently much effort is invested to the studies of nanoscale superconducting samples with the help of scanning tunneling microscopy (STM) techniques [@RoditchevPRL2011; @RoditchevPRL2009] which have been achieved only recently and allows for the direct probe of the structure of vortex cores through measurement of the electronic states LDOS distribution modified by vortices.
Such STM measurements have been proven to be an effective tool of experimental study of electronic structure of vortices in bulk superconductors[@STSHess; @STSmore1; @STSRoditchev; @STSmore2; @RMP-2007]. Indeed for the temperatures much lower than the typical energy scale in superconductors $T\ll T_c$ the local differential conductance of the contact between STM tip and superconductor as a function of voltage $V$: $$\label{Eq:CondLT}
\frac{dI}{dV} (V)= \frac{dI}{dV}_{N} \frac{N({\bf r},E=eV)}{N_0}.$$ where $(dI/dV)_{N}$ is a conductance of the normal metal junction and $N_0$ is the electronic density of states at the Fermi level. The observation of the zero-bias anomaly of tunneling conductance at the center of singly quantized vortices[@STSHess; @STSRoditchev; @STSmore1; @STSmore2; @RMP-2007] clearly confirmed the existence of bound vortex core states predicted by Caroli, de Gennes and Matricon (CdGM)[@CdGM]. In clean superconductors for each individual vortex the energy $\varepsilon(\mu)$ of a subgap electronic state varies from $-\Delta_0$ to $+\Delta_0=\Delta(r=\infty)$ as one changes the angular momentum $\mu$ defined with respect to the vortex axis. At small energies $|\varepsilon|\ll\Delta_0$ the spectrum is a linear function of $\mu$: $$\label{Eq:SinglyQuantized}
\varepsilon(\mu)=\omega\mu$$ Here $\omega \sim \Delta_0/(k_F\xi)$ where $\xi=\hbar V_F/\Delta_0$ is coherence length, $k_F$ is Fermi momentum and $V_F$ is Fermi velocity. The wave functions of the subgap states are localized inside the vortex core because of the Andreev reflection of quasiparticles at the core boundary and determine the low energy LDOS singularity at the vortex center.
In multiquantum vortices the spectrum of electronic states bound in the vortex with the winding number $M$ contains $M$ anomalous branches degenerate by electronic spin [@VolovikAnBr; @multi-spectrum-num; @Melnikov-Vinokur-2002; @TanakaMultiquantum; @SalomaaMultiquantum]: $$\label{Volovik-spectr}
\varepsilon_j(\mu)=\omega_j(\mu-\mu_{j})\,,$$ where $\omega_j \sim \Delta_0/(k_F\xi)$, index $j$ enumerates different spectral branches ($1<j<M$), $-k_F\xi\lesssim\mu_{j}\lesssim k_F\xi$. Each anomalous branch intersects the Fermi level and contributes to the low-energy LDOS. The spectrum of localized electronic states in mesoscopic superconductors with several vortices have been shown to be very sensitive to the mutual vortex position[@Silaev2009]. It has been suggested that testing the properties of electronic spectrum by means of the heat conductivity measurement one can directly observe the transition to the multiquantum vortex state in mesoscopic superconductor [@Silaev2008]. An alternative route is to use STM measurement of local tunnelling conductance being proportional to the LDOS provided $T\ll T_c$. Thus to provide the evidence of multiquantum vortex formation revealed by STM experiments one should find distinctive features of the order parameter structures and LDOS profiles occurring especially in the low temperature regime $T\ll T_c$.
Previously the low temperature properties of multiquantum vortices have not been investigated much. The results of theoretical studies are known only for the particular case of vortices in clean superconductors when the electronic mean free path is much larger than the coherence length. In this regime the contribution of anomalous branches produces singularities of the order parameter distribution near the vortex core in the limit $T\ll
T_c$. In particular the singly quantized vortex features an anomalous increase of the order parameter slope at the vortex center which is known as Kramer-Pesch effect [@KramerPesch; @Gygi]. The generalization to the multiquantum vortex case was suggested in Ref.([@VolovikAnomaly]) where it was analytically predicted that doubly quantized vortex should have square root singularity of the order parameter distribution $\Delta=\Delta(r)$ in the limit $T\ll T_c$. Although the structures of mutliquantum vortices have been calculated self-consistently in the framework of Bogolubov-de Gennes theory the vortex core anomalies have not been discussed yet [@TanakaMultiquantum; @SalomaaMultiquantum]. Moreover multiple anomalous branches of electronic spectrum have been shown to produce complicated patterns in the LDOS distributions investigated in the framework of Bogolubov- de Gennes theory [@TanakaMultiquantum; @SalomaaMultiquantum]. Here we employ an alternative approach of quasiclassical Eilenberger theory [@Eilenberger] to check the predictions of vortex core anomalies and the LDOS patterns in multiquantum vortices in clean superconductors.
Notwithstanding the interesting physics taking place in the clean regime the experimental realization of STM measurements of multiquantum vortex states was implemented on Pb superconductor [@RoditchevPRL2011; @RoditchevPRL2009] with short mean free path being much smaller than the coherence length. This dirty superconductor is more adequately described within the diffusive approximation of the electronic motion resulting in the Usadel equations for the electronic propagators and the superconducting order parameter [@Usadel]. Singly quantized vortex states in dirty superconductors were investigated in detail[@KramerJLTP74] and were shown to lack the low temperature singularity of the $\Delta (r)$ distribution being smoothed out by the impurity scattering of quasiparticle states. Moreover the LDOS distribution inside vortex core does not feature zero bias anomaly since the spectral weight of bound electronic states is distributed smoothly between all energy scales up to the bulk energy gap $\Delta_0$. On the other hand the multiquantum vortex states have not been investigated in the framework of the Usadel theory nor the LDOS distributions around multiquantum vortices in dirty superconductors have been ever calculated.
It is the goal of the present paper to study both the peculiarities of the multiquantum vortex structures especially at low temperatures and the distinctive features of the electronic LDOS near the vortices which would allow unambiguous identification of giant vortices both in clean and dirty regimes. This paper is organized as follows. In Sec. \[theory\] we give an overview of the theoretical framework namely the quasiclassical Eilenberger theory in clean superconductors and Usadel equation in the dirty regime. We discuss the results of self-consistent calculations of the order parameter distributions for multiquantum vortex configurations in Sec. \[op\] and address the LDOS profiles in Sec. \[ldos\]. We give our conclusions in Sec. \[summary\].
Theoretical framework {#theory}
=====================
Clean limit: Eilenberger formalism
----------------------------------
Within quasiclassical approximation [@Eilenberger; @Maki; @schopohl_cm] the band parameters characterizing the Fermi surface is the Fermi velocity $V_{F}$ and the density of states $N_0$. We normalize the energies to the critical temperature $T_c$ and length to $\xi_C= \hbar
V_{F}/T_c$. The magnetic field is measured in units $\phi_0/2\pi\xi_C^2$ where $\phi_0=2\pi\hbar c/e$ is magnetic flux quantum. The system of Eilenberger equations for the quasiclassical propagators $f,f^+,g$ reads $$\begin{aligned}
\label{Eq:EilenbergerF}
&{\bf n_p}\left(\nabla+i {\bf A}\right) f +
2\omega f - 2 \Delta g=0, \\ \nonumber
&{\bf n_p}\left(\nabla-i {\bf A}\right) f^+ -
2\omega f^+ + 2\Delta^* g=0.
\end{aligned}$$ Here ${\bf A}$ is a vector potential of magnetic field, the vector ${\bf n_p}$ parameterizes the Fermi surface and $\omega$ is a real quantity which should be taken at the discrete points of Matsubara frequencies $\omega_n=(2n+1)\pi T$ determined by the temperature $T$. The quasiclassical propagators obey normalization condition $g^2+ff^+=1$. The self-consistency equation for the gap is $$\label{Eq:SelfConsClean}
\Delta ({\bf r})=2\pi T \Lambda\sum_{n=0}^{N_d} S_F^{-1} \oint_{FS}
f (\omega_n, {\bf r}, {\bf n_p}) d^2S_p.$$ where $\Lambda$ is coupling constant, $S_F$ is a Fermi surface area and the integration is performed over the Fermi surface. Hereafter to simplify the calculations we assume the Fermi surface to be cylindrical and parameterized by the angle $\theta_p$ so that ${\bf
n_p}=(\cos\theta_p,\sin\theta_p)$. In Eq.(\[Eq:SelfConsClean\]) $N_d (T) = \omega_d/(2\pi T)$ is a cutoff at the Debye energy $\omega_d$ which is expressed through physical parameter $T_c$ and $\Lambda$ as follows $$\label{Eq:SelfConsistencyHom}
\sum^{N_d(T_c)}_{n=0}\frac{\Lambda}{n+1/2}=1.$$ The LDOS is expressed through the analytical continuation of quasiclassical Green’s function to the real frequencies $$\label{Eq:LDOSclean}
N({\bf r})=N_0 S_F^{-1} \oint_{FS}
Re [ g (\omega=-iE+0, {\bf r}, {\bf n_p}) ] d^2S_p.$$
Assuming the vortex line to be oriented along the ${\bf z}$ axis we choose the following ansatz of the superconducting order parameter corresponding to axially symmetric vortex bearing $M$ quanta of vorticity $\Delta(x,y)=|\Delta|(r) e^{iM\varphi}$ where $r=\sqrt{x^2+y^2}$ is the distance from the vortex center, $\varphi= \arctan(y/x)$ is the polar angle. Below we neglect the influence of the magnetic field on the vortex structure which is justified for superconductors with large Ginzburg-Landau parameter.
For numerical treatment of the Eqs.(\[Eq:EilenbergerF\]) we follow the Refs. [@Maki; @schopohl_cm] and introduce a Ricatti parametrization for the propagators. The essence of this method is a mathematical trick which allows to solve two first order Ricatti equations instead of second-order system of Eilenberger equations. Starting with some reasonable ansatz for the order parameter the first order Ricatti equations are solved by the standard procedure. Then the corrected order parameter is calculated according to Eq. (\[Eq:SelfConsClean\]). The badly converging sum in Eq.(\[Eq:SelfConsClean\]) is renormalized in a usual way with the help of Eq.(\[Eq:SelfConsistencyHom\]). Then one should take into account only several terms in the sum (\[Eq:SelfConsClean\]). E.g. $\omega_n< 10 T_c$ is enough for the temperature range $T>0.05 T_c$ considered at the present paper. The iteration of this procedure repeats until convergence of the order parameter is reached with an accuracy $10^{-4} T_c$.
Dirty limit: Usadel equations
-----------------------------
In the presence of impurity scattering the Eilenberger Eqs. (\[Eq:EilenbergerF\]) contain an additional diagonal self-energy term[@Eilenberger]. When the scattering rate exceeds the corresponding energy gap (dirty limit) the Eilenberger theory allows for significant simplification. In this case the quasiclassical Usadel equations [@Usadel] are applicable. The structure of singly-quantized vortices with $M=\pm
1$ in dirty superconductors was studied extensively in the framework of the Usadel equations [@KramerJLTP74] $$\omega F -\left[
G(\mathbf{\nabla}-i\mathbf{A})^{2}F-
F\mathbf{\nabla}^{2}G
\right]
=\Delta G \label{UsadelFG}$$ where $G$ and $F$ are normal and anomalous quasiclassical Green’s functions averaged over the Fermi surface satisfying the normalization condition $G^2+F^*F=1$. To facilitate the analysis, we introduce reduced variables: we use $T_{c}$ as a unit of energy and $\xi_{D}=\sqrt{\mathcal{D}/2 T_{c}}$ where $\mathcal{D}$ is a diffusion constant as a unit of length. The Usadel equation is to be supplemented with the self-consistency equation for the order parameter $$\label{Eq:SelfConsDirty}
\Delta ({\bf r})= 2\pi T\Lambda\sum_{n=0}^{N_d} F (\omega_n,{\bf r}).$$ We again neglect the influence of the magnetic field on the vortex structure. It is convenient to introduce the vector potential in Eq.(\[UsadelFG\]) corresponding to a pure gauge field which removes the phase of the order parameter $$\label{Eq:VPgauge}
{\bf A}=M \frac{{\bf z\times r}}{r^2}.$$ Using $\theta-$ parametrization [@KramerTeta] ($F=\sin\theta$, $G=\cos\theta$) the Usadel equation can be rewritten in the form $$\frac{1}{r}\frac{d}{dr} \left(r \frac{d}{dr} \theta\right)-\frac{M^{2}}{2r^2}
\sin (2\theta)+
\left(\Delta\cos\theta-\omega\sin\theta\right) =0. \label{Usadel-Reduced}$$ Performing the renormalization of summation by $\omega_n$ in self-consistency Eq.(\[Eq:SelfConsDirty\]) we need to solve Eq.(\[Usadel-Reduced\]) for a limited range of frequencies. We take $\omega_n\leq 10 T_c$ which allows to obtain very good accuracy. The nonlinear Eq.(\[Usadel-Reduced\]) was solved iteratively. At first we choose a reasonable initial guess and linearize the equation to find the correction. The corresponding boundary problem for non-homogeneous second-order linear equation was solved by the sweeping method and the procedure was repeated untill convergence was reached. With the help of obtained solutions of Eq.(\[Usadel-Reduced\]) we calculated the corrected order parameter (\[Eq:SelfConsDirty\]). We repeated the whole procedure to find the order parameter profile with an accuracy $10^{-4} T_c$.
Local density of states (LDOS) $N(E,r)$, which is accessible in tunneling experiments, can be obtained from $\theta(\omega,r)$ using analytic continuation $$N(E,r)=Re\left[ \cos\theta ( \omega\rightarrow -i
E+\delta,r)\right] \label{Eq:DOSdirty}$$ To calculate the LDOS we solve the Eq.(\[Usadel-Reduced\]) for $\omega=-iE$. In this case it is in fact a system of two coupled second order equations for the real and imaginary parts of $\theta$. We use the iteration method again by solving repeatedly the linearized system for the corrections of $\theta$. The corresponding boundary problems for second-order linearized equations for ${\rm Re} \theta$ and ${\rm Im} \theta$ were solved in turns by the sweeping method.
Order parameter structures of multiquantum vortices {#op}
===================================================
To determine the behavior of gap functions $\Delta=\Delta (r)$ in multiquantum vortices we solved numerically the sets of Eilenberger Eqs. (\[Eq:EilenbergerF\],\[Eq:SelfConsClean\]) and Usadel Eqs. (\[Eq:SelfConsDirty\],\[Usadel-Reduced\]) which describe the clean and dirty regimes correspondingly. At first let us consider the clean regime. The order parameter profiles in vortices with winding numbers $M=1,\;2,\;3,\;4$ are shown in Fig.\[Fig:CleanOP\](a,b,c,d) for the temperatures $T/T_c= 0.1;\;0.5;\;0.9$. One can see that at elevated temperatures $T=0.9 T_c$ (red dashed curves) and $T=0.5 T_c$ (green dash-dotted curve) the order parameter follows Ginzburg-Landau asymptotic $\Delta (r) \sim r^M$ at small $r$.
At low temperature $T=0.1 T_c$ the order parameter distribution inside vortex core is drastically different from the Ginzburg-Landau behaviour as shown by blue solid lines in Fig.(\[Fig:CleanOP\]). In particular the singly quantized vortex in Fig.\[Fig:CleanOP\](a) features the Kramer-Pesch effect [@KramerPesch] when the order parameter slope at $r=0$ grows as $d\Delta/dr \sim 1/T$ when $T\rightarrow 0$. In case of mutiquantum vortices with $M>1$ the gapless branches of electronic spectrum (\[Volovik-spectr\]) produce anomalies in the multiquantum vortex core structures[@VolovikAnomaly]. To observe the vortex core anomalies we plot in Fig.(\[Fig:VortexCoreAnomaly\]) the derivatives $d\Delta(r)/dr$ obtained self consistently for the vortex winding numbers $M=1,2,3,4$. In accordance with the analytical consideration[@VolovikAnomaly] the vortex core anomalies result in the singular behavior of $d\Delta(r)/dr$ at low temperatures. We find that at $T=0.1 T_c$ in multiquantum vortices with $M>1$ the calculated dependencies $d\Delta(r)/dr$ have sharp maxima at finite $r\neq 0$. According to the analytical predictions these maxima originate from the square root singularity of the order parameter which is produced by the contribution of the anomalous energy branches of electronic spectrum [@VolovikAnomaly].
In general for higher values of winding numbers $M>1$ in the limit $T\rightarrow 0$ one should have $M/2$ singularities of $d\Delta
(r)/dr$ for even $M$ and $(M+1)/2$ singularities for odd $M$. For the particular examples of $M=2,4$ there are one and two peaks of $d\Delta /dr$ at $T=0.1 T_c$ shown by blue solid line in Fig.\[Fig:VortexCoreAnomaly\] (b,d). We found that the order parameter of $M=3$ vortex has linear asymptotic $\Delta(r)\sim r$ at small $r$ shown in the Fig.\[Fig:CleanOP\](c). The slope of this linear dependence grows at decreasing temperature which analogously to the Kramer-Pesch effect in single-quantum vortex[@KramerPesch]. This behaviour is demonstrated by the dotted black line in Fig.\[Fig:CleanOP\]c corresponding to $T=0.05 T_c$. This effect is featured by all vortices with odd winding numbers originating from the anomalous energy branch crossing the Fermi level at $\mu=0$ in the Eq.(\[Volovik-spectr\]).
![\[Fig:CleanOP\] The distribution of the order parameter around vortex cores in clean superconductor at different temperatures. The panels (a,b,c,d) correspond to the winding numbers $M=1,2,3,4$. Blue solid, green dash-dotted and red dashed lines correspond to the temperature $T/T_c=0.1;\;0.5;\; 0.9$.](OpClean.eps){width="1.0\linewidth"}
![\[Fig:VortexCoreAnomaly\] The vortex core anomaly revealed at the sharp peak of radial dependence of the order parameter profile derivative $d\Delta (r)/dr$ normalized to the value $\xi_C/T_c$ around vortex cores in clean superconductor at different temperatures. The panels (a,b,c,d) correspond to the winding numbers $M=1,2,3,4$ correspondingly. Blue solid, green dash-dotted and red dashed lines correspond to the temperature $T/T_c=0.1;\;0.5;\; 0.9$. Dotted black line if panel (c) is for $T=0.05 T_c$; together with the blue solid curve in the panel (a) it demonstrates the peaked order parameter slope at the vortex center $d\Delta (r=0)/dr$ for odd winding numbers $M$. ](VortexCoreAnomaly.eps){width="1.0\linewidth"}
Next consider the case of dirty superconductor and calculate the core structures of multiquantum vortices. The results of calculation are shown in Fig.(\[Fig:DirtyOp\]) for the winding numbers $M=1,2,3,4$ and temperatures $T/T_c= 0.1;\;0.5;\;0.9$. As expected the vortices in dirty regime do not feature singularities in the order parameter distribution in contrast to the clean case considered above.
![\[Fig:DirtyOp\] The distribution of the order parameter around vortex core in dirty superconductor at different temperatures. The panels (a,b,c,d) correspond to the winding numbers $M=1,2,3,4$ correspondingly. Blue solid, green dash-dotted and red dashed lines correspond to the temperature $T/T_c=0.1;\;0.5;\; 0.9$. ](OpDirty.eps){width="1.0\linewidth"}
The comparison of vortex core structures in clean and dirty superconductors at $T/T_c=0.1$ is presented in Fig.\[Fig:CleanDirtyOp\] for the winding numbers $M=1,2,3,4$. To demonstrate the difference between clean and dirty cases we plot the dependencies $\Delta=\Delta(r)$ in logarithmic scale in Figs.\[Fig:CleanDirtyOp\](b,d) correspondingly. In the dirty case the order parameter has Ginzburg-Landau power law asymptotic $\Delta(r)= \alpha r^M$ which takes place at $r\rightarrow 0$ even at very low temperatures $T\ll T_c$. In Fig.\[Fig:CleanDirtyOp\](a,b) the low-temperature behavior $\Delta (r)$ in the clean case is drastically different from Ginzdurg-Landau regime. In particular the multiquantum vortex with $M=3$ shown by blue dash-dotted line in Fig.\[Fig:CleanDirtyOp\]a has linear asymptotic at $r=0$. The slope of linear asymptotic for $M=3$ should grow with decreasing temperature featuring an analog of Kramer-Pesch effect for multiquantum vortices. Furthermore the order parameter in $M=4$ vortex shown by red dashed line in Fig.\[Fig:CleanDirtyOp\]a is almost zero at finite region $r<R_c$ where $R_c\sim \xi_C/2$. This behavior is caused by the dominating contribution of the electronic states corresponding to anomalous branches (\[Volovik-spectr\]) to the superconducting order parameter at $r<R_c$. Thus contribution is zero at $r<min
(\mu_{01},\mu_{02})/k_F$ in the limit $T\rightarrow 0$ [@VolovikAnomaly]. Thus the multiquntum vortices with even winding numbers $M$ are well described by the step-wise vortex core model used previously for the analytical analysis of the vortex core spectrum [@Melnikov-Vinokur-2002].
![\[Fig:CleanDirtyOp\] The distribution of the order parameter around multiquantum vortex core at $T/T_c=0.1$ in (a) dirty superconductor and (b) clean superconductor. Black solid, green dotted, blue dash-dotted and red dashed lines correspond to the winding numbers $M=1,2,3,4$. ](CleanDirtyOp.eps){width="1.0\linewidth"}
LDOS profiles of multiquantum vortices {#ldos}
======================================
Having in hand the order parameter structures calculated self-consistently is Sec.(\[op\]) we calculate the LDOS distributions formed by the electronic states localized at the vortex core. We start with the case of clean superconductor which is known to demonstrate peculiar profiles of LDOS originating from multiple anomalous energy branches of localized electrons [@TanakaMultiquantum; @SalomaaMultiquantum]. Here we calculate the LDOS distributions for the winding numbers $M=1,2,3,4$ shown in Fig.(\[Fig:CleanLDOS\]). The LDOS plots are similar to that obtained in the framework of Bogolubov- de Gennes theory[@TanakaMultiquantum; @SalomaaMultiquantum].
Introducing a polar coordinate system $(r,\varphi)$ and defining the $z$ projection of quasiparticle angular momentum through the impact parameter of quasiclassical trajectory[@Silaev2008] $\mu=- [{\bf r},{\bf
k}_F]\cdot{\bf z}_0$ the LDOS inside the singly quantized vortex core can be found with the help of Eq.(\[Eq:SinglyQuantized\]) as follows: $N (E,r) =(k_F/2\pi\xi_C)\int_0^{2\pi}\delta[E-\omega
k_Fr\sin(\varphi-\theta_p)]d\theta_p$. Here we evaluate the LDOS summing up over the quasiparticle states at the trajectories characterized by the direction of the quasiparticle linear momentum ${\bf k_F}= k_F (\cos\theta_p,\sin\theta_p)$. This expression yields a singular behaviour of zero energy LDOS at $r>r_0$[@Ullah; @Maki; @IchiokaStar]: $N(E,r)=1/(2\pi\omega \sqrt{r^2-r_0^2}\xi_C)\approx
N_0 \xi_C/\sqrt{r^2-r_0^2}$, where $N_0=(1/2\pi)m/\hbar^2$ is a normal metal LDOS and $r_0=E/(\omega k_F )$. Thus the LDOS profile of singly quantized vortex has the ring form with the radius $r_0$ being a function of energy. The dependence $N=N (E,r)$ is shown in Fig.(\[Fig:CleanLDOS\])a for a singly quantized vortex.
In multiquantum vortices the spectrum of low energy states (\[Volovik-spectr\]) contains several anomalous branches which intersect the Fermi level and contribute to the low-energy DOS. The LDOS profile corresponding to the spectrum (\[Volovik-spectr\]) consists of a set of axially symmetric ring structures[@Melnikov-Vinokur-2002; @TanakaMultiquantum; @SalomaaMultiquantum]. Note that for an even winding number the anomalous branch crossing the Fermi level at $\mu=0$ (i.e. at zero impact parameter) is absent and, as a result, the LDOS peak at the vortex center disappears. Using the same procedure as for the singly quantized vortices and the spectrum (\[Volovik-spectr\]) we obtain the LDOS in the form $N(E,r)=\sum_{i=1}^M \vartheta(r-r_{0i})/(2\pi\omega_i \sqrt{r^2-r_{0i}^2}\xi_C)$ where $r_{0i}=[\mu_{0i}+E/\omega_i]/k_F$ and the step function $\vartheta(r)=0(1)$ at $r<(>)r_{0i}$. At $E=0$ the spectrum is symmetric so that the LDOS profile has $M/2$ peaks for even $M$ and $(M+1)/2$ for odd $M$. At $E\neq 0$ the degeneracy is removed and each peak splits by two as can be seen from the LDOS plots in Fig.(\[Fig:CleanLDOS\]).
![\[Fig:CleanLDOS\] The distribution of the LDOS around vortex cores at $T/T_c=0.1$ in clean superconductor as function of energy and distance from the vortex core $N=N(r,E)$. The panels (a,b,c,d) correspond to the values of vorticity $M=1,2,3,4$ correspondingly. ](LDOSclean2.eps){width="1.0\linewidth"}
Smearing of energy levels due to scattering effects leads to a reduction of LDOS peak at the vortex center. However, the LDOS peak survives even in “dirty” limit when a mean free path is smaller than a coherence length $l<\xi$. To find the form of LDOS peak at the vortex core we consider the dirty case described by Usadel Eq.(\[Usadel-Reduced\]). The LDOS distributions around multiquantum vortices calculated according to Eqs.(\[Usadel-Reduced\],\[Eq:DOSdirty\]) are shown in Fig.(\[Fig:DirtyLDOS\]). The profiles of LDOS at zero energy level $N=N(r)$ in multiquantum vortices $M>1$ feature plateau near the vortex center . This is in high contrast to the case of singly quantized vortex $M=1$. The cross sections $N=N(E)$ at different values of distance from the vortex center are shown in Fig.(\[Fig:DirtyLDOS(E)\]) for $T/T_c=0.1$ and the winding numbers $M=1,2,3,4$. These plots clearly demonstrate that with tunneling spectroscopy measurements it is hard to determine the center of the multiquantum vortex core for $M>2$. Indeed for $M=3$ the dependencies $N=N(E)$ for $r=0$ and $r=2\xi_D$ are very close to each other. For $M=4$ the same is true up to $r=3\xi_D$.
In fact the discussed LDOS plateau occur due to the very slow spatial dependence of $\delta N (r) = 1-N(r)/N_0$ at small $r$ which can deduced directly from Eqs.(\[Usadel-Reduced\],\[Eq:DOSdirty\]). Indeed linearizing the Eq.(\[Usadel-Reduced\]) for $\omega=0$ we obtain $$\label{Eq:asymptotic}
\left[\frac{1}{r}\frac{d}{dr} \left(r \frac{d}{dr} \theta\right) -\frac{M^2}{r^2} + \Delta (r)\right]\theta = 0$$ which define the asymptotic $\theta (r) = \alpha r^M$. Next the Eq. (\[Eq:DOSdirty\]) yields the LDOS deviation $\delta N = \theta^2/2 = \alpha^2 r^{2M}/2$. This analytical asymptotic perfectly agrees with the numerical results which can be seen from the logarithmic scale plot of $N(r)$ in Fig.(\[Fig:DirtyLDOS\]) b. An interesting feature of such LDOS plateau is that they survive at the distances compared to the size of the multiquantum vortex core which is much larger than the coherence length $\xi_D$. That is we find that the size of the plateau shown in Fig.(\[Fig:DirtyLDOS1D\]) is approximately given by $R_p= M\xi_D/2$ for $M>1$.
![\[Fig:DirtyLDOS\] The distributions of LDOS around vortex cores at $T/T_c=0.1$ in dirty superconductor as functions of energy and distance from the vortex core $N=N(r,E)$. The panels (a,b,c,d) correspond to the values of winding number $M=1,2,3,4$. ](LDOSdirtyNew.eps){width="1.0\linewidth"}
![\[Fig:DirtyLDOS(E)\] The cross sections $N=N(E)$ at different values of distance from the vortex center $r$ in dirty superconductor at $T/T_c=0.1$. The panels (a,b,c,d) correspond to the values of winding number $M=1,2,3,4$. Blue dotted, dash-dotted, dashed and solid lines correspond to the distances $r/\xi_D=2;3;5;10$. Thin solid red line indicates the flat LDOS at the vortex center $r=0$. ](LDOS1DE.eps){width="1.0\linewidth"}
![\[Fig:DirtyLDOS1D\] (a) The LDOS profiles for zero energy $E=0$ around vortices at $T/T_c=0.1$ in dirty superconductor as function of the distance from the vortex center $N=N(r)$. (b) The logarithmic plot of $\delta N (r)=1-N(r)/N_0$ demonstrating the power law asymptotic $\delta N (r)\sim r^{2M}$ at $r\rightarrow 0$. Black solid, green dotted, blue dash-dotted and red dashed lines correspond to the winding numbers $M=1,2,3,4$. ](LDOS1D.eps){width="1.0\linewidth"}
Conclusion {#summary}
==========
To summarize we have calculated self-consistently in the framework of quasiclassical Eilenberger theory the order parameter structures of multiquantum vortices together with the local density of electronic states both in clean and dirty superconductors. We have fond that at the temperatures near $T_c$ the order parameter profiles of vortices are qualitatively similar in clean and dirty regimes (compare the dependencies $\Delta (r)$ for $T=0.9 T_c$ shown by red dashed curves in Figs.\[Fig:CleanOP\] and \[Fig:DirtyOp\] for clean and dirty cases correspondingly). In this temperature regime the order parameter asymptotic at $r\rightarrow 0$ is determined by the power law $\Delta (r) =
\alpha r^M$ which is consistent with the result of Gizburg-Landau theory valid at $|T/T_c-1| \ll 1$.
On the other hand in the low temperature limit $T=0.1 T_c$ vortices in clean superconductor demonstrate the anomalies in the order parameter distribution - the singularities of the derivative $d\Delta/dr$ predicted in Ref.([@VolovikAnomaly]) and shown in Fig.(\[Fig:VortexCoreAnomaly\]). Such singularities occur due to the contribution of anomalous electronic spectral branches to the order parameter. The singular behavior of $d\Delta/dr$ in multiquantum vortices is analogous to the Kramer-Pesch effect[@KramerPesch] taking place for singly quantized vortex $M=1$ which has steep order parameter slope $d\Delta/dr (r=0) \sim 1/T $ at $T\rightarrow 0$. In dirty superconductors the asymptotic $\Delta(r\rightarrow 0)$ at the vortex core obeys the Ginzburg-Landau power law behavior even at low temperature $T=0.1 T_c$ which is clearly demonstrated in logarithmic scale plots in Fig.\[Fig:CleanDirtyOp\]d.
In the framework of quasiclassical theory we calculated the LDOS distributions in muliquantum vortices with winding numbers $M=1,2,3,4$. The LDOS profiles in the clean regime are similar to that obtained previously with the help of Bogolubov-de Gennes theory[@TanakaMultiquantum; @SalomaaMultiquantum]. Most importantly we determined the LDOS profiles in dirty regime which directly correspond to the modern experiments on scanning tunneling microscopy of multiquantum vortices in mesoscopic superconductors. The zero energy LDOS profile near the vortex center is shown to be $N(r)/N_0=1-\alpha r^{2M}$ which holds with good accuracy at $r<M \xi_D/2$. Thus for the values of $M>2$ the LDOS profile is almost flat at the sizable region near the vortex center $r<M
\xi_D/2$ (see Fig.\[Fig:DirtyLDOS1D\]). Such LDOS plateau can be considered as a hallmark of multiquantum vortex formation revealed by STM in dirty mesoscopic superconductors[@RoditchevPRL2011; @RoditchevPRL2009].
Acknowledgements
================
This work was supported, in part by Russian Foundation for Basic Research Grant N 13-02-01011 and Russian President Foundation (SP- 6811.2013.5). Discussion with Dr. Vasily Stolyarov is greatly acknowledged .
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---
abstract: 'It is shown how cosmological perturbation theory arises from a fully quantized perturbative theory of quantum gravity. Central for the derivation is a non-perturbative concept of gauge-invariant local observables by means of which perturbative invariant expressions of arbitrary order are generated. In particular, in the linearised theory, first order gauge-invariant observables familiar from cosmological perturbation theory are recovered. Explicit expressions of second order quantities are presented as well.'
author:
- 'Romeo Brunetti$^{1,a}$, Klaus Fredenhagen$^{2,b}$, Thomas-Paul Hack$^{3,c}$, Nicola Pinamonti$^{4,d}$ and Katarzyna Rejzner$^{5,e}$'
title: Cosmological perturbation theory and quantum gravity
---
$^1$Dipartimento di Matematica, Università di Trento,\
$^2$II Institute für Theoretische Physik, Universität Hamburg,\
$^3$Institute für Theoretische Physik, Universität Leipzig,\
$^4$Dipartimento di Matematica, Università di Genova, and INFN, Sezione di Genova,\
$^5$Department of Mathematics, University of York.
Email: $^A$brunetti@science.unitn.it, $^B$klaus.fredenhagen@desy.de, $^C$thomas-paul.hack@itp.uni-leipzig.de, $^D$pinamont@dima.unige.it, $^E$kasia.rejzner@york.ac.uk
Introduction
============
The fluctuations of the cosmological microwave background provide a deep insight into the early history of the universe. The most successful theoretical explanation is inflationary cosmology where a scalar field (the inflaton) is coupled to the gravitational field. Usually, the theory is considered in linear order around a highly symmetric background, typically the spatially flat Friedmann-Lema\^ itre-Robertson-Walker spacetime.
Extending the theory to higher orders is accompanied by severe obstacles. Already in a classical analysis the definition of gauge-invariant observables turns out to be rather complicated; moreover, one is immediately confronted with the problem of constructing a theory of quantum gravity. Previous treatments of higher-order cosmological perturbation theory include [@Bartolo:2006fj; @Bruni:1996im; @Langlois:2010vx; @Maldacena:2002vr; @Malik:2008im; @Nakamura:2004rm; @Noh:2004bc; @Hwang:2012aa]; many further references on the subject can be found e.g. in [@Langlois:2010vx].
In a recent paper [@BFR] three of us reanalysed the field theoretical construction of quantum gravity from the view point of locally covariant quantum field theory. This analysis was based on the methods of perturbative Algebraic Quantum Field Theory (pAQFT), see [@FR-Advances; @in; @AQFT] and references therein, and on an adapted version of the Batalin-Vilkovisky formalism for the treatment of local gauge symmetries [@Hollands; @FR-BVQ]. The result was that a consistent theory (in the sense of an expansion into a formal power series) exists and is independent of the background. Due to non-renormalisability, however, in each order of perturbation theory new dimensionful coupling constants occur, which have to be fixed by experiments; hence the theory should be interpreted as an effective theory that is valid at scales where these new constants are irrelevant. One might hope that non-perturbative effects improve the situation in the sense of Weinberg’s concept of asymptotic safety, since there are encouraging results supporting this perspective; see for example [@Reuter; @Reuter2]. Furthermore, it is difficult to observe any effects of quantum gravity, so it seems reasonable to start from the hypothesis that at presently accessible scales the influence of these higher order contributions is small.
One of the main questions addressed by [@BFR] in the construction of the theory was the existence of local observables. It was answered, in a way familiar from classical general relativity, by using physical scalar fields, e.g. curvature scalars, as coordinates, and by expressing other fields as functions of these coordinates. Since quantization in the framework of pAQFT relies on a field theoretical version of deformation quantization of classical theories (first introduced in [@DF]), the classical construction can be transferred to the quantum realm.
The procedure works as follows. One selects $4$ scalar fields $X_{\Gamma}^a, a=1,\ldots 4$, which are functionals of the field configuration $\Gamma$ which includes the spacetime metric $g$, the inflaton field $\phi$ and possibly other fields. The fields $X_{\Gamma}^a$ are supposed to transform under diffeomorphisms $\chi$ as $$\label{equivariance}
X_{\chi^*\Gamma}^a=X_{\Gamma}^a\circ\chi\ ,$$ where $\chi^*$ denotes the pullback (of sections of direct sums of tensor products of the cotangent bundle) via $\chi$. We choose a background $\Gamma_0$ such that the map $$X_{\Gamma_0}:x\mapsto (X_{\Gamma_0}^1,\ldots,X_{\Gamma_0}^4)$$ is injective. In order to achieve injectivity on cosmological backgrounds $\Gamma_0$, we shall be forced to include the coordinates $x$ in the construction of $X_\Gamma$ in a way which is compatible with . We then consider $\Gamma$ sufficiently near $\Gamma_0$ and set $$\alpha_{\Gamma}=X_{\Gamma}^{-1}\circ X_{\Gamma_0}\,.$$ We observe that $\alpha_{\Gamma}$ transforms under diffeomorphisms – which leave the background $\Gamma_0$, that is by definition fixed, invariant – as $$\alpha_{\chi^*\Gamma}=\chi^{-1}\circ\alpha_{\Gamma}\,.$$ Let now $A_{\Gamma}$ be any other scalar field which is a local functional of $\Gamma$ and transforms under diffeomorphisms as in . Then the field $$\mathcal {A}_{\Gamma}:=A_{\Gamma}\circ\alpha_{\Gamma}$$ is invariant under diffeomorphisms and may be considered as a local observable. Note that invariance is obtained by shifting the argument of the field in a way which depends on the configuration.
The physical interpretation of this construction is as follows: the fields $X^a_\Gamma$ are configuration-dependent coordinates such that $[A_{\Gamma}\circ X_{\Gamma}^{-1}](Y)$ corresponds to the value of the quantity $A_{\Gamma}$ provided that the quantity $X_{\Gamma}$ has the value $X_{\Gamma}=Y$. Thus $A_\Gamma \circ X_\Gamma^{-1}$ is a partial or relational observable [@Rovelli:2001bz; @Dittrich:2005kc; @Thiemann:2004wk], and by considering $\mathcal {A}_{\Gamma} = A_\Gamma \circ X_\Gamma^{-1} \circ X_{\Gamma_0}$ we can interpret this observable as a field on the background spacetime.
Clearly, to make things precise, one also has to characterise the region in the configuration space where all the maps are well defined and restrict oneself to configurations $\Gamma$ in the appropriate neighbourhood of the background $\Gamma_0$, see [@BFR; @Igor] for details.
Fortunately, in formal deformation quantization as well as in perturbation theory, only the Taylor expansion of observables around some background configuration enters, hence it is sufficient to establish the injectivity of $X_{\Gamma_0}$ in order for the expansion of $\mathcal{A}_{\Gamma_0+\delta\Gamma}$ around $\Gamma_0$ to be well-defined. As an example we compute this expansion up to the first order. We obtain $$\mathcal{A}_{\Gamma_0+\delta\Gamma}=A_{\Gamma_0}+\left\langle\frac{\delta A_{\Gamma}}{\delta \Gamma}(\Gamma_0),\delta\Gamma\right\rangle+\frac{\partial A_{\Gamma_0}}{\partial x^{\mu}}\left\langle\frac{\delta\alpha_{\Gamma}^{\mu}}{\delta\Gamma}(\Gamma_0),\delta\Gamma\right\rangle + O(\delta \Gamma^2)\ .$$ The third term on the right hand side is necessary in order to get gauge-invariant fields (up to first order). We calculate $$\frac{\delta\alpha_{\Gamma}^{\mu}}{\delta\Gamma}(\Gamma_0)=-\left(\left(\frac{\partial X_{\Gamma_0}}{\partial x}\right)^{-1}\right)^{\mu}_ a\frac{\delta X_{\Gamma}^a}{\delta\Gamma}(\Gamma_0)\ .$$
In this work we apply this general idea to inflationary cosmology. In contrast to other systematic or covariant attempts to define gauge-invariant quantities in higher-order cosmological perturbation theory, see for example [@Langlois:2010vx; @Malik:2008im; @Nakamura:2014kza; @Hwang:2012aa], our construction works off-shell, is based on a clear and simple concept which is applicable to general backgrounds such that cosmological perturbation theory may be viewed as a particular application of perturbative quantum gravity [@BFR]. Moreover, we construct non-perturbative gauge-invariant quantities whose perturbative expansion to arbitrary orders may be computed algorithmically without the need for additional input at each order.
This paper is organised as follows: In the second section we recall a few basic facts about perturbation theory of the Einstein-Klein-Gordon system on cosmological backgrounds. In the third section we describe the general method to obtain gauge invariant observables at all orders on generic backgrounds. We furthermore discuss how to treat the case of a FLRW background where the large symmetry prevents us from using coordinates constructed from the dynamical fields alone. The fourth section contains the analysis of two gauge invariant observables at second order. The steps necessary for the construction of a full all-order quantum theory are briefly sketched in Section 5. Finally a number of conclusions are drawn in the last section.
Perturbations of the Einstein-Klein-Gordon system on a FLRW spacetime {#sec:perturbationsintro}
=====================================================================
We consider the Einstein-Klein-Gordon system, namely a minimally coupled scalar field $\tilde\phi$ with potential $V(\tilde\phi)$ propagating on a Lorentzian spacetime $(M,\tilde g)$ with field equations $$\label{eq:EKG}
R_{ab}-\frac{1}{2}R \tilde g_{ab} = T_{ab} ,\qquad - \Box \tilde\phi + V^{(1)}(\tilde\phi) = 0,$$ where $T_{ab}$ is the stress tensor of $\tilde\phi$, $R_{ab}$ the Ricci tensor and $R$ the Ricci scalar. We discuss perturbations of this system around a background. A linearised theory is obtained starting from a one-parameter family of solutions $\lambda \mapsto \Gamma_\lambda := (\tilde g_{\lambda},\tilde\phi_{\lambda})$ and considering $$\delta \Gamma := (\gamma,\varphi) := \left. \frac{d}{d\lambda}(\tilde g_{\lambda},\tilde\phi_{\lambda}) \right|_{\lambda = 0},$$ hence $\Gamma_0:=( g,\phi):=(\tilde g_0,\tilde\phi_0)$ is the background configuration while $\delta \Gamma = (\gamma,\varphi)$ is the linearised perturbation.
The background solution we choose consists of a flat Friedmann-Lema\^ itre-Robertson-Walker (FLRW) spacetime $(M,g)$ together with a scalar field $\phi$ which is constant in space. We recall that a flat FLRW spacetime is conformally flat and that $$\label{eq:FRW}
M = I \times \mathbb{R}^3, \qquad g= a^2(\tau)(-d\tau\otimes d\tau+\sum_i d{x}^i\otimes d{x}^i ),$$ where $I\subset \mathbb{R}$ is an open interval, the scale factor $a(\tau)$ is a function of the conformal time $\tau$ and where $x^i$ are three-dimensional Cartesian (comoving) coordinates. The background equations of motion of the system are best displayed in terms of the auxiliary function $$\mathcal{H} := \frac{a'}{a} ,$$ where $a'$ indicates the derivative with respect to the conformal time. $\mathcal{H}$ is related to the Hubble parameter $H=\mathcal{H}a^{-1}$ and to the Ricci scalar $R = 6(\mathcal{H}'+ \mathcal{H}^2) a^{-2}$. The background equations of motion are $$\begin{gathered}
\mathcal{H}^2= (\phi')^2+2a^2 V(\phi), \qquad
2(\mathcal{H}'+ 2\mathcal{H}^2) = - (\phi')^2+2a^2 V(\phi),\\
\phi'' + 2\mathcal{H} \phi'+ a^2V^{(1)}(\phi) = 0.\end{gathered}$$
A generic perturbation $\gamma$ of the FLRW metric $g$ can be decomposed in the following way $$\label{eq:perturbation}
\gamma =
a(\tau)^2\begin{pmatrix}
-2A && (-\partial_i B + V_i)^t \\
-\partial_i B + V_i && 2(\partial_i\partial_j E+\delta_{ij}D+\partial_{(i}W_{j)}+T_{ij})
\end{pmatrix}$$ where $A,B,D,E$ are scalars, $V, W$ are three dimensional vectors and $T$ is a tensor on 3-dimensional Euclidean space. The decomposition is unique if all these perturbations vanish at infinity and if $${T_i}^i = 0 , \qquad \partial_i {T^i}_j = 0 , \qquad \partial_i {V^i} = 0 , \qquad \partial_i {W^i} = 0$$ (see e.g. Proposition 3.1 in [@Hack]).
Under an infinitesimal first order gauge transformation the linear perturbations transform in the following way $$\gamma_{ab} \mapsto \gamma_{ab} + \mathcal{L}_{\xi} g_{ab} = \gamma_{ab} + 2\nabla_{(a} \xi_{b)} , \qquad
\varphi \mapsto \varphi + \mathcal{L}_{\xi} \phi = \varphi + \xi(\phi).$$ In particular $$\begin{gathered}
A\mapsto A+(\partial_\tau+\mathcal{H})r, \qquad
B\mapsto B+r-s', \qquad
D\mapsto D+\mathcal{H}r, \qquad
E\mapsto E+s, \\
\varphi\mapsto \varphi+\phi'r, \qquad
V_i\mapsto V_i + v_i', \qquad
W_i\mapsto W_i + v_i, \qquad
T_{ij}\mapsto T_{ij}, \qquad \end{gathered}$$ where the generator $\xi$ of one-parameter gauge transformations is also decomposed as $$\label{eq:perturbation_diffeo}
\xi^0 = r,\qquad \xi^i = \partial_i s + v_i, \qquad \partial_i v^i = 0.$$ Notice that the gauge transformations do not mix scalar, vector or tensor perturbations at linear order.
Furthermore, we observe that tensor perturbations are gauge-invariant and that gauge-invariant vector perturbations can be obtained considering $X_i := W'_i-V_i$. Regarding the scalar perturbations we see that the following fields are gauge-invariant $$\label{eq:Bardeen}
\Phi:= A-(\partial_t+\mathcal{H})(B+E'), \qquad
\Psi:= D- \mathcal{H}(B+E'), \qquad \chi := \varphi - \phi' (B+E').$$ The first two of them are called Bardeen potentials.
Let us recall the form of the linearised equations of motions satisfied by the gauge-invariant perturbations. The first observation is that the equations of motion respect the decomposition in scalar, vector and tensor perturbations. In particular, for the vector and tensor perturbations, it holds that $$\label{eq:linear-equations}
\Delta X_i =0, \qquad (\partial_t + 2\mathcal{H})X_i =0, \qquad
\frac{1}{a^2} (\partial_t^2 +2 \mathcal{H} \partial_t - \Delta ) T_{ij} =0.$$ For the scalar part the equations of motion are better displayed in terms of the Mukhanov-Sasaki variable $$\label{eq:muk-sas}
\mu := \chi - \frac{\phi'}{\mathcal{H}} \Psi = \varphi - \frac{\phi'}{\mathcal{H}} D.$$ The equation of motion for this variable is decoupled also from the other scalars of the theory, in fact $$\left( -\Box +\frac{R}{6} - \frac{z''}{z a^2} \right) \mu = 0 , \qquad z := \frac{a \phi'}{\mathcal{H}}.$$ The other scalar perturbations can be obtained in terms of $\mu$. In particular the Bardeen potential $\Phi$ is the unique solution of $$\label{eq:onshellfirstorder}
\Delta \Phi = \frac{\phi^\prime}{2}\left(\mu^\prime + \left(\frac{\mathcal{H}^\prime}{\mathcal{H}}-\frac{\phi^{\prime\prime}}{\phi^\prime}\right)\mu\right)
$$ while the other scalar perturbations are given by $$\label{eq:onshellfirstorder2}
\Psi = -\Phi , \qquad
\chi = \frac{2}{\phi'} (\partial_\tau + \mathcal{H}) \Phi.$$
We briefly discuss the situation beyond linear order. According to [@Sonego:1997np], infinitesimal diffeomorphisms may be approximated by so-called knight diffeomorphisms, which are of the form $\exp \mathcal{L}_{\xi}$ with $\xi = \lambda \xi_1 + \frac12 \lambda^2 \xi_2 + O(\lambda^3)$. Analogously we may expand a configuration $\Gamma$ as $ \Gamma = \Gamma_0+\delta \Gamma=\Gamma_0 + \lambda \delta\Gamma_1 + \frac12 \lambda^2 \delta\Gamma_2 + O(\lambda^3)$, and determine the transformation behaviour of separate orders by considering $\exp \mathcal{L}_{\xi} \, \Gamma$ at fixed order in $\lambda$, see for example [@Bartolo:2006fj; @Bruni:1996im; @Malik:2008im; @Nakamura:2004rm; @Noh:2004bc]. Assuming that $\xi$ and $\delta \Gamma$ vanish at spatial infinity, each order $\xi_i$ and $\delta \Gamma_i$ may be uniquely decomposed as in and . The transformation behaviour of the components of the latter decomposition becomes more complicated than at linear order, since higher-order gauge transformations mix scalar, vector and tensor quantities in a non-local fashion, as do the higher-order equations of motion. We shall not be concerned with the explicit form of higher-order gauge transformations in this work, as our constructions do not rely on these details and the quantities we consider are manifestly all-order gauge-invariant from the outset.
For the remainder of this work we shall use the following notation motivated by the fact that the space of configurations is an affine space. We decompose a general configuration $\Gamma$ as $\Gamma := (\tilde g, \tilde \phi) :=\Gamma_0 + \delta \Gamma $, where $\tilde g := g + \gamma$, $\tilde \phi := \phi + \varphi$ and $\delta \Gamma := (\gamma,\varphi)$ effectively subsumes linear and higher orders of the perturbation of the background $\Gamma_0 := (g,\phi)$. This applies analogously to the components of the decomposition of $\gamma$.
For later use we recall a useful observation regarding Bardeen potentials. The linear Bardeen potentials $\Phi$, $\Psi$ and the gauge-invariant scalar field perturbation $\chi$ in have the advantage that they coincide with $A$, $D$, and $\varphi$ respectively in the so-called longitudinal or conformal gauge where the components $B$ and $E$ of the metric perturbation $\gamma$ vanish. This gauge and the definition of the gauge-invariant quantities $\Phi$, $\Psi$ and $\chi$ may be extended to higher orders, such that also at higher orders $\Phi=A$, $\Psi = D$, $\chi=\varphi$ if $B=E=0$, see for example [@Malik:2008im].
All-order gauge-invariant observables on FLRW backgrounds {#sec_covcoords}
=========================================================
In this section we provide details on the general construction of all-order gauge-invariant quantities on general and FLRW backgrounds before discussing examples in the next section.
In perturbative Algebraic Quantum Field Theory (pAQFT) – the conceptual framework underlying perturbative quantum gravity in [@BFR] – observables of a field theory are described as functionals of smooth field configurations $\Gamma=(\tilde g,\tilde\phi)$. For the purpose of cosmological perturbation theory, we need the additional restriction that configurations vanish at spatial infinity. In order to be able to operate on the functionals, some regularity is required: the functional derivatives to all orders should exist as distributions of compact support.
Moreover, we restrict our attention to local functionals, i.e. those functionals whose $n-$th order functional derivatives are supported on the diagonal of $M^n$ for every $n$. Examples of objects of this form are $$\label{eq:field}
A_{\Gamma}(f):= \int_M A_{\Gamma} f$$ where $A_{\Gamma}$ is a smooth scalar function which is a polynomial in the derivatives of the field configuration $\Gamma=(\tilde g,\tilde\phi)$ (i.e. $A_{\Gamma}(x)=F(j_x(\Gamma))$ with $F$ a smooth function on the appropriate jet bundle) and where $f$ is a smooth compactly supported test density. However, later on in this work we are forced to consider also functionals which violate this locality condition as well as the condition of compact support. The diffeomorphisms $\chi$ of the spacetime act on configurations via pullback $\Gamma\mapsto\chi^*\Gamma$, and candidates for gauge-invariant fields are equivariant in the sense that $$\label{eq_equivariant}
A_{\chi^*\Gamma}=A_{\Gamma}\circ\chi\ .$$ Thus in order to exhibit gauge-invariant functionals one has to consider test densities $f_{\Gamma}$ which depend on the field configuration $\Gamma$ such that $$f_{\chi^*\Gamma}=\chi_*f_{\Gamma}\,,$$ where $\chi_*$ is the pushforward of test densities via $\chi$.
As described in the Introduction, in the general case we solve the problem by choosing four scalar fields $X_{\Gamma}^a$ which constitute a coordinate system $X_{\Gamma}$ for a given background $\Gamma_0$, and define the $\Gamma$-dependent diffeomorphism $$\alpha_{\Gamma}=X_{\Gamma}^{-1}\circ X_{\Gamma_0}\ .$$ For arbitrary test densities $f$, we may now consider the $\Gamma$-dependent test densities $f_{\Gamma}=\alpha_{\Gamma}{}_*f$ in order to obtain gauge-invariant observables $A_\Gamma(f_\Gamma)$ by means of . Equivalently, we may directly consider the gauge-invariant field $$\label{eq:ref3}
\mathcal{A}_{\Gamma}=A_{\Gamma}\circ\alpha_{\Gamma}\ .$$
Scalars that can be used as coordinates on generic backgrounds $\Gamma_0$ are e.g. traces of powers of the Ricci operator ${\mathbf{R}}$ $$\label{eq:ricci-scalars}
X_\Gamma^a:=\text{Tr} ({\mathbf{R}}^{a}), \qquad a\in\{1,2,3,4\}$$ (the operator which maps one forms to one forms and whose components are given in terms of the Ricci tensor ${R_{a}}^b$). When other (matter) fields are present in the considered model, also these can serve as coordinates, e.g., in the case of a Einstein-Klein-Gordon system, the scalar field $\tilde \phi$.
In view of renormalisation it is advisable to use coordinates $X_\Gamma$ which are local functionals of the configuration $\Gamma$. As we shall discuss in the following, this does not seem to be possible in cosmological perturbation theory on account of the symmetries of FLRW backgrounds $\Gamma_0$.
Perturbative expansion up to second order {#sec:generalexpansion2}
-----------------------------------------
To illustrate the general procedure we compute the second order expansion of the gauge-invariant field $\mathcal{A}_\Gamma$ which was to first order described in the Introduction.
We observe that we have to calculate the functional derivatives of the diffeomorphisms ${\alpha_{\Gamma}}$ with respect to $\Gamma$. We use the notation $$\left\langle\frac{\delta^n}{\delta \Gamma^n}X_{\Gamma}(\Gamma_0),\delta\Gamma^{\otimes n}\right\rangle=:X_n\ ,\quad \left\langle\frac{\delta^n}{\delta\Gamma^n}\alpha_{\Gamma}(\Gamma_0),\delta\Gamma^{\otimes n}\right\rangle=:x_n$$ and find up to second order $$x_0^{\mu}(x)=x^{\mu}\,,\quad x_1^{\mu}=-J^{\mu}_aX^a_1\,,$$ where $J$ is the inverse of the Jacobian of $X_{\Gamma_0}$, and $$x_2^{\mu}=-J^{\mu}_aX_2^a-J^{\mu}_aJ^{\nu}_bJ^{\rho}_c\frac{\partial^2 X_0^a}{\partial x^{\nu}\partial x^{\rho}}X_1^bX_1^c+2J^{\mu}_aJ^{\nu}_b\frac{\partial X_1^a}{\partial x^\nu}X_1^{b}\ .$$ We use an analogous notation for the Taylor expansions of the fields $A_{\Gamma}$ and $\mathcal{A}_{\Gamma}$ and find $$\label{eq_gaugeinvexp1}
\mathcal{A}_0=A_0\,,\quad \mathcal{A}_1=A_1+\frac{\partial A_0}{\partial x^{\mu}}x_1^{\mu}\,,$$ and $$\label{eq_gaugeinvexp2}
\mathcal{A}_2=A_2+2\frac{\partial A_1}{\partial x^{\mu}}x_1^{\mu}+\frac{\partial A_0}{\partial x^{\mu}}x_2^{\mu}+\frac{\partial^2 A_0}{\partial x^{\mu}\partial x^{\nu}}x_1^{\mu}x_1^{\nu}\ .$$
Non-degenerate covariant coordinates on FLRW backgrounds {#sec:coordinates}
--------------------------------------------------------
In order to obtain these expansions we need a $4$-tuple of equivariant fields which define a non-degenerate coordinate system on the background $\Gamma_0$. This is possible in the generic case, e.g. by using the ansatz , but creates problems, if the background metric possesses non-trivial symmetries. This applies to the case of FLRW backgrounds $\Gamma_0$ where only time functions can be constructed out of the background metric $g$ and the background scalar field $\phi$. In the following we present a construction of non-degenerate coordinates which solves the above-mentioned problem at the expense of being non-local, albeit in a controlled way. Note that introducing additional external fields as reference coordinates like in the Brown-Kuchař model [@Brown:1994py] is not useful in the context of cosmological perturbation theory because these fields would appear in the final gauge-invariant expressions and thus an interpretation of these in terms of only the fundamental dynamical fields is difficult. The construction we present in the following does involve the comoving spatial coordinates $x^i$ of the FLRW spacetime as an external input. However the explicit dependence on $x^i$ disappears from the final expressions because these depend on $X_{\Gamma_0}$ only via its Jacobian.
The simplest choice of the time coordinate is provided by the inflaton field itself, so we set $$\label{eq_coord0}
X_\Gamma^0 = \tilde \phi = \phi + \varphi\,.$$ The construction of the spatial coordinates $X^i_\Gamma$ needs a bit of preparation. To this end, we consider the unit time-like vector $$\label{eq_nphi}
n_\phi = \frac{\tilde g^{-1}(d\tilde \phi,\cdot)}{\sqrt{|\tilde g^{-1}(d\tilde \phi,d\tilde \phi)}|} = \frac{1}{a}(1-A)\partial_\tau + \frac{1}{a}\left(\partial^i B -\frac{\partial^i \varphi}{\phi^\prime}\right) \partial_i + O(\delta \Gamma^2)$$ and the tensor $$\label{eq_hphi}
h_\phi = \tilde g + \tilde g(n_\phi,\cdot) \otimes \tilde g(n_\phi,\cdot)\,,$$ where $\partial^i := \partial_i := \partial/\partial x^i$ and $x^i$ for $i\in\{1,2,3\}$ are comoving spatial coordinates on the FLRW spacetime $(M,g)$. $n_\phi$ is a unit normal on the hypersurfaces of constant $\tilde \phi$ and $h_\phi$ is the induced metric on these hypersurfaces.
Let $\Delta_\phi$ denote the Laplacian for $h_\phi$ and $G_\phi$ its inverse, which we choose by imposing the boundary condition that the background value of $G_\phi$ is the Coulomb potential $G_\Delta$ with suitable factors of the scale factor $a$. We define and compute $$\Delta_\phi := \Delta_0 + \delta\Delta\,,\qquad \Delta_0 := \frac{\Delta}{a^2}\,,\qquad \Delta := \sum^3_{i=1} \partial^2_i$$ $$\delta\Delta = -\lambda\left( \frac{ 2( D + \Delta E)\Delta
-(\partial^i ( D - \Delta E))\partial_i}{a^2}+\frac{(\Delta \varphi ) \partial_\tau+(\partial^i \varphi)(2\partial_\tau + {{\mathcal H}}) \partial_i}{a^2 \phi^\prime}\right)+O(\delta \Gamma^2)$$ $$G_\phi := G_0 + \delta G\,,\qquad G_0 := a^2 G_\Delta\,, \qquad G_\Delta \circ \Delta = \1\quad \text{on functions that vanish at spatial infinity}\,,$$ $$\delta G = \sum^\infty_{n=1} (-1)^n G_0 \circ (\delta \Delta \circ G_0)^{\circ n} = - G_0 \circ \delta \Delta \circ G_0 + O(\delta\Gamma^2)\,.$$ Using these objects, we obtain $$\label{eq:Ycoords}
Y_\Gamma^i := \left(1-G_\phi \circ \Delta_\phi \right)x^i = x^i + \partial_i ( E+G_\Delta\mathfrak{R})+O(\delta \Gamma ^2)\,,\qquad \mathfrak{R} := \frac{{{\mathcal H}}}{\phi^\prime}\mu\,.$$ We observe that $Y_\Gamma^i$ are harmonic coordinates for $\Delta_\phi $ that we have constructed by means of $x^i$, i.e. harmonic coordinates for $\Delta_0$. The construction of $Y_\Gamma^i$ makes sense for all configurations $\Gamma$ which vanish at spatial infinity, but not in general. The restriction to this set of configurations from the outset is natural in the context of cosmological perturbation theory – recall that the decomposition is unique only in this case – and does not create problems for the pAQFT framework. For consistency, we have to restrict the class of infinitesimal diffeomorphisms we consider in the same manner. In fact, a straightforward computation reveals that the functionals $Y_\Gamma^i$ are equivariant with respect to all diffeomorphisms $\chi$ that vanish at spatial infinity $$\chi^* Y_\Gamma^i = Y_{\chi^*\Gamma}^i + (1-G_{\chi^* \phi}\circ \Delta_{\chi^* \phi})(\chi^* x^i - x^i ) = Y_{\chi^*\Gamma}^i\,,$$ but not with respect to arbitrary diffeomorphisms. Here $\Delta_{\chi^* \phi}$ denotes the Laplacian constructed analogous to $\Delta_{\phi}$ but with $\chi^* \tilde \phi$ instead of $\tilde \phi$ and $G_{\chi^* \phi}$ denotes its inverse with the discussed boundary condition. Consequently, the observables constructed by means of the equivariant coordinates and via are gauge-invariant with respect to diffeomorphisms which vanish at spatial infinity. As anticipated, the coordinates $Y_\Gamma^i$ are non-local, but the non-locality of $G_\phi $ is relatively harmless since its wave front set is that of the $\delta$-function, and renormalisation of expressions involving such objects is well under control, cf. Section \[sec:quantization\].
The coordinates are not entirely well-suited for practical computations because of the fact that the rescaled Mukhanov-Sasaki variable $\mathfrak{R}$ appears convoluted with the Coulomb potential. In order to remedy this we use a different family of spatial hypersurfaces and a corresponding modification of the spatial Laplacian and its inverse. To this end we consider a number of additional quantities related to the slicing induced by the time-function $\tilde \phi$: the lapse function $N_\phi$, the extrinsic curvature $K_{\phi,ab}$, and the spatial Ricci scalar $R^{(3)}_\phi $ which are defined and computed respectively as $$\begin{gathered}
\label{eq_lapsenongi}
N_\phi:= |\tilde{g}^{-1}_\lambda(d\tilde\phi,d\tilde \phi)|^{-1/2} = \frac{a}{\phi^\prime}\left(1 - \frac{\varphi'}{\phi'} + A\right) + O(\delta\Gamma^2)\,,\\
K_{\phi,ab} := {h_{\phi,a}}^c\nabla_{c} n_{\phi,b}\,,\qquad K_\phi := {K_{\phi,a}}^a = \frac{3{{\mathcal H}}}{a} + O(\delta \Gamma)\,,\label{eq_traceK}\\
R^{(3)}_\phi := K_{\phi,ab}K_\phi ^{ba}-K_\phi ^2+2\left(R_{ab}-\frac12 R \tilde g_{ab}\right)n_\phi^an_\phi^b = \frac{4}{a^2} \Delta {{\mathfrak R}}+ O(\delta\Gamma^2)\,, \label{eq_spatialCurv}\end{gathered}$$ where $n_\phi$ and $h_\phi$ are defined respectively in and . Using these quantities, we define a new time function $$\mathfrak{t}:=\tilde \phi - \frac{3 N_\phi}{4 K_\phi } G_\phi R^{(3)}_\phi = \phi + \frac{\phi^\prime}{{{\mathcal H}}} D+O(\delta\Gamma ^2)\,,$$ If we define the spatial metric $h_\mathfrak{t}$, the Laplacian $\Delta_\mathfrak{t}$ and its inverse $G_\mathfrak{t}$ in analogy to $h_\phi$, $\Delta_\phi $ and $G_\phi $ by replacing $\tilde \phi$ with $\mathfrak{t}$ we obtain $$\label{eq:Xcoords}
X_\Gamma^i := \left(1-G_\mathfrak{t} \circ \Delta_\mathfrak{t}\right)x^i = x^i + \partial_i E+O(\delta \Gamma ^2)\,,$$ and the spatial coordinates $X_\Gamma^i$ share the qualitative properties of the initially defined $Y_\Gamma^i$.
Examples of gauge-invariant observables at second order {#sec_obs}
=======================================================
In the previous sections we have developed a principle to construct gauge-invariant perturbative observables from non-gauge-invariant ones. In the following we demonstrate this principle at the example of two observables which are relevant in Cosmology. To this end we use the covariant coordinates and .
Despite the mild non-locality inherent in the covariant spatial coordinates $\eqref{eq:Xcoords}$, we are interested in observables $A_\Gamma$ which are local functionals of the configuration $\Gamma$. The non-locality of $\mathcal{A}_\Gamma = A_\Gamma \circ \alpha_\Gamma$ implied by the non-locality of $X^i_\Gamma$ in appears only because we consider the local functional $A_\Gamma$ relative to the non-local functional $X_\Gamma$. Since the background $\Gamma_0$ depends only on time the same applies to the background value of any local functional $A_\Gamma$. Consequently, at first order only the field $X^0_\Gamma$ chosen as time coordinate enters the formula for gauge-invariant fields. At second order also the fields used as spatial coordinates $X^i_\Gamma$ enter the expression.
The inverse $J$ of the Jacobi matrix of the coordinate transform $X_{\Gamma_0}$ on the background is $$J=\left(\begin{array}{cccc}
\frac{1}{\phi'}&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&1
\end{array}\right).$$ The field dependent shifts from Section \[sec:generalexpansion2\] with respect to these coordinates up to second order are $$x_1^0=-\frac{\varphi}{\phi^{\prime}}\,,\qquad \ x_1^i=-\partial_i E\,,$$ and $$x_2^{0}=-\frac{\phi''{\varphi}^2}{(\phi')^3}+\frac{2}{\phi'}\left(\frac{\varphi'\varphi}{\phi'}+(\partial_i \varphi)\partial^i E\right),$$ $$x_2^{i}=\frac{2\varphi}{\phi'}\partial_i E^\prime +2(\partial^i \partial^j E)\partial_j E - (X^i_\Gamma - x^i - \partial_i E)\,.$$ Thus, for a field $A_\Gamma$ whose value on the background depends only on time the contributions up to second order for the gauge-invariant modification $\mathcal{A}_\Gamma = A_\Gamma \circ \alpha_\Gamma$ are $$\mathcal{A}_0=A_0\,,\qquad \mathcal{A}_1=A_1-\frac{A_0'\varphi}{\phi'}\,,$$ $$\mathcal{A}_2=A_2-\frac{2A_1'\varphi}{\phi'}-2(\partial_iA_1)\partial^i E+A_0'\left(-\frac{\phi''{\varphi}^2}{(\phi')^3}+\frac{2}{\phi'}\left(\frac{\varphi'\varphi}{\phi'}+(\partial_i \varphi)\partial^i E\right)\right)+\frac{A_0''\varphi^2}{(\phi')^2}\,.$$ If we were to use the fields $Y^i_\Gamma$ as spatial coordinates rather than the fields $X^i_\Gamma$ , then the corresponding expression for $\mathcal{A}_1$ would remain unchanged whereas $\mathcal{A}_2$ would change by replacing all occurrences of $\partial_i E$ by $\partial_i E + G_\Delta \partial_i \mathfrak{R}$. This demonstrates the dependence of the gauge-invariant constructions on the chosen covariant coordinate system.
The lapse function {#sec_obs_lapse}
------------------
The Sachs-Wolfe effect is one of the main building blocks of the current understanding of the Cosmic Microwave Background (CMB). A rough estimate of this effect can be obtained using the Tolman idea, see e.g. [@Mukhanov:2003xr]. Given a spacetime with a (conformal) timelike Killing field $\kappa$ and a state in equilibrium relative to the $\kappa$-flow with absolute temperature $T$, an observer with four-velocity $u\propto \kappa$ measures the temperature $\widetilde T=T/N$ with $N$ denoting the lapse function $N=\sqrt{|g(\kappa,\kappa)|}$.
In the context of Cosmology we use the Klein-Gordon field $\tilde\phi$ as a time coordinate and consider the vector $$\kappa_\phi := N_\phi n_\phi = \frac{1}{\phi'}\partial_\tau + O(\delta \Gamma)$$ with $N_\phi$, $n_\phi$ defined in and respectively as an approximate conformal Killing vector – in the sense that $\mathcal{L}_{\kappa_\phi} \tilde g - 2 {{\mathcal H}}/\phi' \tilde g = O(\phi'', \delta \Gamma)$. The corresponding lapse function is $N_\phi = a/\phi^\prime + O(\delta \Gamma)$. Its background value is not vanishing and thus it is not automatically gauge-invariant at linear order.
As described in Section \[sec\_covcoords\], we may obtain a non-perturbatively gauge-invariant version of the lapse function by setting and computing $$\begin{aligned}
{{\mathcal N}}_\phi:= & N_\phi \circ \alpha_\Gamma = \frac{a}{\phi^\prime}\left(1-\left((\partial_\tau + \mathcal{H})\frac{\varphi}{\phi^\prime} - A\right)\right)+O(\delta\Gamma^2)\label{eq_lapsegi}\\
=&\frac{a}{\phi^\prime}\left(1-\left((\partial_\tau + \mathcal{H})\frac{\chi}{\phi^\prime} - \Phi\right)\right)+O(\delta\Gamma^2)\,,\notag\end{aligned}$$ where $\Phi$ and $\chi$ are the gauge-invariant fields reviewed in Section \[sec:perturbationsintro\]. Using the on-shell identities , and the definition of the Mukhanov-Sasaki field $\mu$ we can rewrite the linear term as $$\begin{aligned}
{{\mathcal N}}_{\phi,1}=& \lambda\frac{a}{\phi^\prime}\left((\partial_\tau + \mathcal{H})\frac{\chi}{\phi^\prime} - \Phi\right)=\frac{a}{(\phi^\prime)^2}\left(\mu^\prime + \left(\frac{\mathcal{H}^\prime}{\mathcal{H}}-\frac{\phi^{\prime\prime}}{\phi^\prime}\right)\mu\right)\\
=&\frac{2a}{(\phi^\prime)^3} \Delta \Phi = -\frac{2a}{(\phi^\prime)^3} \Delta \Psi\,.\end{aligned}$$ Using the quantities introduces in Section \[sec:coordinates\], we may extract the Bardeen potential on-shell from $N_{\phi}$ as $$\left[\frac{1}{2 N_{\phi}^3} G^2_\phi \Delta_\phi N_{\phi}\right]\circ \alpha_\Gamma = \Phi + O(\delta \Gamma^2)\,.$$ In fact, one could use the above equation as a covariant, gauge-invariant, all-order (and on shell) definition of $\Phi$; however, we shall refrain from doing so.
In order to display second order expressions in a readable form we omit terms containing the metric perturbation components $V_i$, $W_j$ and $T_{ij}$ and use once more the Bardeen potentials $\Phi$, $\Psi$ and the gauge-invariant scalar field perturbation $\chi$. We stress that the particular expressions of these fields at linear and higher order are not needed for the actual computations but just for a compact display of the result. Using this, we arrive at the following second order form of the gauge-invariant lapse function $$\begin{aligned}
{{\mathcal N}}_{\phi,2} &= \frac{a}{\phi'}\left(-\Phi^2 -2 \left(\frac{\Phi \chi}{\phi'}\right)' -2 \mathcal{H}\frac{\Phi \chi}{\phi'} +2 \left(\left(\frac{\chi}{\phi'}\right)'\right)^2+\left( \frac{\phi''}{\phi'}+2\mathcal{H} \right)\left(\frac{\chi^2}{\phi'^2}\right)'
+
\right.
\\
&\qquad \left.+\left(\mathcal{H}^2+\mathcal{H}'+\frac{\phi'''}{\phi'}+\mathcal{H}\frac{\phi''}{\phi'}- \frac{\phi''^2}{\phi'^2}\right)\frac{\chi^2}{\phi'^2} + \sum^3_{i=1}\left(\partial_i\left(\frac{\chi}{\phi'}\right)\right)^2
+2\frac{\chi}{\phi'}\left(\frac{\chi}{\phi'}\right)''
\right),\end{aligned}$$ where, as before, we use the notation that e.g. $\Phi = \lambda \Phi_1 + \frac12 \lambda^2 \Phi_2+O(\lambda^3)$ and omit the second order terms linear in $\Phi$, $\chi$ displayed already in .
The spatial curvature
---------------------
A further observable of interest is the scalar curvature of the spatial metric induced by a particular slicing because for a large class of slicings this quantity vanishes in the background and thus is automatically gauge-invariant at linear order. Moreover, for the slicing defined by the inflation field it is related to the Mukhanov-Sasaki field $\mu$ which has a very simple dynamical equation.
We have already discussed the spatial curvature relative to the slicing induced by $\tilde \phi$. It may be computed as $$R^{(3)}_\phi = \frac{4}{a^2} \Delta {{\mathfrak R}}+ O(\delta\Gamma^2)\,,\qquad \mathfrak{R} = \frac{{{\mathcal H}}}{\phi^\prime}\mu =\frac{{{\mathcal H}}}{\phi^\prime}\varphi - D \,.$$ In the literature, the quantity ${{\mathfrak R}}$ is usually called the *comoving curvature perturbation*. This is due to the fact that the $\tilde \phi$-slicing may be equivalently characterised by the condition that $$T(\tilde \phi)_{ab} n_\phi^a = -\tilde g_{ab} n_\phi^a T(\tilde \phi)_{cd} n_\phi^c n_\phi^d\,,$$ i.e. that the energy flux of $\tilde \phi$ is parallel to $n_\phi$, where $T(\tilde \phi)_{ab}$ is the stress tensor of $\tilde \phi$.
An alternative slicing considered in the literature is the one defined by the energy density $\tilde\rho$ of $\tilde \phi$ $$\tilde\rho := T(\tilde \phi)_{ab} n_\phi^a n_\phi^b = \rho + \varrho\,,$$ $$\rho := \frac{(\phi')^2}{2 a^2}\,,\qquad \varrho := V^{(1)}(\phi)\varphi + \frac{\phi'(\varphi' - \phi' A)}{a^2}+O(\delta \Gamma^2)\,.$$ The spatial curvature $R^{(3)}_\rho$ with respect to this slicing, defined in analogy to $R^{(3)}_\phi$, reads $$R^{(3)}_\rho = \frac{4}{a^2}\Delta \zeta + O(\delta \Gamma)\,,\qquad \zeta := \frac{{{\mathcal H}}}{\rho'}\varrho-D \,,$$ where $\zeta$ is called *uniform density perturbation* because $\tilde \rho$ is by definition constant on the hypersurfaces in the slicing relative to $\tilde \rho$. The global sign in the definition of $\zeta$ is conventional.
As anticipated, the background contributions of $R^{(3)}_\phi$ and $R^{(3)}_\rho$ vanish and thus $${{\mathcal R}}^{(3)}_\phi := R^{(3)}_\phi \circ \alpha_\Gamma = R^{(3)}_\phi + O(\delta\Gamma^2)\,,\qquad {{\mathcal R}}^{(3)}_\rho := R^{(3)}_\rho \circ \alpha_\Gamma = R^{(3)}_\rho + O(\delta\Gamma^2)\,,$$ cf. , . In order to display the second order contribution to ${{\mathcal R}}^{(3)}_\phi$, we make the simplifications discussed for the lapse function in Section \[sec\_obs\_lapse\]. Proceeding like this, we find $$\begin{aligned}
\label{eq_spatialgi}
{{\mathcal R}}^{(3)}_{\phi,2}=&\frac{8}{a^2}\left(\Delta\left(2{{\mathfrak R}}^2 - \frac{\chi}{\phi'}(\partial_\tau + 2{{\mathcal H}}){{\mathfrak R}}+ \frac12\left({{\mathcal H}}'+2{{\mathcal H}}^2-\frac{{{\mathcal H}}\phi''}{\phi'}\right)\left(\frac{\chi}{\phi'}\right)^2\right)\right.\\
&\qquad\qquad \left.- \frac{5(\partial_i {{\mathfrak R}}) \partial^i {{\mathfrak R}}}{2}\right).\notag\end{aligned}$$ We omit the result for ${{\mathcal R}}^{(3)}_{\rho,2}$ computed with the coordinate system $X_\Gamma$ defined in and , because it is rather long due to the “mismatch” between the time coordinate $\tilde \phi$ used in $X^0_\Gamma$ and the time coordinate $\tilde \rho$ used in the definition of $R^{(3)}_\rho$. Clearly, using $\tilde \rho$ as a time coordinate in both aspects we would obtain a second order expression ${{\mathcal R}}^{(3)}_{\rho,2}$ which is of the form up to the replacements $$\label{eq_replacements}
{{\mathfrak R}}\mapsto \zeta\,,\qquad \phi\mapsto \rho\,,\qquad \chi\mapsto \pi:= V^{(1)}(\phi)\chi + \frac{\phi'(\chi'-\phi'\Phi)}{a^2}\,,$$ where $\pi$ is gauge-invariant with $\pi = \varrho + O(\delta \Gamma^2)$ in the longitudinal gauge.
On shell and at first order, $\mu$, and thus ${{\mathfrak R}}$, are preferred observables because they have canonical equal-time Poisson brackets and thus in the quantized theory they commute at spacelike separations, in contrast to $\Psi$, $\Phi$ and $\chi$ [@Eltzner; @Hack]. Moreover, again on shell and at first order, one may compute $$\zeta = {{\mathfrak R}}-\frac{2 \Delta \Phi}{3 (\phi')^2} = {{\mathfrak R}}- \frac{{{\mathfrak R}}'}{3 {{\mathcal H}}}\,.$$ Consequently, $\zeta$ shares the causality properties of $\mu$ and ${{\mathfrak R}}$.
Apart from the phenomenological relevance of an all-order definition of ${{\mathfrak R}}$, $\mu$ and $\zeta$, it is interesting on conceptual grounds to investigate whether the causality property of these fields persists at higher orders. To this end, we need a fully covariant and gauge-invariant all-order definition of ${{\mathfrak R}}$, $\mu$ and $\zeta$. Such a definition may be given by means of covariant quantities introduced in Section \[sec:coordinates\]: $$\begin{aligned}
\label{eq_higherorderR}
\left[\frac{1}{4} G_\phi R^{(3)}_\phi\right]\circ \alpha_\Gamma &= \frac{a^2}{4} G_\Delta \mathcal{R}^{(3)}_\phi- a^2 G_\Delta \delta\Delta {{\mathfrak R}}+ O(\delta\Gamma^3)\notag\\
&={{\mathcal H}}\frac{\chi}{\phi'}-\Psi + {{\mathfrak R}}^2 - 2 {{\mathcal H}}\frac{\chi}{\phi'}{{\mathfrak R}}+ \frac12\left({{\mathcal H}}'+2{{\mathcal H}}^2-\frac{{{\mathcal H}}\phi''}{\phi'}\right)\left(\frac{\chi}{\phi'}\right)^2 + \\
\notag&\qquad + G_\Delta\left(\frac{(\partial_i {{\mathfrak R}}) \partial^i {{\mathfrak R}}}{2}\right) + O(\delta\Gamma^3)\,,\end{aligned}$$ $$\label{eq_higherordermu}
\left[\frac{3 N_\phi}{4 K_\phi } G_\phi R^{(3)}_\phi\right]\circ \alpha_\Gamma = \mu + O(\delta\Gamma^2)\,,\qquad \left[\frac{1}{4} G_\phi R^{(3)}_\rho\right]\circ \alpha_\Gamma = \zeta + O(\delta\Gamma^2)\,.$$ In we wrote the $O(\delta \Gamma)$ term as ${{\mathcal H}}\chi/\phi' - \Psi$ instead of ${{\mathfrak R}}$ because the fields $\chi$, $\Psi$ are defined in such a way that they are invariant also with respect to second order gauge transformations (cf. the end of Section \[sec:perturbationsintro\]), whereas ${{\mathfrak R}}={{\mathcal H}}\varphi/\phi' - D$ is only gauge-invariant up to the first order.
In analogy to our discussion of $R^{(3)}_\rho$, using $\tilde \rho$ rather than $\tilde \phi$ both as the time coordinate $X^0_\Gamma$ and as the time function defining a foliation of spacetime, we obtain a higher order definition of $\zeta$ which is of the form up to the replacements in (whereby a second order generalisation of $\pi$, which can be constructed in analogy to the second order Bardeen potentials, is needed).
In the literature, several possible second order gauge-invariant corrections to ${{\mathfrak R}}$ are considered. One often encounters constructions where in a gauge with $\varphi=0$ (or $D=0$), the second order corrections to ${{\mathfrak R}}$ vanish – at least in situations where spatial derivatives can be neglected in comparison to temporal ones, see e.g. [@Maldacena:2002vr; @Malik:2008im; @Prokopec:2012ug; @Vernizzi:2004nc]. In fact ${{\mathfrak R}}$ is often defined by the condition ${{\mathfrak R}}= -D$ in a gauge where $\varphi=0$. A quick analysis reveals that this is not the case in our construction . In [@Vernizzi:2004nc] it is argued that expressions for ${{\mathfrak R}}$ valid up to second order that are not of this form, e.g. the one in [@Acquaviva:2002ud], are potentially physically ill-behaved because they are not conserved on “super-Hubble scales”. Here, conservation of a function $f(\tau,\vec x)$ on “super-Hubble scales” means that the Fourier transform $\hat f(\tau,\vec k)$ of $f$ with respect to $\vec x$ satisfies $\partial_\tau \hat f(\tau,\vec k) = O(|\vec k|/{{\mathcal H}})$. This property, whose relevance is explained e.g. in [@Maldacena:2002vr; @Vernizzi:2004nc], usually holds only on-shell. It would be interesting to check whether our result for ${{\mathfrak R}}$ as given in (and the analogous result for $\zeta$) is conserved in this sense; however, this is beyond the scope of the present work.
Quantization {#sec:quantization}
============
In the previous sections we have prepared the ground for an all-order perturbative quantization of the Einstein-Klein-Gordon system on FLRW backgrounds, i.e. for a conceptually clear higher-order generalisation of quantized cosmological perturbation theory. In this section we would like to sketch the steps necessary for a full construction of the quantum theory. A detailed account will be given in a future work [@longpaper].
BRST quantization
-----------------
It is known that a direct quantization of non-linear gauge-invariant observables in a theory with local gauge symmetries is difficult. The standard way out is to perform a gauge fixing in the sense of the BRST method, or more generally, the BV formalism, as treated in [@Hollands; @Fredenhagen:2011an; @FR-BVQ]. There one adds a Fermionic vector field $c^{\mu}$ (the ghost field), which describes the infinitesimal gauge transformations, auxiliary scalar fields $b_{\mu}, \bar{c}_{\mu}$, where $b_{\mu}$ (the Nakanishi-Lautrup field) is Bosonic and $\bar{c}_{\mu}$ (antighost) is Fermionic, $\mu=0,\ldots,3$. Infinitesimal coordinate transformations are described by the BRST operator $s$, which acts on scalar local functionals $A$ of the metric, the inflaton and the $b$ fields by $$s(A)(x)=c^{\mu}(x)\partial_{\mu}A(x)\ ,$$ on the components of the ghost field by $$s(c^{\mu})(x)=c^{\nu}(x)\partial_{\nu}c^{\mu}(x)\ ,$$ on antighosts by $$s(\bar{c}_{\mu})(x)=ib_{\mu}(x)-c^{\nu}(x)\partial_{\nu}\bar{c}_{\mu}(x)$$ and satisfies on products the graded Leibniz rule so that $s^2=0$. One can characterise the classical observables as functionals in the kernel of $s$ modulo those in the image of $s$ (i.e. classical observables belong to the $0$-th cohomology group of $s$).
The field equations for the extended system are the usual field equation for $\tilde\phi$ as well as $$R_{\mu\nu}=T(\tilde\phi)_{\mu\nu}-\frac12 T(\tilde\phi)\tilde g_{\mu\nu}+s(i\partial_{(\mu}\bar{c}_{\nu)})$$ $$\square_{\tilde g} c^{\mu}=0$$ $$\square_{\tilde g} \bar{c}_{\mu}=0$$ $$|\mathrm{det}\tilde g|^{-\frac12}\partial_{\mu}|\mathrm{det}\tilde g|^{\frac12}\tilde g^{\mu\nu}=\kappa^{\mu\nu}b_{\mu}\,.$$ Here $\kappa$ is a non-degenerate fixed tensor.
The quantization of the extended system now proceeds largely analogous to the pure gravity treatment in [@BFR]. The main idea is to use deformation quantization to deform the algebra of functionals as well as the BRST operator $s$. Elements of the cohomology of the quantized (i.e. deformed) BRST operator $s$ are then interpreted as quantized versions of the gauge-invariant fields discussed in the previous sections.
Renormalisation
---------------
A conceptual and technical difference to the pure gravity case treated in [@BFR] arises because of the fact that we have introduced a mild non-locality via the non-local spatial coordinates $X^i_\Gamma$ . In [@BFR] renormalisation was treated in the Epstein-Glaser framework which is initially only suitable for local functionals. As we have to deal with non-local expressions, we need to extend this framework from local quantities to non-local ones. Recall that $$X_{\Gamma}^i=(1-G_\phi \Delta_\phi )x^i = \sum_{k=0}^\infty (-G_0\,\delta \Delta)^k x^i\,,$$ where $\Delta_\phi =\Delta_0+\delta \Delta $ is the Laplacian relative to the $\tilde \phi$-slicing and $G_\phi =\sum_{k=0}^\infty (-G_0\,\delta \Delta)^kG_0$ is its Green’s function for suitable boundary conditions, cf. Section \[sec:coordinates\].
Our gauge-invariant observables can be expanded as Taylor series in $X_\Gamma^a$, so in order to discuss the renormalisation of non-local contributions it is sufficient to discuss the kind of singularities that arise from considering the time-ordered products involving $X_\Gamma^i$. The general strategy is similar to the standard setting. We start with non-renormalised expressions where the $n$-fold time-ordered product involving $X_\Gamma^i$ and local functionals $F_1$,…, $F_{n-1}$ is given by $$\mathcal{T}_n(X_\Gamma^i,F_1,\dots,F_{n-1}):= m\circ e^{\hbar\sum_{0\leq k<l\leq n-1}D_{\mathrm{F}}^{kl}}(X_\Gamma^i\otimes F_1\otimes\dots\otimes F_{n-1})\,.$$ where $m$ denotes pointwise multiplication and $D_{\mathrm{F}}^{kl}\doteq \langle\Delta_{S_0}^{\mathrm{F}},\frac{\delta^2}{\delta\Gamma_k\delta\Gamma_l}\rangle$ with $\Delta_{S_0}^{\mathrm{F}}$ denoting the Feynman propagator of the full linearised theory. For simplicity, we suppress all indices. This expression is then expanded into graphs. The non-locality is expressed by the fact that our graphs have now two kinds of vertices and two kinds of propagators. Namely, there are the “usual” Feynman propagators of the theory (for simplicity all denoted by
(-0.07,0) rectangle +(1.2,-0.02); (0,0.1) – (1,0.1);
), but also the “internal” propagators $G_0$ corresponding to lines
(-0.07,0) rectangle +(1.2,-0.05); (0,0.13) – (1,0.13); (0,0.08) – (1,0.08);
.
As for the vertices, there are the external vertices
(-0.07,0) rectangle +(0.1,-0.1); circle (1.5pt);
arising from local functionals $F_{1},\dots,F_{n-1}$ and from the vertex corresponding to the explicit spacetime dependence of $X^i_\Gamma$, but also the internal vertices
(-0.07,0) rectangle +(0.1,-0.1); circle (1.5pt);
obtained from the $\delta \Delta$ operators. An example contribution would be
(36.57,47.12) rectangle +(66.49,25.85); (39.40,50.07) circle (0.84mm); (59.92,49.95) circle (0.84mm); (80.07,50.20) circle (0.84mm); (100.22,50.07) circle (0.84mm); (40.24,50.28) – (59.14,50.23); (39.94,49.62) – (59.18,49.63); (60.77,50.31) – (79.30,50.31); (60.87,49.69) – (79.36,49.71); (80.88,50.28) – (99.45,50.24); (80.58,49.62) – (99.52,49.64); (39.73,70.23) circle (0.74mm); (39.63,69.62) – (39.63,50.74); (39.83,70.23) – (59.33,50.64); (39.83,70.15) .. controls (47.90,71.01) and (60.00,57.62) .. (59.83,50.93) .. controls (59.83,50.89) and (59.37,50.64) .. (59.40,50.67); (70.01,70.00) circle (0.74mm); (70.01,69.78) – (60.44,50.58); (69.90,69.89) – (79.76,51.04);
To see that such graphs can be renormalised, consider the simplest divergent case, namely
(36.57,47.12) rectangle +(66.49,25.85); (39.40,50.07) circle (0.84mm); (59.92,49.95) circle (0.84mm); (80.07,50.20) circle (0.84mm); (40.24,50.28) – (59.14,50.23); (39.94,49.62) – (59.18,49.63); (60.77,50.31) – (79.30,50.31); (60.87,49.69) – (79.36,49.71); (39.73,70.23) circle (0.74mm); (39.63,69.62) – (39.63,50.74); (39.83,70.23) – (59.33,50.64);
The kernel of $G_0$ considered as a distribution on $M^2$ is of the form $$G_0(x,y) = c(\tau_x) \delta(\tau_x,\tau_y) \frac{1}{|\vec {x}-\vec{y}|}$$ with a smooth function $c$. The wave front set of $G_0(x,y)$ is the one of $\delta(x,y)$ and its scaling degree is 2. The vertex operators $\delta \Delta$ are differential operators of at most second order. By direct inspection we thus see that the only singularity of the loop in the above example is at the total diagonal and by power counting we find that the degree of divergence of this loop is at most 2, so that the appropriately renormalised expression is unique up to at most two derivatives of $\delta$ distributions of the three loop vertices. In general the degree of divergence of a loop containing “internal” propagators may be higher or lower than in the above example depending on the number of Feynman propagators appearing in the loop; the same applies to the renormalisation freedom of general loops.
These arguments indicate that the new types of graphs do not create new problems in the UV regime. We briefly sketch why we do not expect additional IR problems. We have already pointed out that our setup is only meaningful if we restrict the admissible classical configurations to those which vanish at spatial infinity. By consistency we need the same behaviour for the correlation functions of the quantized theory, in particular for the Feynman propagators of the linearised model. Provided quantum states (or more general Hadamard parametrices) with this property exist – this is not obvious and needs to be proven – we expect that the integrals corresponding to the “internal” vertices will converge.
The remaining problem is to deal with the combinatorics of such graphs and ensure that the renormalisation can be performed systematically order by order. This can be done by a slight generalisation of the standard framework and will be discussed in detail in our forthcoming paper [@longpaper]. In the same publication we will also prove the validity of Ward identities analogous to the ones proven by Hollands for the Yang-Mills theory [@Hollands].
Conclusions
===========
We described how cosmological perturbation theory may be derived from a full theory of perturbative quantum gravity. This demonstrates that perturbative quantum gravity can already be tested by present observations. Moreover, on a more practical side, our definition of gauge-invariant observables provides a conceptually simple way of extending the observables which are relevant for the interpretation of cosmological observations to arbitrary high orders.
However, even in linear order, our discussion clarifies the choice of good observables, as we have indicated at the example of the lapse function $N_\phi$ with respect to the spatial hypersurfaces of constant inflaton field. Initially $N_\phi$ is not gauge-invariant, but our construction yields a gauge-invariant version which at linear order and on shell may be expressed in terms of the Bardeen potential $\Phi$ that is related to the temperature fluctuations of the CMB via the Sachs-Wolfe effect.
We computed examples of gauge-invariant observables beyond linear order and found a second-order expression for the comoving curvature perturbation which seems to differ from constructions in other works. As in the literature there is some debate about whether some constructions are physically well-behaved, see. e.g. [@Vernizzi:2004nc], it would be interesting to investigate the physical properties of our result, even though it is clear from the outset that it has a transparent geometric interpretation.
Finally we have sketched the details of the quantization of the Einstein-Klein-Gordon system on cosmological backgrounds beyond linear order. We believe that the strategy outlined here leads to a full renormalised all-order theory of cosmological perturbations by means of which higher order corrections to standard results in cosmology may be computed.
[**Acknowledgements**]{}\
K.F., N.P. and K.R. would like to thank the Erwin Schrödinger Institute in Vienna, where part of the research reported here was carried out, for the kind hospitality.
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---
abstract: 'X-ray spectroscopy of Seyfert 1 galaxies often reveal absorption edges resulting from photoionized gas along the line-of-sight to the central engine, the so-called warm absorber. I discuss how recent [*ASCA*]{} observations of warm absorber variability in MCG$-$6-30-15 can lead us to reject a one-zone model and, instead, have suggested a multi-zone warm absorber. The evidence for dust within the warm absorbers of MCG$-$6-30-15 and IRAS 13349+2438 is also addressed. These dusty warm absorbers reveal themselves by significantly reddening the optical flux without heavily absorbing the soft X-ray photons. Thermal emission from this warm/hot dust may be responsible for the infra-red bump commonly seen in the broad band spectrum of many Seyfert galaxies.'
author:
- 'Christopher S. Reynolds'
title: Warm absorbers in Seyfert 1 galaxies
---
Introduction
============
X-ray observations of Seyfert 1 nuclei have provided direct evidence for significant quantities of optically-thin photoionized gas along our line-of-sight to the central X-ray source. The first indications of such material, which has become known as the [*warm absorber*]{}, were found with low resolution X-ray spectroscopy of the luminous Seyfert nucleus MR 2251-178 using [*EXOSAT*]{} (Halpern 1984). The next significant advancement came several years later when [*ROSAT*]{} PSPC observations of the nearby Seyfert 1 galaxy MCG$-$6-30-15 revealed a broad absorption feature at $\sim 0.8$keV (Nandra & Pounds 1992). This was interpreted as being a blend of the photoelectric K-shell edges of [Ovii]{} and [Oviii]{} (at 0.74keV and 0.87keV, respectively) which are imprinted upon the primary X-ray spectrum when the line-of-sight to the central engine passes through the warm absorber.
The CCDs on board the US/Japanese X-ray satellite [*ASCA*]{} provide an order of magnitude improvement in spectral resolution over the [*ROSAT*]{} PSPC and allow us to study warm absorbers in detail. [*ASCA*]{} observations can separate and independently measure the properties of the [Ovii]{}/[Oviii]{} edges. Detailed scrutiny of the [*ASCA*]{} spectra of bright Seyfert 1’s can also reveal the presence of other edges and lines, although oxygen edges usually dominate. [*ASCA*]{} has shown us that approximately half of all Seyfert 1 nuclei display a warm absorber (Reynolds 1997).
In this contribution, I shall describe two studies that address the physical nature of the warm absorber. Due to the fact that it has been most extensively observed by [*ASCA*]{}, I will focus on results for the Seyfert 1 galaxy MCG$-$6-30-15 ($z=0.008$).
Warm absorber variability
=========================
During Performance Verification observations, [*ASCA*]{} found large variations in the depths of the oxygen edges in MCG$-$6-30-15 (Fabian et al. 1994; Reynolds et al. 1995). The timescales of this variability were found to be as short as $\sim 10\,000$s. However, the physical nature of this variability was unclear. Within the context of a one-zone photoionization model (characterized by the column density $N_{\rm W}$ and the ionization parameter $\xi$), a drop in the primary X-ray flux was accompanied by an increase in both the column density and ionization parameter. It is difficult to imagine a situation in which the total absorber column density is correlated to the intrinsic X-ray flux on such short timescales (Reynolds et al. 1995).
A long [*ASCA*]{} observation of MCG$-$6-30-15 resolved this confusion. Figure 1 shows the intrinsic X-ray luminosity and the warm oxygen edge depths as a function of time for this observation. It can be seen that the [Ovii]{} edge depth remains essentially constant whereas the [Oviii]{} edge depth is strongly anti-correlated with the primary X-ray flux (Otani et al. 1996). The fact that the warm absorber variability is directly related to the primary X-ray variability is direct evidence that the warm absorber is photoionized. However, the constancy of the [Ovii]{} edge is inconsistent with a one-zone absorber model. To see this, we suppose that the warm absorber can be described as a single uniform slab and note that the recombination and photoionization timescales of this material must be short in order to see any variability of the edge depths. The relative depths of the oxygen edges fixes the ionization parameter of this slab to be $\xi\sim 20\,{\rm erg}\,{\rm cm}\,{\rm s}^{-1}$. However, for this value of $\xi$, photoionization models clearly show that the [Ovii]{} edge depth should decrease, and the [Oviii]{} edge depth should increase, as the primary ionizing flux increases. This is contrary to observations.
The simplest extension of the one-zone model, a two-zone model, can nicely account for the observed behaviour (Otani et al. 1996). In this model, much of the [Ovii]{} edge results from a distant, tenuous warm absorber which has a long recombination timescale and so is not seen to react to changes in the ionizing flux. Spectral fitting shows this absorber to have a column density of $N_{\rm W}\approx 5\times 10^{21}\,{\rm cm}^{-2}$, an ionization parameter of $\xi\approx17$, and to be at a distance of $R>1$pc from the central engine. By contrast, much of the [Oviii]{} edge originates from an inner warm absorber which is highly ionized and has a short recombination timescale. A drop in the ionizing flux leads to recombination of fully-stripped oxygen into [Oviii]{} ions and thus an increase in the [Oviii]{} edge depth. This region is constrained to have $N_{\rm W}\approx 1\times 10^{22}\,{\rm cm}^{-2}$, $\xi\approx 70$ and $R<10^{17}$cm. It is tempting to identify this component with an optically-thin component of the broad line region.
It is important to account for the two-zone nature of this absorber when discussing the thermal stability of the warm material. The thermal stability of photoionized plasma to isobaric perturbations can be studied by examining the thermal equilibrium curve on the ($\xi/T$, $T$) plane (McCray 1979; Krolik, McKee and Tarter 1981). Parts of this curve which have negative slope and are associated with a multi-valued regime correspond to thermally unstable equilibria. A one-zone model for the warm absorber implies that it is either thermally unstable or exists in extremely small pockets of stable parameter space (Reynolds & Fabian 1995). However, given a two-zone parameterization, the situation is rather different, as shown in Fig. 2. We find that the outer warm absorber can be understood as material at the extreme (hot) end of the stable cold branch, whereas the inner warm absorber is material at the extreme (cold) end of the stable intermediate/hot branch. Whilst the exact form of this curve is sensitive to the ionizing continuum shape assumed, it is clear that the two-zone nature of the absorber must be accounted for in any discussion of thermal stability.
Optical/UV extinction and dusty warm absorbers
==============================================
Whilst reporting the results of the first optical spectroscopy of MCG$-$6-30-15, Pineda et al. (1980) noted that the X-ray spectrum does [*not*]{} display the neutral absorption expected on the basis of its optical reddening. This issue has recently been investigated in more detail by Reynolds et al. (1997). Both the optical emission line spectrum and the continuum form suggest that this Seyfert nucleus is reddened by $E(B-V)\sim
0.6$ or more. Assuming a standard LMC extinction law and using a Galactic dust/gas ratio implies that a gas column of at least $N_{\rm H}\sim 3\times
10^{21}\,{\rm cm}^{-2}$ should accompany this reddening. However, X-ray spectroscopy with both [*ASCA*]{} and [*ROSAT*]{} can set stringent limits on the cold (i.e., neutral) gas column of $N_{\rm H}<2\times 10^{20}\,{\rm
cm}^{-2}$.
A similar situation has been found for the quasar IRAS 13349+2438 (Brandt et al. 1996). In this object, spectropolarimetry reveals a heavily reddened direct continuum and an unreddened scattered component (Wills et al. 1992). X-ray observations reveal a variable X-ray source (indicating that we are viewing the X-ray source directly, rather than through scattered flux) with a warm absorber but no detectable neutral absorption. The X-ray upper limits on the neutral column are an order of magnitude lower than that expected on the basis of the reddening (Brandt et al. 1996, 1997).
Since the X-ray flux is thought to originate from deeper within the central engine than the optical flux, this discrepancy is difficult to understand as a purely geometric effect. A dust/gas ratio which is 10 times the Galactic value would reconcile these studies, but is difficult to understand physically (we might expect dust grains to be preferentially destroyed near an AGN, but not preferentially created). The most reasonable resolution of this problem is that the dust responsible for the optical reddening resides in the warm absorber. In both MCG$-$6-30-15 and IRAS 13349+2438, the inferred dust/warm-gas ratio is similar to the Galactic dust/cold-gas ratio. In other words, the warm absorber appears to represent Galactic-like dusty gas that has been ionized with little destruction of the dust grains.
Under the conditions envisaged, the dust grains are thermally decoupled from the relatively tenuous gas. They come into a thermal equilibrium such that the thermal radiation emitted by each grain balances the incident AGN flux on that grain. Grains can survive provided two conditions are satisfied. First, the grains must be sufficiently far from the central engine so that they do not sublime. In the case of MCG$-$6-30-15, this translates into a distance limit of $R>10^{17}$cm and suggests that it is the outer warm absorber, as opposed to the inner warm absorber, which is dusty. Secondly, the gas temperature cannot exceed $10^6$K or else the grains would be destroyed via thermal sputtering. This is easily satisfied if the warm absorber is photoionized rather than collisionally ionized. This dust may be responsible for a significant part of the infra-red bump seen in the broad-band spectrum of many Seyfert nuclei.
Conclusions
===========
Two studies relevant to the physical nature of the warm absorber have been described. To summarize the results of these investigations:
1. The temporal variability of the warm absorber in MCG$-$6-30-15 as seen by [*ASCA*]{} suggests that it is comprised of at least two zones. The inner zone may be related to the broad line region whereas the outer zone is at radii characteristic of the putative molecular torus and the narrow line region.
2. The absence of neutral absorption, which is naively expected to accompany the optical reddening in both MCG$-$6-30-15 and IRAS 13349+2438, suggests that the warm absorber may be dusty. In MCG$-$6-30-15, this dust must lie in the outer warm absorber or else it would be sublimated by the intense AGN radiation field.
I am indebted to Andy Fabian, Chiko Otani and Martin Ward with whom much of this work was performed in collaboration. This work was supported by a PPARC (UK) studentship and the National Science Foundation under grant AST-9529175.
[N. Arav]{} How well can you constrain the properties of the two-zone model for explaining the [Ovii]{}/[Oviii]{} edges?
[C. Reynolds]{} MCG$-$6-30-15 shows a fairly clean pattern of variability – the [Ovii]{} edge remains essentially constant whilst only the [Oviii]{} edge varies. In this case we can decouple the two zones quite easily. In a more general case (i.e., both oxygen edges varying) it is rather more difficult. This may be the case in some other Seyfert galaxies (e.g., NGC 3227) in which a complex pattern of warm absorber variability is seen.
[C. Foltz]{} What is known about the ultraviolet spectrum of MCG$-$6-30-15? Is Mg[ii]{} absorption seen?
[C. Reynolds]{} Not much is known. [*IUE*]{} shows it to be a rather faint UV source and I’m not aware of a [*HST*]{} UV spectrum of this object. To my knowledge, the only UV feature that has definitely been detected is the [Civ]{}$\lambda 1549$ line (even the continuum is barely detected). This is all consistent with it being heavily reddened.
[E. Agol]{} What happens to the Fe K$\alpha$ line as the warm absorber varies? Is there any evidence for iron edge absorption?
[C. Reynolds]{} As the X-ray flux is seen to enter a low state, the warm absorber is seen to change in the manner I described, and the iron line is seen to get broader and stronger. However, the iron line is thought to originate from the innermost regions of the accretion flow and so is probably not directly related to any changes in the warm absorber. [*ASCA*]{} does not detect any iron edge, but the constraints aren’t very strong due to the limited sensitivity of [*ASCA*]{} at energies above the iron line.
Brandt W.N., Fabian A.C., Pounds K.A., 1996, MNRAS, 278, 326 Brandt W.N., Mathur S., Reynolds C.S., Elvis M., 1997, submitted Fabian A.C. et al. 1994, PASJ, 46, L59 Halpern J.P., 1984, ApJ, 281, 90 Krolik J.H., McKee C.F., Tarter C.B., 1981, ApJ, 249, 422 McCray R.A., 1979, in Hazard C.R., Mitton S., eds, Active Galactic Nuclei. Cambridge Univ. Press, Cambridge, p.227 Nandra K., Pounds K.A., 1992, Nat, 359, 215 Otani C. et al., 1996, PASJ, 48, 211 Pineda F.J., Delvaille J.P., Grindlay J.E., Schnopper H.W., 1980, 237, 414 Reynolds C.S., 1997, MNRAS, in press Reynolds C.S., Fabian A.C., 1995, MNRAS, 273, 1167 Reynolds C.S., Fabian A. C. Makishima K., Fukazawa Y., Tamura T., 1994, MNRAS, 268, 55 Reynolds C.S., Ward M.J., Fabian A.C., Celotti A., 1997, submitted. Wills B. J., Wills D., Evans N. J., Natta A., Thompson K. L., Breger M., Sitko M. L., 1992, ApJ, 400, 96
|
---
abstract: 'A possible solution to the observed baryon asymmetry in the universe is described, based on the physics of the standard model of electroweak interactions. At temperatures high enough electroweak physics provides violation of baryon number, while $C$ and $CP$ symmetries are not exactly conserved, although in the context of the minimal electroweak model with one Higgs doublet the rate of $CP$ violation is not sufficient enough to generate the observed asymmetry. The condition that the universe must be out of thermal equilibrium requires the electroweak phase transition (EWPT) to be first order. The dynamics of the phase transition in the minimal model is investigated through the effective potential, which is calculated at the one loop order. Finite temperature effects on the effective potential are treated numerically and within the high temperature approximation, which is found to be in good agreement with the exact calculation. At the one loop level the phase transition was found to be of the first order, while the strength of the transition depends on the unknown parameters of the theory which are the Higgs boson and top quark masses [@top] .'
author:
- Nicholas Petropoulos
date: '29 April 2003[^1]'
title: Baryogenesis at the electroweak phase transition
---
*“Why does the whole world have $<\phi>=+v$? Why doesn’t it have $<\phi>=-v$ somewhere? Suppose that God created the universe in the state $<\phi>=0$ and then the universe discovered that it could lower its energy; Where it puts its energy is none of my business, but it gets rid of it – gives it back to God or something;”*
R. Feynman [@Feynman:1976]
[^1]:
|
---
abstract: 'Density matrix embedding theory (Phys. Rev. Lett. [**109**]{}, 186404 (2012)) and density embedding theory (Phys. Rev. [**B 89**]{}, 035140 (2014)) have recently been introduced for model lattice Hamiltonians and molecular systems. In the present work, the formalism is extended to the [*ab initio*]{} description of infinite systems. An appropriate definition of the impurity Hamiltonian for such systems is presented and demonstrated in cases of 1, 2 and 3 dimensions, using coupled cluster theory as the impurity solver. Additionally, we discuss the challenges related to disentanglement of fragment and bath states. The current approach yields results comparable to coupled cluster calculations of infinite systems even when using a single unit cell as the fragment. The theory is formulated in the basis of Wannier functions but it does not require separate localization of unoccupied bands. The embedding scheme presented here is a promising way of employing highly accurate electronic structure methods for extended systems at a fraction of their original computational cost.'
author:
- 'Ireneusz W. Bulik'
- Weibing Chen
- 'Gustavo E. Scuseria'
title: Electron correlation in solids via density embedding theory
---
Introduction
============
Electron correlation plays a crucial role in understanding most physical phenomena in molecules and extended systems. While highly accurate many-body approaches can be nowadays routinely employed in molecular systems, solid state applications remains dominated by density functional theory (DFT). Despite the undeniable success of DFT in extended systems, [@ParrYang; @JCP.2014.140; @JCP.2012.136] there are significant limitations appearing from the approximate form of the exchange-correlation functional [@Science.298.2002] and much work remains to be done in order to ensure systematically improvable predictions. [@Science.321.2008; @PRL.100.136406.2008; @PRB.83.035119.2011; @JP.CM.2013.43.435503; @PRB.87.035117.2013; @JCP.2004.121.1187; @JCP.2005.123.174101; @PRB.83.205128; @JP.CM.2012.14.145504; @PRB.87.035107] An alternative route for incorporating electron correlations is to employ wavefunction-based methods. Recently, significant progress in applying such many-body theories for solid state problems has been made. [@JCP.2008.129.204104; @PhysRevB.80.085118; @JCP.132.2010.151101; @Nature.2013.493.365; @JCTC.2014.10.1698]
Size-extensive, wavefunction-based approaches to solids treat the system as a whole, imposing translational symmetry and Brillouin zone integration. Finite order perturbation theory [@CP.178.1993; @PRB.50.14791; @PRB.51.16553; @JCP.1996.104.8553; @JCP.106.1997.5554; @JCP.109.1998; @JCP.2001.115.9698; @JCP.130.2009.184103; @JCP.133.2010.0741067] and coupled cluster (CC) methods [@JCP.106.1997.10248; @TCA.104.2000.426; @JCP.120.2004.2581; @RodsBook] have been formulated and implemented for infinite systems. Alternatively, the numerical complexity associated with numerous electronic degrees of freedom in solids has been simplified, for example, by means of the method of increments. [@PRB.46.6700.1992; @CPL.1992.548; @JCP.1992.97.8449; @CP.1997.224.121; @PhysRep.2006.428.1]
In the present work, electron correlation in extended systems is accounted for via an embedding approach. The infinite periodic problem is transformed into one of a small fragment (unit cell) entangled with an effective bath. In order to define the bath, we employ the recently introduced density embedding theory (DET), [@PhysRevB.89.035140] which is a simplification of density matrix embedding theory (DMET). [@PhysRevLett.109.186404; @JCTC.9.1428] Here, an approximate solution to the infinite periodic system, typically Hartree-Fock, is used to construct two basis sets. The first basis is associated with a small part of the lattice (fragment) whereas the second is used to describe the excluded complement (bath). Subsequently, one solves the many-body problem for the fragment plus bath, a so-called impurity problem. In this way, correlations between the fragment and the rest of the system are represented by a many-electron environment, not a single-particle potential. [@JCTC.9.1428] The Hartree-Fock (HF) choice for an approximate solution of the infinite solid affords the desired bases in a trivial algebraic manner via diagonalization. Moreover, the resulting Hamiltonian describing the fragment-bath interaction is defined in a much smaller single-particle Hilbert space as compared to the full lattice Hamiltonian. Thus, this construction opens the possibility of employing highly accurate many-body techniques to tackle extended systems.
Indeed, DMET and DET with exact diagonalization[@PhysRevLett.109.186404; @JCTC.9.1428; @Booth2013; @PhysRevB.89.035140] and density matrix renormalization group[@PRB.89.165134] as impurity solvers have been shown to provide high-quality descriptions of model Hamiltonians and molecular systems. In the present work we extend the applicability of this novel approximation to realistic extended systems. Such problems seem a promising niche for this embedding scheme. The definition of fragment is naturally dictated by the periodicity of the system which greatly simplifies the procedure. We propose a way of defining the local Hamiltonian that guarantees exactness of mean-field in mean-field embedding and facilitates coping with the Coulomb problem in solids. Additionally, we discuss practical issues related to preparing the fragment and bath basis for an infinite number of electrons and the problem of disentangled states.
While in the current work we benchmark the DET scheme by describing the fragment-bath interaction at the coupled cluster level of theory, we stress that the methodology is flexible enough to accommodate any correlated wavefunction method. Indeed, the purpose of DET is to provide a finite, small dimension effective Hamiltonian which should account for locally important degrees of freedom. Then, any many-body technique may be employed to solve the impurity problem at hand. In particular, very accurate and computationally affordable schemes [@JCP.2013.139; @PRB.87.235129] may be used to study extended systems with realistic unit cell sizes.
Theory and formalism
====================
In order to keep this paper self-contained, we present in this section a basic introduction to density matrix and density embedding theories. More details can be found in the original papers. [@PhysRevLett.109.186404; @JCTC.9.1428; @Booth2013; @PhysRevB.89.035140; @PRB.89.165134] Subsequently, we discuss the changes required to apply the formalism to the [*ab initio* ]{} Hamiltonian of extended systems. This includes the Schmidt decomposition of Slater determinants for infinite systems and the formulation of an impurity Hamiltonian that allows us to deal with the Coulomb divergence in solids. Unless otherwise specified, single-particle indices denote both spin and spatial coordinates. In the context of periodic systems, cell coordinates are implicitly included except when confusion may arise.
Mean-field based embedding theories
-----------------------------------
Density matrix embedding theory and its simplification, density embedding theory, are projections of the exact Hamiltonian onto a basis obtained by Schmidt decomposition of the ground state wavefunction $|\Psi\rangle$. To be precise, one may cast $|\Psi\rangle$ into [@PhysRevLett.109.186404] $$\begin{aligned}
|\Psi\rangle = \sum_i \lambda_i |\alpha_i\rangle |\beta_i\rangle ,\end{aligned}$$ where $|\alpha\rangle$ represents the part of the system of interest, the fragment, whereas $|\beta\rangle$ represents the rest of the system, the bath. With such states at hand, an impurity Hamiltonian is defined, $$\begin{aligned}
\hat{H}_{\textrm{imp}} = \sum_{ijkl}
|\alpha_i\rangle |\beta_j\rangle \langle \alpha_i | \langle \beta_j |
\hat{H}
|\alpha_k \rangle | \beta_l \rangle \langle \alpha_k | \langle \beta_l | .\end{aligned}$$ which has the same ground state as the exact Hamiltonian.[@PhysRevLett.109.186404] The fragment and bath basis states are, in principle, many-electron states.
In order to make calculations practical, DMET and DET replace the exact ground state with a mean-field approximation, i.e. $|\Psi\rangle$ is replaced with $|\Phi\rangle = \Pi_{p} a{}^\dagger_p |0\rangle$, where $a{}^\dagger_p$ creates a hole state $|\phi_p\rangle$ and $|0\rangle$ is the bare vacuum. The hole creation operators are obtained from a mean-field approximation, here the Hartree-Fock transformation $\mathbb{D}$ of bare fermion operators $c^\dagger$: $$\begin{aligned}
a{}^\dagger_p = \sum_{\mu} \mathbb{D}_{\mu p} c{}^\dagger_{\mu}.\end{aligned}$$ The mean-field solution is obtained for the Hamiltonian of interest augmented with an effective one-body potential $v$, $$\begin{aligned}
\label{FullHam}
\hat{H} = \sum_{\mu \nu} h_{\mu \nu} c{}^\dagger_\mu c_\nu +
\frac{1}{4} \sum_{\mu \nu \lambda \sigma} V_{\mu \nu \lambda \sigma}
c{}^\dagger_\mu c{}^\dagger_\nu c_\sigma c_\lambda
+ \sum_{\mu \nu} v_{\mu \nu} c{}^\dagger_\mu c_\nu .\end{aligned}$$ The meaning of this potential will soon become apparent.
With the above approximation, the task of performing the Schmidt decomposition of the wavefunction describing the whole system amounts to a rather trivial algebraic problem [@JPA.39.L85]. Defining single particle basis associated with a chosen subsystem of the entire problem, $|F\rangle$, one constructs a projection operator onto the fragment $\hat{P}_F = \sum_{i} |F_i\rangle \langle F_i|$ and its complement $\hat{P}_B =
\hat{\mathbb{I}} - \hat{P}_F$. The latter projects onto the bath states. With such tools at hand, one may construct an overlap matrix $\mathbb{M}$, $$\begin{aligned}
\label{MMat}
\mathbb{M}_{pq} = \langle \phi_q | \hat{P}_F | \phi_p \rangle ,\end{aligned}$$ where indices $p$ and $q$ denote the hole states. Diagonalizing the above overlap matrix $\mathbb{V} d \mathbb{V}^\dagger = \mathbb{M}$ yields at most min($n_e$,$n_F$) eigenvalues $d_i$ different from zero where $n_e$ and $n_F$ denote the number of electrons in the system and the number of single-particle states associated with the fragment, respectively. The eigenvectors corresponding to such eigenvalues are then normalized to construct fragment ($|f\rangle$) and bath ($|b\rangle$) states $$\begin{aligned}
\label{ConFrag}
|f_i\rangle &= \sum_{p} \frac{\mathbb{V}{}^\star_{p i}}{\sqrt{d_i}} \hat{P}_F |\phi_p\rangle \nonumber \\
|b_i\rangle &= \sum_{p} \frac{\mathbb{V}{}^\star_{p i}}{\sqrt{1-d_i}} \hat{P}_B |\phi_p\rangle .\end{aligned}$$ The states that correspond to zero eigenvalue are called [*cores*]{} and are discarded from consideration in the impurity problem.
In practical applications, one would like to retain all the fragment states. This requires special care when the eigenvalues $d$ are close to 1 or 0. Such complications are addressed in Appendix \[LinDep\]. Right now, let us stress the consequences of the above approximation which is a key step in the present work. The most prominent result is that the Schmidt decomposition yields single-particle bases that can be further employed in the construction of the impurity Hamiltonian. This fact greatly facilitates computations. Additionally, the number of entangled states (or equivalently the number of non-zero eigenvalues $d$) is defined by the fragment single-particle basis. If one chooses the fragment to be small, accurate many-body techniques can be applied to solve the impurity problem, which we now proceed to introduce.
Let us define creation operators in the impurity basis: $f^{\dagger}$, $b^{\dagger}$ and $e^\dagger$; $f^{\dagger}$ and $b^{\dagger}$ are associated with fragment and bath states, respectively. We will use $e^\dagger$ for general states which can be either fragment or bath. The impurity problem is defined by the Hamiltonian, $$\begin{aligned}
\hat{H}_{\textrm{imp}} &= \sum_{e e^\prime} \tilde{h}_{e e^\prime} e{}^\dagger e^\prime +
\frac{1}{4}\sum_{e e^\prime e^{\prime\prime} e^{\prime\prime\prime}}
\tilde{V}_{e e^\prime e^{\prime\prime} e^{\prime\prime\prime}}
e{}^\dagger e{}^{\prime\dagger} e^{\prime\prime\prime} e^{\prime\prime} \nonumber \\
&+
\sum_{b b^\prime} \tilde{v}_{b b^\prime} b{}^\dagger b^\prime ,\end{aligned}$$ where $\tilde{h}$ and $\tilde{V}$ are one- and (antisymmetrized) two-body terms of the Hamiltonian projected onto the embedding basis. The additional potential $\tilde{v}$, acting only in the bath subspace of the impurity, is introduced to enforce a suitable chosen convergence criterion. In the present work, we make a diagonal ansatz for the effective potential ($\tilde{v}_{ij} = v \delta_{ij}$ hence $v_{\mu \nu} = v \delta_{\mu \nu}$ in Eq. \[FullHam\]), which corresponds to a chemical potential in the bath. We find such a potential by requiring the proper number of electrons in the fragment, on average. Let us stress that, for periodic systems, the average number of electrons per unit cell is known and well defined. The reader is referred to Ref. for other possible choices of the effective potential.
At this point, let us make a few remarks concerning the meaning of the impurity Hamiltonian. Assuming that $\mathbb{M}$ contains $n_F$ eigenvalues different from 1 or 0, this Hamiltonian describes a system of $n_F$ particles in $2 n_F$ spin orbitals. Solving the Hamiltonian in the Hilbert space of the impurity corresponds to a Fock space calculation in the fragment subspace. The bath can be considered as a reservoir of electrons or holes.
Having solved the impurity Hamiltonian, the energy density (energy per fragment) is subsequently computed as $$\begin{aligned}
\label{EnExpr}
E = \sum_{f e} \tilde{h}_{f e} \gamma_{e f}
+ \frac{1}{4} \sum_{f e e^{\prime} e^{\prime \prime}}
\tilde{V}_{f e e^{\prime} e^{\prime \prime}} \Gamma_{e^{\prime} e^{\prime \prime} f e} ,\end{aligned}$$ with $\gamma_{e e^\prime} = \langle e^{\prime\dagger} e \rangle$ and $\Gamma_{e e^\prime e^{\prime\prime} e^{\prime \prime \prime}} =
\langle e^\dagger e^{\prime\dagger} e^{\prime \prime \prime} e^{\prime \prime} \rangle$ being one- and two-particle density matrices of the impurity wavefunction. Again, index $f$($e$) denotes fragment (fragment and bath) single-particle states.
We note that the DET energy is not an expectation value of the true Hamiltonian with an N-particle wavefunction. It is therefore not an upper bound of the true ground state energy.
Schmidt decomposition for periodic systems
------------------------------------------
Density embedding calculations on a truly infinite system require a suitable single-particle basis associated with a fragment. In the present work, we employ the maximally localized Wannier functions [@RevModPhys.84.1419; @WannWC1; @WannWC2; @Zicovich-Wilson] obtained by localization of canonical mean-field crystalline orbitals. In other words, we form a unitary transformation of mean-field basis $|\psi_{n\vec{k}}\rangle$, where $\vec{k}$ labels irreducible representation of the translational group [@pisani1996quantum; @CPL.289.611.1998] and $n$ is a band index. This yields an orthonormal set $|F_{i \vec{G}}\rangle$, where $i$ labels a basis in given cell $\vec{G}$. The orthonormality condition reads, [@RevModPhys.84.1419] $$\begin{aligned}
\langle F_{i \vec{G}} | F_{j \vec{G}^{\prime}} \rangle = \delta_{ij} \delta_{\vec{G}\vec{G}^\prime} .\end{aligned}$$
A few comments are called for at this point. Firstly, during the localization process, we must allow for mixing of hole and particle states. Therefore, there is no need for localizing the particle (unoccupied) orbitals by themselves. In our numerical approach, we did not encounter serious difficulties converging the Wannier basis required by the present formalism. Secondly, as we explain in more details in Appendix \[BandTrun\], one may desire to truncate the space treated in the impurity Hamiltonian only to levels around the Fermi energy. This is accomplished simply by choosing a subset of energy bands for the localization. In other words, only a limited set of the highest valence bands and the lowest conduction bands may be employed while forming $|F_{i\vec{G}}\rangle$ bases. As the number of bands used during the localization process is equal to the number of fragment states per unit cell, a suitable truncation criterion may be used to further limit the single-particle Hilbert space of the impurity Hamiltonian without sacrificing relevant physics.
Analogously, one needs to perform the localization of the hole states, yielding $|\phi_{p\vec{G}}\rangle$, which constitutes the $p^{\textrm{th}}$ hole state associated with cell $\vec{G}$. Whenever a truncation of the conduction band occurs while forming the fragment states, the same truncation should be done during the formation of the hole states.
Now, one is in a position to compute the overlap matrix of Eq. \[MMat\], $$\begin{aligned}
\label{MMatPBC}
\mathbb{M}{}_{p q}^{\vec{G}\vec{G}^\prime} = \langle \phi_{q\vec{G}^\prime} | \hat{P}_F | \phi_{p\vec{G}} \rangle =
\sum_{i} \langle \phi_{q\vec{G}^\prime} | F_{i\vec{0}} \rangle \langle F_{i\vec{0}} | \phi_{p\vec{G}} \rangle ,\end{aligned}$$ needed to define the fragment and bath states of the impurity basis. In the above formula, the summation over the fragment states is limited to the reference cell $\vec{0}$. If embedding of more than one cell is needed, the summation over the entire embedded cluster must be performed. With the aid of the above matrix, fragment and bath states are formed according to Eq. \[ConFrag\] keeping in mind that index $p$ in those equations includes a cell coordinate.
At this point we would like to note that the localization of the hole states and the local nature of the fragment single-particle basis allows for effective truncation of the formally infinite summation over the entire crystal. Indeed, one may limit the summation over cells when $\langle F_{i\vec{0}} | \phi_{p\vec{G}}
\rangle \to 0$ as $|\vec{G}|$ increases. The localization of the hole states therefore dictates a natural length scale one has to consider during DET calculations.
Definition of the impurity Hamiltonian
--------------------------------------
Having established a formalism for constructing the embedding basis, let us turn our attention to the definition of the impurity Hamiltonian. A Hamiltonian for a realistic crystalline material can be written as $$\begin{aligned}
\hat{H} = E_{NN} + \hat{V}_{Ne} + \hat{V}_{ee} + \hat{T} = H_0 + \hat{h} + \hat{V},\end{aligned}$$ where $E_{NN}$ is the nuclear repulsion energy, $\hat{V}_{Ne}$ and $\hat{V}_{ee}$ are the electrostatic electron-nucleus and electron-electron interactions, respectively; $\hat{T}$ is the kinetic energy operator. Those terms can be then arranged into constant ($H_0$) and one- and two-body Hamiltonians ($\hat{h}$ and $\hat{V}$, respectively). As described in the literature, [@PhysRevB.61.16440; @pisani1996quantum; @JCP-105-10983-1996; @PhysRevB.28.5781] the summation of an infinite number of electrostatic terms has to be handled with care in order to avoid divergences and loss of accuracy. Analogous problems may arise while projecting the Hamiltonian onto the embedding basis. In order to deal with such complications, we propose to first recast the Hamiltonian into second-quantized form with the aid of the mean-field Fock matrix $$\begin{aligned}
F_{\mu \nu} = h_{\mu \nu} + \sum_{\lambda \sigma}
V_{\mu \lambda \nu \sigma} \gamma_{\sigma \lambda} ,\end{aligned}$$ as $$\begin{aligned}
\hat{H} & = E_0 - \sum_{\mu \nu} F{}_{\mu\nu} \gamma_{\nu\mu}
+ \frac{1}{2} \sum_{\mu \nu \lambda \sigma}
V_{\mu \lambda \nu \sigma} \gamma_{\sigma \lambda} \gamma_{\nu\mu} \nonumber \\
& + \sum_{\mu \nu}
\Big(F_{\mu\nu} - V_{\mu \lambda \nu \sigma} \gamma_{\sigma \lambda} \Big) c{}^\dagger_\nu c_\nu
+ \frac{1}{4} \sum_{\mu \nu \lambda \sigma}
V_{\mu \nu \lambda \sigma} c{}^\dagger_\mu c{}^\dagger_\nu c_\sigma c_\lambda\end{aligned}$$ In the expression above, $E_0 = E N$ where $E$ is the mean-field energy per unit cell and $N$ is the number of cells. Again, the individual terms in the summation are not necessarily convergent. For example, the constant $\frac{1}{2} \sum_{\mu \nu \lambda \sigma} V_{\mu \lambda \nu \sigma} \gamma_{\sigma \lambda} \gamma_{\nu\mu}$ describing the electron-electron interaction energy is divergent and has no meaningful thermodynamic limit, if evaluated separately. Similarly, $\sum_{\lambda \sigma} V_{\mu \lambda \nu \sigma} \gamma_{\sigma \lambda}$, which contributes to the one-body Hamiltonian above, gives rise to divergent matrix elements. For these reasons, we propose to express all quantities in the embedding basis, before the summation is performed. In other words, we separately project the mean-field potential, the density matrix and the two-body interaction onto the embedding basis, i.e. $F_{\mu \nu} \to \tilde{F}_{e
e^\prime}$, $\gamma_{\mu \nu} \to \tilde{\gamma}_{e e^\prime}$ and $V_{\mu \nu \lambda \sigma} \to \tilde{V}_{e e^\prime e^{\prime \prime} e^{\prime \prime \prime}}$. While the two-body interaction is projected without any modifications, let us explicitly write the one-body part of the impurity Hamiltonian, $\tilde{h}$, and the constant term, $\tilde{E_0}$, $$\begin{aligned}
\label{ReDefOne}
\tilde{h}_{e e^\prime} &= \tilde{F}_{e e^\prime} -
\sum_{e^{\prime \prime} e^{\prime \prime \prime}} \tilde{V}_{e e^{\prime \prime} e^\prime e^{\prime \prime \prime}}
\tilde{\gamma}_{e^{\prime \prime \prime} e^{\prime \prime}}\end{aligned}$$ $$\begin{aligned}
\label{ReDefZero}
\tilde{E_0} &= E N_F - \sum_{f e} \Big( \tilde{F}_{f e} - \frac{1}{2}\sum_{e^\prime e^{\prime \prime}}
\tilde{V}_{f e^\prime e e^{\prime \prime}} \tilde{\gamma}_{e^{\prime \prime} e^\prime} \Big)\tilde{\gamma}_{e f} ,\end{aligned}$$ where again, $E$ denotes the mean-field energy per unit cell, whereas $N_F$ is the number of unit cells in the fragment. As the reader may readily notice, the summation restriction to the fragment basis only has been imposed on the constant term above. The reason for such truncation shall become clear soon.
The construction above constitutes an approximate way of projecting the Hamiltonian. Let us therefore discuss the physical motivation behind it. As shown in Ref. and expanded upon in the Appendix below, the mean-field one-particle density matrix and mean-field Fock matrix commute with each other after projection onto the embedding basis, i.e., $[\tilde{\gamma},\tilde{F}]=0$. Moreover, $\tilde{\gamma}$ is idempotent. Inserting $\tilde{\gamma}$ as an initial guess for the impurity Hamiltonian as define above yields a Fock matrix that is equal to the Fock matrix of the whole system projected onto the embedding basis, $$\begin{aligned}
F{}^{\textrm{imp}}_{e e^\prime} = \tilde{h}_{e e^\prime} + \sum_{e^{\prime \prime} e^{\prime \prime \prime}}
\tilde{V}_{e e^{\prime \prime} e^\prime e^{\prime \prime \prime}}
\tilde{\gamma}_{e^{\prime \prime \prime} e^{\prime \prime}} = \tilde{F}_{e e^\prime} .\end{aligned}$$ The mean-field solution of the impurity problem is therefore the crystal density matrix in the embedding basis. Furthermore, computing the energy according to Eq. \[EnExpr\] (with a constant term defined by Eq. \[ReDefZero\]) reveals that the mean-field energy of the fragment is just the energy per unit cell multiplied by the number of cells taken as fragment constituents. In the above, we have set the effective potential, present in Eq. \[FullHam\] to zero. We conclude that the current definition of the impurity problem ensures exactness of mean-field in mean-field embedding. Furthermore, the solution corresponds to a vanishing effective potential $v$. We would like to stress that the exactness of the mean-field in mean-field embedding has been numerically demonstrated for DMET in Ref. .
Finally, let us note that for nontrivial calculations, that is when a correlated theory is used as an impurity solver, the effective potential has to be optimized and included in the impurity Hamiltonian as well as the full crystal Hamiltonian. In the present work, the diagonal ansatz for the effective potential allows us to eliminate these terms from the mean-field Fock matrix of the crystal as it cannot change the mean-field solution. Therefore, the construction of the embedding basis and the impurity Hamiltonian is performed only once during the calculation. The value of the effective potential is determined in the embedding basis only.
Computational details
=====================
The construction of Wannier functions has been implemented in the Gaussian Developement Version [@gdv] that has also been used to perform the periodic Hartree-Fock calculations. The crystalline orbital localization has been performed by adapting the scheme the of Ref. , where the Boys localization is replaced by the Pipek-Mezey localization [@WannWC2] with the L[ö]{}wdin population. [@WannWC1] For 1D systems, we have used a $\vec{k}$-point mesh of at least 400 points; for 2D and 3D, 4000 and 70000 $\vec{k}$-points have been used, respectively. The hermitized density matrices for coupled cluster with double (CCD) and single and double (CCSD) excitations were obtained using the linear response formalism. [@JCP.1995.103.3561; @JCP.1987.87.5361]
The most diffuse basis functions of the 6-31G basis [@JCP.1972.56.2257; @JCP.1975.62.2921] were changed to 0.35, 0.30, and 0.20 for the carbon, nitrogen and boron atoms, respectively.
In all calculations, eigenvalue thresholds of the Schmidt decomposition for retaining the bath states was set to 10$^{-6}$. The fragment states corresponding to eigenvalues that were closer to 0 or 1 than this threshold were constructed according to the formalism outlined in Appendix \[LinDep\].
The number of cells used in the Schmidt decomposition has been decided by a commutation criterion between mean-field Fock and density matrices after projection onto the embedding basis, $\sum_{ee^\prime}|(\tilde{F}\tilde{\gamma})_{ee^\prime} - (\tilde{\gamma}\tilde{F})_{ee^\prime}|$. The values of this norm are reported for the calculations in the subsequent section.
Results and discussion
======================
1D carbon systems
-----------------
In this section, we asses the performance of DET on three carbon polymers, polyyne (C$\equiv$C)$_\infty$, polyacetylene (CH$=$CH)$_\infty$, and polyethylene (CH$_2$$-$CH$_2$)$_\infty$. In the present work, we adopt the geometries from Ref. . The geometrical parameters are $r_{\mathrm{C}\equiv\mathrm{C}} = 1.263 \AA$, $r_{\mathrm{C}-\mathrm{C}} = 1.132 \AA$ for polyyne, $r_{\mathrm{C}=\mathrm{C}} = 1.369 \AA$, $r_{\mathrm{C}-\mathrm{C}} = 1.426 \AA$, $r_{\mathrm{C}-\mathrm{H}} = 1.091 \AA$, $\angle_{\mathrm{C}=\mathrm{C}-\mathrm{C}} = 124.5^\circ$, $\angle_{\mathrm{C}=\mathrm{C}-\mathrm{H}} = 118.3^\circ$ for polyacetylene, and $r_{\mathrm{C}-\mathrm{C}} = 1.534 \AA$, $r_{\mathrm{C}-\mathrm{H}} = 1.100 \AA$, $\angle_{\mathrm{C}-\mathrm{C}-\mathrm{C}} = 113.7^\circ$, $\angle_{\mathrm{H}-\mathrm{C}-\mathrm{H}} = 106.1^\circ$ for polyethylene. In order to gain better insight into the performance of the DET approximation, we have additionally deformed the above systems by keeping all variables, apart from the carbon-carbon bonds, fixed, while scaling the carbon-carbon bonds uniformly with a parameter $\alpha$. In all calculations, the $1s$ orbitals of carbon were eliminated from consideration in DET and coupled cluster calculations. DET($n$) denotes calculations with $n$ unit cells used as a fragment.
Let us begin the discussion with the most challenging system for the embedding calculation, polyyne. In this case, one expects that the correlation energy contribution to the unit cell would have the slowest decay. [@JCP.120.2004.2581] The results are shown in Fig. \[Fig.C2.STO3G\] and Fig. \[Fig.C2.631G\]. The extrapolated CCD and CCSD results were obtained according to $$\begin{aligned}
E_{\textrm{Extr}}(n) = E_{HF} + E_{\textrm{corr}}(n) - E_{\textrm{corr}}(n-1)
\label{Extrapolation}\end{aligned}$$ where $E_{HF}$ is HF energy per unit cell of infinite system and $E_{\textrm{corr}}(n)$ is the correlation energy of the $n$-unit oligomer with a hydrogen atom as the terminal group. For the case of STO-3G basis, extrapolations for $n=8$ and $n=7$ differ by no more than 0.1 $mE_h$; in the case of the 6-31G basis, we have used $n=9$ which differs from $n=8$ by at most 0.2 $mE_h$. We deem these results sufficiently converged for the purpose of the presented figures. Our extrapolated CCSD correlation energy value for the STO-3G basis and $\alpha=1$, of -155.45 $mE_h$ agrees well with -155.53 $mE_h$ reported in Ref. . In the DET calculations, the number of cells used for the Schmidt decomposition guaranteed that the norm of the commutator of full-system Fock and density matrices projected onto the embedding basis to be at most $3\times 10^{-6}$.
As is clear from Fig. \[Fig.C2.STO3G\], the DET calculations with a single unit cell chosen as a fragment agree well with the extrapolated thermodynamic limit values both for CCSD and CCD as the impurity solver. The maximum discrepancy is below 10 $mE_h$; this translates to an energy difference on the level of $5\%$. Investigating the shape of the energy profile as a function of the uniform stretching parameter $\alpha$, we find the overall agreement satisfactory. The inclusion of electron correlation clearly favors a more stretched configuration. Apparently, DET calculations appropriately capture this trend. One may also notice that including two unit cells as the fragment yields results that are closer to the extrapolated CCSD and CCD results.
Increasing the size of the basis set to 6-31G does not lead to a deterioration of the DET results. As is clear from Fig. \[Fig.C2.631G\], the single-cell DET calculations are again in good agreement with the extrapolated values. The absolute difference does increase slightly but so does the correlation energy. The overall shape of the energy profile is well reproduced by DET. The bonds elongation caused by correlation is well captured.
Let us comment on the size of the impurity problem for polyyne. In the case of the STO-3G basis, there are 8 fragment and 8 bath orbitals for the single cell case, with the impurity bearing 16 electrons. This illustrates how effective the present embedding scheme is in truncating the size of the single-particle Hilbert space of the problem.
The next polymeric system under investigation is polyacetylene. For this example, the extrapolated CCSD and CCD correlation energy from 8 and 7 cells differed by less than 0.1 $mE_h$ for both STO-3G and 6-31G bases. The value of $E_{\textrm{corr}}$ for CCSD with $\alpha=1$, -146.4 $mE_h$, coincides with the one reported in Ref. . The number of cells used in the Schmidt decomposition guaranteed that the norm of the commutator between the mean-field density and Fock matrices in the embedding basis is below $10^{-6}$.
Analogously to polyyne, one notices in Fig. \[Fig.CH.STO3G\] and Fig. \[Fig.CH.631G\] that correlation favors a more elongated carbon-carbon bond. Both DET and extrapolated oligomeric results agree quantitatively. For the STO-3G basis, even the DET(1) calculation yields results within 4 $mE_h$ from extrapolated values, a result that is greatly improved by enlarging the embedded fragment to two cells. Regardless, the DET approximation with both CCD and CCSD as impurity solvers yield results that are rather parallel to the thermodynamic limit ones. With the increased size of the basis set, the agreement remains satisfactory. Though the curvature of the energy profile obtined with DET(1) deviates slightly from the extrapolated data, especially for contracted systems ($\alpha \le 0.95$), the difference is not large. Again, one has to keep in mind that DET calculations are done employing a significantly truncated single-particle basis. For the STO-3G basis, the impurity problem with single cell models the infinite system with a Hamiltionian that describes merely 20 electrons in 20 orbitals (10 fragment and 10 bath states).
The last 1D polymer studied is polyethylene. Just as in the case of polyacetylene, the difference between the extrapolated CCD and CCSD energy using 8 and 7 cells was well below 0.1 $mE_h$. For STO-3G and $\alpha=1$, our extrapolated CCSD correlation energy per unit cell, -135.7 $mE_h$, coincides with the value reported by Hirata. [@JCP.120.2004.2581] The number of cells included in the Schmidt decomposition guaranteed that the norm of the commutator between the mean-field Fock and density matrices is below $10^{-6}$ after projection onto embedding basis.
For polyethylene, the STO-3G DET(1) results (Fig. \[Fig.CH2.STO3G\]) coincide very well with the extrapolated oligomeric data. The small difference is almost constant over the studied values of the $\alpha$ stretching parameter. Increasing the embedded fragment to two cells brings the discrepancy almost to zero. With the bigger basis, 6-31G, once again we observe very good overall agreement between DET and extrapolated data. The maximum difference occurs for the more contracted geometry. Just as in all previous systems, one observes stabilization of a more elongated structure due to correlation effects. Again, let us stress that, within the DET approximation, modeling the infinite system with an impurity problem of 24 electrons in 24 orbitals (12 fragment states and 12 bath states), for the example of DET(1) STO-3G calculations, allows one to obtain a high degree of agreement with the full periodic CCD and CCSD calculations. One should however keep in mind, that the physical interpretation of the impurity problem differs from the true Hamiltonian. In the former, the CC(S)D method is used to effectively perform a Fock space calculation in the unit cell with the aid of an entangled bath. On the other hand, for the full Hamiltonian one considers excitations of the electrons of the entire periodic system.
2D and 3D: boron nitride and diamond
------------------------------------
In this section, we proceed to investigate prototypical 2D and 3D systems. The 2D structure was obtained assuming infinitely separated boron nitride sheets of hexagonal BN [@Science.2014.343.163; @JCPCM.1997.1.1] yielding a graphene-like honeycomb lattice. We performed a single unit embedding with CCSD and CCD as an impurity solver. The number of cells included in the Schmidt decomposition was chosen to provide the norm of the commutator of the mean-field and Fock matrices in the embedding basis below 10$^{-6}$. The two lowest bands were excluded from consideration, which corresponds to freezing 1$s$ orbitals of boron and nitrogen in the CC calculations.
In Fig \[Fig.BN\], we present the dependence of the energy per unit cell with respect to the translation vector defining the underlying honeycomb lattice. As the reader may readily notice, our DET calculations predict noticeable impact from the inclusion of the single excitation in the CC impurity solver. For both, the STO-3G and 6-31G bases, the CCSD energy is consistently below the CCD energy by about 10-20 $mE_h$. Nonetheless, the shape of the curves around the equilibrium point are rather similar. The impact of the electron correlation clearly shifts the position of the optimal structure towards longer translation vectors as compared to mean-field calculations. We also note that the lattice constant obtained with single cell DET embedding is larger than that reported one for the hexagonal BN. [@Science.2014.343.163; @JAC.1994.2216.5] While this elongation is due to the inclusion of only a single sheet, lack of solid substrate or the deficiency of the employed basis set is beyond the scope of the present work. However, we would like to stress the key point of current work. Namely, the size of the Hilbert space of impurity Hamiltonian is small and independent of the dimensionality of the problem. The calculations for the single cell with STO-3G basis required explicit correlated treatment of 16 electrons in 16 orbitals while the 6-31G basis 32 electrons in 32 orbitals.
As a model 3D system, we have selected diamond. The single cell embedding with the STO-3G basis is shown in Fig. \[Fig.Diamond\]. While in this example, the norm of the commutator of Fock and density matrices in the embedding basis is on the level of $3\times10^{-5}$, we have verified that even with larger real space truncation, the DET correlation energy is stable to 0.1 $mE_h$. As the reader may notice, CCD- and CCSD-based DET calculations provide very similar descriptions. The equilibrium geometry occurs for longer translational vector as compared to Hartree-Fock calculations. We do not attempt a quantitative discussion of the data presented, which would require larger bases. However, we point out that the size of the impurity problem involves 16 electrons in 16 orbitals (1$s$ orbitals were removed from the impurity). We believe that this illustrates the key point of the current work: the dimension of the impurity Hamiltonian is independent of the dimensionality of the lattice.
Conclusions
===========
In the present work, we have reported the first application of density embedding theory for realistic periodic systems. We have proposed a practical way of defining the impurity problem and an extension of the Schmidt decomposition to infinite systems. Practical aspects of calculation, including the problem of disentangled bath states has been outlined and assessed. We believe that this point is important for extending DET calculations to larger basis sets and bigger fragments.
Our proposed formalism for realistic Hamiltonians has been quantitatively assessed for several periodic systems with the aid of coupled cluster theory as an impurity solver. The data presented shows good agreement between the coupled cluster DET calculations and the coupled cluster thermodynamic limit, even when using a single cell as fragment. While more extended benchmarks are certainly called for, the current tests are a promising starting point. Indeed, employing more sophisticated many-body techniques to tackle the impurity problem is an interesting option especially as the size of the impurity Hamiltonian does not depend on the dimensionality of the lattice. Furthermore, as the impurity problem in DMET and DET is always finite, application of accurate but not necessarily size-extensive tools becomes feasible. Nonetheless, one should bear in mind that DMET and DET provide a local, finite Hamiltonian solution which may be interpreted as a Fock space calculation performed on the fragment. As such, the philosophy of the calculation differs from the Hilbert space approach to the full lattice.
Despite being a relatively recent model, density matrix embedding theory and its simplification employed here, are accurate and computationally feasible approaches to deal with the numerous electronic degrees of freedom of large systems. We believe that the results presented here support confidence in the predictive power of this approximation.
Acknowledgements
================
I. W. B. would like to acknowledge Thomas M. Henderson for helpful discussions. John Gomez and Jacob Wahlen-Strothman are thanked for carefully reading the manuscript. This work was supported by Department of Energy, Office of Basic Energy Sciences, Grant No. DE-FG02-09ER16053. G. E. S. is a Welch Foundation Chair (C-0036).
Handling band truncation and disentangled states in DET
=======================================================
In this Appendix, we present a formalism for dealing with the Schmidt decomposition of a Slater determinant for truncated particle and hole states as well as cases where fragment and bath states become disentangled. The discussion is similar to the one provided in Ref. but more general.
Fragment states with truncated bands {#BandTrun}
------------------------------------
Let us start by specifying the notation. In the present work, the crystaline orbitals, both in Bloch and Wannier representations are normalized according to $$\begin{aligned}
\langle \psi_{i\vec{k}} | \psi_{j\vec{k}^\prime} \rangle &= \delta_{ij} \delta_{\vec{k}\vec{k}^\prime} \nonumber \\
\langle \phi_{i\vec{G}} | \phi_{j\vec{G}^\prime} \rangle &= \delta_{ij} \delta_{\vec{G}\vec{G}^\prime} ,\end{aligned}$$ and are related by the discrete Fourier relation [@RevModPhys.84.1419] $$\begin{aligned}
| \phi_{i\vec{G}} \rangle &= \frac{1}{\sqrt{N}} \sum_{\vec{k}} e^{-i \vec{k}\cdot\vec{G}}
\sum_{j} \mathbb{U}{}^{\vec{k}}_{ji} |\psi_{j\vec{k}} \rangle ,\end{aligned}$$ where $N$ is the number of unit cells. The idempotent density matrix can then be expressed as $$\begin{aligned}
\hat{\gamma} = \sum_{p\vec{k}} |\psi_{p\vec{k}}\rangle \langle \psi_{p\vec{k}} | =
\sum_{p\vec{G}} | \phi_{p\vec{G}} \rangle \langle \phi_{p\vec{G}} | =
\sum_{p^\prime} | \phi_{p^\prime} \rangle \langle \phi_{p^\prime} | ,\end{aligned}$$ where in the second term, index $p$ denotes hole states at given $\vec{k}$ or labeled by cell index $\vec{G}$, whereas in the last term $p^\prime=(p\vec{G})$ denotes all particle states in all cells.
The orthonormal single-particle basis $|F_{i\vec{G}}\rangle$ becomes $$\begin{aligned}
|F_{i\vec{G}}\rangle = \frac{1}{\sqrt{N}} \sum_{\vec{k}} e^{-i \vec{k}\cdot\vec{G}}
\sum_{j}\mathbb{U}{}^{\vec{k}}_{ji}|\psi_{j\vec{k}}\rangle ,\end{aligned}$$ with index $j$ running over the chosen subset of bands. Because the states $F_{i\vec{G}}$ are orthogonal to Wannier functions obtained by unitary transformation of Bloch functions within the complementary subset of bands, such states have zero overlap with $|F_{i\vec{G}}\rangle$. Therefore, for the sake of argument, one may include these in the definition of the overlap matrix $\mathbb{M}$ (Eq. \[MMatPBC\]). Such states will simply have vanishing amplitude in eigenvectors corresponding to non-zero eigenvalue. Therefore, in the rest of this Appendix, the summations over the indices $p$ and $q$ in Eq. \[MMatPBC\] are formally done over the whole valence band. One just has to impose proper block-diagonal structure of $\mathbb{U}^{\vec{k}}$ while preparing states $|\phi_{p\vec{G}}\rangle$. It now follows directly that the one particle density matrix in the embedding basis has the blocked structure $$\begin{aligned}
\tilde{\gamma} =
\begin{pmatrix}
\gamma^{\textrm{FF}} & \gamma^{\textrm{FB}} \\
\gamma^{\textrm{BF}} & \gamma^{\textrm{BB}}
\end{pmatrix}
=
\begin{pmatrix}
d & \sqrt{d(1-d)} \\
\sqrt{d(1-d)} & 1-d
\end{pmatrix},\end{aligned}$$ where $d$ is the diagonal matrix of non-zero eigenvalues of $\mathbb{M}$. Finally, let us show the commutation relation between the density matrix and the Fock matrix projected onto the embedding basis. To do so, we define matrix $t = \tilde{F}\tilde{\gamma}$ and show it is Hermitian. The fragment-fragment block reads, $$\begin{aligned}
t{}^{\textrm{FF}}_{ij} &= \tilde{F}{}^{FF}_{ij} d_j + \tilde{F}{}^{FB}_{ij}\sqrt{d_j(1-d_j)} \nonumber \\
&= \sqrt{d_i d_j}\sum_{pq} \mathbb{V}_{p i} \langle \psi_p | \hat{F} | \psi_q \rangle \mathbb{V}{}^\star_{q j}
= [t^{\textrm{FF}}_{ji}]^\star ,\end{aligned}$$ where $\hat{F}$ is the crystal Fock operator. We have used the fact that a vector $\hat{F}|\phi_q\rangle$ does not contain contributions from particle states. Hence it can be written as $\sum_{r} |\phi_r\rangle \langle \phi_r |
\hat{F} | \phi_{q} \rangle$ with indices $q$ and $r$ being the hole states. Similarly, $$\begin{aligned}
t{}^{\textrm{BB}}_{ij} &= \sqrt{(1-d_i)(1-d_j)}\sum_{p q} \mathbb{V}_{pi} \langle \phi_p | \hat{F} | \phi_q \rangle
\mathbb{V}{}^\star_{qj} = [ t{}^{\textrm{BB}}_{ji} ]^\star \nonumber \\
t{}^{\textrm{FB}}_{ij} &= \sqrt{d_i(1-d_j)} \sum_{pq} \mathbb{V}_{pi} \langle \phi_p | \hat{F} | \phi_q \rangle
\mathbb{V}{}^\star_{qj} = [ t^{\textrm{BF}}_{ji} ]^\star\end{aligned}$$
Handling disentangled states {#LinDep}
----------------------------
In the following section we suggest a route for dealing with the eigenvalues of $\mathbb{M}$ that are close to 1 or 0. As the reader may easily note, whenever a situation like this occurs, one may face numerical problems with the normalization of embedding basis. The trivial solution, i.e. removing the couple of bath and fragment states that are disentangled, cannot be easily done; one would eliminate possibly important degrees of freedom in the fragment.
Let us consider that there exists a set of eigenvalues $d$ that are close to 1. In such a case, we propose to remove the bath state and retain a modified fragment state that takes the form $$\begin{aligned}
\label{ZeroState}
|\tilde{f}_i\rangle = \sum_{p} \mathbb{V}{}^\star_{pi} | \phi_p \rangle.\end{aligned}$$ Due to the unitarity of $\mathbb{V}$, these states are orthonormal and orthogonal to all other fragment and bath states. Moreover, the density matrix in such a basis takes the form, $$\begin{aligned}
\tilde{\gamma} =
\begin{pmatrix}
d & \sqrt{d(1-d)} & 0 \\
\sqrt{d(1-d)} & 1 -d & 0 \\
0 & 0 & 1 \\
\end{pmatrix},\end{aligned}$$ where the last block is expressed in the basis $|\tilde{f}\rangle$. As is clear, $\tilde{\gamma}$ remains idempotent. It is also straightforward to show that the commutation relation between the one-body density matrix and Fock matrix is preserved.
Let us now turn our attention to the situation when there exists a set of eigenvalues $d$ that are close to 0. We propose to construct an auxiliary matrix $\mathbb{N}$, $$\begin{aligned}
\mathbb{N}_{kl} = \langle F_{l} | \big( \mathbb{I} - \sum_{p} |\phi_p\rangle \langle \phi_p | \Big) F_{k} \rangle .\end{aligned}$$ This matrix admits eigendecomposition $\mathbb{N} = \mathbb{U} \lambda \mathbb{U}^\dagger$. Let us show that for every eigenvalue $d_i$ of $\mathbb{M}$ different from 0, $\mathbb{N}$ has eigenvalue $1-d_i$. We define a column vector $\mathbb{U}^\prime_{ki} = \sum_{q} \langle F_k| \phi_q \rangle \mathbb{V}^\star_{qi}$ with a norm $\sum_{k} \mathbb{U}^{\prime\star}_{ki} \mathbb{U}^\prime_{ki} = d_i$. Hence it is a non-trivial vector whenever $d_i$ is not an exact zero. One may now explicitly verify that $\sum_{k}\mathbb{N}_{lk}\mathbb{U}^\prime_{ki} = (1-d_i) \mathbb{U}^\prime_{li}$
We show that one may replace a fragment state $|f_i\rangle$ with eigenvalue close to 0, with the state $$\begin{aligned}
|\bar{f}_i \rangle = \sum_{k} \frac{\mathbb{U}^{\star}_{ki}}{\sqrt{\lambda_i}} \big(\mathbb{I} - \sum_p
|\phi_p\rangle \langle \phi_p | \big) |F_k\rangle ,\end{aligned}$$ with eigenvalue $\lambda_i = 1 - d_i$ and remove a bath state entangled with $|f_i\rangle$. Let us now demonstrate that $\bar{f}_i$ corresponding to eigenvalue $\lambda_i$ is orthogonal to all fragment states $|f_j\rangle$ corresponding to $d_j$ not equal to $1-\lambda_i$. Namely, $$\begin{aligned}
\langle f_j | \bar{f}_i \rangle &=
\frac{1-d_j}{\sqrt{d_j \lambda_i}} \sum_{kp} \mathbb{V}_{pj}
\langle \phi_p | F_{k} \rangle \mathbb{U}{}^\star_{ki} \nonumber \\
&=
\frac{\lambda_i}{\sqrt{d_j \lambda_i}} \sum_{kp} \mathbb{V}_{pj}
\langle \phi_p | F_{k} \rangle \mathbb{U}{}^\star_{ki}\end{aligned}$$ must vanish whenever $\lambda_i \neq 1 - d_j$. Similarly, for the bath states not associated with fragment $|f_j\rangle$, $$\begin{aligned}
\langle b_j | \bar{f}_i \rangle = -\sqrt{\frac{d_j}{1-d_j}} \langle f_j | \bar{f}_i \rangle .\end{aligned}$$ The orthogonality to states $|\tilde{f}_i\rangle$ (Eq. \[ZeroState\]) also follows. Finally, the density matrix in the embedding basis as define above takes the form $$\begin{aligned}
\tilde{\gamma} =
\begin{pmatrix}
d & \sqrt{d(1-d)} & 0 & 0 \\
\sqrt{d(1-d)} & 1 -d & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
\end{pmatrix}.\end{aligned}$$ One may also verify that the product of the Fock and density matrices in the embedding basis remains Hermitian (one again uses the fact that $\hat{F}|\phi_p\rangle = \sum_r|\phi_r\rangle\langle \phi_r | \hat{F} | \phi_p \rangle$ to show that $\langle \bar{f}_i | \hat{F} | \phi_p \rangle = 0$).
As is clear, the density matrix above is idempotent and traces to an integer number of particles. Therefore, it dictates the number of electrons we include in the impurity problem. However, as in the fragment space we replace eigenvalues that differ from one and zero by a preset value, the total number of particles in the fragment may deviate from the actual with an error proportional to chosen threshold.
Let us finally note that in the derivation above we assumed that eigenvalues of fragment states are arbitrarily small but nonzero. In practical calculations, we did not encounter any problems while forming the fragment states from eigenvectors of $\mathbb{M}$ above the preset threshold and filling the missing fragment states with the eigenvectors of $\mathbb{N}$ to complete the set.
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http://dx.doi.org/10.1016/0925-8388(94)91027-8)
|
---
abstract: |
We consider an open quantum system which contains unstable states. The time evolution of the system can be described by an effective non-hermitian Hamiltonian $H_\mathrm{eff}$, in accord with the Wigner–Weisskopf approximation, and an additional term of the Lindblad form, the socalled dissipator. We show that, after enlarging the original Hilbert space by states which represent the decay products of the unstable states, the non-hermitian part of $H_\mathrm{eff}$ —the “particle decay”— can be incorporated into the dissipator of the enlarged space via a specific Lindblad operator. Thus the new formulation of the time evolution on the enlarged space has a hermitian Hamiltonian and is probability conserving. The equivalence of the new formulation with the original one demonstrates that the time evolution which is governed by a *non-hermitian* Hamiltonian and a dissipator of the Lindblad form is nevertheless completely positive, just as systems with hermitian Hamiltonians.\
PACS numbers: 03.65.-w, 03.65.Yz, 13.25.Es\
Key-Words: Open quantum system, Lindblad operator, particle decay, Wigner–Weisskopf approximation
author:
- |
Reinhold A. Bertlmann,[^1] Walter Grimus,[^2] and Beatrix C. Hiesmayr[^3]\
Institut für Theoretische Physik, Universität Wien\
Boltzmanngasse 5, A–1090 Wien, Austria\
date: 'February 14, 2006'
title: |
UWThPh-2006-1\
An open–quantum–system formulation\
of particle decay
---
#### Introduction:
In reality a quantum system is not isolated but always interacting with its environment and has to be considered as an open quantum system [@BreuerPetruccione; @AlickiFannes]. It leads to a mixing of the states in the system— decoherence—and to an energy exchange between system and environment—dissipation [@Joos; @KublerZeh; @JoosZeh; @Zurek]. The decoherence/dissipation weakens or destroys the typical quantum phenomena, the interferences.
There is great interest to study such decoherence/dissipation processes for elementary particles like the “strange” K-mesons (neutral kaons) and the “beauty” B-mesons [@EllisHagelinNS; @HuetPeskin; @EllisLopezMN], in particular, the decoherence of entangled meson pairs [@BertlmannDurstbergerHiesmayr2002; @BG3; @BernabeuMavromatosP] (for an overview see Ref. [@BertlmannSchladming]). However, these particles are decaying, which leads in the framework of quantum mechanics to some difficulties. In the usual framework the decaying states evolve according to the Wigner–Weisskopf approximation (WWA) where the probability of detecting the particle is not conserved and the corresponding Hamiltonian is, for this reason, non-hermitian.
The master equation for such an open quantum system described by the density matrix $\rho(t)$ is given by $$\label{Lindbladequation}
\frac{\mathrm{d}\rho}{\mathrm{d}t} =
-\,i H_\mathrm{eff} \rho \,+\,i\rho H_\mathrm{eff}^\dagger\,-\,D[\rho] \,,$$ where the dissipator has the following general structure of operators [@Lindblad; @GoriniKossakowskiSudarshan]: $$\label{dissipator}
D[\rho] = \frac{1}{2} \,\sum_j \left(
A^\dagger_j A_j \,\rho + \rho A^\dagger_j A_j
\,-\, 2 A_j \rho A_j^\dagger \right).$$ The operators $A_j$ are usually called Lindblad operators. The effective Hamiltonian of the system is given, according to the WWA, by $$\label{hamiltonian}
H_\mathrm{eff} \; = \; H - \frac{i}{2} \,\Gamma,$$ where $H$ and $\Gamma$ are both hermitian and, in addition, $\Gamma \geq 0$.
In terms of open quantum systems, the time evolution of the density matrix represents a dynamical map which transform initial density matrices $\rho(0)$ on the Hilbert space of states, ${\mathbf{H}}_s$, to final density matrices $\rho(t) \,=\, V_t[\rho(0)]$, while the system is interacting with an environment. Such dynamical maps are i) trace conserving, ii) convex linear, iii) completely positive. Complete positivity is a rather strong and important property. It is defined by demanding that all extensions $V_t \otimes \mathbbm{1}_n$ on the Hilbert space ${\mathbf{H}}_s \otimes \mathbbm{C}^{n}$ are positive, i.e. $$(V_t \otimes \mathbbm{1}_n)[\rho] \geq 0 \quad
\forall \,t \geq 0, \;\,\forall \,n=0,1,2,\dots,
\;\, \mbox{and}\;\,
\forall \,\rho \geq 0,$$ where $\rho$ is a density matrix on ${\mathbf{H}}_s \otimes \mathbbm{C}^{n}$. Physically, it is a reasonable condition since the extension $V_t \otimes \mathbbm{1}_n$ can be considered in quantum information as an operator on the composite quantum system Alice and Bob, which acts locally on Alice’s system without influencing Bob. Furthermore, complete positivity essentially ensures that tensor products of maps $V_t$ remain positive, an important property especially when considering entangled states.
However, extracting from a dynamical map the Hamiltonian term in the usual way [@Preskill-notes] leads to a hermitian Hamiltonian in a master equation like Eq. (\[Lindbladequation\]). In this sense the authors of Ref. [@Caban-etal] investigated recently decoherence *and* decay of the neutral kaons by using a dynamical map corresponding to a master equation with hermitian Hamiltonian. Nevertheless, many authors work with a non–hermitian Hamiltonian—which is a standard procedure in particle physics (see, e.g., Ref. [@QuangPham])—in order to include the decay property of the particle.
#### The formalism:
We are going to show in this letter that both approaches—each one having its own appeal—are indeed equivalent to each other for quite general quantum systems. To be more precise, we ask the following question:\
*Can we work with a hermitian Hamiltonian and incorporate the decay as Lindblad operator such that the time evolution of the system represents a completely positive map? Moreover, does it describe the WWA properly without effecting decoherence and/or dissipation?*\
The answer will be yes, it is possible! But we have to enlarge the Hilbert space and include formally the decay products of the unstable states.
Thus we assume that the total Hilbert space ${\mathbf{H}}_\mathrm{tot}$ is the direct sum $$\label{Hilbertspace}
{\mathbf{H}}_\mathrm{tot} = {\mathbf{H}}_s \oplus {\mathbf{H}}_f,$$ where ${\mathbf{H}}_s$ contains the states of the system we are interested in whereas ${\mathbf{H}}_f$ is the space of the “decay states,” defined in the following way. We assume that $d_s
\equiv \dim {\mathbf{H}}_s < \infty$. Then the non-hermitian part $\Gamma$ of the effective Hamiltonian in Eq. (\[hamiltonian\]) can be decomposed as $$\label{gamma}
\Gamma = \sum_{j=1}^r \gamma_j\, |\varphi_j \rangle \langle \varphi_j |
\quad \mbox{with} \quad \gamma_j > 0 \;\,\forall\, j,$$ with an orthonormal system $\{ \varphi_j \}$ in ${\mathbf{H}}_s$ and $r = \dim {\mathbf{H}}_s - n_0$, where $n_0$ denotes the degeneracy of the eigenvalue zero of $\Gamma$. We demand that $\dim {\mathbf{H}}_f \equiv d_f
\geq r$, otherwise ${\mathbf{H}}_f$ is arbitrary.
In the following, $\rho$ denotes a density matrix on ${\mathbf{H}}_\mathrm{tot}$, where it has the following decomposition: $$\label{densitymatrix}
\rho =
\left( {\begin{array}}{cc} \rho_{ss} & \rho_{sf} \\
\rho_{fs} & \rho_{ff} {\end{array}}\right) \qquad \mathrm{with} \qquad
\rho_{ss}^\dagger = \rho_{ss}, \quad
\rho_{ff}^\dagger = \rho_{ff}, \quad
\rho_{fs}^\dagger = \rho_{sf}.$$ Now we have to define the time evolution $\rho$ on ${\mathbf{H}}_\mathrm{tot}$. The Hamiltonian $H$ and the Lindblad operators $A_j$ are easily extended to the total Hilbert space by $$\label{operators-H-A}
\mathcal{H} = \left( {\begin{array}}{cc} H & 0 \\ 0 & 0 {\end{array}}\right), \qquad
\mathcal{A}_j =
\left( {\begin{array}}{cc} A_j & 0 \\ 0 & 0 {\end{array}}\right) \,.$$ Note that for the definition of $\mathcal{H}$ we have used the *hermitian* part $H$ of $H_\mathrm{eff}$.
Now we turn to the decay. We need a Lindblad operator $\mathcal{B}$ on the full space ${\mathbf{H}}_\mathrm{tot}$ which describes the decay in the subspace $\mathcal{H}_s$. As we will see in the next paragraph, the decay is described by $$\label{operator-B}
\mathcal{B} = \left( {\begin{array}}{cc} 0 & 0 \\ B & 0 {\end{array}}\right)
\quad \mbox{with} \quad
B: \, {\mathbf{H}}_s \to {\mathbf{H}}_f$$ and $$\label{BB}
\Gamma = B^\dagger B.$$ Let $\{ f_k \}$ be an orthonormal basis of ${\mathbf{H}}_f$. Then we can decompose $B$ as $$B = \sum_{k=1}^{d_f}\, \sum_{j=1}^r b_{kj} \,
| f_k \rangle \langle \varphi_j |.$$ In order to fulfill Eq. (\[BB\]), we require $$\sum_{k=1}^{d_f} b_{ki}^* b_{kj} = \delta_{ij}\, \gamma_j.$$ For $d_f \geq r$, such a $d_f \times r$ matrix $\left( b_{kj} \right)$ always exists. The simplest case is $d_f = r$, where we can choose $$B = \sum_{j=1}^{d_f} \sqrt{\gamma_j}\,
| f_j \rangle \langle \varphi_j |.$$ In that case, each unstable decaying state $| \varphi_j \rangle$ would decay into just one specific decay state $| f_j \rangle$.
Then we can write the following master equation for the density matrix on ${\mathbf{H}}_\mathrm{tot}$: $$\label{Lindbladequation-onHtot}
\frac{\mathrm{d}\rho}{\mathrm{d}t} =
-\,i \,\big [\mathcal{H},\rho \big ] \,-\,\mathcal{D}[\rho],$$ with the dissipator $$\label{dissipator-onHtot}
\mathcal{D}[\rho] = \frac{1}{2} \,\sum_j
\left( \mathcal{A}^\dagger_j \mathcal{A}_j\,
\rho \,+\, \rho \,\mathcal{A}^\dagger_j \mathcal{A}_j \,-\, 2 \,\mathcal{A}_j
\rho \mathcal{A}_j^\dagger \right) +
\frac{1}{2} \, \left( \mathcal{B}^\dagger
\mathcal{B}
\,\rho \,+\, \rho \,\mathcal{B}^\dagger \mathcal{B} \,-\, 2 \,\mathcal{B} \rho
\mathcal{B}^\dagger \right).$$ To prove that this master equation contains Eq. (\[Lindbladequation\]), we decompose it with respect to the components of $\rho$: $$\begin{aligned}
\dot{\rho}_{ss} & = & -i \,\big[ H,\rho_{ss} \big] \,-\, \frac{1}{2}
\,\big\{
B^\dagger B, \rho_{ss} \big\} \,-\, \frac{1}{2} \sum_j \left( A_j^\dagger A_j
\,\rho_{ss}
+ \rho_{ss} A_j^\dagger A_j - 2\, A_j \rho_{ss} A_j^\dagger \right),
\label{ss} \\
\dot \rho_{sf} & = & -i \,H \rho_{sf} \,-\, \frac{1}{2} \,B^\dagger B
\,\rho_{sf}
\,-\, \frac{1}{2} \sum_j A_j^\dagger A_j \,\rho_{sf} \,,
\label{sf} \\
\dot \rho_{ff} & = & B \rho_{ss} B^\dagger. \label{ff}\end{aligned}$$ Indeed, with Eq. (\[BB\]), we immediately see that Eq. (\[ss\]) for $\rho_{ss}$ reproduces the original equation (\[Lindbladequation\]). Furthermore, it is obvious from Eqs. (\[Lindbladequation-onHtot\]) and (\[dissipator-onHtot\]) that $\mbox{Tr}\, \rho(t) = 1
\quad \forall\, t \geq 0\,$.
At this point, some general remarks are at order. By construction, the time evolution of $\rho_{ss}$ is independent of $\rho_{sf}$, $\rho_{fs}$, and $\rho_{ff}$. Actually, the time evolution of $\rho_{sf}$ or $\rho_{fs}$ completely decouples from that of $\rho_{ss}$ and $\rho_{ff}$. The time evolution of $\rho_{ff}$, the density matrix of the decay states, is determined solely by $\rho_{ss}(t)$ and is the characteristic for a decay process!
With the initial condition $\rho_{ff}(0) = 0$, it is simply given by $$\label{rho-ff}
\rho_{ff} (t) = B \int_0^t \mathrm{d}t' \rho_{ss}(t') \,B^\dagger.$$ If we choose the initial condition $\rho_{sf}(0) = 0$, then $\rho_{sf}$ remains zero for all times and the same applies to $\rho_{fs}$. Anyway, these parts of $\rho$ are totally irrelevant for our discussion.
Generally, for decaying systems the following properties of the time evolution (\[Lindbladequation-onHtot\]) are most important:
- $\rho_{ss}(t)$ and $\rho_{ff}(t)$ are positive $\forall\,t \geq 0$;
- The time evolution of $\rho_{ss}$ is *completely positive*.
**Proof:** The density matrix $\rho(t)$ obeys the time evolution given by the master equation (\[Lindbladequation-onHtot\]), therefore, $\rho(t)\geq 0$ $\forall \,t \geq 0$. Thus, confining ourselves to vectors $v \in \mathcal{H}_s$, we evidently have $0 \leq \langle v |
\rho(t) v \rangle \equiv \langle v | \rho_{ss}(t) v \rangle$ and, therefore, $\rho_{ss}(t) \geq
0$. The same reasoning holds for $\rho_{ff}(t)$. As for complete positivity, we note that on ${\mathbf{H}}_\mathrm{tot} \otimes \mathbbm{C}^n$ a general operator has the structure $R = \sum_k \sigma_k
\otimes L_k$, with $\sigma_k$ operating on ${\mathbf{H}}_\mathrm{tot}$ and $L_k$ on $\mathbbm{C}^n$. We know that the dynamical map $V_t$ induced by Eq. (\[Lindbladequation-onHtot\]) is completely positive. Therefore, if $R \geq 0$, then we have $(V_t \otimes \mathbbm{1}_n)[R] \geq 0$. We also know from Eqs. (\[ss\]), (\[sf\]), and (\[ff\]), that $V_t$ has a well defined restriction to operators acting purely on ${\mathbf{H}}_s$. Thus, $R \geq 0$ on ${\mathbf{H}}_s \otimes \mathbbm{C}^n$ is mapped into a positive operator on the same space by $V_t \otimes \mathbbm{1}_n$. This concludes the proof.
#### The case of a non-singular $\Gamma$:
Finally, it is interesting to consider the special case of a non-singular $\Gamma$, where all states of ${\mathbf{H}}_s$ decay. Then, $\dim {\mathbf{H}}_f \geq \dim {\mathbf{H}}_s$ and, in addition to what we discussed before, the following properties of the time evolution (\[Lindbladequation-onHtot\]) hold:
1. $\lim_{t \to \infty} \rho_{sf}(t) = 0$,
2. $\lim_{t \to \infty} \rho_{ss}(t) = 0$,
3. $\lim_{t \to \infty} \mathrm{Tr}\, \rho_{ff}(t) = 1$.
**Proof:**
1. We consider the equation $$\label{omega}
\frac{\mathrm{d}}{\mathrm{d}t} \,\omega =
-iH \,\omega \,-\, \frac{1}{2} \,\Gamma' \,\omega \,,
\quad \mbox{with} \quad
\Gamma' = \Gamma + \sum_j A^\dagger_j A_j$$ and $\omega(t) \in {\mathbf{H}}_s$. From Eq. (\[omega\]) we derive $$\frac{\mathrm{d}}{\mathrm{d}t} \,|\omega|^2 =
- \langle \omega | \Gamma' \omega \rangle \leq - \gamma'_0 \,|\omega|^2,$$ where $\gamma'_0 > 0$ is the smallest eigenvalue of $\Gamma'$. Since $|\omega|^2 \geq 0$, it follows from this equation that $|\omega(t)|^2 \leq |\omega(0)|^2 \exp(-\gamma'_0 t)$ and, consequently, $\lim_{t \to \infty} \omega(t) = 0$. This leads to $\lim_{t \to \infty} \rho_{sf}(t) = 0$, because $\rho_{sf}$ is a linear combination of operators $| \omega \rangle \langle f |$ with $\omega \in {\mathbf{H}}_s$, $f \in {\mathbf{H}}_f$.
2. Taking the trace of Eq. (\[ss\]), we obtain $$\label{trss}
\frac{\mathrm{d}}{\mathrm{d}t} \,\mathrm{Tr}\, \rho_{ss}(t) = -
\mathrm{Tr}\,\big(\Gamma \rho_{ss}(t)\big) \leq -
\gamma_0 \,\mathrm{Tr}\,\rho_{ss}(t).$$ Here, $\gamma_0$ is the smallest eigenvalue of $\Gamma$. From Eq. (\[trss\]) we derive the inequality $\mathrm{Tr}\, \rho_{ss}(t) \leq \mathrm{Tr}\, \rho_{ss}(0)
\,\exp (-\gamma_0 t)$, whence we find $\lim_{t \to \infty} \mathrm{Tr}\,\rho_{ss}(t) = 0$. Since $\rho_{ss}(t)\geq 0$, the whole matrix must vanish in the limit $t \to \infty$.
3. Point ii) and $\mathrm{Tr} \,\rho(t) =
\mathrm{Tr}\, \rho_{ss}(t) + \mathrm{Tr}\, \rho_{ff}(t) = 1$ imply $\lim_{t \to \infty} \mathrm{Tr}\, \rho_{ff}(t) = 1$.
#### The simplest possible example:
We choose the minimal dimensions $d_s = d_f = 1$, the Hamiltonian $$\label{H-example}
\mathcal{H} = \left( {\begin{array}}{cc} m & 0 \\ 0 & 0 {\end{array}}\right),$$ and set $A_j=0$, i.e. we neglect any decoherence part. Then, Eqs. (\[ss\]), (\[ff\]) and (\[rho-ff\]) simplify considerably, providing the result $$\label{densitymatrix-explicit}
\rho(t) = \left( {\begin{array}}{cc} e^{-\Gamma t} & 0 \\
0 & 1 - e^{-\Gamma t} {\end{array}}\right),$$ where we have chosen the initial conditions $\rho_{ss}(0) = 1$, $\rho_{ff}(0) = \rho_{sf}(0) = 0$. This example illustrates the general fact that, for non-singluar $\Gamma$, the probability loss of the system is balanced by the probability increase of the decay states.
Considering the mixedness of the quantum states, we can use as measure $$\delta(t) \equiv \mbox{Tr}\, \rho^2(t) =
1 - 2 \,e^{-\,\Gamma\,t} + 2 \,e^{-\,2\,\Gamma\,t}.$$ We have $\delta(0) = 1$, $\lim_{t\to\infty} \delta(t) = 1$ and $\delta(t) < 1$ for $0 < t < \infty$, i.e. in the beginning, $\rho$ represents a pure state, then the state is mixed, whereas for large times it approaches a pure state again, when the system definitely has changed into the decay state.
The case of neutral kaons, where the Hamitonian $H$ and the operator $B$ are $2\times 2$ matrices, has been investigated in detail in Ref. [@Caban-etal].
#### Kraus operators:
Finally, in the case of our example, we would like to consider the time evolution of the density matrix as a dynamical map $V_t$. In our case, $V_t$ can be represented by a sum of operators [@BreuerPetruccione; @AlickiFannes]: $$\label{dynmap}
\rho(0) \;\longrightarrow\; V_t[\rho(0)] =
\sum_j M_j(t) \,\rho(0) M_j^\dagger(t) = \rho(t),$$ with the normalization $\sum_j M_j^\dagger(t) M_j(t) = \mathbbm{1}$ for the Kraus operators $M_j(t)$. For the correspondence between the Lindblad and Kraus operators see Ref. [@Preskill-notes].
The operator sum representation (\[dynmap\]) is a very useful approach in quantum information to describe the quantum operations or the specific quantum channels. In our example, the decay of a particle corresponds to the amplitude damping channel of a quantum operation [@NielsenChuang], e.g., the spontaneous emission of a photon, and we can calculate the Kraus operators needed: $$M_0(t) = \left( {\begin{array}}{cc}
\sqrt{1-p(t)} & 0\\
0 & 1\\
{\end{array}}\right) \quad \mathrm{and} \quad
M_1(t) = \left( {\begin{array}}{cc}
0 & 0\\
\sqrt{p(t)} & 0\\
{\end{array}}\right),$$ with the probability $p(t) = 1 - e^{-\Gamma t}\,$.
#### Conclusions:
In this letter we have considered a time evolution given by Eq. (\[Lindbladequation\]), with a non-hermitian Hamiltonian $H_\mathrm{eff}$ and a dissipator of the Lindblad form (\[dissipator\]). Assuming that the non-hermitian part of $H_\mathrm{eff}$ describes “particle decay,” we have shown that such a time evolution is completely positive. Our strategy was to add the space of “decay states” ${\mathbf{H}}_f$ to the space of states ${\mathbf{H}}_s$ and to extend the time evolution in a straightforward way to the full space ${\mathbf{H}}_s \oplus {\mathbf{H}}_f$, such that this full time evolution is probability–conserving. With the initial condition that the density matrix is only non-zero on ${\mathbf{H}}_s$, the Lindblad operator $\mathcal{B}$ of Eq. (\[operator-B\]), which is responsible for the decay, shifts states from ${\mathbf{H}}_s$ to ${\mathbf{H}}_f$, whereas at the same time the $\mathcal{A}_j$ terms cause decoherence and dissipation on ${\mathbf{H}}_s$. Thus, particle decay and decoherence/dissipation are related phenomena, which can be described by the same formalism of a completely positive time evolution.
*Acknowledgement:* The authors acknowledge financial support of EURIDICE HPRN-CT-2002-00311.
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[^1]: E-mail: reinhold.bertlmann@univie.ac.at
[^2]: E-mail: walter.grimus@univie.ac.at
[^3]: E-mail: beatrix.hiesmayr@univie.ac.at
|
---
abstract: 'Filamentary structures are ubiquitous from large-scale molecular clouds (few parsecs) to small-scale circumstellar envelopes around Class 0 sources ($\sim$1000 AU to $\sim$0.1 pc). In particular, recent observations with the *Herschel Space Observatory* emphasize the importance of large-scale filaments (few parsecs) and star formation. The small-scale flattened envelopes around Class 0 sources are reminiscent of the large-scale filaments. We propose an observationally derived scenario for filamentary star formation that describes the evolution of filaments as part of the process for formation of cores and circumstellar envelopes. If such a scenario is correct, small-scale filamentary structures (0.1 pc in length) with higher densities embedded in starless cores should exist, although to date almost all the interferometers have failed to observe such structures. We perform synthetic observations of filaments at the prestellar stage by modeling the known Class 0 flattened envelope in L1157 using both the Combined Array for Research in Millimeter-wave Astronomy (CARMA) and the Atacama Large Millimeter/Submillimeter Array (ALMA). We show that with reasonable estimates for the column density through the flattened envelope, the CARMA D-array at 3mm wavelengths is not able to detect such filamentary structure, so previous studies would not have detected them. However, the substructures may be detected with CARMA D$+$E array at 3 mm and CARMA E array at 1 mm as a result of more appropriate resolution and sensitivity. ALMA is also capable of detecting the substructures and showing the structures in detail compared to the CARMA results with its unprecedented sensitivity. Such detection will confirm the new proposed paradigm of non-spherical star formation.'
author:
- 'Katherine Lee, Leslie Looney'
- Doug Johnstone
- John Tobin
bibliography:
- 'paper.bib'
title: 'Filamentary Star Formation: Observing the Evolution toward Flattened Envelopes'
---
|
---
abstract: 'The aim of this paper is to characterize the nonnegative functions $\varphi$ defined on $(0,\infty)$ for which the Hausdorff operator $$\Ha_\varphi f(z)= \int_0^\infty f\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}dt$$ is bounded on the Hardy spaces of the upper half-plane $\mathcal H_a^p(\mathbb C_+)$, $p\in[1,\infty]$. The corresponding operator norms and their applications are also given.'
address:
- 'High School for Gifted Students, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam'
- 'Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet Nam'
- 'Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet Nam'
author:
- Ha Duy Hung
- Luong Dang Ky
- Thai Thuan Quang
title: Hausdorff operators on holomorphic Hardy spaces and applications
---
[^1]
\[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\]
Introduction and the main result
================================
Let $\varphi$ be a locally integrable function on $(0,\infty)$. The [*Hausdorff operator*]{} $H_\varphi$ is then defined for suitable functions $f$ on ${{\mathbb R}}$ by $$\label{real Hausdorff operator}
H_\varphi f(x)=\int_0^\infty f\left(\frac{x}{t}\right) \frac{\varphi(t)}{t} dt,\quad x\in{{\mathbb R}}.$$
The Hausdorff operator is an interesting operator in harmonic analysis. There are many classical operators in analysis which are special cases of the Hausdorff operator if one chooses suitable kernel functions $\varphi$, such as the classical Hardy operator, its adjoint operator, the Cesàro type operators, the Riemann-Liouville fractional integral operator,... See the survey article [@Li] and the references therein. In the recent years, there is an increasing interest in the study of boundedness of the Hausdorff operator and its commuting with the Hilbert transform on the real Hardy spaces and on the Lebesgue spaces, see for example [@An; @AS; @BBMM; @BHS; @HKQ; @Li07; @Li; @LM; @LM2; @Xi].
Let $\mathbb C_+$ be the upper half-plane in the complex plane. For $0<p\leq \infty$, the Hardy space $\mathcal H_a^p(\mathbb C_+)$ is defined as the set of all holomorphic functions $f$ on $\mathbb C_+$ such that $$\|f\|_{\mathcal H_a^p(\mathbb C_+)}:= \sup_{y>0} \left(\int_{-\infty}^{\infty} |f(x+iy)|^p dx\right)^{1/p}<\infty$$ if $0<p<\infty$, and if $p=\infty$, then $$\|f\|_{\mathcal H_a^\infty(\mathbb C_+)}:= \sup_{z\in\mathbb C_+} |f(z)|<\infty.$$
It is classical (see [@Du; @Ga]) that if $f\in \H_a^p(\mathbb C_+)$, then $f$ has a [*boundary value function*]{} $f^*\in L^p({{\mathbb R}})$ defined by $$f^*(x)= \lim_{y\to 0} f(x+iy),\quad \mbox{a.e.}\; x\in{{\mathbb R}}.$$
Let $p\in [1,\infty]$ and let $\varphi$ be a nonnegative function in $L^1_{\rm loc}(0,\infty)$ for which $$\label{main inequality}
\int_0^\infty t^{1/p-1}\varphi(t)dt<\infty.$$ Then it is well-known (see [@An]) that $H_\varphi$ is bounded on $L^p({{\mathbb R}})$, and thus $H_\varphi(f^*)\in L^p({{\mathbb R}})$ for any boundary value function $f^*$ of a function $f$ in $\H_a^p(\mathbb C_+)$. A natural question arises is that whether the transformed function $H_\varphi(f^*)$ is also the boundary value function of a function in $\H_a^p(\mathbb C_+)$? In some special cases of $\varphi$ and $1<p<\infty$, using the spectral mapping theorem and the Hille-Yosida-Phillips theorem, Arvanitidis-Siskakis [@AS] and Ballamoole-Bonyo-Miller-Millerstudied [@BBMM] studied and gave affirmative answers to this question.
In the present paper, we give an affirmative answer to the above question by studying a complex version of $H_\varphi$ defined by $$\Ha_\varphi f(z)= \int_0^\infty f\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}dt,\quad z\in \mathbb C_+.$$
Our main result reads as follows.
\[main theorem\] Let $p\in [1,\infty]$ and let $\varphi$ be a nonnegative function in $L^1_{\rm loc}(0,\infty)$. Then $\Ha_\varphi$ is bounded on $\H_a^p(\mathbb C_+)$ if and only if (\[main inequality\]) holds. Moreover, in that case, we obtain $$\|\Ha_\varphi\|_{\H^p_a(\mathbb C_+)\to \H^p_a(\mathbb C_+)}=\int_0^\infty t^{1/p-1}\varphi(t)dt$$ and, for any $f\in \H_a^p(\mathbb C_+)$, $$(\Ha_\varphi f)^*= H_\varphi(f^*).$$
It should be pointed out that some main results in [@AS; @BBMM] (see [@AS Theorems 3.1, 3.3 and 4.1] and [@BBMM Theorem 3.4]) can be viewed as special cases of Theorem \[main theorem\] by choosing suitable kernel functions $\varphi$. In the setting of Hardy spaces $\mathcal H^p(\mathbb D)$ on the unit disk $\mathbb D=\{z\in \mathbb C: |z|<1\}$, Galanopoulos and Papadimitrakis ([@GP Theorems 2.3 and 2.4]) studied and obtained some similar results to Theorem \[main theorem\] for $1<p<\infty$ while it is slightly different at the endpoints $p=1$ and $p=\infty$ (see also the survey article [@Li]).
Furthermore, if we denote by $\H^1({{\mathbb R}})$ the real Hardy space in the sense of Fefferman-Stein (see the last section), then by using Theorem \[main theorem\], we obtain the following result.
Let $\varphi$ be a nonnegative function in $L^1_{\rm loc}(0,\infty)$ such that $H_\varphi$ is bounded on $\H^1({{\mathbb R}})$. Then, $$\int_0^\infty \varphi(t)dt \leq \|H_\varphi\|_{\H^1({{\mathbb R}})\to \H^1({{\mathbb R}})}<\infty.$$
The above corollary is not only give an answer to a question posted by Liflyand [@Li07 Problem 4], but also give a lower bound for the norm of $H_\varphi$ on $\H^1({{\mathbb R}})$. Another corollary of Theorem \[main theorem\] is:
Let $p\in (1,\infty)$ and let $\varphi$ be as in Theorem \[main theorem\]. Then $H_\varphi$ is bounded on $L^p({{\mathbb R}})$ if and only if (\[main inequality\]) holds. Moreover, in that case, $$\|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}= \int_0^\infty t^{1/p-1}\varphi(t)dt$$ and $H_\varphi$ commutes with the Hilbert transform $H$ on $L^p({{\mathbb R}})$.
Throughout the whole article, we use the symbol $A \lesssim B$ (or $B\gtrsim A$) means that $A\leq C B$ where $C$ is a positive constant which is independent of the main parameters, but it may vary from line to line. If $A \lesssim B$ and $B\lesssim A$, then we write $A\sim B$. For any $E\subset {{\mathbb R}}$, we denote by $\chi_E$ its characteristic function.
Proof of Theorem \[main theorem\]
=================================
In the sequel, we always assume that $\varphi$ is a nonnegative function in $L^1_{\rm loc}(0,\infty)$. Also we remark that, for any $f\in \mathcal H_a^p(\mathbb C_+)$, the function $\Hau f$ is well-defined and holomorphic on $\mathbb C_+$ provided (\[main inequality\]) holds, since $$\label{a pointwise estimate for f}
|f(x+iy)|\leq \left(\frac{2}{\pi y}\right)^{1/p} \|f\|_{\mathcal H_a^p(\mathbb C_+)}$$ and $$|f'(x+iy)| \leq \frac{2}{y}\left(\frac{4}{\pi y}\right)^{1/p}\|f\|_{\mathcal H_a^p(\mathbb C_+)}$$ for all $z=x+iy\in\mathbb C_+$. See Garnett’s book [@Ga p. 57].
Given an holomorphic function $f$ on $\mathbb C_+$, we define the [*nontangential maximal function*]{} of $f$ by $$\mathcal M(f)(x)= \sup_{|t-x|<y} |f(t+iy)|, \quad x\in {{\mathbb R}}.$$
The following lemma is classical and can be found in [@Du; @Ga].
\[nontangential maximal function characterization\] Let $0<p<\infty$. Then:
1. For any $f\in \mathcal H_a^p(\mathbb C_+)$, we have $$\|f^*\|_{L^p({{\mathbb R}})}= \|f\|_{\mathcal H_a^p(\mathbb C_+)}\quad\mbox{and}\quad \lim_{y\to 0}\|f(\cdot+iy)- f^*(\cdot)\|_{L^p({{\mathbb R}})}=0.$$
2. $f\in \mathcal H_a^p(\mathbb C_+)$ if and only if $\mathcal M(f)\in L^p({{\mathbb R}})$. Moreover, $$\|f\|_{\mathcal H_a^p(\mathbb C_+)}\sim \|\mathcal M(f)\|_{L^p({{\mathbb R}})}.$$
\[a lemma for p is infinity\] Theorem \[main theorem\] is true for $p=\infty$.
Suppose that $\int_0^\infty t^{-1}\varphi(t)dt$ is finite. Then, for any $f\in \H^\infty_a(\mathbb C_+)$, $$\|\Ha_\varphi f\|_{\mathcal H_a^\infty(\mathbb C_+)}=\sup_{z\in\mathbb C_+}\left|\int_0^\infty f\left(\frac{z}{t}\right) \frac{\varphi(t)}{t} dt\right|\leq \int_0^\infty t^{-1}\varphi(t)dt \|f\|_{\mathcal H^\infty_a(\mathbb C_+)}.$$ Therefore, $\Ha_\varphi$ is bounded on $\mathcal H^\infty_a(\mathbb C_+)$, moreover, $$\label{a lemma for p is infinity, 1}
\|\Hau\|_{\H^\infty_a(\mathbb C_+)\to \H^\infty_a(\mathbb C_+)}\leq \int_0^\infty t^{-1}\varphi(t)dt.$$
On the other hand, we have $$\|\Hau\|_{\H^\infty_a(\mathbb C_+)\to \H^\infty_a(\mathbb C_+)}\geq \frac{\|\Hau(1)\|_{\H^\infty_a(\mathbb C_+)}}{\|1\|_{\H^\infty_a(\mathbb C_+)}}=\int_0^\infty t^{-1}\varphi(t)dt.$$ This, together with (\[a lemma for p is infinity, 1\]), implies that $$\|\Hau\|_{\H^\infty_a(\mathbb C_+)\to \H^\infty_a(\mathbb C_+)}= \int_0^\infty t^{-1}\varphi(t)dt.$$ Moreover, by the dominated convergence theorem, for any $x\ne 0$, $$(\Ha_\varphi f)^*(x)=\lim_{y\to 0}\int_0^\infty f\left(\frac{x}{t}+ \frac{y}{t}i\right) \frac{\varphi(t)}{t}dt = \int_0^1 f^*\left(\frac{x}{t}\right) \frac{\varphi(t)}{t}dt = H_\varphi(f^*)(x).$$
Conversely, suppose that $\Hau$ is bounded on $\H^\infty_a(\mathbb C_+)$. As the function $f(z)\equiv 1$ is in $\H^\infty_a(\mathbb C_+)$, we obtain that $\Hau f= \int_0^1 t^{-1}\varphi(t)dt<\infty$.
\[key lemma\] Let $p\in [1,\infty)$ and let $\varphi$ be such that (\[main inequality\]) holds. Then
1. $\Hau$ is bounded on $\Hd$, moreover, $$\|\Hau\|_{\Hd\to \Hd}\leq \int_0^\infty t^{1/p-1}\varphi(t)dt.$$
2. If supp $\varphi\subset [0,1]$, then $$\|\Hau\|_{\Hd\to \Hd} = \int_0^1 t^{1/p-1}\varphi(t)dt.$$
3. For any $f\in \Hd$, we have $$(\Ha_\varphi f)^*= H_\varphi(f^*).$$
\(i) For any $f\in \mathcal H^p_a(\mathbb C_+)$, we have $$\begin{aligned}
\mathcal M(\Ha_\varphi f)(x) &=& \sup_{|u-x|<y} \left|\int_0^\infty f\left(\frac{u+iy}{t}\right)\frac{\varphi(t)}{t}dt\right|\\
&\leq& \int_0^\infty \sup_{|\frac{u}{t}-\frac{x}{t}|<\frac{y}{t}} \left|f\left(\frac{u}{t}+\frac{y}{t}i\right)\right|\frac{\varphi(t)}{t}dt = H_{\varphi}(\mathcal M f)(x)
\end{aligned}$$ for all $x\in \mathbb R$. Therefore, by the Minkowski inequality and Lemma \[nontangential maximal function characterization\](ii), $$\begin{aligned}
\|\Ha_\varphi f\|_{\mathcal H^p_a(\mathbb C_+)}\lesssim \|\mathcal M(\Ha_\varphi f)\|_{L^p(\mathbb R)} &\leq& \|H_{\varphi}(\mathcal M f)\|_{L^p(\mathbb R)}\\
&\leq& \int_0^\infty \left(\int_{{{\mathbb R}}} \left|\mathcal M f\left(\frac{x}{t}\right)\right|^p dx\right)^{1/p} \frac{\varphi(t)}{t} dt \\
&=& \int_0^\infty t^{1/p-1}\varphi(t) dt \|\mathcal M f\|_{L^p(\mathbb R)}\\
&\lesssim& \int_0^\infty t^{1/p-1}\varphi(t) dt \|f\|_{\mathcal H^p_a(\mathbb C_+)}.
\end{aligned}$$ This proves that $\Hau$ is bounded on $\mathcal H^p_a(\mathbb C_+)$, moreover, $$\label{key lemma, 0}
\|\Hau\|_{\Hd\to \Hd}\lesssim \int_0^\infty t^{1/p-1}\varphi(t) dt.$$
In order to show $$\label{key lemma, 1}
\|\Hau\|_{\Hd\to \Hd}\leq \int_0^\infty t^{1/p-1}\varphi(t) dt,$$ let us first assume that (iii) is proved. Then, by Lemma \[nontangential maximal function characterization\](i) and the Minkowski inequality, we get $$\begin{aligned}
\left\|\Hau f\right\|_{\mathcal H_a^p(\mathbb C_+)} = \left\|(\Hau f)^*\right\|_{L^p(\mathbb R)} &=&\left\|H_\varphi (f^* )\right\|_{L^p(\mathbb R)}\\
&\leq& \int_0^\infty \left(\int_{{{\mathbb R}}} \left|f^*\left(\frac{x}{t}\right)\right|^p dx\right)^{1/p} \frac{\varphi(t)}{t} dt\\
&=& \|f^*\|_{L^p(\mathbb R)} \int_0^\infty t^{1/p-1}\varphi(t)dt \\
&=& \|f\|_{\mathcal H_a^p(\mathbb C_+)} \int_0^\infty t^{1/p-1}\varphi(t)dt.
\end{aligned}$$ This proves that (\[key lemma, 1\]) holds.
0.3cm
\(ii) Let $\delta\in (0,1)$ be arbitrary and let $\varphi_\delta(t)= \varphi(t)\chi_{[\delta,\infty)}(t)$ for all $t\in (0,\infty)$. Since (\[key lemma, 1\]) holds, we see that $$\|\mathscr H_{\varphi_\delta}\|_{\mathcal H^p_a(\mathbb C_+) \to \mathcal H^p_a(\mathbb C_+)} \leq \int_0^\infty t^{1/p-1}\varphi_\delta(t)dt = \int_\delta^1 t^{1/p-1}\varphi(t)dt <\infty$$ and $$\label{the truncation}
\|\Hau -\mathscr H_{\varphi_\delta}\|_{\mathcal H^p_a(\mathbb C_+) \to \mathcal H^p_a(\mathbb C_+)} \leq \int_0^\infty t^{1/p-1}[\varphi(t)-\varphi_\delta(t)] dt= \int_0^\delta t^{1/p-1}\varphi(t)dt.$$
For any $\varepsilon>0$, we define the function $f_\varepsilon: \mathbb C_+ \to\mathbb C$ by $$f_\varepsilon(z)= \frac{1}{(z+i)^{1/p+\varepsilon}},$$ where, and in what follows, $\zeta^{1/p+\varepsilon}=|\zeta|^{1/p+\varepsilon} e^{i(1/p+\varepsilon)\arg \zeta}$ for all $\zeta\in\mathbb C$. Then $$\label{norm of f}
\|f_\varepsilon\|_{\mathcal H^p_a(\mathbb C_+)}= \left(\int_{-\infty}^{\infty}\frac{1}{\sqrt{x^2+1}^{1+p\varepsilon}} dx\right)^{1/p}<\infty.$$
For all $z= x+iy\in\mathbb C_+$, we have $$\mathscr H_{\varphi_\delta}(f_\varepsilon)(z) - f_\varepsilon(z)\int_0^\infty t^{1/p-1}\varphi_\delta(t)dt = \int_{\delta}^{1}[\phi_{\varepsilon,z}(t)-\phi_{\varepsilon,z}(1)] t^{1/p-1} \varphi(t)dt,$$ where $\phi_{\varepsilon,z}(t):= \frac{t^{\varepsilon}}{(z+ ti)^{1/p+\varepsilon}}$. For any $t\in [\delta,1]$, a simple calculus gives $$\begin{aligned}
|\phi_{\varepsilon,z}(t)-\phi_{\varepsilon,z}(1)| &\leq& |t-1|\sup_{s\in [\delta,1]}|\phi_{\varepsilon,z}'(s)|\\
&\leq& \frac{\varepsilon \delta^{-1-1/p}}{\sqrt{x^2+ 1}^{1/p+\varepsilon}} + \frac{(1/p+\varepsilon) \delta^{-1-1/p}}{\sqrt{x^2+1}^{1+1/p+\varepsilon}}.\end{aligned}$$ This, together with (\[norm of f\]), yields $$\begin{aligned}
\label{an estimate for the norm}
&&\frac{\left\|\mathscr H_{\varphi_\delta}(f_\varepsilon) - f_\varepsilon \int_0^\infty t^{1/p-1}\varphi_\delta(t)dt \right\|_{\mathcal H^p_a(\mathbb C_+)}}{\|f_\varepsilon\|_{\mathcal H^p_a(\mathbb C_+)}}\\
&\leq& \int_{\delta}^{1} t^{1/p-1}\varphi(t)dt\left[\varepsilon \delta^{-1-1/p}+ \frac{(1/p+\varepsilon) \delta^{-1-1/p}\left(\int_{-\infty}^{\infty} \frac{1}{\sqrt{x^2+1}^{p+1}}\right)^{1/p}}{\left(\int_{-\infty}^{\infty}\frac{1}{\sqrt{x^2+1}^{1+p\varepsilon}}dx\right)^{1/p}}\right] \to 0\nonumber\end{aligned}$$ as $\varepsilon\to 0$. As a consequence, $$\int_{\delta}^{1} t^{1/p-1}\varphi(t)dt=\int_0^\infty t^{1/p-1}\varphi_\delta(t)dt \leq \|\mathscr H_{\varphi_\delta}\|_{\mathcal H^p_a(\mathbb C_+)\to \mathcal H^p_a(\mathbb C_+)}.$$ This, combined with (\[the truncation\]), allows us to conclude that $$\|\Hau\|_{\mathcal H^p_a(\mathbb C_+)\to \mathcal H^p_a(\mathbb C_+)}\geq \int_0^1 t^{1/p-1}\varphi(t)dt - 2 \int_0^\delta t^{1/p-1}\varphi(t)dt\to \int_0^1 t^{1/p-1}\varphi(t)dt$$ as $\delta \to 0$ since $\int_0^1 t^{1/p-1}\varphi(t)dt<\infty$. Hence, by (\[key lemma, 1\]), $$\|\Hau\|_{\mathcal H^p_a(\mathbb C_+)\to \mathcal H^p_a(\mathbb C_+)}= \int_0^1 t^{1/p-1}\varphi(t)dt.$$
\(iii) For any $\sigma>0$, it follows from (\[a pointwise estimate for f\]) that the function $$f_\sigma(z):= f(z+ i\sigma)$$ is in $\mathcal H_a^p(\mathbb C_+)\cap \mathcal H^\infty_a(\mathbb C_+)$. Let $\delta$ and $\varphi_\delta$ be as in (ii). Noting that $$\int_0^\infty t^{-1}\varphi_\delta(t)dt\leq \delta^{-1/p}\int_{\delta}^{\infty}t^{1/p-1}\varphi(t)dt<\infty,$$ Lemma \[a lemma for p is infinity\](ii) gives $(\mathscr H_{\varphi_\delta}(f_\sigma))^*= H_{\varphi_\delta}(f_\sigma^*)$. Therefore, by Lemma \[nontangential maximal function characterization\](i), [@An Theorem 1] and (\[key lemma, 0\]), we obtain that $$\begin{aligned}
&&\|(\Hau f)^* - H_\varphi(f^*)\|_{L^p({{\mathbb R}})}\\
&\leq& \|\mathscr H_{\varphi-\varphi_\delta} f\|_{\Hd} + \|H_{\varphi-\varphi_\delta}(f^*)\|_{L^p({{\mathbb R}})} + \\
&& +\|\mathscr H_{\varphi_\delta}(f- f_\sigma)\|_{\Hd} + \|H_{\varphi_\delta}(f^* - f_\sigma^*)\|_{L^p({{\mathbb R}})}\\
&\lesssim& \|f\|_{\Hd} \int_0^\infty t^{1/p-1}[\varphi(t)-\varphi_\delta(t)]dt + \|f^*-f_\sigma^*\|_{L^p({{\mathbb R}})}\int_0^\infty t^{1/p-1}\varphi_\delta(t)dt\\
&\lesssim& \|f\|_{\Hd} \int_0^\delta t^{1/p-1}\varphi(t)dt + \|f^*-f_\sigma^*\|_{L^p({{\mathbb R}})}\int_0^\infty t^{1/p-1}\varphi(t)dt\to 0
\end{aligned}$$ as $\sigma\to 0$ and $\delta\to 0$. This completes the proof of Lemma \[key lemma\].
Now we are ready to give the proof of Theorem \[main theorem\].
By Lemmas \[a lemma for p is infinity\] and \[key lemma\], it suffices to prove that $$\label{main theorem, 0}
\int_0^\infty t^{1/p-1} \varphi(t)dt \leq \|\Hau\|_{\Hd \to \Hd}$$ whenever $\Hau$ is bounded on $\Hd$ for $1\leq p<\infty$.
Indeed, we first claim that $$\label{necessary condition}
\int_0^\infty t^{1/p-1} \varphi(t)dt <\infty.$$
Assume (\[necessary condition\]) holds for a moment.
For any $m>0$, we define $\varphi_m(t)= \varphi(mt)\chi_{(0,1]}(t)$ for all $t\in (0,\infty)$. Then, by Lemma \[key lemma\](i), we see that $$\begin{aligned}
\label{main theorem, 1}
\left\|\Hau -\mathscr H_{\varphi_m\left(\frac{\cdot}{m}\right)}\right\|_{\Hd \to \Hd} &=& \left\|\mathscr H_{\varphi-\varphi_m\left(\frac{\cdot}{m}\right)}\right\|_{\Hd \to \Hd}\nonumber\\
&\leq& \int_0^\infty t^{1/p-1} \left[\varphi(t)- \varphi_m\left(\frac{t}{m}\right)\right] dt \nonumber\\
&=& \int_m^\infty t^{1/p-1}\varphi(t)dt<\infty.\end{aligned}$$
Noting that $$\left\|f\left(\frac{\cdot}{m}\right)\right\|_{\Hd}= m^{1/p} \|f(\cdot)\|_{\Hd}\quad\mbox{and}\quad \mathscr H_{\varphi_m\left(\frac{\cdot}{m}\right)} f= \mathscr H_{\varphi_m} f\left(\frac{\cdot}{m}\right)$$ for all $f\in\Hd$, Lemma \[key lemma\](ii) gives $$\begin{aligned}
\left\|\mathscr H_{\varphi_m\left(\frac{\cdot}{m}\right)}\right\|_{\Hd \to \Hd} &=& m^{1/p} \left\|\mathscr H_{\varphi_m}\right\|_{\Hd \to \Hd}\\
&=& m^{1/p} \int_0^1 t^{1/p-1} \varphi_m(t)dt = \int_0^m t^{1/p-1} \varphi(t)dt.\end{aligned}$$ Combining this with (\[main theorem, 1\]) allows us to conclude that $$\|\Hau\|_{\Hd\to \Hd}\geq \int_0^\infty t^{1/p-1} \varphi(t)dt - 2 \int_m^\infty t^{1/p-1} \varphi(t)dt \to \int_0^\infty t^{1/p-1} \varphi(t)dt$$ as $m\to\infty$ since $\int_0^\infty t^{1/p-1} \varphi(t)dt<\infty$. This proves (\[main theorem, 0\]).
0.3cm
Now we return to prove (\[necessary condition\]). Indeed, we consider the following two cases.
[**Case 1:**]{} $p=1$. Take $f(z)= \frac{1}{(z+i)^2}$ for all $z\in\mathbb C_+$. Then $$\|f\|_{\mathcal H^1_a(\mathbb C_+)}= \int_{-\infty}^{\infty} \frac{1}{x^2+1} dx<\infty.$$ Therefore, by the Fatou lemma, we get $$\begin{aligned}
\infty > \|\Hau f\|_{\mathcal H^1_a(\mathbb C_+)} &=& \sup_{y>0} \int_{-\infty}^{\infty} \left|\int_0^\infty \frac{1}{\left[\frac{x}{t} + i\left(\frac{y}{t}+1\right)\right]^2} \frac{\varphi(t)}{t}dt\right|dx\\
&\geq& 2 \sup_{y>0} \int_{0}^{\infty} dx \int_{0}^{\infty}\frac{\frac{x}{t}\left(\frac{y}{t}+1\right)}{\left[\left(\frac{x}{t}\right)^2 + \left(\frac{y}{t}+1\right)^2\right]^2} \frac{\varphi(t)}{t}dt\\
&\geq& 2 \int_{0}^{\infty} dx \int_{0}^{\infty} \frac{\frac{x}{t}}{\left[\left(\frac{x}{t}\right)^2 + 1\right]^2} \frac{\varphi(t)}{t}dt\\
&=& 2 \int_{0}^{\infty}\frac{u}{[u^2+1]^2} du \int_0^\infty \varphi(t)dt.\end{aligned}$$ This proves (\[necessary condition\]).
0.15cm
[**Case 2:**]{} $1<p<\infty$. For any $0<\varepsilon<1-1/p$, take $$f_\varepsilon(z)= \left(\frac{1}{z+ i \varepsilon }\right)^{1/p+\varepsilon}$$ for all $z\in \mathbb C_+$. Then $$\label{norm of f for p>1}
\|f_\varepsilon\|_{\mathcal H^p_a(\mathbb C_+)}= \left(\int_{-\infty}^{\infty} \frac{1}{{\sqrt{x^2+\varepsilon^2}}^{1+p\varepsilon}} dx\right)^{1/p} <\infty$$ and $$\begin{aligned}
\infty>\|\Hau(f_\varepsilon)\|^p_{\mathcal H^p_a(\mathcal C_+)} &=& \sup_{y>0}\int_{-\infty}^{\infty}\left|\int_0^\infty \left(\frac{1}{\frac{x}{t}+ i\left(\frac{y}{t}+\varepsilon\right)}\right)^{1/p+\varepsilon} \frac{\varphi(t)}{t} dt \right|^p dx\\
&\geq& \int_0^\infty \left|\int_0^\infty \frac{\frac{x}{t}}{\sqrt{\left(\frac{x}{t}\right)^2+\varepsilon^2}} \frac{1}{\sqrt{\left(\frac{x}{t}\right)^2+ \varepsilon^2}^{1/p+\varepsilon}}\frac{\varphi(t)}{t} dt\right|^p dx,\end{aligned}$$ where we used the Fatou lemma and the fact that $$\mbox{Re} \left(\frac{1}{\frac{x}{t}+ i\left(\frac{y}{t}+\varepsilon\right)}\right)^{1/p+\varepsilon}\geq \frac{\frac{x}{t}}{\sqrt{\left(\frac{x}{t}\right)^2+\left(\frac{y}{t}+\varepsilon\right)^2}} \frac{1}{\sqrt{\left(\frac{x}{t}\right)^2+\left(\frac{y}{t}+\varepsilon\right)^2}^{1/p+\varepsilon}}$$ for all $x,y,t>0$ since $0<1/p+\varepsilon<1$. This, together with (\[norm of f for p>1\]), gives $$\begin{aligned}
\|\Hau\|^p_{\mathcal H^p_a(\mathbb C_+)\to \mathcal H^p_a(\mathbb C_+)} &\geq& \frac{\|\Hau(f_\varepsilon)\|^p_{\mathcal H^p_a(\mathbb C_+)}}{\|f_\varepsilon\|^p_{\mathcal H^p_a(\mathbb C_+)}}\\
&\geq& \frac{\int_1^\infty \left|\int_0^{1/\varepsilon} \frac{\frac{x}{t}}{\sqrt{\left(\frac{x}{t}\right)^2+\varepsilon^2}} \frac{1}{\sqrt{\left(\frac{x}{t}\right)^2+ \varepsilon^2}^{1/p+\varepsilon}}\frac{\varphi(t)}{t} dt\right|^p dx}{2 \int_0^\infty \frac{1}{{\sqrt{x^2+\varepsilon^2}}^{1+p\varepsilon}} dx}\\
&\geq& \frac{1}{2^{\frac{3+ p(1+\varepsilon)}{2}}} \left(\int_{0}^{1/\varepsilon} t^{1/p-1+\varepsilon} \varphi(t)dt\right)^p \frac{\int_1^\infty \frac{1}{x^{1+p\varepsilon}}dx}{\varepsilon^{-p\varepsilon}\int_0^\infty \frac{1}{\sqrt{x^2+1}^{1+p\varepsilon}}dx}.\end{aligned}$$ Hence, $$\int_{0}^{1/\varepsilon} t^{1/p-1+\varepsilon} \varphi(t)dt \leq 2^{\frac{3+ p(1+\varepsilon)}{2p}} \varepsilon^{-\varepsilon} \left(\frac{\int_0^\infty \frac{1}{\sqrt{x^2+1}^{1+p\varepsilon}}dx}{\int_1^\infty \frac{1}{x^{1+p\varepsilon}}dx}\right)^{1/p} \|\Hau\|_{\mathcal H^p_a(\mathbb C_+)\to \mathcal H^p_a(\mathbb C_+)}.$$ Letting $\varepsilon \to 0$, we obtain $$\int_0^\infty t^{1/p-1} \varphi(t)dt \leq 2^{\frac{3+p}{2p}}\|\Hau\|_{\mathcal H^p_a(\mathbb C_+)\to \mathcal H^p_a(\mathbb C_+)}<\infty.$$ This proves (\[necessary condition\]), and thus ends the proof of Theorem \[main theorem\].
Some applications
=================
Let $1\leq p<\infty$, we define (see [@St]) the [*Hilbert transform*]{} of $f\in L^p({{\mathbb R}})$ by $$H(f)(x):= \frac{1}{\pi}{\rm p. v.}\int_{-\infty}^{\infty} \frac{f(y)}{x-y}dy, \quad x\in{{\mathbb R}}.$$
\[commutes with the Hilbert transform\] Let $p\in (1,\infty)$ and let $\varphi$ be as in Theorem \[main theorem\]. Then $H_\varphi$ is bounded on $L^p({{\mathbb R}})$ if and only if (\[main inequality\]) holds. Moreover, in that case, $$\|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}= \int_0^\infty t^{1/p-1}\varphi(t)dt$$ and $H_\varphi$ commutes with the Hilbert transform $H$ on $L^p({{\mathbb R}})$.
In order to prove Theorem \[commutes with the Hilbert transform\], we need the following lemmas.
\[boundary value characterization 1\] Let $1< p<\infty$. Then:
1. If $g\in L^p({{\mathbb R}})$, then $f^*:= g + iH(g)$ is the boundary value function of some function $f\in \mathcal H_a^p(\mathbb C_+)$.
2. Conversely, if $f^*$ is a boundary value function of $f\in \mathcal H_a^p(\mathbb C_+)$, then there exists a real-valued function $g\in L^p({{\mathbb R}})$ such that $f^*= g+iH(g)$.
Moreover, in those cases, $$\|g\|_{L^p({{\mathbb R}})} \sim \|g+ iH(g)\|_{L^p({{\mathbb R}})}=\|f^*\|_{L^p({{\mathbb R}})}=\|f\|_{\mathcal H_a^p(\mathbb C_+)}.$$
\[key lemma for commuting relation with the Hilbert transform\] Let $p\in (1,\infty)$ and let $\varphi$ be such that (\[main inequality\]) holds. Then:
1. $H_\varphi$ is bounded on $L^p({{\mathbb R}})$, moreover, $$\|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}\leq \int_0^\infty t^{1/p-1}\varphi(t)dt.$$
2. If supp $\varphi\subset [0,1]$, then $$\|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}= \int_0^1 t^{1/p-1}\varphi(t)dt.$$
Suppose that (\[main inequality\]) holds. By Lemma \[key lemma for commuting relation with the Hilbert transform\](i), $$\label{commutes with the Hilbert transform, 1}
\|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}\leq \int_0^\infty t^{1/p-1}\varphi(t)dt.$$
Conversely, suppose that $H_\varphi$ is bounded on $L^p({{\mathbb R}})$. We first claim that $$\label{commutes with the Hilbert transform, 2}
\int_0^\infty t^{1/p-1}\varphi(t)dt<\infty.$$
Assume (\[commutes with the Hilbert transform, 2\]) holds for a moment.
For any $m>0$, take $\varphi_m$ is as in the proof of Theorem \[main theorem\]. Then, by a similar argument to the proof of Theorem \[main theorem\], we get $$\begin{aligned}
\|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})} &\geq& \|H_{\varphi_m\left(\frac{\cdot}{m}\right)}\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})} - \|H_\varphi - H_{\varphi_m\left(\frac{\cdot}{m}\right)}\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}\\
&\geq& \int_0^\infty t^{1/p-1}\varphi(t)dt - 2 \int_m^\infty t^{1/p-1}\varphi(t)dt\to \int_0^\infty t^{1/p-1}\varphi(t)dt
\end{aligned}$$ as $m\to\infty$. This, together with (\[commutes with the Hilbert transform, 1\]), yields $$\|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})} = \int_0^\infty t^{1/p-1}\varphi(t)dt.$$
Now let us return to prove (\[commutes with the Hilbert transform, 2\]). Indeed, for any $\epsilon\in (0,1)$, take $$f_\epsilon(x)= |x|^{-1/p-\epsilon}\chi_{\{y\in{{\mathbb R}}: |y|>1\}}(x)$$ and $$g_\epsilon(x)= |x|^{-1/p+\epsilon}\chi_{\{y\in{{\mathbb R}}: |y|<1\}}(x)$$ for all $x\in{{\mathbb R}}$. Then some simple computations give $$\|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}\geq \frac{\|H_\varphi(f_\epsilon)\|_{L^p({{\mathbb R}})}}{\|f_\epsilon\|_{L^p({{\mathbb R}})}} \gtrsim \epsilon^\epsilon \int_1^{1/\epsilon} t^{1/p-1}\varphi(t)dt$$ and $$\|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}\geq \frac{\|H_\varphi(g_\epsilon)\|_{L^p({{\mathbb R}})}}{\|g_\epsilon\|_{L^p({{\mathbb R}})}} \gtrsim \epsilon^\epsilon \int_\epsilon^{1} t^{1/p-1}\varphi(t)dt.$$ Letting $\epsilon\to 0$, we get $$\int_1^\infty t^{1/p-1}\varphi(t)dt \lesssim \|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}<\infty$$ and $$\int_0^1 t^{1/p-1}\varphi(t)dt \lesssim \|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}<\infty.$$ This proves (\[commutes with the Hilbert transform, 2\]).
0.2cm
Finally, we need to show that $H_\varphi$ commutes with the Hilbert transform $H$ on $L^p({{\mathbb R}})$. To this ends, it suffices to show $$\label{commutes with the Hilbert transform, 3}
H_\varphi(H(f))= H(H_\varphi(f))$$ for all real-valued functions $f$ in $L^p({{\mathbb R}})$. Indeed, by Theorem \[main theorem\] and Lemma \[boundary value characterization 1\], there exists a real-valued function $g$ in $L^p({{\mathbb R}})$ such that $$g + i H(g)= H_\varphi(f+ i H(f)).$$ This proves (\[commutes with the Hilbert transform, 3\]), and thus completes the proof of Theorem \[commutes with the Hilbert transform\].
Let $1<p<\infty$, we denote by $H^p_+({{\mathbb R}})$ and $H^p_-({{\mathbb R}})$ the subspaces of $L^p({{\mathbb R}})$ consisting of those functions whose Poisson extensions to the upper half-plane $\mathbb C_+$ are holomorphic and anti-holomorphic, respectively.
It is well-known (see [@Du; @Ga; @St]) that $$\label{upper Hardy spaces}
H^p_{+}({{\mathbb R}})= \{f+ iH(f): f\in L^p({{\mathbb R}})\}$$ and $$\label{lower Hardy spaces}
H^p_{-}({{\mathbb R}})= \{f- iH(f): f\in L^p({{\mathbb R}})\}.$$ Moreover, $L^p({{\mathbb R}})= H^p_{+}({{\mathbb R}}) \oplus H^p_{-}({{\mathbb R}})$.
\[first corollary\] Let $p\in (1,\infty)$ and let $\varphi$ be such that (\[main inequality\]) holds. Then $H_\varphi$ is bounded on the space $H^p_+({{\mathbb R}})$, moreover, $$\|H_\varphi\|_{H^p_+({{\mathbb R}})\to H^p_+({{\mathbb R}})}= \int_0^\infty t^{1/p-1}\varphi(t)dt$$ and $H_\varphi$ commutes with the Hilbert transform $H$ on $H^p_+({{\mathbb R}})$.
It follows from Theorem \[commutes with the Hilbert transform\] that $H_\varphi(f)$ belongs to $H^p_+({{\mathbb R}})$ for all $f\in H^p_+({{\mathbb R}})$, and thus $$\label{a corollary, 1}
\|H_\varphi\|_{H^p_+({{\mathbb R}})\to H^p_+({{\mathbb R}})}\leq \|H_\varphi\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}= \int_0^\infty t^{1/p-1}\varphi(t)dt.$$
For any $\varepsilon >0$, by Theorem \[main theorem\], there exists $f_\varepsilon\in \Hd$ for which $$\frac{\|H_\varphi(f_\varepsilon^*)\|_{L^p({{\mathbb R}})}}{\|f_\varepsilon^*\|_{L^p({{\mathbb R}})}}= \frac{\|(\Hau f_\varepsilon)^*\|_{L^p({{\mathbb R}})}}{\|f_\varepsilon^*\|_{L^p({{\mathbb R}})}} = \frac{\|\Hau f_\varepsilon\|_{\Hd}}{\|f_\varepsilon\|_{\Hd}}\geq \int_0^\infty t^{1/p-1}\varphi(t)dt -\varepsilon.$$ This, together with (\[a corollary, 1\]), allows us to conclude that $$\|H_\varphi\|_{H^p_+({{\mathbb R}})\to H^p_+({{\mathbb R}})}=\int_0^\infty t^{1/p-1}\varphi(t)dt.$$
Finally, $H_\varphi$ commutes with the Hilbert transform $H$ on $H^p_+({{\mathbb R}})$ is followed from Theorem \[commutes with the Hilbert transform\] and (\[upper Hardy spaces\]).
\[second corollary\] Let $p\in (1,\infty)$ and let $\varphi$ be such that (\[main inequality\]) holds. Then $H_\varphi$ is bounded on the space $H^p_-({{\mathbb R}})$, moreover, $$\|H_\varphi\|_{H^p_-({{\mathbb R}})\to H^p_-({{\mathbb R}})}= \int_0^\infty t^{1/p-1}\varphi(t)dt$$ and $H_\varphi$ commutes with the Hilbert transform $H$ on $H^p_-({{\mathbb R}})$.
It follows from Theorem \[first corollary\] and the fact that $f\in H^p_+({{\mathbb R}})$ if and only if $\bar f\in H^p_-({{\mathbb R}})$.
Let $\Phi$ be in the Schwartz space $\mathcal S({{\mathbb R}})$ satisfying $\int_{{{\mathbb R}}}\Phi(x)dx\ne 0$. For any $t>0$, set $\Phi_t(x):= t^{-1}\Phi(x/t)$. Following Fefferman and Stein [@FS; @St], we define the [*real Hardy space*]{} $\H^1({{\mathbb R}})$ as the set of all functions $f\in L^1({{\mathbb R}})$ such that $$\|f\|_{\H^1({{\mathbb R}})}:= \left\|M_{\Phi}(f)\right\|_{L^1({{\mathbb R}})}<\infty,$$ where $M_{\Phi}(f)$ is the [*smooth maximal function*]{} of $f$ defined by $$M_{\Phi}(f)(x)= \sup_{t>0}|f*\Phi_t(x)|,\quad x\in{{\mathbb R}}.$$
Remark that the norm $\|\cdot\|_{\H^1({{\mathbb R}})}$ depends on the choice of $\Phi$, but the space $\H^1({{\mathbb R}})$ does not depend on this choice (see Proposition \[some equivalent characterizations of H1\] below).
The following lemma is well-known.
\[boundary value characterization\]
1. If $g\in \H^1({{\mathbb R}})$, then $f^*:= g + iH(g)$ is the boundary value function of some function $f\in \mathcal H_a^1(\mathbb C_+)$.
2. Conversely, if $f^*$ is a boundary value function of $f\in \mathcal H_a^1(\mathbb C_+)$, then there exists a real-valued function $g\in \H^1({{\mathbb R}})$ such that $f^*= g+iH(g)$.
Moreover, in those cases, $$\|g\|_{\H^1({{\mathbb R}})} \sim \|g+ iH(g)\|_{\H^1({{\mathbb R}})}=\|f^*\|_{\H^1({{\mathbb R}})}\sim \|f^*\|_{L^1({{\mathbb R}})}=\|f\|_{\mathcal H_a^1(\mathbb C_+)}.$$
Let $P_t(x)=\frac{1}{\pi}\frac{t}{x^2+t^2}$ be the Poisson kernel on ${{\mathbb R}}$. For any $f\in L^1({{\mathbb R}})$, we denote $u(y,t)= f*P_t(y)$. Then, set $$\mathcal M_P(f)(x)= \sup_{|y-x|<t} |u(y,t)|\quad\mbox{and}\quad S(f)(x)=\left[\iint_{|y-x|<t}\left(|u_t(y,t)|^2 +|u_y(y,t)|^2\right) dy dt\right]^{1/2}.$$
A function $a$ is called an $\H^1$-atom related to the interval $B$ if
1. supp $a\subset B$;
2. $\|a\|_{L^\infty({{\mathbb R}})}\leq |B|^{-1}$;
3. $\int_{{{\mathbb R}}} a(x)dx=0$.
We define the Hardy space $\H^1_{\rm at}({{\mathbb R}})$ as the space of functions $f\in L^1({{\mathbb R}})$ which can be written as $f=\sum_{j=1}^\infty \lambda_j a_j$ with $a_j$’s are $\H^1$-atoms and $\lambda_j$’s are complex numbers satisfying $\sum_{j=1}^\infty |\lambda_j|<\infty$. The norm on $\H^1_{\rm at}({{\mathbb R}})$ is then defined by $$\|f\|_{\H^1_{\rm at}({{\mathbb R}})}:= \inf\left\{\sum_{j=1}^\infty |\lambda_j|: f= \sum_{j=1}^\infty \lambda_j a_j\right\}.$$
The following proposition is classical and can be found in Stein’s book [@St].
\[some equivalent characterizations of H1\] Let $f\in L^1({{\mathbb R}})$. Then the following conditions are equivalent:
1. $f\in \H^1({{\mathbb R}})$.
2. $\mathcal M_\Phi(f)(\cdot)= \sup_{|y-\cdot|<t}|f*\Phi_t(y)|\in L^1({{\mathbb R}})$.
3. $\mathcal M_P(f)\in L^1({{\mathbb R}})$.
4. $S(f)\in L^1({{\mathbb R}})$.
5. $f\in \H^1_{\rm at}({{\mathbb R}})$.
6. $H(f)\in L^1({{\mathbb R}})$.
Moreover, in those cases, $$\begin{aligned}
\|f\|_{\H^1({{\mathbb R}})} &\sim& \|\mathcal M_\Phi(f)\|_{L^1({{\mathbb R}})}\sim \|\mathcal M_P(f)\|_{L^1({{\mathbb R}})}\sim \|S(f)\|_{L^1({{\mathbb R}})}\\
&\sim & \|f\|_{\H^1_{\rm at}({{\mathbb R}})}\sim \|f\|_{L^1({{\mathbb R}})} + \|H(f)\|_{L^1({{\mathbb R}})}.
\end{aligned}$$ Of course, the above constants are depending on $\Phi$.
The following gives a lower bound for the norm of $H_\varphi$ on $\H^1({{\mathbb R}})$.
\[a lower bound\] Let $\|\cdot\|_*$ be one of the six norms in Proposition \[some equivalent characterizations of H1\]. Assume that $H_\varphi$ is bounded on $(\H^1({{\mathbb R}}),\|\cdot\|_*)$. Then, $$\int_0^\infty \varphi(t)dt \leq \|H_\varphi\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)}<\infty$$ and $H_\varphi$ commutes with the Hilbert transform $H$ on $\H^1({{\mathbb R}})$.
It should be pointed out that, when supp $\varphi\subset [1,\infty)$ and $\|\cdot\|_{*}=\|\cdot\|_{\H^1_{\rm at}({{\mathbb R}})}$, the above theorem is due to Xiao [@Xi p. 666] (see also [@LL; @Li08]).
In order to prove Theorem \[a lower bound\], we need the following lemma.
\[a key lemma for real Hausdorff operators\] Let $\varphi$ be such that $\int_0^\infty \varphi(t)dt<\infty$ and supp $\varphi\subset [0,1]$. Then, $$\int_0^1 \varphi(t)dt\leq\|H_\varphi\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)}<\infty.$$
It is well-known (see [@An; @HKQ; @LM]) that if $\int_0^\infty \varphi(t)dt<\infty$, then $H_\varphi$ is bounded on $\H^1({{\mathbb R}})$, moreover, $$\label{an upper bound for the norm}
\|H_\varphi\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)} \lesssim \int_0^\infty \varphi(t)dt= \int_0^1 \varphi(t)dt.$$
We now show that $$\int_0^1 \varphi(t)dt\leq\|H_\varphi\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)}.$$ Indeed, let $\delta\in (0,1)$ and $\varphi_\delta$ be as in the proof of Lemma \[key lemma\](ii). For any $\varepsilon>0$, define the function $f_\varepsilon: \mathbb C_+\to\mathbb C$ by $$f_\varepsilon(z)=\frac{1}{(z+i)^{1+\varepsilon}}.$$ Then, by Lemma \[key lemma\](iii), Lemma \[boundary value characterization\], Proposition \[some equivalent characterizations of H1\] and (\[an estimate for the norm\]), $$\begin{aligned}
&& \frac{\left\|H_{\varphi_\delta}(f_\varepsilon^*) - f_\varepsilon^* \int_0^\infty \varphi_\delta(t)dt \right\|_{*}}{\|f_\varepsilon^*\|_{*}}\\
&\lesssim& \frac{\left\|\mathscr H_{\varphi_\delta}(f_\varepsilon) - f_\varepsilon \int_0^\infty \varphi_\delta(t)dt \right\|_{\mathcal H^1_a(\mathbb C_+)}}{\|f_\varepsilon\|_{\mathcal H^1_a(\mathbb C_+)}}\to 0
\end{aligned}$$ as $\varepsilon\to 0$. This implies that $$\int_{\delta}^{1} \varphi(t)dt= \int_{0}^{\infty} \varphi_\delta(t)dt\leq \|H_{\varphi_\delta}\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)},$$ and thus $$\int_0^1 \varphi(t)dt\leq \|H_{\varphi}\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)}$$ since $$\|H_{\varphi}- H_{\varphi_\delta}\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)}\lesssim \int_0^\infty (\varphi(t)-\varphi_\delta(t))dt= \int_0^\delta \varphi(t)dt\to 0$$ as $\delta \to 0$. This ends the proof of Lemma \[a key lemma for real Hausdorff operators\].
It follows from [@HKQ Theorem 3.3] that $$\int_0^\infty \varphi(t)dt <\infty.$$
For any $m>0$, let $\varphi_m$ be as in the proof of Theorem \[main theorem\]. Then, by (\[an upper bound for the norm\]), $$\begin{aligned}
\label{approximates}
&&\left\|H_\varphi - H_{\varphi_m(\frac{\cdot}{m})}\right\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)} \\
&=& \left\|H_{\varphi- \varphi_m(\frac{\cdot}{m})}\right\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)}\nonumber\\
&\lesssim& \int_0^\infty \left[\varphi(t)-\varphi_m(\frac{t}{m})\right]dt =\int_m^\infty \varphi(t)dt,\nonumber
\end{aligned}$$ where the constant is independent of $m$.
Noting that $$\left\|f\left(\frac{\cdot}{m}\right)\right\|_{*}= m \|f(\cdot)\|_{*}\quad\mbox{and}\quad H_{\varphi_m\left(\frac{\cdot}{m}\right)} f= H_{\varphi_m} f\left(\frac{\cdot}{m}\right)$$ for all $f\in \H^1({{\mathbb R}})$, Lemma \[a key lemma for real Hausdorff operators\] gives $$\begin{aligned}
\left\|H_{\varphi_m\left(\frac{\cdot}{m}\right)}\right\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)} &=& m \left\|H_{\varphi_m}\right\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)}\\
&\geq& m \int_0^1 \varphi_m(t)dt = \int_0^m \varphi(t)dt.
\end{aligned}$$ This, together with (\[approximates\]) and $\lim\limits_{m\to\infty} \int_m^\infty \varphi(t)dt=0$, allows us to conclude that $$\int_0^\infty \varphi(t)dt\leq \left\|H_\varphi\right\|_{(\H^1({{\mathbb R}}),\|\cdot\|_*)\to (\H^1({{\mathbb R}}),\|\cdot\|_*)}.$$
Using the Fourier transform, Liflyand and Móricz proved in [@LM2] that $H_\varphi$ commutes with the Hilbert transform $H$ on $\H^1({{\mathbb R}})$. However, we also would like to give a new proof of this fact here. It suffices to prove $$\label{commutes with the Hilbert transform at p=1}
H_\varphi(H(f))= H(H_\varphi(f))$$ for all real-valued functions $f$ in $\H^1({{\mathbb R}})$. Indeed, by Theorem \[main theorem\] and Lemma \[boundary value characterization\], there exists a real-valued function $g$ in $\H^1({{\mathbb R}})$ such that $$g + i H(g)= H_\varphi(f+ i H(f)).$$ This proves (\[commutes with the Hilbert transform at p=1\]), and thus completes the proof of Theorem \[a lower bound\].
Let $a: (0,\infty) \to [0,\infty)$ be a measurable function. Following Carro and Ortiz-Caraballo [@CO], we define $$\mathscr S_a F(z)=\int_0^\infty F(tz)a(t)dt,\quad z\in \mathbb C_+,$$ for all holomorphic functions $F$ on $\mathbb C_+$; and define $$S_a f(x)=\int_0^\infty f(tx) a(t)dt, \quad x\in {{\mathbb R}},$$ for all measurable functions $f$ on ${{\mathbb R}}$.
It is easy to see that $$\mathscr S_a F= \H_{\varphi}F \quad\mbox{and}\quad S_a f= H_\varphi f,$$ where $\varphi(t)= t^{-1}a(t^{-1})$ for all $t\in (0,\infty)$. Hence, it follows from Theorems \[main theorem\], \[commutes with the Hilbert transform\] and \[a lower bound\] that:
\[a theorem for adjointed operators\] Let $p\in [1,\infty]$ and let $a: (0,\infty) \to [0,\infty)$ be a measurable function. Then $\mathscr S_a$ is bounded on $\H_a^p(\mathbb C_+)$ if and only if $$\label{an inequality for adjoint operators}
\int_0^\infty t^{-1/p} a(t)dt<\infty.$$ Moreover, when (\[an inequality for adjoint operators\]) holds, we obtain $$\|\mathscr S_a\|_{\H^p_a(\mathbb C_+)\to \H^p_a(\mathbb C_+)}=\int_0^\infty t^{-1/p} a(t)dt$$ and, for any $f\in \H_a^p(\mathbb C_+)$, $$(\mathscr S_a f)^*= S_a(f^*).$$
\[commutes with the Hilbert transform for adjointed operator\] Let $p\in (1,\infty)$ and let $a$ be as in Theorem \[a theorem for adjointed operators\]. Then $S_a$ is bounded on $L^p({{\mathbb R}})$ if and only if (\[an inequality for adjoint operators\]) holds. Moreover, in that case, $$\|S_a\|_{L^p({{\mathbb R}})\to L^p({{\mathbb R}})}= \int_0^\infty t^{-1/p} a(t)dt$$ and $S_a$ commutes with the Hilbert transform $H$ on $L^p({{\mathbb R}})$.
\[commutes with the Hilbert transform on real Hardy space for adjointed operator\] Let $a$ be as in Theorem \[a theorem for adjointed operators\]. Then $S_a$ is bounded on $\H^1({{\mathbb R}})$ if and only if $\int_0^\infty t^{-1} a(t)dt<\infty.$ Moreover, in that case, $$\int_0^\infty t^{-1} a(t)dt\leq \|S_a\|_{\H^1({{\mathbb R}})\to \H^1({{\mathbb R}})}<\infty$$ and $S_a$ commutes with the Hilbert transform $H$ on $\H^1({{\mathbb R}})$.
Also it is easy to see that if (\[main inequality\]) holds for $1<p<\infty$, then $$\int_{{{\mathbb R}}} H_\varphi f(x) g(x)dx= \int_{{{\mathbb R}}} f(x) S_\varphi g(x)dx$$ whenever $f\in L^p({{\mathbb R}})$ and $g\in L^q({{\mathbb R}})$, $q=p/(p-1)$. Namely, $S_\varphi$ can be viewed as the Banach space adjoint of $H_\varphi$ and vice versa. Therefore, by Theorem \[a lower bound\], Theorem \[commutes with the Hilbert transform on real Hardy space for adjointed operator\] and [@LL Theorem 1], a duality argument gives:
1. $S_\varphi$ is bounded on $BMO({{\mathbb R}})$ if and only if $\int_0^\infty \varphi(t)dt<\infty$. Moreover, in that case, $$\|S_\varphi\|_{BMO({{\mathbb R}})\to BMO({{\mathbb R}})}=\int_0^\infty \varphi(t)dt.$$
2. $H_\varphi$ is bounded on $BMO({{\mathbb R}})$ if and only if $\int_0^\infty t^{-1}\varphi(t)dt<\infty$. Moreover, in that case, $$\|H_\varphi\|_{BMO({{\mathbb R}})\to BMO({{\mathbb R}})}=\int_0^\infty t^{-1}\varphi(t)dt.$$
Here the space $BMO({{\mathbb R}})$ (see [@FS; @JN]) is the dual space of $\H^1({{\mathbb R}})$ defined as the space of all functions $f\in L^1_{\rm loc}({{\mathbb R}})$ such that $$\|f\|_{BMO({{\mathbb R}})}:=\sup_{B}\frac{1}{|B|}\int_B \left|f(x)-\frac{1}{|B|}\int_B f(y)dy\right|dx<\infty,$$ where the supremum is taken over all intervals $B\subset{{\mathbb R}}$.
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[**Acknowledgements.**]{} The authors would like to thank the referees for their carefully reading and helpful suggestions.
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[^1]: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.22.
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abstract: |
The estimation of time-varying networks for functional Magnetic Resonance Imaging (fMRI) data sets is of increasing importance and interest. In this work, we formulate the problem in a high-dimensional time series framework and introduce a data-driven method, namely *Network Change Points Detection (NCPD)*, which detects change points in the network structure of a multivariate time series, with each component of the time series represented by a node in the network. NCPD is applied to various simulated data and a resting-state fMRI data set. This new methodology also allows us to identify common functional states within and across subjects. Finally, NCPD promises to offer a deep insight into the large-scale characterisations and dynamics of the brain.
Keywords: Spectral clustering; Change point analysis; Network change points; Stationary bootstrap; fMRI; Resting-state data.
author:
- |
Ivor Cribben and Yi Yu\
Alberta School of Business, Canada\
Statistical Laboratory, University of Cambridge, U.K.
bibliography:
- 'jrssc-final.bib'
title: Estimating whole brain dynamics using spectral clustering
---
Introduction
============
\[sec:int\]
In the ‘Big Data’ era, time-varying network models are used to solve many important problems. In particular, a great current challenge in neuroscience is the reconstruction of the dynamic manner in which brain regions interact with one another in both task-based and resting-state functional magnetic resonance imaging (fMRI) studies. This reconstruction has the ability to have a major impact on the understanding of the functional organisation of the brain. fMRI is a neuroimaging technique that indirectly measures brain activity. [@Ogawa] introduced the blood-oxygen-level dependent (BOLD) contrast, which is currently the primary form of fMRI due to its high spatial resolution and its non-invasiveness. BOLD is based on the dependence between blood flow in the brain and neuronal activation. In other words, when a certain brain region is active, extra blood flows to this region. In particular, the brain is usually parcellated into small cubic regions roughly a few millimetres in size called voxels and the brain activity is measured in each voxel across a sequence of scans with time series as outputs.
Functional connectivity (FC) analysis examines functional associations between time series pairs in specified voxels or regions [@biswal]. The simplest methods for estimating FC include the use of covariance, correlation and precision matrices [@cummine]. FC can also be estimated using spectral measures such as coherence and partial coherence [@cribbenfiecas; @fiecas]. In addition, the FC between brain voxels or regions can be represented by an interconnected brain network. Here, vertices and edges represent brain region time series and their FC, respectively. The idea of studying the brain as an FC network is helpful as it can be viewed as a system with various interacting regions which produce complex behaviours. Similar to other biological networks, understanding the complex network organisation of the brain can lead to profound clinical implications [@sporns; @bullmore].
All of these methods, however, assume that the data are stationary over time, that is, the dependence or the FC between brain regions remains constant throughout the experiment. Although this assumption is convenient for both statistical estimation and computational reasons, it presents a simplified version of a highly integrated and dynamic phenomenon. Evidence of the non-stationary behaviour of time series from brain activity has been observed not only in task-based fMRI experiments [@cribben2; @cribben3; @debener; @fox; @eichele; @sadaghiani], but also prominently in resting-state data [@delamillieure; @doucet].
This evidence has led to an increase in the number of statistical methods that estimate the time-varying or dynamic connectivity. The covariance, correlation and precision matrix approaches discussed above all have a natural time-varying analogue. Using a moving-window, these approaches begin at the first time point, then a block of a fixed number of time points are selected and all the data points within the block are used to estimate the FC. The window is then shifted a certain number of time points and the FC is estimated on the new data set. By shifting the window to the end of the experimental time course, researchers can estimate the time-varying FC. Many research papers have considered this approach. [@chang], [@kiviniemi] and [@hutchison2] investigated the non-stationary behaviour of resting-state connectivity using a moving-window approach, based on a time-frequency coherence analysis with wavelet transforms, an independent component analysis and a correlation analysis, respectively. [@leonardi] studied whole brain dynamic FC using a moving-window and a principal component analysis technique that is applied to resting-state data. [@leonardi2] introduced a data-driven multivariate method, namely higher-order singular value decomposition, which models whole brain networks from group-level time-varying FC data using a moving-window based on a tensor decomposition. [@allen], [@handwerker], [@jones] and [@sakouglu] considered a group independent component analysis [@calhoun] to decompose multi-subject resting-state data into functional brain regions, and a moving-window and $k$-means clustering of the windowed correlation matrices to study whole brain time-varying networks.
While the moving-window approach can be used to observe time-varying FC, and is computationally feasible, it also has limitations [@hutchison]. For example, the choice of block size is crucial and sensitive, as different block sizes can lead to different FC patterns. Another pitfall is that the technique gives equal weight to all $k$ neighbouring time points and 0 weight to all the others [@lindquist:yuting]. In order to estimate time-varying FC without the use of sliding windows, [@zhang] proposed the dynamic Bayesian variable partition model that estimates and models multivariate dynamic functional interactions using a unified Bayesian framework. This method first detects the temporal boundaries of piecewise quasi-stable functional interaction patterns, which are then modelled by representative signature patterns and whose temporal transitions are characterised by finite-state transition machines.
There are a number of methods that utilize change point procedures for estimating the time-varying connectivity between brain imaging signals, including Dynamic Connectivity Regression (DCR: [@cribben1; @cribben2]), FreSpeD [@schroder], and a novel statistical method for detecting change points in multivariate time series [@kirch]. [@schroder] employ a multivariate cumulative sum (CUSUM)-type procedure to detect change points in autospectra and coherences for multivariate time series. Their methods allows for the segmentation of the multivariate time series but also for the direct interpretation of the change in the sense that the change point can be assigned to one or multiple time series (or Electroencephalogram channels) and frequency bands. [@kirch] consider the At Most One Change (AMOC) setting and the epidemic setting (two change points, where the process reverts back to the original regime after the second change point) and provide some theoretical results.
All of these techniques are based on different methodologies but each of them performs very well in their own right. However, they have limitations. The most obvious is that they are all restricted by the number of time series from either the channels or brain regions. For DCR, the algorithm begins to slow after 50 time series. In addition, FreSpeD considers only 21 time series channels in its application and, like the DCR method, it is limited by the minimum separability assumption, which means that there has to be a certain distance between change points. Finally, the method of [@kirch] is also restricted by the number of time series they can include because the proposed test statistics require the estimation of the inverse of the long run auto-covariance, which is particularly difficult in higher-dimensional settings and even more problematic in the multivariate case because of the number of entries in the positive-definite weight matrix. Their method also focuses on changes in the model parameters, which is limiting as it is difficult to interpret a change in a parameter.
In this paper, our aim is not only to detect the network structural changes along the experimental time course, but also to represent the high-dimensional brain imaging data in a low dimensional clustering structure; in other words, we are interested in combining the research areas of change point detection in time series analysis and community detection in network analysis. Recently, both change point detection in time series and community detection in network statistics have become topical areas (see e.g. [@FrickEtal2014], [@ChoFryzlewicz2015], [@WangSamworth2016], for some up-to-date work on change point detection, and e.g. [@Newman2006], [@WangBickel2015], [@Jin2015] for some recent work on community detection). The essence of these two areas is to partition the data set into different clusters/segments that share some fundamental similarities but differ from the other clusters/segments. Specifically, change point detection is the segmentation of non-stationary time series into several stationary segments while community detection is the partitioning of complex networks into several tightly-knit clusters.
To this end, we introduce a data-driven method, namely *network change point detection (NCPD)*, the detailed methodology and algorithm of which are explained in Section \[sec:modelal\]. NCPD’s strength, unlike the other change point methods above, is that it can consider thousands of time series and in particular the case where the number of brain regions well exceeds the number of time points in the experiment, i.e., $p \gg T$, where $p$ is the number of regions of interest or voxels and $T$ is the number of time points. Using NCPD, one can, therefore, consider whole brain dynamics, which departs from the moving window technique and promises to offer deeper insight into the large scale characterisations of the whole brain. We apply the new method to a resting-state fMRI experiment. Dynamic FC is prominent in the resting-state when mental activity is unconstrained. This analysis has led to the robust identification of cognitive states at rest. NCPD not only allows for the estimation of time-varying connectivity but also finds common cognitive states that recur in time and across subjects in a group study. By unveiling the time-varying cognitive states of both controls and subjects with neuropsychiatric diseases such as Alzheimer’s, dementia, autism and schizophrenia using NCPD, we can compare their FC patterns and endeavour to develop new understandings of these diseases. NCPD is, to the best of our knowledge, the first paper to consider estimating change points for time evolving community network structure in a multivariate time series context. Although this paper is inspired by and developed for brain connectivity studies, our proposed method pertains to a general setting and can also be used in a variety of situations where one wishes to study the evolution of a high dimensional network over time. The rest of this paper is organised as follows. The required notation for this paper is introduced in Section \[sec:notation\]. NCPD is outlined in Section \[sec:modelal\], with simulations and real data analysis presented in Sections \[sec:sims\] and \[sec:rsfmridata\], respectively. We conclude this paper with a discussion in Section \[sec:discussion\].
Notation {#sec:notation}
--------
In this subsection, we introduce the standard graph-theoretic notation. We do not distinguish between the terms ‘network’ and ‘graph’ in this paper. Let $G := (V,E)$ denote a $p$-node undirected simple graph, where $V := \{1, \ldots, p\}$ and $E := \{(i, j),\, 1\leq i< j\leq p\}$ are the collections of vertices and edges, respectively. A $K$-*partition* is a pairwise disjoint collection $\{V_k: k =1,\ldots,K\}$ of non-empty subsets of $V$ such that $V = V_1 \sqcup \ldots \sqcup V_K$, where $\sqcup$ denotes disjoint union.
The *adjacency matrix* of $G$ is denoted by $A:= (A_{ij})_{1\leq i,j \leq p}$, where $A_{ij} = 1$ if $(i, j) \in E$ or $(j, i) \in E$, otherwise $A_{ij} = 0$. The *degree* of vertex $i\in V$ is $d_i := \sum_{j=1}^p A_{ij}$, and the degree matrix is $D := \mathrm{diag}(d_1,\ldots,d_p)$. The vital tool in spectral clustering is the *Laplacian matrix* [@Chung1997], which is defined as $$\label{eq-laplacian}
L := D - A,$$ with eigenvalues $0 = \lambda_1 \leq \ldots \leq \lambda_p$ and corresponding unit-length eigenvectors $\boldsymbol{v}_1, \ldots, \boldsymbol{v}_p$.
In the rest of this paper, we denote true covariance matrices and sample correlation matrices by $\Sigma$ and $R$ respectively. For a matrix $M$, denote $M_{(i)}$ as the $i$th row and $M_{(a:b)}$ as the sub-matrix of $M$ consisting of the $a$th to $b$th rows of $M$, where $a < b$.
Methodology {#sec:modelal}
===========
In this section, we introduce the NCPD method. Motivated by, but not restricted to, the brain dynamics analysis, we use ‘nodes’ instead of ‘voxels’ or ‘regions’. To start this section, we illustrate the algorithm and then elaborate more on the details. The input of the following algorithm includes:
- a data matrix $Y \in \mathbb{R}^{T\times p}$, where $T$ and $p$ are the numbers of time points and nodes respectively;
- the pre-specified number of communities $K$;
- a collection of candidate change points $\boldsymbol{\delta} := \{\delta_1, \ldots, \delta_m\} \subset \{1, \ldots, T\}$;
- a pre-specified significance thresholding $\alpha \in (0, 1)$.
$(Y_{\mathrm{L}}, Y_{\mathrm{R}}) \gets$ $(Y_{(1: \delta_j)}, Y_{(\delta_j + 1 : T)})$ $(R_{\mathrm{L}}, R_{\mathrm{R}}) \gets (\mathrm{corr}(Y_{\mathrm{L}}), \mathrm{corr}(Y_{\mathrm{R}}))$ $(\boldsymbol{z}_{\mathrm{L}}, C_{\mathrm{L}}) \gets \mathrm{SpectralClustering}(R_{\mathrm{L}}, K)$; $(\boldsymbol{z}_{\mathrm{R}}, C_{\mathrm{R}}) \gets \mathrm{SpectralClustering}(R_{\mathrm{R}}, K)$ $(U_{\mathrm{L}}^i, U_{\mathrm{R}}^i) \gets (C_{\mathrm{L}}^{\boldsymbol{z}_{\mathrm{L}},i}, C_{\mathrm{R}}^{\boldsymbol{z}_{\mathrm{R}},i})$ $\gamma_j \gets$ sum of the singular values of $U_{\mathrm{L}}^{\top}U_{\mathrm{R}}$ $\mathrm{FLAG}_j \gets \mathrm{StationaryBootstrap}(\alpha, \gamma_j)$ **return** $\{\delta_j: j\in\{1, \ldots, m\}, \mathrm{FLAG}_j = \mathrm{significant}\}$
The work-flow and the pseudo-code of NCPD are given in Figure \[Fig:work-flow\] and Algorithm \[Alg:main\], respectively.
Spectral clustering
-------------------
Spectral clustering [@DonathHoffman1973] is a computationally feasible and nonparametric method widely used in community detection in network statistics. In an undirected simple network $G = (V, E)$, we believe that the vertices are tightly-knit within the communities and loosely-connected between communities. A natural criterion in the recovery of the community structure is to minimise the number of between community edges with the sizes of the communities as the normalisation, namely the ratio cut [@WeiCheng1991], which is defined as $$\mathrm{RCut}(V_1,\ldots,V_K) := \sum_{k = 1}^K \frac{W(V_k, V_k^c)}{|V_k|},$$ where $W(V_k,V_k^c) := \sum_{i \in V_k, j \in V_k^c} A_{ij}$ is the total number of edges connecting $V_k$ and its complement $V_k^c$. However, seeking the partition minimising $\mathrm{RCut}$ is an NP-hard problem [@GJS1976], while the spectral clustering is its convex relaxation [@VonLuxburg].
In a high-dimensional time series data set, $Y\in\mathbb{R}^{T\times p}$, with each component (each column of $Y$) a node in a collaboration network, the connectivity network in the given time period is therefore captured by its correlation matrix $R \in \mathbb{R}^{p \times p}$. Treating the correlation matrix $R$ as the adjacency matrix, its corresponding Laplacian matrix $L$ can be computed following Equation .
Spectral clustering unveils the community structure by exploiting the eigen-structure of the Laplacian matrix $L$. Let $V$ consist of the unit-length eigenvectors that are associated with the $K$ smallest eigenvalues of $L$, namely $V = (\boldsymbol{v}_1, \ldots, \boldsymbol{v}_K)$, which is a $K$-dimensional embedding of the $p$-dimensional network. The information of each node is therefore captured by a point in $\mathbb{R}^K$. To discover the community structure, $k$-means clustering is applied to the rows of $V$ and returns the community labels $\boldsymbol{z} := (z_1, \ldots, z_p) \in \{1, \ldots, K\}^p$ and $K$ centroids. A generic spectral clustering algorithm is provided in Algorithm \[Alg:SC\] with the input being the adjacency matrix $A$ and the pre-specified community number $K$, and the outputs are the estimated labels $\boldsymbol{z}$ and centroids of the communities $C$.
$d_i \gets \sum_{j = 1}^p A_{ij}$ $D \gets \mathrm{diag}\{d_1, \ldots, d_p\}$ $L \gets D - A$ $\{\boldsymbol{v}_1, \ldots, \boldsymbol{v}_K\} \gets$ unit-length eigenvectors of $L$ which are associated with the $K$ smallest eigenvalues of $L$ $V \gets (\boldsymbol{v}_1, \ldots, \boldsymbol{v}_K)$ cluster labels for all nodes and centroids of $K$ communities $(\boldsymbol{z}, C) \in \mathbb{R}^p \otimes \mathbb{R}^{K\times K} \gets$ results of $k$-means clustering on the rows of $V$ with $K$ centres **return** $(\boldsymbol{z}, C)$
Note that, in high-dimensional data analysis, penalised precision matrices are often used to study the underlying graphs, when the assumption is only a few pairs out of $p(p-1)/2$ pairs are correlated, for a $p$-node network. However, in this paper, we propose that sparsity means that a low dimension formation appears in the community structure. A $p$-node and $K$-community network can have related pairs of order $O(p^2)$, which is not achievable by penalising precision matrices.
Singular values
---------------
Recall that the main purpose of this paper is to detect change points in terms of the network structure. Spectral clustering unveils the community structure and reduces the dimension of the data sets from $p$ – the number of nodes – to $K$ – the number of communities. The next task is to evaluate the deviance between the network before and after a certain candidate change point.
A natural measurement of the difference between two spaces spanned by the columns of two matrices respectively is the *principal angles*: if $V, W \in \mathbb{R}^{p \times K}$ both consist of orthonormal columns, then the $K$ principal angles between their column spaces are $\cos^{-1} \sigma_1,\ldots,\cos^{-1}\sigma_K$, where $\sigma_1 \geq \cdots \geq \sigma_K$ are the singular values of $V^\top W$. Principal angles between pairs of subspaces can be regarded as natural generalisations of acute angles between pairs of vectors.
The rationale behind principal angles in community detection is the community label invariance. Since the columns of matrix $U$ represent the communities, the measurement should be invariant in terms of the rotation of the columns, i.e. right multiplied by any orthogonal matrix $O \in \mathbb{R}^{K \times K}$ . For any orthogonal matrix $O \in \mathbb{R}^{K \times K}$, matrices $V^\top W$ and $V^\top WO$ have the same singular values.
In our problem, we are interested in network structure changes. Spectral clustering on the Laplacian matrix provides the community information. For each candidate change point, we then construct new matrices $U_{\mathrm{L}}$ and $U_{\mathrm{R}}$, whose rows are the corresponding centroids. The column spaces of $U_{\mathrm{L}}$ and $U_{\mathrm{R}}$ encode the averaged location information, so we do not impose the condition that the columns have to be orthonormal. However, for a certain change point candidate, the singular values of $U_{\mathrm{L}}^{\top}U_{\mathrm{R}}$ still unveil the deviance in terms of the network structure between the two networks separated by the candidate change point. We denote by $\gamma = \gamma(U_{\mathrm{L}}, U_{\mathrm{R}}) := \sum_{k = 1}^K \sigma_k$, where $\{\sigma_1, \ldots, \sigma_K\}$ are the singular values of $U_{\mathrm{L}}^{\top}U_{\mathrm{R}}$. In the sequel, the subscript of $\gamma$ indicates the corresponding candidate change point.
Since the singular values are the cosine values of the principal angles, the smaller $\gamma$ is, the more prominent the difference between the two subspaces is; therefore, a change point is expected to have the smallest $\gamma$ value.
Selection and stopping criteria {#sec:selec-cri}
-------------------------------
In principal, we treat every time point as a candidate change point and compare the deviance of the networks before and after it. However, in practice, we need enough data points to construct a network. Denote $n_{\min}$ as the lower limit of the number of time points needed to construct a network and recall that $T$ is the total number of time points available, we calculate the $\gamma$ criterion values associated with all the points from $n_{\min}$ to $T - n_{\min}$ and pick the time point with the smallest $\gamma$ after deleting the *outliers*, the definition of which we will now specify.
We use an example to illustrate the necessity of the exclusion of the outliers. In Figure \[fig:cri-value\], the $\gamma$ criterion values of all candidate change points are presented. The upper and lower panels represent the first and second change points, respectively. The true change points occur at time points 300 and 200 respectively in these two plots. In addition, ideally the change along the time-axis should be smooth. In practice, as we can see in Figure \[fig:cri-value\], there are some points that have very different values from those of their neighbours, i.e. those coloured in red.
For candidate change points $j$, $2\leq j \leq m-1$, define the outlier detection value $\eta_j := \max\{|\gamma_j - \gamma_{j-1}|, |\gamma_j - \gamma_{j+1}|\}$, $\eta_1 := |\gamma_2 - \gamma_1|$ and $\eta_m := |\gamma_m - \gamma_{m-1}|$. The outliers are those points that have extremely large values of $\eta$, i.e. those which are associated with the largest 5% of the $\eta$ values and are denoted by red points in Figure \[fig:cri-value\].
We run the algorithm exhaustively until the available time points are fewer than the pre-specified threshold, and construct networks for each segment; an illustration is given in Figure \[fig:split\].
Inference on change points {#sec:inference}
--------------------------
In this subsection we discuss an inferential procedure for the cosine of the principal angles between the two subspaces at the candidate change points. As the candidate change points are found, we estimate confidence bounds for the $\gamma$ criterion values at every candidate change point using the stationary bootstrap [@politis]. An assumption of the proposed methodology is the presence of autocorrelation in the individual time series of the data matrix $Y$. Hence, by using the stationary bootstrap or resampling blocks of data points, the dependence structure inherent in the data remains intact and the correct confidence bounds are calculated. The stationary bootstrap is an adaptation of the block bootstrap [@Liu:Singh] but it resamples blocks of data that are of varying block sizes.
The stationary bootstrap procedures aim to detect whether the smallest criterion value $\gamma$, over the time period being studied after outlier deletion, is significant. Without loss of generality, we assume the time period being studied is from $1$ to $T$, and the first change point occurs at time point $\delta$, which has the smallest criterion value $\gamma$ after outlier deletions. The procedure then generates pseudo-samples and conducts statistical inference based on these. To describe the algorithm, we adopt the method in [@politis], letting $B_{i, b} := \{Y_{(i)}, Y_{(i+1)}, \ldots, Y_{(i+b-1)}\}$ be the block consisting of $b$ consecutive time points starting from the $i$th one. In the case $j > T$, set $Y_{(j)} = Y_{(i)}$ with $i := j(\mathrm{mod} T)$ and $Y_{(0)} := Y_{(T)}$. A pseudo-sample $\bigl\{Y_{(1)}^{*}, Y_{(2)}^{*}, \ldots, Y_{(T)}^{*}\bigr\}$ is generated as follows:
1. independently generate $M$ realisations $L_1, \ldots, L_M$ from the geometric distribution with parameter $p \in (0, 1)$, such that $\sum_{i= 1}^{M-1}L_i < T$ and $\sum_{i=1}^M L_i \geq T$;
2. independently generate $M$ realisations $I_1, \ldots, I_M$ from the discrete uniform distribution on $\{1, \ldots, T\}$;
3. the pseudo-sample is the first chunk of $T$ realisations in $\{B_{I_1, L_1}, \ldots, B_{I_M, L_M}\}$.
To test whether the $\gamma$ criterion value at the candidate change point $\delta$ is significant, we generate many, say 1000, pseudo-samples $Y_{(1:T)}^{*}$, for each of which, a new $\gamma_{\delta}$ – the criterion value at $\delta$ – is calculated. The null hypothesis is that the time point $\delta$ is not a change point; therefore for a pre-specified $p$-value $\alpha \in (0, 1)$ ($\alpha = 0.05$ in the numerical studies in this paper), we calculate $c_\alpha$ as the $100\alpha$th empirical quantile of the stationary bootstrap distribution of $\gamma_\delta$. If the observed $\gamma_\delta$ is smaller than $c_\alpha$, we conclude that $\delta$ is a significant change point, indicating a change in network structure; otherwise, it is not a significant change point. This procedure is repeated for each candidate change point.
If the data are assumed to be independent and identically distributed, we perform a permutation inference procedure. The permutation procedure is identical to the stationary bootstrap procedure above except we permute the data instead of resampling blocks of data.
Choice of $K$
-------------
In the spectral clustering step, the number of communities $K$ is a pre-specified parameter as the choice of $K$ in this framework remains an open problem. However, recently progress has been made on this topic [e.g. @FrancoEtal2014; @ChenLei2014]. Unfortunately, all the existing methods require extra computational resources. In our method, the true number of communities is $K_o$ and we pre-specify an over-estimated $K$. We will show in the numerical results that our method is robust with respect to $K$ and will perform uniformly well when $K$ is over-estimated.
We can understand this phenomenon from a dimension-reduction perspective. Spectral clustering embeds a $p$-dimensional data set into a $K$-dimensional space. When $K$ is under-estimated, important directions are missing. One of the steps in our algorithm calculates the singular values between two modified dimension-reduced matrices. Instead of using the matrices consisting of the principal components, we replace the rows by the centroid of the cluster it is in. This, on one hand, makes the community structure more prominent; on the other hand, it further reduces the dimension from $p \times K$ to $K \times K$. Ideally, for a $K_o$-community network, the principal component matrix is of rank $K_o$; therefore, if $K = K_o$, the two orthonormal matrices expand the basis of $\mathbb{R}^{K_o}$ space and the singular values are all 1.
Simulations {#sec:sims}
===========
In this section we examine the performance of NCPD through various simulation settings. For each setting, we perform 100 repetitions, provide a diagram to illustrate how the network structure changes over time and a quantified description of the distributional aspect. To summarise the results in each setting, we plot the Gaussian kernel smoothed empirical density of the occurrences of the detected change points. As we noted in Section \[sec:selec-cri\], we require a certain number of points at the beginning of the time series to initiate the algorithm ($n_{\min}$). During this period, we assume that the network structure is the same. However, the time points close to the two ends of the time axis (data close to $n_{\min}$ and $T-n_{\min}$) tend to capture some network structural changes. We call the occurrence of change points close to the points $n_{\min}$ and $T-n_{\min}$ the *edge phenomenon*, and we delete change points that are significant but close to ends, and report the remaining false positive change points as *modified false positives*.
To visualise the networks, we set a threshold for the correlation between two nodes. If the absolute value of the correlation between two time series is larger than our pre-specified threshold (0.3 is used in the numerical results in this paper), we present the two corresponding nodes connected by an edge; otherwise, disconnected. This is only for the sake of visualisation, while the weighted networks are decided without this threshold.
Descriptions of the settings {#subsec:description}
----------------------------
### Network structure changes
In Figure \[design\], we illustrate how the network structure changes in different settings. The left, middle and right panels are for settings 1, 2 and 3, respectively.
- In Setting 1, the change point occurs in the middle of the time series, with the true number of communities being $K_o = 2$ both before and after the change point. At the change point, the vertices labels are randomly reshuffled.
- In Setting 2, there are three change points and they are located at the first, second and third quarters of the whole time line, respectively. In the first time segment, the true number of communities $K_o = 3$, i.e., there are three clusters, one of which is equally merged into the other two clusters at the first change point. Vertex labels are randomly shuffled at the second change point, while keeping $K_o = 2$. The true number of communities $K_o$ returns to 3, by moving one third of each community into a third community.
- In Setting 3, two change points occur, with the true number of communities $K_o = 2$ remaining constant for the whole time course. At each change point, half of the vertices in each community are moved to the other community.
In terms of changing nature of the network structure, the easiest is Setting 1, where only one change point occurs and the community labels are reshuffled at the change point. Setting 2 covers the situation where the true number of communities $K_o$ changes. Setting 3 is the most challenging setup, where the structural changes only involve separating or merging existing communities.
### Distributional description
- In Setting 1, $(p, T) = (400, 200)$ and the data are generated from the multivariate Gaussian distribution $\mathcal{N}(0, \Sigma)$, where $$\Sigma_{ij} = \begin{cases}
0.75, & \mbox{if } i\neq j \mbox{ and } i, j \mbox{ are in the same cluster};\\
1, & \mbox{if } i = j;\\
0.20, & \mbox{otherwise}.
\end{cases}$$
- In Setting 2, $(p, T) = (600, 400)$ and the data are generated from the multivariate Gaussian distribution $\mathcal{N}(0, \Sigma)$, where $$\Sigma_{ij} = \begin{cases}
0.75, & \mbox{if } i\neq j \mbox{ and } i, j \mbox{ are in the same cluster};\\
1, & \mbox{if } i = j;\\
0.20^{|i-j|}, & \mbox{if } i, j \mbox{ are not in the same cluster}.\\
\end{cases}$$
- In Setting 3, $(p, T) = (800, 600)$ and the data are generated from the same distribution as that in Setting 2.
Results
-------
In this subsection, we present results in various formats. Bearing in mind the fact that the detected change point may differ by a few time points from the true change point, we define those which are at most 10 time points away (either before or after) from the true change point as the *true positives (TP)*. We present the average number of TP across all 100 repetitions in each setting, along with the standard error (in brackets). In addition, we present the frequency of the *false positives (FP)*, as well as that of the modified FP, which excludes all the detected change points that are at most 10 time points away from $n_{\min}$, which is 50 in the simulations, and $T - n_{\min}$.
In Figure \[fig:density\], we plot the Gaussian-kernel smoothed empirical densities of the change points for each of the simulation settings, with the red vertical lines indicating the true change points. Notice that the true change point occurs at or near the peaks of the density curves in all the settings. More quantitative results are collected in Table \[tab:simulations\]. As we can see from the results, NCPD preforms well across all settings, with the true TPs all lying in the detected TP intervals. Modified FP frequencies are significantly smaller than those of the FPs. It is also worthwhile to point out that as long as $K$ is overestimated, NCPD performs robustly.
In addition, we present the network graphs. In Figure \[fig:networks\], we pick one realisation for each setting. The left, middle and right panels are representatives of settings 1, 2 and 3, respectively. In the lower panel, we plot the networks before and after the detected change point. The specific number of communities in the lower panel are $K = 4, 5$ and 4, respectively, i.e. $K$ different colours indicating $K$ different communities. Let us take the left panel, which represents Setting 1 as an example. It is obvious that in the lower-left panel, which represents the network before the change point, the blue and green nodes belong to the same group, and most of the black and red nodes belong to the other. There are no connections between these two groups. After the change point, the network is visualised in the lower-right panel, with the same layout of the vertices. In this case, these two groups are well-connected and the four different colours are mixed between the two groups. In the upper panels, we present the true change point and plot the network using the true number of communities $K_o$ in each part. In the left panel, we can see that the red and black nodes are well-separated in the network prior to the change point, while they are mixed between communities in the network after the change point.
Resting-state fMRI data {#sec:rsfmridata}
=======================
We apply NCPD to a resting-state fMRI data set, as described in [@habeck]. Participants ($n=45$) are instructed to rest in the scanner for 9.5 minutes, with the instruction to keep their eyes open for the duration of the scan. We apply the Anatomical Automatic Labeling [@tzourio] atlas to the adjusted voxel-wise time series and produce time series for 116 Regions of Interest (ROIs) for each subject by averaging the voxel time series within the ROIs. In total, each time series contained 285 time points (9.5 minutes with TR = 2).
Table \[tab:real\] shows the significant change point locations for all 45 subjects. Every subject except one (subject 16) has at least two significant change points in their community network structure with the maximum number of change points being 4. The table indicates that not only does the number of community network change points differ across subjects, but the location of the change points is also variable. In addition, some subjects remain in states for long periods while others transition more quickly. Hence, the method has a major advantage over moving-window type methods as we do not have to choose the window length, which can have significant consequences on the estimated FC.
We used 1,000 stationary bootstrap resamples of the data for inference on the cosine values of the principal angles at each candidate change point with an average block size of 57 (or 20% of the data set) and the minimum distance between change points, $n_{\min}=50$. Previous work [@allen] found the existence of 7 resting-state networks; hence, we specified $K=7$ clusters or communities in our algorithm. The time taken to run the algorithm on each subject ($T=285$, $p=116$) using a Intel(R) Core(TM) i5-3210M 2.50GHz CPU was on average 132s.
In Figure \[fig:caplets\], we mimic the idea in the simulation study to plot the empirical density of the detected change points. In order to get repeated samplings, every time we delete 10% of the data sequentially and therefore have 10 repetitions for each data set. We can see that these is a bump around time point 100 in both subjects (taking the edge phenomenon into consideration), which is a change point we should be cautious about.
Figure \[fig:netplots\] shows the estimated community network graphs for data between each pair of significant change points shown in Table \[tab:real\]. The graphs on the first, second and third rows represent the graphs for subjects 3, 1, and 15, respectively. The colour of the node represents which community the ROI time series belongs to.
Cross subject comparisons
-------------------------
As the data are from a resting-state study when mental activity is unconstrained, we do not expect the community network structure for each subject to match along the same partitions in all cases, that is, we do not anticipate that subject 3’s community network in their first partition to be similar to subject 15’s community network in their first partition. However, we do foresee similarities across the subjects’ partition plots. In particular, we assume that subjects will enter some common stable functional ‘states’ or community network patterns.
In Section \[sec:modelal\], we use the singular values of the product matrix, $U_{\mathrm{L}}^{\top}U_{\mathrm{R}}$, consisting of the matrices (networks) before and after a certain time point, to detect the change point. The rationale for this is that the criterion value represents the similarity of the networks. In the same spirit, when estimating the similarity of the networks across different subjects, we can use the singular values of $U_{i,j}^{\top}U_{k,l}$ where $U_{i,j}$ is the transformed network from subject $i$, partition $j$ and $U_{k,l}$ is the transformed network from subject $k$, partition $l$. In Figure \[fig:partnetplots\], we calculate the criterion values, $U_{i,j}^{\top}U_{k,l}$, between a small number of cross-subject functional state pairs. The higher the criterion values in the matrix, the more similar the networks within or across subjects. For example, the criterion value for the two networks for subject 2, partition 2 and subject 20, partition 2 (Figures A and B) is located in element (2,5) of the matrix and represents two of the most similar across subject state-pairs.
To show (graphical) evidence of common functional states across subjects, we also plot in Figure \[fig:partnetplots\] the community network structure for subject 2 (partition 2), subject 20 (partition 2), subject 16 (partition 1) and subject 30 (partition 3). We compare the similarity of Figures \[fig:partnetplots\] A and B and Figures \[fig:partnetplots\] C and D. In particular, we compare the colour patterns of the graphs. For example, in Figure \[fig:partnetplots\], the green, yellow and red nodes in A and the green, yellow and red nodes in B are very similar. There are also similar crossovers between the aqua, pink and blue nodes. Hence, we conclude that the subjects enter a similar cognitive or functional state. There is also similarity across Figures \[fig:partnetplots\] C and D; here, the aqua nodes in C are very similar to the aqua nodes in D. This is also true for the yellow and blue nodes in both C and D. There are many more examples of this in the data set.
$$\begin{blockarray}{cccccccc}
& S2(1) & S2(2) & S16(1) & S20(1) & S20(2) & S30(2) & S30(3) \\
\begin{block}{c(ccccccc)}
S2(1) & & 1.85 & 1.70 & 1.58 & 1.59 & 1.12 & 1.49 \\
S2(2) & & & 2.16 & 1.80 & \textbf{2.19} & 1.36 & 1.96 \\
S16(1) & & & & 1.43 & 1.35 & 1.38 & \textbf{2.15} \\
S20(1) & & & & & 1.65 & 1.35 & 1.68 \\
S20(2) & & & & & & 1.48 & 1.97 \\
S30(2) & & & & & & & 1.50 \\
S30(3) & & & & & & & \\
\end{block}
\end{blockarray}$$
The resting-state data set in this paper contains regional time series from several networks including the Default Mode Network, Dorsal Attentional Network, Executive Control Network, Senorimotor Network, Visual Network, Auditory Network and Salience Network. However, the common cognitive states or structured patterns found across subjects do not directly link up with these networks, therefore, providing us with observed states that relate differently to previous findings. Moreover, the common structured patterns or functional states found by NCPD include communities between regions not attributed to the list of networks and communities with regions from a few of the networks. In particular, some of the common functional states found show that some communities have strong synchrony across the different networks and weak synchrony with other regions from the same network. Hence, the features found are significant and meaningful given the fact that this is the first study to consider over a hundred fMRI resting-state time series. However, we remain cautious because resting-state fMRI is unconstrained in nature and the functional roles of dynamics and their relationship to subjects’ cognitive state remains unknown. We believe that further investigations into the specificity and consistency of the fMRI functional states or features will be beneficial and that work to elucidate spatiotemporal dynamics associated with spontaneous cognition and behavioural transitions is very important. We hope that NCPD will add to this endeavour.
Discussion {#sec:discussion}
==========
In this paper, we develop a new approach, NCPD, for analysing and modelling multivariate time series from an fMRI study which consists of realisations of complex and dynamic brain processes. The method adds to the literature by improving understanding of the brain processes measured using fMRI. NCPD is, to the best of our knowledge, the first paper to consider estimating change points for time evolving community network structure in a multivariate time series context.
NCPD is an innovative approach for finding psychological states or changes in FC in both task-based and resting-state brain imaging studies. There are several novel aspects of NCPD. Firstly, it allows for estimation of dynamic functional connectivity in a high-dimensional multivariate time series setting, in particular, in situations where the number of brain regions is greater than the number of time points in the experimental time course ($p>T$). Hence, it can consider the dynamics of the whole brain or a very large number of ROI or voxel time series, thereby providing deeper insights into the large-scale functional architecture of the brain and the complex processes within. Secondly, it is not restricted by the situations that commonly occur in change point settings, such as the at most one change (AMOC) setting and the epidemic setting (two change points, where the process reverts back to the original regime after the second change point). Indeed, NCPD is flexible as there is no *a priori* assumption on the number of changes and where the changes occur. Finally, NCPD is, to the best of our knowledge, the first piece of work to consider estimating change points of time evolving community network structure in a multivariate time series context. We introduced a novel metric to find the candidate change points, i.e. the singular values of the product matrices formed by the before and after change point networks. However, NCPD is restricted by the minimum distance between change points (the $\delta$ parameter in the algorithm).
It has been shown that neurological disorders disrupt the functional connectivity pattern or structural properties of the brain [@greicius; @menon]. Future work entails applying NCPD to subjects with brain disorders such as depression, Alzheimer’s disease and schizophrenia and to control subjects who have been matched using behavioural data. NCPD may lead to the robust identification of cognitive states at rest for both controls and subjects with these disorders. It is hoped that the large-scale temporal features resulting from the accurate description of functional connectivity from our novel method will lead to better diagnostic and prognostic indicators of the brain disorders. More specifically, by comparing the change points and the community network structures of functional connectivity of healthy controls to patients with these disorders, we may be able understand the key differences in functional brain processes. In particular, NCPD allows us to find common cognitive states that recur in time, across subjects, and across groups in a study.
While NCPD is applied to resting-state fMRI data in this work, it could seamlessly be applied to an Electroencephalography (EEG) or Magnetoencephalography (MEG) data set. Moreover, NCPD pertains to a general setting and can also be used in a variety of situations where one wishes to study the evolution of a high dimensional network over time.
NCPD appears to have a large computational cost with the binary segmentation of the data and the stationary bootstrap procedure for inference on the candidate change points. However, the resting-state fMRI data set shows how fast the algorithm is. Both the binary segmentation and the stationary bootstrap procedures use parallel computing and on a dual core processor the average time to run the algorithm on one subject ($T=285$, $p=116$) was 132s. Obviously, with access to more cores this time will decrease significantly. The code for the algorithm can already be downloaded from http://www.statslab.cam.ac.uk/$\sim$yy366/.
Acknowledgement
===============
We sincerely thank the Editor, the Guest Editor and two referees for their constructive comments. Funding for Ivor Cribben was provided by the Pearson Faculty Fellowship (Alberta School of Business) and a Alberta Health Services (AHS) Grant. Yi Yu is supported by the Professor Richard Samworth’s Engineering and Physical Sciences Research Council Early Career Fellowship EP/J017213/1.
|
---
abstract: 'Some aspects of hadron production in collisions remain unresolved, including the low-hadron-momentum structure of high-parton-energy dijets, separation of triggered dijets from the [*underlying event*]{} (UE), the systematics of [multiple parton interactions]{} and possible systematic underestimation of dijet contributions to high-energy nuclear collisions. In this study we apply a minimum-bias [*trigger-associated*]{} (TA) correlation analysis to collisions. We extract a hard component from TA correlations that can be compared with measured jet fragment systematics derived from collisions. The kinematic limits on jet fragment production may be determined. The same method may be extended to å collisions where the role of minimum-bias jets in spectra and correlations is strongly contested.'
address: 'CENPA 354290, University of Washington, Seattle, USA'
author:
- 'Thomas A. Trainor and Duncan J. Prindle\'
title: 'Improved isolation of the p-p underlying event based on minimum-bias trigger-associated hadron correlations'
---
Introduction
============
Several open issues for hadron production in collisions relate to dijet production, both the frequency of hard parton scattering and the subsequent fragmentation to jets. In this study we infer the hard scattering rate from the two-component multiplicity systematics of single-particle spectra and introduce a trigger-associated correlation analysis to extract minimum-bias jet fragment distributions. We wish to determine the momentum correlation structure of minimum-bias jets down to the kinematic limits.
Two-component model of p-p single-particle $\bf y_t$ spectra
============================================================
The two-component model of single-particle (SP) spectra is defined by [@ppprd] \[ppspec\] [dn\_[ch]{}]{}/[y\_t dy\_t ]{} &=& \_s( n\_[ch]{}) S\_0(y\_t) + \_[h]{}( n\_[ch]{}) H\_0(y\_t), where $n_{ch}$ is integrated within some acceptance $\Delta \eta$ and $\rho_x = n_x / \Delta \eta$.
![ First: Single-particle spectra for seven $n_{ch}$ classes; Second: Scaled spectrum hard components $H(y_t,n_{ch})$; Third: Event distributions on $n_{ch}$ for several energies; Fourth: Hard-component multiplicity $n_h$ (dijet) trend on soft component $n_s$. $n_h$ is the integral (end-point amplitude) of measured $H(y_t,n_{ch})$ independent of shape. \[ppspec\]](ppcms110a "fig:"){width="0.24\columnwidth"} ![ First: Single-particle spectra for seven $n_{ch}$ classes; Second: Scaled spectrum hard components $H(y_t,n_{ch})$; Third: Event distributions on $n_{ch}$ for several energies; Fourth: Hard-component multiplicity $n_h$ (dijet) trend on soft component $n_s$. $n_h$ is the integral (end-point amplitude) of measured $H(y_t,n_{ch})$ independent of shape. \[ppspec\]](ppcms110b "fig:"){width="0.24\columnwidth"} ![ First: Single-particle spectra for seven $n_{ch}$ classes; Second: Scaled spectrum hard components $H(y_t,n_{ch})$; Third: Event distributions on $n_{ch}$ for several energies; Fourth: Hard-component multiplicity $n_h$ (dijet) trend on soft component $n_s$. $n_h$ is the integral (end-point amplitude) of measured $H(y_t,n_{ch})$ independent of shape. \[ppspec\]](ppcms20aspec "fig:"){width="0.24\columnwidth" height="0.242\columnwidth"} ![ First: Single-particle spectra for seven $n_{ch}$ classes; Second: Scaled spectrum hard components $H(y_t,n_{ch})$; Third: Event distributions on $n_{ch}$ for several energies; Fourth: Hard-component multiplicity $n_h$ (dijet) trend on soft component $n_s$. $n_h$ is the integral (end-point amplitude) of measured $H(y_t,n_{ch})$ independent of shape. \[ppspec\]](ppcomm12cc "fig:"){width="0.24\columnwidth" height="0.247\columnwidth"}
Figure \[ppspec\] (first) shows rescaled $y_t$ spectra for seven multiplicity classes with $n_{ch} / \Delta \eta \approx 1.7,\ldots,19$. Fixed soft-component model $S_0$ is the asymptotic limit of spectra scaled by soft-component multiplicity $n_s$. Subtraction of $S_0$ and a second rescaling reveals hard components $H(y_t,n_{ch})$ scaled by $(n_s / \Delta \eta)^2$ (second panel) nearly independent of $n_{ch}$ approximated by fixed hard-component model $H_0(y_t)$. Soft-component multiplicity $n_s$ may serve as a proxy for [*participant partons*]{} (low-$x$ gluons) with substantial event-wise fluctuations (third panel). We observe (fourth panel) that $n_h \propto n_s^2$ (points), a trend inconsistent with that expected for the [*eikonal model*]{} (dashed curve $\propto n_s^{4/3}$) typically invoked in Monte Carlo models [@pythia; @herwig]. These 1D spectrum results provide the model functions and dijet systematics required to analyze and interpret the trigger-associated correlations presented below.
Systematics of minimum-bias p-p angular correlations
====================================================
Combinatoric minimum-bias (MB) angular correlations on angle differences $\eta_\Delta = \eta_1 - \eta_2$ and $\phi_\Delta = \phi_1 - \phi_2$ accepting all particle pairs (no $p_t$ cuts) can be described by a 2D model function including only a few elements [@porter2; @porter3; @anomalous]. The principal correlation components are jet-related same-side (SS) 2D peak and away-side (AS) 1D peak on azimuth (back-to-back jets) and nonjet (NJ) quadrupole $\cos(2\phi_\Delta)$.
![ First, second: Jet-related and non-jet quadrupole angular correlations for multiplicity classes $n = 1$ and 6; Third: Scaled amplitudes of jet-related structure vs soft multiplicity $n_s$; Fourth: Scaled nonjet quadrupole amplitude vs $n_s$ \[angcorr\]](ppcms23-0d "fig:"){width="0.24\columnwidth"} ![ First, second: Jet-related and non-jet quadrupole angular correlations for multiplicity classes $n = 1$ and 6; Third: Scaled amplitudes of jet-related structure vs soft multiplicity $n_s$; Fourth: Scaled nonjet quadrupole amplitude vs $n_s$ \[angcorr\]](ppcms23-5d "fig:"){width="0.24\columnwidth"} ![ First, second: Jet-related and non-jet quadrupole angular correlations for multiplicity classes $n = 1$ and 6; Third: Scaled amplitudes of jet-related structure vs soft multiplicity $n_s$; Fourth: Scaled nonjet quadrupole amplitude vs $n_s$ \[angcorr\]](ppcms10anewer "fig:"){width="0.24\columnwidth" height="0.22\columnwidth"} ![ First, second: Jet-related and non-jet quadrupole angular correlations for multiplicity classes $n = 1$ and 6; Third: Scaled amplitudes of jet-related structure vs soft multiplicity $n_s$; Fourth: Scaled nonjet quadrupole amplitude vs $n_s$ \[angcorr\]](ppcms10d2 "fig:"){width="0.24\columnwidth" height="0.22\columnwidth"}
Figure \[angcorr\] (first, second) shows angular correlations for multiplicity classes $n = 1$ and 6. Minor elements of the 2D model fits (proton fragment correlations, Bose-Einstein correlations, conversion electron pairs) have been subtracted leaving the jet-related components and the NJ quadrupole. The third panel shows trends on $n_s$ for jet-related amplitudes consistent with dijet number $n_j= 0.03 (n_s/2.5)^2$ (within $\Delta \eta = 2$) corresponding to pQCD dijet total cross section $\sigma_{dijet} = 2.5$ mb [@fragevo]. The NJ quadrupole trend on $n_s$ can be predicted. The observed centrality trend for collisions is $A_Q(b) \equiv \rho_0(b)v_2^2(b) \approx B\, N_{bin}(b) \epsilon_{optical}^2(b)$ [@davidhq]. For the non-eikonal case $N_{bin} \rightarrow N_{part}^2$ and impact parameter $b$ is not an observable, so $n_{ch} A_Q(b) \propto N_{part} N_{bin} \langle\epsilon_{optical}^2\rangle \propto N_{part}^3 \propto n_s^3$. Based on dijet systematics we expect $(n_{ch}/n_s) A_Q(n_s) \propto n_s^2$, which is confirmed in the fourth panel.
Trigger-associated (TA) two-component model (TCM)
=================================================
Based on SP spectrum and 2D MB dijet angular correlation systematics we can construct a TCM for trigger-associated correlations [@pptrig]. For each collision [*event type*]{} (soft or hard) the hadron with the highest transverse rapidity $y_{tt}$ is the [*trigger*]{} particle. All other hadrons are [*associated*]{}, with rapidities $y_{ta}$. Definition of the TA TCM is an exercise in compound probabilities. The unit-normal 1D trigger spectrum for multiplicity class $n_{ch}$ denoted by $T(y_{tt},n_{ch}) \equiv [{1}/{N_{evt}(n_{ch})}] {dn_{trig}}/{y_{tt}dy_{tt}}$ is modeled by \[trigspec\] T(y\_[tt]{},n\_[ch]{}) &=& P\_s(n\_[ch]{})G\_s(y\_[tt]{}) n\_[ch]{} F\_s(y\_[tt]{}) + P\_h(n\_[ch]{})G\_h(y\_[tt]{}) n\_[ch]{} F\_h(y\_[tt]{}), where $P_x(n_{ch})$ is an [event-type]{} probability, $G_x(y_{tt})$ is a void (above $y_{tt}$) probability and $F_x(y_{tt})$ is a unit-normal SP spectrum for event-type $x = s$ (soft, no dijets) or $h$ (hard, at least one dijet), with $G_x(y_{tt})\, n_{ch} F_x(y_{tt}) \equiv T_x(y_{tt},n_{ch})$. The Poisson event-type probabilities are defined by $P_s = \exp(-n_j)$ and $P_h = 1 - P_s$. The void probabilities are defined by $G_x = \exp(- n_{x\Sigma})$, where $n_{x\Sigma}$ is the appropriate spectrum integral above $y_{tt}$.
![ Trigger spectrum data (points) and TCM (curves) for n = 1, 3, 5, 7. \[tatcm\]](ppcms118a1 "fig:"){width="0.24\columnwidth" height="0.22\columnwidth"} ![ Trigger spectrum data (points) and TCM (curves) for n = 1, 3, 5, 7. \[tatcm\]](ppcms118a3 "fig:"){width="0.24\columnwidth" height="0.22\columnwidth"} ![ Trigger spectrum data (points) and TCM (curves) for n = 1, 3, 5, 7. \[tatcm\]](ppcms118a5 "fig:"){width="0.24\columnwidth" height="0.22\columnwidth"} ![ Trigger spectrum data (points) and TCM (curves) for n = 1, 3, 5, 7. \[tatcm\]](ppcms118a7 "fig:"){width="0.24\columnwidth" height="0.22\columnwidth"}
Figure \[tatcm\] shows trigger spectra (points) for four multiplicity classes. Solid curves $T(y_{tt},n_{ch})$ are defined by Eq. (\[trigspec\]). The other curves refer to TCM trigger-spectrum components. The TCM describes the trigger spectra well.
The unit-normal 2D TA distribution for event-type $x$ and multiplicity class $n_{ch}$ is [*joint probability*]{} $F_x(y_{ta},y_{tt}) \equiv T_x(y_{tt}) A_x(y_{ta}|y_{tt})$, where the [*chain rule*]{} for compound probabilities has been invoked. $A_x(y_{ta}|y_{tt})$ is the [*conditional probability*]{} that an associated particle is emitted at $y_{ta}$ in an event of type $x$ given a trigger at $y_{tt}$ with probability $T_x(y_{tt})$. The TA TCM is then \[tadist\] F(y\_[ta]{},y\_[tt]{}, n\_[ch]{}) &=& P\_s (n\_[ch]{}) T\_s(y\_[tt]{})A\_s(y\_[ta]{}|y\_[tt]{}) + P\_h (n\_[ch]{}) T\_h(y\_[tt]{}) A\_h(y\_[ta]{}|y\_[tt]{}), where the TCM $A_x$ are formed from the SP-spectrum TCM elements with certain [marginal constraints]{} [@pptrig]. Hard component $H_h(y_{ta}|y_{tt})$ of $A_h(y_{ta}|y_{tt})$ represents the sought-after momentum correlation structure of MB jets.
Measured trigger-associated correlations
========================================
Trigger-associated correlations can be presented both as joint probabilities $F(y_{ta},y_{tt},n_{ch})$ and as conditional probabilities $A(y_{ta}|y_{tt},n_{ch}) = F / T$ using the chain rule for joint probabilities.
![ Left: TA correlations F for multiplicity class $n = 6$ and for data and TCM (first and second respectively); Right: Same for conditional correlations $A = F/T$. \[tadata\]](ppcms116a6 "fig:"){width="0.24\columnwidth"} ![ Left: TA correlations F for multiplicity class $n = 6$ and for data and TCM (first and second respectively); Right: Same for conditional correlations $A = F/T$. \[tadata\]](ppcms117a6 "fig:"){width="0.24\columnwidth"} ![ Left: TA correlations F for multiplicity class $n = 6$ and for data and TCM (first and second respectively); Right: Same for conditional correlations $A = F/T$. \[tadata\]](ppcms116b6 "fig:"){width="0.24\columnwidth" height="0.23\columnwidth"} ![ Left: TA correlations F for multiplicity class $n = 6$ and for data and TCM (first and second respectively); Right: Same for conditional correlations $A = F/T$. \[tadata\]](ppcms117b6 "fig:"){width="0.24\columnwidth" height="0.23\columnwidth"}
Figure \[tadata\] (left) shows the measured joint distribution $F$ for $n = 6$ (first) and its corresponding TCM (second). Figure \[tadata\] (right) shows the measured conditional distribution $A$ (third) and its TCM (fourth). In both cases the agreement is good below $y_{ta} \approx 2.5$. TCM hard component $H_0'$ is based on a simple factorization approximation and plays no role in extraction of the data hard components described below. The jet-related hard component dominates TA structure for $y_{ta}$, $y_{tt} > 2.5$. The data and TCM hard components differ substantially.
Extracting the TA hard component
================================
Dividing Eq. (\[tadist\]) by Eq. (\[trigspec\]) we obtain the total conditional distribution \[tacond\] A(y\_[ta]{}|y\_[tt]{}, n\_[ch]{}) &=& R\_s (y\_[tt]{},n\_[ch]{}) A\_s(y\_[ta]{}|y\_[tt]{}) + R\_h (y\_[tt]{},n\_[ch]{}) A\_h(y\_[ta]{}|y\_[tt]{}), where the $R_x \leq 1$ are [*trigger fractions*]{}. The TCM conditional distributions are $A_s = S_0''$ and $A_h = p_s' S_0' + p_h' H_0'$ for $y_{ta} < y_{tt}$, with primes on $X_0'$ denoting the effects of marginal constraints as described in Ref. [@pptrig], and $p_x' = n_x' / (n_{ch} - 1)$. Given that expression we can isolate the hard component of the TA conditional distribution by subtracting the TCM soft components \[hardcomp\] H\_h’(y\_[ta]{}|y\_[tt]{},n\_[ch]{}) &=& \[A(y\_[ta]{}|y\_[tt]{}) - R\_s S”\_0(y\_[ta]{}|y\_[tt]{}) - R\_h p’\_s S’\_0(y\_[ta]{}|y\_[tt]{})\], the hard component (dijet momentum structure) per hard event. All subtractions use the same soft-component models derived from SP spectra.
![ Left: Per-hard-event hard component $H_h'(y_{ta}|y_{tt},n_{ch})$ for multiplicity classes $n = 2$ and 6; Right: The same data scaled by number of dijets per hard event $n_j / P_h$ to yield the per-dijet hard component. Lines are discussed in the text. \[hardcomp\]](ppcms116c3 "fig:"){width="0.24\columnwidth"} ![ Left: Per-hard-event hard component $H_h'(y_{ta}|y_{tt},n_{ch})$ for multiplicity classes $n = 2$ and 6; Right: The same data scaled by number of dijets per hard event $n_j / P_h$ to yield the per-dijet hard component. Lines are discussed in the text. \[hardcomp\]](ppcms116c6 "fig:"){width="0.24\columnwidth"} ![ Left: Per-hard-event hard component $H_h'(y_{ta}|y_{tt},n_{ch})$ for multiplicity classes $n = 2$ and 6; Right: The same data scaled by number of dijets per hard event $n_j / P_h$ to yield the per-dijet hard component. Lines are discussed in the text. \[hardcomp\]](ppcms116c3no-lines "fig:"){width="0.24\columnwidth"} ![ Left: Per-hard-event hard component $H_h'(y_{ta}|y_{tt},n_{ch})$ for multiplicity classes $n = 2$ and 6; Right: The same data scaled by number of dijets per hard event $n_j / P_h$ to yield the per-dijet hard component. Lines are discussed in the text. \[hardcomp\]](ppcms116c6no "fig:"){width="0.24\columnwidth"}
Figure \[hardcomp\] (left) shows hard components $H_h'(y_{ta}|y_{tt})$ for multiplicity classes $n = 2$, 6 (first and second respectively). The jet structure [*per hard event*]{} increases substantially with $n_{ch}$ because the probability that one or more additional dijets accompanies a triggered dijet (multiple parton interactions or MPI) becomes substantial. We can divide the left panels by the number of dijets per hard event $n_j / P_h$ to obtain the right panels. The resulting [*per-dijet*]{} structure appears to be approximately independent of $n_{ch}$ (universal). Universality is consistent with the $n_j(n_s)$ trend inferred from SP spectra.
TA azimuth dependence and the transverse region or TR
=====================================================
The azimuth structure of TA correlations relative to the trigger direction is of interest for several reasons including “underlying event” (UE) studies.
![ First: Conventional definition of azimuth regions for underlying event analysis; Second, Third, Fourth: TA hard components for toward T, transverse TR and away A averaged over multiplicity classes $n = 2$, 3, 4 to minimize MPI. \[azimuth\]](underlying "fig:"){width="0.24\columnwidth"} ![ First: Conventional definition of azimuth regions for underlying event analysis; Second, Third, Fourth: TA hard components for toward T, transverse TR and away A averaged over multiplicity classes $n = 2$, 3, 4 to minimize MPI. \[azimuth\]](ppcms116etwd "fig:"){width="0.24\columnwidth"} ![ First: Conventional definition of azimuth regions for underlying event analysis; Second, Third, Fourth: TA hard components for toward T, transverse TR and away A averaged over multiplicity classes $n = 2$, 3, 4 to minimize MPI. \[azimuth\]](ppcms116etrns "fig:"){width="0.24\columnwidth"} ![ First: Conventional definition of azimuth regions for underlying event analysis; Second, Third, Fourth: TA hard components for toward T, transverse TR and away A averaged over multiplicity classes $n = 2$, 3, 4 to minimize MPI. \[azimuth\]](ppcms116eawy "fig:"){width="0.24\columnwidth"}
Figure \[azimuth\] (first) shows the conventional azimuth partition relative to trigger direction (arrow) into three equal regions: “toward” (T), “transverse” (TR) and “away” (A). In some studies the A region is split into two parts $A_1$ and $A_2$ as shown. In conventional UE analysis it is assumed that the triggered dijet does not contribute to the TR, which region should therefore permit unbiased access to the UE [*complementary to the triggered dijet*]{} [@cdfue; @rick].
Figure \[azimuth\] (second, third, fourth) shows TA hard components per hard event for T, TR and A regions respectively, averaged over lower multiplicity classes $n = 2$, 3, 4 to reduce dijet pileup (MPI) to less than 15%. Those data averaged over azimuth are equivalent to Fig. \[hardcomp\] (right). Most notable is the substantial triggered-dijet contribution to the TR region (third panel), contradicting a common UE assumption. Compared to the T region (second) the A region (fourth) is both significantly softer [*and*]{} harder. The A region must be harder on average to compensate the trigger particle excluded from conditional distribution $A$ in the T region. The A region is also softer on average because of trigger bias to lower-energy jets due to initial-state $k_t$ effects and toward a softer fragmentation cascade within those jets.
The underlying event and multiple parton interactions
=====================================================
Other issues emerge for conventional UE analysis. Based on MB dijet angular correlations as in Fig. \[angcorr\] (left) we expect a substantial contribution to the TR from any dijet [@pptheory]. Figure \[uesys\] (first) shows a projection onto azimuth of the model fit to Fig. \[angcorr\] (first) approximating MB jet structure from non-single-diffractive collisions. There is a substantial overlap of SS and AS jet peaks and resulting strong jet contribution to the TR. Figure \[uesys\] (second) shows $N_\perp$ spectra from the TR described by the TCM of Eq. (\[ppspec\]) with the amplitude of (jet) hard-component $H$ as expected for hard (triggered) events.
![ First: Model fit to 2D MB jet angular correlations (curves) projected onto 1D azimuth showing substantial jet contribution to the TR (hatched); Second: Spectrum of $N_\perp$ yield in the TR (points, [@cdfue]) showing jet-related hard component (curve $H$); Third: Simulated $N_\perp$ density vs jet trigger condition showing increase to saturation due to selection of low-multiplicity hard events [@pptheory]; Fourth: Number of jets per hard event $n_j / P_h$ vs $n_{ch}$ inferred from SP spectrum systematics [@ppprd; @pptheory]. \[uesys\]](ppcms90i2 "fig:"){width="0.24\columnwidth" height="0.23\columnwidth"} ![ First: Model fit to 2D MB jet angular correlations (curves) projected onto 1D azimuth showing substantial jet contribution to the TR (hatched); Second: Spectrum of $N_\perp$ yield in the TR (points, [@cdfue]) showing jet-related hard component (curve $H$); Third: Simulated $N_\perp$ density vs jet trigger condition showing increase to saturation due to selection of low-multiplicity hard events [@pptheory]; Fourth: Number of jets per hard event $n_j / P_h$ vs $n_{ch}$ inferred from SP spectrum systematics [@ppprd; @pptheory]. \[uesys\]](ppcms90k2 "fig:"){width="0.24\columnwidth" height="0.23\columnwidth"} ![ First: Model fit to 2D MB jet angular correlations (curves) projected onto 1D azimuth showing substantial jet contribution to the TR (hatched); Second: Spectrum of $N_\perp$ yield in the TR (points, [@cdfue]) showing jet-related hard component (curve $H$); Third: Simulated $N_\perp$ density vs jet trigger condition showing increase to saturation due to selection of low-multiplicity hard events [@pptheory]; Fourth: Number of jets per hard event $n_j / P_h$ vs $n_{ch}$ inferred from SP spectrum systematics [@ppprd; @pptheory]. \[uesys\]](ppcms90e1 "fig:"){width="0.24\columnwidth" height="0.233\columnwidth"} ![ First: Model fit to 2D MB jet angular correlations (curves) projected onto 1D azimuth showing substantial jet contribution to the TR (hatched); Second: Spectrum of $N_\perp$ yield in the TR (points, [@cdfue]) showing jet-related hard component (curve $H$); Third: Simulated $N_\perp$ density vs jet trigger condition showing increase to saturation due to selection of low-multiplicity hard events [@pptheory]; Fourth: Number of jets per hard event $n_j / P_h$ vs $n_{ch}$ inferred from SP spectrum systematics [@ppprd; @pptheory]. \[uesys\]](ppcms116f "fig:"){width="0.24\columnwidth" height="0.233\columnwidth"}
Fig. \[angcorr\] (third) shows the TR $N_\perp$ density vs trigger condition $y_{t,trig}$. The increase to a saturation value is conventionally attributed to MPI. However, a study based on the TCM for SP spectra reveals that the $N_\perp$ increase results from a dijet contribution to the TR for hard events with low ($\approx$ NSD) multiplicities where the incidence of MPI is negligible [@pptheory]. Fig. \[angcorr\] (fourth) shows the calculated dijet number per hard event vs multiplicity. For NSD collisions ($n_{ch} / \Delta \eta \approx 2.5$) the MPI rate is only a few percent. From TA and angular-correlation analysis we conclude that application of a trigger $y_{t,trig}$ (jet) condition in UE analysis selects for jets within mainly low-multiplicity ($\approx$ NSD) hard events. Applying an $n_{ch}$ condition instead would select for multiple MB dijets (MPI) in higher-multiplicity events.
Kinematic limits on physical MB jet fragment production
=======================================================
The results in Figs. \[hardcomp\] (lines in third) and \[azimuth\] (second and fourth panels) reveal the kinematic limits of minimum-bias jet fragment production: Trigger hadrons extend down to $\approx 1$ GeV/c ($y_{tt} \approx 2.7$), and associated hadrons extend down to $\approx 0.35$ GeV/c (AS, $y_{ta} \approx 1.5$) or 0.5 GeV/c (SS, $y_{ta} \approx 2$). We conjecture that this also represents the low-hadron-momentum (and large-angle) structure of high-parton-energy jets, the common base of any dijet. Higher-energy jets contain a few additional high-momentum hadrons located close to the dijet axis and therefore outside the TR. TA correlation analysis could be extended to å collisions to verify the strong contribution from MB jets (minijets) even in more-central collisions [@anomalous; @fragevo; @jetspec].
These TA results are consistent with measured FFs from LEP, HERA and CDF and with a pQCD parton spectrum that predicts measured dijet production [@eeprd; @pptheory] and the shape of the MB spectrum hard component [@fragevo]. The MB-jet-related SS 2D peak volume is also consistent with pQCD predictions [@jetspec]. Conventional trigger-associated $p_t$ [*cuts*]{} invoked in å dihadron correlation analysis accept only a small fraction of the actual dijet number and jet fragments and, combined with so-called ZYAM subtraction of a combinatorial background, produce an unphysical picture of dijets in nuclear collisions in which jet structure is minimized and distorted [@tzyam].
Summary
=======
The two-component (soft + hard) model (TCM) of hadron production in high-energy nuclear collisions works remarkably well. Based on various comparisons with theory the soft component represents fragments from projectile nucleons (their gluon constituents), and the hard component represents dijet fragments from large-angle-scattered partons (gluons).
In this study the TCM has been applied to MB trigger-associated (TA) correlations for several charge multiplicity classes of 200 GeV collisions. A conditional hard component $H_h(y_{ta}|y_{tt})$ has been extracted by analogy with TCM analysis of single-particle spectra. The TA hard component reveals the kinematic limits of jet fragment production and is directly comparable with measured jet fragmentation functions from collisions.
These TA correlation results have implications for underlying-event (UE) analysis. Consistent with MB angular-correlation analysis the TA results confirm that the triggered dijets make a strong contribution to the transverse region or TR, contradicting conventional UE assumptions. The increase of the $N_\perp$ charge multiplicity in the TR with jet trigger $y_{t,trig}$ results not from multiple parton interactions (MPI) but from increased probability of low-multiplicity hard events including only a single dijet. The MPI rate is increased by selecting instead higher event multiplicities $n_{ch}$. The physical UE is then the MPI rate determined by $n_{ch}$ plus the TCM soft component.
.1in This work was supported in part by the Office of Science of the U.S. DOE under grant DE-FG03-97ER41020.
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---
abstract: 'The combination of edge caching and coded multicasting is a promising approach to improve the efficiency of content delivery over cache-aided networks. The global caching gain resulting from content overlap distributed across the network in current solutions is limited due to the increasingly personalized nature of the content consumed by users. In this paper, the cache-aided coded multicast problem is generalized to account for the correlation among the network content by formulating a source compression problem with distributed side information. A correlation-aware achievable scheme is proposed and an upper bound on its performance is derived. It is shown that considerable load reductions can be achieved, compared to state of the art correlation-unaware schemes, when caching and delivery phases specifically account for the correlation among the content files.'
author:
- 'P. Hassanzadeh, A. Tulino, J. Llorca, E. Erkip [^1] [^2] [^3]'
bibliography:
- 'ISTC\_arxiv\_v1.bib'
title: ' Cache-Aided Coded Multicast for Correlated Sources'
---
Introduction
============
\[sec:Introduction\] Proper distribution of popular content across the network caches is emerging as one of the promising approaches to address the exponentially growing traffic in current wireless networks. Recent studies [@maddah14fundamental; @maddah14decentralized; @ji14average; @ji15order; @ji15groupcast; @ji15efficient] have shown that, in a cache-aided network, exploiting globally cached information in order to multicast coded messages that are useful to a large number of receivers exhibits overall network throughput that is proportional to the aggregate cache capacity. The fundamental rate-memory trade-off in a broadcast caching network has been characterized in [@maddah14fundamental; @maddah14decentralized; @ji14average; @ji15order; @ji15groupcast; @ji15efficient]. While these initial results are promising, these studies treat the network content as independent pieces of information, and do not account for the additional potential gains arising from further compression of correlated content distributed across the network. , In this paper, we investigate how the correlations among the library content can be explored in order to further improve the performance in cache-aided networks. ,We consider a network setup similar to [@maddah14fundamental; @maddah14decentralized; @ji14average; @ji15order; @ji15groupcast; @ji15efficient], but assume that the files in the library are correlated. Such correlations are especially relevant among content files of the same category, such as episodes of a TV show or same-sport recordings, which, even if personalized, may share common backgrounds and scene objects.
As in existing literature on cache-aided networks, we assume that the network operates in two phases: a caching (or placement) phase taking place at network setup followed by a delivery phase where the network is used repeatedly in order to satisfy receiver demands. The design of the caching and delivery phases forms what is referred to as a [*caching scheme*]{}. During the caching phase, caches are filled with content from the library according to a properly designed [*caching distribution*]{}. During the delivery phase, the sender compresses the set of requested files into a multicast codeword by computing an [*index code*]{} while exploring all correlations that exist among the requested and cached content. In [@timo2015rate], the rate-memory region for a correlated library was characterized for lossy reconstruction in a scenario with only two receivers and a single cache. In this paper, we consider correlation-aware lossless reconstruction in a more general setting with multiple caches, and with a nonuniform demand distribution.
We propose an achievable correlation-aware scheme, named Correlation-Aware RAP caching and Coded Multicast delivery (CA-RAP/CM), in which receivers store content pieces based on their popularity as well as on their correlation with the rest of the file library during the caching phase, and receive compressed versions of the requested files according to the information distributed across the network and their joint statistics during the delivery phase. The scheme consists of: 1) exploiting file correlations to store the [*more relevant bits*]{} during the caching phase such that expected delivery rate is reduced, and 2) optimally designing the coded multicast codeword based on the joint statistics of the library files and the aggregate cache content during the delivery phase. Additional refinements are transmitted, when needed, in order to ensure lossless reconstruction of the requested files at each receiver. Given the exponential complexity of CA-RAP/CM, we then provide an algorithm which approximates CA-RAP/CM in polynomial time, and we derive an upper bound on the achievable expected rate. We numerically compare the rates achieved by the proposed correlation-aware scheme with existing correlation-unaware schemes, and our results confirm the additional gains achievable, especially for small memory size.
The paper is organized as follows. The problem formulation is presented in Sec. \[sec:Problem Formulation\]. In Sec. \[sec:Achievable Scheme\] we describe CA-RAP/CM and its polynomial-time approximation, and we quantify the associated rate-memory trade-off. We provide numerical simulations and concluding remarks in Secs. \[sec:Simulations\] and \[sec:Conclusion\], respectively.
Network Model and Problem Formulation {#sec:Problem Formulation}
=====================================
[**Notation:**]{} For ease of exposition, we use $\{A_i\}$ to denote the set of elements $\{A_i:i\in\mathcal I\}$, with $\mathcal I$ being the domain of index $i$, and we define $[n]\triangleq\{1,\dots,n\}$. $\FF_2^*$ denotes the set of finite length binary sequences.
Similar to previous work [@ji14average; @ji15order], we consider a broadcast caching network composed of one sender (e.g., base station) with access to a file library composed of $m$ files, each of entropy $F$ bits. Without loss of generality, we assume that each file $f\in [m]$ is represented by a vector of i.i.d. random binary symbols of length $F$, ${\sf W}_f \in \FF_2^F$, [ and hence $H({\sf W}_f) = F$ for all $f\in [m]$.]{}[^4] We assume that the symbols belonging to different files can be correlated, i.e., $H({\sf W}_1,\dots,{\sf W}_m)\leq mF$, and we denote their joint distribution by $P_{\mathcal W}$. The sender communicates with $n$ receivers (e.g., access points or user devices) $\mathcal U=\{1,\dots,n\}$ through a shared error-free multicast link. Each receiver has a cache of size $MF$ bits and requests files in an independent and identically distributed (i.i.d.) manner according to a demand distribution $\qbf = (q_1,\dots,q_m)$, where $q_f$ denotes the probability of requesting file $f\in [m]$. A caching scheme for this network consists of:
- [**Cache Encoder:**]{} Given a library realization $\{W_f\}$, the cache encoder at the sender computes the cache content $Z_u(\{W_f\})$ at receiver $u \in \mathcal U$, using the set of functions $\{Z_u: \FF_2^{m F} \rightarrow \FF_2^{MF} : u \in \Uc\}$. The encoder designs the cache configuration $\{Z_u\}$ jointly across receivers, taking into account global system knowledge such as the number of receivers and their cache sizes, the number of files, their aggregate popularity, and their joint distribution $P_{\mathcal W}$.
- [**Multicast Encoder:**]{} After the caches are populated, the network is repeatedly used for different demand realizations. At each use of the network, the receivers place a random demand vector $\bold{f} \in[m]^{n}$ at the sender, according to $\qbf$. The demand realization is denoted by $\fbf =(f_1,\dots,f_n)$. The multicast encoder at the sender computes codeword $X(\fbf, \{W_f\},\{Z_u\})$, which will be transmitted over the shared link, using the function $X : [m]^{n} \times \FF_2^{mF} \times \FF_2^{nMF} \rightarrow \FF_2^*$. In this work, we consider a fixed-to-variable almost-lossless framework.
- [**Multicast Decoders:**]{} Receivers recover their requested files using their cached content and the received multicast codeword. More specifically, receiver $u$ recovers $W_{f_u}$ using its decoding function $\zeta_u : [m]^n \times \FF_2^* \times \FF_2^{MF} \rightarrow \FF_2^{F}$, as $\widehat{W}_{f_u} = \zeta_u(\fbf, X,Z_u)$.
The worst-case (over the file library) probability of error of the corresponding caching scheme is defined as $$\begin{aligned}
\label{perr}
& P_e^{(F)} = \sup_{\{W_f : f \in [m]\}} \PP \left(\widehat{W}_{f_u} \neq W_{f_u} \right). \notag\end{aligned}$$ In line with previous work [@ji14average; @ji15order], the (average) rate of the overall [caching scheme]{} is defined as $$\label{average-rate}
R^{(F)} = \sup_{\{ W_f : f \in [m] \}} \; \frac{\EE[J(X)]}{F}.$$ where $J(X)$ denotes the length (in bits) of the multicast codeword $X$.
\[def:achievable-rate\] A rate $R$ is [*achievable*]{} if there exists a sequence of caching schemes for increasing file size $F$ such that [$\lim_{F \rightarrow \infty} P_e^{(F)} = 0 \notag$, and $\limsup_{F \rightarrow \infty} R^{(F)} \leq R.$]{}
The goal of this paper is to design a caching scheme that results in a small achievable rate $R$.
Correlation-Aware RAP Caching and Coded Multicast Delivery (CA-RAP/CM) {#sec:Achievable Scheme}
======================================================================
In this section, we introduce CA-RAP/CM, a correlation-aware caching scheme, which is an extension of a RAndom Popularity-based (RAP) caching policy followed by a Chromatic-number Index Coding (CIC) delivery policy [@ji14average; @ji15order]. In CA-RAP/CM, both the cache encoder and the multicast encoder are designed according to the joint distribution $P_{\mathcal W}$, in order to exploit the correlation among the library files. First, consider the following motivating example.
\[ex:CA RAP\] Consider a file library with $m=4$ uniformly popular files $\{W_1,W_2,W_3,W_4\}$ each with entropy $F$ bits. We assume that the pairs $\{W_1 ,W_2\}$ and $\{W_3, W_4\}$ are independent, while correlations exist between $W_1$ and $W_2$, and between $W_3$ and $W_4$. Specifically, $H(\sf W_1|\sf W_2)=H(\sf W_2|\sf W_1)=F/4$ and $H(\sf W_3|\sf W_4)=H(\sf W_4|\sf W_3)=F/4$. The sender is connected to $n=2$ receivers $\{u_1, u_2\}$ with cache size $M=1$. While a correlation-unaware scheme (e.g., [@maddah14decentralized; @ji14average]) would first compress the files separately and then cache $1/4^{th}$ of each file at each receiver, existing file correlations can be exploited to cache the more relevant bits. For example, one could split files $W_2$ and $W_4$ into two parts $\{ W_{2,1}, W_{2,2}\}$ and $\{ W_{4,1}, W_{4,2}\}$, each with entropy $F/2$, and cache $\{ W_{2,1}, W_{4,1}\}$ at $u_1$ and $\{ W_{2,2}, W_{4,2}\}$ at $u_2$, as shown in Fig. \[fig:Examples\]. During the delivery phase, consider the worst case demand, e.g., $\fbf = (W_3,W_1)$, the sender first multicasts the XOR of the compressed parts $W_{2,1}$ and $W_{4,2}$. Refinement segments, with refinement rates $H(\sf W_3|\sf W_4)$ and $H(\sf W_1|\sf W_2)$ can then be transmitted to enable lossless reconstruction, resulting in a total rate $R=1$. Note that a correlation-unaware scheme would need a total rate $R=1.25$ regardless of the demand realization [@maddah14decentralized; @ji14average].
![Network setup and cache configuration of Example \[ex:CA RAP\].[]{data-label="fig:Examples"}](Examples.png){width="2.7in"}
The main components of the CA-RAP/CM scheme are: $i$) a [*Correlation-Aware Random Popularity Cache Encoder*]{} (CA-RAP) and $ii$) a [*Correlation-Aware Coded Multicast Encoder*]{} (CA-CM).
Correlation-Aware Random Popularity Cache Encoder {#subsec:CA RAP}
-------------------------------------------------
The CA-RAP cache encoder is a correlation-aware random fractional cache encoder whose key differentiation from the cache encoder RAP, introduced in [@ji14average; @ji15order], is to choose the fractions of files to be cached according to both their popularity as well as their correlation with the rest of the library. Similar to the cache encoder RAP, each file is partitioned into $B$ equal-size packets, with packet $b\in[B]$ of file $f\in[m]$ denoted by $W_{f,b}$. The cache content at each receiver is selected according to a *caching distribution*, $\pbf = (p_{1}, \dots, p_{m})$ with $0 \leq p_{f} \leq 1/M$ $\forall f \in[m]$ and $\sum_{f = 1}^{m} p_{f} = 1$, which is optimized to minimize the rate of the corresponding index coding delivery scheme. For a given caching distribution $\pbf$, each receiver caches a subset of $p_{f}MB$ distinct packets from each file $f \in [m]$, independently at random. We denote by $\Cbf=\{ \Cbf_1,\dots,\Cbf_n \}$ the packet-level cache configuration, where $\Cbf_{u}$ denotes the set of file-packet index pairs, $(f,b)$, $f\in[m]$, $b\in[B]$, cached at receiver $u$. In Example \[ex:CA RAP\], $B=2$, the caching distribution corresponds to $p_{W_2}=p_{W_4}=1/2$, $p_{W_1}=p_{W_3}=0$, and the packet-level cache configuration is $\Cbf=\{\{(2,1),(4,1)\},\{(2,2),(4,2)\}\}$. While the caching distribution of a correlation-unaware scheme prioritizes the caching of packets according to the aggregate popularity distribution (see [@ji14average; @ji15order]), the CA-RAP caching distribution accounts for both the aggregate popularity and the correlation of each file with the rest of the library when determining the amount of packets to be cached from each file.
Correlation-Aware Coded Multicast Encoder
-----------------------------------------
[For a given demand realization $\fbf$, the packet-level demand realization is denoted by $\Qbf=[\Qbf_1,\dots,\Qbf_n]$, where $\Qbf_{u}$ denotes the file-packet index pairs $(f,b)$ associated with the packets of file $W_{f_u}$ requested, but not cached, by receiver $u$. ]{}
The CA-CM encoder capitalizes on the additional coded multicast opportunities that arise from incorporating cached packets that are, not only equal to, but also correlated with the requested packets into the multicast codeword. The CA-CM encoder operates by constructing a [*clustered conflict graph*]{}, and computing a linear index code from a valid[^5] coloring of the conflict graph, as described in the following.
### Correlation-Aware Clustering {#subsubsec:correlation-Aware Source Clustering}
For each requested packet $W_{f, b}$, $(f,b)\in\Qbf$, the correlation-aware clustering procedure computes a [*$\delta$-ensemble*]{} $\mathcal G_{f,b}$, where $\mathcal G_{f,b}$ is the union of $W_{f,b}$ and the subset of all cached and requested packets that are $\delta$-correlated with $W_{f,b}$, as per the following definition.[^6]
([**$\delta$-Correlated Packets**]{})\[def:delta-cor\] For a given threshold $\delta\leq1$, packet ${\sf W}_{f,b}$ is $\delta$-correlated with packet ${\sf W}_{f',b'}$ if $H({\sf W}_{f,b},{\sf W}_{f',b'})\leq (1+\delta)F $ bits, for all $f,f' \in [m]$ and $b,b' \in [B]$.
[This classification [ (clustering)]{} is the first step for constructing the clustered conflict graph.]{}
### Correlation-Aware Cluster Coloring {#subsubsec:Correlation-Aware Chromatic Cluster Covering}
The clustered conflict graph $\mathcal H_{\Cbf,\Qbf}=({{\cal V}}, {{\cal E}})$ is constructed as follows:
- Vertex set ${{\cal V}}$: The vertex (node) set $ {{\cal V}}=\widehat{\mathcal V}\cup\widetilde{\mathcal V}$ is composed of root nodes $\widehat{\mathcal V}$ and virtual nodes $\widetilde{\mathcal V}$.
- Root Nodes: There is a root node $ \hat v\in\widehat{\mathcal V}$ for each packet $W_{f,b}$, requested by each receiver, uniquely identified by the pair $\{ \rho(\hat v),\mu(\hat v)\}$, with $\rho(\hat v)$ denoting the packet identity, i.e., file-packet index pair $(f,b)$, and $\mu(\hat v)$ denoting the receiver requesting it.
- Virtual Nodes: For each root node $\hat v \in \widehat{\mathcal {{\cal V}}}$, all the packets in the $\delta$-ensemble $\mathcal G_{\rho(\hat v)}$ other than $\rho(\hat v)$ are represented as virtual nodes in $\widetilde{\mathcal V}$. We identify virtual node $\tilde v\in\widetilde{\mathcal V}$, having $\hat v$ as a root note, with the triplet $\{ \rho(\tilde v), \mu(\hat v),r(\tilde v)\}$, where $\rho(\tilde v)$ indicates the associated with vertex $\tilde v$, $\mu(\hat v)$ indicated the receiver requesting $\rho(\hat v)$, and $r(\tilde v) = \hat v$ is the root of the $\delta$-ensemble that $\tilde v$ belongs to.
We denote by ${{\cal K}}_{\hat v} \subseteq {{\cal V}}$ the set of vertices containing root node $\hat v \in \widehat{{{\cal V}}}$ and the virtual nodes corresponding to the packets in its $\delta$-ensemble $\mathcal G_{\rho(\hat v)}$, and we refer to ${{\cal K}}_{\hat v}$ as the [*cluster*]{} of root node $\hat v$.
- Edge set ${{\cal E}}$: For any pair of vertices $v_1, v_2 \in {{\cal V}}$, there is an edge between $v_1$ and $v_2$ in $\mathcal E$ if both $v_1$ and $v_2$ are in the same cluster, or if the two following conditions are jointly satisfied 1) $\rho(v_1) \neq \rho(v_2)$, 2) packet $\rho(v_1) \notin \Cbf_{\mu(v_2)}$ or packet $\rho(v_2) \notin \Cbf_{\mu(v_1)}$.
Given a valid coloring of $\mathcal H_{\Cbf,\Qbf}$, a valid *cluster coloring* of $\mathcal H_{\Cbf,\Qbf}$ consists of assigning to each cluster ${{\cal K}}_{\hat v},\; \forall \hat v\in \widehat{{{\cal V}}}$, one of the colors assigned to the vertices inside that cluster. For each color in the cluster coloring, only the packets with the same color as the color assigned to their corresponding cluster, are XORed together and multicasted to the users. Using its cached information and the received XORed packets, each receiver is able to reconstruct a (possibly) distorted version of its requested packet, due to the potential reception of a packet that is $\delta$-correlated with its requested one. The encoder transmits refinement segments, when needed, to enable lossless reconstruction of the demand at each receiver. The coded multicast codeword results from concatenating: 1) for each color in the cluster coloring, the XOR of the packets with the same color, and, 2) for each receiver, if needed, the refinement segment. The CA-CM encoder selects the valid cluster coloring corresponding to the shortest coded multicast codeword.
[Note that if correlation is not considered or non-existent, the clustered conflict graph is equivalent to the conventional index coding conflict graph [@ji14average; @ji15order], i.e., the subgraph of $\mathcal H_{\Cbf,\Qbf}$ resulting from considering only the root nodes $\widehat{\mathcal V}$.]{}
[It is important to note that the number of colors in the cluster coloring chosen by CA-CM is always smaller than or equal to the chromatic number[^7] of the conventional index coding conflict graph. [ Such reduction in the number of colors is obtained by considering correlated packets that are cached in the network, which possibly results in less conflict edges and provides more options for coloring each cluster. Intuitively, CA-CM allows for the requested packets that had to be transmitted by themselves otherwise, to be represented by correlated packets that can be XORED together with other packets in the multicast codeword.]{} ]{}
Greedy Cluster Coloring (GClC) {#subsec: algorithms}
------------------------------
Given that graph coloring, and by extension cluster coloring, is NP-Hard, in this section we propose [*Greedy Cluster Coloring (GClC)*]{}, a polynomial-time approximation to the cluster coloring problem. GClC extends the existing Greedy Constraint Coloring (GCC) scheme [@ji15order], to account for file correlation in cache-aided networks and consists of a combination of two coloring schemes, such that the scheme resulting in the lower number of colors (i.e., shorter multicast codeword) is chosen. Uncoded refinement segments are transmitted to ensure lossless reconstruction of the demand.
[ In GClC, we assume that any vertex (root node or virtual node) $v\in\mathcal V$ is identified by the triplet $\{\rho(v),\mu(v), r(v)\}$, which is uniquely specified by the associated with $v$ and by the cluster to which $v$ belongs.]{} Specifically, given a vertex $v \in {{\cal K}}_{\hat v}$, then $\rho(v)$ indicates the associated with vertex $v$, while $\mu(v)=\mu(\hat v)$ and $r(v)=\hat v$. Further define $\eta(v)\triangleq\{u\in\mathcal U: \rho(v) \in \Cbf_u\}$ for any $v\in {{\cal V}}$. The unordered set of receivers $\{\mu(v), \eta(v)\}$, corresponding to the set of receivers either requesting or caching packet $\rho(v)$, is referred to as the [*receiver label*]{} of vertex $v$.
Algorithm GClC$_1$ starts from a root node $\hat v\in\widehat{{\cal V}}$ among those not yet selected, and searches for the node $v_t \in {{\cal K}}_{\hat v}$ which forms the largest independent set ${{\cal I}}$ with all the vertices in $ {{\cal V}}$ having its same receiver label.[^8] Next, vertices in set ${{\cal I}}$ are assigned the same color (see lines 20-23). Algorithm GClC$_2$ is based on a correlation-aware extension of GCC$_2$ in [@ji15order], and corresponds to a generalized uncoded (naive) multicast: For each root node $\hat v \in \widehat{{\cal V}}$, whose cluster has not yet been colored, only the vertex $ v_t\in \mathcal K_{\hat v}$ whom is found among the nodes of more clusters, i.e., correlated with a larger number of requested packets, is colored and its color is assigned to $\mathcal K_{\hat v}$ and all clusters containing $v_t$. For both GClC$_1$ and GClC$_2$, when the graph coloring algorithm terminates, only a subset of the graph vertices, $\mathcal V$, are colored such that only one vertex from each cluster in the graph is colored. This is equivalent to identifying a valid cluster coloring where each cluster is assigned the color of its colored vertex. Between GClC$_1$ and GClC$_2$, the cluster coloring resulting in the lower number of colors is chosen. For each color assigned during the coloring, the packets with the same color are XORed together, and multicasted.
Performance of CA-RAP/CM
------------------------
\[subsec:correlation aware rate\]
In this section, we provide an upper bound of the rate achieved with CA-RAP/CM. For a given $\delta$, we define the [*match matrix*]{} $\Gbf$ as the matrix whose element $\Gbf_{f'f}$ $(f, f') \in [m]^2$ is the largest value such that for each packet $W_{f,b}$ of file $f$, there are at least $\Gbf_{f'f}$ packets of file $f'$ that are $\delta$-correlated with $W_{f,b}$, and are distinct from the packets correlated with packet $W_{f,b'}$, $\forall \, b'\in[B]$.
\[thm:general CA-RAP/CM\] Consider a broadcast caching network with $n$ receivers, cache capacity $M$, demand distribution $\qbf$, a caching distribution $\pbf$, library size $m$, correlation parameter $\delta$, and match matrix $\Gbf$. Then, the achievable expected rate of CA-RAP/CM, $R(\delta , \pbf)$, is upper bounded, as $F \rightarrow \infty$, with high probability as $$\begin{aligned}
R(\delta , \pbf) \leq \min \left \{\psi(\delta , \pbf) + \Delta R(\delta , \pbf),\bar m \right \}, \end{aligned}$$ where $$\begin{aligned}
& \psi(\delta , \pbf) \triangleq \sum\limits_{\ell =1 }^{n} \binom{n}{\ell} \sum\limits_{f =1}^m \rho_{\ell,f} \lambda_{\ell,f} ,\notag \\
& \Delta R(\delta , \pbf) \leq \sum\limits_{\ell =1 }^{n} \ell\binom{n}{\ell} \sum\limits_{f =1}^m \rho_{\ell,f}^* \lambda_{\ell,f}^* \delta \notag \\
& \quad +
n \sum\limits_{f =1}^m q_f (1-p_{f}M)\Big(1 - \prod\limits_{f' =1}^{m} (1-p_{f'}M)^{{(\Gbf-{\bf I})}_{f'f}}\Big) \delta,
\notag \\
&\bar m \triangleq \sum_{f =1}^m ( 1- (1-q_f)^{n} ), \notag\end{aligned}$$ with $$\begin{aligned}
&\lambda_{\ell,f} \triangleq \Big(\prod\limits_{f'=1}^m (1-p_{f'}M)^{(n-\ell+1)\Gbf_{f'f}} \Big)\notag \\
& \qquad\qquad\qquad\qquad\quad\; {\scalebox{0.99}{$ \bigg(1-\prod\limits_{f'=1}^m\Big(1-(p_{f'}M)^{\ell-1}\Big)^{\Gbf_{f'f}} \bigg) $}} ,\notag\end{aligned}$$ $$\begin{aligned}
& \rho_{\ell,f} \triangleq \mathbb P \Big\{ f = \argmax\limits_{d \in \mathcal D} \lambda_{\ell,d} \Big\} ,\notag\\
& \lambda_{\ell,f}^* \triangleq \Big( \prod\limits_{f' =1}^m (1-p_{f'}M)^{(n-\ell+1)\Gbf_{f'f}} \Big) \notag \\
& \quad {\scalebox{0.99}{$ \Big(1-(p_{f}M)^{\ell-1}\Big) \bigg(1-\prod\limits_{f'=1}^m\Big(1-(p_{f'}M)^{\ell-1}\Big)^{(\Gbf-{\bf I} )_{f'f} } \bigg)$}} ,\notag \\
&\rho_{\ell,f}^* \triangleq \mathbb P \Big\{ f = \argmax\limits_{d \in \mathcal D} \lambda^*_{\ell,d} \Big\} , \notag \end{aligned}$$ $\mathcal D$ denoting a random set of $\ell$ elements selected in an i.i.d. manner from $[m]$, and $\bf I$ denoting the identity matrix.
For proof of Theorem \[thm:general CA-RAP/CM\] see [@longerVersion].
The CA-RAP caching distribution is computed as the minimizer of the corresponding rate upper bound, $\pbf^*=\argmin_{\pbf} R({\delta},\pbf)$, resulting in the optimal CA-RAP/CM rate $R({\delta},\pbf^*)$. The resulting distribution $\pbf^{*}$ may not have an analytically tractable expression in general, but can be numerically optimized for the specific library realization. We remark that the rate upper bound is derived for a given correlation parameter $\delta$, whose value can also be optimized to minimize the achievable expected rate $R({\delta},\pbf)$.
Simulations and Discussions
===========================
\[sec:Simulations\] We numerically compare the performance of the polynomial-time CA-RAP/CM scheme given in Sec. \[subsec:correlation aware rate\], with existing correlation-unaware schemes: Local Caching with Unicast Delivery (LC/U), Local Caching with Naive Multicast Delivery (LC/NM), and RAP Caching with Coded Multicast Delivery (RAP/CM) [@ji15order]. LC/U consists of the conventional LFU caching policy, in which the $M$ most popular files are cached at each receiver, followed by unicast delivery. LC/NM combines LFU caching with naive multicast delivery, where a common uncoded stream of data packets is simultaneously received and decoded by multiple receivers. RAP/CM is the combination of RAP caching and coded multicasting as presented in [@ji15order], and is equivalent to CA-RAP/CM when $\delta=0$.
We consider a broadcast caching network with $n=10$ receivers and $m=100$ files requested according to a Zipf distribution with parameter $\alpha$ given by $q_{f} = {f^{-\alpha}}/{\sum_{j=1}^{m}j^{-\alpha}}, \; \forall f \in [m]$. Under demand distribution $\qbf$, we assume a match matrix $\Gbf$ such that, for each $(f, f') \in [m]^2$, $\Gbf_{f,f'}= B_{\rm match}$. Fig. \[fig:alpha 0.8\] displays the rates for $\alpha=0.8$, $\delta = 0.2$, and $mB_{\rm match}=4$, as $M$ varies from $0$ to $100$. We observe that the correlation-aware scheme is able to achieve rate reductions that go well beyond the state of the art correlation-unaware counterpart. Specifically, CA-RAP/CM achieves a $2.7\times$ reduction in the expected rate compared to LC/U and a $2.4\times$ reduction compared to RAP/CM for $M=10$.
![Performance of CA-RAP/CM in a network with Zipf parameter $\alpha = 0.8$, $n = 10$, $m = 100$, $\delta = 0.2$, and $mB_{\rm match}= 4$.[]{data-label="fig:alpha 0.8"}](General_KG24_Delta02_A08.pdf){width="2.8in"}
Conclusion
==========
\[sec:Conclusion\] In this work, we have shown how exploiting the correlation among the library files can result in more efficient content delivery over cache-aided networks. We proposed a correlation-aware caching scheme in which receivers store content pieces based on their popularity as well as on their correlation with the rest of the file library in the caching phase, and receive compressed versions of the requested files according to the information distributed across the network and their joint statistics during the delivery phase. The proposed scheme is shown to significantly outperform state of the art approaches that treat library files as mutually independent. Ongoing and future work entail investigating the effect of relevant system parameters that characterize the correlated library, such as $\delta$ and $\Gbf$, on the achievable expected rate, as well as providing order-optimality results. An alternative correlation-aware scheme is proposed in [@Hassanzadeh2016ITW] where the more relevant content from the jointly compressed library is stored during the caching phase, and the transmissions during the delivery phase ensure perfect reconstruction of the requested content.
[^1]: P. Hassanzadeh and E. Erkip are with the ECE Department of New York University, Brooklyn, NY. Email: {ph990, elza}@nyu.edu
[^2]: J. Llorca and A. Tulino are with Bell Labs, Nokia, Holmdel, NJ, USA. Email: {jaime.llorca, a.tulino}@nokia.com
[^3]: A. Tulino is with the DIETI, University of Naples Federico II, Italy. Email: {antoniamaria.tulino}@unina.it
[^4]: The results derived in this paper can be easily extended to the case in which the symbols within each file are not necessarily binary and independent.
[^5]: A valid coloring of a graph is an assignment of colors to the vertices of the graph such that no two adjacent vertices are assigned the same color.
[^6]: In practice, the designer shall determine the level at which to compute and exploit correlations between files based on performance-complexity tradeoffs. For example, in video applications, a packet may represent a video block or frame.
[^7]: The chromatic number of a graph is the minimum number of colors over all valid colorings of the graph.
[^8]: An independent set is a set of vertices in a graph, no two of which are adjacent.
|
---
abstract: 'As in the case of the hydrogen atom, bound-state wave functions are needed to generate hadronic spectra. For this purpose, in 1971, Feynman and his students wrote down a Lorentz-invariant harmonic oscillator equation. This differential equation has one set of solutions satisfying the Lorentz-covariant boundary condition. This covariant set generates Lorentz-invariant mass spectra with their degeneracies. Furthermore, the Lorentz-covariant wave functions allow us to calculate the valence parton distribution by Lorentz-boosting the quark-model wave function from the hadronic rest frame. However, this boosted wave function does not give an accurate parton distribution. The wave function needs QCD corrections to make a contact with the real world. Likewise QCD needs the wave function as a starting point for calculating the parton structure function.'
---
[**QCD’s Partner needed for Mass Spectra and Parton Structure Functions**]{}\
Y. S. Kim\
Center for Fundamental Physics, University of Maryland, College Park, Maryland 20742, U.S.A., yskim@umd.edu
Presented at the Excited QCD, Zakopane, Poland (February 2009).
At the 1965 meeting of the American Physical Society held in Washington, DC, U.S.A. Freeman Dyson stated that quantum electrodynamics can become more effective if combined with other theories [@dyson65]. He was right, but he gave a wrong example. He mentioned the calculation of the neutron-proton mass difference by Dahsen and Frautchi as an example. It is still believed that the neutron and proton have the same mass, and the mass difference comes from an electromagnetic perturbation. However, their perturbation calculation uses a non-localized a wave function which increases exponentially for large distance [@kim66].
Dyson was still right in saying that QED needs a partner to be most effective. The partner is the localized bound-state wave function. Let us look at the Lamb shift. QED gives to the Coulomb potential a delta function correction at the origin. The S state gets affected by this potential, while the P state is insensitive to this correction at the origin. This results in the shifts between and P and S states. The Lamb shift is regarded as one of the triumphs of quantum electrodynamics.
Indeed, in order to calculate the Lamb shift, we need hydrogen wave functions, but quantum electrodynamics cannot produce localized wave functions with probability interpretation. We still have to solve the wave equation with the standing-wave boundary condition to get the Rydberg energy levels and corresponding wave functions.
QED with Feynman diagrams is designed to address scattering problems in the Lorentz-covariant world. The situation is the same in QCD, which is an extension of QED with gluon instead of photons. QCD can make corrections to the existing mass spectra and structure functions, but cannot produce wave functions with proper boundary conditions. Thus, QCD alone cannot produce hadronic mass spectra or parton distributions. It needs a partner.
![Quantum mechanics in Galilei and Einstein systems. It is possible to construct a Lorentz-covariant model of bound states. Feynman and his students in 1971 wrote down a Lorentz-invariant differential equation which contains both running and standing waves.[]{data-label="comet77"}](comet77.eps)
In 1971, Feynman and his students noted that harmonic oscillator wave functions with their three-dimensional degeneracy can explain the main features of the hadronic spectra [@fkr71]. Earlier in 1969 [@fey69], Feynman proposed his parton picture where a fast-moving hadrons appears like a collection of partons with properties quite different from those of the quarks inside a static hadron.
In their 1971 paper [@fkr71], Feynman [*et al.*]{} wrote down the Lorentz-invariant equation which can be separated into the Klein-Gordon equation for a free hadron, and a harmonic-oscillator equation for the quarks inside the hadron, which determines the hadronic mass. Feynman’s equation of 1971 contains both running waves for the hadron and the standing waves for the quarks inside the hadron, as indicated in Fig. \[comet77\].
The oscillator equation takes the form $$\label{fkr11}
\frac{1}{2} \left[\left(\frac{\partial}{\partial x_{\mu}}\right)^2
- x_{\mu}^2 \right] \psi\left(x_{\mu}\right) =
\lambda \psi \left(x_{\mu}\right) ,$$ where $x_{\mu}$ is the four-vector specifying the space-time separation between the quarks. For convenience, we ignore all physical constants such as $c, \hbar$, as well as the spring constant for the oscillator system.
In the hadronic rest frame, if the time-like excitations are suppressed, this equation produces hadronic mass spectra [@fkr71]. If the hadron starts moving along the $z$ direction, we can separate out the transverse coordinates $x$ and $y$, and write the differential equation of Eq.(\[fkr11\]) as $$\label{fkr22}
\frac{1}{2} \left[-\left(\frac{\partial}{\partial z}\right)^2 + z^2
+\left(\frac{\partial}{\partial t}\right)^2 - t^2 \right]
\psi(z,t) = \lambda \psi(z,t) ,$$ where $t$ is the time-separation variable between the quarks. From this equation, Feynman [*et al.*]{} wrote down their solution $$\label{fkr33}
\psi(z,t) = \exp{\left\{-\frac{1}{2}\left(z^2 - t^2\right)\right\}}.$$ This form is a Gussied function for the space-like $z$ coordinate if the time-like variable $t$ is ignored. It is also invariant under Lorentz boosts along the $z$ direction. However, due to its non-local time-like distribution, this expression cannot be regarded as a physically meaningful wave function.
On the other hand, this equation also has a solution of the form $$\label{kn11}
\psi(z,t) = \exp{\left\{-\frac{1}{2}\left(z^2 + t^2\right)\right\}}.$$ This solution is Gaussian in both the $z$ and $t$ variables. Is it then possible to attach a physical interpretation to this wave function.
First, the time-separation $t$ exists wherever there is a space separation, according to Einstein. According to quantum mechanics, there is a time-energy uncertainty relation associated with this variable, as shown in Fig. \[kn73\].
As Dirac noted in 1927 [@dir27], this time-energy uncertainty does not cause excitations, while Heisenberg’s uncertainty generate excitations along the space-like $z$ axis. However, this space-time asymmetry is quite consistent the internal space-time symmetries dictated by Wigner’s little group [@wig39; @knp86]. According to Wigner, the internal space-time symmetry of massive particles is that of the three-dimensional rotation group without the time variable. We can summarize these in terms of the circle given in Fig. \[kn73\].
How about the Lorentz invariance? The form given in Eq.(\[fkr33\]) is invariant under Lorentz boosts as $\left(z^2 - t^2\right)$ is. However, the expression $\left(z^2 + t^2\right)$ in Eq.(\[kn11\]) is not invariant. Is this the end of the story? No! Let us boost this form using Dirac’s light-cone system [@dir49].
![Space-time picture of quantum mechanics. In his 1927, Dirac noted that there is a c-number time-energy uncertainty relation, in addition to Heisenberg’s position-momentum uncertainty relations, with quantum excitations. This idea is illustrated in the first figure. In 1949, Dirac produced his light-cone coordinate system as illustrated in the second figure. It is then not difficult to produce the third figure, for a Lorentz-covariant picture of quantum mechanics.[]{data-label="kn73"}](kn73.eps)
If the hadron moves along the $z$ direction with the velocity parameter $\beta$, the wave function of Eq.(\[kn11\]) becomes $$\label{kn22}
\exp{\left\{-\frac{1}{4}\left[\frac{1 - \beta}{1 + \beta}(z + t)^2
+ \frac{1 + \beta}{1 - \beta}(z - t)^2\right] \right\}},$$ This is an elliptic distribution given in Fig. \[kn73\], where the circular distribution is modulated by Dirac’s light-cone picture of Lorentz boosts. The circle is “squeezed” into the ellipse.
The question is whether we can see the effects of this Lorentz squeeze in the real world. In 1973 [@kn73ff], in terms of Lorentz-squeezed hadrons, Kim and Noz were able to explain the form factor calculation of Fujimura, Kobayashi, and Namiki who derived the dipole cut-off of the proton form factor for large momentum transfers [@fuji70].
According to Fig. \[kn73\], the quark distribution becomes concentrated along the immediate neighborhood of one of the light cones as the hadronic speed becomes closer to that of light. In 1977 [@knp86; @kn77par], Kim and Noz were able to explain the peculiarities of Feynman’s parton picture. Partons have the following peculiar properties.
- Partons are like free particles, unlike the quarks inside a hadron.
- The parton distribution function becomes wide-spread as the hadron moves faster. The width of the distribution is proportional to the hadron momentum.
- The number of partons appears to be infinite.
In the ellipse given in Fig. \[kn73\], one of the axis becomes longer while the other becomes shorter. In 2005 [@kn05job], Kim and Noz were able to associate these axes as the interaction time between the quarks and the interaction time of one of the quarks with the external signal, respectively. Thus, the external signal is not able to sense other quarks in the hadron. This is what Feynman said in his original papers on the parton model [@fey69].
Kim and Noz indeed explained all the peculiarities of Feynman’s parton picture, and proved that the quark model and the parton model are two different manifestations of one Lorentz-covariant entity. However, is it possible to calculate the parton distribution function by boosting the quark wave function from the rest frame? The hadron, when it moves fast, contains both valence partons and gluonic partons. We should therefore obtain the valence parton distribution by boosting the rest-frame wave function.
In 1980 [@hwa80], Hwa observed that the external signals do not directly interact with the quarks, but with dressed quarks called valons. Thus, if we remove the valon effect, we should be able to measure the distribution of valence quarks. With this point in mind, Hussar in 1981 compared the parton distribution from the boosted oscillator wave function and the experimentally measured distribution [@hussar81]. Hussar’s result is given in Fig. \[hussar\].
As we can see in this figure, there is a general agreement between the experimental data and the theoretical curve derived from the static quark distribution. Yet, the disagreement is substantial, and this is the gap QCD has to feel in. This work is yet to be carried out. The wave function needs QCD to make contacts with the real world. Likewise, QCD needs the wave function as a starting point for calculating the parton distribution. They are need each other. They are the partners.
![Parton distribution function from Hussar’s paper [@hussar81]. Although there is a general agreement between theory and experiment, the disagreement is substantial. This difference could be corrected by QCD.[]{data-label="hussar"}](hussar.eps)
[99]{}
F. J. Dyson, [*Physics Today*]{}, June 1965, page 21.
Y. S. Kim, [*Phys. Rev.*]{} [**142**]{}, 1150 (1966).
R. P. Feynman, M. Kislinger, and F. Ravndal, [*Phys. Rev. D*]{} [**3**]{}, 2706 (1971).
R. P. Feynman, [*Phys. Rev. Lett.*]{} [**23**]{}, 1415 (1969). See also R. P. Feynman, in [*Proceedings of the Third International Conference, Stony Brook, NY, U.S.A.*]{} (ed. C. N. Yang, et al.), pp 237 (Gordon and Breach, New York, 1969).
P. A. M. Dirac, [*Proc. Roy. Soc. (London)*]{} [**A114**]{}, 243 (1927).
E. Wigner, [*Ann. Math.*]{} [**40**]{}, 149 (1939).
Y. S. Kim and M. E. Noz, [*Theory and Applications of the Poincaré Group*]{} (Reidel, Dordrecht, The Netherlands 1986).
P. A. M. Dirac, [*Rev. Mod. Phys.*]{} [**21**]{}, 392 (1945).
Y. S. Kim and M. E. Noz, [*Phys. Rev. D*]{} [**8**]{}, 3521 (1973).
K. Fujimura, T. Kobayashi, and M. Namiki, [*Prog. Theor. Phys.*]{} [**43**]{}, 73 (1970).
Y. S. Kim, and M. E. Noz, [*Phys. Rev. D*]{} [**15**]{}, 335 (1977); Y. S. Kim, [*Phys. Rev. Lett.*]{} [**63**]{}, 348 (1989).
Y. S. Kim and M. E. Noz, [*J. Opt. B: Quantum and Semiclass. Opt.*]{} [**7**]{}, S458 (2005).
R. Hwa, [*Phys. Rev. D*]{}[**22**]{}, 759 (1980); R. Hwa and M. S. Zahir, [*Phys. Rev. D*]{}[**23**]{}, 2539 (1981);
P. E. Hussar, [*Phys. Rev. D*]{} [**23**]{}, 2781 (1981).
|
---
abstract: 'The present lectures were prepared for the Faro International Summer School on Factorization and Integrable Systems in September 2000. They were intended for participants with the background in Analysis and Operator Theory but without special knowledge of Geometry and Lie Groups. The text below represents a sort of compromise: it is certainly impossible not to speak about Lie algebras and Lie groups at all; however, in order to make the main ideas reasonably clear, I tried to use only matrix algebras such as $\frak{gl}(n)$ and its natural subalgebras; Lie groups used are either $GL(n)$ and its subgroups, or *loop groups* consisting of matrix-valued functions on the circle (possibly admitting an extension to parts of the Riemann sphere). I hope this makes the environment sufficiently easy to live in for an analyst. The main goal is to explain how the factorization problems (typically, the matrix Riemann problem) generate the entire small world of Integrable Systems along with the geometry of the phase space, Hamiltonian structure, Lax representations, integrals of motion and explicit solutions. The key tool will be the *classical r-matrix* (an object whose other guise is the well-known Hilbert transform). I do not give technical details, unless they may be exposed in a few lines; on the other hand, all motivations are given in full scale whenever possible. I hope that this choice agrees with the spirit of the Faro School and will help to bridge the gap between different branches of Mathematical Analysis.'
address: |
Mathématique Physique\
Université de Bourgogne\
Dijon\
France
---
The story of the discovery of the modern theory of Integrable Systems is certainly too long (and too well-known), and I can hardly add anything new; so let me tell just a few words before addressing the bulk of the subject. The study of completely integrable systems goes back to the classical papers of Euler, Lagrange, Jacobi, Liouville and others on analytical mechanics. By the end of the XIX-th century all interesting examples seemed to have been exhausted, and the interest has shifted to the qualitative study of chaotic behaviour. The new age in the study of integrable systems has begun with the famous paper [@GGKM] on the KdV equation, which was the first example of an *infinite dimensional* dynamical system with nontrivial but highly regular behaviour and with a rich excitation spectrum. Amplifying the earlier results of Gardner, Greene, Kruskal and Miura and of Peter Lax [@PL] Faddeev and Zakharov [@ZF] have shown in 1971, exactly 30 years ago, that the KdV equation is in fact a *completely integrable Hamiltonian system* in a technical sense. Within a short time, many more examples have been discovered, notably, the sine–Gordon equation, the first ever example of a *relativistic* completely integrable system [@ZTF]. These discoveries were particularly exciting in view of the possible physical applications: while it was of course clear that “generic” nonlinear equations are non-integrable, it has been argued that Fundamental Physics always deals with highly *non-generic* equations which might be integrable in some sense or other. These initial hopes have been fulfilled only partially; one major obstacle is that the new technique does not apply to (natural) non-linear equations in realistic space-time dimension. On the other hand, the mathematics of complete integrability has proved to be extraordinarily rich, bringing together Functional Analysis, Algebraic Geometry, Lie Groups, Representation Theory, Symplectic Geometry and much more. The analytic machinery used in the initial papers was based on the Inverse Scattering Problem; the subsequent developments allowed to single out the basic geometric ideas of the theory and to provide a unified basis for different examples. One of the key ingredients of this geometric approach are Infinite Dimensional Lie Groups; in some loose sense, one can say that Integrable Systems always possess some rich *hidden symmetry*. One may recall that one of the original motivations of S.Lie has been the use of continuous transformation groups for the study of differential equations. With the modern methods at hand, we came closer to that goal; it has now become clear, however, that almost all classical examples of integrable mechanical systems from the XIX century textbooks, as well as the infinite dimensional systems associated with integrable PDE’s are related not to finite-dimensional Lie groups, but rather to their *infinite-dimensional analogs*.
The general geometric construction that we shall discuss below allows to unify the following characteristic features that are typical for all examples known so far:
1. The equations of motion are compatibility conditions for a certain auxiliary system of linear equations.
2. They are Hamiltonian with respect to a natural Poisson bracket.
3. Integrals of motion are spectral invariants of the auxiliary linear operator. They are in involution with respect to the Poisson bracket referred to above.
4. The solution of the equations of motion reduces to some version of the Riemann-Hilbert problem.
Depending on the nature of the auxiliary linear problem, the associated nonlinear equations may be divided into the following three groups:
a\) Finite-dimensional systems,
b\) Infinite-dimensional systems with one or two spatial variables,
c\) Integrable systems on one-dimensional lattices.
In case (a) the auxiliary linear problem is the eigenvalue problem for a finite-dimensional matrix (possibly depending on an additional parameter). In case (b) the associated linear operator is differential. In case (c) it is a difference operator.
As it happens, the key properties 1–4 referred to above are corollaries of a single general theorem which may be adapted to numerous concrete applications. The original idea of this theorem is due to B.Kostant and M.Adler; its important amplification brings in the notion of *classical r-matrix* which provides a link between abstract Riemann-Hilbert problems and the ideas of Quantum Group Theory. The statement and proof of this theorem are particularly simple for systems of types (a) and (b). (Lattice systems require a special treatment, since the associated Poisson brackets belong to a different and more sophisticated class. We shall discuss this case later on, but it is natural to begin with the simpler cases (a) and (b)).
One more word of caution: while it is aesthetically very attractive to deduce a large variety of examples together with their explicit solutions from a simple general construction, there is one important disadvantage: for a given dynamical system (even if it is known to be completely integrable!) it is very difficult to tell *a priori,* what is the underlying Lie group, or Lie algebra. The practical way around this difficulty is to look at various examples associated with different Lie algebras; with some skill, one manages to recognize among these examples both classical and new integrable systems which admit physical interpretation. The list of interesting Lie algebras includes:
1. Finite dimensional semisimple Lie algebras. Associated integrable systems include open Toda lattice and other finite dimensional Hamiltonian systems which may be integrated in *elementary functions* (rational functions of $\exp t$, or $t$, where $t$ is the time variable).
2. Loop algebras, or affine Lie algebras. Associated integrable systems are finite dimensional Hamiltonian systems which may be integrated in *Abelian functions* of time. Integrable tops, as well as almost all classical examples from the XIXth century Analytical Mechanics find their place here.
3. Double loop algebras and their central extensions. This class of algebras accounts for integrable PDE’s admitting the zero curvature representation (such as the Nonlinear Schroedinger equation, the Sine-Gordon equation and many others).
4. Algebras of pseudodifferential operators. The KdV equation and its higher analogs come from this example, although it is more practical to derive them from double loop algebras.[^1]
5. Algebra of vector fields on the line. This algebra, or rather its central extension *(the Virasoro algebra)* and the associated loop algebra again are related to the KdV equation.
As we see from this list, the choice of the Lie algebra determines not only the possible kinematics (i.e., the structure of the phase space) of the dynamical systems which admit a realization based on this algebra, but also the functional class of the possible solutions. In all cases, it is very important to examine *central extensions* of the algebras in question (if any), as well as their non-trivial automorphisms: they usually lead to new examples. Non-trivial central extensions exist in cases 2, 3, 4, 5, each one leading to a non-trivial theory. As for outer automorphisms of these algebras, they serve, for instance, to define *twisted loop algebras* with important range of applications (see Section 7.1) and are also used in the study of nonlinear finite-difference equations.
It is virtually impossible to provide these lectures with a comprehensive list of references. The point of view adopted here is certainly rather subjective. I have included a few references to the old original papers as well as to several reviews.
The first question that is prior to the study of the dynamics of a mechanical system is that of its *kinematics*, i.e. of the structure of its phase space. Typically, the phase space of an individual dynamical system should be a symplectic manifold. However, an important conclusion which may be drawn from the practical study of numerous examples is that integrable systems associated with auxiliary linear problems always arise in *families.* The appropriate geometrical setting for the study of such families is provided by the theory of *Poisson manifolds*. Let me recall that a Poisson bracket on a smooth manifold $M$ is the structure of a Lie algebra on the space $C^{\infty }(M)$; moreover, the Poisson bracket satisfies the Leibniz rule, i.e., it is a derivation with resect to each argument. In local coordinates, a Poisson bracket is written as $$\left\{ \varphi ,\psi \right\} (x)=\sum_{i,j}\pi _{ij}(x)\frac{\partial
\varphi }{\partial x_{i}}\frac{\partial \psi }{\partial x_{j}},$$ where $\pi _{ij}$ is an antisymmetric tensor *(Poisson tensor)* satisfying a quadratic differential constraint which assures the Jacobi identity. When the manifold $M$ is symplectic, i.e., admits a nondegenerate closed 2-form $\omega =\sum \omega ^{ij}dx_{i}\wedge dx_{j}$, the associated Poisson tensor $\pi _{ij}=\left( \omega ^{ij}\right) ^{-1}$. Reciprocally, whenever the Poisson tensor is nondegenerate, its inverse is a symplectic form. In general, a Poisson manifold is not symplectic; the fundamental theorem which goes back to Lie asserts that it always admits a stratification whose strata are already symplectic manifolds (*the symplectic leaves* of our Poisson manifold). The geometrical meaning of this decomposition is very simple. Any $H\in C^{\infty }(M)$ defines on $M$ a Hamiltonian vector field which acts on $\varphi \in C^{\infty }(M)$ via $$X_{H}\varphi =\left\{ H,\varphi \right\} ;$$ for a given point $x\in M$ the tangent vectors $X_{H}\left( x\right) $ span a linear subspace in the tangent space $T_{x}M$; this is precisely the tangent space to the symplectic leave passing through $x$.[^2] By construction, Hamiltonian vector fields are tangent to symplectic leaves, and hence the Hamiltonian flows are preserving each leaf separately. A closely related property of general Poisson manifolds is the existence of *Casimir functions.* By definition, a function $H\in C^{\infty }(M)$ is called a Casimir function if it lies in the center of the Poisson bracket; equivalently, Casimir functions define trivial Hamiltonian equations on $M$. Restrictions of Casimir functions to symplectic leaves in $M$ are constants; reciprocally, the common level surfaces of Casimir functions define a stratification of $M$; typically, this stratification is more coarse than the stratification into symplectic leaves (i.e., symplectic leaves are not completely separated by the values of the Casimirs), but in many applications the knowledge of Casimir functions yields a sufficiently accurate description.
A very typical example of a Poisson manifold is the *dual space of a Lie algebra*. I shall briefly recall the corresponding construction, since it proved to be very important for the study of integrable systems. Let $\frak{g}$ be a Lie algebra, $\frak{g}^{*}$ its dual space, $P\left( \frak{g}^{*}\right) $ the space of polynomial functions on $\frak{g}^{*}$. By the Leibniz rule, a Poisson bracket on $P\left( \frak{g}^{*}\right) $ is completely determined by its value on the subspace of linear functions $\frak{g}\subset P\left( \frak{g}^{*}\right) $; for $X,Y\in \frak{g}$ let us set simply $$\left\{ X,Y\right\} \left( L\right) =\left\langle L,\left[ X,Y\right]
\right\rangle ,\;L\in \frak{g}^{*}. \label{lp}$$ The Jacobi identity for the bracket (\[lp\]) follows from that for the Lie bracket in $\frak{g}$; since $P\left( \frak{g}^{*}\right) $ is dense in $C^{\infty }\left( \frak{g}^{*}\right) $, it canonically extends to all smooth functions. Explicitly, we have $$\{\varphi _{1},\varphi _{2}\}\left( L\right) =\left\langle L,\left[ d\varphi
_{1}\left( L\right) ,d\varphi _{2}\left( L\right) \right] \right\rangle .
\label{lpg}$$ (Note that $d\varphi _{i}\left( L\right) \in ({\frak{g}}^{*})
^{*}\simeq \frak{g}$, and hence the Lie bracket is well defined). The bracket (\[lpg\]) is usually called the *Lie–Poisson bracket*. Its properties are closely related to the distinguished representation of the associated Lie group, the *coadjoint representation.* Let $G$ be a Lie group with Lie algebra $\frak{g};\;\exp :\frak{g} \rightarrow G$ the exponential map. The adjoint and coadjoint representations of $G$ acting in $\frak{g}$ and $\frak{g}^{*}$, respectively, are defined by $$\begin{aligned}
Ad\,
\,g\cdot X &=&\left( \frac{d}{dt}\right) _{t=0}g\cdot \exp tX\cdot
g^{-1},\;X\in \frak{g}, \\
\left\langle Ad\,^{*}g\cdot L,X\right\rangle &=&\left\langle L,Ad\,
\,g^{-1}\cdot X\right\rangle ,\;X\in \frak{g},\;L\in \frak{g}^{*}.\end{aligned}$$ Set $$ad\,X\cdot Y=\left( \frac{d}{dt}\right) _{t=0}Ad\,
\exp tX\cdot Y,\;ad^{*}\,X\cdot L=\left( \frac{d}{dt}\right)
_{t=0}Ad^{*}\exp tX\cdot L.$$ Clearly, one has $ad\,X\cdot Y=\left[ X,Y\right] , ad^{*}\,X=-\left(
ad\,X\right) ^{*}$. The following fundamental theorem again goes back to Lie; it was rediscovered by Kirillov and Kostant in 1960’s:
\(i) Symplectic leaves of the Lie-Poisson bracket coincide with $G$-orbits in $\frak{g}^{*}$ *(coadjoint orbits).* (ii) Casimir functions of the Lie-Poisson bracket are precisely the coadjoint invariant functions on $\frak{g}^{*}$.
It is very easy to verify a somewhat weaker property.
\[h\]Let $\varphi \in C^{\infty }(\frak{g}^{*})$ be an arbitrary function; the Hamiltonian equation of motion defined by $\varphi $ with respect to the Lie-Poisson bracket may be written in the following form: $$\frac{dL}{dt}=-ad^{*}\,d\varphi \left( L\right) \cdot L,\;L\in \frak{g}^{*};
\label{H}$$ in other words, the velocity vector, associated with any Hamiltonian equation on $\frak{g}^{*}$ is automatically tangent to the coadjoint orbit passing through $L$.
In the context of integrable systems coadjoint orbits are of particular importance: in many applications, the phase spaces of integrable systems *are* coadjoint orbits for some appropriate Lie group[^3]. Classification of coadjoint orbits for particular Lie groups is a good exercise (which may be quite involved depending on the nature of the Lie group); let us just quote a few examples which will be useful in the sequel.
Let $\frak{g}=\frak{gl}(n)$ be the full matrix algebra; its dual space $\frak{g}^{*}$ may be canonically identified with $\frak{g}$ by means of the invariant inner product[^4] $$\left\langle X,Y\right\rangle ={\rm tr}\,XY. \label{tr}$$ Thus the adjoint and the coadjoint representations of the corresponding Lie group $G=GL(n)$ are identical; we have $Ad\,
^{*}g\cdot L=gLg^{-1}$; the coadjoint orbits consist of conjugate (isospectral) matrices; their classification is given by the Jordan normal form. Casimir functions are spectral invariants of matrices; their level surfaces consist of a finite number of coadjoint orbits.
Let $\frak{b}_{+}\subset \frak{g}$ be the subalgebra of upper triangular matrices; the pairing (\[tr\]) allows to identify its dual with the space $\frak{b}_{-}$ of lower triangular matrices. The coadjoint representation of the corresponding Lie group $B_{+}$ of upper triangular invertible matrices is given by $$Ad\,
^{*}b\cdot F=P_{-}\left( b\cdot F\cdot b^{-1}\right) ,\;b\in B_{+},\;F\in
\frak{b}_{-},$$ where $P_{-}:\frak{g}\rightarrow \frak{b}_{-}$ is the projection operator which replaces by zeros all matrix coefficients above the principal diagonal.
In this example the adjoint and the coadjoint representations are *inequivalent.*
After this brief discussion of Poisson geometry and coadjoint orbits let us return to the study of integrable systems. The use of linear Poisson brackets and of coadjoint orbits seems a good guess to get a proper kinematical description of our future examples; this suggestion is further supported by the following simple observation which specializes proposition \[h\] above.
\[one\]Assume that $\frak{g}=\frak{gl}(n)$ is identified with its dual space and equipped with the Lie–Poisson bracket. For any $\varphi \in
C^{\infty }(\frak{g})$ the Hamiltonian equation of motion is written in the form $$\label{ham}
\frac{dL}{dt}=-\left[ d\varphi \left( L\right) ,L\right] ;$$ hence all Hamiltonian flows on $\frak{g}$ preserve spectral invariants of matrices.
Equation looks exciting: one is tempted to compare it with the famous Lax equations. A closer look on the picture reveals, however, that proposition leads to a deception. Indeed, spectral invariants of matrices are *Casimir functions* for the Lie-Poisson bracket; their conservation is a trivial fact which has nothing to do with integrability of equation . There is very little chance that this equation with arbitrary Hamiltonian $\varphi $ will be completely integrable; on the other hand, the spectral invariants themselves which seem to be natural candidates to provide integrable systems, generate *trivial* flows, in view of the following simple fact:
\[casimirs\]For any Lie algebra $\frak{g}$ a function $\varphi \in
C^{\infty }(\frak{g}^{*})$ is a Casimir function for the Lie–Poisson bracket on $\frak{g}^{*}$ if and only if $$ad^{*}\,d\varphi \left( L\right) \cdot L=0$$ for any $L\in \frak{g}^{*}$. (Note that $d\varphi \left( L\right) \in (\frak{g}^{*})^{*}\simeq \frak{g}$; when $\frak{g}$ and $\frak{g}^{*}$ are identified, this relation is reduced to $\left[ d\varphi \left( L\right)
,L\right] =0.)$
Despite this initial setback, the original idea to use Lie–Poisson brackets and coadjoint orbits can be saved. However, instead of the initial Lie-Poisson bracket which provides the set of spectral invariants but does not yield any nontrivial dynamics associated with them, we must find a *different* one. It’s at this point that the classical r-matrix is brought into play.
Let $\frak{g}$ be a Lie algebra We shall say that $r\in {\mathrm {End}}\,
\frak{g}$ is a *classical r-matrix* if the bracket
$$\left[ X,Y\right] _{r}=\frac{1}{2}\left( \left[ rX,Y\right] +\left[
X,rY\right] \right) \label{rbr}$$
is a Lie bracket, i.e. if it satisfies the Jacobi identity. The skew symmetry of (\[rbr\]) is obvious for any $r$. We denote the Lie algebra with the bracket (\[rbr\]) by $\frak{g}_{r}$ and say that $\left( \frak{g},\frak{g}_{r}\right) $ is a *double Lie algebra.*
If $\frak{g}$ is a double Lie algebra, there are *two* different Poisson brackets in the space $\frak{g}^{*}$, namely, the Lie-Poisson brackets of $\frak{g}$ and $\frak{g}_{r}$. The latter bracket will be referred to as the* r-bracket*, for short.
A class of double Lie algebras which is of great importance for applications is constructed as follows. Assume that there is a vector space decomposition of $\frak{g}$ into a direct sum of two Lie subalgebras, $\frak{g}=\frak{g}_{+}\dotplus \frak{g}_{-}$. Let $P_{\pm }$ be projection operators onto $\frak{g}_{\pm }$ parallel to the complementary subalgebra; set $$\label{r}
r=P_{+}-P_{-}.$$ In this case, bracket (\[rbr\]) is given by
$$\lbrack X,Y]_{r}=[X_{+},Y_{+}]-[X_{-,}Y_{-}], \label{stand}$$
where $X_{\pm }=P_{\pm }X,Y_{\pm }=P_{\pm }Y$. In other words, the bracket (\[rbr\]) is the difference of Lie brackets in $\frak{g}_{+}$ and $\frak{g}_{-}$. The Jacobi identity for $\frak{g}_{r}$ is obvious from (\[stand\]).
As discussed in Section 6, in typical applications the Lie algebra $\frak{g}$ is a *loop algebra*, i.e., an algebra of matrix-valued functions on the circle, and $\frak{g}_{\pm}$ its subalgebra consisting of functions which are analytic inside (resp., outside) the circle. In that case, the classical r-matrix (\[rbr\]) is precisely the Hilbert transform. Of course, general classical r-matrices need not have this simple form (although (\[rbr\]) is by far the most important example of all). We shall discuss the general theory of r-matrices a little later; let us first state the key theorem which motivates the definition.
Let $I(\frak{g}^{*})$ be the ring of Casimir functions on $\frak{g}^{*}$ (with respect to the original Lie-Poisson bracket); equivalently, $I(\frak{g}^{*})\subset C^{\infty }(\frak{g}^{*})$ is the set of coadjoint invariants.
\[AKS\](i) Functions in $I(\frak{g}^{*})$ are in involution with respect to the r-bracket on $\frak{g}^{*}$. (ii) The equations of motion induced by $h\in I(\frak{g}^{*})$ with respect to the r-bracket have the form
$$\frac{dL}{dt}=-ad_{\frak{g}}^{*}M\cdot L,\ M=r(dh(L))\ \label{lax}$$
If $\frak{g}$ admits a nondegenerate invariant bilinear form, so that $ad_{\frak{g}}^{*}\simeq ad_{\frak{g}}$, equations (2.4) have the *Lax form* $$\frac{dL}{dt}=\left[ L,M\right] .$$
\(i) Let $h_{1},h_{2}\in I(\frak{g}^{*})$; set $dh_{i}\left(
L\right) =X_{i}$ for brevity. By definition, $$\arr{2.0}{l} {
\left\{ h_{1},h_{2}\right\} _{r}\left( L\right) \ds =\left\langle L,\left[
X_{1},X_{2}\right] _{r}\right\rangle \\
\ds =\frac{1}{2}\left\langle L,\left[ rX_{1},X_{2}\right] +\left[
X_{1},rX_{2}\right] \right\rangle \\
\ds =\frac{1}{2}\left\langle ad_{\frak{g}}^{*}X_{2}\cdot L,rX_{1}\right\rangle
-\frac{1}{2}\left\langle ad_{\frak{g}}^{*}X_{1}\cdot L,rX_{2}\right\rangle
=0,}$$ since, by proposition \[casimirs\], $ad_{\frak{g}}^{*}X_{2}\cdot L=ad_{\frak{g}}^{*}X_{1}\cdot L=0$. (ii) We have $$\frac{dL}{dt}=-ad_{\frak{g}_{r}}^{*}dh\left( L\right) \cdot L;$$ (\[rbr\]) implies that $$ad_{\frak{g}_{r}}^{*}X\cdot L=\frac{1}{2}\left( ad_{\frak{g}}^{*}rX\cdot
L+r^{*}\left( ad_{\frak{g}}^{*}X\cdot L\right) \right) ; \label{co}$$ since $h\in I(\frak{g}^{*})$, the second term in (\[co\]) vanishes.
The matrix $M$ in is not uniquely defined: one can always add to it something which commutes with $L$. Here is a useful option: set $$M_{\pm }=\pm \frac{1}{2}\left( r\pm Id\right) \left( dh\left( L\right)
\right) ; \label{m}$$ equation holds with *any* of these two operators. Below, we shall see that this form of M-operator appears naturally from the global formula for the solutions.
Theorem has a transparent geometrical meaning: it shows that the trajectories of the dynamical systems with Hamiltonians $h\in I(\frak{g}^{*})$ lie in the intersection of *two families of orbits* in $\frak{g}^{*}$, the coadjoint orbits of $\frak{g}$ and $\frak{g}_{r}$. Indeed, the coadjoint orbits of $\frak{g}_{r}$ are preserved by all Hamiltonian flows in $\frak{g}_{r}$. On the other hand, because of (2.4), the flow is always tangent to the $\frak{g}$-orbits in $\frak{g}^{*}$.
In many applications the intersections of orbits are precisely the “Liouville tori” for our dynamical systems.
The scheme outlined so far incorporates only two of the three main features of the inverse scattering method: the Poisson brackets and the Lax form of the equations of motion. As it happens, the most important feature of this method, the reduction of the equations of motion to the Riemann problem, is already implicit in our scheme. An abstract version of the Riemann problem is provided by the *factorization problem* in Lie groups.
We shall state a factorization theorem, which is the global version of Theorem \[AKS\], for the simplest r-matrices of the form (\[r\]). Let $G$ be a connected Lie group with Lie algebra $\frak{g}$, and let $G_{\pm }$ be its subgroups corresponding to $\frak{g}_{\pm }$.
\[fact\]Let $h\in I\left( \frak{g}^{*}\right) ,X=dh\left( L\right) $. Let $g_{\pm }(t)$ be the smooth curves in $G_{\pm }$ which solve the factorization problem $$\exp tX=g_{+}(t)\cdot g_{-}(t)^{-1},\;g_{+}(0)=e. \label{exp}$$ Then the integral curve $L(t)$ of equation (2.4) with $L(0)=L$, is given by any of the two formulae, $$L(t)=Ad\,
_{G}^{*}\,g_{+}(t)^{-1}\cdot L=Ad\,
_{G}^{*}\,g_{-}(t)^{-1}\cdot L. \label{two}$$
Differentiating with respect to $t$ we get $$\frac{dL}{dt}=-ad^{*}\left(g_{+}^{-1}\dot{g}_{+}\right) \cdot
L=-ad^{*}\left( g_{-}^{-1}\dot{g}_{-}\right) \cdot L.$$ We shall check that $g_{\pm }^{-1}\dot{g}_{\pm }=M_{\pm }$, where $M_{\pm }$ are the M-operators from (\[m\]). Due to our special choice of $r$ we have $$M_{\pm }\left( t\right) =\pm P_{\pm }X\left( t\right) ,$$ where $X\left( t\right) =dh\left( L\left( t\right) \right) $. The $Ad^{*}G$-invariance of $h$ implies that $$X(t)=Ad\,
_{G}g_{\pm }(t)^{-1}\cdot X.$$
Check this formula in matrix case (“gradients of invariant functions are covariant”).
Writing in the form $g_{+}(t)\exp tX=g_{-}(t)$ and differentiating with respect to $t$, we get $$g_{+}^{-1}\dot{g}_{+}+Ad\,
_{G}g_{-}(t)^{-1}\cdot X=\dot{g}_{-}g_{-}^{-1}$$ Since $g_{\pm }^{-1}\dot{g}_{\pm }\in \frak{g}_{\pm }$, this implies $g_{\pm
}^{-1}\dot{g}_{\pm }=\pm P_{\pm }X\left( t\right) $, as desired.
Note that the two possible choices of sign in are equivalent precisely because $X(t)$ belongs to the centralizer of $L(t)$, i.e., $ad^{*}X(t)\cdot L(t)=0$ (in fact, this is the characteristic property of Casimir functions).
By the implicit function theorem, the factorization exists for $t $ sufficiently small; note that in our proof we need *not* assume that this factorization exists globally for all $t$. Geometrically, this means that the solution of the Lax equation exists as long as the curve $\exp tX$ remains in the “big cell” $G_{+}\times G_{-}\subset G$; in other words, the flow associated with the Lax equation is not necessarily complete. One can show in typical examples that the curve intersects “complementary” cells of positive codimension transversally and returns back to the big cell; for the exceptional values of $t$ the solution “escapes to infinity”, i.e., displays a pole in $t$.
A more geometric proof of Theorem is based on *Hamiltonian reduction*. Recall that the Hamiltonian reduction applies to Hamiltonian dynamical systems with high degree of symmetry; it allows to exclude certain redundant degrees of freedom. Classically, the use of reduction is to simplify multidimensional systems getting quotient, or reduced, systems of lower dimension. However, as pointed out in [@KKS], one can reverse this reasoning and use Hamiltonian reduction in the opposite direction, starting with a simple multidimensional system with high symmetry (“free dynamics”) and getting a complicated reduced system as an output. In order to apply this idea, one has to answer the following questions:
1. Find a “big” phase space and suggest the “free dynamics” which will yield Lax equations as the quotient system.
2. Make sure there is an expected high degree of symmetry for the free system.
3. Perform the reduction.
Although the proof based on this approach is much longer than the elementary computation presented above, it is more transparent and explains the origin of the result. A simple candidate for the big phase space is the *cotangent bundle* $T^{*}G$ equipped with the canonical symplectic structure. Let us first of all describe the “free dynamics” on $T^{*}G$. The group $G$ acts on itself by left and right translations; these actions naturally lift to $T^{*}G$; both actions are Hamiltonian with respect to the canonical symplectic structure. Let us identify $T^{*}G$ with $G\times \frak{g}^{*}$ by means of *left translations*.
In left trivialization the action of $G$ on $T^{*}G\simeq G\times \frak{g}^{*}$ by left (right) translations is given by $$\begin{aligned}
\lambda \left( g\right) &:&\left( x,L\right) \longmapsto \left( gx,L\right) ,
\label{act} \\
\rho \left( g\right) &:&\left( x,L\right) \longmapsto \left(
xg^{-1},Ad^{*}g\cdot L\right) . \nonumber\end{aligned}$$
(In *left* trivialization the action of $G$ by *right* translations induces a nontrivial action in the fiber $\frak{g}^{*}$; it is easy to check that it is precisely the coadjoint action.) Left-invariant functions on $T^{*}G$ are identified with functions on $\frak{g}^{*}$. Since the canonical Poisson bracket of left-invariant functions is also left-invariant, this induces a Poisson structure on $\frak{g}^{*}$; it is easy to check that it coincides with the *Lie-Poisson bracket*. Casimir functions on $\frak{g}^{*}$ canonically lift to smooth functions on $T^{*}G$ which are $G$-*biinvariant*. For $h\in I(\frak{g}^{*})$ let us denote by $\hat{h}\in C^{\infty }\left( T^{*}G\right) $ the corresponding biinvariant Hamiltonian on $T^{*}G$.
The Hamiltonian flow on $T^{*}G$ defined by $\hat{h}$ is given (in left trivialization) by $$F_{t}:\left( x,L\right) \longmapsto \left( x\cdot \exp tdh(L),L\right)
,\;x\in G,\;L\in \frak{g}^{*}. \label{F}$$ In other words, integral curves of $\hat{h}$ project to left translates of one-parameter subgroups in $G$; the choice of $h$ determines the (constant) velocity vector $dh(L)$ which depends only on the initial data.
Since the “free Hamiltonian” $\hat{h}$ is $G$-biinvariant, the flow $F_{t}$ admits reduction with respect to *any* subgroup $U\subset
G\times G$. There is, at this stage, a very large freedom in the choice of such a subgroup which all lead to different but meaningful quotient systems. The particular choice which is imposed by the choice the r-matrix (\[r\]) is $U=G_{+}\times G_{-}$. By (\[act\]), with our choice of trivialization, the action of $G_{+}\times G_{-}$ on $T^{*}G\simeq G\times \frak{g}^{*}$ is given by $$\left( g_{+},g_{-}\right) :\left( x,L\right) \longmapsto \left(
g_{+}xg_{-}^{-1},Ad\,
_{G}^{*}g_{-}\cdot L\right) . \label{a}$$
We now turn to the reduction procedure. In textbooks the reduction is usually described in a rather complicated way (which involves the moment map and a good deal of symplectic geometry) (see, e.g., [@arn]). Here is an elementary substitute. Let us ask what is bad about the naive suggestion: consider a group action $G\times M\rightarrow M$ on a symplectic manifold and take the projection $\pi :M\rightarrow M/G$ onto the quotient space? The answer is: the quotient space is no longer symplectic, since symplectic forms transform by pullback, and there is no natural symplectic form on $M/G$ (it in not even in general even dimensional!) But on the other hand, the quotient space *does* carry a Poisson bracket (which transforms by push-forward!). The difficult part of reduction consists in the description of the particular *symplectic leaves* of this quotient Poisson bracket; it’s at this stage that one needs the moment map and all other machinery. If we do not want a too detailed description, or if we manage to guess the symplectic leaves in some other way, everything becomes simple! In the present case, we can display a map $\pi $ which is constant on the orbits of $G_{+}\times G_{-}$ in $M$ and hence its image yields a *model* of the quotient space; the Poisson structure on this quotient is easy to recognize.
For $x\in G$ we denote by $x_{\pm }$ the solution of the factorization problem $$x=x_{+}\cdot x_{-}^{-1},\;x_{+}\in G_{+},\;x_{-}\in G_{-}. \label{f}$$
\(i) The map $\pi :T^{*}G\longrightarrow \frak{g}^{*}:\left( x,L\right)
\longmapsto Ad\,
_{G}^{*}x_{-}^{-1}\cdot L$ is constant on the orbits of $G_{+}\times G_{-}$ in $T^{*}G$. (ii) If $G$ is globally diffeomorphic to $G_{+}\times G_{-}$, i.e., the factorization problem (\[f\]) is always solvable, $\pi $ is a global cross-section of this action. (iii) The induced Poisson structure on $T^{*}G/G_{+}\times G_{-}\simeq \frak{g}^{*}$ coincides with the Lie-Poisson bracket for $\frak{g}_{r}\simeq \frak{g}_{+}\oplus \frak{g}_{-}$.
The check of (i) and (ii) is immediate; the proof of (iii) requires a little knowledge of symplectic geometry (or else a three-line computation); we shall not present it here (see [@RS]).
\[q\]The flow (\[F\]) *factorizes over* ${\frak{g}}\,_{r}^{*}$; in other words, there exists a natural flow $\bar{F}_{t}:\frak{g}\,_{r}^{*}{\longrightarrow \frak{g}}\,_{r}^{*}$ (called the *quotient* flow) which makes the following diagram
(4818,1904)(1501,-3635) (4600,-1900)[(0,0)\[lb\][$T^{*}G$]{}]{} (4600,-3000)[(0,0)\[lb\][${\frak{g}}\,_{r}^{*}$]{}]{} (2200,-1900)[(0,0)\[lb\][$T^{*}G$]{}]{} (2200,-3000)[(0,0)\[lb\][${\frak{g}}\,_{r}^{*}$]{}]{} (3400,-2900)[(0,0)\[lb\][${\bar{F_t}}$]{}]{} (2400,-2400)[(0,0)\[lb\][${\pi}$]{}]{} (4800,-2400)[(0,0)\[lb\][${\pi}$]{}]{} (3400,-2050)[(0,0)\[lb\][${F_t}$]{}]{} (2600,-1850)[(1,0)[1900]{}]{} (2600,-3000)[(1,0)[1900]{}]{} (4700,-2100)[(0,-1)[700]{}]{} (2300,-2100)[(0,-1)[700]{}]{}
commutative; the quotient flow $\bar{F}_{t}:\frak{g}\,_{r}^{*}{\longrightarrow \frak{g}}\,_{r}^{*}$ is given by $$\bar{F}_{t}:L\longmapsto
Ad\,
\,_{G}^{*}\,g_{-}\left( t\right) ^{-1}\cdot L, \label{minus}$$ where, as in (\[exp\]), $g_{+}\left( t\right) g_{-}\left( t\right)
^{-1}=\exp t\,dh(L)$.
The flow $\bar{F}_{t}$ is precisely the result of the reduction procedure. Note that (\[minus\]) involves only $g_{-}$; this is due to our choice of trivialization of $T^{*}G$; trivialization by *right* translations yields the equivalent formula for the quotient flow $\bar{F}_{t}:L\longmapsto
Ad_{G}^{*}g_{+}\left( t\right) ^{-1}\cdot L$.
In general, of course, $G$ need not be diffeomorphic to $G_{+}\times G_{-}$; still, $\frak{g}^{*}$ provides a model for a “big cell” in the quotient space $T^{*}G/G_{+}\times G_{-}$; one can show that under very mild restrictions the action (\[a\]) is proper and hence the quotient space is a well-defined manifold; the quotient flow induced on this manifold is also well defined and may be regarded as the natural completion of the incomplete flow associated with Lax equations.
The choice of the subgroup $G_{+}\times G_{-}\subset G\times G$ as the symmetry group for the “free system” may seem arbitrary; indeed, there are many other possible choices leading to meaningful examples. (Among the dynamical systems which may be obtained in this way, there are, e.g., the *Calogero-Moser systems*, cf. [@KKS].) The key property which characterizes our special situation is the simple description of the quotient space: as we see from Lemma \[q\], in this case the quotient space is simply the dual of a Lie algebra with its Lie-Poisson bracket, and its symplectic leaves (which are the* symplectic quotients* of $T^{*}G) $ are the coadjoint orbits of $G_{r}$; for other choices of the symmetry group the description of the quotient space will be more complicated and it may no longer be a homogeneous space.
We have already mentioned that the most interesting r-matrices are associated with decompositions of the Lie algebra into complementary Lie subalgebras. However, it is worth examining the general conditions on $r$ which follow from the Jacobi identity for the r-bracket. These conditions are known as the *classical Yang-Baxter equations*; they were first derived as a semi-classical approximation to the *quantum Yang-Baxter equations* which arise in the study of quantum completely integrable systems.
The restrictions on $r$ which follow from the Jacobi identity are quite easy to establish. For $r\in
End\,\frak{g}$ set $$B_{r}\left( X,Y\right) =[rX,rY]-r([rX,Y]+[X,rY]). \label{Obstr}$$
The r-bracket satisfies the Jacobi identity if and only if, for any $X,Y,Z\in \frak{g}$, $$\lbrack B_{r}(X,Y),Z]+[B_{r}(Y,Z),X]+[B_{r}(Z,X),Y]=0. \label{B}$$
The proof is straightforward: just substitute into the Jacobi identity and regroup the terms. The necessary and sufficient condition is usually replaced by sufficient conditions which are *bilinear* rather than trilinear. The simplest sufficient condition is the so-called *classical Yang-Baxter equation* (CYBE) $$B_{r}(X,Y)=0. \label{cybe}$$ Another important sufficient condition is the *modified classical Yang-Baxter equation* (mCYBE) $$B_{r}\left( X,Y\right) =-c\,[X,Y],\;c=const. \label{mcybe}$$ By rescaling, we may always assume that $c=1$. Note that the r-matrices satisfy mCYBE with $c=1$.
The reason for the study of classical r-matrices satisfying the modified classical Yang-Baxter identity is that although they do not in general have the simple form , one can still associate with them a factorization problem. By contrast, the ordinary classical Yang-Baxter identity represents a degenerate case and does not lead to a factorization problem.
Let us briefly describe the corresponding construction. Given an r-matrix which satisfies mCYBE, set $$r_{\pm }=\frac{1}{2}\left( r\pm Id\right) . \label{pm}$$
For each $X,Y\in \frak{g}$ $$\lbrack r_{\pm }X,R_{\pm }Y] = r_{\pm }([X,Y]_{r}),$$ i.e., $r_{\pm }:\frak{g}_{R}\rightarrow \frak{g}$ are Lie algebra homomorphisms.
Set $\frak{g}_{\pm }=\mathrm{Im}\,r_{\pm }$. Clearly, $\frak{g}_{\pm }$ is a Lie subalgebra of $\frak{g}$. If $r$ has the form (\[r\]), then $r_{\pm }=\pm P_{\pm }$ and the subalgebras $\frak{g}_{\pm }=P_{\pm }(\frak{g})$ are complementary. In general case this is no longer true. However, this difficulty may be resolved by passing to the *double* of $\frak{g}$. By definition, $$\frak{d}=\frak{g}\oplus \frak{g}$$ is the direct sum of two copies of $\frak{g}$.
The mapping $i_{r}:\frak{g}_{r}\rightarrow \frak{g}\oplus \frak{g}:X\longmapsto
(r_{+}X,r_{-}X)$ is a Lie algebra embedding, and each $Y\in \frak{g}$ has a unique decomposition, $Y=Y_{+}-Y_{-}$, where $(Y_+,Y_{-})\in
\mathrm{Im}\, i_{r}$.
The last assertion follows immediately from the obvious identity $r_{+}-r_{-}=Id$.
Now, let $G,G_{r}$ be (local) Lie groups which correspond to $\frak{g}$, $\frak{g}_{r}$. The homomorphisms $r_{\pm }$ give rise to the Lie group homomorphisms which we denote by the same letters. Put $G_{\pm }=r_{\pm}(G_{R})$. The composition of maps $$i_{r}:G_{r}\longrightarrow G\times G:x\longmapsto (r_{+}x,r_{-}x)$$ is a Lie group embedding. Consider the map $$m:G\times G\longrightarrow G:(u,v)\longmapsto uv^{-1}.$$ Then $f=m\circ i_{r} : G_{r}\longrightarrow G$ is a local homeomorphism, and therefore an arbitrary element $y\in G$ which is sufficiently close to unity admits a unique representation $y=y_{+}y_{-}^{-1}
$ with $\left( y_{+},y_{-}\right) \in \mathrm{Im}\,i_{r}$.
The proof of Theorem \[fact\] extends to the present setting with only minor changes.
It is probably worth giving examples of r-matrices satisfying which are *not* of the form . Let $\frak{g}=\frak{gl}(n)$ be the matrix algebra; let us consider its decomposition $\frak{g}=\frak{n}_{+}\dot{+}\frak{h}\dot{+}\frak{n}_{-}\ $ into direct sum of upper triangular, diagonal, and lower triangular matrices. Let $P_{\pm },P_{0}$ be the corresponding projection operators. Let $r_{0}$ be the *partially defined* linear operator on $\frak{g}$ with domain $\mathrm{Dom}(r_{0})=\frak{n}_{+}\dot{+}\frak{n}_{-}$ given by $$r_{0} X = \left\{ \arr{1.0}{l}{
{\phantom -} X, \, \hbox{if } X \in \frak{n}_{+}, \\
-X , \, \hbox{if } X \in \frak{n}_{-}. }
\right.$$ We want to extend $r_{0}$ to the entire linear space $\frak{g} $ in such a way that it satisfies the modified classical Yang-Baxter identity. Let us first drop the latter condition and consider *all* linear operators $r\supset r_{0},\;\mathrm{Dom}(r)=\frak{g}$. Let us set again $r_{\pm }=\frac{1}{2}\left( r\pm Id\right) ,i_{r}=r_{+}\oplus r_{-}$ and consider the subspace $\frak{g}_{r}=\mathrm{Im}\,i_{r}$. It is easy to see that this subspace is transversal to the diagonal subalgebra $\frak{g}_{d}=\{\left( X,X\right) \in \frak{g}\oplus \frak{g}\}$; conversely, all extensions $r \supset r_{0}$ are in bijective correspondence with linear subspaces $\frak{g}_{r}\subset \frak{g}\oplus
\frak{g}$ which contain $\frak{n}_{+}\oplus \frak{n}_{-}$ and are transversal to the diagonal. All such subspaces may be parametrized in the following way: $$\frak{g}_{r}=\left\{ \left( X_{+}+X_{0},X_{-}+\theta X_{0}\right) ;X\in
\frak{g}\right\} ,$$ where $X_{\pm }=\pm P_{\pm }X,X_{0}=P_{0}X$ and $\theta \in \mathrm{End}\,\frak{h}$ is a linear operator; equivalently, the extensions $r\supset r_{0}$ are described by the von Neumann formulae, $$\ \left\{ \arr{1.1}{l}{
{\phantom z} _{\phantom z} X = X_{+}-X_{-}+\left( Id-\theta \right) X_{0}, \\
r_{\theta }X =X_{+}+X_{-}+\left( Id+\theta \right) X_{0}.}
\right.$$ The transversality condition is equivalent to the non-degeneracy of $Id-\theta $. Since $\frak{h}\subset \frak{g}$ is an abelian subalgebra which normalizes both $\frak{n}_{+}$ and $\frak{n}_{+}$, it is easy to see that $\frak{g}_{r}\subset \frak{g}\oplus \frak{g}$ is a Lie subalgebra (and not merely a linear subspace) for any $\theta $, and hence all $r_{\theta }\supset r_{0}$ satisfy mCYBE. In more general examples, when the relevant subalgebra is not abelian, there are additional algebraic constraints on $\theta $ which make the construction partially rigid.
*The moral of the story:* One can construct new classical r-matrices from the standard ones of the form by assuming that the simple formula applies not globally, but only on some subspace of positive codimension and then using the extension theory of linear operators. The same trick works for loop algebras; in that case, the relevant codimensions are finite. Let us set, for example, $\frak{G}=L\frak{g},\;\frak{g}=\frak{gl}\left( n\right) $; by definition, $\frak{G}$ is the algebra of Laurent polynomials with matrix coefficients. Set $\frak{N}_{+}=\{X(z)=X_{0}+X_{1}z+X_{2}z^{2}+...;X_{0}\in \frak{n}_{+}\},\;\frak{N}_{-}=\{X(\lambda )=X_{0}+X_{1}z^{-1}+X_{2}z^{-2}+...;X_{0}\in \frak{n}_{-}\}$. Then $\frak{G}=\frak{N}_{+}\dot{+}\frak{h}\dot{+}\frak{N}_{-}$, where $\frak{h}$ is the subalgebra of constant diagonal matrices. Hence we may apply the previous construction without any modification. A classification theorem, due to Belavin and Drinfeld, assures that all r-matrices on semisimple Lie algebras and their loop algebras satisfying mCYBE and certain additional conditions arise in a similar way; of course, the most difficult part of the construction is the case when $\frak{h}$ is replaced by a non-abelian subalgebra. (Cf. [@BD], [@What].)
Let again $\frak{g}=\frak{gl}(n)$ be the full matrix algebra. There are several natural decompositions of $\frak{g}$ into direct sum of complementary subalgebras, e.g., $$\frak{g}=\frak{b}_{+}+\frak{n}_{-}, \label{g}$$ where $\frak{n}_{-}$ is the subalgebra of lower triangular nilpotent matrices and $\frak{b}_{+}=\frak{h}+\frak{n}_{+}$ the complementary subalgebra of upper triangular matrices. Another decomposition is $$\frak{g}=\frak{b}_{+}+\frak{k}, \label{i}$$ where $\frak{k}=so(n)$ is the Lie algebra of skew symmetric matrices. We can associate two standard classical r-matrices with these decompositions: $$\begin{aligned}
r_{\mathrm{Gauss}} &=&P_{\frak{b}_{+}}-P_{\frak{n}_{-}}, \\
r_{\mathrm{Iwasawa}} &=&\tilde{P}_{\frak{b}_{+}}-P_{\frak{k}},\end{aligned}$$ where $P_{\frak{b}_{+}},P_{\frak{n}_{-}},\tilde{P}_{\frak{b}_{+}},P_{\frak{k}}$ are the respective projection operators (mind that of course $P_{\frak{b}_{+}}\neq \tilde{P}_{\frak{b}_{+}})$. The Lie groups $G_{r_{\mathrm{Gauss}}}$ and $G_{r_{\mathrm{Iwasawa}}}$ are isomorphic to $B_{+}\times N_{-}$ and $B_{+}\times K$, respectively. The associated factorization problems in the general linear group $G=GL(n)$ are the Gauss decomposition of matrices, $$g=bn^{-1},b\in B_{+},n\in N_{-}, \label{G}$$ in the former case, and the Iwasawa decomposition $$g=bk^{-1},b\in B_{+},k\in K, \label{I}$$ in the latter one. (Here $N_{-}\subset GL(n)$ denotes the subgroup of lower triangular unipotent matrices, $B_{+}\subset GL(n)$ the subgroup of upper triangular matrices, and $K=SO(n)\subset GL(n)$ the subgroup of orthogonal matrices.) Note that in the Iwasawa case the product map $B_{+}\times
K\rightarrow G:\left( b,k\right) \mapsto bk^{-1}$ is a bijection onto $G$ and hence the factorization problem is always solvable. For the Gauss decomposition the image of $B_{+}\times N_{-}\rightarrow G:(b,n)\longmapsto
bn^{-1}$ is an open dense subset in $G$, and hence the factorization problem is solvable for almost all (though not for all) initial data.
The dual space $\frak{g}^{*}$ is canonically identified with $\frak{g}$ by means of the invariant inner product $$\left\langle X,Y\right\rangle ={\rm tr}\, XY;$$ the decompositions , give rise to the biorthogonal decompositions $$\frak{g}^{*}=\frak{b}_{+}^{\bot }+\frak{n}_{-}^{\bot }=\frak{b}_{+}^{\bot }+\frak{k}^{\bot },$$ which provide the models for dual spaces of the subalgebras, $$\frak{b}_{+}^{*}\simeq \frak{n}_{-}^{\bot }=\frak{b}_{-},\frak{n}_{-}^{*}\simeq \frak{b}_{+}^{\bot }=\frak{n}_{+}$$ in the case of the Gauss decomposition and $$\frak{b}_{+}^{*}\simeq \frak{k}^{\bot }=\frak{p},\frak{k}^{*}\simeq \frak{b}_{+}^{\bot }=\frak{n}_{+}$$ in the case of the Iwasawa decomposition (here $\frak{p}\subset Mat(n)$ denotes the subspace of symmetric matrices).[^5] Let us focus on coadjoint orbits of $B_{+}$. Denote by $\frak{d}_{p}\subset Mat(n),p\in \Bbb{Z}$, the set of all matrices supported on p-th diagonal, $$\frak{d}_{p}=\left\{ X\in Mat(n);X_{ij}=0\;\hbox{for }j-i\neq p\right\}$$
\(i) Let us model the dual space $\frak{b}_{+}^{*}$ on $\frak{b}_{-}$; for all $q\geq 0$ the subspaces $\frak{b}_{-}^{q}=\oplus _{p=0}^{q}\frak{d}_{-p}\subset \frak{b}_{-}$ are invariant with respect to the coadjoint action of $B_{+}$. (ii) Similarly, if $\frak{b}_{+}^{*}$ is modelled on $\frak{p}$, the subspaces $\frak{p}^{q}=\oplus _{p=0}^{q}\left( \frak{d}_{p}+\frak{d}_{-p}\right) \cap \frak{p}$ are also invariant.
Orbits in the subspace $\frak{b}_{-}^{0}$ are all trivial (i.e., each point in this subspace is stable with respect to the coadjoint action and hence is a separate orbit); the subspace $\frak{b}_{-}^{1}$ contains orbits of maximal dimension $2n-2$ (over $\C$ there is just one such orbit, over the reals there is a finite number of them); the typical example is the orbit ${\mathcal{O}}_{f}$ which contains the matrix $f\in \frak{d}_{-1}\subset \frak{b}_{-}$, $$f=\left(
\begin{array}{llll}
0 & 0 & \cdots & 0 \\
1 & \ddots & \ddots & \vdots \\
\vdots & \ddots & 0 & 0 \\
0 & \cdots & 1 & 0
\end{array}
\right) ;$$
\(i) ${\mathcal{O}}_{f}$ consists of all matrices of the form $$l=\left(
\begin{array}{llll}
p_{1} & 0 & \cdots & 0 \\
b_{1} & \ddots & \ddots & \vdots \\
\vdots & \ddots & p_{n-1} & 0 \\
0 & \ddots & b_{n-1} & p_{n}
\end{array}
\right) ,b_{i}\neq 0,\sum p_{i}=0.$$ (Over the reals, in addition, $\mathrm{sign}\,b_{i}=+1.\footnote{It is easy to see that in the real case the signs of all matrix coefficients
below the principal diagonal are preserved by the coadjoint action; hence there are
exactly $2^{n-1}$ open orbits in our subspace.})$ (ii) Lie-Poisson brackets of coordinate functions $p_{i},b_{j}$ on ${\mathcal O}_{f}$ are given by $$\left\{ p_{i},p_{j}\right\} =\left\{ b_{i},b_{j}\right\} =0,\left\{
p_{i},b_{i}\right\} =-\left\{ p_{i+1},b_{i}\right\} =b_{i}.$$ (iii) If we set $b_{i}=\exp \left( q_{i}-q_{i+1}\right) $, the coordinates $q_{i}$ have canonical Poisson brackets with momenta, $\left\{
p_{i},q_{j}\right\} =\delta _{ij}$.
Thus, as a symplectic manifold, ${\mathcal{O}}_{f}$ is isomorphic to the standard phase space $\R ^{2n-2}$. In the Iwasawa model, the same orbit is realized by symmetric matrices, $$L=\left(
\begin{array}{llll}
p_{1} & b_{1} & \cdots & 0 \\
b_{1} & \ddots & \ddots & \vdots \\
\vdots & \ddots & p_{n-1} & b_{n-1} \\
0 & \ddots & b_{n-1} & p_{n}
\end{array}
\right) . \label{toda}$$ Recall that our main theorem associates dynamical systems to coadjoint orbits in $\frak{g}_{r}^{*}$; since $\frak{g}_{r_{\mathrm{Gauss}}}$ and $\frak{g}_{r_{\mathrm{Iwasawa}}}\ $are direct sums, $$\frak{g}_{r_{\mathrm{Gauss}}}\simeq \frak{b}_{+.}\oplus \frak{n}_{-},\;\frak{g}_{r_{\mathrm{Iwasawa}}}\simeq \frak{b}_{+.}\oplus \frak{k},$$ respectively. Coadjoint orbits of this bigger algebra are Cartesian products of the coadjoint orbits of the factors. In the Iwasawa case the simplest meaningful choice is to take the *zero orbit* of $K$ in $\frak{k}^{*}\simeq \frak{n}_{+}$; with this choice, becomes the Lax matrix for the associated dynamical systems. The Hamiltonians are spectral invariants of $L$; taking, for instance, $$H=\frac{1}{2}{\rm tr}\,
L^{2}$$ and expressing it in terms of the canonical variables $p_{i},q_{i}$, we get $$H=\frac{1}{2}\sum_{i}p_{i}^{2}+\sum_{i}\exp 2\left( q_{i}-q_{i+1}\right) ,$$ i.e., the Hamiltonian of the open Toda lattice. In the Gauss case, it is also possible to take the zero orbit of the complementary subalgebra $\frak{n}_{-}$; with this choice the Lax matrix $l$ will be lower triangular and hence its spectral invariants will depend only on momenta $p_{i}$ and the corresponding Hamiltonians will be trivial. Luckily, in this case there is another option: we may take any one-point orbit $\{e\}$ of $N_{-}$ in $\frak{n}_{-}^{*}\simeq \frak{n}_{+}$; the constant matrix $e$ is simply added to the Lax matrix. This procedure does not add any new degrees of freedom to our system, but it modifies the embedding of the “little orbit” ${\mathcal{O}}_{f}$ into the big algebra and hence the spectral invariants of the Lax matrix. Specifically, set $$e=\left(
\begin{array}{llll}
0 & 1 & \cdots & 0 \\
0 & \ddots & \ddots & \vdots \\
\vdots & \ddots & 0 & 1 \\
0 & \cdots & 0 & 0
\end{array}
\right) ,\; L_{\mathrm{Gauss}}=l+e.$$
The reader familiar with semisimple Lie algebras will notice that the Jordan matrices $e,f$ are *principal nilpotent elements *of the general linear algebra, which suggests the way to generalize the above construction to other algebras.
\(i) $\left\{ e\right\} \subset \frak{d}_{1}\subset \frak{n}_{+}$ is a one-point coadjoint orbit of $N_{-}$. (ii) In canonical coordinates, the Hamiltonian $h=\frac{1}{2}{\rm tr}\,
L_{\mathrm{Gauss}}^{2}$ is given by $$h=\frac{1}{2}\sum_{i}p_{i}^{2}+\sum_{i}\exp \left( q_{i}-q_{i+1}\right) .$$
Note that the Hamiltonians $H$ and $h$ (which are defined on the same manifold ${\mathcal{O}}_{f}$) are *different* (although, in this particular case, they happen to be related by a simple canonical change of variables); so are the associated factorization problems which solve the Toda equations.
Other spectral invariants, e.g., $h_{p}={\rm tr}\,L_{\mathrm{Gauss}}^{p},\,p=2,3,...$, form a system of integrals of motion in involution. Obviously, the number of independent integrals does not exceed $n-1$ (mind that $h_{1}$ reduces to a constant on ${\mathcal{O}}_{f}$).
\(i) Write down explicitly the two factorization problems. (ii) Find the explicit relation between them. (iii) Show that over the reals the group element $\exp tL_{\mathrm{Gauss}}(p,q)$ lies in the “big cell” for all $\left( p,q\right) \in \Bbb{R}^{2n-2}$ and for all $t\in \Bbb{R}$, and hence the Gauss factorization is always possible. (iv) Check that the integrals $h_{2},...,h_{n}$ remain functionally independent after restriction to ${\mathcal{O}}_{f}$.
In both cases, the entries of the factors are *rational functions* of $\exp t$ (with coefficients depending on the initial data, i.e., on $p,q$); thus the functional dependence of the solutions on the time variable is fairly simple. It is easy to see that this simple behaviour is characteristic for all Lax equations associated with factorization problems in finite-dimensional Lie groups.
The subspaces $\frak{d}_{p}\subset \frak{g}$ satisfy $\left[ \frak{d}_{p},\frak{d}_{q}\right] \subset \frak{d}_{p+q}$, i.e., they form a *grading *of the matrix algebra. As a consequence, the subspaces $\oplus _{p\geq n}\frak{d}_{p},\;n=0,1,2,...$, are Lie subalgebras and form a decreasing filtration of $\frak{b}_{+}$; by duality, the subspaces $\frak{b}_{-}^{q}=\oplus _{p\leq -n}\frak{d}_{-p}\subset \frak{b}_{-}\simeq \frak{b}_{+}^{*},\;n=0,-1,-2,...$, form an increasing filtration of $\frak{b}_{+}^{*}$ by $Ad\, _{B_{+}}^{*}$-invariant subspaces. The orbit ${\mathcal{O}}_{f}$ is the “biggest” orbit in the subspace $\frak{v}_{1}$; its choice is quite natural, since $\dim {\mathcal{O}}_{f}=2n-2$ is twice the number of independent Hamiltonians, which is precisely the amount needed for complete integrability. Dynamical systems associated with coadjoint orbits of higher dimension (which are abundant) have not got enough “obvious” integrals of motion to assure their *Liouville integrability;* on the other hand, all these systems are explicitly solvable by means of the factorization problem. This queer situation is due to the resonance behaviour of these systems: their trajectories span a submanifold in the phase space with codimension higher than in the generic case.
An important general conclusion to be held from this example is the role of *grading*: it provides a natural decomposition into subalgebras, as well as plenty of invariant subspaces for the coadjoint action.
As already mentioned, loop algebras provide a natural environment for the study of numerous finite-dimensional systems. In this Section we shall briefly outline the corresponding constructions.
Let $\frak{g}$ be a semisimple Lie algebra with invariant inner product ($\frak{g}=\frak{sl}(n)$ with the inner product $\left( X,Y\right) ={\rm tr}\,
XY$ is a good example; in the sequel we shall mainly deal with this standard matrix case). Its loop algebra, $L\frak{g}$ is the Lie algebra of Laurent polynomials with coefficients in $\frak{g}$, $L\frak{g=g}\left[
z,z^{-1}\right] $ with pointwise commutator, $\left[ Xz^{n},Yz^{m}\right] =\left[ X,Y\right] z^{n+m}$, or $\left[ x,y\right] \left(z\right)=
\left[ x\left( z\right) ,y\left( z\right) \right] $. We may regard an element $x\in \frak{G}$ as a polynomial mapping from the unit circle $\Bbb{T}$ into $\frak{g}$. An invariant inner product on $\frak{G}$ is given by $$\left\langle x,y\right\rangle =\int_{\Bbb{T}}{\mathrm tr}\,x\left( z\right)
y\left( z\right) \frac{dz}{2\pi iz}. \label{inn}$$ Let $\frak{g}_{n}=\frak{g}\cdot z^{n}$; clearly, $\left[ \frak{g}_{n},\frak{g}_{m}\right] \subset \frak{g}_{n+m}$, and hence the decomposition$L\frak{g=\oplus }_{n\in {\Z}}\,\frak{g}_{n}$ defines a grading (the so called *standard grading*) of the loopalgebra. Set $$L\frak{g}_{+}=\frak{\oplus }_{n\geq 0}\frak{g}_{n}=\frak{g}\left[ z\right] ,L\frak{g}_{-}=\frak{\oplus }_{n<0}\frak{\frak{g}_{n}}=z^{-1}\frak{g}\left[
z^{-1}\right] . \label{split}$$
\[loops\](i) $L\frak{g}_{+}$ and $L\frak{g}_{-}$ are graded subalgebras of $L\frak{g}$, and $$L\frak{g} = L\frak{g}_{+}{\dot{+}}L\frak{g}_{-}.$$ (ii) The inner product (\[inn\]) sets $L\frak{g}$ into duality with itself; in particular, $L\frak{g}_{\pm }$ is set into duality with $L\frak{g}_{\mp }$. (iii) The ring of Casimir functions on $L\frak{g}^{*}\simeq L\frak{g}^{*}$ is generated by the functionals $$\Phi _{n,m}\left[ L\right] =\frac{1}{n}\mathrm{ Res}\,_{z=0}\,{\mathrm{tr}}\,L\left( z\right) ^{n}z^{-m-1}.$$
Note that these functionals are smooth in the sense of ordinary calculus of variations and ${\mathrm{grad}}\,\Phi _{n,m}=\frac{1}{z^{m}}L^{n-1}$.
Let $P_{\pm }$ be the projection operators onto $L\frak{g}_{\pm }$ parallel to the complementary subalgebra, $r=P_{+}-P_{-}$. In analytic terms, $L\frak{g}$ consists of trigonometric polynomials on the circle, and $r$ is the standard Hilbert transform. The Lie algebra $\left( L\frak{g}\right) _{r}$ is isomorphic to the direct sum $L\frak{g}_{+}\oplus L\frak{g}_{-}$. For $X\in L\frak{g}$ we set $X_{\pm }=\pm P_{\pm }X$.
The coadjoint action of $\left( L\frak{g}\right) _{r}$ on its dual is given by $$ad^{*}X\cdot L=P_{+}\left[ X_{-},L\right] -P_{-}\left[ X_{+},L\right] ;$$ this action leaves invariant the subspace of polynomial loops $L\frak{g}\subset \left( L\frak{g}\right) _{r}^{*}$.
Notice that both linear operators $L\mapsto P_{+}\left[ X_{-},L\right] $ and $L\mapsto P_{-}\left[ X_{+},L\right] $ are Toeplitz.
Since $L\frak{g}$ is infinite-dimensional, the choice of the associated Lie group becomes non-obvious. A reasonable option is to take the group $\mathcal{G}_{W}$ of all Wiener maps $g:\Bbb{T}\rightarrow G$. The Lie algebra $L\frak{g}$ may also be replaced by its appropriate completion, e.g. the Wiener algebra $L\frak{g}_{W}$. Of course, with this choice the full dual space $L\frak{g}_{W}^{*}$ becomes a rather complicated object; the point is that the set of polynomial loops $L\frak{g}\subset L\frak{g}_{W}^{*} $ is invariant with respect to the coadjoint action of $\left( L\frak{g}\right) _{r}$ (though of course not with respect to the coadjoint action of $L\frak{g}_{W}$!). Let ${\mathcal{G}}_{W}^{+}\subset {\mathcal{G}}_{W}$ be the subgroup of Wiener maps $g:\Bbb{T}\rightarrow G$ which are holomorphic in the unit circle, and ${\mathcal{G}}_{W}^{\_}\subset {\mathcal{G}}_{W}$ the subgroup of maps which are holomorphic outside the unit circle and satisfy the normalization condition $\lim_{z\rightarrow \infty }g\left(
z\right) =id. $ The Lie group which corresponds to $\left( L\frak{g}\right)
_{r}$ may be identified with ${\mathcal{G}}_{W}^{r}={\mathcal{G}}_{W}^{+}\times {\mathcal{G}_{W}}^{-}$; its coadjoint action is given by $$\begin{aligned}
\left( Ad^{*}h\cdot L\right) &=&P_{+}\left( h_{-}\left( z\right) L\left(
z\right) h_{-}^{-1}\left( z\right) \right) -P_{-}\left( h_{+}\left( z\right)
L\left( z\right) h_{+}^{-1}\left( z\right) \right) ;\; \\
h &=&\left( h_{+},h_{-}\right) \in {\mathcal{G}}_{W}^{+}\times {\mathcal{G}}_{W}^{-}.\end{aligned}$$
For $\frak{g}=\frak{sl}(n)$ describe all coadjoint orbits of ${\mathcal{G}}_{W}^{r}{\mathcal{\ }}$in the subspace of Laurent polynomials of the form $$L\left( z\right) =l_{-1}z^{-1}+l_{0}+l_{1}z.$$
An analyst may feel disappointed with our choice of the ‘restricted dual’ $L\frak{g}\subset L\frak{g}_{W}^{*}$ of the Wiener algebra; indeed, the coadjoint orbits which are contained in this space are modelled on (matrix) polynomial functions; even general rational functions are not allowed, to say nothing of more interesting classes of analytic functions. The reason for this deliberate restriction is very simple: we are willing to get dynamical systems which admit a simple parametrization and (possibly) some physical interpretation; practical experience shows that examples which are physically interesting are usually associated with coadjoint orbits of the lowest possible dimensions. (By contrast, most of the orbits which lie in the ‘exotic’ part of the full dual space are infinite-dimensional.) This does not mean of course that dynamical systems which are modelled on analytic functions of a more complicated nature are totally uninteresting; but once again, ‘good’ examples are associated not with generic coadjoint orbits in the ‘very big dual’, but rather with well-embedded finite-dimensional ones. (Below we shall see how to construct such examples using a different choice of the basic Lie algebra.) One more reason to single out the finite-dimensional orbits is the possibility to bring into play the highly powerful machinery of Algebraic Geometry: we shall see below that polynomial (or, more generally, rational) Lax matrices give rise to algebraic curves of *finite genus* and Lax equations are linearized on their Jacobians. Lax matrices associated with infinite-dimensional orbits will lead to curves of *infinite genus*.
The specialization of our main theorem to the present situation may be stated as follows:
\[Riemann\] (i) Invariant functionals $\Phi _{n,m}$ give rise to Hamiltonian equations of motion on $L\frak{g}\subset \left( L\frak{g}\right)
_{r}^{*}$ with respect to the Lie-Poisson bracket of $L\frak{g}_{r}$; these equations may be written in the Lax form, $$\frac{dL}{dt}=\left[ L,M_{\pm }\right] ,M_{\pm }=\pm P_{\pm }\left( \mathrm{grad}\,\Phi _{n,m}\right) . \label{laxeq}$$ (ii) The integral curve of with origin $L_{0}$ is given by $$L\left( t,z\right) =g_{\pm }\left( t,z\right) ^{-1}L_{0}g_{\pm }\left(
t,z\right) , \label{curve}$$ where $g_{+}\left( t\right) ,g_{-}\left( t\right) $ solve the matrix Riemann problem $$\label{riem}
\arr{2.0}{c}{\ds
\exp t\,{\mathrm {grad}}\,\Phi _{n,m}\left[ L_{0}\right] \left( z\right)
= g_{+}\left( t,z\right) g_{-}\left( t,z\right) ^{-1},\\
\ds g_{+}\left( t\right)
\in {\mathcal {G}}_{W}^{+}, \;g_{-}\left( t\right) \in {\mathcal {G}}_{W}^{-}.}$$
Note that ${\mathrm{grad}}\,\Phi _{n,m}$ is a Laurent polynomial and hence is regular in the punctured Riemann sphere ${\C}P_{1}\backslash \left(
\left\{ 0\right\} \cup \left\{ \infty \right\} \right) =\C \backslash
\left\{ 0\right\} $. Hence the factorization problem (\[riem\]) has the following geometric meaning. The projective line ${\C}P_{1}$ is covered by two domains $U_{+}={\C}P_{1}\backslash \left\{ \infty \right\} $ and $U_{-}={\C}P_{1}\backslash \left\{ 0\right\} $. The function $\exp tdh(L)$ is regular in $U_{+}\cap U_{-}={\C}\backslash \left\{ 0\right\} $ and may be regarded as the transition function of a holomorphic vector bundle over ${\C}P_{1}$. Factorization problem amounts to an analytic trivialization of this bundle. It is well known (see [@SegalPressley]) that not all vector bundles over $\Bbb{C}P_{1}$ are analytically trivial: each $n$-dimensional bundle breaks up into a sum of line bundles, and their degrees, $d_{1},...,d_{n}\in \Z$ form a full system of holomorphic invariants of the given bundle. In the language of transition functions this means that $\exp tdh(L)$ admits a factorization of the form $$\exp tdh(L)=g_{+}\left( z,t\right) d\left( z\right) g_{-}\left( z,t\right) ^{-1},$$ where $d\left( z\right) =diag\left( z^{d_{1}},...,z^{d_{n}}\right) $. Thus formula (\[curve\]) requires that all partial indices $d_{1},...,d_{n}$ are zero. One can prove that this is true at least for $t$ sufficiently small [@Gohberg]. The exceptional values of $t\in \C$ for which problem does not admit a solution form a discrete set in $\C$; at these points the solution $L\left( t\right) $ has a pole: the trajectory of the Lax equation goes off to infinity.
Theorem provides a link between the Hamiltonian scheme of Section 3 and the algebro-geometric methods of the finite-band integration theory (see [@alg-geom]). Namely, the formula for the trajectories immediately implies that the Lax equations linearize on the Jacobian of the spectral curve associated with the Lax matrix. The proof is so short and simple that I would like to reproduce it here.
Let $\frak{g}=\frak{gl}\left( n,\C \right) $ and $L\left( z\right) =\sum
X_{i}z^{i},X_{i}\in \frak{g}$, a matrix-valued Laurent polynomial. Let us consider the algebraic curve $\Gamma _{0}\subset {\C}\backslash
\left\{ 0\right\} \times C$ defined by the characteristic equation $$\det (L(z)-\lambda )=0; \label{curv}$$ we may regard $z,\lambda $ as meromorphic functions defined on $\Gamma _{0}$. Assume that the spectrum of $L\left( z\right) $ is simple for generic $z$ (this key technical assumption is satisfied in most applications). For each nonsingular point $P\subset \Gamma _{0}$ which is not a branching point of $\lambda $ there is a one-dimensional eigenspace $E\left( P\right) \subset
\C ^{n}$ of $L\left( z\left( P\right) \right) $ with eigenvalue $\lambda
\left( P\right) $. This gives a holomorphic line bundle on $\Gamma _{0}$ defined everywhere except for singular points and branching points. Let $\Gamma $ be the nonsingular, compact model of $\Gamma _{0}$. One can show that the eigenvector bundle extends to a holomorphic line bundle $E\longrightarrow \Gamma $ on the whole smooth curve $\Gamma $.[^6]
The spectral curve $\Gamma $ with two distinguished meromorphic functions $z$ and $\lambda $ and the line bundle $E\longrightarrow \Gamma $ constitute the set of spectral data for $L\left( z\right) $. The evolution determined by a Lax equation of motion leaves the spectral curve (7.3) invariant, and the dynamics of the line bundle $E$ is easy to describe. Let $h$ be the Hamiltonian of our Lax equation; set $M=dh\left( L\right) $; since $[L,M]$ = 0 pointwise and the spectrum of $L$ is simple, the eigenvectors of $L\left(
z\right) $ are also the eigenvectors of $M\left( z\right) $, $$M(z\left( P\right) )v=\mu \left( P\right) v,\;P\in \Gamma ,\;v\in
E(P)\subset \C ^{n},$$ where $\mu \left( P\right) $ is a meromorphic function on $\Gamma $. Define the domains $V_{\pm }\subset \Gamma $ by $V_{\pm }=\left\{ P\in \Gamma
;z\left( P\right) ^{\pm 1}\neq \infty \right\} $. Clearly, $V_{+}\cup
V_{-}=\Gamma $ and $\mu $ is regular in the intersection $V_{+}\cap
V_{-}=\Gamma _{0}$. Recall that a line bundle over a curve is specified by its *transition function* (with values in the multiplicative group $\C^{*}$) with respect to some covering; tensor product of bundles corresponds to the ordinary product of transition functions. The equivalence classes of line bundles form an abelian group ${\mathrm{Pic\,}}\Gamma $ with respect to the tensor product. Let $F_{t}$ be the line bundle on $\Gamma $ determined by the transition function $\exp t\mu \left( P\right) $ with respect to the covering $\left\{ V_{+},V_{-}\right\} $ For all $t\in \C$ the bundles $F_{t}$ have degree zero and form a 1-parameter subgroup in the Picard group ${\mathrm{Pic}}_{0}\,\Gamma $ of equivalence classes of holomorphic line bundles of degree zero on $\Gamma $ which, by Abel’s theorem, is canonically isomorphic to ${\mathrm{Jac}}\,\Gamma $, the Jacobian of $\Gamma $. (See, for instance, [@Griffiths].)
\[th7.1\] The line bundle $E$ regarded as a point of ${\mathrm{Pic\,}}\Gamma $ evolves linearly with time, $E\left( t\right) =E\otimes F_{-t}$
Since the Lax matrix evolves by similarity transformation, its eigenvectors evolve linearly. Let $g_{\pm }\left( t\right) $ be the solution of the Riemann problem . In view of , the moving eigenspace $E_{t}\left( P\right) $ regarded as a subspace of $\Gamma
\times \C ^{n}$, is expressed as $$E_{t}\left( P\right) =g_{+}\left( t,z\left( P\right) \right) ^{-1}E\left(
P\right) \; \label{eig}$$ over$\;V_{+}$, and $$E_{t}\left( P\right) =g_{-}\left( t,z\left( P\right) \right) ^{-1}E\left(
P\right) \; \label{eigmin}$$ over$\;V_{-}$. In other words, $g_{\pm }\left( t,z\right) ^{-1}$ define isomorphisms between $E\left( P\right) $ and $E_{t}\left( P\right) $ over $V_{\pm }$. The transition function in $V_{+}\cap V_{-}$ which matches these two isomorphisms is $g_{+}\left( t,z\left( P\right) \right) ^{-1}g_{-}\left(
t,z\left( P\right) \right) |_{E_{t}\left( P\right) }$. It is easy to check that $$g_{+}\left( t,z\right) g_{-}\left( t,z\right) ^{-1}=\exp tdh\left(
L_{0}\right)$$ implies $$g_{-}\left( t,z \right) g_{+}\left(
t,z \right) ^{-1}=\exp tdh\left( L\left( t\right) \right) ;$$ hence, $$\label{pic}
g_{+}\left( t,z\left( P\right) \right) ^{-1}g_{-}\left( t,z\left( P\right)
\right) |_{E_{t}\left( P\right) }=e^{-tM\left( t,z\left( P\right) \right)
}|_{E_{t}\left( P\right) }=e^{-t\mu \left( P\right) },$$ or $E_{t}\left( P\right) =E\left( P\right) \otimes F_{-t}$, as claimed.
The eigenvector $\psi \left( t,P\right) =\left( \psi _{1},...,\psi
_{n}\right) \in E\left( t,P\right) \subset \Bbb{C}^{n}$ is called the *Baker-Akhiezer function* of $L\left( z,t\right) $. From (\[eig\], \[eigmin\]) it follows that the Baker-Akhiezer function $\psi \left(
t,P\right) $ in the domains $V_{\pm }\subset \Gamma $ may be written in the form $$\psi _{\pm }\left( t,P\right) =g_{\pm }\left( t,z\left( P\right) \right)
^{-1}\psi \left( P\right) ,$$ so that $$\psi _{+}\left( t,P\right) =e^{-t\mu \left( P\right) }\psi _{-}\left(
t,P\right) .$$ Since $\partial _{t}g_{\pm }\cdot g_{\pm }^{-1}=M_{\pm }$ (see the proof of theorem \[fact\]), we have $$\frac{d}{dt}\psi _{\pm }\left( t,P\right) =-M_{\pm }\left( t,z\left(
P\right) \right) \psi _{\pm }\left( t,P\right) .$$ Using the machinery of Algebraic Geometry, it is possible to construct the Baker-Akhiezer function explicitly, in terms of the Riemann theta functions and Abelian integrals. This, in turn, allows to obtain an explicit solution of the Riemann problem. Let us explain how the matrices $g_{\pm }\left(
t,z\right) $ may be reconstructed from $\psi _{\pm }\left( t,P\right) $. Suppose that $z\in \Bbb{C}$ is not a ramification point of $\Gamma $ (i.e., all eigenvalues of $L(z)$ are distinct); let $P_{1},...,P_{n}$ be the points of $\Gamma $ which lie over $z$. Let us arrange the column vectors $\psi
_{\pm }\left( t,P_{1}\right) ,...,\psi _{\pm }\left( t,P_{n}\right) $ in a $n\times n$ matrix $\hat{\psi}_{\pm }\left( t,z\right) $. Put $$g_{\pm }\left( t,z\right) =\hat{\psi}_{\pm }\left( t,\lambda \right) \hat{\psi}_{\pm }\left( 0,\lambda \right) ^{-1}. \label{gplus}$$ Note that if we change the ordering of branches $P_{1},...,P_{n},\hat{\psi}_{\pm }$ is multiplied on the right by a permutation matrix and hence $g_{\pm }$ remains invariant.
\(i) $g_{\pm }\left( t,z\right) $ satisfies the differential equation $$\frac{dg_{\pm }\left( t,z\right) }{dt}=-M_{\pm }\left( t,z\right) g_{\pm
}\left( t,z\right) ,\;M_{\pm }\left( t,z\right) =\left(
{\mathrm {grad\,}}h\left[ L\left( t,z\right) \right] \right) _{\pm }. \label{ev}$$ (ii) $g_{\pm }$ are entire functions of $z^{\pm 1}$. (iii) $g_{\pm }$ solve the factorization problem $$g_{+}\left( t,z\right) g_{-}\left( t,z\right) ^{-1}=\exp t{\mathrm {grad\,}}h\left[ L\left( 0,z\right) \right] .$$
The key assertion (ii) is an easy consequence of the differential equation (\[ev\]); indeed, by (\[gplus\]) $g_{\pm }$ is the fundamental solution of (\[ev\]) normalized by $g_{\pm }\left( 0,z\right) =Id$; hence it is holomorphic in the domain where $M_{\pm }\left( t,z\right) $ is nonsingular, i.e., in ${\C}P_{1}\backslash \left\{ \infty \right\} $ and ${\C}P_{1}\backslash
\left\{ 0\right\} $, respectively.
The decomposition of the loop algebra we used so far is based on its *standard grading* (i.e., grading by the powers of the loop parameter $z)$. This grading is by no means unique, and it is possible to use other gradings to produce new examples of Lax representations. Another possibility (which actually absorbs the former one) is to bring into play *twisted loop algebras.* Let $\sigma $ be an automorphism of $\frak{g}$ of order $n$. The twisted loop algebra $L(\frak{g},\sigma )$ is the subalgebra of $L\frak{g}$ defined by $$L(\frak{g},\sigma )=\left\{ x\in L\frak{g};\sigma \left( x\left(z
\right) \right) =x\left( \epsilon z \right) \right\} , \label{tw}$$ where $\epsilon =\exp \frac{2\pi i}{n}$ is the root of unity. Equivalently, $L(\frak{g},\sigma )\subset L\frak{g}$ is the stable subalgebra of the automorphism ${\sigma }:L\frak{g\rightarrow }L\frak{g}$ such that $x^{{\sigma }}\left( z \right) =\sigma \left( x\left( \epsilon
z \right) \right) $ for all $x\in L\frak{g.}$ The isomorphism class of the twisted loop algebra depends on the properties of $\sigma $; one can show that when $\sigma $ is an *inner* automorphism, the stable subalgebra $L(\frak{g},\sigma )$ is isomorphic to $L\frak{g}$ as an abstract Lie algebra; however, the grading which is induced on $L(\frak{g},\sigma
)\subset L\frak{g}$ by the standard grading in $L\frak{g}$ is different. The classical theorem due to V.Kac asserts that all different gradings on $L\frak{g}$ may be obtained in this way. (Among the integrable systems that may be very naturally constructed along these lines one may quote *periodic Toda lattices* which are associated with the decomposition of loop algebras derived from their* principal grading*, see [@RS] for details.)
The most interesting case for applications is when $\sigma $ is an outer automorphism of order 2 (an involution). In this case we may assume that $\frak{g}$ and $L\frak{g}$ are real. Here is the key example: $\frak{g}=\frak{gl}(n),\sigma \left( X\right) =-X^{t}$ ( $t$ denotes transposition). The loop algebra $L\frak{g}^{\sigma }$ consists of Laurent polynomials $$X\left( z\right) =\sum X_{k}z^{k},$$ where $X_{2p}=-X_{2p}^{t},X_{2p+1}=X_{2p+1}^{t}$. Antisymmetric matrices belong to the Lie algebra $so(n)$ of the orthogonal group, which describes kinematics of the rigid body; on the other hand, symmetric matrices are reminiscent of the quadratic form associated with kinetic energy ( inertia tensor). Thus, the twisted loop algebra seems to be a good candidate to set up the stage for applications to the mechanics of the rigid body. Let us equip $L\frak{g}^{\sigma }$ with the inner product $$\left\langle X,Y\right\rangle =-{\rm Res}\,_{z=0}\frac{1}{z}{\rm tr}\,
X\left( z\right) Y\left( z\right) \label{inntw}$$ which sets $L\frak{g}^{\sigma }$ into duality with itself (mind the difference with (\[inn\]): the factor $z^{-1}$ makes the coupling respect parity; the minus sign makes the inner product positive on $so(n)$). Let $r$ be the classical r-matrix associated with the standard decomposition $L\frak{g}^{\sigma }=L\frak{g}_{+}^{\sigma }{\dot{+}}L\frak{g}_{-}^{\sigma }$, as in . Obviously, $$\left( L\frak{g}_{+}^{\sigma }\right) ^{*}\simeq \left( L\frak{g}_{-}^{\sigma }\right) ^{\perp }=\oplus _{k\leq 0}\frak{g}\cdot
z^{k},\;\left( L\frak{g}_{-}^{\sigma }\right) ^{*}\simeq \left( L\frak{g}_{+}^{\sigma }\right) ^{\perp }=\oplus _{k>0}\frak{g}\cdot z^{k}.$$ Coadjoint orbits of $L\frak{g}_{r}^{\sigma }$ are direct products of orbits of $L\frak{g}_{+}^{\sigma }$ lying in $\left( L\frak{g}_{-}^{\sigma }\right)
^{\perp }$ and orbits of $L\frak{g}_{-}^{\sigma }$ lying in $\left( L\frak{g}_{+}^{\sigma }\right) ^{\perp }$. Lax matrices describing the motion of the rigid body and related mechanical systems belong to the simplest $ad^{*}$-invariant subspace $L\frak{g}_{-1,1}^{\sigma }$ consisting of matrices $$L\left( z\right) =az^{-1}+l+bz,l\in so\left( n\right) ,a=a^{t},b=b^{t}.$$
\(i) All monomials $bz\in L\frak{g}^{\sigma }$ are 1-point orbits of $L\frak{g}_{-}^{\sigma }$. (ii) Coadjoint representation of $L\frak{g}_{+}^{\sigma }$ in the subspace $\left\{ az^{-1}+l\right\} \subset \left( L\frak{g}_{+}^{\sigma }\right) ^{*}$ factors through its finite-dimensional quotient $L\frak{g}_{+}^{\sigma }/z^{2}L\frak{g}_{+}^{\sigma }$. (iii) The quotient algebra $L\frak{g}_{+}^{\sigma }/z^{2}L\frak{g}_{+}^{\sigma }$ is isomorphic to the semidirect product $\frak{g}_{0}^{\sigma }=so\left( n\right) \ltimes sym(n)$ of the orthogonal algebra $so\left( n\right) $ and the space of symmetric matrices (with zero Lie bracket).
The associated Lie group $G_{0}^{\sigma }$ is the semidirect product of $SO\left( n\right) $ and the additive group of the linear space $sym(n)$ of symmetric $n\times n$-matrices. Its coadjoint orbits are easy to describe; here is a simple example:
Fix a unit vector $e\in {\R}^{n}$ and let $a=e\otimes e$ be the rank one orthogonal projection operator onto $\Bbb{R\cdot }e\subset {\R}^{n}$. Let $T^{*}S^{n-1}\ $be the cotangent bundle of the sphere $S^{n-1}=SO\left(
n\right) \cdot e\subset {\R}^{n}$ realized as the subbundle of $S^{n-1}\times {\R}^{n}$, $$T^{*}S^{n-1}=\left\{ \left( x,p\right) \in S^{n-1}\times \Bbb{R}^{n};\left\langle p,x\right\rangle =0.\right\}$$ There is natural map $\pi :T^{*}S^{n-1}\rightarrow \mathcal{O}_{a}$ onto the coadjoint orbit of $G_{0}^{\sigma }$ passing through the monomial $z^{-1}a$, $$\pi :\left( x,p\right) \longmapsto z^{-1}x\otimes x+p\wedge x.$$ ($\pi $ is actually a double covering.)
The Lax matrix associated with this orbit has the form[^7] $$L\left( z\right) =z^{-1}x\otimes x+p\wedge x+z\,b,\;x\in SO\left( n\right) ,\;p\in
{\R}^{n},\; \left\langle p,x\right\rangle =0;$$ the associated phase space describes the point moving on a sphere. The constant matrix $b\in sym\left( n\right) $ does not affect kinematics, but is quite useful to produce interesting Hamiltonians. The simplest Hamiltonian is $$H=-\frac{1}{4}{\mathrm {Res}}\,_{z=0}\,{\mathrm {tr}}\,
\frac{1}{z}L\left( z\right) ^{2}=\frac{1}{2}\left\langle p,p\right\rangle -\frac{1}{2}\left\langle bx,x\right\rangle ;$$ it describes the so called *Neumann problem* (point moving on a sphere in a quadratic potential).
Describe the set of commuting integrals of motion for the Neumann problem which are *quadratic* in momenta.
We may generalize this example in various ways; a useful remark is that we may avoid a too detailed description of the coadjoint orbits. Instead, we may produce a map $\pi $ onto such orbit, or a union of orbits, which is compatible with the Poisson structure but need not be a bijection (and so possibly introduces some extra variables). This idea is implemented in the following statement. Set $\;K=SO\left( n\right) ,\frak{k}=so(n)$ and let $T^{*}K\simeq K\times \frak{k}$ be the cotangent bundle (equipped with its standard Poisson bracket). Fix $a\in sym\left( n\right) $ and consider the mapping $$\pi :T^{*}K\longrightarrow \left( \frak{g}_{0}^{\sigma }\right) ^{*}:\left(
k,\rho \right) \longmapsto k\left( \rho +a\right) k^{-1}$$
$\pi $ is a Poisson mapping (i.e., maps canonical Poisson brackets in $T^{*}K $ onto Lie-Poisson brackets in $\left( \frak{g}_{0}^{\sigma }\right)
^{*}$); its image is a union of coadjoint orbits of the semidirect product $G_{0}^{\sigma }=K\times sym\left( n\right) $.
\[In this statement we identified $\frak{g}_{0}^{\sigma }$ with $Mat\left( n\right) $ as a linear space and also used the inner product $\left\langle X,Y\right\rangle =-{\rm tr}\,
XY$ to identify the dual space $\frak{g}_{0}^{\sigma }$ with $Mat\left(
n\right) $. \]
The cotangent bundle $T^{*}K$ is naturally interpreted as the phase space of a rigid body in $\Bbb{R}^{n}$; we get a family of Lax matrices parametrized by points of $T^{*}K:$$$L\left( z\right) =z^{-1}kak^{-1}+k\rho k^{-1}+zb; \label{manakov}$$ Hamiltonians which may be derived from include the so called *Manakov case* of the motion of a free top in ${\R}^{n}$ (for $n=3$ this is the classical *Euler top*), or more generally, the Manakov top in a quadratic potential.
One may wonder, what is the relation of the low-dimensional Neumann system to the “big” phase space $T^{*}K$ (we have $\dim T^{*}S^{n-1}=2n-2$ and $\dim T^{*}K=n\left( n-1\right) $). The answer is that, for special choices of $a\in sym\left( n\right) $, the Hamiltonians associated with the Lax matrix possess high symmetry (resulting from the redundancy introduced by $\pi )$. The Neumann system is the result of Hamiltonian reduction of the “big system” with respect to this symmetry group.
One may notice that the use of the twisted loop algebra was indeed crucial: the built-in symmetry of the Lax matrix accounts both for the correct kinematics of the rigid body (antisymmetric matrices) and for the symmetry of the related quadratic forms (notably, of the kinetic energy). Further generalization is straightforward: we must scan the list of semisimple Lie algebras and their involutions and look for nice-looking opportunities. In this way we get the following list:
----------------------------------------------------------------------------------------------- -- --
**[Algebra]{} & **[Involution]{} & **[Related systems]{}\
$\frak{gl}(n)$ & $X\mapsto -X^{t}$ & Manakov top, Neumann system, etc.\
$so(n,1)$ & $X\mapsto -X^{t}$ & Lagrange top, spherical pendulum, etc.\
$so\left(p,q\right) ,p>q\geq 2$ & $X\mapsto -X^{t}$ & Kowalevski top and its generalizations\
$so(n,n)$ & $X\mapsto -X^{t}$ & Interacting Manakov tops\
$G_2\subset so(4,3)$ & $X\mapsto -X^{t}$ & Exotic integrable top on $SO(4)$\
******
----------------------------------------------------------------------------------------------- -- --
In all cases, Lax matrices belong to the subspace $L\frak{g}_{-1,1}^{\sigma }$; to get particular examples (for instance, the Kowalevski top) one sometimes has to perform additional Hamiltonian reduction; we refer the reader to [@RS] for details.
Let us finally discuss the implications of the twisting automorphism for the geometry of the spectral curve and for the linearization theorem.
\(i) Let us assume that the Lax matrix $L\left( z\right) \in L\frak{g}^{\sigma }$ and $\sigma $ is an *inner automorphism* of $\frak{g},\mathrm{ord\,}\sigma =m$. In that case the spectral curve $\Gamma =\left\{
\left( z,\lambda \right) \in \Bbb{C}^{2};\det \left( L\left( z\right)
-\lambda \right) =0\right\} $ admits an automorphism $\hat{\sigma}:\left(
z,\lambda \right) \mapsto \left( \epsilon z,\lambda \right) $ (here $\epsilon =
\exp 2\pi i/n$ is the root of unity); this automorphism lifts to $\mathrm{Pic\,}\Gamma $ and the transition function (\[pic\]) which determines the evolution of the eigenbundle of $L$ is *invariant* under $\hat{\sigma}$. (ii) Suppose that $\sigma $ is an *outer involution,* $\sigma \left( X\right) =-X^{t}$. Then the spectral curve admits an automorphism $\hat{\sigma}:\left( z,\lambda \right) \mapsto \left(
-z,-\lambda \right) $; the transition function is *anti-invariant* under $\hat{\sigma}$, i.e., $\hat{\sigma}:\exp t\mu (z,\lambda )\mapsto \exp
\left( -t\mu (z,\lambda )\right) $.
The check of both assertions is obvious: An inner automorphism preserves the eigenvalues of a matrix; by contrast, $\sigma \left( X\right) =-X^{t}$ changes the sign of the eigenvalues. The logarithm of the transition function $\mu \left( z,\lambda \right) $ is the eigenvalue of the gradient ${\mathrm {grad\,}} H\left[ L\right] \left( z\right) $ (more precisely, one of its branches associated with the eigenvector of $L\left( z\right) $ which corresponds to $\lambda )$. Since ${\mathrm {grad\,}}H\left[ L\right] \in L\frak{g}^{\sigma }$, $\mu \left( z,\lambda \right) $ is invariant when $\sigma $ is inner and changes sign when it’s derived from transposition.
Let us now describe how to deal with Lax matrices which are *rational functions* of the spectral parameter. As we mentioned, it is possible to trace down the corresponding coadjoint orbits inside the dual space of an appropriate completion of the standard loop algebra, but it is more practical to choose our basic Lie algebra in a different way.
Let $D=\left\{ z_{1},...,z_{N}\right\} \subset {\C}P_{1}$ be a finite set; we assume that $\infty \in D$. For $z_{j}\in D$ let $\lambda _{j}$ be the local parameter on ${\C}P_{1}$ at $z=z_{j}$, i.e., $\lambda
_{j}=z-z_{j}$ if $z_{j}\neq \infty $ and $\lambda _{\infty }=z^{-1}$ for $z_{j}=\infty $. We define the *local algebra* $\frak{G}_{z_{j}}$ as the algebra of formal Laurent series in local parameter with coefficients in a little Lie algebra $\frak{g},\frak{G}_{z_{j}}=\frak{g}\left( \left(
\lambda _{j}\right) \right) $. (We may assume that $\frak{g}$ is the matrix algebra with the standard inner product.) If $z_{j}\neq \infty $, let $\frak{G}_{z_{j}}^{+}$ be the algebra of formal Taylor series in local parameter; for $z_{j}=\infty $ we set $\frak{G}_{\infty }^{+}=\lambda _{\infty }\frak{g}\left[ \left[ \lambda _{\infty }\right] \right] $ (in other words, $\frak{G}_{\infty }^{+}$ consists of formal Taylor series without constant term). Put $$\frak{G}_{D}=\bigoplus_{z_{j}\in D}\frak{G}_{z_{j}},\;\frak{G}_{D}^{+}=\bigoplus_{z_{j}\in D}\frak{G}_{z_{j}}^{+}$$ (direct sum of Lie algebras). Let $\frak{g}(D)$ be the algebra of rational functions on ${\C}P_{1}$ with coefficients in $\frak{g}$ which are regular outside $D$; it is naturally embedded into $\frak{G}_{D}$ (the embedding assigns to each $X\in \frak{g}(D)$ the collection of its Laurent series at each point of $D$).
\[mittag\](i) $\frak{G}_{D}=\frak{g}(D){\dot{+}}\frak{G}_{D}^{+}$ (direct sum of linear spaces). (ii) The $\Bbb{C}$-bilinear inner product on $\frak{G}_{D}$$$\left\langle X,Y\right\rangle =\sum_{z_{j}\in D}{\mathrm{Res}}_{z_{j}}{\mathrm {tr}}\,
X_{j}Y_{j}d\lambda _{j} \label{res}$$ is invariant and nondegenerate. (iii) $\frak{g}(D)$ and $\frak{G}_{D}^{+}$ are isotropic subspaces with respect to (\[res\]); moreover, $\frak{g}(D)\frak{\simeq }\left( \frak{G}_{D}^{+}\right) ^{*}$. (iv) Coadjoint orbits of $\frak{G}_{D}^{+}$ in $\frak{g}(D)$ are finite-dimensional.
*Sketch of a proof.* An element $X=\left(
X_{j}\right) _{z_{j}\in D}$ is a finite collection of Laurent series; stripping each of them of its positive part we get a set of *principal parts* at $z_{j}\in D$; let $X^{0}$ be the unique rational function with these principal parts; by construction, $X-X^{0}\in \frak{G}_{D}^{+}$ (mind the special role of $\infty $ which fixes the normalization condition!). In brief, we can say that the decomposition $\frak{G}_{D}=\frak{g}(D){\dot{+}}\frak{G}_{D}^{+}$ is equivalent to the Mittag-Leffler theorem for rational functions. Isotropy of $\frak{g}(D)$ and $\frak{G}_{D}^{+}$ means that the inner product restricted to these subspaces is identically zero; this condition assures that the associated classical r-matrix is skew symmetric. For $\frak{G}_{D}^{+}$ this assertion is immediate, since (for $z\neq\infty$) the product of two Taylor series has zero residue; for $z=\infty$ the residue disappears because of the normalization condition. The isotropy of $\frak{g}(D)$ is a reformulation of the classical theorem: the sum of residues of a rational function is zero.
Since $\frak{G}_{D}^{+}$ is a direct sum of local algebras, its coadjoint orbits are direct products of the coadjoint orbits of each local factor; it is easy to see that the coadjoint orbits of the local algebra $\frak{G}_{z_{j}}^{+}$ are modelled on rational functions with a single pole at $z=z_{j}$; moreover, the subspace of rational functions with prescribed order of singularity at this point is stable under the coadjoint action of $\frak{G}_{z_{j}}^{+}$. Clearly, this subspace is finite-dimensional, which proves (iv).
Describe coadjoint orbits in the subspace of functions admitting only simple poles.
Our main theorem immediately applies in this setting and provides an ample set of integrals of motion in involution for Lax equations with rational Lax matrix.
\(1) Since we are interested only in Lax operators which are global rational functions on the Riemann sphere, we consider only coadjoint orbits of $\frak{G}_{D}^{+}\subset \left( \frak{G}_{D}\right) _{r}$; this is legitimate, since $\left( \frak{G}_{D}\right) _{r}$ splits into direct sum of two complementary subalgebras, $$\left( \frak{G}_{D}\right) _{r}\simeq \frak{G}_{D}^{+}\oplus \frak{g}(D),$$ and hence its orbits are direct products of orbits lying in $\frak{g}(D)$and in $\frak{G}_{D}^{+}$; in other words, we take orbits which project into zero in $\frak{G}_{D}^{+}$. (2) The global algebra $\frak{G}_{D}^{+}$ is decomposed into direct sum of local factors, $\oplus _{z_{j}\in D}\frak{G}_{z_{j}}^{+}$; coadjoint orbits of each local algebra are the same that we encountered for the ordinary loop algebra. What makes things different, is the way these orbits are embedded into the bigger algebra; this embedding affects the choice of the invariant Hamiltonians as well as the formulation of the factorization problem.
The use of formal series is well adapted for the study of coadjoint orbits in $\frak{g}(D)$; in order to be able to define Lie groups associated with our Lie algebras, we must change the topology by replacing formal series with convergent ones. Let ${\mathcal{G}}_{z_{j}}^{W}$ be the group of germs of functions with values in $G$ which are regular in some punctured disc around $z_{j}\in {\C}P_{1}$ (with topology of uniform absolute convergence), $({\mathcal{G}}_{z_{j}}^{+})^{W}\subset {\mathcal{G}}_{z_{j}}^{W}$ its subgroup consisting of functions regular in the entire small disc, and $G(D)$ the group consisting of holomorphic mappings $\Bbb{C}P_{1}\backslash
D\rightarrow G$. The infinitesimal decomposition described in proposition \[mittag\] (i) corresponds to the following multiplicative problem:
*Given a set of local meromorphic functions* $g_{1}$, * *$...,g_{N} $*,* $g_{j}\in \mathcal{G}_{z_{j}}^{W}$, * find a global meromorphic function* $g_{0}$* which is regular in the punctured sphere* ${\C}P_{1}\backslash D$* such that* $g_{0}g_{j}^{-1}$* is regular in some small disc around* $z_{j}$.
This is the standard *multiplicative Cousin problem*; its geometrical meaning is the same as for the matrix Riemann problem discussed above: it corresponds to the trivialization of a vector bundle over ${\C}P_{1}$ (defined with respect to a different covering of the sphere).
Reformulate the global factorization theorem in this setting.
One is of course tempted to generalize the above construction replacing ${\C}P_{1}$ with an arbitrary Riemann surface. There is an obvious obstruction which comes from the *Mittag-Leffler theorem for curves:* a global meromorphic function on a curve $\Gamma $ with prescribed principal parts exists if and only if these principal parts satisfy a set of linear constraints; roughly, the sum of residues $$\sum_{z_{j}\in D}{\mathrm {Res}}\,_{z_{j}}{\mathrm {tr}}\,
X_{j}\omega$$ must be zero for all holomorphic differentials $\omega \in H^{1}(\Gamma
)\otimes \frak{g.}$ The trouble is that the constrained data do not form a Lie subalgebra inside the global algebra $\frak{G}_{D}$, and hence one cannot find a complement of $\frak{G}_{D}^{+}$ which is a Lie subalgebra. When $\Gamma $ is elliptic, this obstruction may be overcome by imposing additional automorphy conditions, see e.g. [@RS].
In applications to integrable PDE’s, Lax matrices are replaced by first order matrix differential operators. The systematic treatment of these applications is based on the use of *double loop algebras*, or, more precisely, of their central extensions. Let us start with discussion of the central extension of the ordinary loop algebra.
Set $\frak{g}=\frak{gl}(n)$ and let $\frak{G}=C^{\infty }\left( S^{1};\frak{g}\right) $ be the Lie algebra of smooth functions on the circle with values in $\frak{g}$ and with pointwise commutator. (Mind that in the present setting we choose topology in our loop algebra in a different way! This is because we are willing to treat functions of $x\in S^{1}$ as dynamical variables for our future evolution equations.) We equip $\frak{G}$ with the invariant inner product $$\left\langle X,Y\right\rangle =\int_{0}^{2\pi }{\rm tr}\,
XYdx; \label{int}$$ accordingly, we get an embedding $\frak{G}\subset \frak{G}^{*}$ which defines the *smooth dual* of $\frak{G}$. Put $$\omega \left( X,Y\right) =\int_{0}^{2\pi }{\rm tr}\,
X\cdot \frac{dY}{dx}dx; \label{maur}$$
$\omega $ is a skew symmetric bilinear form which satisfies the cocycle condition $$\omega \left( \left[ X,Y\right] ,Z\right) +\omega \left( \left[ Y,Z\right]
,X\right) +\omega \left( \left[ Z,X\right] ,Y\right) =0. \label{cocycl}$$
Put $\widehat{\frak{G}}=\frak{G}\oplus {\C}$ and define the Lie bracket in $\widehat{\frak{G}}$ by $$\left[ \left( X,c\right) ,\left( Y,c^{\prime }\right) \right] =\left( \left[
X,Y\right] ,\omega \left( X,Y\right) \right) ,\;X,Y\in \frak{G},\;c,c^{\prime }\in \Bbb{C}.$$ (The Jacobi identity is equivalent to .) Notice that $\frak{c}=\{\left( 0,c\right) ;c\in \Bbb{C}\}\subset \widehat{\frak{G}}$ is central in $\widehat{\frak{G}}$ and $\frak{G}$ may be identified with the quotient algebra, $\frak{G}=\widehat{\frak{G}}/\frak{c}$. The (smooth) dual of $\widehat{\frak{G}}$ may be identified with $\frak{G}\oplus {\R}$.
The coadjoint representation of $\widehat{\frak{G}}$ is given by $$ad^{*}X\cdot \left( L,e\right) =\left( \left[ X,L\right] +e\partial
_{x}X,0\right) . \label{coad}$$ (Note that this representation is trivial on the center $\frak{c}\subset
\widehat{\frak{G}}$ and therefore it may be regarded as a representation of $\frak{G}.)$
Let $\Bbb{G}=C^{\infty }\left( S^{1};G\right) $ be the Lie group associated with the Lie algebra $\frak{G}$. The coadjoint representation of $\frak{G}$ in $\widehat{\frak{G}}^{*}$ may easily be integrated to a representation of $\Bbb{G}$.
\[Ad\]The coadjoint representation of $G$ in $\widehat{\frak{G}}^{*}$ is given by $$Ad\,
^{*}g\cdot \left( L,e\right) =\left( gLg^{-1}+e\partial _{x}g\cdot
g^{-1},e\right) . \label{Coad}$$
Proposition admits a remarkable geometric interpretation. Consider the *auxiliary linear differential equation* $$e\frac{d\psi }{dx}=L\psi \label{aux}$$ (we regard it as a differential equation on the line with periodic potential $L$).
Coadjoint representation (\[Coad\]) leaves invariant the hyperplanes $e=const$ in $\widehat{\frak{G}}^{*}$; on each hyperplane $e=const\neq 0$ it is equivalent to the gauge transformation of the potential $L$ in the linear equation (\[aux\]) induced by the natural action of $\Bbb{G}=C^{\infty
}\left( S^{1};G\right) $ on its solutions by left multiplication, $g:\psi
\mapsto g\cdot \psi $.
Let $\psi _{0}\in C^{\infty }\left( {\R};G\right) $ be the fundamental solution of (\[aux\]) normalized by $\psi _{0}\left( 0\right) =id$; $T_{L}=\psi _{0}\left( 2\pi \right) \in G$ is called the *monodromy matrix* of $L$.
*(Floquet)* Two potentials $L,L^{\prime }\in C^{\infty }\left( S^{1};\frak{g}\right) $ lie on the same coadjoint orbit in $\widehat{\frak{G}}^{*}$ (with fixed $e\neq 0)$ if and only $T_{L}$ and $T_{L^{\prime }}$ are conjugate in $G$.
*Sketch of a proof.* Without restricting the generality we may assume that $T_{L}=T_{L^{\prime }}$. Let $\psi _{L},\psi _{L^{\prime }}$ be the fundamental solutions normalized by $\psi _{L}\left( 0\right) =\psi
_{L^{\prime }}\left( 0\right) =id$; put $g(x)=\psi _{L}\left( x\right) \psi
_{L^{\prime }}\left( x\right) ^{-1}$. Clearly, $g$ is $2\pi $-periodic on the line and $g\cdot \psi _{L^{\prime }}=\psi _{L}$, hence $Ad\,
^{*}g\cdot L^{\prime }=L$.
All coadjoint orbits lying in the hyperplanes $e=const\neq 0$ have finite codimension; the ring of Casimir functions is generated by spectral invariants of the monodromy.
The Hamiltonian mechanics in $\widehat{\frak{G}}^{*}$ may be defined with the help of the elementary calculus of variations. Let $\varphi \left[
L\right] $ be a smooth functional of the potential $L,X_{\varphi }=\mathrm{grad}\varphi \left[ L\right] \in \frak{G}$ its Frechet derivative defined by $$\frac{d}{ds}\varphi \left[ L+s\eta \right] =\int_{0}^{2\pi }{\mathrm {tr}}\,
X_{\varphi }\left( x\right) \eta \left( x\right) dx.$$ The Lie-Poisson bracket of two functionals $\varphi _{1},\varphi _{2}$ is given by $$\left\{ \varphi _{1},\varphi _{2}\right\} \left[ L\right] =\int_{0}^{2\pi }{\mathrm {tr}}\,
\left( L\left( x\right) \left[ X_{\varphi _{1}}\left( x\right) ,X_{\varphi
_{2}}\left( x\right) \right] +eX_{\varphi _{1}}\partial _{x}X_{\varphi
_{2}}\right) dx.$$
The Hamiltonian equation of motion on $\widehat{\frak{G}}^{*}$ with Hamiltonian $\varphi $ is equivalent to the following differential equation for the potential $L:$$$\frac{\partial L}{\partial t}=-\left[ X_{\varphi },L\right] -e\frac{\partial
X_{\varphi }}{\partial x}. \label{zero}$$
Equation has a nice geometrical meaning. Let us consider the $\frak{g}$-valued differential form $$Ldx+X_{\varphi }dt;$$ it may be regarded as a connection form of a connection on ${\R}^{2}$ (with values in $\frak g$); equation then means that this connection has zero curvature (hence the term “zero curvature equation”.) We would like to use the central extension $\widehat{\frak{G}}$ as a building block to construct integrable equation; there are already two reassuring points:
1. The description of coadjoint orbits in $\widehat{\frak{G}}^{*}$ automatically leads to the auxiliary linear problem .
2. Equations of motion associated with $\widehat{\frak{G}}$ are zero curvature equations, as desired.
However, there are also two major drawbacks:
1. There is only a finite number of independent Casimirs (one can take e.g. the coefficients of the characteristic polynomial $\det \left(
T_{L}-\lambda \right) )$.
2. The Casimirs are highly nonlocal functionals of the potential.
By contrast, in order to get integrable PDE’s we need an *infinite* number of conservation laws; these conservation laws are usually expressed as integrals of* local densities* which are polynomial in the matrix coefficients of the potential $L$ and its derivatives in $x$. The way to resolve these difficulties is suggested by the auxiliary linear equation (\[aux\]): in order to characterize the potential, we need to know the monodromy *for all energies*; in other words, we must introduce into spectral parameter. Algebraically, this means that we have to modify the choice of our basic Lie algebra.
Let us put $\frak{G}=C^{\infty }\left( S^{1};\frak{g}\right) $ as before; let ${\mathbf{g=}} \frak{G}\otimes {\C}\left( \left( z\right) \right) $ be the algebra of formal Laurent series with values in $\frak{G}$. (In other words we can say that $\mathbf{g}$ is the *double loop algebra* of $\frak{g}$; the different roles of the variables $x,z$ are imposed by our choice of its central extension.) We equip $\mathbf{g}$ with the inner product $$\left\langle X,Y\right\rangle ={\mathrm {Res}}\,_{z=0}\int
{\mathrm {tr}}\,
X(x,z)Y\left( x,z\right) dx=\frac{1}{2\pi i}\int
{\mathrm {tr}}\,
X(x,z)Y\left( x,z\right) dxdz. \label{restr}$$
The 2-cocycle $\omega $ on $\mathbf{g}$ is defined by $$\omega \left( X,Y\right) =\left\langle X,\frac{dY}{dx}\right\rangle .
\label{omega}$$
Let $\widehat{\mathbf{g}}$ be the central extension of $\mathbf{g}$ defined by this cocycle.[^8] As before, we may identify the dual of $\widehat{\mathbf{g}}$ with ${\mathbf{g}}\oplus \Bbb{C}$. Formula for the coadjoint representation remains valid. We conclude that the coadjoint representation for $\widehat{\mathbf{g}}$ coincides with (infinitesimal) gauge transformation associated with the auxiliary linear problem , where this time $L\in \mathbf{g,}$ i.e., it is a formal series in $z$ with coefficients in $\frak{G}$. In other words, $z$ plays the role of spectral parameter in the auxiliary linear problem, as desired. There are some troubles with the definition of the associated Lie group, but let us ignore them for the moment. Notice that if $L$ is a polynomial in $z,z^{-1}$, the monodromy matrix $T_{L}$ is a well-defined analytic function of $z$ (with values in $G=GL\left( n\right) $) which is holomorphic in ${\C}\backslash \left\{ 0\right\} $.
Our choice of the basic Lie algebra makes it easy to define the other key element of our scheme, the classical r-matrix. Set ${\mathbf{g}}_{+}=\frak{G}\otimes {\C}\left[ \left[ z\right] \right] ,{\mathbf{g}}_{-}=\frak{G}\otimes z^{-1}{\C}\left[ z^{-1}\right] $. Clearly, ${\mathbf{g}}={\mathbf{g}}_{+}{\dot{+}}{\mathbf{g}}_{-}$ as a linear space. Both subalgebras ${\mathbf{g}}_{+}$ and ${\mathbf{g}}_{-}$ are isotropic with respect to the inner product which sets them into duality. As before, we put $$r=P_{+}-P_{-}, \label{skew}$$ where $P_{+},P_{-}$ are the projection operators onto ${\mathbf{g}}_{+}$ and ${\mathbf{g}}_{-}$, respectively, and define the r-bracket on $\mathbf{g}$ by $\left[ X,Y\right] _{r}=\frac{1}{2}\left( \left[ rX,Y\right] +\left[
X,rY\right] \right) $. In this way we get the algebra ${\mathbf{g}}_{r}$, but this is still not quite what we need to apply our main theorem, since our basic algebra is $\widehat{\mathbf{g}}$, not $\ {\mathbf{g}!}$ As it happens, the theorem survives central extension.
\[or\]Let $\omega $ be a 2-cocycle on $\mathbf{g}\ $ and $r\in End\,\mathbf{g}$ a linear operator which satisfies the modified Yang-Baxter identity. Set $$\omega _{r}\left( X,Y\right) =\frac{1}{2}\left( \omega \left( rX,Y\right)
+\omega \left( X,rY\right) \right) .$$ Then $\omega _{r}$ is a 2-cocycle on $\mathbf{g}_{r}$.
Prove lemma \[or\] (the proof is abstract and uses only manipulation with the Jacobi and the Yang-Baxter identities).
Let $\widehat{\mathbf{g}_{r}}$ be the central extension of ${\mathbf{g}}_{r}$ associated with $\omega _{r}$. It is easy to see that $\left( \widehat{\mathbf{g}},\widehat{{\mathbf{g}}_{r}}\right) $ is a double Lie algebra and we may apply our main idea: *Casimirs of* $\widehat{\mathbf{g}}$*regarded as Hamiltonians with respect to the Lie-Poisson bracket of* $\widehat{{\mathbf{g}}_{r}}$* give rise to generalized Lax equations.* Actually, there is one more simplification, which is due to our choice of $\omega $ (see ):
Suppose that $r\in End\,\mathbf{g}$ is skew symmetric with respect to the inner product on $\mathbf{g}$; then $\omega _{r}=0$.
The r-matrix clearly satisfies this condition; hence the algebra $\widehat{{\mathbf{g}}_{r}}={\mathbf{g}}_{r}\oplus {\C}$ splits and the Lie-Poisson brackets for $\widehat{{\mathbf{g}}_{r}}$ and ${\mathbf{g}}_{r}$ coincide. Explicitly, this means that the Poisson bracket of two smooth functionals $\varphi _{1},\varphi _{2}$ defined on ${\mathbf{g}}_{r}^{*}\simeq
\mathbf{g}$ is given by $$\label{local}
\left\{ \varphi _{1},\varphi _{2}\right\} _{r}\left[ L\right] =\frac{1}{2\pi
i}\int \mathrm{tr\,}\left( \left[ \mathrm{grad\,}\varphi _{1},\mathrm{grad\,}\varphi _{1}\right] _{r}\left( x,z\right) \cdot L\left( x,z\right) \right)
dxdz, \label{varbr}$$ where ${\mathrm{grad\,}}\varphi _{i}\left[ L\right] \left( x,z\right) \in
\mathbf{g}$, $i=1,2$, is the Frechet derivative.
The skew symmetry of $r$ makes the above discussion of the cocycle $\omega
_{r}$ void; however, as already noticed (see footnote ), we may modify the cocycle $\omega $ by a weight factor $\phi \left( z\right) $, and in that case our Poisson bracket will contain derivatives $\partial _{x}$ of the gradients.[^9]
The antisymmetry of $r$ makes the description of coadjoint orbits very simple. In the absence of cocycle we must deal with the orbits of $\mathbf{g}_{r}$; note that the r-matrix is acting only on the variable $z$ in the double loop algebra, and hence the other variable $x$ becomes a parameter. Let us consider the ‘little’ loop algebra $L\frak{g}=\frak{g}\otimes {\C}\left( \left( z\right) \right) $ and the associated algebra $L\frak{g}_{r}$ which we have already discussed in Section 6. The ‘big’ algebra ${\mathbf{g}}_{r}$ consists of smooth periodic functions on the line with values in $L\frak{g}_{r}$ and with pointwise commutator.
Fix an orbit ${\mathcal{O}}\subset \left( L\frak{g}_{r}\right) ^{*}$; then ${\mathbf{O}}=Map(S^{1},{\mathcal{O}})$ is an orbit of ${\mathbf{g}}_{r}$.
More generally, we may vary orbits $\mathcal{O}$ lying over different points of $S^{1}$ (i.e., consider smooth families of orbits in $\left( L\frak{g}_{r}\right) ^{*}$ parametrized by $S^{1}$).
\[Heis\] Let $\frak{g}=sl\left( 2\right) $; then the matrices $s\in
\frak{g}$, $$s=\left(
\begin{array}{ll}
s_{3} & s_{1}+is_{2} \\
s_{1}-is_{2} & -s_{3}
\end{array}
\right) ,s_{j}\in \Bbb{C},s_{1}^{2}+s_{2}^{2}+s_{3}^{2}=1, \label{spin}$$ form a coadjoint orbit ${\mathcal{S}}_{1}\subset sl(2)$. Check that ${\mathcal{O}}_{H}=\left\{ z^{-1}\ s,s\in {\mathcal{S}}_{1}\right\} \subset L\frak{g}\simeq L\frak{g}^{*}$ is a coadjoint orbit of $L\frak{g}_{+}\subset L\frak{g}_{r}$. The corresponding orbit $\mathbf{O}_{H}\subset \mathbf{g}$ is parametrized by a triple of $2\pi $-periodic functions $s_{j},j=1,2,3$, satisfying the constraint . The associated linear differential operator is $$\label{wh}
\frac{d}{dx}-\frac{1}{z}s\left( x\right) .$$ One can show that the simplest local Hamiltonian associated with is $$H=-\frac{1}{2}\mathrm{tr}\int s_x^2\, dx;$$ the corresponding nonlinear equation describes the Heisenberg ferromagnet: $$s_t=\left[s,s_{xx}\right].$$
\[Schroed\]Set $$\sigma =\left(
\begin{array}{ll}
1 & 0 \\
0 & -1
\end{array}
\right) .$$ The matrices $$U=\left(
\begin{array}{ll}
0 & u \\
v & 0
\end{array}
\right) +z\sigma ,u,v\in \Bbb{C},$$ form a coadjoint orbit of the Lie algebra $L\frak{g}_{-}\subset L\frak{g}_{r} $. The corresponding orbit ${\mathbf{O}}_{S}\subset {\mathbf{g}}$ is parametrized by a pair of functions $u,v\in C^{\infty }(S^{1}, \C )$; the associated linear operator is $$\frac{d}{dx}-U;$$ this is (essentially) the Lax operator for the so called split nonlinear Schroedinger equation[^10] $$i\partial_t u=-u_{xx}+u^2v,\\
-i\partial_t v=-v_{xx}+v^2u.$$
In a more general way, for $\frak{g}=\frak{gl}(n)$ we can easily construct coadjoint orbits associated with linear differential operators of the form $$\frac{d}{dx}-U\left( x,z\right) , \label{lin}$$ where the potential $U$ is a Laurent polynomial in $z$.
Examples , show that our approach is indeed reasonable: starting with double loop algebras, we arrived at the natural auxiliary linear problems with spectral parameter of the form which is familiar in spectral theory. The next question is to construct an appropriate class of Hamiltonians. Formally, the Hamiltonians are spectral invariants of the auxiliary linear operator, i.e., the spectral invariants of its monodromy matrix. The monodromy matrix $M_{U}$ of is a holomorphic function in ${\C}P_{1}\backslash \left( \left\{ 0\right\}
\cup \left\{ \infty \right\} \right) $; any functional of the monodromy which is invariant under conjugation gives rise to a zero curvature equation on the coadjoint orbits of ${\mathbf{g}}_{r}$. The mapping ${\mathbf{M}}:U\rightsquigarrow M_{U}$ is the *direct spectral transform* for . We may regard $\mathbf{M}$ as a mapping from $\mathbf{g}$ into the group of matrix-valued functions which are regular in the punctured Riemann sphere. When the Poisson structure is *ultralocal* (i.e., the r-matrix is skew with respect to the inner product in $\mathbf{g}$), the spectral transform $\mathbf{M}$ has an important property: the target space carries a natural Poisson bracket and $\mathbf{M}$ is a Poisson mapping (i.e., it preserves Poisson brackets). We shall briefly outline this computation, since it plays an important role in the theory and brings into play an important class of Poisson brackets (the so called *Sklyanin brackets*). To simplify the notation, let us start with ordinary loop algebras.
Let $\frak{g}=\frak{gl}(n)$, and $r\in {\mathrm {End}}\,\frak{g}$ a linear operator which satisfies the modified Yang-Baxter identity and is skew symmetric with respect to the inner product on $\frak{g}$. Let $\frak{G}=C^{\infty }(S^{1},\frak{g})$; we equip $\frak{G}$ with the inner product ; the Poisson bracket of functionals on $\frak{G}^{*}\simeq \frak{G}$ is given by $$\left\{ \varphi _{1},\varphi _{2}\right\} _{r}\left[ L\right]
=\int_{0}^{2\pi }{\rm tr}\,
\left( \left[
{\mathrm {grad}}\,\varphi _{1}\left[ L\right] \left( x\right) ,{\mathrm {grad}}\,\varphi _{2}\left[ L\right] \left( x\right) \right] _{r}L\left( x\right)
\right) dx. \label{rbrack}$$
For $L\in \frak{G}$ let $\psi _{L}$ be the fundamental solution of normalized by $\psi _{L}(0) =id$ and $M_{L}\in GL(n)$ the monodromy matrix. Fix a smooth function $\varphi \in
C^{\infty }\left( GL(n) \right) $ and put $h_{\varphi }\left[
L\right] =\varphi \left( M_{L}\right) $. Let $\nabla \varphi ,\nabla
^{\prime }\varphi \in \frak{g}$ be the *left* and *right gradients* of $\varphi $ on $G=GL\left( n\right) $ defined by $$\frac{d}{ds}_{s=0}\varphi (e^{sX}x)={\mathrm{tr\,}}\left( X\cdot \nabla
\varphi \left( x\right) \right) ,\;\frac{d}{ds}_{s=0}\varphi (xe^{sX})={\mathrm {tr}}\,
\left( X\cdot \nabla ^{\prime }\varphi \left( x\right) \right) ,X\in \frak{g}.$$
The Frechet derivative of the functional $h_{\varphi}$ is given by $${\mathrm {grad}}\,h_{\varphi }\left[ L\right] (x)=\psi \left( x\right) \nabla
^{\prime }\varphi \left( M_{L}\right) \psi \left( x\right) ^{-1}.
\label{varder}$$
The Frechet derivative satisfies the differential equation $$\frac{dX}{dx}=\left[ L,X\right] \label{grad}$$ and the boundary conditions $$X\left( 0\right) =\nabla ^{\prime }\varphi \left( M_{L}\right) ,X\left( 2\pi
\right) =\nabla \varphi \left( M_{L}\right) . \label{bound}$$
*Sketch of a proof.* Taking variation of the both sides of , we get $$\partial _{x}\delta \psi _{L}=L\delta \psi _{L}+\left( \delta L\right)
\psi _{L}. \label{v}$$ Set $\delta \psi _{L}=\psi _{L}Y$, where $Y$ is an unknown function, $Y\in
C^{\infty }\left( {\R},\frak{g}\right) ,Y\left( 0\right) =0$; yields $\partial _{x}Y=\psi ^{-1}\delta L\psi $, whence $$Y\left( x\right) =\int_{0}^{x}\psi _{L}^{-1}\left( y\right) \delta L\left(
y\right) \psi _{L}\left( y\right) \,dy.$$ Since $M_{L}=\psi _{L}\left( 2\pi \right) $, we get $$M_{L}^{-1}\delta M_{L}=\int_{0}^{2\pi }\psi _{L}^{-1}\cdot \delta L\cdot
\psi _{L}\,dy.$$ Now, $$\delta \varphi \left( M_{L}\right) ={\mathrm {tr}}\,
\left( \nabla ^{\prime }\varphi \left( M_{L}\right) \cdot M_{L}^{-1}\delta
M_{L}\right) = \int_{0}^{2\pi } {\mathrm {tr}} \,
\psi _{L}(y) \nabla ^{\prime }\varphi \left( M_{L}\right) \psi
_{L}^{-1}(y) \cdot \delta L(y) \,dy,$$ which implies . Taking derivatives of the both sides of yields .
The Poisson bracket of two functionals $h_{\varphi _{1}},h_{\varphi _{2}}$ on $\frak{G}_{r}^{*}$ is given by $$\left\{ h_{\varphi _{1}},h_{\varphi _{2}}\right\} \left[ L\right]
=h_{\left\{ \varphi _{1},\varphi _{2}\right\} }\left[ L\right] , \label{b}$$ where the Poisson bracket of $\varphi _{1},\varphi _{2}\in C^{\infty }\left(
G\right) $ is defined by $$\left\{ \varphi _{1},\varphi _{2}\right\} _{G}=\frac{1}{2}{\mathrm {tr}}\,
\left( r\left( \nabla \varphi _{1}\right) \nabla \varphi _{2}-r\left( \nabla
^{\prime }\varphi _{1}\right) \nabla ^{\prime }\varphi _{2}\right) .
\label{skl}$$
Let us equip the group $G=GL\left( n\right) $ with the Poisson bracket . Then the monodromy map $\mathbf{M}:\frak{G}_{r}^{*}\rightarrow
G:L\rightsquigarrow M_{L}$ preserves Poisson brackets.
Set $X_{i}={\mathrm {grad}}\,h_{\varphi _{i}}$, $i=1,2$. We have $$\arr{2.0}{ll}{\ds
\left\{ h_{\varphi _{1}},h_{\varphi _{2}}\right\} \left[ L\right]
=\int_{0}^{2\pi }{\mathrm {tr}}\,
\left( \left[ X_{1},X_{2}\right] _{r}L\right) dx \\
\qquad \ds =\frac{1}{2}\int_{0}^{2\pi }{\mathrm {tr}}\,
\left( \left[ rX_{1},X_{2}\right] +\left[ X_{1},rX_{2}\right] \right) L\,dx
\\
\qquad \ds =\frac{1}{2}\int_{0}^{2\pi }{\mathrm {tr}}\,
\left( \left[ L,X_{2}\right] rX_{1}+\left[ L,X_{1}\right] rX_{2}\right) \,dx
\\
\qquad \ds =\frac{1}{2}\int_{0}^{2\pi }\frac{d}{dx}{\mathrm {tr}}
\left( rX_{1}\cdot X_{2}\right) \,dx,}$$ where we used the definition of the r-bracket, the cyclic invariance of trace, the differential equation satisfied by $X_{i}$ and, finally, the skew symmetry of $r$. Evaluating the last integral and taking into account the boundary conditions for $X_{i}$, we get .
Formula defines a remarkable Poisson bracket (*the Sklyanin bracket*) on the group manifold itself. The Jacobi identity for this bracket is not obvious (though it follows from our computation). Its properties will be discussed in some detail in Section 9.
Show that central functions on $G$ (i.e., functions which satisfy $\varphi
\left( xy\right) =\varphi \left( yx\right) $ for all $x,y\in G)$ commute with respect to the Sklyanin bracket.
The above discussion applies to loop algebra $\frak{G}=C^{\infty }(S^{1},\frak{g})$; the generalization to the double loop algebra is straightforward: in our computation, we must replace the finite dimensional algebra $\frak{g}$ with its loop algebra $L\frak{g}$; accordingly, smooth functions $\varphi _{1},\varphi _{2}\in C^{\infty }\left( G\right) $ are replaced by *smooth functionals* on the corresponding loop group, their left and right gradients are replaced by the left and right variational derivatives, etc. Spectral invariants of the auxiliary linear problem correspond to *central functionals* on the loop group. An example of such a functional is given by *evaluation functionals* $H_{n,w}\left[ U\right] ={\mathrm {tr}}\,
M_{U}^{n}(w), w\in \Bbb{C}P_{1}\backslash \left( \left\{ 0\right\} \cup
\left\{ \infty \right\} \right) $.
Compute the variational derivative of $H_{n,w}$ with respect to $U$.
The fundamental drawback of these functionals is, however, their *nonlocality.* The description of *local* functionals is outlined in the next paragraph.
In contrast to evaluation functionals, local conservation laws are related to the asymptotic expansions of the monodromy matrix at its essential singularities, i.e., for $z=0,\infty $. This implies some additional difficulties:
1. These functionals are not defined everywhere on the double loop algebra.
2. The associated formal series are in most cases divergent.
Let us assume that the potential $U\left( x,z\right) $ in the auxiliary linear problem is a Laurent polynomial, $$U=\sum_{-N}^{M}U_{k}z^{k},U_{k}\in C^{\infty }\left( S^{1},\frak{gl}(n)\right) ;$$ let $J_{0}=U_{-N},J_{\infty }=U_{M}$ be its lowest and highest coefficients..
\[reg\]$U$ is called *regular* if
1. the matrices $J_{0}\left( x\right) ,J_{\infty }\left( x\right) $ are semisimple,
2. the centralizers of $J_{0}(x),J_{\infty }(x)$ in $\frak{g}=\frak{gl}(n)$ are conjugate for all $x\in S^{1}$.
We have seen that the set of Laurent polynomials of fixed degree is a Poisson subspace for the r-bracket. It is easy to check that the regularity condition holds for entire coadjoint orbits in this subspace, and hence it is a characteristic of our phase space. Our next theorem allows to construct for regular Lax operators two series of local Hamiltonians which are associated with the poles of $U$ on ${\C}P_{1}$. For concreteness, we shall describe the construction of the series associated with the pole at infinity. Performing a suitable gauge transformation we may assume that the leading coefficient at infinity $J_{\infty }(x)$ satisfies a stronger condition:
*(ii*$^{\prime }$*) the centralizer of* $J_{\infty }$* in* $\ \frak{g}=\frak{gl}(n)$* is a fixed subalgebra* $\frak{g}^{J_{\infty
}}\subset \frak{g}$* which does not depend on* $x$.
(Note that since, by construction, local Hamiltonians are gauge invariant, this stronger condition does not restrict generality.) Let $\frak{g}_{J_{\infty }}\subset \frak{g}^{J_{\infty }}$ be the commutant of $\frak{g}^{J_{\infty }}$, $$\frak{g}_{J_{\infty }}=\left\{ X\in \frak{g};\left[ X,Y\right] =0\hbox{ for
all }Y\in \frak{g}^{J_{\infty }}\right\} .$$
\[NF\](On normal form at infinity) There exists a formal gauge transformation $$\Phi ^{\infty }=Id+\sum_{m=1}^{\infty }\Phi _{m}z^{-m},\Phi _{m}\in
C^{\infty }\left( S^{1},Mat\left( n\right) \right) ,$$ which transforms the differential operator $\partial _{x}-U$ into normal form, $$\left( \Phi ^{\infty }\right) ^{-1}\circ \left( \frac{d}{dx}-U\right) \circ
\Phi ^{\infty }=\frac{d}{dx}-D^{\infty }, \label{inter}$$ where $$D^{\infty }=\sum_{m=-M}^{\infty }D_{m}^{\infty }z^{-m},\,D_{m}^{\infty }\in
C^{\infty }\left( S^{1},\,\frak{g}^{J_{\infty }}\right) ,\,D_{-M}^{\infty
}=J_{\infty };$$ matrix coefficients of $\Phi _{m}^{\infty },\,D_{m}^{\infty }$ are expressed as polynomials of the coefficients of $U$ and its derivatives in $x$. [^11]
*Sketch of a proof.* The intertwining relation $\left( \ref{inter}\right) $ is equivalent to the differential equation $$\left( \frac{d}{dx}-U\right) \Phi ^{\infty }=-\Phi ^{\infty }D^{\infty };
\label{entre}$$ which may be solved recurrently in powers of the local parameter $z^{-1}$. The first nontrivial coefficients $\Phi _{1},D_{-M+1}$ satisfy $$J_{\infty }\Phi _{1}^{\infty }-\Phi _{1}^{\infty }J_{\infty
}=D_{-M+1}^{\infty }-U_{M-1}. \label{recur}$$ This equation for $\Phi _{1}$ admits a solution if and only if the r.h.s. is in the image of $ad\,J_{\infty }\in
End\,\frak{g}$. Assumption (i) above implies that $$\frak{g}=\mathrm{Im}\,ad\,J_{\infty }{\dot{+}}\ker ad\,J_{\infty },$$ moreover, by (ii$^{\prime }$) $\mathrm{Im}\,ad\,J_{\infty }$ and $\ker
ad\,J_{\infty }=\frak{g}^{J_{\infty }}$ do not depend on $x$. Hence, $D_{-M+1}^{\infty }\in \frak{g}^{J_{\infty }}$ is uniquely determined from the solvability condition of $\left( \ref{recur}\right) $ and $$\Phi _{1}^{\infty }=\left( ad\,J_{\infty }\right) ^{-1}\left(
D_{-M+1}^{\infty }-U_{M-1}\right) .$$ If the coefficients $\Phi _{1}^{\infty },...,\Phi _{m}^{\infty
},D_{-M+1}^{\infty },...,D_{-M+m}^{\infty }$ are already determined, we get for $\Phi _{m+1}^{\infty }$ the relation of the form $$ad\,J_{\infty }\cdot \Phi _{m+1}^{\infty }=-F_{m}\left( U,\Phi _{1}^{\infty
},...,\Phi _{m}^{\infty },D_{-M+1}^{\infty },...,D_{-M+m}^{\infty }\right) ,
\label{rec}$$ where $F_{m}$ depends on $U$ and on the already determined coefficients and their derivatives. By the same argument, allows to determine $D_{-M+m+1}^{\infty },\Phi _{m+1}^{\infty }$.
The coefficients $\Phi _{m}^{\infty },D_{m}^{\infty }$ are determined from not completely canonically, since we must fix in some way the operator $\left( ad\,J_{\infty }\right) ^{-1}$. One can show that this freedom corresponds to the possibility to perform gauge transformations $$\begin{aligned}
\frac{d}{dx}-D^{\infty } &\rightsquigarrow &\exp \left( -\phi \right) \circ
\left( \frac{d}{dx}-D^{\infty }\right) \circ \exp \phi , \label{jau} \\
\phi &=&Id+\sum_{m=1}^{\infty }\phi _{m}z^{-m},\phi _{m}\in \frak{g}^{J_{\infty }}. \nonumber\end{aligned}$$
The formal series $\Phi ^{\infty }$ is sometimes called the *formal Baker function at infinity* of the operator $L=\partial _{x}-U$. For $\alpha
\in \frak{g}_{J_{\infty }}\otimes \Bbb{C}\left[ z,z^{-1}\right] $ and put $$H_{\alpha }^{\infty }\left[ U\right] ={\mathrm {Res}}\,_{z=0}\int_{0}^{2\pi }{\mathrm {tr}}\,
\alpha \left( z\right) D^{\infty }(x,z)\,dx. \label{halpha}$$
\[zer\]
1. Functionals $H_{\alpha }^{\infty }$ do not depend on the freedom in the definition of the normal form.
2. All functionals $H_{\alpha }^{\infty }$ are in involution with respect to the Poisson bracket on ${\mathbf{g}}_{r}^{*}$.
3. Hamiltonian equation of motion defined by $H_{\alpha }^{\infty }$ on ${\mathbf{g}}_{r}^{*}$ have the form of zero curvature equations.
Gauge transformations leave the density ${\mathrm {tr}}\,\alpha \left( z\right) D^{\infty }(x,z)$ invariant up to a total derivative.
*Sketch of a proof.* Gauge transformations map $D^{\infty }$ into $e^{-\phi} D^{\infty }e^{\phi} - e^{-\phi}\partial _{x} e^{\phi} $. By a standard formula, $$\partial _{x}\left( \exp \phi \right)=
\frac{e^{-ad\phi}-Id}{-ad\phi}\cdot \partial_x\phi=\\
\left(Id-\frac{1}{2}ad\phi+\frac{1}{3!}(ad\phi)^2+...\right)\cdot\partial_x\phi.$$ Hence $$\begin{aligned}
{\mathrm {tr}}\,
\alpha \exp \left( -\phi \right) \partial _{x}\left( \exp \phi \right) &=&{\mathrm {tr}}\,
\alpha \cdot \left( \frac{e^{-ad\phi }-Id}{-ad\phi }\cdot \partial _{x}\phi
\right) = \\
{\mathrm {tr}}\,
\left( \frac{e^{-ad\phi }-Id}{-ad\phi }\cdot \alpha \right) \cdot \partial
_{x}\phi &=&{\mathrm {tr}}\,
\alpha \cdot \partial _{x}\phi =\partial _{x}\left(
{\mathrm {tr}}\,
\alpha \phi \right) ;\end{aligned}$$ where we also used the invariance of trace, the condition $\alpha \in \frak{g}_{J_{\infty }}$ which assures that it commutes with $\phi $ and, finally, the condition $\partial _{x}\alpha =0$.
The Frechet derivative of $H_{\alpha }^{\infty }$ is given by $${\mathrm {grad}}\,H_{\alpha }^{\infty }=\Phi ^{\infty }\alpha \left( \Phi ^{\infty }\right)
^{-1}, \label{freche}$$ where $\Phi ^{\infty }$ is the formal Baker function.
*Sketch of a proof.* Taking variations of both sides of (\[entre\]), we get $$\delta D^{\infty }=\left( \Phi ^{\infty }\right) ^{-1}\delta U\Phi ^{\infty
}+\left[ D^{\infty },\left( \Phi ^{\infty }\right) ^{-1}\delta \Phi ^{\infty
}\right] -\partial _{x}\left( \left( \Phi ^{\infty }\right) ^{-1}\delta \Phi
^{\infty }\right) .$$ Hence $$\begin{aligned}
\lefteqn{\delta H_{\alpha }^{\infty }= } \\
&&{\mathrm {Res}}\,_{z=0}\int_{0}^{2\pi }\left\{
{\mathrm {tr}}\,\Phi ^{\infty }\alpha \left( \Phi ^{\infty }\right) ^{-1}\delta U+{\mathrm {tr}}\,\left( \partial _{x}\alpha -\left[ D^{\infty },\alpha \right] \right) \left(
\Phi ^{\infty }\right) ^{-1}\delta \Phi ^{\infty }\right\} dx,\end{aligned}$$ where we used the invariance of trace and integrated by parts; the contribution of the second term vanishes, since $\partial _{x}\alpha =\left[
D^{\infty },\alpha \right] =0$. $\Box $
The Frechet derivative $X={\mathrm {grad}}\, H_{\alpha }^{\infty }$ satisfies the differential equation $$\partial _{x}X=\left[ U,X\right] . \label{di}$$
Indeed, and imply $$\partial _{x}X=\left( U\Phi ^{\infty }-\Phi ^{\infty }D^{\infty }\right)
\alpha \left( \Phi ^{\infty }\right) ^{-1}
-\Phi ^{\infty }\alpha \left( \Phi ^{\infty }\right) ^{-1}\left( U\Phi
^{\infty }-\Phi ^{\infty }D^{\infty }\right) \left( \Phi ^{\infty }\right)
^{-1}$$ $$=\left[ U,X\right] -\Phi ^{\infty }\left[ D^{\infty },\alpha \right] \left(
\Phi ^{\infty }\right) ^{-1}=\left[ U,X\right] .$$ Note that geometrically is equivalent to $$ad\,_{\widehat{\mathbf{g}}}^{*}\,
{\rm grad}\,H_{\alpha }^{\infty }\left[ U\right] \cdot U=0, \label{casim}$$ where $ad\,_{\widehat{\mathbf{g}}}^{*}$ is the coadjoint representation of the Lie algebra $\widehat{\mathbf{g}}$ (the central extension of $\mathbf{g}$); this is precisely the property which characterizes the Casimirs of a Lie algebra (cf. proposition ). In the present setting $H_{\alpha
}^{\infty }$ is not a true Casimir function: it is defined only for regular elements $U\in \mathbf{g}$ with fixed highest coefficient. However, a short inspection of the proof of theorem shows that it uses only ; the last assertion of theorem now follows.
In a similar way, we may define the second series of local Hamiltonians which is associated with the pole at $z=0$; one can show that the Hamiltonians from these two families mutually commute (this does not follow immediately from the arguments above, but may be proved in a similar way). In a more general way, if the potential $U$ is a rational function on ${\C}P_{1}$(cf. Section 7.2), we may associate a series of local Hamiltonians to each of its poles; the corresponding Frechet derivatives are formal Laurent series in local parameter at the pole.
In applications, it is quite common to deal with Lax representations which contain higher order differential operators; the most famous example is the KdV equation associated with the Schroedinger operator on the line $$D_{2}=-\frac{d^{2}}{dx^{2}}+u\left( x\right) .$$ In order to put these operators into our framework, one needs extra work. We shall outline the procedure without going into details. First of all, an $n$-th order differential equation $$D_{n}=\frac{d^{n}\psi }{dx^{n}}+u_{n-2}\frac{d^{n-2}\psi }{dx^{n-2}}+...+u_{0}\psi +z\psi =0 \label{sc}$$ may be written as a first order matrix equation, $$\frac{d}{dx}\varphi +L\varphi =0, \label{first}$$ where $$L=\left(
\begin{array}{llll}
0 & 1 & ... & 0 \\
\vdots & \ddots & \ddots & \vdots \\
0 & \ddots & 0 & 1 \\
u_{0}+z & \ldots & u_{n-2} & 0
\end{array}
\right) ,\ \varphi =\left(
\begin{array}{l}
\psi \\
\psi ^{\prime } \\
\vdots \\
\psi ^{\left( n-1\right) }
\end{array}
\right) . \label{n}$$ However, the companion matrix in (\[n\]) contains “too much zeros” and cannot be directly associated with a coadjoint orbit of the loop algebra. Let us observe first of all that choosing the column vector $\varphi $ in (\[n\]) in this particular form is not quite canonical: we may add to $\psi ^{\left( k\right) }$, $k=1,2,...,n-1$, an arbitrary linear combination (possibly, with variable coefficients) of $\psi
,\psi ^{\prime },...,\psi ^{\left( k-1\right) }$; this freedom amounts to a gauge transformation $\varphi \left( x\right) \mapsto n\left( x\right)
\cdot \varphi \left( x\right) $, where $n$ is a lower triangular (unipotent) matrix. The potential $L$ in (\[n\]) becomes an arbitrary matrix of the form $$L=\left(
\begin{array}{lllll}
\ast & 1 & 0 & \cdots & 0 \\
\ast & * & 1 & \ddots & \vdots \\
\vdots & \ddots & * & \ddots & 0 \\
& & & \ddots & 1 \\
\ast & \cdots & & * & *
\end{array}
\right) \label{tri}$$ The companion matrix in (\[n\]) is the result of *gauge fixing;* indeed, we have
For each potential $L$ of the form (\[tri\]) there exists a unique lower triangular gauge transformation which transforms it into the canonical form (\[n\]).
Conclusion: The space of n-th order differential operators is the *quotient space of the set of all potentials $ L $ of the form modulo the gauge action of the lower triangular group*. This quotient space is modelled on the set of companion matrices. With a little skill in symplectic geometry one may describe this quotient space in terms of Hamiltonian reduction (the key point is to observe that potentials of the form (\[tri\]) form a level surface for the moment map associated with our gauge action).
There is one more difficulty: the term of highest degree in $z$ in the potential $L\left( z\right) $ is a *nilpotent matrix*, and so the expansion procedure which yields local integrals of motion does not work. To tackle with this trouble let us recall that the loop parameter $z$ is in fact associated with a special grading *(the standard grading)* of the loop algebra; in this grading, the constant matrix in (\[first\]) is $$\left(
\begin{array}{lllll}
0 & 1 & 0 & \cdots & 0 \\
& 0 & 1 & \ddots & \vdots \\
\vdots & & \ddots & \ddots & 0 \\
0 & & & \ddots & 1 \\
z & \cdots & & 0 & 0
\end{array}
\right) . \label{circ}$$ If we use the so called *principal grading* of the loop algebra instead, (\[circ\]) is replaced, after rescaling, with $$\zeta \cdot \left(
\begin{array}{lllll}
0 & 1 & 0 & \cdots & 0 \\
& 0 & 1 & \ddots & \vdots \\
\vdots & & \ddots & \ddots & 0 \\
0 & & & \ddots & 1 \\
1 & \cdots & & 0 & 0
\end{array}
\right) ,$$ which is already a semisimple matrix with different eigenvalues; now local conservation laws may constructed by expansion in $\zeta ^{-1}$ in the usual way.
Formula (\[freche\]) for the Frechet derivative of a local Hamiltonian makes it clear that the direct use of the global factorization theorem (theorem \[fact\]) to solve zero curvature equations is impossible. Indeed, ${\mathrm {grad}}\, H_{\alpha }^{\infty }\left[ U\right] $ is a *formal* series in local parameter $z^{-1}$; it is therefore impossible even to define the 1-parameter subgroup $\exp t\,{\mathrm {grad}}\,\/H_{\alpha }^{\infty }\left[ U\right] $; this reflects real analytic difficulties which exist in the study of the initial value problem with arbitrary initial data for integrable PDE’s. Our way around this difficulty is to introduce the following definition.
A differential operator $\partial _{x}-U$ is called *strongly regular at zero (at infinity)* if $U$ satisfies the conditions imposed in definition \[reg\] and, moreover, its formal Baker function at zero (at infinity) is convergent.
One may of course strongly doubt the merits of this definition. The point is, however, that strongly regular operators form a* homogeneous space:* there is a natural action of the loop group (called *dressing transformations*) on the set of potential which preserves strong regularity. Moreover, commuting Hamiltonian flows generated by local Hamiltonians are naturally included into the dressing group as a maximal commutative subgroup. Examples of strongly regular operators include solitonic and finite-band solutions; on the other hand, it is very difficult to give their complete characteristics in local terms (i.e., to tell which initial data in the space $C^{\infty }(S^{1},\frak{G})$ correspond to strongly regular operators). The group action in question was discovered by Zakharov and Shabat [@ZSh] (though at the time they did not notice the composition law), and later by Sato and his school [@Sato]. A subtle question in the theory of dressing transformations is the treatment of boundary conditions. In our discussion above we used *periodic* boundary conditions; the main motivation was the Floquet theorem which gives an accurate description of coadjoint orbits and Casimir functions for the central extension of the loop algebra. Dressing transformations do *not* preserve periodicity.[^12] One more interesting point is the relation between dressing and the Poisson brackets: since the dressing action is defined in terms of the Riemann problem and, on the other hand, the Poisson structure is derived from the classical r-matrix related to the *same* Riemann problem, one anticipates some relation between the two matters. However, the simplest guess appears to be wrong: dressing transformations do *not* preserve Poisson brackets on the phase space of integrable PDE’s. The exact relation is more subtle: the loop group itself carries a natural Poisson bracket, again derived from the classical r-matrix, and dressing is an example of a *Poisson group action* [@rims]. The same Poisson structure on the loop group plays an important role in the study of difference Lax equations, and also yields a semiclassical approximation in the theory of Quantum groups.
In this paragraph we shall discuss the simplest facts about dressing transformations. We start with the motivation of the main definitions.
Let $\Bbb{G}$ be the double loop group consisting of maps $$g:\Bbb{R}\times \Bbb{C}P_{1}\backslash \left( \left\{ 0\right\} \cup \left\{
\infty \right\} \right) \rightarrow GL\left( n\right)$$ which are holomorphic with respect to the 2nd argument. The Lie algebra $\mathbf{g}$ of $\Bbb{G}$ consists of maps $$U:\Bbb{R}\times \Bbb{C}P_{1}\backslash \left( \left\{ 0\right\} \cup \left\{
\infty \right\} \right) \rightarrow \frak{gl}\left( n\right)$$ which are holomorphic with respect to the 2nd argument. Informally, we may call elements of $\Bbb{G}$ *wave functions*. Define the mapping $$p:\Bbb{G}\rightarrow \mathbf{g:}\psi \longmapsto U_{\psi }=\partial _{x}\psi
\cdot \psi ^{-1};$$ $\Bbb{G}$ acts on itself by left multiplications; we have $$g\cdot U_{\psi }\stackrel{\rm def}{=}U_{g\psi }=gU_{\psi }g^{-1}+\partial
_{x}g\cdot g^{-1}, \label{jauge}$$ in other words, left multiplication on $\Bbb{G}$ induces gauge transformations on the set of “potentials” $U$. Conversely, if $U\in \mathbf{g}$, let $\psi
_{U}$ be the fundamental solution of the differential equation on the line $$\frac{d\psi }{dx}=U(x,z)\psi , \label{d}$$ normalized by $\psi _{U}\left( 0\right) =Id$. The mapping $$\mathbf{\psi }:{\mathbf{g}}\rightarrow \Bbb{G}:U\mapsto \psi _{U}$$ is a right inverse of $p$. Let $\mathcal{G}\subset \Bbb{G}$ be the subgroup consisting of maps which do not depend on $x\in {\R}$.
\[loopgr\](i) ${\mathcal{G}}\subset \Bbb{G}$ is the stationary subgroup of the zero potential on the line. (ii) ${\mathcal{G}}^{U}=\psi _{U}{\mathcal{G}}\psi _{U}^{-1}\subset \Bbb{G}$ is the stationary subgroup of $U\in \mathbf{g.}$
Let $\Bbb{G}_{+}\subset \Bbb{G}$ be the subgroup consisting of functions which are regular in ${\C}P_{1}\backslash \left\{ \infty \right\} $ with respect to the second argument, and $\Bbb{G}\_\subset \Bbb{G}$ the subgroup of functions which are regular in ${\C}P_{1}\backslash \left\{ 0\right\} $ and satisfy the normalization condition $g_{-}\left( \infty \right) =Id$. The *factorization problem* in $\Bbb{G}$ consists in representing $g\in
\Bbb{G}$ as $g=g_{+}g_{-}^{-1},g_{\pm }\in \Bbb{G}_{\pm }$; the first argument of $g(x,z)$ is regarded as a parameter.
Formula $$dr_{\left( x,y\right) }\psi =\left( \psi xy^{-1}\psi ^{-1}\right)
_{+}^{-1}\psi x=\left( \psi xy^{-1}\psi ^{-1}\right) _{-}^{-1}\psi y
\label{dr}$$ defines a *right* group action $dr:\left( {\mathcal{G}}\times {\mathcal{G}}\right) \times \Bbb{G}\longrightarrow \Bbb{G}$.
The definition looks rather exotic; in particular, the composition law for dressing transformations is not at all obvious. Note that the equality in is closely related to the fact that $\psi xy^{-1}\psi ^{-1}\in {\mathcal{G}}^{U_{\psi }}$ and hence the two factors $\left( \psi xy^{-1}\psi ^{-1}\right) _{\pm }$ define the same gauge transformation of the potential $U_{\psi }$. To check the composition law we shall give a geometric interpretation of . To avoid lengthy notation we shall use a model example.
Let $K$ be a group admitting a factorization into product of its subgroups $K_{\pm }$; set $D(K)=K\times K$. Let $K^{\delta }\subset K\times K$ be the diagonal subgroup, $K^{\delta }=\left\{ \left( x,x\right) ;x\in K\right\} $, and $K_{r}=K_{+}\times K_{-}$.
$D(K)=K_{r}\cdot K^{\delta }$; in other words, the factorization problem in $D(K)$, $$\left( x,y\right) =\left( \eta _{+},\eta _{-}\right) \cdot \left( \xi ,\xi
\right) ,\,\eta _{\pm }\in K_{\pm },\,\xi \in K, \label{dd}$$ is uniquely solvable.[^13]
Indeed, we have $$\eta _{\pm }=\left( xy^{-1}\right) _{\pm },\;\xi =\left( xy^{-1}\right)
_{+}^{-1}x=\left( xy^{-1}\right) _{-}^{-1}y.$$
The quotient space $K_{r}\backslash D(K)$ of left coset classes $\mathrm{mod}\,K_{r}$ is modelled on the diagonal subgroup $K^{\delta }$; the projection $\pi :D\left( K\right) \rightarrow K^{\delta }$ is given by $$\pi (x,y)=\left( xy^{-1}\right) _{+}^{-1}x=\left( xy^{-1}\right) _{-}^{-1}y.$$
The group $D(K)$ acts on itself by right translations. Consider the commutative diagram
(150,100)(10,10) (152,110)[(0,0)\[lb\][$D(K)$]{}]{} (150,65)[(0,0)\[lb\][$K_r\backslash D(K)$]{}]{} (157,10)[(0,0)\[lb\][$K$]{}]{} (22,110)[(0,0)\[lb\][$D(K)\times D(K)$]{}]{} (10,65)[(0,0)\[lb\][${K_r\backslash D(K)}\times D(K)$]{}]{} (28,10)[(0,0)\[lb\][$K\times D(K)$]{}]{} (120,115)[(0,0)\[lb\][$m$]{}]{} (165,90)[(0,0)\[lb\][${\pi}$]{}]{} (58,90)[(0,0)\[lb\][${\pi}\times id$]{}]{} (120,16)[(0,0)\[lb\][${\mathrm dr}$]{}]{} (100,112)[(1,0)[45]{}]{} (100,68)[(1,0)[45]{}]{} (100,13)[(1,0)[45]{}]{} (161,105)[(0,-1)[30]{}]{} (53,105)[(0,-1)[30]{}]{} (162,55)[(0,-1)[30]{}]{} (160,55)[(0,-1)[30]{}]{} (53,55)[(0,-1)[30]{}]{} (51,55)[(0,-1)[30]{}]{}
The right action $K\times D(K)\stackrel{dr}{\longrightarrow }K$ induced by the identification of $K^{\delta }\subset D\left( K\right) $ with the coset space $K_{r}\backslash D(K)$ is given by $$dr(x,y):k\longmapsto \left( kxy^{-1}k^{-1}\right) _{+}kx=\left(
kxy^{-1}k^{-1}\right) _{-}ky. \label{dres}$$
A comparison of and explains the mystery around the definition. In our model setting we assumed for simplicity that the factorization problem is globally solvable. In general this is of course not true; however, under reasonable conditions it is solvable on an open dense subset of the big group, and hence the diagonal subgroup may be identified with a “big cell” in the quotient space. Thus the situation is not much different from the treatment of e.g. fractional linear transformations on the line.
Returning back to , note that the diagonal subgroup $\mathcal{G} ^{\delta }\subset D\left( {\mathcal{G}}\right) $ acts by $dr\left( g,g\right):\psi \mapsto \psi g$; this action amounts to a simple change of the normalization of the wave function and does nor affect the potential $U=\partial _{x}\psi \psi ^{-1}$. On the other hand, the subgroup ${\mathcal{G}} _{r}={\mathcal{G}}_{+}\times {\mathcal{G}}_{-}$ preserves the normalization condition $\psi \left( 0\right) =Id$. Hence we may define an action ${\mathcal{G}}_{r}\times
{\mathbf{g}}\rightarrow {\mathbf{g}}$ on the space of potentials with the help of commutative diagram $$\label{DR}
\begin{picture}(100,50)(10,20)
\put(110,70){\makebox(0,0)[lb]{$\Bbb{G}$}}
\put(110,20){\makebox(0,0)[lb]{$\mathbf{g}$}}
\put(10,70){\makebox(0,0)[lb]{${\mathcal {G}}_r\times \Bbb{G}$}}
\put(10,20){\makebox(0,0)[lb]{${\mathcal {G}}_r\times \mathbf{g}$}}
\put(70,75){\makebox(0,0)[lb]{$\mathrm {dr}$}}
\put(118,46){\makebox(0,0)[lb]{$\mathbf{\psi }$}}
\put(32,46){\makebox(0,0)[lb]{${id\times \mathbf{\psi }}$}}
\put(70,28){\makebox(0,0)[lb]{$\mathrm {dr}$}}
\put(45,72){\vector(1,0){55}}
\put(45,23){\vector(1,0){55}}
\put(112,30){\vector(0,1){35}}
\put(25,30){\vector(0,1){35}}
\end{picture}$$ Let ${\mathbf{g}}_{M,N}\subset {\mathbf{g}}$ be the subspace of Laurent polynomials with the degrees of pole at zero (at infinity) not exceeding $M$ (resp., $N)$
Dressing action on $\mathbf{g}$ preserves ${\mathbf{g}}_{M,N}$.
*Sketch of a proof.* Compare two equivalent formulae for dressing which follow from ; the first one shows that dressing does not increase the degree of pole at $0$, the second one, that it does not affect infinity.
This argument explains the key idea of the dressing method: indeed, the most striking property of dressing is the fact that it preserves the structure of poles of the Lax operator. A slight refinement of the same argument shows that *dressing preserves symplectic leaves of the r-bracket* in ${\mathbf{g}}_{M,N}\subset {\mathbf{g}}\simeq {\mathbf{g}}_{r}^{*}$ (here $r$ is the standard r-matrix associated with the factorization problem in $\Bbb{G}$).
Dressing preserves strong regularity.
*Sketch of a proof.* Formal Baker functions at zero and at infinity of the dressed operator are given by $$\begin{aligned}
\Phi _{0}^{g} &=& \left( \psi g_{+}g_{-}^{-1}\psi ^{-1}\right) _{+}^{-1}\Phi
_{0}, \\
\Phi _{\infty }^{g} &=& \left( \psi g_{+}g_{-}^{-1}\psi ^{-1}\right)
_{-}^{-1}\Phi _{\infty }.\end{aligned}$$ Clearly the gauge factors $\left( \psi g_{+}g_{-}^{-1}\psi ^{-1}\right)
_{+}^{-1},\left( \psi g_{+}g_{-}^{-1}\psi ^{-1}\right) _{-}^{-1}$ expand into convergent series in local parameter around zero and infinity, respectively; hence the same is true for the dressed wave functions.
In applications, dressing is usually applied to *trivial,* or free, Lax matrices. Let us assume that the leading coefficient at infinity is a diagonal matrix with distinct eigenvalues. By definition, free Lax operator has the form $$L_{free}=\frac{d}{dx}-D(z),$$ where $D(z)$ is a constant diagonal matrix (with coefficients which are polynomial in $z)$. Our next assertion shows that the factorization theorem survives for regular potentials; moreover, the dynamical flows associated with all Lax equations (derived from a given Lax operator) correspond to the action of an abelian subgroup of the “big” dressing group (essentially, the group of diagonal loops which are regular at infinity).
Assume that $L$ is obtained from $L_{free}$ by dressing, $L=L_{free}^{g}$. The integral curve of the Hamiltonian equation of motion with the Hamiltonian $H_{\alpha }$ defined (\[halpha\]) which starts at $L$ is given by $$L(t) =g_{\pm }(t) ^{-1}\circ L\circ g_{\pm }(t) ,$$ where $g_{\pm }(t) $ are regarded as multiplication operators on the line and $g_{+}(t,x)$, $g_{-}(t,x)$ are the solutions of the factorization problem[^14] $$\begin{aligned}
g_{+}\left( t,x\right) g_{-}\left( t,x\right) ^{-1} &=& \psi _{free}\left(
x\right) \exp t\alpha \left( z^{-1}\right) \cdot g\cdot \psi _{free}\left(
x\right) ^{-1}, \\
\psi _{free}\left( x\right) &=& \exp xD(z).\end{aligned}$$
We have already mentioned that integrable systems which are associated with *difference operators* require a special treatment; in this case the underlying Poisson structures are *nonlinear*, and hence the geometric setting we considered so far, based on the use of the Lie-Poisson brackets, must be generalized. Nonlinear equations associated with a finite difference operator may be regarded as lattice analogues of zero curvature equations. They are usually written in the form
$$\frac{dL_{m}}{dt}=L_{m}M_{m+1}-M_{m}L_{m},m\in \Bbb{Z}. \label{dlax}$$
Equation is the compatibility condition for the linear system $$\begin{aligned}
\psi _{m+1} &=&L_{m}\psi _{m}, \label{daux} \\
\frac{d\psi _{m}}{dt} &=&M_{m}\psi _{m},\; m\in {\Z}. \nonumber\end{aligned}$$ This system of equations is covariant under the gauge transformations of the form $$\psi_n\mapsto g_m\psi_m,\; L_m\mapsto g_{m+1}L_mg_m^{-1},\;M_m\mapsto g_mM_mg_m^{-1}
+\partial_tg_m\cdot g_m^{-1}.$$
The use of difference operators associated with a 1-dimensional lattice is particularly well-suited for the study of “multi-particle” problems. Let us assume that “potentials” $L_{m}$ are periodic, $L_{m+N}=L_{m}$; the period $N$ may be interpreted as the number of copies of an elementary system. In this way we get families of Hamiltonians which remain integrable for all $N$. The phase spaces for such systems are direct products of “one-particle” phase spaces.
It is natural to suppose that the dynamics associated with difference Lax equations develops on submanifolds of a matrix Lie group $G$ (or of a loop group, if there is a spectral parameter), rather than on Lie algebras or their duals.[^15] As before, we are looking for a geometric theory which should simultaneously account for the Poisson structure of the phase space, the origin of conservation laws, and the reduction of dynamics to factorization problems. An extension of the geometric scheme described in Section 3 to lattice systems is based on the theory of *Poisson Lie groups* introduced by Drinfeld [@dr] following the pioneering work of Sklyanin [@skl].[^16] Unlike the Lie-Poisson brackets discussed before, this new class of Poisson brackets was virtually unknown in geometry.
Very briefly, the motivation for the formal definitions we are going to discuss is as follows. As in the continuous case, the natural Hamiltonians associated with the zero curvature equations should be *gauge invariant.* Let us assume that $L_{m+N}=L_{m}$. Consider the monodromy mapping which assigns to the set of local Lax matrices their ordered product, $$T:G^{N}\rightarrow G:\left( L_{0},...,L_{N-1}\right) \longmapsto T_{L}=\stackrel{\curvearrowleft }{\prod_{k}}L_{k}.$$ A version of the Floquet theorem asserts that two difference operators with periodic coefficients are gauge equivalent if and only if their monodromy matrices are conjugate. Thus one expects the integrals of motion of equation to be of the form $$h_{k}={\rm tr}\,
T_{L}^{k}.$$ This will hold if the monodromy itself satisfies a Lax equation, $$\frac{dT_{L}}{dt}=\left[ T_{L},A_{L}\right] \label{dnov}$$ In more formal terms, let $F_{t}:G^N\longrightarrow G^N $ be the dynamical flow associated with and $\bar{F}_{t}:G\mathbf{\longrightarrow }G$ the corresponding flow associated with ; then the following diagram should be commutative: $$\label{fl}
\begin{picture}(100,50)(10,20)
\put(110,70){\makebox(0,0)[lb]{$G^N$}}
\put(110,20){\makebox(0,0)[lb]{$G$}}
\put(10,70){\makebox(0,0)[lb]{$G^N$}}
\put(10,20){\makebox(0,0)[lb]{$G$}}
\put(60,28){\makebox(0,0)[lb]{${\bar{F_t}}$}}
\put(100,46){\makebox(0,0)[lb]{$T_L$}}
\put(20,46){\makebox(0,0)[lb]{$T_L$}}
\put(60,61){\makebox(0,0)[lb]{${F_t}$}}
\put(23,72){\vector(1,0){80}}
\put(23,23){\vector(1,0){80}}
\put(115,65){\vector(0,-1){35}}
\put(15,65){\vector(0,-1){35}}
\end{picture}$$ We want to equip our phase spaces with Poisson structures which are compatible with all mappings in this diagram. Moreover, we would like to keep to our geometric picture suggested by theorem \[AKS\]; this means that we must find *two* Poisson brackets in each space, so that
1. *Spectral invariants of the monodromy are Casimir functions for the first structure.*
2. *They are in involution with respect to the second one and generate difference Lax equations (respectively, ordinary Lax equations for the monodromy).*
3. *The flows* $F_{t},\bar{F}_{t}$* preserve* intersections* of symplectic leaves for the two brackets.*
4. *Vertical arrows in the diagram (\[fl\]) are Poisson mappings with respect to both structures.*
5. *Finally, the equations of motion (both upstairs and downstairs) should reduce to a factorization problem.*
It is remarkable that all these conditions may be satisfied with the help of a single ingredient, the classical r-matrix, the same one which is responsible for the factorization problem. As compared to the previous case (that of Lie algebras), we need only one extra property (which actually is satisfied in most of the examples we considered beforehand): *the Lie algebra* $\frak{g}$* of our Lie group* $G$* must be equipped with an invariant inner product and the r-matrix* $r\in End\,\frak{g}$*must be a skew-symmetric operator.* The construction which provides Poisson brackets satisfying all these conditions is rather nontrivial (in fact, an important message is that this is possible at all!); it may be naturally divided into two separate problems:
1. *Given an r-matrix,* find the brackets on ${\mathbf{G}}=G^{N}$ and on $G$ which have spectral invariants of the monodromy as their Casimir functions.
2. Find a Poisson bracket on ${\mathbf{G}}=G^{N}$ which yields zero curvature equations as the equations of motion and assures that the monodromy map is compatible with the Poisson brackets.
The key point in both questions is that the r-matrix is fixed *in advance* and we must arrange the brackets with its help (otherwise, there are too many options and the problem is not well posed!).
The second question is better known than the first one; in fact, it is this question that has led to the theory of *Poisson groups.* The key step is the following simplifying assumption:
- *Dynamical variables associated with different factors in ${\mathbf{G}}=\underbrace{G\times ...\times G}_{N}$ commute with each other.*
By induction, the monodromy $T:G^{N}\longrightarrow G$ is a Poisson mapping if *the product map* $m:G\times G\rightarrow G$* preserves Poisson brackets.*
Poisson bracket on a Lie group $G$ satisfying the property above is called *multiplicative;* a Lie group equipped with multiplicative bracket is called a *Poisson Lie group.*
[Let us explain this condition in a more explicit way. Let $\varphi,\psi \in C^{\infty}(G)$; put $\Phi(x,y)=\varphi(xy), \Psi(x,y)=\psi(xy)$, $\Phi,\Psi \in C^{\infty}(G\times G)$. In order to compute the Poisson bracket $\{\Phi, \Psi\}$ we regard them as functions of *two* variables, that is, we compute derivatives of $\Phi, \Psi$ with respect to $x$ for fixed $y$ and with respect to $y$ for fixed $x$ and take the sum of these two terms; on the other hand we may compute the bracket $\{\varphi,\psi \}$ for functions of *one* variable $z\in G$ and then insert $z=xy$. Multiplicativity means that the two results coincide.]{}
Let us assume that the Lie algebra $\frak{g}$ of $G$ carries an invariant inner product and $r\in {\mathrm{End}}\,\frak{g}$ is skew and satisfies the modified Yang-Baxter identity. For $\varphi \in C^{\infty }\left( G\right) $ let $\nabla \varphi ,\nabla ^{\prime }\varphi \in \frak{g}$ be its *left and right gradients* defined by $$\begin{aligned}
\left\langle \nabla \varphi \left( x\right) ,X\right\rangle &=& \left( \frac{d}{ds}\right) _{s=0}\varphi \left( e^{sX}\cdot x\right) , \\
\left\langle \nabla ^{\prime }\varphi \left( x\right) ,X\right\rangle &=&
\left( \frac{d}{ds}\right) _{s=0}\varphi \left( x\cdot e^{sX}\right) ,X\in
\frak{g.}\end{aligned}$$
The bracket on $G\ $$$\left\{ \varphi ,\psi \right\} =\frac{1}{2}\left( \left\langle r\left(
\nabla \varphi \right) ,\nabla \psi \right\rangle -\left\langle r\left(
\nabla ^{\prime }\varphi \right) ,\nabla ^{\prime }\psi \right\rangle
\right) \label{skl1}$$ is multiplicative and satisfies the Jacobi identity.[^17]
Formula coincides with , which we deduced from the study of the monodromy map in the continuous case.
[This is of course not a coincidence. To explain why the Poisson bracket for the monodromy on the circle should be multiplicative let us consider the auxiliary problems with potentials $L$ consisting of two separate patches, so that ${\mathrm supp}\,L$ is the union of two disjoint intervals, ${\mathrm supp}\,L=I'\cup I''$. Let us denote by ${\frak G}_{I'},\;{\frak G}_{I''}$ the set of all potentials supported on $I',\,I''$, respectively; then ${\frak G}_{I'},\;{\frak G}_{I''}\subset{\frak G}$ are Poisson submanifolds, and moreover, ${\frak G}_{I'\cup I''}={\frak G}_{I'}\times {\frak G}_{I''}$, again as Poisson manifolds (which means that $L'\in{\frak G}_{I'},L'\in{\frak G}_{I'}$ may be treated as independent variables with respect to our Poisson structure).[^18] Clearly, for $L=L'+L'',\; L'\in {\frak G}_{I'}, L''\in {\frak G}_{I''}$ we have $M_L=M_{L''}M_{L'}$ and the Poisson bracket for the monodromy matrix may be computed in two different ways: either by computing the Poisson brackets for the monodromy matrices $M_{L''}, M_{L'}$ regarded as independent variables, or alternatively by computing directly the monodromy $M_L$ for the potential $L=L'+L''$ decomposed into two separate patches. The two results should of course coincide, and this means precisely that the Poisson bracket for the monodromies should be multiplicative. ]{}
Note that $\nabla \varphi \left( x\right) =Ad\,x\cdot \nabla ^{\prime
}\varphi \left( x\right) $, or, in the matrix case, $\nabla \varphi \left(
x\right) =x\cdot \nabla ^{\prime }\varphi \left( x\right) \cdot x^{-1}$, so we may rewrite (\[skl1\]) using only left gradients: $$\left\{ \varphi ,\psi \right\} \left( x\right) ={\mathrm {tr}}\,
\left( \eta _{r}\left( x\right) \cdot \left( \nabla \varphi \wedge \nabla
\psi \right) \right) ,$$ where we set $$\eta _{r}\left( x\right) =\frac{1}{2}\left( r-Ad\,x^{-1}\circ r\circ
Ad\,x\right)$$ and identify $\nabla \varphi \wedge \nabla \psi \in \frak{g}\wedge \frak{g}$ with an antisymmetric linear operator on $\frak{g}$, using the inner product. The function $\eta _{r}:\frak{g}\rightarrow \mathrm{End}\,\frak{g}$ satisfies the following remarkable functional equation:[^19] $$\eta _{r}\left( xy\right) =\eta _{r}\left( x\right) +Ad\,x^{-1}\circ \eta
_{r}\left( y\right) \circ Ad\,x, \label{cocy}$$ One may check that this functional equation is exactly equivalent to the multiplicativity of .
Assume that $G=GL\left( n\right) $ is a matrix group. Let us consider “tautological” functions $\phi _{ij}$ on $G$ which assign to a matrix $L\in G$ its matrix coefficients, $\phi _{ij}\left( L\right) =L_{ij}$; clearly, the ring of polynomials ${\C}\left[ \phi _{ij}\right]$ is dense in $C^{\infty }\left( G\right) $, and the Poisson bracket on $G$ is completely specified by its values on the “generators” $\phi _{ij}$. Let us identify $r\in {\mathrm {End}}\,\frak{gl}(n)$ with an element of $\frak{gl}(n)\otimes \frak{gl}(n)\simeq
Mat\left( n^{2}\right) $.
The Poisson bracket of matrix coefficients is given by $$\left\{ \phi _{ij},\phi _{km}\right\} \left( L\right) =\left[ r,L\otimes
L\right] _{ikjm}. \label{matr}$$
The commutator in the r.h.s is computed in $Mat\left( n^{2}\right) $. Usually people do not distinguish $\phi _{ij}$ and its values and write this formula (with suppressed matrix indices!) as
$$\label{rll}
\left\{ L \, \stackrel{\otimes }{,} \, L \right\} =\left[ r, L \otimes L \right] .$$
Formula has served as the original definition of the *Sklyanin bracket.* Note that the r.h.s. in is a *quadratic* expression in matrix coefficients (this is to be compared with the Lie-Poisson bracket which is *linear*).
Let us note some important properties of the Sklyanin bracket.
1. * The bracket is identically zero at the unit element of the group.* (Indeed at $x=e$ right and left gradients coincide).
2. * Linearizing the bracket at the origin of group, we get the structure of a Lie algebra in the cotangent space* $T_{e}^{*}G=\frak{g}^{*}:$ if $\xi ,\zeta \in \frak{g}^{*},X\in \frak{g}$, choose $\varphi ,\psi \in
C^{\infty }\left( G\right) $ in such a way that $\nabla \varphi \left(
e\right) =\xi ,\nabla \psi \left( e\right) =\zeta $ and set $$\left\langle \left[ \xi ,\zeta \right] _{*},X\right\rangle =\left( \frac{d}{ds}\right) _{s=0}\{\varphi ,\psi \}\left( e^{sX}\right) =\left( \frac{d}{ds}\right) _{s=0}{\mathrm {tr}}\,
\left( \eta _{r}\left( e^{sX}\right) \cdot \left( \xi \wedge \zeta \right)
\right)$$ (the second formula checks that the bracket $\left[ \xi ,\zeta \right] _{*}$ does not really depend on the choice of $\varphi ,\psi $ and so is well defined).
\[bial\]The bracket $\left[ \xi ,\zeta \right] _{*}$ coincides with the *r-bracket* (up to the identification of* *$\frak{g}$ and $\frak{g}^{*}$ induced by the inner product[^20]: $$\left[ \xi ,\zeta \right] _{*}=\frac{1}{2}\left( \left[ r\xi ,\zeta \right]
+\left[ \xi ,r\zeta \right] \right) . \label{rr}$$
Up to dualization, coincides with which was our starting point in Section 3. In the present setting we get some extra properties which follow from multiplicativity of the bracket. Set $$\delta _{r}\left( X\right) =\left( \frac{d}{ds}\right) _{s=0}
\eta _{r}\left( e^{sX}\right) .$$ Explicitly, we get $$\delta _{r}\left( X\right) =adX\circ r-r\circ adX.$$
\[coc\](i) We have $${\mathrm {tr}}\,
\left( \delta _{r}\left( X\right) \circ \left( \xi \wedge \zeta \right)
\right) =\left\langle \left[ \xi ,\zeta \right] _{*},X\right\rangle .
\label{du}$$ (ii) The mapping $\delta _{r}:\frak{g}\rightarrow {\mathrm {End}}\,
\frak{g}$ satisfies the functional equation $$\delta _{r}\left( \left[ X,Y\right] \right) =\left[ adX,\delta _{r}\left(
Y\right) \right] -\left[ adY,\delta _{r}\left( X\right) \right] .
\label{cocycle}$$
Equation shows that $\delta _{r}$ is the dual of the commutator map $\frak{g}^{*}\wedge \frak{g}^{*}\rightarrow \frak{g}^{*}$.[^21]
\[big\]A pair $\left( \frak{g},\frak{g}^{*}\right) $ is called a *Lie bialgebra* if (i) $\frak{g}$ and $\frak{g}^{*}$ are set in duality as linear spaces, (ii) both $\frak{g}$ and $\frak{g}^{*}$ are Lie algebras, (iii) the dual of the commutator map $\left[ ,\right] _{*}:\frak{g}^{*}\wedge \frak{g}^{*}\rightarrow \frak{g}^{*}$ satisfies the functional equation .
One can show that (iii) implies that in, the dual way, the mapping $\delta
_{*}:\frak{g}^{*}\rightarrow \frak{g}^{*}\wedge \frak{g}^{*}$ which dualizes the commutator $\left[ ,\right] :\frak{g}\wedge \frak{g}\rightarrow \frak{g}$ is a 1-cocycle on $\frak{g}^{*}$, and so this definition is symmetric.
It is instructive to compare the definitions of Lie bialgebras and of the double Lie algebras introduced in Section 3. These definitions are *different* and use different notions of the classical r-matrix. In the case of double Lie algebras there are two Lie brackets on the *same underlying linear space*; the classical r-matrix is a linear operator $r\in
{\mathrm {End}}\, \frak{g}$; in the case of Lie bialgebras there are two Lie brackets which are defined on *dual spaces* $\frak{g}$ and $\frak{g}^{*}$. The motivation for these definitions are very much different as well: as we saw, double Lie algebras provide a natural setting for the Involutivity theorem (theorem ); Lie bialgebras naturally arise in the study of multiplicative Poisson brackets on Lie groups. Proposition specifies the setting in which these two notions merge together: we must assume that $\frak{g}$ carries an* invariant inner product* which allows to identify $\frak{g}$ and $\frak{g}^{*}$ and that $r\in {\mathrm {End}}\, \frak{g}$ is *skew*. One more natural condition is the *modified Yang-Baxter equation* (which assures that there is an underlying factorization problem). When all three conditions are satisfied, we say that $\left( \frak{g},\frak{g}^{*}\right) $ is a *factorizable Lie bialgebra.* Factorizable Lie bialgebras and the associated Poisson Lie groups provide a natural environment for all applications to lattice integrable systems.
Before turning to lattice zero curvature equations let us discuss ordinary Lax equations on Lie groups. Here is a version of the factorization theorem (theorem ) which applies in this setting. Let $G$ be a matrix Lie group; we assume that the Poisson bracket on $G$ is given by and that its tangent Lie bialgebra $\left( \frak{g},\frak{g}^{*}\right) $ is factorizable. Let $I\left( G\right) \in C^{\infty }\left( G\right) $ be the algebra of central functions on $G$ ($\varphi \in C^{\infty }\left( G\right) $ is central if $\varphi \left( xy\right) =\varphi \left( yx\right) $ for all $x,y\in G$).
\[Gr\](i) All central functions are in involution with respect to the Sklyanin bracket . (ii) Hamiltonian equation on $G$ with Hamiltonian $h\in I\left( G\right) $ may be written in Lax form$\footnote{In this formula the velocity vector $dL/dt$ belongs to the tangent space $T_{L}G$; in order to be more accurate, we may rewrite \reff{la} as an
equality in the Lie algebra:
\[ L^{-1}dL/dt=M_{\pm }-Ad\,L^{-1}\cdot M_{\pm } \]
}$ $$\frac{dL}{dt}=LM_{\pm }-M_{\pm }L, \label{la}$$ where $M_{\pm }=r_{\pm }\left( \nabla h\left( L\right) \right) $. (iii) The integral curve $L\left( t\right) $ of (\[la\]) with $L\left( 0\right)
=L_{0}$ is given by $$L\left( t\right) =g_{\pm }\left( t\right) ^{-1}L_{0}g_{\pm }\left( t\right) ,
\label{sol}$$ where $g_{+}\left( t\right) ,g_{-}\left( t\right) $ are the solutions of the factorization problem in $G$ $$g_{+}\left( t\right) g_{-}\left( t\right) ^{-1}=\exp t\nabla h\left(
L_{0}\right) \label{pr}$$ associated with $r$.
As before, the direct proof of theorem is easy: to check that is an integral curve of just compute the derivative of the r.h.s in . As in Section 3.3, there exists also a geometric proof which explains the background machinery. Below, we shall briefly outline the corresponding construction.
As already noted, Lie bialgebras possess a remarkable symmetry: if $\left( \frak{g},\frak{g}^{*}\right) $ is a Lie bialgebra, the same is true for $\left(
\frak{g}^{*},\frak{g}\right)$. Hence the *dual group* $G^{*}$ (which corresponds to $G^*$) also carries a multiplicative Poisson bracket. In the case of factorizable Poisson groups this dual bracket may be pushed forward to $G$ by means of the factorization map. Thus we get *two* brackets on $G$ which fit into our geometric treatment of Lax equations. The best way to understand this duality is to notice that both $G$ and $G^{*}$ are Poisson subgroups of a bigger Poisson group, the *double of $G$*.
Let $\left( \frak{g},\frak{g}^{*}\right) $be a Lie bialgebra; the linear space $\frak{d}=\frak{g}\oplus \frak{g}^{*}$ carries a natural inner product $$\left\langle \left\langle \left( X,F\right) ,\left( X^{\prime },F^{\prime
}\right) \right\rangle \right\rangle =\left\langle F,X^{\prime
}\right\rangle +\left\langle F^{\prime },X\right\rangle . \label{dub}$$ The following key theorem was discovered by Drinfeld.
There exists a unique structure of the Lie algebra on $\frak{d}$ such that: (i) $\frak{g},\frak{g}^{*}\subset \frak{d}$ are Lie subalgebras. (ii) The inner product is invariant.
Let $P_{\frak{g}},P_{\frak{g}^{*}}$ be the projection operators onto $\frak{g},\frak{g}^{*}$ in the decomposition $\frak{d}=\frak{g}\oplus \frak{g}^{*}$. Set $r_{\frak{d}}=P_{\frak{g}}-P_{\frak{g}^{*}}$; then $r_{\frak{d}}$ defines on $\frak{d}$ the structure of a factorizable Lie bialgebra.
The pair $\left( \frak{d},\frak{d}^{*}\right) $ is called the *Drinfeld double* of $\left( \frak{g},\frak{g}^{*}\right)$. When the initial Lie bialgebra $\left( \frak{g},\frak{g}^{*}\right) $ is itself factorizable, the description of the double is very simple. Consider the Lie algebra $\frak{d}=\frak{g}\oplus \frak{g}$ (direct sum of two copies of $\frak{g}$) and equip it with the inner product $$\left\langle \left\langle \left( X,Y\right) , \left( X^{\prime}, Y^{\prime }\right) \right\rangle \right\rangle = \left\langle X,X^{\prime }\right\rangle
-\left\langle Y,Y^{\prime }\right\rangle , \label{mi}$$ where $\left\langle ,\right\rangle $ is the invariant inner product on $\frak{g}$.
The double of a factorizable Lie algebra is canonically isomorphic to $\frak{d}=\frak{g}\oplus \frak{g.}$
*Sketch of a proof.* We have already seen in Section 4 that there are two natural homomorphisms $r_{\pm }:\frak{g}^{*}\rightarrow \frak{g}$ given by (\[pm\]); their combination yields an embedding $\frak{g}^{*}\subset \frak{g}\oplus \frak{g}$. Let $\frak{g}^{\delta }\subset \frak{g}\oplus \frak{g}$ be the diagonal subalgebra, $\frak{g}^{\delta }=\{(X,X); \, X\in \frak{g}\}$. As discussed in Section 4, ${\frak{d}}={\frak{g}}^{\delta }{\dot{+}}{\frak{g}}^{*}$; it is easy to check that the skew symmetry of $r$ and the choice of the inner product in $\frak{d}$ imply that $\frak{g}^{*}$ and $\frak{g}^{\delta }$ are isotropic with respect to the inner product (\[mi\]); this is equivalent to the skew symmetry of $r_{\frak{d}}=P_{\frak{g}}-P_{\frak{g}^{*}}$. In matrix notation, $r_{\frak{d}}\in {\mathrm {End}}\,\left( \frak{g}\oplus \frak{g}\right)$ is given by a $2\times 2$ block matrix: $$r_{\frak{d}}=\left(
\begin{array}{ll}
r & 2r_{+} \\
2r_{-} & -r
\end{array}
\right) . \label{block}$$
The Lie group which corresponds to $\frak{d}$ is $D\left( G\right) =G\times
G$. Let $G^{\delta },G^{*}$ be its subgroups which correspond to $\frak{g}^{\delta },\frak{g}^{*}$. Clearly, $G^{\delta }\subset D\left( G\right) $ is the diagonal subgroup. As in (\[dd\]), we may associate with the r-matrix $r_{\frak{d}}$ a factorization problem in $D\left( G\right) $. Let us assume for simplicity that it is globally solvable, i.e. $D\left( G\right) \simeq
G\cdot G^{*}$.
Let us equip $D\left( G\right) $ with the Sklyanin bracket associated with $r_{\frak{d}}$. Then $G^{\delta },G^{*}\subset D\left( G\right) $ are Poisson subgroups (i.e. they are Poisson submanifolds and the induced Poisson structure is multiplicative).[^22]
The bracket induced on $G^{\delta }$ coincides with the original Sklyanin bracket associated with $r$; the Poisson bracket on the dual group $G^{*}$ is described in the following way. Consider the mapping $m:D\left( G\right)
\rightarrow G:\left( x,y\right) \mapsto xy^{-1}$; its restriction to $G^{*}\subset D\left( G\right) $ is a diffeomorphism.
The Poisson bracket on $G$ induced by $m:G^{*}\rightarrow G$ is given by $$\begin{aligned}
\left\{ \varphi ,\psi \right\} _{*}&=&1/2\left\langle r\nabla \varphi ,\nabla
\psi \right\rangle +1/2\left\langle r\nabla ^{\prime }\varphi ,\nabla
^{\prime }\psi \right\rangle \nonumber \\
&& -\left\langle r_{+}\nabla \varphi ,\nabla ^{\prime }\psi \right\rangle
-\left\langle r_{-}\nabla ^{\prime }\varphi ,\nabla \psi \right\rangle ,
\label{dualbr}\end{aligned}$$ where $\nabla \varphi ,\nabla \psi $ and $\nabla ^{\prime }\varphi ,\nabla
^{\prime }\psi $ are left and right gradients of $\varphi ,\psi $.
Formula looks rather complicated; however, the bracket $\left\{ ,\right\} _{*}$ is very remarkable.
\[dgroup\](i) Symplectic leaves of $\left\{ ,\right\} _{*}$ coincide with conjugacy classes in $G$. (ii) Casimir functions of $\left\{
,\right\} _{*}$ are precisely the central functions on $G$. (iii) The bracket (\[dualbr\]) vanishes at the unit element $e\in G$; the induced Lie bracket on the tangent space coincides with the original Lie bracket on $\frak{g}$.
Thus the bracket provides the missing ingredient of our geometric picture: we have got now *two Poisson structures* on the same underlying manifold $G$ and Lax equations preserve intersections of two systems of symplectic leaves.
The symplectic leaves of the Sklyanin bracket also admit a description in terms of the factorization problem. Let us identify $G$ with the quotient space $D\left( G\right) /G^{*}$ using the factorization $D\left( G\right)
=G^{\delta }\cdot G^{*}$ Let us denote by $\pi $ the canonical projection $D\left( G\right) \rightarrow D\left( G\right) /G^{*}$; define the action $G^{*}\times G\rightarrow G$ using the commutative diagram
(150,100)(10,10) (152,110)[(0,0)\[lb\][$D(G)$]{}]{} (150,65)[(0,0)\[lb\][$D(G)/G^*$]{}]{} (157,10)[(0,0)\[lb\][$G$]{}]{} (30,110)[(0,0)\[lb\][$G^*\times D(G)$]{}]{} (24,65)[(0,0)\[lb\][$G^*\times D(G)/G^*$]{}]{} (33,10)[(0,0)\[lb\][$G^*\times G$]{}]{} (120,115)[(0,0)\[lb\][$m$]{}]{} (148,90)[(0,0)\[lb\][${\pi}$]{}]{} (23,90)[(0,0)\[lb\][${id\times \pi}$]{}]{} (108,16)[(0,0)\[lb\][$Dress$]{}]{} (100,112)[(1,0)[45]{}]{} (100,68)[(1,0)[45]{}]{} (100,13)[(1,0)[45]{}]{} (161,105)[(0,-1)[30]{}]{} (53,105)[(0,-1)[30]{}]{} (162,55)[(0,-1)[30]{}]{} (160,55)[(0,-1)[30]{}]{} (53,55)[(0,-1)[30]{}]{} (51,55)[(0,-1)[30]{}]{}
(Here $m$ is the group multiplication in $D\left( G\right) $ restricted to the subgroup $G^{*}\subset D\left( G\right) .)$ By analogy with the definition of dressing transformations, this action is called *dressing action.*
Symplectic leaves of the Sklyanin bracket in $G$ coincide with the orbits of $G^{*}$ in $G$ with respect to the dressing action.
[More explicitly, the dressing prescription is as follows: given $x\in G$, $\left( h_{+},h_{-}\right) \in G^{*}$ solve the factorization problem in $D\left( G\right) $;$$\left( h_{+}x,h_{-}x\right) =\left( x^{\prime }g_{+},x^{\prime }g_{-}\right)
,x^{\prime }\in G,\left( g_{+},g_{-}\right) \in G^{*};$$ then $Dress\left( h_{+},h_{-}\right) \cdot x=x^{\prime }$. This immediately yields the following formula in terms of the factorization problem in $G:$$$Dress\left( h_{+},h_{-}\right) \cdot x=h_{+}x\left(
x^{-1}h_{+}^{-1}h_{-}x\right) _{+}=h_{-}x\left(
x^{-1}h_{+}^{-1}h_{-}x\right) _{-}, \label{dract}$$ where $x^{-1}h_{+}^{-1}h_{-}x=\left( x^{-1}h_{+}^{-1}h_{-}x\right) _{+}\cdot
\left( x^{-1}h_{+}^{-1}h_{-}x\right) _{-}^{-1}$ is the factorization in $G$ associated with the original r-matrix.]{}
Check this formula using the definition of dressing action.
Dressing action may be regarded as a nonlinear analog of the coadjoint representation, as it is clear from the following simple assertion.
Dressing action leaves the unit $e\in G$ invariant; the linearization of the dressing action in the tangent space $T_{e}G\simeq \frak{g}$ coincides with the coadjoint representation of $G^{*}$ in $\frak{g}^{**}\simeq \frak{g}$.
In applications it is natural to assume that $G$ is an *algebraic loop group* consisting of matrices whose coefficients are rational functions of $z$. Orbits of the dressing action of $G^{*}$ in this loop group are finite-dimensional, and we get a description of phase spaces for Lax equations which is largely parallel to the case of coadjoint orbits of $G_{r}
$ discussed in Section 6.
One more application of the dual Poisson structure described by is the accurate description of the Poisson properties of the dressing transformations from Section 8.6. We keep to the notation introduced in lemma \[loopgr\]. Let $\mathcal{G}$ be the loop group; its Lie algebra $L\frak{g} =\frak{g}[z,z^{-1}]$ is equipped with the inner product $$\left\langle X,Y\right\rangle ={\mathrm{Res}}_{z=0}{\mathrm{tr}}
\left( X\left(z\right) Y\left( z\right) \right)$$ and with the standard r-matrix $r=P_{+}-P_{-}$ associated with the Riemann problem. The loop group $\mathcal{G}$ equipped with the corresponding Sklyanin bracket becomes a Lie-Poisson group. Let ${\mathcal{G}}_{r}\simeq {\mathcal{G}}^{*} = {\mathcal{G}}_{+}\times {\mathcal{G}}_{-}$ be the dual group equipped with its natural Poisson structure .
The dressing action described by the commutative diagram is a Poisson group action.
(We shall not reproduce the proof here; see [@rims], [@Uhl].)
One more important ingredient of the geometric picture outlined in Section 3.3 is the “big phase space” with “free” dynamical flow. Its counterpart in the present setting is provided by the so called *symplectic double* of $G$. Let again $D\left( G\right) =G\times G$ be the double of $G$; the Sklyanin bracket on $D\left( G\right) $ is given by $$\left\{ \varphi ,\psi \right\} =\left\langle \left\langle r_{\frak{d}}\nabla
\varphi ,\nabla \psi \right\rangle \right\rangle -\left\langle \left\langle
r_{\frak{d}}\nabla ^{\prime }\varphi ,\nabla ^{\prime }\psi \right\rangle
\right\rangle \stackrel{def}{=}\left\{ \varphi ,\psi \right\} ^{g}-\left\{
\varphi ,\psi \right\} ^{d} \label{skl-d}$$ As noticed in footnote \[Ob\], the terms with left and right gradients separately do not satisfy the Jacobi identity; the obstructions cancel when the two are combined together. Explicitly, $$\begin{array}{l}
\{ \{ \varphi _{1},\varphi _{2}\} ^{g},\varphi _{3}\}
^{g}+c.p. = \left\langle \left\langle \left[ \nabla \varphi _{1},\nabla
\varphi _{2}\right] ,\nabla \varphi _{3}\right\rangle \right\rangle ,
\label{Ja} \\
\\
\{ \{ \varphi _{1},\varphi _{2}\} ^{d},\varphi _{3}\}
^{d}+c.p. = -\left\langle \left\langle \left[ \nabla ^{\prime }\varphi
_{1},\nabla ^{\prime }\varphi _{2}\right] ,\nabla ^{\prime }\varphi
_{3}\right\rangle \right\rangle .
\end{array}$$ The two terms cancel, since $\nabla \varphi \left( x\right) =x\cdot \nabla
^{\prime }\varphi \left( x\right) \cdot x^{-1}$ and the inner product is $Ad$-invariant. It is important to notice that the crucial minus sign in is due not to the minus in , but rather to the fact that the action of a group by right translations is its *anti-*representation. (There are no terms of “mixed chirality” in the obstruction, since left and right translations commute with each other.) Thus we get the following assertion:
\[twor\]Let $r,r^{\prime }\in {\mathrm {End}}\,
\frak{d}$ be two arbitrary classical r-matrices satisfying the modified Yang-Baxter equation; the bracket $$\left\{ \varphi ,\psi \right\} _{r,r^{\prime }}=\left\langle \left\langle
r\nabla \varphi ,\nabla \psi \right\rangle \right\rangle +\left\langle
\left\langle r^{\prime }\nabla ^{\prime }\varphi ,\nabla ^{\prime }\psi
\right\rangle \right\rangle \label{two r}$$ satisfies the Jacobi identity.
Specifically, let us take $r=r^{\prime }=r_{\frak{d}}$; the resulting Poisson structure is *nondegenerate* (at least if we assume – as we always do in this Section – that the factorization problem in $D$ is globally solvable), and hence defines a *symplectic structure* on $D\left( G\right) $.
The manifold $D\left( G\right)$ with the Poisson bracket $$\left\{ \varphi ,\psi \right\} _{+}=\left\langle \left\langle
r_{\frak{d}}\nabla \varphi ,\nabla \psi \right\rangle \right\rangle +\left\langle
\left\langle r_{\frak{d}}\nabla ^{\prime }\varphi ,\nabla ^{\prime }\psi
\right\rangle \right\rangle \label{plus}$$ is called the *symplectic double* of $G$. We shall denote it $D\left(
G\right) _{+}$ in order to distinguish it from the *Drinfeld double* equipped with the Sklyanin bracket.
The bracket is *not* multiplicative, so from the point of view of the theory of Poisson groups $D\left( G\right) _{+}$ is not a group! Instead, we have the following property which shows that $D\left( G\right)
_{+}$ is a *principal homogeneous space* for $D\left( G\right) :$
\[poiss\]Left and right multiplication in $D\left( G\right) $ induce Poisson mappings $D\left( G\right) \times D\left( G\right)
_{+}\longrightarrow D\left( G\right) _{+},D\left( G\right) _{+}\times D\left( G\right) \longrightarrow D\left( G\right) _{+}$.
Proposition paves the way to use the reduction technique in our present setting. Let us recall the point of view on reduction adopted in Section 3.3: if $M$ is symplectic and $K\times M\longrightarrow M$ is a group action, the reduction is the natural projection map $\pi
:M\longrightarrow M/K$ onto the space of $K$-orbits in $M$. The key property which we need to get a Poisson bracket on $M/K$ is this: *Poisson bracket of two* $G$*-invariant functions on *$M$* is again* $G$*-invariant*. Let us discuss briefly how can one control this property. For $X\in \frak{k}$ let us denote by $\hat{X}\in Vect\,M$ the vector field on $M$ generated by the 1-parameter transformation group $\exp tX$. We have: $$\varphi \in C^{\infty }\left( M\right) \emph{\ is\ }G\emph{-invariant\ }\Longleftrightarrow \hat{X}\varphi =0\emph{\ for\ all\ }X\in \frak{k.}$$ When vector fields $\hat{X}\in Vect\,M$ are Hamiltonian we have simply $$\hat{X}\{ \varphi ,\psi \} =\{ \hat{X}\varphi ,\psi \}
+\{ \varphi ,\hat{X}\psi \} =0.$$
In the case of Poisson group actions vector fields $\hat{X}$ are no longer Hamiltonian; however, the rate of nonconservation of Poisson brackets by these vector fields may be characterized very sharply. For $\varphi \in
C^{\infty }\left( M\right) ,x\in M$, let us denote by $\xi _{\varphi
}\left( x\right) \in \frak{k}^{*}$ the linear functional defined by $$\left\langle \xi _{\varphi }\left( x\right) ,X\right\rangle =\frac{d}{dt}_{t=0}\varphi \left( \exp tX\cdot x\right) .$$
Let us assume that $K$ is a Poisson Lie group with Lie bialgebra $\left(
\frak{k},\frak{k}^{*}\right) $. The mapping $K\times M\longrightarrow M$ is a Poisson mapping if and only if $$\hat{X}\{ \varphi ,\psi \} - \{ \hat{X}\varphi ,\psi \}
-\{ \varphi ,\hat{X}\psi \} =\left\langle \left[ \xi _{\varphi
},\xi _{\psi }\right] _{*},X\right\rangle .$$
When $\hat{X}\varphi =\hat{X}\psi =0$ for all $X\in \frak{k},\xi
_{\varphi }=\xi _{\psi }\equiv 0$, and hence $\hat{X}\left\{ \varphi ,\psi
\right\} =0$, which assures the possibility of reduction. In a more general way, let us say that a subgroup $H\subset K$ is *admissible* if $\hat{X}\varphi =\hat{X}\psi =0$ for all $X\in \frak{h}\Longrightarrow \hat{X}\left\{ \varphi ,\psi \right\} =0$.
$H\subset K$ is admissible $\Leftrightarrow \frak{h}^{\bot }\subset
\frak{k}^{*}$ is a Lie subalgebra.
*Sketch of a proof.* $\hat{X}\varphi =\hat{X}\psi =0$ for all $X\in
\frak{h}$ implies that $\xi _{\varphi },\xi _{\psi }\in \frak{h}^{\bot }$. When $\frak{h}^{\bot }\subset \frak{k}^{*}$ is a Lie subalgebra, $\left\langle \left[ \xi _{\varphi },\xi _{\psi }\right] _{*},X\right\rangle
=0$ for all $X\in \frak{h}$ and hence $\hat{X}\left\{ \varphi ,\psi \right\}
=0$.
As a first example of reduction, let us derive from the dual Poisson bracket on $G$.
\[quot\]Consider the action $G\times D\left( G\right)
_{+}\longrightarrow D\left( G\right) _{+}:h:\left( x,y\right) \longmapsto
\left( hx,hy\right) $. This action is admissible, and the projection map $p:$ $D\left( G\right) _{+}\longrightarrow G:\left( x,y\right) \longmapsto
x^{-1}y $ is constant on its orbits. The quotient Poisson bracket on $D\left( G\right) _{+}/G\simeq G$ coincides with .
[Here is a simple check of this assertion: Let $\varphi \in
C^{\infty }\left( G\right) $; set $\Phi \left( x,y\right) =\varphi \left(
x^{-1}y\right) $. Clearly, $\Phi $ is left $G^{\delta }$-invariant; hence its left gradient $\nabla \Phi $ is in $\left( \frak{g}^{\delta }\right)
^{\bot }\subset \frak{d}$; but the inner product in $\frak{d}$ is so chosen that $\frak{g}^{\delta }\subset \frak{d}$ is a *maximal isotropic subspace*, i.e., $\left( \frak{g}^{\delta }\right)
^{\bot }=\frak{g}^{\delta }$. Let us compute the Poisson bracket $\left\{
\Phi _{1},\Phi _{2}\right\} _{+}$ for two functions of such type. Since $\nabla \Phi _{1},\nabla \Phi _{2}\in \frak{g}^{\delta }$ and $r_{\frak{d}}|_{\frak{g}^{\delta}}=id$, we have $$\left\langle \left\langle r_{\frak{d}}\nabla \Phi _{1},\nabla \Phi
_{2}\right\rangle \right\rangle =\left\langle \left\langle \nabla \Phi
_{1},\nabla \Phi _{2}\right\rangle \right\rangle =0.$$ On the other hand[^23], $$\ \nabla ^{\prime }\Phi \left( x,y\right) \ =\left(
\begin{array}{l}
-\nabla\phantom{\prime } \varphi \left( x^{-1}y\right) \\
\phantom{-}\nabla ^{\prime }\varphi \left( x^{-1}y\right)
\end{array}
\right) \in \frak{g} \oplus \frak{g};$$ substituting this expression into $\left\langle \left\langle r_{\frak{d}}\nabla ^{\prime }\Phi _{1},\nabla ^{\prime }\Phi _{2}\right\rangle
\right\rangle $ and using , we get .]{}
We can now state the nonlinear version of reduction theorem described in Section 3.3. Let $\varphi \in I\left( G\right) $ be a central function. Define the Hamiltonian $h_{\varphi }$ on $D\left( G\right) _{+}$ by $h_{\varphi }\left( x,y\right) =\varphi \left( x^{-1}y\right) $.
(On free dynamics) The Hamiltonian flow on $D\left( G\right) _{+}$ defined by $h_{\varphi }$ is given by $$F_{t}:\left( x,y\right) \longmapsto \left( xe^{tX},ye^{tX}\right)
,\;X=\nabla \varphi \left( x^{-1}y\right) . \label{Free}$$
\(i) The Hamiltonian $h_{\varphi }$ is invariant with respect to the group $G^{*}$ which is acting on $D\left( G\right) _{+}$ via $$\left( h_{+},h_{-}\right) :\left( x,y\right) \longmapsto \left(
h_{+}xh_{-}^{-1},h_{+}yh_{-}^{-1}\right)$$ (ii) $G^{*}$ is an admissible subgroup in $D\left( G\right) \times D\left(
G\right) $ (which acts on $D\left( G\right) _{+}$ by left and right translations). (iii) The mapping $$\pi :D\left( G\right) _{+}\longrightarrow G:\left( x,y\right) \longmapsto
y_{+}^{-1}xy_{-}$$ is constant on $G^{*}$-orbits in $D\left( G\right) _{+}$ and allows to identify the quotient space $D\left( G\right) _{+}/G^{*}$ with the subgroup $G=\left\{ \left( x,e\right) ; x\in G\right\} \subset D\left( G\right) _{+}$.[^24] (iv) The quotient Poisson structure on $D\left( G\right) _{+}/G^{*}\simeq G$ coincides with the Sklyanin bracket. (v) The quotient flow $\bar{F}_{t}\ $on $G$ is given by $\bar{F}_{t}:x\longmapsto g_{\pm }\left( t\right) ^{-1}xg_{\pm }\left(
t\right) $, where $g_{+}\left( t\right) ,g_{-}\left( t\right) $ solve the factorization problem $\exp t\nabla \varphi \left( x\right) =g_{+}\left(
t\right) g_{-}\left( t\right) ^{-1}$, and satisfies the Lax equation (\[la\]).
So far our geometric construction is restricted to ordinary Lax equations on a *single copy* of $G$. In order to put lattice zero curvature equations into our framework we need one more effort. First of all, let us state the factorization theorem which applies in this case (cf. Section 9.1). Let again $$T:G^{N}\rightarrow G:\left( L_{0},...,L_{N-1}\right) \longmapsto T_{L}=\stackrel{\curvearrowleft }{\prod_{k}}L_{k}$$ be the monodromy map; choose $\varphi \in I\left( G\right) $ and set $H_{\varphi }=\varphi \circ T$. We define the “wave function” $\psi _{m}$ associated with the auxiliary linear problem (\[daux\]) by $$\psi _{m}=\stackrel{\curvearrowleft }{\prod_{0\leq k\leq m-1}}L_{k}$$ The Poisson structure on $G^{N}$ is defined as the direct product of Sklyanin brackets on each factor. As usual, we assume that our basic r-matrix is skew and satisfies the modified Yang-Baxter equation, so that $\left( \frak{g},\frak{g}^{*}\right) $ is a factorizable Lie bialgebra.
\[fdcase\](i) The Hamiltonian equation of motion on $G^{N}$ with Hamiltonian $H_{\varphi }$ may be written as $$\frac{dL_{m}}{dt}=L_{m}M_{m+1}^{\pm }-M_{m}^{\pm }L_{m}, \label{lzcurv}$$ where $$M_{m}^{\pm }=r_{\pm }\left( \psi _{m}\nabla \varphi \left( T_{L}\right) \psi
_{m}^{-1}\right) . \label{mop}$$ (ii) Its integral curve with origin $L^{0}=\left(
L_{0}^{0},...,L_{N-1}^{0}\right) \ $ is given by $$L_{m}\left( t\right) =g_{m}^{\pm }\left( t\right) ^{-1}L_{m}^{0}g_{m+1}^{\pm
}\left( t\right) , \label{evo}$$ where $g_{m}^{+}\left( t\right) ,g_{m}^{-}\left( t\right) $ are the solutions of the factorization problem $$g_{m}^{+}\left( t\right) ,g_{m}^{-}\left( t\right) ^{-1}=\psi _{m}^{0}\cdot
\exp t\nabla \varphi \left( T_{L^{0}}\right) \cdot \left( \psi
_{m}^{0}\right) ^{-1}. \label{evol}$$
As usual, a direct proof of theorem is easy. One point worth to be mentioned is the computation of the gradients of the Hamiltonian: for that end we may use the “variation of constants” in the auxiliary linear problem , similarly to the case of differential operators on the circle discussed in Section 8.3. A geometric derivation is not so straightforward. Let us introduce the following notation in order to simplify the bulky formulae. We set $\mathbf{G}=G^{N},\frak{G}=\oplus ^{N}\frak{g},{\mathbf{L}}=\left( L_{0},...,L_{N-1}\right) \in \mathbf{G}$. Let $\tau $ be the automorphism of $\mathbf{G}$ induced by cyclic permutation of indices, $\left( L_{0},...,L_{N-1}\right) ^{\tau }=\left(
L_{1},L_{2},...,L_{N-1},L_{0}\right) $; we denote the corresponding automorphism of $\frak{G}$ by the same letter.[^25] Equations , , may be rewritten as $$\label{Quot}
\begin{array}{l}
\quad \frac{d\mathbf{L}}{dt} =\mathbf{LM}_{\pm }^{\tau }-\mathbf{M}_{\pm }
\mathbf{L}, \\
\mathbf{L}\left( t\right) =\mathbf{g}_{\pm }\left( t\right) ^{-1}
\mathbf{L}^{0}\mathbf{g}_{\pm }\left( t\right) ^{\tau },
\end{array}$$ where $$\mathbf{g}_{+}\left( t\right) \mathbf{g}_{-}\left( t\right) ^{-1} =\mathbf{\psi }^{0}\exp t\nabla \varphi \left( T_{L^{0}}\right) \left( \mathbf{\psi }^{0}\right) ^{-1}.$$ (in the last expression $\exp t\nabla \varphi $ is embedded into $\mathbf{G}=G^{N}$ diagonally). The important feature of these formulae is the presence of the *twisting automorphism* $\tau $. It plays the key role in the definition of two more objects: (1) *The second Poisson bracket* on $\mathbf{G}$ which has got the spectral invariants of the monodromy as its Casimirs and (2) *The twisted Poisson structure* on the “big double” $\mathbf{D}\left( \mathbf{G}\right) =\mathbf{G}\times \mathbf{G}$ which is responsible for the free dynamics. We shall start with the latter question.
Let $r\in {\mathrm {End}}\, \frak{g}$ be the classical r-matrix associated with the factorization problem in a single copy of $G$. The corresponding r-matrix in the “big” algebra $\mathbf{g}$ is the direct sum of $r$’s acting in each copy of $\frak{g}$, $$\mathbf{r}=\oplus _{m=0}^{N-1}r.$$ Note that $\mathbf{r}\circ \tau =\tau \circ \mathbf{r.}$ In a similar way, the r-matrix of the “big double” $\frak{D}=\frak{G}\oplus \frak{G}$ is $$\mathbf{r}_{\frak{D}}=\oplus _{m=0}^{N-1}r_{\frak{d}};$$ in block notation we have $$\mathbf{r}_{\frak{D}}=\left(
\begin{array}{ll}
\mathbf{r} & 2\mathbf{r}_{+} \\
2\mathbf{r}_{-} & -\mathbf{r}
\end{array}
\right)$$ (cf. ). Let us define the automorphism $\mathbf{\tau }\in
Aut\left( \mathbf{G}\times \mathbf{G}\right) $ by $\left( \mathbf{x},\mathbf{y}\right) ^{\mathbf{\tau }}=\left( \mathbf{x},\mathbf{y}^{\tau }\right) $; the corresponding automorphism of $\frak{D}=\frak{G}\oplus \frak{G}$ is again denoted by the same letter. Put $\mathbf{r}_{\frak{D}}^{\mathbf{\tau }}=\mathbf{\tau }\circ \mathbf{r}_{\frak{D}}\circ \mathbf{\tau }^{-1}$; in block notation we have $$\mathbf{r}_{\frak{D}}^{\mathbf{\tau }}=\left(
\begin{array}{ll}
\mathbf{r} & 2\mathbf{r}_{+}\circ \mathbf{\tau }^{-1} \\
2\mathbf{\tau \circ r}_{-} & -\mathbf{r}
\end{array}
\right) .$$ Let us define the *twisted Poisson structure* on $\mathbf{D}\left(
\mathbf{G}\right) $ by $$\left\{ \varphi ,\psi \right\} _{\mathbf{\tau }}=\left\langle \left\langle
\mathbf{r}_{\frak{D}}\nabla \varphi ,\nabla \psi \right\rangle \right\rangle
+\left\langle \left\langle \mathbf{r}_{\frak{D}}^{\mathbf{\tau }}\nabla
^{\prime }\varphi ,\nabla ^{\prime }\psi \right\rangle \right\rangle .
\label{twist}$$ The Jacobi identity for follows from proposition . Our next assertion is the twisted version of proposition .
The action $\mathbf{G}\times \mathbf{D}\left( \mathbf{G}\right) _{\mathbf{\tau }}\longrightarrow \mathbf{D}\left( \mathbf{G}\right) _{\mathbf{\tau }}:h:\left( x,y\right) \longmapsto \left( hx,hy\right) $ is admissible; the projection map $p:\mathbf{D}\left( \mathbf{G}\right) _{\mathbf{\tau }}\longrightarrow \mathbf{G}:\left( \mathbf{x},\mathbf{y}\right) \longmapsto
\mathbf{x}^{-1}\mathbf{y}$ is constant on its orbits. The quotient Poisson bracket on $\mathbf{D}\left( \mathbf{G}\right) _{+}\mathbf{/G}\simeq \mathbf{G}$ is given by $$\left\{ \varphi ,\psi \right\} =\left\langle \left\langle \mathbf{r}_{\frak{D}}^{\mathbf{\tau }}\left(
\begin{array}{l}
\nabla \varphi \\
-\nabla ^{\prime }\varphi
\end{array}
\right) ,\left(
\begin{array}{l}
\nabla \psi \\
-\nabla ^{\prime }\psi
\end{array}
\right) \right\rangle \right\rangle . \label{twbr}$$
The proof is parallel to that of proposition . The properties of the quotient bracket are also quite remarkable. (Note that unlike the Sklyanin bracket on $\mathbf{G}=G\times ...\times G$ the bracket is *non-local*, due to presence of the twist $\tau)$. Let us consider the monodromy map $T:\mathbf{G}\longrightarrow G$. Assume that $\mathbf{G}$ carries the Poisson bracket and $G$ is equipped with the bracket .
\[twstr\](i) The monodromy $T:\mathbf{G}\longrightarrow G$ is a Poisson mapping. (ii) The spectral invariants of the monodromy are the Casimirs of the quotient bracket. (iii) Define the gauge action $\mathbf{G}\times
\mathbf{G}\longrightarrow \mathbf{G}:\mathbf{x}:\mathbf{g}\longmapsto
\mathbf{xg}\left( \mathbf{x}^{\tau }\right) ^{-1}$; when $N$ is odd, its orbits coincide with symplectic leaves of the quotient bracket.
In brief, we see that the bracket provides us with the missing ingredient for our geometric picture: it is the sought for *second bracket* which we need for a geometric treatment of lattice zero curvature equations.
(on free dynamics) Let $\varphi \in I\left( G\right) $; set $h_{\varphi
}=\varphi \, \circ T \, \in C^{\infty }\left( \mathbf{G}\right)$, $H_{\varphi
}=h_{\varphi }\circ p\in C^{\infty }\left( \mathbf{D}\left( \mathbf{G}\right) \right) $. The integral curves of the Hamiltonian $H_{\varphi }$ in $\mathbf{D}\left( \mathbf{G}\right) _{\mathbf{\tau }}$ are given by $$\left( \mathbf{x}\left( t\right) ,\mathbf{y}\left( t\right) \right) =\left(
x_{0}e^{tX},y_{0}e^{tX}\right) ,X=\nabla \varphi \left( T(\mathbf{x}^{-1}y)\right) . \label{freefl}$$
\[last\]Set $\mathbf{G}^{*}=\left( G^{*}\right) ^{N}$ and consider the action $\mathbf{G}^{*}\times \mathbf{D}\left( \mathbf{G}\right) _{\mathbf{\tau }}\longrightarrow \mathbf{D}\left( \mathbf{G}\right) _{\mathbf{\tau }}$ given by $$\left( \mathbf{h}_{+},\mathbf{h}_{-}\right) :\left( \mathbf{x},\mathbf{y}\right) \longmapsto \left( \mathbf{h}_{+}\mathbf{xh}_{-}^{-1},\mathbf{h}_{+}\mathbf{y}\left( \mathbf{h}_{-}^{\tau }\right) ^{-1}\right) .$$ This action is admissible; the quotient space $\mathbf{D}\left( \mathbf{G}\right) _{\mathbf{\tau }}/\mathbf{G}^{*}$ may be identified with $\mathbf{G\
}$by means of the map $$\pi :\mathbf{D}\left( \mathbf{G}\right) \longrightarrow \mathbf{G:}\left(
\mathbf{x},\mathbf{y}\right) \longmapsto \mathbf{y}_{+}^{-1}\mathbf{xy}_{-}^{\tau ^{-1}}$$ which is constant on the orbits of ${\mathbf{G}}^{*}$, and the quotient Poisson structure coincides with the Sklyanin bracket. The flow admits reduction with respect to this action; the quotient flow is coincides with .
Theorem fills the last gap in our geometric picture and shows that the qualitative behaviour of difference zero curvature equations is the same as in the case of linear phase spaces.
It is my personal pleasure to thank the Organizing Committee and in particular Professor A. Ferreira dos Santos and Dr. N. Manojlovic for their kind invitation and for the inspiring atmosphere they have created during the School.
The present work was partially supported by the INTAS Open 00-00055 grant.
[STUV]{}
,[Vol. 4]{}[2001]{}
[^1]: A further generalization is possible: we can add one more spatial variable and consider the loop algebra based on the algebra of pseudodifferential operators; this yields the so called KP equation for functions of *three* variables.
[^2]: The integrability condition which assures the local existence of the submanifold with the given tangent distribution immediately follows from the Jacobi identity; the subtle part of the proof consists in checking that the possible jumps of the rank of the Poisson tensor $\pi $ do not lead to singularities of the leaves; one can check that the Lie derivative of $\pi $ along any Hamiltonian vector field is zero, and hence its rank is constant along the leaves (but may jump in the transversal direction).
[^3]: It’s probably worth making some precisions: when we deal with concrete equations, coadjoint orbits are almost allways a good starting point, but it may be quite useful to enrich our tools. In some cases, an orbit is too *big* for our purpose and it is possible to cancel out some degrees of freedom by passing to the quotient space over some manifest symmetry group. On the other hand, in some cases, an orbit is too *small* and it’s more practical to use a bigger phase space which is mapped onto the orbit in a way which is compatible with its Poisson structure. Finally, there are classes of examples when the context of Lie algebras appears to be too restrictive and we have to deal with *nonlinear* Poisson brackets from the very beginning. We shall comment on these examples later on (see Section 9).
[^4]: Throughout these lectures *inner product* means a nondegenerate symmetric bilinear form ($\Bbb{C}$-bilinear in the case of complex algebras); over the reals we do not impose any positivity condition.
[^5]: Mind that the model for the dual of a subalgebra depends not only on the subalgebra itself but also on the choice of its complement in the big Lie algebra (and eventually also on the choice of the inner product whenever it is not unique).
[^6]: Indeed, the mapping $P\mapsto E\left(
P\right) $ determines a meromorphic mapping $\Gamma \rightarrow {\C}P_{n-1}$ (this is a corollary of the elementary analytic perturbation theory). By a standard theorem, any such mapping is actually holomorphic, and hence the eigenvector bundle extends to $\Gamma $.
[^7]: We denote by $p\wedge x$ the $n\times n$-matrix with entries $p_i x_j-p_jx_i$; in a similar way, the entries of $x\otimes x$ are $x_ix_j$.
[^8]: \[ref\]In the case of the loop algebra the definition of the 2-cocycle $\omega $ is essentially unique; for the double loop algebra there is a possibility to introduce into a weight factor $\phi (z)$ which does not depend on $x$; this weight factor will modify the auxiliary linear problem. This freedom is useful in applications; our choice in is the simplest one possible.
[^9]: Poisson bracket of functionals which does not contain derivatives $\partial
_{x}$ of the gradients is sometimes called *ultralocal*; in more complicated cases, Poisson brackets may contain derivatives *(non-ultralocal case)* or even be non-local, i.e., contain integral operators.
[^10]: For simplicity, in both examples we deal with complex Lie algebras; in order to pass to a real form we must choose an [*anti-involution*]{} of our basic loop algebra; this is of course a necessary step in order to make the auxiliary linear operators genuine skew selfadjoint operators.
[^11]: In various applications $J_{0},J_{\infty }$ are regular matrices with distinct eigenvalues. In that case $\frak{g}^{J_{0}}=\frak{g}_{J_{0}}$ and $\frak{g}^{J_{\infty }}=\frak{g}_{J_{\infty }}$ are abelian subalgebras; hence theorem \[NF\] means that the potential $U$ may be transformed to *diagonal form* by a formal gauge transformation. When the eigenvalues of $J_{0},J_{\infty }$ have multiplicities, the potential may be transformed only to *block diagonal form.*
[^12]: There is a different version of dressing which uses the *Riemann problem in the half-plane;* under some additional restrictions, this version is adapted to the study of rapidly decreasing solutions.
[^13]: $D(K)$ is called *the double* of $K$; we have already used a similar construction for Lie algebras in Section 4. One may notice that the construction below uses only factorization in $D(K)\ $and the subgroups $K_{\pm }$ need not be complementary in $K$. In Section 9 we shall once again encounter this construction in the study of Poisson Lie groups.
[^14]: This problem is nontrivial, since in order to get a nonzero Hamiltonian $H_{\alpha }$, the constant diagonal matrix $\alpha \left( z^{-1}\right) $ must have pole at zero.
[^15]: Typical dynamical systems of this kind are the classical analogues of lattice models in quantum statistics, although some of the systems which we mentioned earlier (e.g., Toda lattices, certain tops, etc.) also admit difference Lax representations.
[^16]: This latter paper was in turn motivated by the Quantum Inverse Scattering Method developed by Faddeev, Takhtajan and Sklyanin (cf. [@Fad]) and by the work of Baxter on Quantum Statistical Mechanics (cf. [@bax]).
[^17]: \[Ob\]Formula seems to be not the simplest bracket which can be arranged using an antisymmetric operator: why not take $\left\{ \varphi
,\psi \right\} ^{g}\left( x\right) =\left\langle r\left( \nabla \varphi
\right) ,\nabla \psi \right\rangle $, or $\left\{ \varphi ,\psi \right\}
^{d}\left( x\right) =\left\langle r\left( \nabla ^{\prime }\varphi \right)
,\nabla ^{\prime }\psi \right\rangle $ ? The reason is this: when $r$ satisfies the modified Yang-Baxter identity, neither of these brackets satisfies Jacobi. However, the obstructions cancel each other when we take the *difference*, or the *sum* of the two (and precisely in these two cases)! We shall return to this question in Section 9.4 below.
[^18]: At this point we use the crucial property of : the Poisson operator is a multiplication operator.
[^19]: This condition is usually expressed by saying that $\eta _{r}$ is a *1-cocycle* (in fact, a coboundary) on $G$. In these lectures we shall not use this language.
[^20]: Formula explains the choice of normalization in : we wanted to get the same thing as in .
[^21]: In formal terms, equation means that $\delta _{r}$ is a *1-cocycle* on $\frak{ g}$ (with values in ${\mathrm {End}}\, \frak{g}$).
[^22]: More precisely, the bracket induced on $G^{*}\subset D\left( G\right) $ has opposite sign, due to the minus sign in $r_{\frak{d}}=P_{\frak{g}}-P_{\frak{g}^{*}}$.
[^23]: Gradients in the l.h.s are computed with respect to two variables $x,y$; in the r.h.s they are computed with respect to a single variable.
[^24]: As usual, for $g\in G$ we denote by $g_{+},g_{-}$ the solutions of the factorization problem $g=g_{+}g_{-}^{-1}$.
[^25]: The twisting automorphism $\tau $ plays the role which is similar to that of the derivation $\partial _{x}$ for ordinary zero curvature equations; we can say that its use allows to reproduce for lattice systems the effects of central extension of loop algebras discussed in Section 8.1.
|
---
abstract: 'Modern society depends on the flow of information over online social networks, and popular social platforms now generate significant behavioral data. Yet it remains unclear what fundamental limits may exist when using these data to predict the activities and interests of individuals. Here we apply tools from information theory to estimate the predictive information content of the writings of Twitter users and to what extent that information flows between users. Distinct temporal and social effects are visible in the information flow, and these estimates provide a fundamental bound on the predictive accuracy achievable with these data. Due to the social flow of information, we estimate that approximately 95% of the potential predictive accuracy attainable for an individual is available within the social ties of that individual only, without requiring the individual’s data.'
author:
- 'James P. Bagrow'
- Xipei Liu
- Lewis Mitchell
date: 'August 15, 2017'
title: Information flow reveals prediction limits in online social activity
---
The flow of information in online social platforms is now a significant factor in protest movements, national elections, and rumor and misinformation campaigns [@shirky2011political; @lotan2011arab; @delvicario2016]. Data collected from these platforms are a boon for researchers [@lazer2009computational] but also a source of concern for privacy, as the social flow of predictive information can reveal details on both users and non-users of the platform [@Garciae1701172]. Information flow on social media has primarily been studied *structurally* (for example, by tracking the movements of keywords [@lotan2011arab; @gruhl2004information; @bakshy2012role; @bakshy2015exposure] or adoptions of behaviors [@aral2009distinguishing; @centola2010spread; @aral2012identifying]) or *temporally*, by applying tools from information theory to quantify the information contained in the timings of user activity, as temporal relationships between user activity reflect underlying coordination patterns [@ver2012information; @borge2016dynamics]. Yet neither approach considers the full extent of information available, both the complete language data provided by individuals and their temporal activity patterns. Here we unify these two primary approaches, by applying information-theoretic estimators to a collection of Twitter user activities that fully incorporate language data while also accounting for the temporal ordering of user activities.
We gathered a dataset of $n = 927$ users of the Twitter social media platform. Users were selected who wrote in English, were active for at least one year, and had comparably sized social networks. We applied both computational tools and human raters to help avoid bots and non-personal accounts. For each user, we retrieved all of their public posts excluding retweets (up to the 3200 most recent public posts, as allowed by Twitter). Examining these texts, we determined each user’s $15$ most frequent Twitter contacts and gathered the texts of those users as well, providing us ego-alter pairs. See () for full details on data collection, filtering, and processing.
The ability to accurately profile and predict individuals is reflected in the predictability of their written text. The predictive information contained within a user’s text can be characterized by three related quantities, the entropy rate $h$, the perplexity $2^{h}$, and the predictability $\Pi$. The entropy rate quantifies the average uncertainty one has about future words given the text one has already observed (Fig. \[fig:introFig\]A). Higher entropies correspond to less predictable text and reflect individuals whose interests are more difficult to predict. In the context of language modeling, it is also common to consider the perplexity. While the entropy rate specifies how many bits $h$ are needed on average to express subsequent words given the preceding text, the perplexity tells us that our remaining uncertainty about those unseen words is equivalent to that of choosing uniformly at random from among $2^{h}$ possibilities. For example, if $h=6$ bits (typical of individuals in our dataset), the perplexity is 64 words, a significant reduction from choosing randomly over the entire vocabulary (social media users have ${\approx}5000$-word vocabularies on average; see ). Lastly, the predictability $\Pi$, given via Fano’s inequality [@CoverThomas], is the probability that an *ideal* predictive algorithm will correctly predict the subsequent word given the preceding text. Repeated, accurate predictions of future words indicate that the available information can be used to build profiles and predictive models of a user and estimating $\Pi$ allows us to fundamentally bound the usefulness of the information present in a user’s writing without depending on the results of specific predictive algorithms.
{width="\textwidth"}
Information theory has a long history of estimating the information content of text [@shannon1951prediction; @brown1992estimate; @schurmann1996entropy; @kontoyiannis_nonparametric_1998]. Crucially, information is available not just in the words of the text but in their order of appearance. We applied a nonparametric entropy estimator that incorporates the full sequence structure of the text [@kontoyiannis_nonparametric_1998]. This estimator has been proved to converge asymptotically to the true entropy rate for stationary processes and has been applied to human mobility data [@song2010limits]. See for details.
The text streams of the egos were relatively well clustered around $h\approx 6.6$ bits, with most falling between 5.5–8 bits (Fig. \[fig:introFig\]B). Equivalently, this corresponds to a perplexity range of ${\approx}45$–$256$ words, far smaller than the typical user’s ${\approx}5000$-word vocabulary, and a mean predictability of ${\approx}53\%$, quite high for predicting a given word out of ${\approx}5000$ possible words on average. We found this typical value of information comparable to other sources of written text, but social media texts were more broadly distributed—individuals were more likely to be either highly predictable or highly unpredictable compared with formally written text (see ).
Next, instead of asking how much information is present in what the ego has previously written regarding what the ego will write in the future, we ask how much information is present on average in what the has previously written regarding what the ego will write in the future (Fig. \[fig:introFig\]A). If there is consistent, predictive information in the alter’s past about the ego’s future, especially beyond the information available in the ego’s own past, then there is evidence of information flow.
Replacing the ego’s past writing with the alter’s past converts the entropy to the *cross-entropy* (see ). The cross-entropy is always greater than the entropy when the alter provides less information about the ego than the ego, and so an increase in cross-entropy tells us how much information we lose by only having access to the alter’s information instead of the ego’s. Indeed, estimating the cross-entropy between each ego and their most frequently contacted alter (Fig. \[fig:introFig\]B), we saw higher cross-entropies spanning from 6–12 bits (equivalently, perplexities from 64–4096 words or predictabilities spread from 0–60%). While less frequently contacted alters provided less predictive information than alters in close contact (see ), even for the closest alters there was a broader range of cross-entropies than the entropies of the egos themselves. This implies a diversity of social relationships: sometimes the ego is well informed by the alter, leading to a cross-entropy closer to the ego’s entropy, while other times the ego and alter exhibit little information flow.
Thus far we have examined the information flow between the ego and individual alters, but actionable information regarding the future of the ego may be embedded in the combined pasts of multiple alters (Fig. \[fig:introFig\]C). To address this, we generalized the cross-entropy estimator to multiple text streams (see ). We then computed the cross-entropies and predictabilities as we successively accumulated alters in order of decreasing contact volume (Fig. \[fig:introFig\]D). As more alters were considered, cross-entropy decreased and predictability increased, which is sensible as more potential information is available. Interestingly, with 8–9 alters, we observed a predictability of the ego given the alters at or above the original predictability of the ego alone. As more alters were added, up to our data limit of 15 alters, this increase continued. Paradoxically, this indicated that there is potentially more information about the ego within the total set of alters than within the ego itself.
To understand this apparent paradox, we need to address a limitation with the above analysis: it does not incorporate the ego’s own past information. It may be that the information provided by the alters is simply redundant when compared with that of the ego. Therefore, we simply included the ego’s past alongside the alters, generalizing the estimator to an entropy akin to a transfer entropy [@schreiber2000measuring; @staniek2008symbolic], a common approach to studying information flow. This entropy is computed in the “Alters and ego” curves in Fig. \[fig:introFig\]D. A single alter provided a small amount of extra information beyond that of the ego, ${\approx}1.9\%$ more predictability. This provided us a quantitative measure of the extent of information flow between individual users of social media Beyond the most frequently contacted alter, as more alters were added this extra predictability grew: at 15 alters and the ego there was ${\approx}6.9\%$ more predictability than via the ego alone. Furthermore, the information provided by the alters without the ego is strictly less than the information provided by the ego and alters together, resolving the apparent paradox.
However, this extra predictability also appeared to saturate, and eventually adding more alters will not provide extra information. This observation is compatible with Dunbar’s number which uses cognitive limits to argue for an upper bound on the number of meaningful ties an ego can maintain (${\approx}150$ alters) [@dunbar1993coevolution]. The question now becomes, given enough ties what is the upper bound for the predictability?
To extrapolate beyond our data window, we fitted a nonlinear saturating function to the curves in Fig. \[fig:introFig\]D, (see for details and validation of our extrapolation procedure). From fits to the raw data, we found a limiting predictability given the alters of $\Pi_{\infty} = 60.8\% \pm 0.691$% (Fig. \[fig:introFig\]E). Of course, egos will not have an infinite number of alters, so a more plausible extrapolation point may be to Dunbar’s number: $\Pi_{150} \approx 60.3$%, within the margin of error for $\Pi_{\infty}$, indicating that saturation of predictive information has been reached. Similarly, extrapolating the predictability including the ego’s past gives $\Pi_{\infty} = 64.0\% \pm 1.54\%$ ($\Pi_{150} = 63.5\%$).
These extrapolations showed that significant predictive information was available in the combined social ties of individual users of social media. In fact, there is so much social information that an entity with access to all social media data will have only slightly more potential predictive accuracy (${\approx}64\%$ in our case) than an entity which has access to the activities of an ego’s alters but not to that ego (${\approx}61\%$). This may have distinct implications for privacy: if an individual forgoes using a social media platform or deletes her account, yet her social ties remain, then that platform owner potentially possesses $95.1\% \pm 3.36\%$ of the achievable predictive accuracy of the future activities of that individual.
Two issues can affect the cross-entropy as a measure of information flow. The first is that the predictive information may be due simply to the structure of English: commonly repeated words and phrases will represent a portion of the information flow. The second is that of a common cause: egos and alters may be independently discussing the same concepts. This is particularly important on social media with its emphasis on current events [@kwak2010twitter].
To study these issues, we constructed two types of controls. The first randomly pairs users together by shuffling alters between egos. The second constructed pseudo-alters by assembling, for each real alter, a random set of posts made at approximately the same times as the real alter’s posts, thus controlling for temporal confounds. Both controls used real posted text and only varied the sources of the text. As shown in Fig. \[fig:introFig\]D, the real alters provided more social information than either control. There was a decrease in entropy as more control alters were added, but the control cross-entropy remained above the real cross-entropy. We also observed that for a single alter the temporal control had a lower cross-entropy than the social control and therefore temporal effects provide more information than social effects (underscoring the role of social media as a news platform [@kwak2010twitter]), although both controls eventually converge to a limiting predictability of ${\approx}51\%$.
Given the importance of temporal information in online activity, to what extent is this reflected in the information flow? Do recent activities contain most of the predictive information or are there long-term sources of information? To estimate recency effects, we applied a censoring filter to the ego’s text stream, removing at each time period the text written in the previous $\Delta T$ hours and measuring how much the mean predictability decreased compared with the mean predictability including the recent text. Increasing $\Delta T$ decreased $\Pi$, especially evident when removing the first 3–4 hours (Fig. \[fig:recency\]A): we found an average decrease in predictability of ${\approx}1.4\%$ at 4 hours. This loss in predictability relative to the uncensored baseline is comparable to the gain from the rank-1 alter we observed in Fig. \[fig:introFig\]D. In other words, close alters tended to contain a quantity of information about the ego comparable to the information within just a few hours of the ego’s own recent past. Beyond 24 hours the predictability loss continued approximately linearly (). We then applied this censoring procedure to the alters alone and the alters combined with the ego, excluding their recent text and measuring how the cross predictability changed on average from their respective baselines. We found a similar drop in predictability during the first few hours, but then a more level trend than when censoring the ego alone. This leveling off showed that less long-term information was present in the alters’ pasts than within the ego’s.
{width="50.00000%"}
Next we studied recency by the activity frequencies of alters and egos. Individuals who post frequently to social media, keeping up on current events, may provide more predictive information about either themselves or their social ties than other, infrequent posters. We found that the self-predictability of users was actually independent of activity frequency () but there were strong associations between activity frequency and social information flow: egos who posted 8 times per day on average were ${\approx}17\%$ more predictable given their alters than egos who posted once per day on average (Fig. \[fig:recency\]B). Interestingly, this trend reversed itself when considering the activity frequencies of the alters: alters who posted 8 times per day on average were ${\approx}23\%$ less predictive of their egos than alters who posted once per day on average. Highly active alters tended to inhibit information flow, perhaps due to covering too many topics of low relevance to the ego.
Information flow reflects the social network and social interaction patterns (Fig. \[fig:mentionsVsKLeaAndKLae\]). We measured information flow for egos with more popular alters compared with egos with less popular alters. Alters with more social ties provided less predictive information about their egos than alters with fewer ties (Fig. \[fig:mentionsVsKLeaAndKLae\]A). The decrease in predictability of the ego was especially strong up to alters with ${\approx}400$ ties, where the bulk of our data lies, but the trend continued beyond this as well. This decreasing trend belies the power of hubs in many ways: while hubs strongly connect a social network topologically [@Reka2000], limited time and divided attention across their social ties bounds the alter’s ability to participate in information dynamics mediated by the social network and this is reflected in the predictability.
Reciprocated contact is an important indicator of social relationships [@wasserman1994social], especially in online social activity where so much communication is potentially one-sided [@kwak2010twitter]. In Fig. \[fig:mentionsVsKLeaAndKLae\]B we investigated how directionality in contact volume, how often the ego mentions the alter and vice versa, related to information flow. We found that the ego was more predictable given the alter for those dyads where the alter more frequently contacted the ego, but there was little change across dyads when the ego mentioned the alter more or less frequently (Fig. \[fig:mentionsVsKLeaAndKLae\]B). We also observed a similar trend for information flow but in reverse, when predicting the alter given the ego (). These trends captured the reciprocity of information flow: an alter frequently contacting an ego will tend to give predictive information about the ego, but the converse is not true: an ego can frequently contact her alter but that does not necessarily mean that the alter will be any more predictive, as evidenced by the flat trend in Fig. \[fig:mentionsVsKLeaAndKLae\]B.
In summary, the ability to repeatedly and accurately predict the text of individuals provides considerable value to the providers of social media, allowing them to develop profiles to identify and track individuals [@de2013unique; @de2015unique] and even manipulate information exposure [@pariser2011filter]. That information is so strongly embedded socially underscores the power of the social network: by knowing who are the social ties of an individual and what are the activities of those ties, our results show that one can in principle accurately profile even those individuals who are not present in the data [@Garciae1701172].
The time-ordered cross-entropy (Fig. \[fig:introFig\]A) applied here to online social activity is a natural, principled information-theoretic measure that incorporates all the available textual and temporal information. While weaker than full causal entailment, by incorporating time ordering we identify social information flow as the presence of useful, predictive information in the past of one’s social tie beyond that of the information in one’s own past. Doing so closely connects this measure with Granger causality and other strong approaches to information flow [@granger1969investigating; @schreiber2000measuring].
We gratefully acknowledge the resources provided by the Vermont Advanced Computing Core. This material is based upon work supported by the National Science Foundation under Grant No. IIS-1447634. LM acknowledges support from the Data To Decisions Cooperative Research Centre (D2D CRC), and the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS).
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---
abstract: 'The interference of the directly emitted photoelectron wave and the wave scattered coherently by neighboring atoms gives holographic fringes in the photoelectron emission intensity $I(\hat{\bf R})$. In the electron emission holography technique in surface physics, $I(\hat{\bf R})$ is inverted holographically to give a 3D-image of the environment of the source atom. Earlier [@usl]–[@us2], we pointed out that the polarization pattern ${\bf P}(\hat{\bf R})$ similarly can be viewed as a hologram of the spin environment of the source atom by virtue of the exchange scattering of the photoelectron by the neighboring atoms. In this paper, we point out that spin-orbit correlations in the photoelectron initial state are responsible for holographic spin-dependent contributions to the intensity hologram $I(\hat{\bf R})$, even if the directly emitted photoelectrons are unpolarized. This remarkable result implies that the emission intensity contains spin information just as the polarization pattern ${\bf P}(\hat{\bf R})$. Although the spin dependent signal in the hologram is rather small ($\sim 5 \% $ in most cases of interest), we show how spin information can be extracted from the intensity hologram, making use of the point symmetry of the environment of the source atom. This way of analyzing photoelectron intensity holograms to extract short-range spin information opens up a new avenue for surface magnetism studies.'
address:
- Institute for Theoretical Atomic and Molecular Physics
- 'Harvard-Smithsonian Center for Astrophysics'
- 'Cambridge, MA 02138'
- 'Physics Department, Rice University'
- 'Houston, TX 77251-1892'
author:
- Eddy Timmermans
- 'G. T. Trammell and J. P. Hannon'
title: Spin holography II
---
Introduction
============
A photoelectron emitted from an atom may be scattered by neighboring atoms. The scattered waves interfere with the unscattered wave and the intensity pattern at large distances, $I(\hat{\bf R})$, forms a hologram which can be analyzed to yield an image of the neighborhood of the electron emitting atom [@Bart]–[@Szoke].
If the neighboring atoms have spins, then exchange scattering results in the polarization of the outgoing electrons and in addition to the intensity $I(\hat{\bf R})$, the polarization pattern ${\bf P}(\hat{\bf R})$ also forms a hologram which can be analyzed to yield an image of the spins of the neighboring atoms. While this scheme of ‘spin holography’ is elegant, it suffers from the fact that present day electron polarimeters are very inefficient (eff. $\sim 10^{-4}$) detracting from the practical utility of this way of imaging the spins. Therefeore, the existence of an alternate method to extract the spin information from the intensity hologram $I(\hat{\bf R})$, such as the scheme that we discuss in this paper, is of interest.
Spin holography is a holography scheme yielding spin information on the near-neighborhood probed by electron holography. This is in contrast to the long range order and bulk magnetism determinations in magnetic diffraction methods using neutrons or X-rays. Since the effect of exchange interaction on the electron-atom scattering becomes less pronounced as the kinetic energy of the electron increases, it is likely that spin holography will prove to be most useful if the photoelectron has relatively low energy ($\sim$ 100-200 eV). In this energy range, the mean-free path of the electron is of the order of a few nanometers and only atoms within that distance of the photoelectron source atom will contribute significantly to the hologram. Therefore, we expect that electron holographic spin determination will be useful as a probe of surface magnetism, with the source atoms lying within the first few surface layers or as adatoms adhering to the surface. As an example where holographic spin determination is useful, we can use adatom sources on the surface of a ferromagnet to determine the spins of the first few layers, while the surface information is difficult to obtain from ‘diffraction’ measurements.
Consider the example of a substrate with the first two layers magnetized, layer $\# 1$ having all its atoms with spin $s_{a}$ and layer $\# 2$ having spin $s_{b}$. Then while the holographic method with the sources, e.g. adatoms on the top surface, would yield a proper image of the neighboring atoms on the two layers, diffraction measurements can only distinguish the actual magnetic surface ($1_{a}$, $2_{b}$) from the surface with the spins of the layers reversed ($1_{b}$, $2_{a}$), by detailed compairison of the diffraction data with numerical multiple scattering calculations.
Clearly, if the photoelectrons are $\it{polarized}$ then the $\it{intensity}$ holograms, $I(\hat{\bf R})$, are sensitive to the spins of the neighboring scatterers. However, a new and important feature of the ‘spherical wave holograms’ is that the photoelectrons need not be polarized for $I(\hat{\bf R})$ to be used to image the spins of the neighboring atoms: unpolarized electrons suffice if there are spin-orbit correlations. As a consequence, if the photoelectrons are ejected from inner core spin-orbit split subshells (e.g. $L_{2}$ or $L_{3}$), $\it{even \; by \; linearly \; polarized \; or \; unpolarized \;
incident \; light}$, then as we show below, their intensity holograms $I(\hat{\bf R})$, can be analyzed to image the spins. This is an interesting and important new result. It is important that linearly polarized synchrotron radiation can be used to image the spins. It is interesting for the same reason – one is surprised (at least at first) that linearly polarized light ejecting electrons from a complete inner atomic subshell gives an intensity pattern with a linear dependence on the spins ${\bf s}_{i}$ of the neighboring atoms. (Indeed, the analysis of these matters lead us to interesting new spinor calculus results [@usp]). Below, we discuss the cases where linearly polarized, circularly polarized and unpolarized light is used to eject the photoelectrons.
The paper is organized as follows. In section II, we calculate the photoelectron hologram, including the exchange scattering contributions which give a linear dependence on the spins of the atoms in the near-neighborhood of the photoelectron source atom. Often the emitting atoms are centers of point symmetry, e.g. $C_{nv}$, for the crystalline surfaces being investigated. In those cases, we can use symmetry to aid in spin determination, as we discuss in section III. We then proceed in section IV to illustrate the technique for a specific example of a $C_{4v}$ environment. In section V, we discuss various schemes to extract spin information from the holograms. Finally, in section VI, we comment on the interesting case of rare earth magnetism, for which the theory of spin holography, as formulated below, needs to be generalized. We conclude and summarize the results in section VII.
Electron Holograms
==================
Initially, to be definite, we consider photoelectrons emitted from a $p_{1/2}$ subshell of a source atom following the absorption of photons linearly polarized in the z-direction. The $p_{1/2}$ subshell consists of two electrons with magnetic quantum numbers $m_{j}$ = $+1/2$ and $m_{j} = -1/2$ respectively. The primary electron wave emitted from the $m_{j}$-state is a spherically outgoing wave of wave number k, $$|\psi^{0}_{m_{j}} \rangle
\sim |\chi_{m_{j}} (\hat{\bf r}) \rangle \; \frac{\exp(ikr)}{r} \; \; ,
\label{e:pw}$$ where $|\chi_{m_{j}} (\hat{\bf r}) \rangle$ is the appropriate angular dependent spinor of the primary electron wave. In the reference frame pictured in Fig.(1), these spinors (in the dipole approximation) are proportional to $$\begin{aligned}
&&|\chi_{1/2} (\hat{\bf R}) \rangle =
\left(
\begin{array}{c}
\cos^{2} (\theta ) + c' \\
\sin (\theta ) \cos(\theta ) \exp (i\phi )
\end{array} \right) \; \; ;
\nonumber \\
\nonumber \\
&&|\chi_{-1/2} (\hat{\bf R}) \rangle =
\left(
\begin{array}{c}
\sin (\theta ) \cos(\theta ) \exp (-i\phi ) \\
\cos^{2} (\theta ) + c' \\
\end{array} \right) \; \; ,
\label{e:sp}\end{aligned}$$ where $\theta$ is the usual polar angle and $\phi$ the azimuthal angle of $\hat{\bf R}$ in Eq.(\[e:sp\]), and in the following, $c'=(c-1)/3$, where c is the ratio of the radial matrix elements $M_{0}$ ($M_{2}$) and corresponding phase shifts $\delta_{0}$ ($\delta_{2}$) for emission into the continuum s- or d-states, c = $(M_{0}/M_{2})
\exp[i(\delta_{0}-\delta_{2})]$.
The primary wave is scattered by an atom i, giving a scattered wave $|\psi^{i}_{m_{j}} \rangle$. We suppose that $\rm{k \; r}_{i} >> 1$ and approximate the scattered wave by that of an incident plane wave with wave vector $k\hat{\bf r}_{i}$, see [@Bart2]–[@Bart4] for a discussion of this approximation in holography. In this case $|\psi^{i}_{m_{j}} \rangle$ takes the form, $$\begin{aligned}
|\psi^{i}_{m_{j}} (\hat{\bf R}) \rangle
= \frac{\exp (ik r_{i}) } { r_{i} } \;
f(\hat{\bf R},\hat{\bf r}_{i}) \; \exp (-ik\hat{\bf R}\cdot \hat{\bf r}_{i})
\; |\chi_{m_{j}} (\hat{\bf r}_{i}) \rangle \; \frac{\exp(ikR)}{R} \; \; ,
\label{e:psimj}\end{aligned}$$ where $f(\hat{\bf R},\hat{\bf r}_{i})$ is the scattering amplitude for the scattering of the photoelectron, incident along the $\hat{\bf r}_{i}$-direction and scattered in the $\hat{\bf R}$-direction.
If now we neglect multiple scattering, then $|\psi_{m_{j}} ({\bf r}) \rangle = |\psi^{0}_{m_{j}} ({\bf r}) \rangle
+ \sum_{i} |\psi^{i}_{m_{j}} ({\bf r}) \rangle$ represents the photoelectrons emitted from the $m_{j}$-state. The number of electrons emitted in direction $\hat{\bf R}$, per unit of solid angle is, up to a constant of proportionality which depends on the intensity of the incident photons and the counting time, equal to the ‘intensity’ $I(\hat{\bf R}) \equiv R^{2} \; \sum_{m_{j}} \langle \psi_{m_{j}} ({\bf R}) |
\psi_{m_{j}} ({\bf R}) \rangle$. Using the above expression we find $$\begin{aligned}
I(\hat{\bf R}) &=& \sum_{m_{j}} \; \left[
\langle \chi_{m_{j}} (\hat{\bf R}) | \chi_{m_{j}} (\hat{\bf R}) \rangle +
\; \sum_{i} 2 \; Re \left( [\exp(ikr_{i})/r_{i}]
\right.
\right.
\nonumber \\
&&
\left.
\left.
\langle \chi_{m_{j}} (\hat{\bf R}) | f(\hat{\bf R},\hat{\bf r}_{i}) |
\chi_{m_{j}} (\hat{\bf r}_{i}) \rangle \exp (-ik \hat{\bf R} \cdot
\hat{\bf r}_{i}) \right) + \cdots \right]
\label{e:i}\end{aligned}$$ In (\[e:i\]) we keep only zero and first order terms in the scattering amplitudes, f, neglecting the effects of multiple scattering and self interference terms (such terms tend to degrade the holographic images and can be partially eliminated by using special kernels in the hologrgaphic transform, as well as sums over holograms collected at different energies, as has been discussed extensively in the literature [@Bart5]–[@Ton]).
The phase factor $\exp(-ik\hat{\bf R}\cdot {\bf r}_{i})$ in Eq.(\[e:i\]) gives holographic fringes in the angular intensity pattern; and transforms of $I(\hat{\bf R})$ by suitable kernels proportional to $\exp(ik\hat{\bf R}\cdot {\bf r})$, where ${\bf r}$ is the parameter of the transform, peak in the vicinities of ${\bf r} = {\bf r}_{i}$, yielding atomic images.
If scattering atom i has a spin, then the scattering amplitude for coherent scattering, $f(\hat{\bf R},\hat{\bf r}_{i})$, depends on the thermally averaged expectation value of the spin vector, ${\bf s}_{i}$, of the scattering atom, $$\begin{aligned}
f(\hat{\bf R},\hat{\bf r}_{i}) = f_{0}(\hat{\bf R},\hat{\bf r}_{i}) +
f_{s}(\hat{\bf R},\hat{\bf r}_{i}) \sigma \cdot {\bf s}_{i} \; \; ,
\label{e:f}\end{aligned}$$ where the first term is the spin independent contribution $f_{0}$ and the second term is the exchange contribution, proportional to the scalar product of ${\bf s}_{i}$ with the Pauli-spin operator $\sigma$ of the photoelectron. If we could treat the scattering atoms as spherically symmetric then $f_{0}$ and $f_{s}$ would only depend on $\hat{\bf R} \cdot
\hat{\bf r}_{i}$. In reality, the valence shell electron distributions are affected by the presence of the neighboring atoms. The charge (and spin) distribution within a Wigner-Seitz cell representing an atom is not spherically symmetric about the cell’s center. This is particularly true for for the valence shell electrons having the uncompensated spins which is our primary concern. In the copper oxides of interest in the high $T_{c}$ materials, for example, the uncompensated spins are believed to occupy $x^{2}-y^{2}$ d-orbitals on the copper ions. Furthermore, we note that the scattering amplitude (\[e:f\]) describes exchange scattering by a magnetic ion if the orbital moments are ‘quenched’ and the spin direction is free to change without changing the spatial distribution of the charge or spin distributions. This description is accurate for transition elements, but fails for rare earth ions (except for ${\rm Gd}^{3+}$, in which case $L = 0$). In that case, there is strong spin-orbit correlation and both $f_{0}$ and $f_{s}$ in Eq.(\[e:f\]) must themselves be taken to be functions of ${\bf s}_{i}$. In the following, we shall assume a scattering amplitude of the form (\[e:f\]) and we will return to the more complicated but important case of rare earth ions later (section VI). In any case, due to the spin dependence of the scattering amplitude, it should be evaluated inside the spinor bracket of Eq.(\[e:i\]). With the scattering amplitude (\[e:f\]) we obtain $$\begin{aligned}
I(\hat{\bf R}) &=& \langle \langle \hat{\bf R} | \hat{\bf R} \rangle \rangle
+ 2 Re \left( \sum_{i} \; \; [\exp(ikr_{i})/r_{i}] \; \times
\right.
\nonumber \\
&&
\left.
\left[
f_{0}(\hat{\bf R},\hat{\bf r}_{i}) \;
\langle \langle \hat{\bf R} | \hat{\bf r}_{i} \rangle \rangle
+ f_{s}(\hat{\bf R},\hat{\bf r}_{i}) \;
\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle \rangle
\cdot {\bf s}_{i} \; \right] \exp (-ik{\bf R} \cdot {\bf r}_{i}) + \ldots
\; \right) \; ,
\label{e:i2}\end{aligned}$$ where we represent the sum over the initial electron states $m_{j}$ of the ‘spinor interference brackets’ by $$\begin{aligned}
&& \langle \langle \hat{\bf R}| \hat{\bf r}_{i} \rangle \rangle
\; \; = \sum_{m_{j}} \; \langle \chi_{m_{j}} (\hat{\bf R}) | \chi_{m_{j}}
(\hat{\bf r}_{i}) \rangle \; \; \; ,
\nonumber \\
&& \langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle \rangle
= \sum_{m_{j}} \; \langle \chi_{m_{j}}(\hat{\bf R}) | \sigma |
\chi_{m_{j}} (\hat{\bf r}_{i}) \rangle \; .
\label{e:db}\end{aligned}$$
Eq.(\[e:i2\]) includes an interference term that is proportional to $f_{s}$ and the spin ${\bf s}_{i}$ of the scattering atom so that the intensity is sensitive to the spins of the neighboring atoms if the ‘spin interference matrix element’, $\langle \langle \hat{\bf R} | \sigma
| \hat{\bf r}_{i} \rangle \rangle$, does not vanish. Generally, even for linearly polarized light (or indeed, unpolarized light) the spin interference matrix element $\it{does \; not \; vanish}$ if the electron is ejected from a spin-orbit split sublevel.
In our example of photoemission from a $p_{1/2}$ level, the interference matrix elements for photons of arbitrary polarization $\hat{\epsilon}$ ($\hat{\epsilon} = \hat{\bf z}$, for z-linearly polarized, $\hat{\epsilon} = (\hat{\bf x} \pm i \hat{\bf y})/\sqrt{2}$ for right- and left-hand polarized light etc ....) are equal to $$\begin{aligned}
&& \langle \langle \hat{\bf R} | \hat{\bf r} \rangle \rangle
= 2 \{ (\hat{\epsilon}^{*} \cdot \hat{\bf R}) \; (\hat{\epsilon} \cdot
\hat{\bf r} ) \; (\hat{\bf r} \cdot \hat{\bf R}) \; + |c'|^{2}
+ c'^{*} \; (\hat{\epsilon} \cdot \hat{\bf r}) \;
(\hat{\epsilon}^{*} \cdot \hat{\bf r})
\; + c' (\hat{\epsilon}^{*} \cdot \hat{\bf R}) \;
(\hat{\epsilon} \cdot \hat{\bf R}) \; \} \; ,
\nonumber \\
&&
\langle \langle \hat{\bf R} | \sigma | \hat{\bf r} \rangle \rangle
= -2 \; i \; \{
|c'|^{2} \; (\hat{\epsilon}^{*} \times \hat{\epsilon}) - c' \;
(\hat{\epsilon} \times \hat{\bf R}) \; (\hat{\epsilon}^{*} \cdot
\hat{\bf R}) + c'^{*} \; (\hat{\epsilon}^{*} \times \hat{\bf r}) \;
(\hat{\epsilon} \cdot \hat{\bf r})
\nonumber \\
&& \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;
\; \; \; \; \; \; \; \; \; \; \; \; \; \; \;
+ (\hat{\epsilon}^{*} \cdot \hat{\bf R}) \;
(\hat{\epsilon} \cdot \hat{\bf r}) \; (\hat{\bf R} \times \hat{\bf r})
\} \; \; .
\label{e:int}\end{aligned}$$
For emission from a filled $p_{3/2}$-shell the intensity interference matrix element, $\langle \langle \hat{\bf R} | \hat{\bf r} \rangle \rangle$, is twice the intensity matrix element of Eq.(\[e:int\]), as a consequence of the $p_{3/2}$-shell having twice as many electrons. Similarly, the $p_{3/2}$ spin interference matrix element, $\langle \langle \hat{\bf R} |\sigma | \hat{\bf r} \rangle \rangle$ is the negative of the spin interference matrix element shown on the last line of Eq.(\[e:int\]) [@spinorb], which follows from the fact that the $p_{3/2}$ and $p_{1/2}$ spin matrix elements add up to the spin matrix element for photoemission from a p-shell which, in the absence of spin-orbit interaction, vanishes.
It is a feature of spherical wave electron holography that the interference between electron waves emitted in $\it{different}$ directions is recorded. As a consequence, it is not the spin polarization of the electrons emitted in the direction of the scattering atom $\hat{\bf r}_{i}$ that gives the holographic spin dependence, but the interference matrix element of the emission directions $\hat{\bf R}$ and $\hat{\bf r}_{i}$, $\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i}
\rangle \rangle$. The occurrence of these quantities in our theory leads to interesting new aspects of angular momentum theory.
That the spin matrix element is of the form of Eq.(\[e:int\]) can be understood from general considerations – the matrix element must be ‘bilinear’ in the photon polarization, meaning that each term contains $\hat{\epsilon}$ and $\hat{\epsilon}^{*}$. Furthermore, the spin matrix element must be a pseudo vector made up of $\hat{\epsilon}$, $\hat{\epsilon}^{*}$, $\hat{\bf R}$, and $\hat{\bf r}_{i}$. In the dipole approximation, the wave emitted from a p-shell is an admixture of an s and a d-wave, from which follows that each term contains $\hat{\bf R}$ and $\hat{\bf r}_{i}$ twice or not at all. Based on these considerations we can deduce the general form of the individual terms that make up the $p_{1/2}$ (or $p_{3/2}$) spin matrix element.
Some remarkable results follow from Eq.(\[e:int\]). For example, the spin interference matrix element for z-linearly polarized light does not vanish : $$\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle
\rangle_{\bf z} =
\; -2 \; i \; \{ - c' (\hat{\bf z} \times \hat{\bf R}) \; (\hat{\bf z} \cdot
\hat{\bf R}) \; + c'^{*} (\hat{\bf z} \times \hat{\bf r}_{i} )
\; (\hat{\bf z} \cdot \hat{\bf r}_{i})
+ (\hat{\bf z} \cdot \hat{\bf R}) \; (\hat{\bf z} \cdot \hat{\bf r}_{i})
\; (\hat{\bf R} \times \hat{\bf r}_{i}) \} \; ,
\label{e:spinlin}$$ implying that linearly polarized photons give spin dependent photoelectron intensities. The expected spin of the directly emitted photoelectrons in the ${\bf R}$-direction is proportional to $\langle \langle
\hat{\bf R} | \sigma | \hat{\bf R} \rangle \rangle$, which can be obtained from Eq.(\[e:spinlin\]), putting $\hat{\bf r}_{i} \rightarrow \hat{\bf R}$. This gives an electron spin polarization that is proportional to $$\langle \langle \hat{\bf R} | \sigma | \hat{\bf R} \rangle \rangle_{\bf z} =
- \; \frac{4}{3} \; \sin (\delta_{0}-\delta_{2}) \; \frac{M_{0}}{M_{2}} \;
(\hat{\bf z} \times \hat{\bf R}) \; (\hat{\bf z} \cdot \hat{\bf R}) \; \; ,
\label{e:spind}$$ where we replaced the imaginary part of the $c'$-parameter by its expression in terms of the phase shifts and the radial dipole matrix elements. The result of Eq.(\[e:spind\]) makes it clear that the resulting spin polarization is in fact caused by the interference of the outgoing s and d-waves, as was pointed out previously [@Heinz]. Nevertheless, even if the spin polarization (\[e:spind\]) of the unscattered photoelectrons vanish, because, for example, the electron is emitted as a pure d-wave, the interference spin matrix element does $\it{not}$ vanish. The pure d-wave photoemission corresponds to $c'=-1/3$ giving $$\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle \rangle _{\bf z}
= -2 \; i \; \{ \frac{1}{3}
(\hat{\bf z} \times \hat{\bf R}) \; (\hat{\bf z} \cdot
\hat{\bf R}) - \frac{1}{3} (\hat{\bf z} \times \hat{\bf r}_{i}) \;
(\hat{\bf z} \cdot \hat{\bf r}_{i}) \;
+ \; (\hat{\bf z} \cdot \hat{\bf R}) \; (\hat{\bf z} \cdot \hat{\bf r}_{i})
\; (\hat{\bf R} \times \hat{\bf r}_{i} ) \} \; \; .
\label{e:spindw}$$
Thus the d-wave photoelectron incident upon and scattered from the scattering atom is not spin polarized, (i.e. $\langle \langle \hat{{\bf r}}_{i}|\sigma|
\hat{{\bf r}}_{i} \rangle \rangle = 0$), nor is the unscattered wave emitted in the $\hat{\bf R}$-direction polarized, (i.e. $\langle \langle \hat{{\bf R}}|\sigma|\hat{{\bf R}}
\rangle \rangle = 0$); but the interference between the photoelectron waves emitted in the ${\bf r}_{i}$-direction and scattered into the direction $\hat{\bf R}$ with the waves emitted directly in that direction does result in a polarization and gives a photoelectron intensity sensitive to the spin of the scattering atom. Another interesting consequence of this angular correlation in the photoelectron spin is that the interference of the directly emitted and scattered electron waves gives a finite spin polarization to the photoelectrons, even though the directly emitted electrons are spin unpolarized and the scattering atom is not magnetic.
Perhaps even more surprising than the fact that incident linearly polarized light gives rise to spin-sensitive holograms is that so does unpolarized light. Note that unpolarized light incident along the z-direction, say, on the source atom gives photoelectron intensities equal to the sum of those for x- and y-linearly polarized beams. Now the sum of x-,y- and z-polarized beams gives $$\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle \rangle
= - \; 2 \; i \; (\hat{\bf R} \times \hat{\bf r}_{i} ) \;
(\hat{\bf r}_{i} \cdot \hat{\bf R}) \; \; ,
\label{e:xyz}$$ thus, $$\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle \rangle
_{\bf x} \; +
\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle \rangle
_{\bf y} =
- \langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle \rangle
_{\bf z} - 2 \; i \; (\hat{\bf R} \times \hat{\bf r}_{i}) \;
({\bf r}_{i} \cdot \hat{\bf R}) \; \; ,
\label{e:z}$$ and substituting from Eq.(\[e:spinlin\]) we see that unpolarized light also gives a non-vanishing spin contribution to the photoelectron intensity and thus can serve to photoemit electrons with an intensity hologram that is sensitive to the spins of the nearby atoms.
These surprising results are caused by the spin-orbit correlation in the initial shell, which is preserved in the emitted d-waves. The remarkable consequences clearly indicate that the interference matrix elements contain novel and interesting physics.
Returning to the intensity hologram of Eq.(\[e:i2\]), it is clear from the subsequent discussion of the interference matrix elements that the spin contributions to the intensity hologram don’t vanish if the electrons are emitted from an atomic spin-orbit split shell. The spin contributions are, however, relatively small, only making an addition of the order of $|f_{s} {\bf s}_{i} / f_{0}|$ ($\simeq 0.05$, e.g., for electron scattering by an iron atom at 100 eV) to the ordinary, spin independent ‘charge’ hologram. [*The challenge then is to extract the spin information from the hologram*]{}.
Spin Analysis
=============
The question becomes how to isolate the spin dependent part of the hologram from the much larger ‘charge’ part background.
Referring back to Eq.(\[e:i2\]), if we were able to reverse all the ${\bf s}_{i}$’s of the sample then the difference hologram would contain only the spin dependent terms. Reversing the sample’s spins, however, would be feasible only for ferro- or ferrimagnetic samples. Alternatively, the difference of holograms below and above the magnetic ordering temperatures can serve to determine the spins in the cases where we can assume that the change of the spin independent factors in Eq.(\[e:i2\]) is small.
Other than those two methods depending on changing the magnetization of the sample by changing the magnetic field or the temperature, which may not be feasible, there are other methods which can serve our purposes:
If the primary wave, (\[e:sp\]), had only an s-wave component rather than the d-s mixture, then only the $|c'|^{2}$ terms in Eq.(\[e:int\]) would be present. In that case $\langle \langle \hat{\bf R} | \hat{\bf r}_{i}
\rangle \rangle \sim |c'|^{2}$, $\langle \langle \hat{\bf R} | \sigma
| \hat{\bf r}_{i} \rangle \rangle _{\pm} \sim \pm |c'|^{2} \; \hat{\bf z}$ for $\hat{\epsilon} = (\hat{\bf x} \pm \hat{\bf y})/\sqrt{2}$, and the difference holograms for incident right and left-hand circularly polarized light around, successively x,y and z-axes would determine all three components of the spin vectors. In fact, however, the s-wave component is almost an order of magnitude smaller than the dominant d-wave contribution in the case of photoemission from a p-shell at the relevant emission energies ($\sim 100 - 200$ eV). Furthermore, we note that the extraction of the spin terms from right-left polarization differences contain spin independent ‘charge’ contributions. Indeed, with Eq.(\[e:int\]) we find $$\langle \langle \hat{\bf R} | \hat{\bf r}_{i} \rangle \rangle _{+}
\; - \langle \langle \hat{\bf R} | \hat{\bf r}_{i} \rangle \rangle _{-}
\; = - \; 2 \; i \; \left[
(\hat{\bf r}_{i} \cdot \hat{\bf R} ) \;
(\hat{\bf R} \times \hat{\bf r}_{i}) \cdot \hat{\bf z}
\; \right] \; \; ,
\label{e:intpm}$$ and referring back to Eq.(\[e:i2\]), we see that the difference hologram does indeed contain ‘charge’ contributions.
We can, however, use $\it{symmetry}$ to aid in spin determination. Often the emitting atoms are centers of point symmetry, e.g. $C_{nv}$, for the crystalline surfaces being investigated. Any differences detected after subjecting the hologram to a point symmetry group operation is then the result of the symmetry breaking spin contributions. Based on this general idea, we work out practical schemes to construct ‘spin holograms’. For the sake of simplicity, we start by illustrating the main idea for the case of reflection symmetry in subsection \[3a\]. In subsection \[3b\], we approach the problem from a more general perspective.
Reflection Spin Holography {#3a}
--------------------------
Let $I(\hat{\bf R},\hat{\epsilon};m)$ represent the photoelectron intensity emitted in the $\hat{\bf R}$-direction, for photon polarization $\hat{\epsilon}$, matter state m, and let $I(\hat{\bf R}',\hat{\epsilon}';m')$ be the corresponding quantities following a spatial reflection. Since the Hamiltonian is invariant under reflections, $$I(\hat{\bf R},\hat{\epsilon};m) =
I(\hat{\bf R}',\hat{\epsilon}';m') \; \; .
\label{e:ref}$$ Also, $$I(\hat{\bf R}',\hat{\epsilon}';m) =
I(\hat{\bf R},\hat{\epsilon};m') \; \; ,
\label{e:ref2}$$ since the square of the reflection is the unit operator. Now ‘m’ represents the initial state of the source atom, centered at the origin, and the various neighboring atoms centered at ${\bf r}_{i}$, with spins ${\bf s}_{i}$, and other properties, including valence electron spatial distributions, $o_{i}$ : m = $[ {\bf r}_{1},{\bf s}_{1}, o_{1} ; {\bf r}_{2},{\bf s}_{2}, o_{2} ,
\ldots , {\bf r}_{i},{\bf s}_{i}, o_{i} ; \ldots ]$, m’ = $[ {\bf r}_{1}',{\bf s}_{1}', o_{1}' ; \ldots ]$. Since ${\bf s}$ is an axial vector, ${\bf s}_{//}' = - {\bf s}_{//}$, ${\bf s}_{\perp}' = {\bf s}_{\perp}$, where the $//$ and $\perp$ subscripts refer to components respectively parallel and perpendicular to the plane of reflection.
If the source atom is in a site with $C_{nv}$ point symmetry, magnetic ordering of the neighboring atoms may break that symmetry, and the difference hologram, $$\begin{aligned}
I_{V} (\hat{\bf R},\hat{\epsilon};m) &\equiv& \frac{1}{2} \;
\left[ I(\hat{\bf R},\hat{\epsilon};m) - I(\hat{\bf R}',\hat{\epsilon}';m)
\right] \; ,
\nonumber \\
&=& \frac{1}{2} \; \left[ I(\hat{\bf R},\hat{\epsilon};m)
- I({\bf R},\hat{\epsilon};m') \right] \; \; ,
\label{e:iv}\end{aligned}$$ contains only the symmetry breaking spin terms if the reflection plane, V, is a symmetry plane. This is so because if atom ‘j’ is carried into atom ‘k’ by the reflection then $o_{j}' = o_{k}$, by symmetry, and the non-spin dependent part of the hologram, the ‘charge’ hologram is unchanged by the reflection.
If atom ‘i’ lies in the reflection plane then ${\bf r}_{i} =
{\bf r}_{i}' \;$,$ \; o_{i} = o_{i}' \;$,$ \; {\bf s}_{i,//} = -{\bf s}_{i,//}'
\;$, $ \; {\bf s}_{i,\perp} = {\bf s}_{i,\perp}' \; $, and the contribution of ‘i’ to the difference hologram $I_{V}$ is (see Eq.(\[e:i2\])) $$I_{V}(\hat{\bf R}) = 2 \; Re \{ [\exp(ikr_{i})/r_{i}] \;
f_{s}({\bf R},{\bf r}_{i}) \langle \langle \hat{\bf R} | \sigma
| \hat{\bf r}_{i} \rangle \rangle \cdot {\bf s}_{i,//} \;
\exp(-ik\hat{\bf R}\cdot {\bf r}_{i}) \} \; \; .
\label{e:iv2}$$
On the other hand, if atom ‘j’ does not lie in the reflection plane ‘V’ and is carried into atom ‘k’ by the reflection then ${\bf s}_{k}
\rightarrow \frac{1}{2} ({\bf s}_{k} -{\bf s}_{j}')$ = $\frac{1}{2}({\bf s}_{k}
+ {\bf s}_{j,//} - {\bf s}_{j,\perp})$, and the contribution of site ‘k’ to $I_{V}$ becomes $$I_{V}(\hat{\bf R}) = 2 \; Re \{ [\exp(ikr_{k})/r_{k}] \;
f_{s}({\bf R},{\bf r}_{k}) \langle \langle \hat{\bf R} | \sigma
| \hat{\bf r}_{k} \rangle \rangle \cdot \frac{1}{2} ({\bf s}_{k}
+{\bf s}_{j,//}-{\bf s}_{j,\perp}) \;
\exp(-ik\hat{\bf R}\cdot {\bf r}_{k}) \} \; \; .
\label{e:iv3}$$ The hologram $I_{V}$ given by summing the in-plane, Eq.(\[e:iv2\]), and out of plane, Eq.(\[e:iv3\]), terms can then be transformed, as we shall discuss later, to determine components of the spins of the various atoms.
Projection Spin Holography {#3b}
--------------------------
The above considerations regarding reflections illustrate the use of symmetry to obtain spin holograms. If G is an element, a reflection or a rotation, of a point symmetry group which leaves the charge or chemical neighborhood invariant, then the difference hologram involving G has only the symmetry breaking spin contributions. The argument is essentially that given above. For example, the difference between the emission intensity $I(\hat{\bf R},\hat{\epsilon};m)$ measured in direction $\hat{\bf R}$ with incident photons of polarization $\hat{\epsilon}$ and the intensity measured in direction $G[\hat{\bf R}]$ with photons of polarization $G[\hat{\epsilon}]$, $I(G[\hat{\bf R}],G[\hat{\epsilon}];m)$ contains only spin contributions because the charge contributions cancel.
To see that, we note that the Hamiltonian is invariant under all of the transformations of the euclidean group (except for the small parity non-conserving terms which are irrelevant for our considerations). Thus, it follows that $$I(\hat{\bf R},\hat{\epsilon};m) = I(G[\hat{\bf R}],G[\hat{\epsilon}];G[m]) \; ,
\label{e:genref}$$ which is the generalization of (\[e:ref\]), and where, as before, m represents the initial state of the source atom and the neighboring atoms, m = $[{\bf r}_{1},{\bf s}_{1},o_{1};{\bf r}_{2},{\bf s}_{2};o_{2}, \ldots ]$. Also, as in Eq.(\[e:ref2\]), $$I(G^{-1}[\hat{\bf R}],G^{-1}[\hat{\epsilon}];m) =
I(\hat{\bf R},\hat{\epsilon} ; G[m]) \; \; ,
\label{e:genref2}$$ where $G[m] = [G[{\bf r}_{1}],G[{\bf s}_{1}], G
[o_{1}]; \ldots , G[{\bf r}_{j}],[G[{\bf s}_{j}], G
[o_{j}]; \ldots ]$. Since $G$ and $G^{-1}$ belong to the symmetry group, for every atom $j$ there is an atom $k$ for which ${\bf r}_{j} =
G[{\bf r}_{k}]$ (and $o_{j} = G[o_{k}]$). If we denote the label $k$ by $k = g[j]$, then $G[{\bf s}_{k}] = G[{\bf s}_{g[j]}]$ and we can represent the transformed initial state as $G[m] = [{\bf r}_{1},G[{\bf s}_{g[1]}], o_{1} ; \ldots
; {\bf r}_{j},G[{\bf s}_{g[j]}],o_{j} ; \ldots ],$ representing a neighborhood which has the same ‘charge’ distribution’ as m, but a different spin arrangement: ${\bf s}_{1}$ apears now as $G[{\bf s}_{1}]$ at the site of $G[{\bf r}_{1}]$ etc... . Consequently, only spin dependent terms do not cancel in the difference hologram $$I_{G} \equiv \frac{1}{2}
\; \left[ I(\hat{\bf R},\hat{\epsilon};m)-I(G^{-1}[{\bf R}],
G^{-1}[\hat{\epsilon}];
m) \right] \; .
\label{e:ig}$$ More generally, any linear combination of the type $\sum_{G} a(G) \;
I(G^{-1}[\hat{\bf R}],G^{-1}[\hat{\epsilon}];m)$, where the sum $\sum_{G}$ extends over all elements G of the symmetry group of the ‘charge’ environment and where the sum of the coefficients $a(G)$ vanishes, $\sum_{G} a(G) = 0$, results in a hologram of which the charge contributions cancel out but not the spin terms. In fact, the resulting angular pattern, $\sum_{G} a(G) \;
I(G[\hat{\bf R}],G[\hat{\epsilon}];m)$ is a hologram of the spin arrangment obtained from the actual spin distribution, by replacing the spin ${\bf s}_{j}$ at site ${\bf r}_{j}$ by $\sum_{G} a(G) G[{\bf s}_{[g(j)]}]$ : $$\begin{aligned}
I_{a}(\hat{\bf R},\hat{\epsilon};m) &\equiv&
\sum_{G} a(G) I(G^{-1}[\hat{\bf R}],G^{-1}[\hat{\epsilon}];m)
\; \; = \; \sum_{G} a(G) I(\hat{\bf R},\hat{\epsilon}; G[m])
\nonumber \\
&=& 2 \; \sum_{j} Re \{
[exp(ikr_{j})/r_{j}]\; f_{s}(\hat{\bf R},\hat{\bf r}_{j})
\; \exp(-ik\hat{\bf R} \cdot \hat{\bf r}_{j}) \;
\nonumber \\
&& \; \; \; \; \; \; \; \; \; \; \;
\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{j} \rangle \rangle
\cdot \sum_{G} a(G) G[{\bf s}_{g[j]}] \} + \cdots \; \; .
\label{e:ia}\end{aligned}$$
Consider then the ‘star’ of an atom at ${\bf r}_{1}$, i.e., the atoms at $({\bf r}_{1}, {\bf r}_{2}, \cdots ,
{\bf r}_{N})$ into which ${\bf r}_{1}$ is carried by the operations of the group. (Thus for the group $C_{nv}$, N=n, if ${\bf r}_{1}$ lies in a reflection plane, and N=2n, the order of the group, if it doesn’t.) Then the spins on the N atoms constitute a 3N dimensional linear vector space which transforms into itself under the group and which can be decomposed into irreducible subspaces forming bases of the irreducible representation of the group by the well known methods of group theory [@LL]. We may represent a vector in this 3N dimensional space as ${\bf s}_{(N)} \equiv \{ {\bf s}_{1},{\bf s}_{2}, \ldots ; {\bf s}_{N}
\}$ = $ ( {\bf s}_{1x},{\bf s}_{1y},{\bf s}_{1z} ; {\bf s}_{2x},{\bf s}_{2y},
{\bf s}_{2z} ; \ldots )$. From the general principles of group theory, we know that the ${\bf s}_{(N)}$-vector can be decomposed into components that transform according to the irreducible representation ‘$\alpha$’ of the symmetry group. The decomposition, ${\bf s}_{(N)}$ = $\sum_{i,\alpha}
{\bf s}_{(N)}^{(\alpha,i)}$, is achieved by means of the idempotent operator, $e^{\alpha}_{ii} = \frac{l_{\alpha}}{h}
\sum_{G} D_{ii}^{\alpha} (G^{-1}) G$, which, when applied to a linear vector space, selects out the $(\alpha,i)$-component of the vector : $${\bf s}_{(N)}^{(\alpha,i)} \; = \; \left( \frac{l_{\alpha}}{h} \right)
\; \sum_{G} D_{ii}^{\alpha}(G^{-1}) \; G[{\bf s}_{(N)}] \; \; ,
\label{e:proj}$$ where $D_{ij}^{\alpha}(G)$ is the matrix representation of G in the irreducible representation $\alpha$, $h$ is the order of the symmetry group and i takes on values from 1 to $l_{\alpha}$, the dimension of the representation.
Returning to the spin hologram of Eq.(\[e:ia\]), it is then clear that identifying the $a(G)$ coefficients with $\left( \frac{l_{\alpha}}{h}
\right) \; D_{ii}^{\alpha} (G^{-1})$, gives a pattern that is the hologram of the total spin ${\bf s}_{(N)}$ projected onto the irreducible $(\alpha,i)$-mode ${\bf s}_{(N)}^{(\alpha,i)}$.
This procedure, projecting out the irreducible modes of the photoelectron hologram, $$I^{(\alpha,i)} (\hat{\bf R},\hat{\epsilon};m)
= \left( \frac{l_{\alpha}}{h} \right) \; \sum_{G} \; D^{\alpha}_{ii} (G^{-1})
\; I(G^{-1}[\hat{\bf R}],G^{-1}[\hat{\epsilon}];m) \; \; ,
\label{e:ialpha}$$ giving a hologram of the irreducible $(\alpha,i)$-mode of the spin pattern, is completely general and gives the maximum information that can be obtained from symmetry.
Now we take $\alpha = 1$ for the identical representation, $D^{1}(G) = 1$, so that the vector ${\bf s}^{1}_{(N)}$, $${\bf s}^{1}_{(N)} = \left(\frac{1}{l}\right)
\sum_{G} \; G^{-1}[{\bf s}_{(N)}] \; \; ,
\label{e:inv}$$ is invariant under all G’s and thus does not appear in the difference holograms as in (\[e:ia\]). Such a vector is invariant under all symmetry operations G and cannot be distinguished from the charge environment using symmetry alone. The number of such linearly independent invariant vectors is given as usual by the compound character averaged over all the group elements. For $C_{nv}$ there is one such vector for the N=n atom star: ${\bf s}_{i} = C \hat{\bf z} \times {\bf r}_{i}$. Consequently, out of the 3n linearly independent vectors $3n-1$ can be determined from the holograms (\[e:ialpha\]). For the N=2n star there are three such vectors : in addition to that for the N=n case, there are also $\pm \hat{\bf z}$, and $\pm \hat{\bf r}_{i}$, where the signs for the two semi-stars, each consisting of n atoms carried into each other by the $C_{n}$ subgroup, differ. In this case, one determines $6n-3$ linearly independent vectors by our method.
An Example with $C_{4v}$ Symmetry
=================================
We illustrate the projection scheme for photoemission from an atom placed in the environment pictured in Fig.(2). Four identical neighboring atoms are located in the same horizontal plane with azimuthal angles $\phi = 0 \;$,$ \; \pi/2 \; $,$ \; \pi$ and $3\pi/2$, placed at equal distance from the emitting atom. The result is a ‘charge’ environment of $C_{4v}$-symmetry.
The $C_{4v}$-symmetry group has 8 elements (h=8): the identity operation E, 2 rotations in the horizontal plane by $\pi/2$ : $\phi \rightarrow
\phi + \pi/2$ ($C_{4}$) and $\phi \rightarrow \phi - \pi/2$ ($C_{4}'$), one rotation by $\pi$ ($C_{2}$), and 4 reflections with respect to the vertical planes of azimuthal angle $\phi = 0 \;$ ($s_{V}'$), $\phi = \pi/4 \;$ ($s_{d}$), $\phi = \pi/2 \;$ ($s_{V}$), $\phi = 3\pi/2 \;$ ($s_{d}'$). There are 5 classes $\{ E, 2 C_{4}, C_{2}, 2 s_{V}, 2 s_{d} \}$ and 5 irreducible representations costumarily denoted by $\{ A_{1}, A_{2}, B_{1},
B_{2}, E \}$. For the reader’s convenience, we show the character table of $C_{4v}$ in table I.
The spin arrangement consists of four spin vectors (12 components) which generate a 12-dimensional representation of $C_{4v}$: $D^{(12)}$. Reducing the 12 components to modes that transform according to the irreducible representations, we find one $A_{1}$, one $B_{1}$, two $A_{2}$, two $B_{2}$ and three $E$-modes: $$D^{(12)} \; = 1 \; A_{1} \oplus \; 1 \; B_{1} \; \oplus \; 2 \;
A_{2} \oplus \; 2 \; B_{2} \oplus 3 \; E \; .
\label{e:red}$$ Each spin environment is a linear combination of these modes, pictured in Fig(3). Of the irreducible modes, only the $A_{1}$-mode, invariant under all $C_{4v}$-operations cannot be determined by means of the projection scheme, the other 11 components are determined. Notice that one of the $A_{2}$-modes represents a ferromagnetic spin arrangement, with the spins in the z-direction. The in-plane (in the x and y-direction) ferromagnetic spin arrangements are E-modes.
In constructing the spin holograms (\[e:ialpha\]), we take linear combinations of transformed emission patterns, $I(G^{-1}[\hat{\bf R}],
G^{-1}[\hat{\epsilon}];m)$. If $\epsilon = \epsilon_{0} (= \hat{\bf z})$, then since $G[\hat{\epsilon}_{0}] = \hat{\epsilon}_{0}$ for all elements $G$ of $C_{4v}$, only a single photoelectron hologram, $I_{0}(\hat{\bf R})$, must be measured. For notational convenience we indicate in what follows the photon polarization by means of a subscript. With circularly polarized light incident along the z-direction, $\hat{\epsilon} = \hat{\epsilon}_{+}, \; \hat{\epsilon}_{-}$, the two intensity holograms, $I_{+}$ and $I_{-}$, suffice to construct the projection holograms (\[e:ialpha\]).
Representing the symmetry operation by the corresponding change in the azimuthal angle dependence $\{$e.g. $I_{0}(s_{V}[\hat{\bf R}])$ is represented by $I_{0}(\pi-\phi)$ etc...$\}$, we can write the spin holograms for the $A_{2}$, $B_{1}$ and $B_{2}$-modes as $$\begin{aligned}
I^{A_{2}} (\hat{\epsilon}_{0} ;\hat{\bf R}) &=&
\frac{1}{8} \; \{
I_{0}(\phi) + I_{0}(\phi+\pi/2) + I_{0}(\phi-\pi/2) + I_{0}(\phi+\pi)
\nonumber \\
&& - I_{0}(\pi-\phi) - I_{0}(-\phi) - I_{0}(\pi/2-\phi) - I_{0}(-\phi-\pi/2)
\} \; \; ,
\nonumber \\
I^{B_{1}} (\hat{\epsilon}_{0}; \hat{\bf R}) &=&
\frac{1}{8} \; \{
I_{0}(\phi) - I_{0}(\phi+\pi/2) - I_{0}(\phi-\pi/2) + I_{0}(\phi+\pi)
\nonumber \\
&& + I_{0}(\pi-\phi) + I_{0}(-\phi) - I_{0}(\pi/2-\phi) - I_{0}(-\phi-\pi/2)
\} \; \; ,
\nonumber \\
I^{B_{2}} (\hat{\epsilon}_{0}, \hat{\bf R}) &=&
\frac{1}{8} \; \{
I_{0}(\phi) - I_{0}(\phi+\pi/2) - I_{0}(\phi-\pi/2) + I_{0}(\phi+\pi)
\nonumber \\
&& - I_{0}(\pi-\phi) - I_{0}(-\phi) + I_{0}(\pi/2-\phi) + I_{0}(-\phi-\pi/2)
\} \; \; ,
\label{e:mod01}\end{aligned}$$ where we suppressed the dependence on the polar angle, $\theta$, which is invariant under $C_{nv}$ symmetry elements. For the one-dimensional $A$ and $B$-modes, the coefficients can simply be read from the character table (table I). For the two-dimensional $E$-representation, on the other hand, we need to construct an actual representation. The result is: $$\begin{aligned}
I^{(E,1)} (\hat{\epsilon}; \hat{\bf R}) &=&
\frac{1}{4} \; \{
I_{0} (\phi) - I_{0} (\phi + \pi) + I_{0}(\pi-\phi) - I_{0}(-\phi) \} \; \; ,
\nonumber \\
I^{(E,2)} (\hat{\epsilon}, \hat{\bf R}) &=&
\frac{1}{4} \; \{
I_{0} (\phi) - I_{0} (\phi + \pi) - I_{0}(\pi-\phi) + I_{0}(-\phi) \} \; \; .
\label{e:mod02}\end{aligned}$$ The analoguous spin holograms for the circularly polarized photons, are $$\begin{aligned}
I^{A_{2}} (\hat{\epsilon}_{+};\hat{\bf R})
&=& \frac{1}{8} \; \{
I_{+}(\phi) + I_{+} (\phi+\pi/2) + I_{+}(\phi-\pi/2) + I_{+} (\phi+\pi)
\nonumber \\
&& - I_{-}(\pi-\phi) - I_{-}(-\phi) - I_{-}(\pi/2 - \phi) - I_{-}(-\phi-\pi/2)
\} \; \; ,
\nonumber \\
I^{B_{1}} (\hat{\epsilon}_{+};\hat{\bf R})
&=& \frac{1}{8} \; \{
I_{+}(\phi) - I_{+} (\phi+\pi/2) - I_{+}(\phi-\pi/2) + I_{+} (\phi+\pi)
\nonumber \\
&& + I_{-}(\pi-\phi) + I_{-}(-\phi) - I_{-}(\pi/2 - \phi) - I_{-}(-\phi-\pi/2)
\} \; \; ,
\nonumber \\
I^{B_{2}} (\hat{\epsilon}_{+};\hat{\bf R})
&=& \frac{1}{8} \; \{
I_{+}(\phi) - I_{+} (\phi+\pi/2) - I_{+}(\phi-\pi/2) + I_{+} (\phi+\pi)
\nonumber \\
&& - I_{-}(\pi-\phi) - I_{-}(-\phi) + I_{-}(\pi/2 - \phi) + I_{-}(-\phi-\pi/2)
\} \; \; .
\nonumber \\
I^{(E,1)} (\hat{\epsilon}_{+};\hat{\bf R}) &=&
\frac{1}{4} \; \{
I_{+}(\phi) - I_{+} (\phi+\pi) + I_{-}(\pi-\phi) - I_{-}(-\phi) \} \; \; ,
\nonumber \\
I^{(E,2)} (\hat{\epsilon}_{+};\hat{\bf R}) &=& \frac{1}{4} \; \{
I_{+}(\phi) - I_{+} (\phi+\pi) - I_{-}(\pi-\phi) + I_{-}(-\phi) \} \; \; .
\label{e:mod2}\end{aligned}$$
Eqs. (\[e:mod01\]), (\[e:mod02\]) and (\[e:mod2\]) constitute an analysis of the measured hologram $I(\hat{\bf R}, \hat{\epsilon},m)$ into its six components (adding the $A_{1}$ component to those given) appropriate to $C_{4v}$-symmetry. The advantage of this mode of analysis is that each component is the hologram of a relatively simple spin configuration ${\bf s}_{(N)}^{(\alpha,i)}$, as given in Fig.(3) for each neighboring star of atoms. Determination of the components ${\bf s}_{(N)}^{(\alpha,i)}$ then gives ${\bf s}_{N} (= \sum_{(\alpha,i)} {\bf s}_{(N)}^{(\alpha,i)})$.
Extracting information from the spin holograms
==============================================
How then, do we get the ${\bf s}^{(\alpha,i)}_{(N)}$ from the hologram $I^{(\alpha,j)}$? There are two main methods. First we can transform the hologram to give images of a sort of the individual atoms. These ‘images’ take the form shown in Eq.(\[e:imias\]) below of standing spherical waves centered on the various atoms with strengths proportional to ${\bf s}^{(\alpha,i)}_{(N)}$. The advantage of this approach is that it gives a 3D ‘picture’ of the spin distribution on the neighboring atoms. In fact, although these holographic images are of interest in particular for obtaining a first qualitative determination of the spins, we think that the most efficient and accurate spin determination will involve iterative least square fitting of the direct hologram $I^{(\alpha,j)}
({\hat{\bf R}})$.
We first discuss the nature of the holographic image. The field of the holographic image of a traditional (optical) spherical wave hologram $I(\hat{\bf R})$, formed by irradiating the negative of the hologram with a spherical ingoing wave, can be shown [@Bart] to be proportional to $${\cal I}({\bf r}) = \int I(\hat{\bf R}) \; \exp(ik \hat{\bf R} \cdot {\bf r})
\; d \Omega_{\hat{\bf R}} \; \; \; \; ,
\label{e:simtr}$$ where $d \Omega_{\hat{\bf R}}$ denotes an infinitesimal solid angle of the emission direction $\hat{\bf R}$. In electron emission holography, transforms of the type of Eq.(\[e:simtr\]) are called images, and as we shall see the atoms contributing to the hologram appear in the image as centers of spherical waves.
Consider the simple example of a hologram formed by a spherically symmetric primary wave, $\exp(ikr)/r$, scattered coherently with scattering amplitude $f$ by atoms i at positions ${\bf r}_{i}$. In the region $k|{\bf r}-{\bf r}_{i}| >> 1$, the wave scattered by atom i is $$[\exp(ikr_{i})/r_{i}] \; f(\widehat{{\bf r}-{\bf r}_{i}};{\bf r}_{i}) \;
\exp(ik|{\bf r}-{\bf r}_{i}|)/|{\bf r}-{\bf r}_{i}| \; \; \; \; ,
\label{e:scatw}$$ where $\widehat{{\bf r}-{\bf r}_{i}}$ denotes $({\bf r}-{\bf r}_{i})/|{\bf r}-{\bf r}_{i}|$. The interference between the primary and scattered waves then gives holographic fringes in the emission intensity in the far-region, corresponding to $$\sum_{i} \;
2 \; Re \{ [\exp(ikr_{i})/r_{i}] \; f(\hat{\bf R};\hat{\bf r}_{i}) \;
\exp(-ik \hat{\bf R} \cdot {\bf r}_{i}) \; \} \; \; .
\label{e:intt}$$ Although the exchange scattering and the angular correlations expressed by the interference matrix elements complicates the expressions, we can still recognize Eq.(\[e:intt\]) in the projection hologram $I^{(\alpha,j)}$ of Eq.(\[e:ia\]), replacing $f(\hat{\bf R};\hat{\bf r}_{i})$ by an ‘effective’ scattering amplitude: $f \rightarrow f^{(eff)}$, where $$f^{(eff)}
(\hat{\bf R};\hat{\bf r}_{i}) =
\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle \rangle
\cdot {\bf s}^{(\alpha,j)}_{i} \;
f_{s} (\hat{\bf R}; \hat{\bf r}_{i}) \; \; \; ,
\label{e:feff}$$ and where ${\bf s}^{(\alpha,j)}_{i}$ is the spin on atom i in the ($\alpha ,j$)-projection of the ${\bf s}_{(N)}$-spin vector. The interference terms (\[e:intt\]) then contribute ‘imaging’ terms $\sum_{i} \; {\cal I}_{i}({\bf r})$ to the transform, where $$\begin{aligned}
{\cal I}_{i} ({\bf r}) =
[\exp(ikr_{i})/r_{i}] \; \int
f^{(eff)}(\hat{\bf R};\hat{\bf r}_{i})
\exp(ik \hat{\bf R} \cdot [{\bf r} - {\bf r}_{i}]) \;
d \Omega_{\hat{\bf R}} \; \; \; \; .
\label{e:imi}\end{aligned}$$
We calculate the image by expanding the exponential factor in partial waves, $\exp(ik\hat{\bf R}\cdot [ {\bf r}-{\bf r}_{i}])$ $ =
4\pi \sum_{l,m} (i)^{l} \; j_{l} (k|{\bf r}-{\bf r}_{i}|)
Y_{lm} (\widehat{{\bf r}-{\bf r}_{i}}) Y^{*}_{lm}(\hat{\bf R})$, where $j_{l}$ denotes the spherical Bessel functions of the second kind and the $Y_{lm}$-functions are the spherical harmonics. Similarly, $f^{(eff)}
(\hat{\bf R};\hat{\bf r}_{i}) = \sum_{LM} F_{LM} Y_{LM}(\hat{\bf R})$, and assuming for the sake of the argument that the hologram is collected over the full $4\pi$ sphere of emission directions, we obtain a closed expression for ${\cal I}_{i}$: $${\cal I}_{i} ({\bf r}) =
[\exp(ikr_{i})/r_{i}] \; 4 \pi \sum_{LM} F_{LM} \; (i)^{L} \;
j_{L}(k|{\bf r}-{\bf r}_{i}|) \; Y_{LM}(\widehat{{\bf r}-{\bf r}_{i}})
\; \; \; ,
\label{e:imicl}$$ where we made use of the orthonormality of the spherical harmonics. From Eq.(\[e:imicl\]), we see that the holographic image is in fact a sum of standing partial waves. Notice that only the spherically symmetric (L=0) contribution in Eq.(\[e:imicl\]) actually peaks at the position of the scattering atom. For nonzero values of L, the $j_{L}$ Bessel functions reach their maximum value away from the origin [@foot1]. It is of some interest to note that in the region $k|{\bf r}-{\bf r}_{i}| >> 1$, using the asymptotic expansion of the Bessel functions, $j_{l}(x) \sim \sin(x-l\pi/2)/x$, and with $(-1)^{L} Y_{LM}(\hat{\bf r}) = Y_{LM}(-\hat{\bf r})$, the image ${\cal I}_{i}$ is a simple superposition of incoming and outgoung waves: $$\begin{aligned}
{\cal I}_{i} ({\bf r}) &\sim& \; \frac{2\pi}{ik} \; \sum_{LM} \;
F_{LM} \; \{ Y_{LM}(\widehat{{\bf r}-{\bf r}_{i}}) \;
[\exp(ik|{\bf r}-{\bf r}_{i}|)
/|{\bf r}-{\bf r}_{i}| ] \nonumber \\
&& \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;
- Y_{LM}(-[\widehat{{\bf r}-{\bf r}_{i}}]) \;
[ \exp(-ik|{\bf r}-{\bf r}_{i}|)
/|{\bf r}-{\bf r}_{i}| ] \}
\nonumber \\
&=& \frac{2\pi}{ik} \; \; \{ f^{(eff)} (\widehat{{\bf r}-{\bf r}_{i}}) ;
\hat{\bf r}_{i}) \; [ \exp(ik|{\bf r}-{\bf r}_{i}|)
/|{\bf r}-{\bf r}_{i}| ]
\nonumber \\
&& \; \; \; \; \; \; \; \; \; \;
- f^{(eff)} (-[\widehat{{\bf r}-{\bf r}_{i}}]) ;
-\hat{\bf r}_{i}) \; [ \exp(-ik|{\bf r}-{\bf r}_{i}|)
/|{\bf r}-{\bf r}_{i}| ] \} \; \; .
\label{e:imias}\end{aligned}$$
Returning to the problem of determining the spins, we assume that the scattering amplitudes $f_{s}$ in Eq.(\[e:feff\]) can be calculated numerically and can be used in the analysis of the hologram. By analyzing the holographic image of the spin holograms and by determining the particular mixture of partial waves in the image (s,p,d etc...), one can then, in principle, find the spin vectors.
However, regarding the use of calculated scattering amplitudes, we should remark that most numerical schemes assume spherical symmetry for the scatterer. While this assumption is mostly justified for the spin independent scattering amplitude $f_{0}$, it is usually not for the spin dependent exchange scattering from the valence shell. As a matter of fact, the exchange scattering amplitude $f_{s}$ is a quantity that is interesting in its own right, as we discuss in the next section. Note that the projection holograms give the possibility of determining $f_{s}(\hat{\bf R},\hat{\bf r}_{i})$ (and not just $|f_{s}|^{2}$). In many cases of interest, the spins are ordered in an arrangement with known projection onto a particular ($\alpha ,j$)-spin mode, or it might be possible to order the spins into such an arrangement, for example by applying an external magnetic field (in the case of $C_{4v}$-symmetry discussed above, one might be able to align the spins along the z-axis which is an $A_{2}$-mode, or along the x-axis, which is an $(E,1)$-mode. Under those circumstances the spin directions of the ($\alpha,j$)-mode are known and only the exchange scattering amplitude $f_{s}(\hat{\bf R},\hat{\bf r}_{i})$ needs to be determined in the expression for the projection hologram, which can be achieved by means of a least-square fit.
In fact, although holographic spin images are of interest, we think that the most efficient and accurate spin determinations will involve fitting the hologram in a scheme that is similar to the procedure to determine $f_{s}$, with the coefficients of the different spin modes of the same ($\alpha,j$)-representation as additional parameters to be determined. One can devise an iterative scheme to determine $f_{s}$ and the spins: Use the calculated $f_{s}$ to estimate the atomic spin vectors. Then, use the obtained spins to refine the scattering amplitude $f_{s}$ through a least-square fit. The resulting $f_{s}$ is then used to obtain a better fit of the spins, and so on, until convergence is reached.
Note on Rare Earths and Spin Holography
========================================
In our discussion in the preceeding sections we assumed that the spin dependent part of the coherent scattering amplitude, see Eq. (\[e:f\]), for an atom ‘j’ was of the form $F_{s} \sim {\bf \sigma} \cdot \langle {\bf S}_{j} \rangle \; f_{s} ({\bf k},
{\bf k}')$. This is a good approximation, however, only when we can assume that the spin density of atom j is of the form ${\bf s}_{j} ({\bf r}) = {\bf S}_{j} \rho_{j}
({\bf r})$; that is, that the orbital angular moments are ‘quenched’. This is usually a good assumption for transition elements in crystals; but not for rare earth (or actinide) elements where there are strong spin-orbit correlations. In that case, we obtain (see appendix) $$F_{s}({\bf k},{\bf k}') = {\bf \sigma} \cdot
\sum_{K = 0, M = - K}^{2l, K} A_{K} \;
T_{K M}^{\ast} (\hat{\bf k},\hat{\bf k}') \langle {\bf S}_{j}
{\cal Y}_{K M} ({\bf L}_{j}) \rangle \; .
\label{e:fgen}$$ In the equation $F_{s}$ is the spin-dependent coherent exchange contribution to the scattering amplitude (${\bf k} \rightarrow {\bf k}'$) from a rare earth ion ‘j’ where Russel-Saunders coupling is assumed to apply with ${\bf S}_{j}$ and ${\bf L}_{j}$ the good spin and orbital angular momentum of the ion. ${\cal Y}_{K M}({\bf L})$ is an irreducible tensor constructed from the angular momentum operator ${\bf L}$, and $T_{K M} (\hat{\bf k},\hat{\bf k}')$ is an irreducible tensor depending on $\hat{\bf k}$ and $\hat{\bf k}'$. $A_{K}$ involves radial exchange integrals as well as other coupling coefficients (see appendix).
For the rare earths l=3 and Eq.(\[e:fgen\]) gives $F_{s}$ in terms of the scattering from the six multipole moment tensors of the uncompensated spin distributions of the rare earth ion.
If the spin and orbits are not coupled then $$F_{s} = {\bf \sigma} \cdot \langle {\bf S}_{j} \rangle
\sum_{K , M} A_{K} T_{K M}^{\ast} (\hat{\bf k},\hat{\bf k}')
\langle {\cal Y}_{K M} ({\bf L}_{j}) \rangle \; ,
\label{e:fquen}$$ and we have the quenched orbital result of Eq.(\[e:f\]).
In Eqs.(\[e:fgen\]) and (\[e:fquen\]), $K = 0$ is the isotopic scattering term, $T_{0} (\hat{\bf k},\hat{\bf k}') =
f_{0} (\hat{\bf k} \cdot \hat{\bf k}')$, $T_{1}$ the vector term, $T_{1}(\hat{\bf k},\hat{\bf k}')
= \hat{\bf k} \; a_{11}(\hat{\bf k} \cdot
\hat{\bf k}') + \hat{\bf k}' \; a_{12} (\hat{\bf k} \cdot {\bf k}')
+ \hat{\bf k} \times
\hat{\bf k}' \; a_{13}(\hat{\bf k} \cdot \hat{\bf k}')$. Similarly, $T_{20} (\hat{\bf k},
\hat{\bf k}') = (\hat{\bf k}_{z}^{2} - 1/3) \; a_{21}(\hat{\bf k}
\cdot \hat{\bf k}')
+ (\hat{\bf k}'^{2}_{z} -1/3) \; a_{22} (\hat{\bf k} \cdot \hat{\bf k}') +
(\hat{\bf k}_{z} \hat{\bf k}_{z}' -1/3 \hat{\bf k} \cdot \hat{\bf k}') \;
a_{23}(\hat{\bf k}
\cdot \hat{\bf k}') , \cdots $.
In the hypothetical case of no spin-orbit coupling then the spin $\langle
{\bf S}_{j} \rangle$ would be free to vary independent of the anisotropic charge distribution and would correspond to our previously discussed theory. In fact, however, magnetic studies show [@Mac] that $J = |{\bf L} + {\bf S}|$ retains its Hunds rule ground state value, $ J = L + S $, for the second half shell, and $ J = L - S $ for the first half shell in the crystal (except $Eu^{3+}$ and $Ce^{3+}$) to a good approximation.
In that case Eq.(\[e:fgen\]) becomes $$F_{s} = \sum_{K'=K-1,K=0,M'}^{K+1,2l}
A_{K} B_{K K'} (\sigma \otimes T_{K}
(\hat{\bf k}, \hat{\bf k}'))^{\ast}_{K'M'}
\langle {\cal Y}_{K' M'} ({\bf J}) \rangle \; \; \; ,
\label{e:fhund}$$ where $B_{K K'}$ is another coupling coefficient (see Appendix) and $$(\sigma \otimes T_{K}(\hat{\bf k},\hat{\bf k}'))_{K'M'}
\equiv
\sum_{\mu,\nu} C_{\mu \; \nu \; M'}^{1 \; K \; K'}
\sigma_{\mu} T_{K \nu} (\hat{\bf k},\hat{\bf k}') \; .
\label{e:t}$$
Now, instead of just $\langle {\bf S} \rangle$ or $\langle {\bf J} \rangle $ to represent the spin dependent coherent exchange scattering, there are seven, 2l+1, multipole moment distributions $\langle {\cal Y}_{K M} ({\bf J}) \rangle$ giving $(2l+1)^{2} = 49$ parameters determining the scattering. $Gd^{3+}$ is uniquely simple because it is spherically symmetric so that in Eq.(\[e:fgen\]) $\langle {\cal Y}_{K M} ({\bf L}) \rangle =
7 \delta_{K 0}$ and thus $F_{s} \sim \sigma \cdot \langle {\bf J} \rangle
f_{0}(\hat{\bf k} \cdot \hat{\bf k}')$. For the other rare earths however, and for $E_{k,k'}
\simeq 100 \; {\rm eV}$, tensor components up to $K \simeq 3,4$ make important contributions to $F_{s}$.
The question now arises, are there good reasons to measure and analyze ‘spin holograms’ of rare earth ions, given their evident complexity? We think there is because of the relation of such scattering measurements to the RKKY mechanism responsible for the rare earth ion interactions in solids. The interaction arises from the exchange scattering of conduction electrons with 4f shell electrons of an ion inducing a stationary spin polarization wave in the surrounding conduction band. The role of the anisotropic, higher moment terms, Eq.(40), in $F_{s}$ on the RKKY-interaction have not been investigated as thoroughly as their importance warrants. Frederick Specht [@Specht] in an early investigation retained only terms up to $K =2$ and found quite large effects. Thus for two nearest neighbor $Tb$ ions whose moments were aligned along the inter-ionic direction he obtained $E(0)
= -6.6 k_{B}$, while if the moments are perpendicular to that direction he obtained $E(\pi/2) = - 8.1 k_{B}$, a $20 \%$ variation; while for Tm ions the numbers were $E(0) =-1.2 k_{B}, E(\pi/2) = -.7 k_{B}$, a $50 \%$ variation. (Note that for the cigar shaped spin distribution of Tb $|E(\pi/2)| > |E(0)|$, while the opposite held for the pancake shaped Tm spin distributions).
We think that spin holographic experiments on a ferromagnetic surface layer of terbium, for example, as a function of temperature, magnetic field, and $k$ of the photoelectron, revealing the effects of the moments of various orders, would be very interesting.
An interesting question is what measurement must be made to determine the quantum state of a system [@Gayle]. Consider a system with angular momentum $J$. In general its state is determined by its density matrix $\rho_{M,M'}$. What measurements can be made to determine the $(2J+1)^{2}-1 = 4J(J+1)$ real numbers required to specify $\rho$? In reference [@Gayle] we outlined one method using ‘Feynman filters’; but we also mentioned that Fano [@Fano] had shown that measurement of the $4J(J+1)$ expected multipole moments $\langle {\cal Y}_{K M} ({\bf J})
\rangle, K =1,....,2J$ also would suffice. In our work here we have seen that the $\langle {\cal Y}_{K M} ({\bf J}) \rangle$’s could be determined (in principle) for $K \leq 2l+1$, or $K \leq 2L+1$ by means of coherent electron scattering, the higher order multipoles vanish. Thus for Gd only the first order moment, $\langle
{\bf J} \rangle$ could be so determined, while for the other rare earths much more state information can be determined, but not enough to fully determine the state.
Conclusions
===========
The main message of this paper is that we can construct holograms of the atomic spins in the near-neighborhood of a photoelectron emitting atom from the angularly resolved photoelectron emission intensities. For the important case of photoelectrons emitted by source atoms with $C_{nv}$-environment, we show how these spin holograms can be constructed using symmetry. Furthermore, we discuss schemes to holographically image the spins and to extract accurate spin information from the spin holograms.
The photons used to emit the electrons that give the intensity holograms for the purpose of spin holography can be, ${\it but \;
do \; not \; have \; to \; be}$, circularly polarized. Incident linearly polarized, or indeed even ${\it unpolarized}$ light would also serve the purpose. This statement could appear somewhat puzzling because it seems to imply that it is unnecessary to polarize the primary photoelectron waves in order to probe the spins of the nearby scattering atoms. We show that the surprising ability of unpolarized light to record spin information in the photoelectron intensity is a consequence of the spin-orbit correlations in the initial inner-core electron states from which the photoelectron is emitted. The spin-orbit correlations give a finite interference contribution to the spin of electrons emitted in different directions. It is the interference of the photoelectron emitted in the $\hat{\bf R}$-direction, and that emitted in the direction of atom i and then scattered into the $\hat{\bf R}$-direction, that gives the holographic fringes in the photoelectron intensity. By virtue of exchange scattering the holographic fringes are then sensitive to the spin of the scattering atom, with contributions proportional to the scalar product of the atomic spin and the interference contribution to the photoelectron spin, $\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i} \rangle
\rangle$. Note that it is not, as one might have surmised, the spin of the photoelectron wave emitted in the ${\bf r}_{i}$-direction, $\langle \langle \hat{\bf r}_{i} | \sigma | \hat{\bf r}_{i} \rangle \rangle$, that is of importance. Indeed, under some conditions, this primary wave spin polarization can vanish, implying unpolarized electrons impinging on the scattering atom, while $\langle \langle \hat{\bf R} | \sigma | \hat{\bf r}_{i}
\rangle \rangle \neq 0$, implying a spin dependent photoelectron intensity hologram. In the paper, we calculate the spin intereference matrix element for photoemission from a $p_{1/2}$ shell, by absorbing photons of arbitrary polarization.
Furthermore, to obtain an actual spin hologram from the photoelectron intensity, it is necessary to ‘separate out’ the spin dependent holographic fringes, which typically contribute 5 $\%$ or less, from the rest of the hologram. In this paper, we point out how this separation can be obtained using symmetry if the source atom is in a site of $C_{nv}$-symmetry. Specifically, we show that a group theoretical projection of the emission intensity onto the irreducible ($\alpha,j$)-representation gives a hologram of the projection of the spin arrangement of the nearby atoms onto the ($\alpha,j$)-spin mode. This way, one can obtain almost all components of the spin vectors (all but one for the n-atom star and all but three for the 2n-atom star, as explained in the text).
The advantage of measuring emission intensities, rather than having to spin analyze the emitted electrons, is considerable: one does not lose the usual factor of $10^{4}$ in intensity due to the low efficiency of the electron polarimeters. Consequently, we expect that the proposed experiments are quite feasible at a synchrotron source and it is possible that the data have already been obtained and only need to be analyzed.
To conclude, we repeat that spin holography is a surface probe which probes the average short-range magnetic environment of the photoelectron source atom. By using atoms adsorbed on a surface as sources, spin holography can reveal the influence of these adsorbates on the spins of the neighboring substrate atoms. It can then provide detailed information on many interesting systems, such as adsorbed oxygen atoms quenching the magnetism of a nickel surface, or iron atoms inducing moments on neighboring palladium substrate atoms (superparamagnetism). Alternatively, substrate atoms can be used as sources to image the spins of objects adsorbed on the surface, such as small molecules. Finally, spin holography should be useful for surface studies of magnetic systems with large and complicated unit cells of N spins, such as garnets. (Spin holography gives information about the N spins as a function of the N positions, whereas the usual diffraction techniques give the $N^{2}$ spin-spin correlation as functions of the $N^{2}$ relative positions).
Acknowledgments {#acknowledgments .unnumbered}
===============
Part of the work was carried out with support from National Science Foundation, Grant No DMR-9013058. The work of E.T. is supported by the National Science Foundation through a grant for the Institute for Atomic and Molecular Physics at Harvard University and Smithsonian Astrophysical Observatory.
Appendix {#appendix .unnumbered}
========
We shall sketch the derivation of Eqs.(38-40).
If ${\bf s}_{c}$ is the spin of the positive energy ‘free’ electron and ${\bf s}_{i}$ that of a bound electron then the spin exchange operator is $1/2 + 2 {\bf s}_{c} \cdot
{\bf s}_{i}$, and the exchange scattering amplitude for free electron momentum ${\bf k}$ and bound electron in orbital $|u_{lm}\rangle $ going to a free electron momentum ${\bf k}'$ and bound electron $\rightarrow |u_{lm'}\rangle $ is $$f_{s} = (1/2 + 2 {\bf s}_{c} \cdot {\bf s}_{i} )
\; {\cal S} ({\bf k},u_{lm};{\bf k}',u_{lm'}) \; \; ,
\label{e:a1}$$ where $${\cal S} = - \int \exp[-i{\bf k}' \cdot {\bf r}_{2}] \; u^{\ast}_{lm'}
({\bf r}_{1})
\frac{e^{2}}{r_{12}} u_{lm}({\bf r}_{2}) \; \exp[i{\bf k} \cdot {\bf r}_{1}] \;
d^{3} r_{1} d^{3} r_{2} \; \; ,
\label{e:a2}$$ is the exchange integral.
Expanding the integral in partial waves we get $${\cal S} = \sum_{p\; p' \; \lambda} E(l,p,k;l,p',k';\lambda)
\sum_{q\; q' \; \mu}
\langle l,m' | Y_{pq} | \lambda \mu \rangle \;
\langle \lambda \mu | Y_{p'q'} | lm \rangle
Y^{\ast}_{p'q'}(\hat{\bf k}') Y^{\ast}_{pq}(\hat{\bf k}) \; \; ,
\label{e:a3}$$ where E involves radial overlap integrals $$\begin{aligned}
&& E(l,p,k;l,p',k';\lambda) =
\nonumber \\
&& -(i)^{p-p'} \frac{(4\pi)^{3} e^{2}}{(2p+1)(2p'+1)(2\lambda+1)}
\int j_{p'}(k'r_{2}) g_{l}(r_{1}) \frac{r^{\lambda}_{<}}{r^{\lambda+1}_{>}}
g_{l}(r_{2}) j_{p}(kr_{1}) r_{1}^{2} r_{2}^{2} dr_{1} dr_{2} \; ,
\label{e:a4}\end{aligned}$$ with $j_{p}$ the spherical bessel function and $u_{lm}({\bf r}) = g_{l} (r)
Y_{lm}(\hat{\bf r})$.
Now making use of the Racah-Wigner techniques ([@Racah]) we obtain $$\begin{aligned}
&&{\cal S} = \sum_{p\; p' \; \lambda} \langle l || Y_{p} || \lambda \rangle
\langle \lambda || Y_{p'} || l \rangle \; \sqrt{2\lambda +1}
\; E(l,p,k;l,p',k';\lambda)
\nonumber \\
&& \; \; \; \; \; \; \; \; \cdot \; \sum_{K M} (-)^{K} \sqrt{2K+1}
\left\{ \matrix{l&p&\lambda \cr p'&l&K \cr } \right\}
\left[ \left( Y_{p'} (\hat{k}') \otimes Y_{p}(\hat{k})
\right)^{K}_{M} \right]^{\ast} C^{l \; \; K \; \; l}_{m \; M \; m'} \; \; .
\label{e:a5}\end{aligned}$$ We now replace the vector addition coefficient in (\[e:a5\]) in accordance with $C^{l \; \; K \; \; l}_{m \; M \; m'} =
\langle l m' | {\cal Y}_{K M} ({\bf l}_{i})
|l m \rangle / \langle l || {\cal Y}_{K} || l \rangle $, where ${\cal Y}_{K
M} ({\bf l})$ is an irreducible tensor of rank $K$ constructed from the angular momentum operators ${\bf l}$.
The exchange scattering operator is now obtained by multiplying ${\cal S}$ by the spin exchange operator and summing over the n electrons in the $4f$ shell: $$\begin{aligned}
F_{x} &=& \sum_{i}^{n} f_{s}(i)
\nonumber \\
&=& \sum_{K = 0, M = -K}^{2l, K}
A_{K} T^{\ast}_{K M} (\hat{\bf k},\hat{\bf k}')
\langle (1/2 + 2 {\bf s}_{c} \cdot
\frac{{\bf S}}{\nu} ) {\cal Y}_{K M} ({\bf L})
\rangle \; \; ,
\label{e:a6}\end{aligned}$$ where ${\bf L}$, ${\bf S}$ are the operators for the total orbital angular momentum and spin of the $4f$ electrons (we assume Russel-Saunders coupling with good L and good S). Also, we have supposed the Russel-Saunders ground state values of $L$ and $S$ and we made the substitution $$\begin{aligned}
&& \sum_{i=1}^{n} (1/2+2 {\bf s}_{c} \cdot
{\bf s}_{i}) \; {\cal Y}_{K M}
({\bf l}_{i})
\nonumber \\
&& \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;
= \pm \frac{ \langle l L || \sum_{i=1}^{\nu} {\cal Y}_{K}
({\bf l}_{i}) || l L \rangle }
{ \langle L || {\cal Y}_{K} ({\bf L}) || L \rangle }
\; \langle (1/2 + 2 {\bf s}_{c} \cdot \frac{{\bf S}}{\nu} )
{\cal Y}_{K M} ({\bf L}) \rangle \; \; ,
\label{e:a7}\end{aligned}$$ where $\nu = n$ for less than half filled shell, $\nu = N - n$ for more than half filled. For more than half filled shell we must choose the minus sign in (\[e:a6\]) and then add $\frac{N}{2} \delta_{K 0} $ to the spin independent term (i.e. rather than $-(\frac{\nu}{2} + 2 {\bf s}_{c} \cdot {\bf S})$ for the $K = 0$ term we get $(\frac{N-\nu}{2} - 2 {\bf s}_{c} \cdot {\bf S})$ for that term). Thus finally $$\begin{aligned}
F_{x} = && \eta \sum_{p,p',\lambda} E(l,p,k;l,p',k';\lambda)
\langle l || Y_{p} | \lambda \rangle \langle \lambda ||
Y_{p'} || l \rangle \sqrt{2\lambda
+1}
\nonumber \\
&& \sum_{K M} (-)^{K}
\sqrt{2K +1}
\left\{ \matrix{l&p&\lambda \cr p'&l&K \cr } \right\}
\left[ Y_{p'} (\hat{\bf k}') \otimes
Y_{p} (\hat{\bf k}) \right]^{\ast}_{K M}
\nonumber \\
&& \; \frac{\langle l L || \sum_{i=1}^{\nu}
{\cal Y}_{K} ({\bf l}_{i}) || l L \rangle }
{ \langle L || {\cal Y}_{K} || L \rangle }
\; (1/2 + 2 {\bf s}_{c} \cdot \frac{{\bf S}}{\nu} )
{\cal Y}_{K M} ({\bf L}) \; \; ,
\label{e:a8}\end{aligned}$$ where $\eta$ is (+/-) for (less than/more than) half filled shell systems. The meaning of $A_{K} T^{\ast}_{K M}
(\hat{\bf k},\hat{\bf k}')$ in Eq.(\[e:a6\]) can then be obtained by compairing with Eq.(\[e:a8\]).
Finally we suppose that $J$ is also a good quantum number (with $J = L - S$ for the first half and $J = L + S$ for the second half rare earths). Then within the manifold of good $L, S$ and $J$ $${\cal Y}_{K M} ({\bf L}) = (-)^{L+S+J+K} \sqrt{(2L+1)(2J+1)}
\left\{ \matrix{J&S&L \cr L&K&J \cr } \right\}
\frac{\langle L || {\cal Y}_{K} ({\bf L}) || L \rangle }{\langle J ||
{\cal Y}_{K} ({\bf J}) || J \rangle } \;
{\cal Y}_{K M}({\bf J}) \; ,
\label{e:a9}$$ and this may be substituted into the spin independent term in Eq.(\[e:a6\]). Similarly, $$\begin{aligned}
&& \sum_{M} T^{\ast}_{K M} (\hat{\bf k},\hat{\bf k}') \; {\bf \sigma}
\cdot {\bf S} \; {\cal Y}_{K M} ({\bf L})
\nonumber \\
&& \; \; \; \; \; = \sum_{K' \nu} \left[ \sigma^{(1)} \otimes
T_{K} \right]^{K' \ast}_{\nu}
\cdot \left[ {\bf S}^{(1)} \otimes
{\cal Y}_{K}({\bf L}) \right]^{K'}_{\nu}
\nonumber \\
&& \; \; \; \; \; = \sum_{K' \nu} \left[ \sigma^{(1)}
\otimes T_{K} (\hat{\bf k},
\hat{\bf k}') \right] ^{K'}_{\nu}
\left[ \matrix{S&L&J \cr 1&K&K' \cr S&L&J \cr } \right]
\langle S || S || S \rangle \langle L ||
{\cal Y}_{K}({\bf L}) || L \rangle
\frac{{\cal Y}_{K' \nu}({\bf J})}
{\langle J || {\cal Y}_{K'} ({\bf J}) || J
\rangle} \; \; ,
\label{e:a10}\end{aligned}$$ and substituting (\[e:a10\]) into Eq.(\[e:a6\]) gives Eq.(\[e:fhund\]) of the text.
--------- --- ---- ---- ---- ----
$A_{1}$ 1 1 1 1 1
$A_{2}$ 1 1 1 -1 -1
$B_{1}$ 1 -1 1 1 -1
$B_{2}$ 1 -1 1 -1 1
$E$ 2 0 -2 0 0
--------- --- ---- ---- ---- ----
: Character table of the group $C_{4v}$[]{data-label="tab1"}
[99]{} E. M. E. Timmermans, G. T. Trammell, and J. P. Hannon, Phys. Rev. Lett. [**72**]{}, 832 (1994). E. M. E. Timmermans, G. T. Trammell, and J. P. Hannon, J. Appl. Phys. [**73**]{}, 6138 (1993). J. J. Barton, Phys. Rev. Lett. 61, 1356 (1988). The idea of atomic source holography was first proposed in the field of X-ray physics: J. P. Hannon and G. T. Trammell, in ${\it
Mossbauer \; Effect \; Methodology}$ (Plenum, New York, 1974), Vol. 9, pp 181-190; J. P. Hannon, N. J. Carron, and G. T. Trammell, Phys. Rev. B [**9**]{}, 2791 (1974); J. T. Hutton, G. T. Trammell, and J. P. Hannon, Phys. Rev. B [**31**]{}, 743 (1985). Phase determination by the reciprocal method of X-ray fluorescence holography, was pointed out by J. T. Hutton, G. T. Trammell, and J. P. Hannon, Phys. Rev. B [**31**]{}, 6420 (1985). A. Szoke, in ${\it Short \; Wavelength}$ ${\it Coherent \; Radiation}$: ${\it Generation \; and \; Applications}$, edited by D. T. Attwood and J. Bokker, AIP Conference proceedings $N_{o}$ 147 (American Institute of Physics, New York, 1986), A scheme of probing the nearby magnetic environment through the emission intensity of photoelectrons polarized by the source atom, which is magnetic and has spin ${\bf s}_{0}$, was proposed by Fadley and coworkers. This scheme can also be used to obtain holograms, which then image $\langle {\bf s}_{0} \cdot {\bf s}_{i} \rangle$ at the position of atom i. B. Sinkovic and C. S. Fadley, Phys. Rev. B [**31**]{}, 4665 (1985); B. Sinkovic, D. J. Friedman, and C. S. Fadley, J. Magn. Magn. Mater. [**92**]{}, 301 (1991), and references therein. Eddy Timmermans, work in progress. Eddy Timmermans, Ph.D. Thesis, Rice University (unpublished) (1995). J. J. Barton, and D. A. Shirley, Phys. Rev. B [**32**]{}, 1906 (1985). J. J. Barton and D. A. Shirley Phys. Rev. B [**32**]{}, 1892 (1985). J. J. Barton, C. C. Bahr, S. W. Robey, Z. Hussain, E. Umbach and D. A. Shirley, Phys. Rev. B [**34**]{}, 3807 (1986). J. J. Barton, Phys. Rev. Lett. [**67**]{}, 3106 (1991). S. Y. Tong, Hua Li, and H. Huang, Phys. Rev. Lett. [**67**]{}, 3102 (1991). B. P. Tonner, Zhi-Lan Han, G. R. Harp, and D. K. Saldin, Phys. Rev. B [**43**]{}, 4423 (1991) and references therein. This is different from the Fano-effect, which spin polarizes the photoelectron by means of the spin-orbit interaction affecting the final continuum electron state. U. Fano, Phys. Rev. [**178**]{},131 (1969). U. Heinzmann, G. Schonhense, and J. Kessler, Phys. Rev. Lett. [**42**]{}, 1603 (1979). Landau and Lifschitz, $\it{Quantum \; Mechanics,}$ ${\it Non-relativistic \; Theory}$, Chapter XII, Pergamon Press (1977). The higher the value of L, the further out this maximum is reached, so that the anisotropy of the scattered wave ‘degrades’ the image for the purposes of determining the positions of the scattering atoms. For the low energy electrons ($ k \sim
1 \AA^{-1}$) this can lead to overlapping images (since the distance between atoms is typically a few $\AA$). In ordinary electron emission holography, problems of this kind are overcome using an imaging transform with a kernel that ‘corrects’ for the anisotropy (for example by dividing the integrand in the holographic transform by the scattering amplitude), and by adding the holograms collected at different emission energies with a ‘phase’ factor $\sim \exp(-ikr_{i})$ that ‘phase locks’ onto the interference term with the wave scattered by atom $i$ (canceling the phase of the $[\exp(ikr_{i})/r_{i}]$ factor in (\[e:imicl\])). To avoid the usual electron holography problems, such as twin images and overlapping images, one can determine the partial wave mixture by holographically transforming with anisotropy-correcting kernels, finding the kernel that gives the most s-wave like image. Alternatively, one can try to ‘isolate’ a single component of the spin vector of atom i, by using a kernel in the holographic transform that gives a peak at ${\bf r}_{i}$ for one spin direction, but vanishes at ${\bf r}={\bf r}_{i}$ for the spin component in the perpendicular direction, see also ref. [@thesis]. J. Jensen and A. R. Mackintosh, [*Rare Earth Magnetism*]{}, Clarendon Press Oxford (1991). F. Specht, Phys. Rev. [**162**]{}, 389 (1967). W. Gayle, E. Guth, G. T. Trammell, Phys. Rev. [**165**]{}, 1434 (1968). U. Fano, Rev. Mod. Phys. [**29**]{}, 74 (1957). The Racah-Wigner calculus is presented in M. E. Rose, [*Elementary Theory of Angular Momentum*]{}, John Wiley (1957); Brian Judd, [*Operator Techniques in Atomic Spectroscopy*]{} Mc. Graw Hill (1963); L. C. Biedenharn and J. D. Louck, [*Angular Momentum in Quantum Physics*]{} in Encyclopedia of Mathematics and its Applications, Gian Carlo Rota Ed. Vol. 8, Addison Wesley (1981). The reduced matrix elements and operator equivalents may be found in A. Abragam and B. Bleony, [*Electronic Paramagnetic Resonance of Transition Ions*]{}, Dover (1970); and in addition to these, expressions for the 6j and 9j coefficients and all other parameters for the rare earths are given in Xiaomin Hu, [*Quasi-elastic Resonant X-Ray Scattering*]{}, Ph. D. thesis, Rice University (1997) (unpublished).
Figure Captions {#figure-captions .unnumbered}
===============
Reference frame used in calculating the spinors of the electrons emitted from a $p_{1/2}$-shell (section II). The photon polarization vector, $\hat{\epsilon}$, is parallel to the $\hat{\bf z}$-direction.
4-atom environment of electron emitting atom (denoted by the shaded sphere). The symmetry-group of the environment is $C_{4v}$, with the $C_{4}$-axis (= $\hat{\bf z}$-axis) perpendicular to the plane of the four atoms.
The group theoretical modes of the spin vectors of the $C_{4v}$-environment shown in Fig.2. The spins are shown in the xy-plane, $\odot$ indicates an up-spin (positive $\hat{\bf z}$-direction), and $\otimes$ indicates a ‘down’-spin (negative $\hat{\bf z}$-direction).
|
---
abstract: 'We present a simple method, combining the density-matrix renormalization-group (DMRG) algorithm with finite-size scaling, which permits the study of critical behavior in quantum spin chains. Spin moments and dimerization are induced by boundary conditions at the chain ends and these exhibit power-law decay at critical points. Results are presented for the spin-$1/2$ Heisenberg antiferromagnet; an analytic calculation shows that logarithmic corrections to scaling can sometimes be avoided. We also examine the spin-$1$ chain at the critical point separating the Haldane gap and dimerized phases. Exponents for the dimer-dimer and the spin-spin correlation functions are consistent with results obtained from bosonization.'
address: 'Department of Physics, Brown University, Providence, RI 02912-1843'
author:
- 'Shan-Wen Tsai and J. B. Marston'
title: |
Density-Matrix Renormalization-Group Analysis of Quantum Critical Points:\
I. Quantum Spin Chains
---
Introduction {#sec:Intro}
============
Quantum critical points are characterized by fluctuations over all length and time scales and by the appearance of power law scaling. In this paper we present a simple but powerful numerical method to access quantum critical points in one-dimensional systems. The method combines the density-matrix renormalization-group (DMRG) algorithm and finite-size scaling ideas. We illustrate the method by applying it to several well-understood quantum spin chains. In a second paper to follow we apply the method to new classes of supersymmetric spin chains which describe various disordered electron systems[@part2].
The development of the density-matrix renormalization-group (DMRG) algorithm by White[@White] represented an important improvement over previous numerical methods for the study of low dimensional lattice models. It has been applied to a wide variety of systems[@proceedings]. The DMRG approach was first used to study the ground state properties and low-energy excitations of one-dimensional chains. It has been extensively applied to the study of various spin chains. Low-lying excited states of the spin-$1$[@White2; @Sorensen1; @Sorensen2] and spin-$1/2$[@Ng] Heisenberg antiferromagnets have been calculated. Likewise, spin-$1$ chains with quadratic and biquadratic interactions[@Bursill; @Fath], a spin-$2$ antiferromagnetic chain[@Schollwock; @Wang1], spin-$1/2$ and spin-$1$ chains with dimerization and/or frustration (next-nearest-neighbor coupling)[@Kato; @Bursill1; @Chitra; @Kolezhuk; @Watanabe], and frustrated spin-$3/2$ and spin-$2$ chains[@Roth] have all been studied. Edge excitations[@Sorensen2; @Schollwock; @Qin; @Polizzi] at the ends of finite spin chains and the effects of perturbations such as a weak magnetic field coupled to a few sites[@Legeza] have been considered. Randomness in the form of random transverse magnetic field in a spin-$1/2$ XY model[@Juoza], random exchange couplings[@Hida1], and random modulation patterns of the exchange[@Schonfeld; @Hida2], has been examined. Finally, alternating spin magnitudes[@Pati], the presence of a constant[@Venuti] or a staggered[@Lou] magnetic field in a spin-$1$ chain, bond doping[@Wang2], the effects of a local impurity[@Sorensen3], and interactions with quantum phonons[@Caron; @Bursill2] have also been considered.
Most of the above work involves systems in which the first excited state is separated from the ground state by a non-zero energy gap as the DMRG works best for gapped systems. First attempts to extract critical behavior of gapless systems used the DMRG to generate renormalization transformations of the coupling constants in the Hamiltonian[@Drzewinski; @Bursill3]. Hallberg et al.[@Hallberg] studied the critical behavior of $S=1/2$ and $S=3/2$ quantum spin chains with periodic boundary conditions through extensive calculations of ground state correlation functions at different separations and different chain sizes $L$. Spin correlation functions in an open chain have also been calculated and compared with results calculated from low-energy field theory, showing that estimates of the amplitudes can also be obtained[@Hikihara]. The approach described in this paper was applied to the spin-$1/2$ Heisenberg chain and a non-Hermitian supersymmetric (SUSY) spin chain[@Kondev]. More recently, critical behavior of classical one-dimensional reaction-diffusion models[@Carlon1] and the two-dimensional Potts model[@Carlon2] has been studied using the finite-size DMRG algorithm. Bulk and surface exponents of the Potts and Ising model have been obtained by using the DMRG to calculate correlation functions at different separations and collapsing curves obtained at different system sizes[@Kaulke]. The SUSY chain describing the spin quantum Hall effect (SQHE) plateau transition was also examined in some detail. Critical exponents were extracted[@Senthil] and compared to exact predictions[@Gruzberg]. Thermodynamic properties of other two-dimensional classical critical systems have also been studied by the DMRG method[@Nishino; @Carlon3; @Honda]. Finally, Andersson et al. investigated the convergence of the DMRG in the thermodynamic limit for a gapless system of non-interacting fermions[@Andersson].
The method described in this paper combines the DMRG algorithm with finite-size scaling analysis, and yields accurate critical exponents. The main advantage of the method is its simplicity. Only the calculation of ground state correlations near the middle of chains with open boundary conditions are required. The relatively simple “infinite-size” DMRG algorithm[@White] is particularly accurate for this job. In Sec. \[sec:method\] we describe the method. The tight-binding model can be solved exactly and in Sec. \[sec:tightbinding\] we use it to illustrate our scaling analysis. DMRG results are presented in Sec. \[sec:s=half\] for the anisotropic $S=1/2$ Heisenberg antiferromagnet and several critical exponents are obtained. An analytical calculation shows that multiplicative logarithmic corrections – which complicate the extraction of accurate critical exponents – may be avoided in some instances. In Sec. \[sec:s=1\], the $S=1$ antiferromagnetic spin chain is studied, focusing on the critical point that separates the Haldane and the dimerized phases. We conclude with a summary in Sec. \[sec:conclusion\].
The DMRG / Finite-Size Scaling Approach {#sec:method}
=======================================
We first describe how critical exponents may be obtained from a finite-size scaling analysis of chains with open or fixed boundary conditions. These boundary conditions are the simplest to implement in DMRG calculations. In the next subsection the DMRG algorithm itself is briefly described.
Finite-Size Scaling {#subsec:finite}
-------------------
To illustrate the sorts of power-law scaling we wish to examine, first consider the case of a spin chain with periodic boundary conditions that is at its critical point. The system can be moved away from criticality by turning on a uniform magnetic field, say in the x-direction, at each site: $$\begin{aligned}
H_B = h~ \sum_{j=1}^{L} S^x_j\ .\end{aligned}$$ This perturbation makes the correlation length finite: $$\begin{aligned}
\xi_B ~\propto~ |h|^{-\nu_B}\ .\end{aligned}$$ Explicit dimerization, breaking the symmetry of translation by one site, also moves the system away from criticality. For a Heisenberg antiferromagnet, this can be realized by the addition of a staggering term $R$ to the Hamiltonian: $$\begin{aligned}
H = \sum_{j=1}^{L-1} [1 + (-1)^j R]~ \vec{S}_j \cdot \vec{S}_{j+1}\ .\end{aligned}$$ The correlation length $\xi$ in this case scales as $$\begin{aligned}
\xi ~\propto~ |R|^{-\nu}\ .\end{aligned}$$ Thus there are two independent exponents which correspond to these two perturbations of critical spin chains. Two-parameter scaling functions can be written for various observables and, for a finite system, these involve two dimensionless variables: the ratios $L/\xi$ and $L/\xi_B$. The induced dimerization, defined for now as the modulation of the $x-x$ and $y-y$ spin-spin correlations on even versus odd links, $$\Delta = (-1)^j ~ \left[ \langle S^x_j S^x_{j+1} + S^y_j S^y_{j+1} \rangle -
\langle S^x_{j-1} S^x_j + S^y_{j-1} S^y_j \rangle \right]\ ,$$ is of course independent of the site index for periodic chains, and scales as a function of the chain length $L$, the field $h$, and the dimerization parameter $R$ as: $$\begin{aligned}
\Delta(L, R, h) = {\rm sgn}(R)~
|R|^{\alpha_{\Delta}} ~f_{\Delta} (L |R|^{\nu}, L |h|^{\nu_B}) \ .\end{aligned}$$ When the applied magnetic field is removed, $h=0$, and this expression simplifies to: $$\begin{aligned}
\Delta(L, R) &=& {\rm sgn}(R)~
|R|^{\alpha_{\Delta}} ~g_{\Delta}(L |R|^{\nu}) \nonumber \\
&\sim& L^{x_{\Delta}}~ R ~~~~ {\rm as} ~~~R \rightarrow 0,
\label{hzero}\end{aligned}$$ where the second line follows from the fact that when the perturbation $R$ is very small, or equivalently when the correlation length is larger than the system size, the net induced dimerization must be an analytic, linear, function of $R$. Therefore, for $|x| \ll 1$, the scaling function $g_{\Delta}(x)$ is given by: $$\begin{aligned}
g_{\Delta}(x) = |x|^{-\alpha_{\Delta}/\nu} ~(a_1 |x|^{1/\nu}
+ a_2 |x|^{2/\nu} + \ldots);\end{aligned}$$ the first term yields linear dependence of $\Delta$ in $R$ in the $R \rightarrow 0$ limit, in agreement with Eq. \[hzero\], and the subsequent terms are higher order corrections. To recover the correct $L$-dependence, we must set $$x_{\Delta} = {{1-\alpha_{\Delta}}\over{\nu}}\ .$$ The exponent $x_{\Delta}$ and the correlation length exponent $\nu$ satisfy the usual relation $$\nu = {{1}\over{2-x_{\Delta}}}\ .$$
The applied magnetic field also polarizes the spins along the chain. The scaling form for the spin moment at each site is given by: $$\begin{aligned}
\langle S^x \rangle = {\rm sgn}(h)~
|h|^{\alpha_B} ~f_B(L |R|^{\nu}, L |h|^{\nu_B})\ . \end{aligned}$$ With no applied dimerization, $R=0$, and we expect the simple power-law: $$\begin{aligned}
\langle S^x \rangle \sim L^{x_B} ~h ~~~~ {\rm as} ~~~h \rightarrow 0\ .\end{aligned}$$ Therefore, $x_B = (1-\alpha_B)/\nu_B$.
Alternatively, dimerization can be induced by open boundary conditions, and we take advantage of this fact to extract critical exponents. As depicted in Fig. \[bc\], open boundary conditions favor enhanced nearest-neighbor spin-spin correlations on the two outermost links. Chains of increasing length $L = 4, 6, 8, \ldots$ exhibit alternating patterns of dimerization on the interior bonds. Likewise, spin moments may be induced in the interior of the chain by applying a magnetic field to the ends of the chain. Strong applied edge magnetic fields completely polarize the end spins and induce non-zero and alternating spin moments along the chain. Alternatively, spin moments can be induced as before by a staggered magnetic field applied along the entire chain. Here however we consider only edge magnetic fields.
We monitor the induced dimerization and spin moments at the center of the chain as the chain length $L$ is enlarged via the DMRG algorithm. This scaling analysis is convenient because the relatively simple infinite-size DMRG algorithm applies to open chains and is most accurate at the center region of the chain where we focus our attention. The induced dimerization and spin moments in the interior of the chain show power-law scaling at the critical point[@deGennes]. Igloi and Rieger demonstrated power-law scaling for a variety of open boundary conditions (free, fixed and mixed)[@Igloi]. At the critical point $R=0$ and $h=0$ the induced dimerization scales as a power-law with possible multiplicative logarithmic corrections: $$\Delta(L/2) = L^{-x_{\Delta}} ~(\ln L)^{y_{\Delta}} ~\big{(} a + \frac{b}{L}
+ \ldots \big{)}\ .
\label{scaling}$$ A similar expression holds for the induced spin moment at the center of the chain, $\langle S^x(L/2) \rangle$, with the replacement of the exponents $x_\Delta \rightarrow x_B$ and $y_\Delta \rightarrow y_B$.
Infinite-Size DMRG Algorithm {#subsec:dmrg}
----------------------------
The name “density-matrix renormalization-group” is something of a misnomer as the method is most accurate away from critical points, when there is an energy gap for excitations. It is helpful to think of the DMRG algorithm as a systematic variational approximation for the calculation of the ground state and/or low-lying excitations, principally in one dimension. The Hilbert space of a quantum chain generally grows exponentially with the chain length, and eventually must exceed available computer memory. The DMRG algorithm is an efficient way to truncate the Hilbert space; as the size of the space retained can be varied (up to machine limits) it is possible to ascertain the size of errors introduced by the truncation.
For simplicity, we use the so-called “infinite-size” DMRG algorithm[@White]. As the algorithm has been described in some detail by White, we just sketch the essentials of the method. It begins with the (numerically exact) diagonalization of an open chain consisting of just four sites, each site having on-site Hilbert space of dimension $D$. For quantum spin chains $D= 2S + 1$, thus $D = 2$ for the spin-1/2 Heisenberg antiferromagnet. The chain is then cut through the middle into two pieces, one half of which is interpreted as the “system” and the other half as the “environment,” the two parts combined being thought of as the entire “universe” of the problem, see Fig. \[dmrg1\]. At this point the reduced density matrix for the system, of size $D M \times
D M$ is constructed by performing a partial trace over the environment half of the chain. It is defined by: $$\rho_{ij} = \sum_{i^\prime = 1}^{DM} \Psi_{i i^\prime} \Psi_{j i^\prime},$$ where $\Psi_{i i^\prime} = \langle i i^\prime | \Psi \rangle$ are the real-valued matrix elements of the eigenstate of interest (the “target” which is often the ground state) projected onto a basis of states labeled by unprimed Roman index $i$ which covers the system half of the chain and primed index $i^\prime$ which covers the environment half of the chain. The eigenvalues of the reduced density matrix are real, positive, and sum up to one; these are interpreted as probabilities. We keep only the $M$ most probable eigenstates corresponding to the largest eigenvalues, and discard the remaining $M (D-1)$ eigenstates. The retained states form a new basis for the problem. Next, two new sites are added to the middle of the chain and the pieces are connected, yielding a chain of size $L = 6$. The process is then repeated by finding the targeted state of this chain, constructing the new reduced density matrix and again projecting onto the $M$ most probable states. As the chain length grows in steps of two, the total Hilbert space dimension grows by a multiplicative factor of $D^2$. None of the Hilbert space is thrown away until the chain grows large enough that its Hilbert space exceeds the space that is held in reserve, in other words until $D^L > D^2 M^2$. The truncation process damages the outer regions of the chain the most, and the central region is treated most accurately.
=4.5in
One advantage of the method presented in this paper is that critical exponents are extracted from ground-state correlations only. Excited states are not needed for these exponents and there is no need to calculate the excitation gap. Furthermore, the finite size analysis described in the previous subsection takes advantage of the fact that the DMRG algorithm works best with open chains and treats the central region of the chain most accurately. The use of the more complicated finite-size algorithm might yield even more accurate results. However, we show below that we can calculate critical exponents to an accuracy of a few percent or better with the infinite-size algorithm.
Tight Binding Model at Half-Filling {#sec:tightbinding}
===================================
As a simple first illustration of our finite-size scaling method we study the ordinary tight binding model of spinless fermions hopping from site-to-site along a chain at half-filling. Obviously, the DMRG algorithm is not needed in this case as we can solve the quadratic problem exactly via a Fourier transform. Due to particle-hole symmetry, at half-filling the chemical potential is zero. The correlation length exponent for this system is $\nu = 1$. A direct way to see this is by introducing the staggering parameter $R$ to modulate the amplitude of the hopping matrix elements on even versus odd links: $$\begin{aligned}
H &=& t \sum_{j=0}^{L-1} [1 + (-1)^j R] \big{(} c^\dagger_j c_{j+1}
+ h.c. \big{)}\ .
\label{stag-tb}\end{aligned}$$
To diagonalize the Hamiltonian, in the case of periodic boundary conditions $c_0 = c_{L}$, we introduce separate fermion operators for even and odd sites as follows: $$\begin{aligned}
c_{2j} = d_{2j} \nonumber \\
c_{2j-1} = e_{2j}
\label{evenodd}\end{aligned}$$ After the Fourier transformation to momentum-space, the Hamiltonian can be written as: $$\begin{aligned}
H = t \sum_k \bigg{\{} [ (1-R) + e^{2ik}(1+R) ] ~d_{k}^{\dagger} e_k +
[ (1-R) + e^{-2ik}(1+R) ] ~e^{\dagger}_{k} d_k \bigg{\}},
\label{fourier-tb}\end{aligned}$$ where the lattice spacing $a = 1$. For each $k$, diagonalization of the $2 \times 2$ matrix yields the dispersion relation: $$\begin{aligned}
\epsilon_k = \pm 2 t \sqrt{ 1 - (1-R^2) \sin^2(k) }.
\label{dispersion-tb}\end{aligned}$$ At half-filling the ground state has all states with $\epsilon_k < 0$ occupied. The left and right Fermi points are, respectively, $k_F = \pm \pi/2$. Hence the gap $m = 2 t |R|$. As the correlation length $\xi \propto m^{-1} \propto |R|^{-1}$ we obtain $\nu = 1$. Since $\nu^{-1} = 2 - x_{\Delta} = 1$, the dimerization exponent $x_\Delta = 1$.
We now reproduce this result using the finite-size scaling method applied to open chains. We consider a finite chain of length $L$ with open boundary conditions and calculate the induced dimerization $\Delta(j) = (-1)^j ~\langle c^\dagger_j c_{j+1}
- c^\dagger_{j+1} c_{j+2} \rangle $ around the chain center $j = L/2$, and extract its leading dependence on $L$. Open boundary conditions are imposed by using the Fourier transform $$\begin{aligned}
c_j &=& \frac{1}{\sqrt{2 (L+1)}} \sum_{m=1}^L c_{k_m}~
( e^{i k_m j} - e^{-i k_m j} ), \nonumber \\
k_m &=& \frac{\pi}{L+1} m, \hskip 1cm m=1, 2, \ldots, L
\label{ft-tb}\end{aligned}$$ as this enforces $c_0 = c_{L+1} = 0$. Filling all of the negative energy states at half-filling, the expectation value of the dimerization at $L/2$ can be found by straightforward calculation: $$\Delta(L/2) \propto \frac{1}{L+1} \sum_{m=\frac{L}{2}+1}^L \big{[}
\cos[k_m (L+3)] - \cos[k_m (L+1)] \big{]}.
\label{dimer-tb}$$ This sum can be evaluated numerically with the result that $x_{\Delta}
\rightarrow 1$ as $L \rightarrow \infty$ as shown in Fig. \[dim-tb\], in agreement with the explicit calculation for the periodic chain. It is also easy to show that open chains with an odd number of sites have vanishing induced dimerization at the center of the chain, as expected by the symmetry of reflection about the central site.
The induced density moment can likewise be obtained either directly by studying the effects of a staggered chemical potential $\mu_{stag}$ (which doubles the size of the unit cell from one to two sites and thus generates a gap $m = 2 |\mu_{stag}|$) or by the inclusion a local chemical potential $\mu$ at the two ends of the chain: $$\begin{aligned}
H \rightarrow H - \mu~ (c^{\dagger}_0 c_0 + c^{\dagger}_{L-1} c_{L-1})\ .\end{aligned}$$ Again the system consists of $L$ sites, the site index running from $0$ to $L-1$, and there are open boundary condition at $j = 0$ and $j = L-1$. For large $\mu \gg 0$, the boundary condition is equivalent to enforcing unit occupancy at the chain ends, $n_0 = n_{L-1} = 1$. This boundary condition is satisfied by the Fourier transform $$\begin{aligned}
c_j &=& \frac{1}{\sqrt{2 (L-1)}} \sum_{m=0}^{L-1} c_{k_m}~
( e^{i k_m j} + e^{-i k_m j} )\end{aligned}$$ with $$\begin{aligned}
k_m &=& \frac{\pi}{L-1} m, \hskip 1cm m=0, 1, \ldots, L-1 \ .\end{aligned}$$ Again it is a simple exercise to calculate the occupancies. At the chain ends we obtain: $\langle c^{\dagger}_0 c_0 \rangle = \langle c^{\dagger}_{L-1} c_{L-1} \rangle
= 1 $ in agreement with the boundary condition. At the center of the chain the occupancy can be evaluated analytically, $$\begin{aligned}
\langle c^{\dagger}(L/2) c(L/2) \rangle =
\frac{1}{L-1} \sum_{m=\frac{L}{2}}^{L-1} \left[ 1 + \cos(k_m L) \right].\end{aligned}$$ It scales as $\langle c^{\dagger} (L/2) c(L/2) \rangle - 1/2 \propto L^{-1}$. Hence $\nu_B
= x_B = 1$ in agreement with the direct calculation of these exponents.
Spin-$1/2$ Antiferromagnet {#sec:s=half}
==========================
We next turn to the study of a richer system: spin-$1/2$ antiferromagnetic chains. We begin with the XY model, which can be solved exactly by a Jordan-Wigner mapping to the tight binding model. We then study the anisotropic XXZ model. The isotropic Heisenberg model is treated separately as there are complicating multiplicative logarithmic corrections to scaling at the isotropic point.
XY model {#subsec:xy}
--------
The Hamiltonian for the spin-$1/2$ XY model, $$H = J \sum_{j=0}^{L-2} \big{[} S^x_j S^x_{j+1} + S^y_j S^y_{j+1} \big{]} \ ,
\label{h-xy}$$ can be written in terms of spinless fermion creation and annihilation operators $c_j^\dagger$ and $c_j$ via the Jordan-Wigner transformation[@Jordan]. An up spin in the z-direction at site $i$ then corresponds to having the site occupied by a fermion, while spin down corresponds to an empty site. The Hamiltonian of Eq. \[h-xy\] is mapped to a nearest-neighbor tight binding Hamiltonian with $t = J$. Based on our analysis in the previous section we can conclude that $\nu = 1$ for the XY model.
Fig. \[xy\] presents our DMRG results for the induced dimerization and induced spin moments, in the x- and in the z-directions, at the center of the chain as a function of the chain length, $L$.
The exponents are obtained from the slopes of the curves shown in Fig. \[xy\]. The induced dimerization exponent for $\Delta(L/2)$ is close to $1$ ($x_{\Delta} = 0.99 \pm 0.01$) as expected from the relation $\nu = 1/(2-x_{\Delta})$. The slope of the log-log plot of the induced spin moment in the z-direction is also close to $1$ ($x_B = 1.01 \pm 0.02$). This result is also expected since it is equivalent to the exponent for the induced density moment in the tight binding model as discussed in the previous section. In the case of the induced spin moment in the x-direction, the exponent is $0.248 \pm 0.003$. This value compares well with the exact number of $1/4$ as derived in the next section.
XXZ model {#subsec:xxz}
---------
Next consider the nearest-neighbor, spin-1/2 XXZ Heisenberg antiferromagnet: $$H = J \sum_{j=0}^{L-2} \big{[} S_j^x S_{j+1}^x + S_j^y S_{j+1}^y +
\gamma S_j^z S_{j+1}^z \big{]}\ .
\label{Heisenberg}$$ Anisotropy in the coupling between the z-components of the spins may be varied by changing $\gamma$. Performing the Jordan-Wigner transformation, the XY terms again yield the tight binding Hamiltonian. Low-energy excitations therefore occur near the two Fermi points at $k = \pm \pi / 2a$. We may treat the non-Gaussian $\gamma$ term as a perturbation and focus on excitations around these Fermi points by defining left and right moving low-energy quasiparticles. Taking the continuum limit and keeping only the low-energy modes, the tight binding term is then effectively described by the massless fermions. The $S_j^z S_{j+1}^z$ term is quartic in the fermion operators. Integrating out the high-energy modes, it will renormalize the fermion velocity and also contain interaction terms.
We then implement Abelian bosonization, with UV cutoff $\alpha$. The effective Hamiltonian is a sine-Gordon model (a derivation can be found in Ref. 49): $$H = H_0 - \frac{y_{\phi}}{2 \pi \alpha^2} \int dx
\cos[\sqrt{8 \pi} \phi(x)]
\label{sine-Gordon}$$ where $$H_0 = u \int dx \bigg{[} K \Pi^2 + \frac{(\partial_x \phi)^2}{K} \bigg{]}.
\label{freeboson1}$$ Here $u = 2 J a = 2 a$ is the bare Fermi velocity and the constant $K \equiv 1 + y_0/2$ depends on the anisotropy $\gamma$. The XY limit correspond to $y_{\phi} = 0$.
The long distance behavior of the staggered part of $S^z$ and $S^-$ are given in terms of the boson fields as: $$\begin{aligned}
S^z(x) &\approx& (-1)^{x/\alpha} \cos [ \phi(x)/R ] \nonumber \\
S^-(x) &\approx& (-1)^{x/\alpha} e^{i 2 \pi R \tilde{\phi}(x)}
\label{spinops}\end{aligned}$$ where the radius $R$ is given by[@Affleck] $$\begin{aligned}
R=\sqrt{ \frac{1}{2\pi} - \frac{\cos^{-1} \gamma}{2 \pi^2} } .
\label{R}\end{aligned}$$ First consider the anisotropic case $\gamma \neq 1$. The isotropic case has logarithmic corrections to scaling that are dealt with in the next section. For $\gamma > 1$ the interaction term is relevant and the system is gapped, and in the Ising universality class. Indeed, in the limit $\gamma \rightarrow \infty$ it is the Ising model. For $\gamma < 1$ the interaction term is irrelevant, the system is gapless and $\Delta(L/2)$ and $\langle S^x(L/2) \rangle$ should exhibit power law decay, with no log corrections as there are no marginal operators. The log-log plots of Fig. \[s12\_sxgamma\] (a) show the induced spin moment in the x-direction at the chain center for different values of the anisotropy $\gamma$. The edge magnetic field in the x-direction is fixed, $h = 1.0$. As expected, for $\gamma > 1$ there is exponential decay and in the cases $\gamma < 1$ the exponents $x_B(\gamma)$ are found by fitting the curves in Fig. \[s12\_sxgamma\] (a) to the form of Eq. \[scaling\]. The exponents $y_B$ are set equal to zero, the higher order corrections are included and give very small deviations from a simple linear fit. In Fig. \[s12\_sxgamma\] (b) the exponents $x_B(\gamma)$ are compared to the exact value $x_B(\gamma) = \pi R^2(\gamma)$ obtained by Affleck[@edgefield]. Agreement is found at the percent level. Affleck derived the exponent as follows. The edge magnetic field in the x-direction applied at $j=0$ corresponds to a term $$H_B = - h S^x(0) = -{\rm constant} \times h \cos[\sqrt{2 \pi} \tilde{\phi}(0)]
\label{edgefield}$$ in the Hamiltonian. For sufficiently large $h$ the energy is minimized by setting $$\tilde{\phi}(0) = 0 \Longrightarrow \phi_R(0) = \phi_L(0) \ .
\label{BC}$$ Regarding $\phi_R$ as an analytic continuation of $\phi_L$, we may identify $$\phi_R(x) = \phi_L(-x) \ .
\label{BCgen}$$ Using this boundary condition, the induced spin moment is given by $$\begin{aligned}
\langle S^x(j) \rangle \approx (-1)^{j/\alpha}
\langle e^{i 2 \pi R \phi_L(j)} e^{-i 2 \pi R \phi_L(-j)} \rangle
\approx \frac{(-1)^{j/\alpha}}{(2j)^{\pi R^2(\gamma)}} \ .\end{aligned}$$ For the XY model ($\gamma = 0$), the induced spin moment in the x-direction therefore decays with exponent $\pi R^2(0) = 1/4$.
A log-log plot of the induced dimerization at the center of the chain for various values of the anisotropy $\gamma$ is shown in Fig. \[s12\_dimgamma\] (a). The free boundary condition at the chain ends corresponds to setting: $$\begin{aligned}
\vec{S}_0 = \vec{S}_{L+1} = 0\ .\end{aligned}$$ This condition translates to $\phi_R(x) = - \phi_L(-x) + \pi R$ in terms of the boson fields[@Eggert] which yields: $$\begin{aligned}
\Delta(j) ~ \approx ~ (-1)^{j/\alpha} \langle
\cos[\phi(j)/R] \rangle
~ \approx ~ \frac{(-1)^{j/\alpha}}{(2j)^{1/4 \pi R^2(\gamma)}}.\end{aligned}$$ In Fig. \[s12\_dimgamma\] (b) the exponents obtained from the slopes of the curves in Fig. \[s12\_dimgamma\] (a) are plotted against the exact values $x_{\Delta} = 1/4 \pi R^2(\gamma)$. Again agreement is found at the percent level.
Another quantity of interest is the sum, instead of the difference, of the spin-spin correlation function on adjacent bonds near the center of the chain: $$\begin{aligned}
\epsilon(L/2) ~ \equiv ~ \frac{1}{2} ~ \big{(} ~ \langle \vec{S}_{L/2} \cdot
\vec{S}_{L/2+1} \rangle + \langle \vec{S}_{L/2-1} \vec{S}_{L/2} \rangle ~
\big{)} ~ ,\end{aligned}$$ which at $\gamma = 1$ equals the energy density per bond and therefore does not vanish in the thermodynamic limit. Fig. \[s12sum\] is a plot of $\epsilon(L/2)$ as a function of the system size $L$ at the isotropic point $\gamma=1$.
As expected, this quantity approaches a constant value $\epsilon(\infty)$ in the thermodynamic limit. After subtracting the extrapolated value at $L \rightarrow \infty$, $\epsilon(L/2)$ too exhibits power law decay of the form of Eq. \[scaling\]. The constant $\epsilon(\infty)$ can be found by an iteration process. Starting with an initial value for $\epsilon(\infty)$ obtained from a rough extrapolation of the curve in Fig. \[s12sum\], we fit the subtracted value $\epsilon(L/2) - \epsilon(\infty)$ to a power-law form. The extrapolated value $\epsilon(\infty)$ is then adjusted slightly until an optimal fit to a pure power law is attained. The extrapolated value found this way is $\epsilon(\infty) = -0.443148$ and Fig. \[s12sumexp\] shows the power law behavior of the subtracted quantity.
We obtain an exponent of $2.1 \pm 0.1$ in the scaling of $\epsilon(L/2)-\epsilon(\infty)$. This is as expected from the linear dispersion relation of Heisenberg antiferromagnets: in a Lorentz-invariant theory the energy density operator has dimension 2.
The DMRG result for the energy per bond is extremely accurate and can be compared with the exact value obtained from the Bethe ansatz solution[@Baxter] of $~\epsilon = 1/4 - \ln 2 = -0.44314718 $. It is crucial to note that the open boundary conditions induce staggering in the strength of the bonds along the chain. To eliminate this effect, the energy per bond must be calculated as the average of the bond energy from two consecutive bonds at the center of the chain. Suggestions that infinite-size DMRG results for the center region of the chain are not very accurate[@Tasaki] appear to have failed to take this effect into account. We have also checked our results at different anisotropies. For the XY case ($\gamma=0.0$), we obtain $\epsilon(\infty) =
-0.318310$ extrapolating from chains up to $L = 200$ and $M = 128$ and the exact result[@Baxter] is $-1/\pi = -0.3183099$.
Logarithmic Corrections to Scaling {#subsec:log}
----------------------------------
In the isotropic XXX limit, the interaction $\cos[\sqrt{8 \pi} \phi(x)]$ in the low-energy effective Hamiltonian Eq. \[sine-Gordon\] becomes marginal and can generate multiplicative logarithmic corrections to scaling. In this section we calculate its effect on the scaling of the induced spin moment $\langle S^x(L/2) \rangle$ when an edge magnetic field $H_B$ in the x-direction is applied. Cancellations occur and in this case there are no multiplicative $\ln(L)$ corrections. As a practical matter, the cancellation of the logarithmic corrections means that numerical calculations of the exponent $x_B$ are particularly precise. We note that finite-size scaling of the spin-spin correlation function has been previously calculated for a spin-$1/2$ chain with periodic boundary conditions[@Barzykin; @exact].
The coupling constants in the sine-Gordon Hamiltonian (Eq. \[sine-Gordon\]) renormalize under a change of the ultraviolet cutoff $\alpha \rightarrow
\alpha e^l$ according to the renormalization group equations[@Giamarchi]: $$\begin{aligned}
\frac{dy_0}{dl} &=& -y_{\phi}^2(l), \nonumber \\
\frac{dy_{\phi}}{dl} &=& -y_{\phi}(l) y_0(l).
\label{RG}\end{aligned}$$ As noted in the previous section, a large edge magnetic field applied at $x=0$ in the x-direction enforces the boundary condition $\phi_R(x) = \phi_L(-x)$ (Eq. \[BCgen\]). Thus $$\langle S^x(x) \rangle \sim (-1)^{x/a} \langle
\cos[\sqrt{2 \pi} \tilde{\phi}(x)] \rangle \sim \langle
e^{i \sqrt{2\pi} \phi_L (x)} e^{-i \sqrt{2\pi} \phi_L (-x)} \rangle.
\label{sx}$$ For the free theory, which corresponds to the XY model $y_{\phi} = 0$, the induced spin moment is simply $$\langle S^x(x) \rangle_0 \sim \exp \big{[} -K U_L (2x) \big{]}
\label{sx0}$$ where $$U_L(x) = \frac{1}{2} \ln ( \frac{\alpha + i x}{\alpha} ).
\label{U_L}$$ But in the general XXZ case we ascertain the effect of the marginal operator by following a procedure similar to one developed by Giamarchi and Schulz[@Giamarchi] who calculated correlation functions for finite [*periodic*]{} chains. We first define the function: $$\begin{aligned}
F(x) \equiv e^{K U_L(2x)} \langle S^x(x) \rangle\ .
\label{fx}\end{aligned}$$ At the XY point $y_{\phi} = 0$ clearly $F(x) = 1$. For small $x$, an expansion of $F$ in powers of $y_{\phi}$ converges, and for sufficiently small coupling $y_{\phi}$, $F(x) \sim 1$. Upon rescaling, the function $F(x)$ also depends on the new length scale and on the rescaled coupling constants $y_0(l)$ and $y_{\phi}(l)$. By an argument similar to the one employed by Kosterlitz[@Kosterlitz], the effect of rescaling $\alpha \rightarrow e^l ~\alpha$ is: $$\begin{aligned}
F(x, \alpha e^l, y(l)) = I(dl, y(l)) F(x, \alpha e^{l+dl}, y(l+dl))\ ,
\label{kost}\end{aligned}$$ where $y(l)$ denotes all the couplings as function of the scaling parameter $l$. The rescaled short distance cutoff is then $\alpha(l) = e^l ~\alpha$, where $\alpha$ is the initial cutoff. Rescaling can be repeated until $\alpha(l) \sim x$, at which point we have: $$\begin{aligned}
F(x, x, y( \ln (x/\alpha))) = O(1)\ .\end{aligned}$$ The contributions to the function $F$ from repeated rescalings, until $\alpha(l)$ reaches $x$, can be written explicitly as: $$\begin{aligned}
F(x, \alpha, y(\alpha)) = \prod_{l=0}^{l=\ln (x/\alpha)} I(dl, y(l)) =
\exp \bigg{\{}
\int_0^{\ln(x/\alpha} \ln \big{[} I(dl, y(l)) \big{]} dl \bigg{\}} \ .
\label{rescaling-f}\end{aligned}$$
We proceed to calculate the function $I$. First we expand $\langle S^x(x) \rangle$ in powers of $y_{\phi}$, writing it in terms of averages with respect to the free Hamiltonian, $$\begin{aligned}
\langle S^x(x) \rangle &\sim& e^{-K U_L(2x)} +
\frac{y_{\phi}}{2 \pi \alpha^2} \int d^2 x'
\langle S^x(x) \cos[\sqrt{8 \pi} \phi(x')] \rangle_0 + \nonumber \\ & &
\frac{1}{2} \big{(} \frac{y_{\phi}}{2 \pi \alpha^2} \big{)}^2
\int d^2 x_1 \int d^2 x_2
\langle S^x(x)
\cos[\sqrt{8 \pi} \phi(x_1)] \cos[\sqrt{8 \pi} \phi(x_2)] \rangle_0
+ \ldots
\label{expansion}\end{aligned}$$ The averages are given by $$\begin{aligned}
\langle S^x(x) \cos[\sqrt{8 \pi} \phi(x')] \rangle_0 = 0 \nonumber\end{aligned}$$ and $$\begin{aligned}
\langle S^x(x)
\cos[\sqrt{8 \pi} \phi(x_1)] && \cos[\sqrt{8 \pi} \phi(x_2)] \rangle_0
\sim \nonumber \\
&& \frac{1}{2} \exp \big{[} -K U_L(2x) + 4 K U_L(2x_1) + 4 K U_L(2x_2) -
4 K U(x_1+x_2) - 4 K U(x_1-x_2) \big{]} \end{aligned}$$ The $O(y_{\phi}^2)$ term can be simplified by assuming that the main contribution comes from configurations where $x_1$ and $x_2$ are very close to each other[@Nelson; @Giamarchi]. Introducing new integration variables $$\begin{aligned}
r &\equiv& x_1 - x_2 \nonumber \\
R &\equiv& \frac{x_1+x_2}{2} \end{aligned}$$ and expanding $U_L$ in powers of $r$, which is assumed to be small, $$\begin{aligned}
U_L(2 x_1) = U_L(2R + r) = U_L(2R) + r ~\partial_R ~U_L(2R) + \ldots \ ,
\nonumber \\
U_L(2 x_2) = U_L(2R - r) = U_L(2R) - r ~\partial_R ~U_L(2R) + \ldots \ , \end{aligned}$$ we obtain the average $$\begin{aligned}
\langle S^x(x) \cos[\sqrt{8 \pi} \phi(x_1)] \cos[\sqrt{8 \pi}
\phi(x_2)] \rangle_0 \sim \frac{1}{2} \exp \left[
- K U_L(2x) - 4 K U(r) \right]\ .\end{aligned}$$ The dependence on $R$ cancels out. The expansion Eq. \[expansion\] becomes $$\begin{aligned}
\langle S^x(x) \rangle &\sim& e^{-K U_L(2x)} \bigg{[} 1 +
\frac{y_{\phi}^2 \Omega}{4 \alpha^2}
\int_{\alpha} dr e^{-4K U(r)} \bigg{]} \ ,\end{aligned}$$ where $\Omega \equiv \int dR$ is a measure of the linear size of the system. Next consider the effect of rescaling $\alpha^{\prime} = \alpha e^{dl}$, where $dl$ is infinitesimal. Using $$\begin{aligned}
\int_{\alpha}^{\infty} dx = \int_{\alpha}^{\alpha^{\prime}} dx +
\int_{\alpha^{\prime}}^{\infty} dx \ \ ,\end{aligned}$$ we obtain: $$\langle S^x(x) \rangle \sim e^{-K U_L(2x)} \bigg{[} 1 +
\frac{y_{\phi}^2}{4 \alpha^2} dl + \frac{y_{\phi}^{\prime 2}}
{4 \alpha^{\prime 2}} \int_{\alpha^\prime} dr e^{-4K U(r)} \bigg{]}
\label{rescaled}$$ Matching this result with Eq. \[kost\], we find: $$I(dl, y_0(l), y_{\phi}(l)) \sim \exp \bigg{[} \frac{y_{\phi}^2 (l)}
{4 \alpha^2} dl \bigg{]}\ ,$$ hence from Eq. \[rescaling-f\] and Eq. \[fx\], we have $$\begin{aligned}
\langle S^x(x) \rangle \sim \exp \bigg{\{} - K U_L(2x) +
\int_0^{\ln(x/\alpha)} \frac{y_{\phi}^2 (l)}{4 \alpha^2} dl \bigg{\}}\ .\end{aligned}$$ Using the RG equations (Eq. \[RG\]), the solution at large $l$ is $y_{\phi}(l) \sim 1/l$ and $$\begin{aligned}
\langle S^x(x) \rangle \sim \bigg{(} \frac{x}{\alpha} \bigg{)}^{-1/2} \exp
\bigg{\{} \int_0^{\ln(x/\alpha)} dl \big{[} ~O(\frac{1}{l^2})~
\big{]} \bigg{\}}\ .
\label{nolog}\end{aligned}$$ There are no multiplicative $\ln(x)$ corrections, as these would require terms of order $O(1/l)$ in the integrand inside the exponential in Eq. \[nolog\]. In our calculation, $O(1/l)$ terms do not appear, only $O(1/l^2)$ and higher-order terms. As a check, we can repeat the same procedure for $\langle S^z(x) \rangle$, with the edge field now oriented in the z-direction. Of course this should give the same result since the system is isotropic, but as the Jordan-Wigner transformation picks the z-direction as the spin quantization axis, the equivalence is not obvious, and the check is non-trivial. Again, explicit calculation shows that $O(1/l)$ terms do not arise. This result is in reasonable agreement with our numerical results. Fitting the DMRG data (see Fig. \[fig-xxx\]) to the form Eq. \[scaling\], we obtain $x_B = 0.485 \pm 0.01$ with a small non-zero value for the log exponent $y_B = 0.06 \pm 0.01$. By contrast, in the case of the induced dimerization we obtain $x_{\Delta} = 0.57 \pm 0.01$ and $y_{\Delta} = 0.10 \pm 0.05$. The error was estimated from deviations obtained by fitting the $M = 128$ data over different ranges of L ($4 \le L \le 600$) and by comparison with $M = 64$ data ($4 \le L \le 300$). Results are systematically improved by increasing the value of $M$.
Finite-size scaling behavior for the XXX model with open boundary conditions[@Hikihara] and periodic boundary conditions[@Hallberg] were obtained from DMRG calculations of ground state energies and correlation functions $\langle S^z(x) S^z(x+r) \rangle$ for different system sizes and separations $r$. In our approach, critical exponents are extracted from expectation values at the center of the chain only. The chain size is increased via the infinite-size DMRG method. It is also advantageous to extract power law exponents when there are no logarithmic corrections.
Conformal Anomaly {#subsec:c-half}
-----------------
Finally, we may calculate the value of the conformal anomaly, $c$. We note that the central charge of the RSOS model[@Sierra] and of the spin-$3/2$ Heisenberg chain[@Hallberg], which is in the same universality class as the spin-$1/2$ chain, have previously been obtained using the DMRG. The conformal anomaly can be extracted by finding the coefficient of the $1/L$ finite-size correction to the free energy, equivalent at zero temperature to the ground state energy. We fit the ground state energy $E_0(L)$ to the following form: $$E_0(L) = A L + B + C / L + \ldots$$ The extensive contribution, proportional to $A$, and the constant term $B$ are non-universal. At the isotropic point $\gamma = 1$ our results for the case of blocksize $M = 128$ and for chain lengths in the range $30 \leq L \leq 100$ yield $C \approx -0.323$. To relate this coefficient to the conformal anomaly we must normalize it by dividing by the speed of low-lying excitations, $v$. The speed can be obtained by extrapolation to the thermodynamic limit of the gap to the lowest-lying excitation multiplied by the chain length: $$v = \lim_{L \rightarrow \infty}~ {{{\rm Gap}(L) \times L}\over{\pi}}\ .$$ We find $v = 2.44$. Now for open boundary conditions[@Cardy], $$c = {{-24 C}\over{\pi v}} \approx 1.01\ .$$ This compares well with the value of $c = 1$ appropriate for a single boson or the pair of left and right moving fermions.
Spin-$1$ chain {#sec:s=1}
==============
As a final example we apply the DMRG / finite-size scaling method to the isotropic spin-$1$ antiferromagnetic chain. This problem is more challenging numerically as the on-site Hilbert space now has dimension $D = 3$ instead of $D = 2$. The most general nearest-neighbor Hamiltonian for the spin-$1$ chain includes the possibility of a biquadratic spin-spin interaction term: $$H = \sum_{j=0}^{L-2} \big{[} \cos \theta ~ \vec{S}_j \cdot \vec{S}_{j+1} +
\sin \theta ~ ( \vec{S}_j \cdot \vec{S}_{j+1} )^2 \big{]}
\label{s1_hamilt}$$ The phase diagram can be represented on a circle parameterized by $\theta$ as depicted in Fig. \[phasediagram1\].
Generically there is a gap to excitations in the antiferromagnetic region of the phase diagram, in accord with the Haldane conjecture[@Haldane]. The point $\theta = 0$ corresponds to the usual pure bilinear Heisenberg antiferromagnet. At the point $\tan \theta = 1/3$ the Hamiltonian can be written as a sum of positive-definite projection operators, and the exact ground state is the AKLT valence bond solid (VBS)[@aklt]. Negative $\sin \theta$ favors dimerization, as the energy is minimized by concentrating singlet correlations on isolated dimers. The dimerized phase also is gapped: a dimer must be broken to generate a spin excitation. The point that separates the dimerized and Haldane phases lies at $\theta = - \pi/4$ and can be solved exactly by the Bethe ansatz[@Takhtajan; @Babudjian]. The chain is quantum critical at this integrable point. The ground state is non-degenerate here as well as in the dimerized and Haldane phases.
DMRG calculations clearly delineate the two massive phases and the critical point separating them, even for relatively small block Hilbert sizes $M$. In Figs. \[s1-sz\] and \[s1-dimer\], the blocksize $M=81$ for the massive phases. Thus the results are numerically exact up only to chain lengths $L=10$. For chain lengths $L > 10$ the Hilbert space is truncated via the DMRG algorithm. To increase accuracy, results at the critical point were obtained with a larger Hilbert size for the blocks, $M=256$.
The induced spin moment at the center of the chain decays exponentially in both the Haldane and the dimerized phases, as expected. The induced dimerization at the chain center also decays exponentially in the Haldane phase, but approaches a non-zero constant in the dimerized phase as it must. Power law decay in both observables occurs at the critical point. Fitting the $M=256$ data shown in Fig. \[s1-dimer\] at the critical point $\theta = - \pi/4$ we obtain dimerization exponents $x_{\Delta} = 0.37 \pm 0.01$ and $y_{\Delta} = 0.3 \pm 0.05$, reflecting the apparent presence of a marginal interaction and consequent multiplicative logarithmic corrections to scaling. Likewise, for a field of $h = 1.0$ applied to the chain ends, the exponents for the spin operator are $x_B = 0.34 \pm 0.01$ and $y_B = 0.23 \pm 0.05$. The values of the exponents compare to the exact values[@spin1a; @spin1b] $x_{\Delta} = 3/8 \approx 0.375$ and $x_B = 3/8$. To the best of our knowledge there are no analytic results at the integrable point $\theta = - \pi/4$ (which corresponds to a $k=2$ SU(2) WZW model) on the size of the logarithmic corrections $y_\Delta$ and $y_B$, at least for open boundary conditions.
Finally, we may repeat the analysis of the conformal anomaly described above in subsection \[subsec:c-half\] for the case of the spin-1 chain at its critical point. For $M = 256$ and fitting over chain lengths $10 \leq L \leq 26$ we find that the speed of excitations is $v = 3.69$, $C = 0.508$, and hence $c = 1.05$. This value is close to its exact value of 1, demonstrating that the conformal anomaly can be reliably extracted even from relatively short chains.
Conclusion {#sec:conclusion}
==========
We have presented a simple method for studying critical behavior of quantum spin chains. Accurate critical exponents can be extracted. For small on-site Hilbert space sizes ($D=2$ for the spin-$1/2$ chain and $D=3$ for spin-$1$ chains) the method does not require supercomputers. Results can be systematically improved by increasing the size of $M$, the dimension of the Hilbert space retained in the blocks, up to limits set by machine memory and speed. The DMRG method works best for massive, non-critical, systems, but it is also quite accurate even at critical points. Critical exponents can be calculated at percent level accuracy. We showed that the leading multiplicative logarithmic correction to the scaling of the induced spin moment cancels out in the case of the isotropic spin-$1/2$ Heisenberg antiferromagnet. Thus accurate exponents can sometimes be found numerically despite the presence of marginal interactions.
Use of the “finite-size” DMRG algorithm might improve the method, but good results were obtained with the relatively simpler “infinite-size” DMRG algorithm. The reason for this is that the finite-size scaling method employed here focuses on the scaling of observables near the center of the chain only, where the “infinite-size” algorithm is particularly accurate. The method can be used to study new systems. For example, several non-interacting but disordered electron systems, like the integer and spin quantum Hall transitions, can be described by supersymmetric Hamiltonians[@LSM; @Tsai1]. In a paper which follows[@part2], we employ the combined DMRG/finite-size method in combination with analytic calculations to understand the behavior of these supersymmetric spin chains.
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abstract: |
We develop a theory of large scale geometry of metrisable topological groups that, in a significant number of cases, allows one to define and identify a unique quasi-isometry type intrinsic to the topological group. Moreover, this quasi-isometry type coincides with the classical notion in the case of compactly generated locally compact groups and, for the additive group of a Banach space, is simply that of the corresponding Banach space. In particular, we characterise the class of separable metrisable groups admitting [*metrically proper*]{}, respectively, [*maximal*]{} compatible left-invariant metrics. Moreover, we develop criteria for when a metrisable group admits metrically proper affine isometric actions on Banach spaces of various degress of convexity and reflexivity.
A further study of the large scale geometry of automorphism groups of countable first order model theoretical structures is separated into a companion paper.
address: |
Department of Mathematics, Statistics, and Computer Science (M/C 249)\
University of Illinois at Chicago\
851 S. Morgan St.\
Chicago, IL 60607-7045\
USA
author:
- Christian Rosendal
title: Large scale geometry of metrisable groups
---
[^1]
Introduction
============
The large scale geometry of finitely generated discrete groups or compactly generated locally compact second countable groups is by now a well-established theory (see [@nowak; @harpe] for recent accounts). In the finitely generated case, the starting point is the elementary observation that the word metrics $\rho_\Sigma$ on a discrete group $\Gamma$ given by finite symmetric generating sets $\Sigma\subseteq \Gamma$ are mutually quasi-isometric and thus any such metric may be said to define the large scale geometry of $\Gamma$. In the locally compact setting, matters have not progressed equally swiftly even though the basic tools have been available for quite some time. Indeed, by a result of R. Struble [@struble] dating back to 1951, every locally compact second countable group admits a compatible left-invariant [*proper*]{} metric, i.e., so that the closed balls are compact. Struble’s theorem was based on an earlier well-known result due independently to G. Birkhoff [@birkhoff] and S. Kakutani [@kakutani] characterising the metrisable topological groups as the first countable topological groups and, moreover, stating that every such group admits a compatible left-invariant metric. However, as is evident from the construction underlying the Birkhoff–Kakutani theorem, if one begins with a compact symmetric generating set $\Sigma$ for a locally compact second countable group $G$, then one may obtain a compatible left-invariant metric $d$ that is quasi-isometric to the word metric $\rho_\Sigma$ induced by $\Sigma$. By applying the Baire category theorem and arguing as in the discrete case, one sees that any two such word-metrics $\rho_{\Sigma_1}$ and $\rho_{\Sigma_2}$ are quasi-isometric, which shows that the compatible left-invariant metric $d$ is uniquely defined up to quasi-isometry by this procedure.
However, thus far, it seems that no one has been able to identify a well-defined large scale geometry of metrisable topological groups beyond the locally compact. Largely, this may be due to the presumed absence of canonical generating sets in general metrisable groups as opposed to the finitely or compactly generated ones.
In the present paper, we offer a solution to this problem, which in many cases allows one to isolate and compute a canonically defined metric on a metrisable topological group $G$ and thus to identify a unique quasi-isometry type of $G$. Moreover, this quasi-isometry type satisfies the main characterics encountered in the finitely or compactly generated settings, namely (i) that it is a topological isomorphism invariant of $G$, (ii) that it is realised on $G$ by some compatible left-invariant metric and (iii) it is non-trivial, witnessed here by capturing all possible large scale behaviour of $G$.
The theory of metrisable topological groups, in particular, Polish groups, has seen tremendous progress in the last twenty years. For example, in the special case of non-Archimedean Polish groups, studies have been done of their topological dynamics and of questions of amenability [@kpt], the structure of conjugacy classes and topological rigidity [@turbulence] and their representation theory [@tsankov]. Our goal here is to add another facet to the theory, by providing the framework for the introduction of more geometrical tools, connecting this with the structure of affine isometric actions on Banach spaces and also by studying in detail the special case of non-Archimedean Polish groups.
The basic idea of our paper is to replace the topological notion of compactness by a metric notion, which we term the [*relative property (OB)*]{}. Namely, a subset $A$ of a metrisable topological group $G$ is said to have [*property (OB) relative to $G$*]{} if $A$ has finite diameter with respect to every left-invariant metric on $G$. Similarly, $G$ is said to have [*property (OB)*]{} [@OB] if it has property (OB) relative to itself, i.e., if $G$ has finite diameter with respect to every compatible left-invariant metric. Whereas, by the theorem of Struble, the relative Property (OB) is equivalent to relative compactness in the case of locally compact second countable groups, this is far from being the case for general metrisable groups. Indeed, a surprising number of these have property (OB) without being compact, e.g., the unitary group $U({\mathcal}H)$ with the strong operator topology [@ricard], homeomorphism groups of spheres ${\rm Homeo}(S^n)$ and of the Hilbert cube ${\rm Homeo}([0,1]^{\mathbb N})$ [@OB], the group of measure-preserving automorphisms of a standard probability space ${\rm Aut}([0,1]^{\mathbb N})$ with the weak topology [@glasner], and, more generally, automorphism groups of separably-categorical metric structures [@OB].
We then define a compatible left-invariant metric $d$ on $G$ to be [*metrically proper*]{} if every set of finite $d$-diameter set has property (OB) relative to $G$ and show that any two such metrics on $G$ will be coarsely equivalent. Moreover, the existence of these metrics is characterised by the following theorem.
A separable metrisable topological group $G$ admits a metrically proper compatible left-invariant metric if and only if $G$ has the [*local property (OB)*]{}, i.e., there is a neighbourhood of the identity having property (OB) relative to $G$.
Though many familiar groups do have the local property (OB), there are counterexamples such as the infinite direct product of groups without property (OB), e.g., ${\mathbb Z}^{\mathbb N}$.
While the metrically proper metrics, when they exist, uniquely define the coarse equivalence class of a metrisable group $G$, they do not suffice to define its quasi-isometry class. For that, we isolate a more restrictive class of metrics. Note first that we may define an ordering on the class of compatible left-invariant metrics on $G$ by letting $\partial\lesssim d$ if there is a constant $K$ so that $\partial\leqslant K\cdot d+K$. We then define a metric $d$ to be [*maximal*]{} if it maximal with respect to this ordering. Note that any two maximal metrics are necessarily quasi-isometric. Moreover, they can be characterised as follows.
The following conditions are equivalent for a metrically proper compatible left-invariant metric $d$ on a metrisable group $G$,
1. $d$ is maximal,
2. $(G,d)$ is large scale geodesic,
3. there is $\alpha>0$ so that $\Sigma=\{g\in G{ \; \big| \;}d(g,1)\leqslant \alpha\}$ generates $G$ and $d$ is quasi-isometric to the word metric $\rho_\Sigma$.
Similarly to the locally compact case, their existence is characterised by $G$ having an appropriate generating set.
A separable metrisable group $G$ admits a maximal compatible left-invariant metric $d$ if and only if $G$ is generated by an open set with property (OB) relative to $G$.
The [*quasi-isometry type*]{} of a metrisable group $G$, when it exists, is then that given by a maximal compatible left-invariant metric. A reassuring fact about this definition is that it is a conservative extension of the existing theory. Namely, as the relative property (OB) in a locally compact group coincides with relative compactness, one sees that our definition of the quasi-isometry type of a compactly generated locally compact group coincides with the classical one given in terms of word metrics for compact generating sets. But, moreover, as will be shown, if $(X,{\lVert\cdot\rVert})$ is a Banach space, then the norm-metric will be maximal on the underlying additive group $(X,+)$, whereby $(X,+)$ will have a well-defined quasi-isometry type, namely, that of $(X,{\lVert\cdot\rVert})$. Thus, nothing essentially new will be said on these two classes of objects.
With the aid of the Milnor–Švarc theorem [@milnor; @svarc] adapted to our setting, we are then capable of computing the quasi-isometry type of a number of different groups. A part from the groups with property (OB) that will be quasi-isometric to the one-point metric space, and the compactly generated locally compact groups and Banach spaces discussed above, we mention the following examples (some of which are studied in the companion paper [@large; @scale]).
- The groups of affine isometries of $X=L^p$ and $X=\ell^p$, $1<p<\infty$, equipped with the topology of pointwise convergence are quasi-isometric to the Banach space $X$.
- The automorphism group ${\rm Aut}(\bf T)$ of the $\aleph_0$-regular tree $\bf T$ is quasi-isometric to $T$. Similarly for the group ${\rm Aut}({\bf T},\mathfrak e)$ of automorphisms of $\bf T$ fixing a given end $\mathfrak e$.
- The isometry group ${\rm Isom}({\mathbb U})$ of the Urysohn metric space ${\mathbb U}$ is quasi-isometric to ${\mathbb U}$.
Since the Banach spaces $L^p$ and $\ell^p$, $1<p<\infty$, are known to all belong to distinct quasi-isometry classes, the same holds for their affine isometry groups, whereby, in particular, these latter cannot be isomorphic as topological groups.
The second part of the paper concerns the interplay between the large scale geometry of a group $G$ and its affine isometric actions on Banach spaces. Possibly the seminal result in this area is due to U. Haagerup [@haagerup], who showed that the free group on finitely many generators admits a proper affine isometric action on Hilbert space, but the subject has since broadened to include less regular types of Banach spaces [@furman]. One motivation for this study is that embeddability into Banach spaces provide an eminent domain for the calibration of various geometric properties of metric spaces, e.g., in terms of the convexity properties of the Banach space.
Our first result uses a well-known construction due to R. F. Arens and J. Eells [@arens] to show that every metrisable group $G$ carrying a metrically proper compatible left-invariant metric admits a proper affine isometric action on some Banach space. We subsequently generalise a construction due to I. Aharoni, B. Maurey and B. S. Mityagin [@maurey] to show that amenable metrisable groups admit metrically proper affine isometric actions on Hilbert space if and only if they have uniformly continuous coarse embeddings into Hilbert space.
Assuming a stronger version of amenability, namely, that the group $G$ is approximately compact, we use techniques originating in work of V. Pestov [@pestov] to show that $G$ admit metrically proper affine isometric actions on super-reflexive spaces, respectively on spaces with Rademacher type $p$ and cotype $q$, if and only if $G$ has a uniformly continuous coarse embedding into a space of the same kind.
Our final result highlights the difference with the theory for locally compact groups. Namely, N. Brown and E. Guentner [@BG] have shown that every finitely generated group admits a proper affine isometric action on a reflexive Banach space and this was generalised in [@haagerup-affine] to compactly generated locally compact groups. However, the generalisation to metrisable groups fails and thus reflexivity becomes a non-trivial restriction. Though we have not been able to determine exactly which metrisable groups act properly on reflexive spaces, a sufficient criterium is given by the following result.
Suppose $d$ is a metrically proper metric on $G$ and that, for all $\alpha>0$, there is a continuous weakly almost periodic function $\phi\in \ell^\infty(G)$ with $d$-bounded support so that $\phi\equiv 1$ on the ball $D_\alpha=\{g\in G{ \; \big| \;}d(g,1)\leqslant \alpha\}$. Then $G$ admits a metrically proper continuous affine isometric action on a reflexive Banach space.
The hypotheses of this theorem are verified whenever the metric $d$ is moreover [*stable*]{} in the sense of J.-L. Krivine and B. Maurey [@KM], whence the following corollary.
Suppose a topological group $G$ carries a compatible left-invariant metrically proper stable metric. Then $G$ admits a metrically proper continuous affine isometric action on a reflexive Banach space.
For a particular instance of this, let us just mention a result of [@large; @scale] stating that if $\bf M$ is a countable atomic model of a stable first-order theory $T$, whose automorphism group ${\rm Aut}(\bf M)$ has the local property (OB), then ${\rm Aut}(\bf M)$ admits a metrically proper stable metric.
The third and last part of our study is separated into a companion paper [@large; @scale], which treats the subcase of non-Archimedean Polish groups, i.e., automorphism groups of countable first-order structures.
[**Acknowledgement.**]{} I wish to thank P. de la Harpe, W. B. Johnson and A. Pillay for helpful comments on and answers to the subject presented here.
Basic theory
============
The relative property (OB)
--------------------------
Let us begin by recalling that a [*metrisable topological group*]{} is a topological group $G$ whose topology can be induced by some metric. Thus, the metric is not part of the given. Also, while separability is occasionally an issue in our study, this is not so for [*complete metrisability*]{}, i.e., that the topology on $G$ can be given by a complete metric. In fact, for our purposes, $G$ can be considered to be equivalent with all of its dense subgroups, as long as these are given the topology induced from $G$.
As mentioned in the introduction, the following definition is central to the rest of the paper. It is a rather trivial generalisation of the concept of groups with property (OB), which was initially considered in [@OB] and also independently by Y. de Cornulier.
Let $A$ be a subset of a metrisable group $G$. We say that $A$ has [*property (OB) relative to $G$*]{} if $A$ has finite diameter with respect to every compatible left-invariant metric on $G$.
Here (OB) is an abbreviation of [*orbites bornées*]{}, i.e., bounded orbits, which refers to the fact that, if $G\curvearrowright (X,d)$ is a continuous isometric action of $G$ on a metric space $(X,d)$ and $A$ has property (OB) relative to $G$, then $A\cdot x$ has finite diameter in $X$ for all $x\in X$. This follows from the fact that, if $\partial$ denotes a compatible left-invariant metric on $G$ (which exists by the Birkhoff–Kakutani metrisation theorem discussed below), then $D(g,f)=d(gx,fx)+\partial(g,f)$ is also compatible and left-invariant.
The following observation will be used repeatedly in the following.
\[triangle ineq\] Let $d$ be a left-invariant metric on a group $G$ and let $A,B\subseteq G$ be subsets of finite $d$-diameter. Then also $A\cdot B=\{ab{ \; \big| \;}a\in A\;\&\; b\in B\}$ and $A{^{-1}}=\{a{^{-1}}{ \; \big| \;}a\in A\}$ have finite $d$-diameter.
Just note that for $a\in A$ and $b\in B$, $$d(1,ab)\leqslant d(1,a)+d(a,ab)=d(1,a)+d(1,b)\leqslant \sup_{g\in A}d(1,g)+\sup_{f\in B}d(1,f)$$ so the products $ab$ have a bounded distance to $1$ and $A\cdot B$ is $d$-bounded. Similarly, $d(a{^{-1}},1)=d(a a{^{-1}},a)=d(1,a)\leqslant \sup_{g\in A}d(1,g)$.
It follows immediately from the Lemma \[triangle ineq\] that the class of subsets having property (OB) relative to $G$ is an ideal, i.e., is hereditary and closed under taking finite unions, and, moreover, is closed under taking products and inverses of sets.
We remark also that, if $d$ is a compatible left-invariant metric on $G$, then $\partial(g,h)=d(g{^{-1}},h{^{-1}})$ defines a compatible right-invariant metric on $G$ and vice versa. Using this observation and Lemma \[triangle ineq\], we see that a subset $A$ has finite diameter with respect every compatible [*left*]{}-invariant metric if and only if $A{^{-1}}$ has finite diameter with respect every compatible [*left*]{}-invariant metric, which again is equivalent to $A$ having finite diameter with respect every compatible [*right*]{}-invariant metric. It follows that property (OB) relative to $G$ is equivalently defined in terms of boundedness with respect to compatible right-invariant metrics on $G$.
The fundamental result on metrisability of topological groups is the metrisation theorem of G. Birkhoff [@birkhoff] and S. Kakutani [@kakutani] stating that a Hausdorff topological group admits a compatible left-invariant metric if and only if it is first countable (henceforth all topological groups considered will be supposed Hausdorff). In particular, every metrisable topological group admits a compatible left-invariant metric. However, not every metrisable group admits a compatible two-sided invariant metric. The class of those that do was determined by V. Klee [@klee] as the so called [*SIN groups*]{} (for [*small invariant neighbourhoods*]{}), namely those that furthermore admit a neighbourhood basis at $1$ consisting of conjugacy invariant sets.
The underlying fundamental fact for Birkhoff’s proof of the metrisation theorem is the following lemma.
\[birkhoff-kakutani\] Let $G$ be a topological group and $(V_n)_{n\in{\mathbb Z}}$ a neighbourhood basis at the identity consisting of open sets satisfying, for all $n\in {\mathbb Z}$,
1. $V_n=V_n{^{-1}}$,
2. $G=\bigcup_{n\in {\mathbb Z}}V_n$,
3. $V_n^3\subseteq V_{n+1}$.
Define $\delta(g_1,g_2)=\inf\big(2^n{ \; \big| \;}g_2{^{-1}}g_1\in V_n\big)$ and put $$d(g_1,g_2)=\inf \Big(\sum_{i=0}^{k-1}\delta(h_i,h_{i+1}){ \; \big| \;}h_0=g_1,
h_k=g_2\Big).$$ Then $$\delta(g_1,g_2)\leqslant 2\cdot d(g_1,g_2)\leqslant 2\cdot\delta(g_1,g_2)$$ and $d$ is a compatible left-invariant metric on $G$.
Nevertheless, the Birkhoff construction will, in general, be too crude for our purposes, as the balls $B_{\epsilon}=\{g\in G{ \; \big| \;}d(g,1)\leqslant {\epsilon}\}$ increase too fast with ${\epsilon}{\rightarrow}\infty$ and, similarly, decrease to fast with ${\epsilon}{\rightarrow}0_+$. We shall mention that, at least for small distances, the more elaborate construction of Kakutani gives a tighter control on the behaviour of the metric.
Reprising this construction, R. Struble [@struble] showed that any metrisable (i.e., second countable) locally compact group admits a compatible left-invariant [*proper*]{} metric, i.e., so that all bounded sets are relatively compact. To do this, one can simply choose the $V_n$ of Lemma \[birkhoff-kakutani\] to be relatively compact.
However, it is not possible to combine the results of Klee and Struble to conclude that a locally compact second countable SIN group admits a two-sided invariant proper metric. For a simple counter-example, take a countable discrete group $\Gamma$ with an infinite conjugacy class ${\mathcal}C\subseteq \Gamma$. Then, if $d$ is a compatible two-sided invariant metric on $\Gamma$, pick $g\in {\mathcal}C$ and note that $d(fgf{^{-1}},1)=d(g,1)$ for all $f\in \Gamma$, i.e., ${\mathcal}C$ lies on a sphere around $1$, showing that $d$ cannot be proper.
Finally, S. Kakutani and K. Kodaira [@kodaira] showed that if $G$ is locally compact, $\sigma$-compact, then for any sequence $U_n$ of neighbourhoods of the identity there is a compact normal subgroup $K\subseteq \bigcap_nU_n$ so that $G/K$ is metrisable.
From Struble’s theorem, the following observation is immediate.
A subset $A$ of a locally compact second countable group $G$ has property (OB) relative to $G$ if and only if $A$ is relatively compact in $G$.
For non-locally compact $G$, the situation is somewhat more interesting.
\[main OB\] Suppose $G$ is a separable metrisable group and $U$ is a symmetric neighbourhood of $1$. Then there is a compatible left-invariant metric $d$ on $G$ so that the following are equivalent for all subsets $A\subseteq G$,
1. $A$ has finite $d$-diameter,
2. there are a finite subset $F\subseteq G$ and some $k\geqslant 1$ so that $A\subseteq (FU)^k$.
Let $F_0=\{1\}\subseteq F_1\subseteq F_2\subseteq F_3\subseteq \ldots \subseteq G$ be an increasing sequence of symmetric finite sets whose union $\bigcup_n F_n$ is dense in $G$. Define $V_n=(F_nUF_n)^{3^n}$ and note that each $V_n$ is symmetric with $$V_n^3=\big((F_nV_0F_n)^{3^n}\big)^3\subseteq (F_{n+1}V_0F_{n+1})^{3^{n+1}}=V_{n+1}.$$ Define also a neighbourhood basis $$V_0\supseteq V_{-1}\supseteq V_{-2}\supseteq \ldots \ni 1$$ at $1$ consisting of symmetric open sets with $V_n^3\subseteq V_{n+1}$ for all $n$. Note that since $\bigcup_nF_n$ is dense in $G$ and $V_0=U$ is non-empty open, we have $G=\bigcup_{n\in {\mathbb Z}} V_n$.
Let now $d$ denote the compatible left-invariant metric defined from $(V_n)_{n\in {\mathbb Z}}$ via Lemma \[birkhoff-kakutani\]. Then, $U$ has finite $d$-diameter, since, by Lemma \[birkhoff-kakutani\], $d(g,1)\leqslant \delta(g,1)\leqslant 2^0=1$ for all $g\in U=V_0$. It follows, by Lemma \[triangle ineq\], that $(FU)^k$ has finite $d$-diameter for all finite $F\subseteq G$ and $k\geqslant 1$, which shows (2)${\Rightarrow}$(1).
For (1)${\Rightarrow}$(2), simply note that if $A\subseteq G$ has finite $d$-diameter, then, by Lemma \[birkhoff-kakutani\], we have $A\subseteq V_n=(F_nUF_n)^{3^n}$ for some $n\geqslant 1$. Since $F_n$ is finite, the result follows.
\[char of rel OB\] The following are equivalent for a subset $A$ of a separable metrisable group $G$,
1. $A$ has property (OB) relative to $G$,
2. for every open $V\ni 1$ there are a finite subset $F\subseteq G$ and some $k\geqslant 1$ so that $A\subseteq (FV)^k$.
(1)${\Rightarrow}$(2): Suppose $A$ has property (OB) relative to $G$ and that $V\ni1$ is open. Then $A$ has finite diameter with respect to the metric $d$ constructed in Lemma \[main OB\] from $U=V\cap V{^{-1}}$ and thus, by (1)${\Rightarrow}$(2) of the same lemma, we have that $A\subseteq (FU)^k\subseteq (FV)^k$ for some finite $F\subseteq G$ and $k\geqslant 1$.
(2)${\Rightarrow}$(1): Suppose that (2) holds and that $d$ is a continuous left-invariant metric. Then the open ball $V=\{g\in G{ \; \big| \;}d(g,1)<1\}$ is a neighbourhood of $1$ and so $A\subseteq (FV)^k$ for some finute $F\subseteq G$ and $k\geqslant 1$. By Lemma \[triangle ineq\], $A$ must have finite $d$-diameter.
We remark that Lemmas \[main OB\] and \[char of rel OB\] may fail when $G$ is no longer required to be separable. To see this, let ${\rm Sym}({\mathbb N})$ denote the group of all (not necessarily finitely supported) permutations of ${\mathbb N}$ and equip it with the discrete topology. Then $U=\{1\}$ is a symmetric neighbourhood of the identity and thus ${\rm Sym}({\mathbb N})\not\subseteq (FU)^k=F^k$ for all finite $F\subseteq {\rm Sym}({\mathbb N})$ and $k\geqslant 1$. Nevertheless, it follows from the main result of G. M. Bergman’s paper [@bergman] that every left-invariant metric on ${\rm Sym}({\mathbb N})$ has bounded diameter and hence ${\rm Sym}({\mathbb N})$ has property (OB) relative to itself.
In a general metrisable (as opposed to locally compact) group, the class of relatively compact sets may be very small and not shed much light on the large or even small scale geometry of the group. In the course of our study we shall see that, for metric purposes, the class of subsets with property (OB) relative to $G$ largely replaces the relatively compact sets to the extent that the analogy is occasionally almost trivial.
Let $(g_n)$ be a sequence in a metrisable group $G$. We say that $(g_n)$ [*tends to infinity*]{} and write $g_n{\rightarrow}\infty$ if there is a compatible left-invariant metric $d$ on $G$ so that $d(g_n,1){\rightarrow}\infty$.
Thus, a subset $A\subseteq G$ has property (OB) relative to $G$ if and only if there is no sequence $(g_n) \subseteq A$ so that $g_n{\rightarrow}\infty$.
Also, in a locally compact second countable group $G$, one has $g_n{\rightarrow}\infty$ if and only if $(g_n)$ eventually leaves every compact set or, equivalently, eventually leaves every set with property (OB) relative to $G$. This, of course, corresponds to the standard use of the notation $g_n{\rightarrow}\infty$.
Similarly, in a metrisable group $G$, we note that, if $g_n{\rightarrow}\infty$, then $(g_n)$ has no subsequence with property (OB) relative to $G$ or, equivalently, that $(g_n)$ eventually leaves every set with property (OB) relative to $G$. It is tempting to believe that this is actually a characterisation of $g_n{\rightarrow}\infty$. Though, as noted above, this is indeed the case in a locally compact second countable group, we shall see that this is not true in general and, e.g., fails in the product $\Gamma_1\times \Gamma_2\times\ldots$ of infinite discrete groups.
\[products\] Let $H=G_1\times G_2\times\ldots$ be the product of metrisable groups $G_n$. Then
1. a subset $A\subseteq H$ has property (OB) relative to $H$ if and only if $A$ is contained in a product $B_1\times B_2\times \ldots$ of subsets $B_i\subseteq G_i$ with property (OB) relative to $G_i$,
2. a sequence $(h_n)$ in $H$ satisfies $h_n{\rightarrow}\infty$ if and only if there is some $k$ so that $${\rm proj}_{G_1\times \ldots\times G_k}(h_n){\mathop{\longrightarrow}\limits_{n{\rightarrow}\infty}}\infty$$ in $G_1\times \ldots\times G_k$.
\(2) Let $(h_n)$ be a sequence in $H$. Assume first that $h_n{\rightarrow}\infty$, as witnessed by some compatible left-invariant metric $d$. Since $d$ is compatible with the topology of $H$, we can find a basic neighbourhood $$U=V_1\times \ldots\times V_k\times G_{k+1}\times G_{k+2}\times \ldots$$ of the identity in $H$ so that $U\subseteq \{h\in H{ \; \big| \;}d(h,1)\leqslant 1\}$. In particular, the subgroup $G_{k+1}\times G_{k+2}\times \ldots$ has finite $d$-diameter. Write now $h_n=a_nb_n$, where $a_n\in G_1\times \ldots\times G_k$ and $b_n\in G_{k+1}\times G_{k+2}\times \ldots$. Then $$d(h_n,1)=d(a_nb_n,1)\leqslant d(a_n,1)+d(b_n,1)\leqslant d(a_n,1)+ {\rm diam}_d(G_{k+1}\times G_{k+2}\times \ldots),$$ whereby, as $d(h_n,1){\rightarrow}\infty$, we have $d(a_n,1){\rightarrow}\infty$ and thus also ${\rm proj}_{G_1\times \ldots\times G_k}(h_n)=a_n{\rightarrow}\infty$.
Conversely, suppose that ${\rm proj}_{G_1\times \ldots\times G_k}(h_n){\mathop{\longrightarrow}\limits_{n{\rightarrow}\infty}}\infty$ for some $k$, as witnessed by some compatible left-invariant metric $\partial_1$ on $G_1\times \ldots\times G_k$. Fix also a compatible left-invariant metric $\partial_2$ on $G_{k+1}\times G_{k+2}\times \ldots$ and let $d$ be defined on $H$ by $d(g_1g_2,f_1f_2)=\partial_1(g_1,f_1)+\partial_2(g_2,f_2)$ for $g_1,f_1\in G_1\times \ldots \times G_k$ and $g_2,f_2\in G_{k+1}\times G_{k+2}\times \ldots$. Then $d(h_n,1){\rightarrow}\infty$, showing that $h_n{\rightarrow}\infty$ in $H$.
\(1) Suppose first that $B_i\subseteq G_i$ for all $i$ and assume that $B_1\times B_2\times \ldots$ fails property (OB) relative to $H$. Then there is a sequence $(h_n)$ in $B_1\times B_2\times \ldots$ so that $h_n{\rightarrow}\infty$ and thus also $${\rm proj}_{G_1\times \ldots\times G_k}(h_n){\mathop{\longrightarrow}\limits_{n{\rightarrow}\infty}}\infty$$ for some specific $k\geqslant 1$. Fix a compatible left-invariant metric $d$ on $G_1\times \ldots\times G_k$ witnessing this. Writing $h_n=b_{n,1}b_{n,2}\cdots b_{n,k}$ with $b_{n,i}\in B_i$, we have that $$d(h_n,1)\leqslant d(b_{n,1},1)+d(b_{n,2},1)+\ldots+d(b_{n,k},1),$$ whereby there must be some $m\leqslant k$ and a subsequence $(b_{n_i,m})_i$ satisfying that $d(b_{n_i,m},1){\mathop{\longrightarrow}\limits_{i{\rightarrow}\infty}} \infty$. Since $d$ restricts to a compatible left-invariant metric on $G_m$, this shows that $B_m$ fails property (OB) relative to $G_m$.
Conversely, suppose $A\subseteq H$ has property (OB) relative to $H$. Then there is no sequence $(h_n)$ in $A$ with $h_n{\rightarrow}\infty$, which by (2) implies that, for every $k$, there can be no sequence $(f_n)$ in ${\rm proj}_{G_k}(A)$ so that $f_n{\rightarrow}\infty$. In other words, $B_k={\rm proj}_{G_k}(A)$ has property (OB) relative to $G_k$ for every $k$ and $A\subseteq B_1\times B_2\times \ldots$.
Let $G_1,G_2,\ldots$ be an infinite sequence of metrisable groups without property (OB) and set $H=G_1\times G_2\times \ldots$. Then there is a sequence $(h_m)$ of elements in $H$ that eventually leaves every set with property (OB) relative to $H$, but so that $h_m\not {\rightarrow}\infty$.
Since each $G_n$ fails property (OB), it contains a sequence $(a_{n,m})_{m\in {\mathbb N}}$ tending to $\infty$. We define $$f_{n,k}=(a_{1,k}, a_{2,k},\ldots, a_{k-1,k},a_{k,k}, a_{k+1,n}, a_{k+2,n},\ldots)\in H$$ and well-order the double sequence $(f_{n,k})_{(n,k)\in {\mathbb N}^2}$ in ordertype ${\mathbb N}$. Now, for every $k$, we have $${\rm proj}_{G_1\times \ldots\times G_k}(f_{n,m})=(a_{1,k}, a_{2,k},\ldots, a_{k-1,k},a_{k,k})$$ for an infinite number of $(n,m)\in {\mathbb N}^2$, which, by Proposition \[products\] (2), shows that $(f_{n,m})_{(n,m)\in {\mathbb N}^2}$ does not tend to infinity in $H$.
On the other hand, to see that $(f_{n,m})_{(n,m)\in {\mathbb N}^2}$ eventually leaves every set with property (OB) relative to $H$, we have to see that, whenever $B_i\subseteq G_i$ have property (OB) relative to $G_i$, then only finitely many $f_{n,m}$ belong to $B_1\times B_2\times\ldots$. So fix $k_0$ large enough so that $a_{1,k}\notin B_1$ for $k\geqslant k_0$ and $n_0$ so that $a_{k_0,n}\notin B_{k_0}$ for all $n\geqslant n_0$. Suppose that $f_{n,k}\in B_1\times B_2\times \ldots$. As the first coordinate of $f_{n,k}$ is $a_{1,k}$, this implies that $k<k_0$ and hence also that the $k_0$th coordinate of $f_{n,k}$ is $a_{k_0,n}$. As we must have $a_{k_0,n}\in B_{k_0}$, we furthermore see that $n<n_0$. So only finitely many $f_{n,m}$ belong to $B_1\times B_2\times \ldots$.
The following definition will be important in the next section.
A metrisable group $G$ is said to have the [*local property (OB)*]{} if there is a neighbourhood $U\ni 1$ with property (OB) relative to $G$.
By Lemma \[products\] (1), we immediately see that an infinite product $H=G_1\times G_2\times \ldots$ of metrisable groups without property (OB), e.g., $H={\mathbb Z}^{\mathbb N}$, cannot have the local property (OB).
Though the relative property (OB) to a large extent is analogous to relative compactness in locally compact groups, there is one major difference which should be kept in mind. Namely, if $G$ is locally compact second countable and $H\leqslant G$ is a locally compact subgroup, then $H$ is closed in $G$ and thus a subset $A\subseteq H$ is relatively compact in $H$ if and only if it is relatively compact in $G$. On the other hand, as shown by V. V. Uspenskiĭ [@uspenski], every Polish group is $H$ isomorphic to a closed subgroup of ${\rm Homeo}([0,1]^{\mathbb N})$ and by [@OB] this latter group has property (OB). Thus, every Polish group $H$ has property (OB) with respect to some ambient Polish group.
Metrically proper metrics
-------------------------
Our aim is now to define and investigate the metric equivalent of proper metrics on locally compact groups. However, in contradistinction to locally compact metrisable groups, that by Struble’s theorem always admit proper metrics, these may not exist on separable metrisable groups.
Let $F\colon (X,d_X) {\rightarrow}(Y,d_Y)$ be a (possibly discontinuous) map between metric spaces $(X,d_X)$ and $(Y,d_Y)$. We define the [*compression modulus*]{} $\kappa_1\colon [0,\infty[{\rightarrow}[0,\infty]$ of $F$ by $$\kappa_1(t)=\inf\big(d_Y(Fx_1, Fx_2){ \; \big| \;}d_X(x_1,x_2)\geqslant t\big)$$ and the [*expansion modulus*]{} $\kappa_2\colon [0,\infty[{\rightarrow}[0,\infty]$ by $$\kappa_2(t)=\sup\big(d_Y(Fx_1, Fx_2){ \; \big| \;}d_X(x_1,x_2)\leqslant t\big).$$
Remark that $\kappa_1$ and $\kappa_2$ are non-decreasing functions and that $F$ is uniformly continuous exactly when $\lim_{t{\rightarrow}0_+}\kappa_2(t)=0$. Also, $F$ is a [*uniform embedding*]{}, i.e., a uniform homeomorphism with its image, if and only if $F$ is uniformly continuous and $\kappa_1(t)>0$ for all $t>0$.
We can now use our compression and expansion moduli to identify several classes of maps between metric spaces by their behaviour on large distances. Unfortunately, there is not general agreement on the terminology in this area, but only on which classes of maps to be studied.
A map $F\colon (X,d_X) {\rightarrow}(Y,d_Y)$ between metric spaces is said to be
1. [*metrically proper*]{} if $F[A]$ has infinite $d_Y$-diameter for every set $A\subseteq X$ of infinite $d_X$-diameter,
2. [*expanding*]{} if $\lim_{t{\rightarrow}\infty}\kappa_1(t)=\infty$,
3. [*bornologous*]{} if $\kappa_2(t)<\infty$ for all $t\in [0,\infty[$,
4. [*Lipschitz for large distances*]{} if there is a constant $K\geqslant 1$ so that $$d_Y(Fx_1, Fx_2)\leqslant K\cdot d_X(x_1,x_2),$$ whenever $d_X(x_1,x_2)\geqslant K$,
5. [*Lipschitz for short distances*]{} if there are constants $K\geqslant 1$ and ${\epsilon}>0$ so that $$d_Y(Fx_1, Fx_2)\leqslant K\cdot d_X(x_1,x_2),$$ whenever $d_X(x_1,x_2)\leqslant {\epsilon}$,
6. [*cobounded*]{} if $\sup_{y\in Y}d_Y(y,F[X])<\infty$,
7. a [*quasi-isometric embedding*]{} if there is a constant $K\geqslant 1$ so that $$\frac 1K d_X(x_1,x_2)\leqslant d_Y(Fx_1, Fx_2)\leqslant K\cdot d_X(x_1,x_2),$$ whenever $d_X(x_1,x_2)\geqslant K$,
8. a [*coarse embedding*]{} if $F$ is both expanding and bornologous,
9. a [*quasi-isometry between $(X,d_X)$ and $(Y,d_Y)$*]{} if $F$ is a cobounded quasi-isometric embedding,
10. a [*coarse equivalence between $(X,d_X)$ and $(Y,d_Y)$*]{} if $F$ is a cobounded coarse embedding.
A few words on equivalent formulations of these concepts are in order.
\(i) Note that $F$ is metrically proper if and only if $F{^{-1}}(B)$ has finite diameter for every set $B\subseteq Y$ of finite diameter, which happens if and only if $d_Y(Fx_0, Fx_n){\rightarrow}\infty$ whenever $(x_n)_{n=0}^\infty$ is a sequence in $X$ satisfying $d_X(x_0,x_n){\rightarrow}\infty$.
\(ii) Expanding is simply a uniform version of being metrically proper. Equivalently, $F$ is expanding if and only if, for all $R>0$, there is $S>0$ so that $$d_X(x_1,x_2)>S{\Rightarrow}d_Y(Fx_1,Fx_2)>R.$$
\(iii) Similarly, $F$ is bornologous if and only if, for all $R>0$, there is $S>0$ so that $$d_X(x_1,x_2)<R{\Rightarrow}d_Y(Fx_1,Fx_2)<S.$$ Thus, a bijection $F$ is bornologous if and only if $F{^{-1}}$ is expanding.
\(iv) $F$ is Lipschitz for large distances if and only if $\kappa_2$ is bounded by an affine function, i.e., if there are constants $K,C$ so that $$d_Y(Fx_1, Fx_2)\leqslant K\cdot d_X(x_1,x_2)+C.$$
\(v) $F$ is Lipschitz for short distances if and only if it is uniformly continuous and its modulus of uniform continuity is bounded below in a neighbourhood of $0$ by a positive linear function.
\(vi) $F$ is a quasi-isometry between $(X,d_X)$ and $(Y,d_Y)$ if and only if $F$ is a quasi-isometric embedding and there is a quasi-isometric embedding $H\colon (Y,d_Y){\rightarrow}(X,d_X)$ so that $H\circ F$ is close to ${\rm id}_X$ and $F\circ H$ is close to ${\rm id}_Y$. Here two maps $\phi, \psi\colon Z{\rightarrow}(V,d)$ are [*close*]{} if $\sup_{z\in Z} d(\phi(z),\psi(z))<\infty$. A similar reformulation is valid for coarse equivalence.
\(vii) Defining metric spaces $(X,d_X)$ and $(Y,d_Y)$ to be [*quasi-isometric*]{} or [*coarsely equivalent*]{} if there is a quasi-isometry, respectively a coarse equivalence, from $(X,d_X)$ to $(Y,d_Y)$, we see using (vi) that these notions are indeed equivalence relations on the class of metric spaces.
A compatible left-invariant metric $d$ on a topological group $G$ is said to be [*metrically proper*]{} if $d(g_n,1){\rightarrow}\infty$ whenever $g_n{\rightarrow}\infty$.
We now have the following characterisation of metrically proper left-invariant metrics.
\[metrically proper\] The following are equivalent for a compatible left-invariant metric $d$ on a metrisable group $G$,
1. $d$ is metrically proper,
2. for all $A\subseteq G$, we have ${\rm diam}_d(A)<\infty$ if and only if $A$ has property (OB) relative to $G$,
3. for every compatible left-invariant metric $\partial$ on $G$, the mapping $${\rm id}\colon (G,\partial){\rightarrow}(G,d)$$ is [*metrically proper*]{},
4. for every compatible left-invariant metric $\partial$ on $G$, the mapping $${\rm id}\colon (G,d){\rightarrow}(G,\partial)$$ is [*bornologous*]{}.
(1)${\Rightarrow}$(2): Suppose $d$ is metrically proper and $A\subseteq G$ is a subset without property (OB) relative to $G$. Then there is some sequence $g_n\in A$ so that $g_n{\rightarrow}\infty$. It follows that also $d(g_n,1){\rightarrow}\infty$, whereby ${\rm diam}_d(A)=\infty$.
(2)${\Rightarrow}$(3): Suppose that (2) holds and $\partial$ is a compatible left-invariant metric on $G$. Then any set $A$ of infinite $\partial$-diameter must fail to have property (OB) relative to $G$ and thus, by (2), must have infinite $d$-diameter.
(3)${\Rightarrow}$(4): Assume that (4) fails, i.e., that there are a compatible left-invariant metric $\partial$ on $G$, a constant $K>0$ and $g_n, f_n\in G$ so that $d(g_n, f_n)<K$, but that $\partial(g_n,f_n){\rightarrow}\infty$. Then, by left-invariance, we have $d(f_n{^{-1}}g_n,1)=d(g_n,f_n)<K$ and $\partial(f_n{^{-1}}g_n,1)=\partial(g_n,f_n){\rightarrow}\infty$, which shows that $\{f_n{^{-1}}g_n\}_{n\in {\mathbb N}}$ has infinite $\partial$-diameter, but finite $d$-diameter, which implies the failure of (3).
(4)${\Rightarrow}$(1): If $d$ is not metrically proper, there is a sequence $g_n{\rightarrow}\infty$ so that $d(g_n,1)\not{\rightarrow}\infty$. Then, if $\partial$ is a compatible left-invariant metric so that $\partial(g_n,1){\rightarrow}\infty$, we see that (4) fails for $\partial$.
We note that, by condition (4) of Lemma \[metrically proper\], if $d$ and $\partial$ are two metrically proper compatible left-invariant metrics on $G$, then the identity map is a coarse equivalence between $(G,d)$ and $(G,\partial)$. It follows that a map into a metric space, $f\colon G{\rightarrow}(X,d_X)$, is bornologous with respect to $d$ if and only if it is bornologous with respect to $\partial$. Similarly with “metrically proper” in place of “bornologous”. Therefore, for groups $G$ admitting such metrics, we can talk of [*bornologous*]{} and [*metrically proper*]{} maps without referring to any specific choice of metric on $G$.
Two compatible left-invariant metrics $d$ and $\partial$ on a group $G$ are said to be [*coarsely equivalent*]{} or [*quasi-isometric*]{} if the identity map is a coarse equivalence, respectively a quasi-isometric equivalence, between $(G,d)$ and $(G,\partial)$.
Since metrically proper metrics are all coarsely equivalent, we define the [*coarse equivalence class*]{} of $G$ to be that determined by any metrically proper compatible left-invariant metric, provided such a metric exists.
Though far from being an equivalent reformulation, the following criterion is useful in identifying metrically proper metrics.
\[geodesic proper\] Suppose that $d$ is a compatible left-invariant [*geodesic*]{} metric on $G$, i.e., so that any two elements $g,f\in G$ are connected by a continuous path of length $d(g,f)$. Then $d$ is metrically proper.
To see this, suppose that $\partial$ is another compatible left-invariant metric on $G$. Since $d$ and $\partial$ are equivalent, there is some ${\epsilon}>0$ so that the $d$-ball $B_{\epsilon}=\{ g\in G{ \; \big| \;}d(g,1)\leqslant {\epsilon}\}$ has finite $\partial$-diameter. As $d$ is geodesic, we see that, for all $n\geqslant 1$ and $f\in B_{n{\epsilon}}$, there are $g_0=1, g_1, g_2, \ldots, g_{n-1}, g_n=f$ so that $d(1, g_i{^{-1}}g_{i+1})=d(g_i,g_{i+1})={\epsilon}$, whence $f= (g_0g_1{^{-1}})\cdot (g_1{^{-1}}g_2)\cdots(g_{n-1}{^{-1}}g_n)\in \big(B_{\epsilon})^n$. In other words, $\big(B_{\epsilon})^n=B_{n{\epsilon}}$. Since $\big(B_{\epsilon})^n$ has finite $\partial$-diameter for all $n$, it follows that every $d$-bounded set is $\partial$-bounded, which verifies condition (3) of Lemma \[metrically proper\].
We mention that, of course, much weaker conditions on $d$ suffice for this argument. Indeed, we only need that, for all ${\epsilon}>0$ and $K>0$, there is some $m=m({\epsilon},K)$ so that every two elements $f,g\in G$ with $d(f,g)\leqslant K$ can be connected by a path $h_0=f, h_1, h_2, \ldots, h_m=g$ where $d(h_i,h_{i+1})\leqslant {\epsilon}$ for all $i$.
\[banach1\] Consider the additive group $(X,+)$ of a Banach space $(X,{\lVert\cdot\rVert})$. Since $X$ is clearly geodesic with respect to the metric induced by the norm, by Example \[geodesic proper\], we see that this latter is metrically proper on $(X,+)$.
It is now time to characterise groups admitting a metrically proper compatible metric.
\[existence of metrically proper\] The following are equivalent for a separable metrisable group $G$,
1. $G$ admits a metrically proper compatible left-invariant metric,
2. $G$ has the local property (OB).
(2)${\Rightarrow}$(1): Suppose $U\ni 1$ is a symmetric open neighbourhood of $1$ with property (OB) relative to $G$ and let $d$ denote the metric on $G$ given by Lemma \[main OB\]. We claim that $d$ is metrically proper. To see this, we verify condition (2) of Lemma \[metrically proper\]. So suppose that $A\subseteq G$ has finite $d$-diameter. Then, by Lemma \[main OB\], there are a finite set $F\subseteq G$ and $k\geqslant 1$ so that $A\subseteq (FU)^k$, which, since $U$ has finite diameter with respect to every compatible left-invariant metric, implies that also $A$ has finite diameter with respect to every compatible left-invariant metric. In other words, $A$ has property (OB) relative to $G$, thus verifying condition (2) of Lemma \[metrically proper\].
(1)${\Rightarrow}$(2): Note that if $d$ is metrically proper, then there is no sequence $g_n$ in $U=\{g\in G{ \; \big| \;}d(g,1)<1\}$ so that $g_n{\rightarrow}\infty$, i.e., $U$ has property (OB) relative to $G$.
Again, Theorem \[existence of metrically proper\] fails without the assumption of separability. To see this, consider the free non-abelian group ${\mathbb F}_{\aleph_1}$ on $\aleph_1$ generators $(a_\xi)_{\xi<\aleph_1}$ ($\aleph_1$ is the first uncountable cardinal number). When equipped with the discrete topology, ${\mathbb F}_{\aleph_1}$ is metrisable and has the local property (OB). On the other hand, if $d$ is any left-invariant metric on ${\mathbb F}_{\aleph_1}$, then some finite-diameter ball $B_k=\{g\in {\mathbb F}_{\aleph_1}{ \; \big| \;}d(g,1)\leqslant k\}$ must contain a denumerable set of generators $\{a_\xi\}_{\xi \in A}$ ordered as ${\mathbb N}$. Noting that ${\mathbb F}_{\aleph_1}={\mathbb F}(A)*{\mathbb F}(\aleph_1\setminus A)$, we can then choose some metric $d_1$ on ${\mathbb F}(A)$ for which $$d_1(a_\xi,1){\mathop{\longrightarrow}\limits_{\xi \in A}}\infty$$ and any other metric $d_2$ on ${\mathbb F}(\aleph_1\setminus A)$ and then define a metric $\partial$ on ${\mathbb F}(\aleph_1)$ by $$\partial(g_1f_1\cdots g_nf_n,1)=\sum_{i=1}^n\big(d_1(g_i,1)+d_2(f_i,1)\big)$$ for $g_i\in {\mathbb F}(A)$ and $f_i\in {\mathbb F}(\aleph_1\setminus A)$. Then $\partial$ witnesses that $a_\xi{\mathop{\longrightarrow}\limits_{\xi \in A}}\infty$ and hence shows that $d$ is not metrically proper.
Nonetheless, every left-invariant metric on the uncountable discrete group ${\rm Sym}({\mathbb N})$ is metrically proper, since any such metric is bounded by the main result of [@bergman].
An isometric action $G\curvearrowright (X,d)$ of a metrisable group $G$ on a metric space $(X,d)$ is said to be [*metrically proper*]{} if, for all $x\in X$, we have $$d(g_nx,x){\rightarrow}\infty$$ whenever $g_n{\rightarrow}\infty$ in $G$.
Setting $x$ to be the identity element $1$ in $G$, one sees that a compatible left-invariant metric $d$ on $G$ is metrically proper if and only if the left-multiplication action $G\curvearrowright (G,d)$ is metrically proper.
The following are equivalent for a separable metrisable group $G$,
1. $G$ admits a metrically proper compatible left-invariant metric,
2. $G$ admits a metrically proper continuous isometric action on a metric space.
(2)${\Rightarrow}$(1): Supose $G\curvearrowright (X,d)$ is a metrically proper continuous isometric action on a metric space and let $D$ be any compatible left-invariant metric on $G$. Fix any $x\in X$ and define $\partial(g,f)=d(gx,fx)+D(g,f)$ and note that $\partial$ is left-invariant, continuous and $\partial \geqslant D$, whereby it is compatible with the topology on $G$. Moreover, since the action is metrically proper, so is the metric $\partial$.
Left and right-invariant geometric structures
---------------------------------------------
As mentioned earlier, even if $G$ has a compatible two-sided invariant metric and a compatible metrically proper left-invariant metric, it may not have a compatible metric that is simultaneously metrically proper and two-sided invariant. However, those groups that do are easily characterised by the following theorem.
\[SIN metrically proper\] The following are equivalent for a separable metrisable group $G$,
1. $G$ admits a metrically proper two-sided invariant compatible metric $d$,
2. $G$ is a SIN group, has the local property (OB) and, if a subset $A$ has property (OB) relative to $G$, then so does $A^G=\{gag{^{-1}}{ \; \big| \;}a\in A\;\&\; g\in G\}$.
The implication (1)${\Rightarrow}$(2) is trivial since every $d$-ball is conjugacy invariant in $G$.
For the implication (2)${\Rightarrow}$(1), note that, since $G$ is a SIN group, we can find a neighbourhood basis $V_0\subseteq V_{-1}\supseteq V_{-2}\supseteq \ldots\ni 1$ at the identity consisting of conjugacy invariant symmetric open sets. Moreover, since $G$ has the local property (OB), some $V_{n}$ has property (OB) relative to $G$. Without loss of generality, we can assume that this happens already for $V_0$. Now, by iteration, define $$V_{n+1}=V_n^3\cup \{g\in G{ \; \big| \;}d(g,1)< n\}^G.$$ Then each $V_n$ is symmetric open, conjugacy invariant, has property (OB) relative to $G$, $G=\bigcup_nV_n$ and $V_n^3\subseteq V_{n+1}$. It follows that the metric defined from $(V_n)_{n\in{\mathbb Z}}$ via Lemma \[birkhoff-kakutani\] is two-sided invariant, metrically proper and compatible with the topology on $G$.
Even for a non-SIN group $G$, the large scale structure may be two-sided invariant, as, for example, is the case with $S_\infty\times {\mathbb Z}$, which is non-SIN, but, as $S_\infty$ has property (OB), is abelian at the large scale. Again, this admits a charaterisation as in Theorem \[SIN metrically proper\].
\[large scale invariant\] The following are equivalent for a metrisable group $G$ and compatible metrically proper left-invariant metric $d$ on $G$,
1. for every compatible right-invariant metric $\partial$ on $G$, the mapping $${\rm id}\colon (G,d){\rightarrow}(G,\partial)$$ is bornologous,
2. if a subset $A$ has property (OB) relative to $G$, then so does $A^G$.
(1)${\Rightarrow}$(2): Suppose that (1) holds and that $A$ is a subset with property (OB) relative to $G$. Without loss of generality, we may assume that $1\in A$. Let $\partial(g,h)=d(g{^{-1}}, h{^{-1}})$, which is a compatible right-invariant metric, whereby ${\rm id}\colon (G,d){\rightarrow}(G,\partial)$ is bornologous. There is therefore some $K>0$ so that $\partial(g,h)\leqslant K$, whenever $d(g,h)\leqslant {\rm diam}_d(A)$. It follows that, for all $a\in A$ and $g\in G$, we have $$d(g,ga{^{-1}})=d(1,a{^{-1}})=d(a,1)\leqslant {\rm diam}_d(A),$$ whereby $$d(1, gag{^{-1}})=d(g{^{-1}}, ag{^{-1}})=\partial(g, ga{^{-1}})\leqslant K.$$ It follows that ${\rm diam}_d(A^G)\leqslant 2K$, which, as $d$ is metrically proper, implies that $A^G$ has property (OB) relative to $G$.
(2)${\Rightarrow}$(1): Assume that (2) holds and that $\partial$ is a compatible right-invariant metric on $G$. Fix $C>0$, whereby $\{g\in G{ \; \big| \;}d(g,1)\leqslant C\}^G$ has property (OB) relative to $G$. Now, since the relative property (OB) is equivalently defined in terms of boundedness for right-invariant metrics, there is a $K$ such that $$\{g\in G{ \; \big| \;}d(g,1)\leqslant C\}^G\subseteq \{h\in G{ \; \big| \;}\partial(h,1)\leqslant K\}.$$ Then, if $d(g,h)\leqslant C$, also $d(h{^{-1}}g,1)\leqslant C$ and thus $\partial(g,h)=\partial(gh{^{-1}},1)=\partial\big(h(h{^{-1}}g)h{^{-1}}, 1\big)\leqslant K$, showing that ${\rm id}\colon (G,d){\rightarrow}(G,\partial)$ is bornologous.
Before stating the next corollary, let us observe that if $d$ is a metrically proper left-invariant metric on $G$, then $\partial$ given by $\partial(g,h)=d(g{^{-1}}, h{^{-1}})$ is metrically proper and right-invariant. This can be seen by noting that $d$-balls and $\partial$-balls coincide.
Suppose that $d_L$ and $d_R$ compatible metrically proper left-, respectively right-invariant, metrics on $G$. Then ${\rm id}\colon (G,d_L){\rightarrow}(G,d_R)$ is a coarse equivalence if and only if, whenever a subset $A$ has property (OB) relative to $G$, then so does $A^G$.
We designate the above situation by saying that the [*left and right geometric structures are coarsely equivalent*]{}.
We should remark that, since the relative property (OB) is equivalently characterised by compatible right-invariant metrics, we have that, if $d$ is metrically proper and left-invariant, then ${\rm id}\colon (G,\partial){\rightarrow}(G,d)$ is metrically proper for every compatible right-invariant $\partial$. However, by the above, ${\rm id}$ may fail to be so uniformly, i.e., it may not be expanding.
Maximal metrics
---------------
Having studied the metric equivalent of proper metrics on locally compact groups, we now focus on those that are quasi-isometric to the word metrics induced by canonical generating sets.
A compatible left-invariant metric $d$ on $G$ is said to be [*maximal*]{} if, for every other compatible left-invariant metric $\partial$ on $G$, the map $${\rm id}\colon (G,d){\rightarrow}(G,\partial)$$ is Lipschitz for large distances,
We remark that, unless $G$ is discrete, this is really the strongest notion of maximality possible for $d$. Indeed, if $G$ is non-discrete and thus $d$ takes arbitrarily small values, then $${\rm id}\colon (G,d){\rightarrow}(G, \sqrt d)$$ fails to be Lipschitz for small distances, while $\sqrt d$ is a compatible left-invariant metric on $G$.
Also, since a map that is Lipschitz for large distances must be bornologous, we see, by Lemma \[metrically proper\], that a maximal metric is always metrically proper. Moreover, any two maximal metrics are clearly bi-Lipschitz equivalent for large distances, i.e., are quasi-isometric.
Recall that, if $\Sigma$ is a symmetric generating set for a group $G$, then we can define an associated [*word metric*]{} $\rho_\Sigma\colon G{\rightarrow}{\mathbb N}$ by $$\rho_\Sigma(g,h)=\min \big(k\geqslant 0{ \; \big| \;}{\exists}s_1, \ldots, s_k\in \Sigma\; \; g=hs_1\cdots s_k\big).$$ Thus, $\rho_\Sigma$ is a left-invariant metric on $G$, but, since it only takes values in ${\mathbb N}$, it will never be a compatible metric on $G$ unless of course $G$ is discrete. However, in certain cases, this may be remedied.
\[construction maximal metrics\] Suppose $d$ is a compatible left-invariant metric on a topological group $G$ so that some ball $B_{\epsilon}=\{g\in G{ \; \big| \;}d(g,1)\leqslant {\epsilon}\}$ generates $G$. Define $\partial$ by $$\partial(f,h)=\inf\Big(\sum_{i=1}^n d(g_i,1){ \; \Big| \;}g_i\in B_{\epsilon}\; \&\; f=hg_1\cdots g_n\Big).$$ Then $\partial$ is a compatible left-invariant metric, quasi-isometric to the word metric $\rho_{B_{\epsilon}}$.
By the continuity of $d$, $\partial$ is a continuous left-invariant [*écart*]{}, i.e., satisfies the triangle inequality. Also, since $\partial\geqslant d$, it is a metric generating the topology on $G$.
To see that $\partial$ is quasi-isometric to $\rho_{B_{\epsilon}}$, note first that $$\partial(f,h)\leqslant {\epsilon}\cdot \rho_{B_{\epsilon}}(f,h).$$ For the other direction, fix $f,h\in G$ and find a shortest sequence $g_1, \ldots, g_n\in B_{\epsilon}$ so that $f=hg_1\cdots g_n$ and $\sum_{i=1}^n d(g_i,1)\leqslant \partial(f,h)+1$. Note that, for all $i$, we have $g_ig_{i+1}\notin B_{\epsilon}$, since otherwise we could coalesce $g_i$ and $g_{i+1}$ into a single term $g_ig_{i+1}$ to get a shorter sequence where $d(g_ig_{i+1}, 1)\leqslant d(g_i,1)+d(g_{i+1},1)$. It thus follows that either $g_i\notin B_{\frac {\epsilon}2}$ or $g_{i+1}\notin B_\frac{\epsilon}2$, whereby there are at least $\frac{n-1}2$ terms $g_i$ so that $g_i\notin B_\frac{\epsilon}2$ and hence $d(g_i,1)>\frac{\epsilon}2$. In particular, $$\frac{n-1}2\cdot \frac {\epsilon}2< \sum_{i=1}^n d(g_i,1)\leqslant \partial(f,h)+1$$ and so, as $\rho_{B_{\epsilon}} (f,h)\leqslant n$, we have $$\frac {\epsilon}4 \cdot \rho_{B_{\epsilon}}(f,h)- \big(1+\frac {\epsilon}4\big)\leqslant \partial(f,h)\leqslant {\epsilon}\cdot \rho_{B_{\epsilon}}(f,h)$$ showing that $\partial$ and $\rho_{B_{\epsilon}}$ are quasi-isometric.
Let us also observe that, if $\Sigma$ and $\Delta$ are two symmetric generating sets for $G$ so that $\Sigma\subseteq (F\Delta)^n$ and $\Delta\subseteq (E\Sigma)^m$ for some finite sets $F,E\subseteq G$ and $n,m\geqslant 1$, then the two word metrics $\rho_\Sigma$ and $\rho_\Delta$ are quasi-isometric. Indeed, it suffices to notice that, in this case, $\Sigma$ is $\rho_\Delta$-bounded and vice versa. This, in particular, applies when $\Sigma$ and $\Delta$ both have property (OB) relative to $G$.
A metric space $(X,d)$ is said to be [*large scale geodesic*]{} if there is $K\geqslant 1$ so that, for all $x,y\in X$, there are $z_0=x,z_1,z_2,\ldots,z_n=y$ so that $d(z_i,z_{i+1})\leqslant K$ and $$\sum_{i=0}^{n-1}d(z_i,z_{i+1})\leqslant K\cdot d(x,y).$$
For example, if $\mathbb X$ is a connected graph, then the shortest path metric $\rho$ on $\mathbb X$ makes $(\mathbb X, \rho)$ large scale geodesic with constant $K=1$.
We should note two well-known facts about large scale geodecity that can easily be checked by hand.
\[facts geodecity\]
1. Large scale geodecity is a quasi-isometric invariant of metric spaces.
2. If $F\colon (X,d_X){\rightarrow}(Y,d_Y)$ is a bornologous map from a large scale geodesic space $(X,d_X)$ to a metric space $(Y,d_Y)$, then $F$ is Lipschitz for large distances.
In our current setup, we can now characterise the maximal metrics among the metrically proper ones.
\[char maximal metric\] The following conditions are equivalent for a metrically proper compatible left-invariant metric $d$ on a metrisable group $G$,
1. $d$ is maximal,
2. $(G,d)$ is large scale geodesic,
3. there is ${\epsilon}>0$ so that $B_{\epsilon}=\{g\in G{ \; \big| \;}d(g,1)\leqslant {\epsilon}\}$ generates $G$ and $d$ is quasi-isometric to the word metric $\rho_{B_{\epsilon}}$.
(3)${\Rightarrow}$(2): We note that the word metric $\rho_{B_{\epsilon}}$ is simply the shortest path metric on the Cayley graph of $(G,B_{\epsilon})$, i.e., the graph whose vertex set is $G$ and whose edges are $\{g,gs\}$, for $g\in G$ and $s\in B_{\epsilon}$. Thus, $(G,\rho_{B_{\epsilon}})$ is large scale geodesic and, since $d$ is quasi-isometric to $\rho_{B_{\epsilon}}$, so is $(G,d)$.
(2)${\Rightarrow}$(1): Assume that $(G,d)$ is large scale geodesic with constant $K\geqslant 1$. Suppose $\partial$ is any other compatible left-invariant metric on $G$. Since $d$ is metrically proper, the identity map from $(G,d)$ to $(G,\partial)$ is bornologous. By Lemma \[facts geodecity\], it follows that it is also Lipschitz for large distances, showing the maximality of $d$.
(1)${\Rightarrow}$(3): Suppose $d$ is maximal. We claim that $G$ is generated by some closed ball $B_k=\{g\in G{ \; \big| \;}d(g,1)\leqslant k\}$. Note that, if this fails, then $G$ is the increasing union of the chain of proper open subgroups $V_n=\langle B_n\rangle$, $n\geqslant 1$. However, it is now easy, using Lemma \[birkhoff-kakutani\], to construct a new metric from the $H_k$ contradicting the maximality of $d$. First, complementing with a neighbourhood basis $V_0\supseteq V_{-1}\supseteq V_{-2}\supseteq \ldots \ni 1$ of symmetric open sets so that $V_{-n}^3\subseteq V_{-n+1}$, and letting $\partial$ denote the metric obtained via Lemma \[birkhoff-kakutani\] from $(V_n)_{n\in {\mathbb Z}}$, we see that, for all $g\in B_n\setminus V_{n-1}\subseteq V_n\setminus V_{n-1}$, we have $$\partial(g,1)\geqslant 2^{n-1}\geqslant n\geqslant d(g,1).$$ Since $B_n\setminus V_{n-1}\neq {\emptyset}$ for infinitely many $n\geqslant 1$, this contradicts the maximality of $d$ and therefore $G=V_k=\langle B_k\rangle$ for some $k\geqslant 1$.
Let now $\partial$ denote the metric obtained from $B_k$ and $d$ via Lemma \[construction maximal metrics\]. Then $d\leqslant \partial$ and, since $d$ is maximal, we have $\partial\leqslant K\cdot d+C$ for some constants $K,C$, showing that $d$ and $\partial$ are quasi-isometric.
Thus, the maximal metrics on a metrisable group $G$ are simply the compatible left-invariant metrics in the quasi-isometry class of word metrics $\rho_\Sigma$, where $\Sigma$ is any symmetric open generating set with property (OB) relative to $G$.
The following are equivalent for a separable metrisable group $G$,
1. $G$ admits a maximal compatible left-invariant metric $d$ ,
2. $G$ is generated by an open set with property (OB) relative to $G$,
3. $G$ has the local property (OB) and is not the union of a chain of proper open subgroups.
(3)${\Rightarrow}$(2): Assume (3) and let $U\ni 1$ be an open set with property (OB) relative to $G$. Since $G$ is separable and is not the union of a chain of proper open subgroups, there must be a finite set $1\in F\subseteq G$ so that $G$ is generated by $UF$, which also has property (OB) relative to $G$.
(2)${\Rightarrow}$(1): Let $U$ be any open generating set with property (OB) relative to $G$ and put $V=(U\cup \{1\})(U\cup\{1\}){^{-1}}$. Then $V$ has property (OB) relative to $G$ and so $G$ has the local property (OB). We can therefore choose some metrically proper compatible left-invariant metric $d$ on $G$ and let ${\epsilon}>0$ be so that $$V\subseteq B_{\epsilon}=\{g\in G{ \; \big| \;}d(g,1)\leqslant {\epsilon}\}.$$ Let now $\partial$ denote the compatible left-invariant metric on $G$ given by Lemma \[construction maximal metrics\]. By Proposition \[char maximal metric\], $\partial$ is maximal.
(1)${\Rightarrow}$(3): If $d$ is maximal, it is metrically proper and thus $G$ has the local property (OB). Moreover, as in the proof of (1)${\Rightarrow}$(3) in Proposition \[char maximal metric\], $G$ cannot be the union of a countable chain of proper open subgroups.
Thus far, we have been able to one the one hand characterise the maximal metrics and also characterise the groups admitting such metrics. However, oftentimes it will be useful to have other criteria that guarantee existence. In the context of finitely generated groups, the main such criteria is the Milnor–Švarc lemma [@milnor; @svarc] of which we will have a close analogue. For this, we will need the following definition.
An isometric group action $G\curvearrowright (X,d)$ on a metric space is said to be [*cobounded*]{} if, for all $x\in X$, $$\sup_{y\in X}d(y,G\cdot x)<\infty,$$ or, equivalently, there is a set $A\subseteq X$ of finite diameter so that $X=G\cdot A$.
\[milnor connected\] Suppose $G$ is a metrisable group with a metrically proper cobounded continuous isometric action $G\curvearrowright (X,d)$ on a connected metric space. Then $G$ admits a maximal compatible left-invariant metric $\partial$.
Since the action is cobounded, there is an open set $A\subseteq X$ of finite diameter so that $G\cdot A=X$. We let $$S=\{g\in G{ \; \big| \;}g\cdot A\cap A\neq {\emptyset}\}$$ and observe that $S$ is an open neighbourhood of $1$ in $G$. Note that, since the action is metrically proper and ${\rm diam}_d(S\cdot x)<\infty$ for all $x\in A$, there is no sequence $g_n{\rightarrow}\infty$ in $S$, or, equivalently, $S$ has property (OB) relative to $G$.
To see that $G$ admits a maximal compatible left-invariant metric $\partial$, it now suffices to verify that $G$ is generated by $S$. For this, observe that, if $g,f\in G$, then $$\begin{split}
\Big(g\langle S\rangle \cdot A\Big)\cap \Big( f\langle S\rangle\cdot A\Big)\neq {\emptyset}&{\Rightarrow}\Big( \langle S\rangle f{^{-1}}g \langle S\rangle\cdot A\Big)\cap A\neq {\emptyset}\\
&{\Rightarrow}\Big( \langle S\rangle f{^{-1}}g \langle S\rangle\Big)\cap S\neq {\emptyset}\\
&{\Rightarrow}f{^{-1}}g\in \langle S\rangle\\
&{\Rightarrow}g\langle S\rangle= f\langle S\rangle.
\end{split}$$ Thus, distinct left cosets $g\langle S\rangle$ and $f\langle S\rangle$ give rise to disjoint open subsets $g\langle S\rangle\cdot A$ and $f\langle S\rangle \cdot A$ of $X$. However, $X=\bigcup_{g\in G}g\langle S\rangle\cdot A$ and $X$ is connected, which implies that there can only be a single left coset of $\langle S\rangle$, i.e., $G=\langle S\rangle$.
\[milnor large scale geodesic\] Suppose $G$ is a metrisable group with a metrically proper cobounded continuous isometric action $G\curvearrowright (X,d)$ on a large scale geodesic metric space. Then $G$ admits a maximal compatible left-invariant metric $\partial$. Moreover, for every $x_0\in X$, the map $$g\in G\mapsto gx_0\in X$$ is a quasi-isometry between $(G,\partial)$ and $(X,d)$.
Fix some $x_0\in X$ and a compatible left-invariant metric $D\leqslant 1$ on $G$. Then, by the continuity of the action, the following defines a compatible left-invariant metric $\partial$ on $G$ $$\partial (g,f)=D(g,f)+d(gx_0,fx_0).$$ Moreover, since the action is metrically proper, so is the metric $\partial$. Now, since $D\leqslant 1$ and the action is cobounded, we see that $g\in G\mapsto gx_0\in X$ is a quasi-isometry between $(G,\partial)$ and $(X,d)$. Since $(X,d)$ is large scale geodesic, so is $(G,\partial)$, whence $\partial$ is maximal by Proposition \[char maximal metric\].
\[banach2\] Consider again the additive group $(X,+)$ of a Banach space $(X,{\lVert\cdot\rVert})$. By Example \[banach1\], the norm-metric is metrically proper on $(X,+)$ and hence the shift-action of $(X,+)$ on $X$ itself is a metrically proper transitive isometric action. Since also the norm-metric on $X$ is geodesic, it follows from Theorem \[milnor large scale geodesic\], that the identity map is a quasi-isometry between the abelian topological group $(X,+)$ and the Banach space $(X,{\lVert\cdot\rVert})$. In other words, the quasi-isometry type of the group $(X,+)$ is none other than the quasi-isometry type of $(X,{\lVert\cdot\rVert})$ itself, showing that, for Banach spaces, our theory coincides with the usual large scale geometry.
\[banach3\] Assume $(X,{\lVert\cdot\rVert})$ is a separable Banach space. Then, by the Mazur–Ulam theorem, every surjective isometry $A\colon X{\rightarrow}X$ is affine, meaning that there is a linear isometry $f\colon X{\rightarrow}X$ and a vector $x\in X$ so that $A(y)=f(y)+x$ for $y\in X$.
Now, let ${\rm Aff}(X,{\lVert\cdot\rVert})$ be the group of affine isometries of $(X,{\lVert\cdot\rVert})$ equipped with the topology of pointwise convergence on $(X,{\lVert\cdot\rVert})$, i.e., $g_i{\rightarrow}g$ if and only if ${\lVertg_i(x)- g(x)\rVert}{\rightarrow}0$ for all $x\in X$, and let ${\rm Isom}(X,{\lVert\cdot\rVert})$ be the closed subgroup consisting of linear isometries. Here the induced topology on ${\rm Isom}(X,{\lVert\cdot\rVert})$ is simply the strong operator topology. Then ${\rm Aff}(X,{\lVert\cdot\rVert})$ is a metrisable topological group and may be written as a topological semi-direct product $${\rm Aff}(X,{\lVert\cdot\rVert})={\rm Isom}(X,{\lVert\cdot\rVert})\ltimes (X,+).$$ That is, ${\rm Isom}(X,{\lVert\cdot\rVert})\ltimes (X,+)$ is the topological product space ${\rm Isom}(X,{\lVert\cdot\rVert})\times (X,+)$ equipped with the group operation $(f,x)*(g,y)=(fg,x+f(y))$.
Suppose now that ${\rm Isom}(X,{\lVert\cdot\rVert})$ has property (OB). We claim that ${\rm Aff}(X,{\lVert\cdot\rVert})$ is quasi-isometric to $(X,{\lVert\cdot\rVert})$. To see this, note that ${\rm Isom}(X,{\lVert\cdot\rVert})$ and $(X,+)$ embed into ${\rm Isom}(X,{\lVert\cdot\rVert})\ltimes (X,+)$ via $f\mapsto (f,0)$ and $x\mapsto ({\rm Id}, x)$ respectively. Thus, as the ball $B_\alpha=\{x\in X{ \; \big| \;}{\lVertx\rVert}\leqslant \alpha\}$ has property (OB) relative to $(X,+)$ (cf. Example \[banach2\]) and ${\rm Isom}(X,{\lVert\cdot\rVert})$ has property (OB), the subsets $\{{\rm Id}\}\times B_\alpha$ and ${\rm Isom}(X,{\lVert\cdot\rVert})\times \{0\}$ have property (OB) relative to ${\rm Isom}(X,{\lVert\cdot\rVert})\ltimes (X,+)$. Therefore, also $${\rm Isom}(X,{\lVert\cdot\rVert})\times B_\alpha= \Big(\{{\rm Id}\}\times B_\alpha\Big)*\Big({\rm Isom}(X,{\lVert\cdot\rVert})\times \{0\}\Big)$$ has property (OB) relative to ${\rm Isom}(X,{\lVert\cdot\rVert})\ltimes (X,+)$ for all $\alpha$. Thus, if $(f_n, x_n)$ is a sequence so that $(f_n,x_n){\rightarrow}\infty$, we see that $(f_n,x_n)$ must eventually leave each set ${\rm Isom}(X,{\lVert\cdot\rVert})\times B_\alpha$, which implies that ${\lVertx_n\rVert}{\rightarrow}\infty$. In particular, this shows that the canonical isometric action $${\rm Isom}(X,{\lVert\cdot\rVert})\ltimes (X,+)\curvearrowright (X,{\lVert\cdot\rVert})$$ given by $(f,x)(y)=f(y)+x$ is metrically proper. Since it is evidently transitive, we conclude by Theorem \[milnor large scale geodesic\], that the mapping $(f,x)\mapsto (f,x)(0)=x$ is a quasi-isometry between ${\rm Isom}(X,{\lVert\cdot\rVert})\ltimes (X,+)$ and $(X,{\lVert\cdot\rVert})$. I.e., ${\rm Aff}(X,{\lVert\cdot\rVert})$ is quasi-isometric to $(X,{\lVert\cdot\rVert})$ as claimed.
Since the unitary group $U({\mathcal}H)$ of separable infinite-dimensional Hilbert space has property (OB) in the discrete topology [@ricard] and thus also in the strong operator topology, by Example \[banach3\], we see that the group of affine isometries of ${\mathcal}H$ is quasi-isometric to ${\mathcal}H$ itself.
Also, S. Banach described the linear isometry groups of $\ell^p$, $1<p<\infty$, as consisting entirely of sign changes and permutations of the basis elements. Thus, the isometry group is the semi-direct product $S_\infty\ltimes \{-1,1\}^{\mathbb N}$ of two groups with property (OB). By Proposition 4.1 [@OB], it follows that ${\rm Isom}(\ell^p)$ has property (OB) and thus, by Example \[banach2\], that the affine isometry group ${\rm Aff}(\ell^p)$ is quasi-isometric to $\ell^p$.
By results due to C. W. Henson [@henson], the $L^p$-lattice $L^p([0,1],\lambda)$, with $\lambda$ being Lebesgue measure and $1<p<\infty$, is $\omega$-categorical in the sense of model theory for metric structures. This also implies that the Banach space reduct $L^p([0,1],\lambda)$ is $\omega$-categorical and hence that the action by its isometry group on the unit ball is approximately oligomorphic. By Theorem 5.2 [@OB], it follows that the isometry group ${\rm Isom}(L^p)$ has property (OB) and thus, as before, that the affine isometry group ${\rm Aff}(L^p)$ is quasi-isometric to $L^p$.
Now, by results of W. B. Johnson, J. Lindenstrauss and G. Schechtman [@johnson] (see also Theorem 10.21 [@lindenstrauss]), any Banach space quasi-isometric to $\ell^p$ for $1<p<\infty$ is, in fact, linearly isomorphic to $\ell^p$. Also, for $1<p<q<\infty$, the spaces $L^p$ and $L^q$ are not coarsely equivalent since they then would be quasi-isometric (being geodesic spaces) and, by taking ultrapowers, would be Lipschitz equivalent, contradicting Corollary 7.8 [@lindenstrauss].
Thus, it follows that all of ${\rm Aff}(\ell^p)$ and ${\rm Aff}(L^p)$ for $1<p<\infty$ have distinct quasi-isometry types and, in particular, cannot be isomorphic as topological groups.
\[semidirect\] Suppose $d$ is a compatible left-invariant metric on a topological group $G$ and $K{\trianglelefteq}G$ is closed normal subgroup of finite diameter. Then the Hausdorff distance $d_H$ on the quotient space $G/K$, defined by $$d_H(gK,fK)=\max\big\{\sup_{a\in gK}\inf_{b\in fK}d(a,b), \sup_{b\in fK}\inf_{a\in gK}d(a,b)\big\},$$ is a compatible left-invariant metric on the group $G/K$ and, moreover, satisfies $$d_H(gK,fK)=\inf_{k\in K}d(g,fk)=\inf_{k\in K}d(gk,f).$$
Suppose now that $G$ is a metrisable group and $K{\trianglelefteq}G$ is a closed normal subgroup with property (OB) relative to $G$ so that the quotient group $G/K$ admits a maximal compatible left-invariant metric $\partial$. Then we claim that the quotient map $g\in G\mapsto gK\in G/K$ is a quasi-isometry between $G$ and $G/K$.
To see this, note first that the left-shift $G\curvearrowright (G/K, \partial)$ is a transitive continuous isometric action of $G$ on a large scale geodesic metric space. By Theorem \[milnor large scale geodesic\], it suffices to show that the action is metrically proper. So suppose $g_n{\rightarrow}\infty$ in $G$ and fix a compatible left-invariant metric on $G$ witnessing this, i.e., $d(g_n,1){\rightarrow}\infty$. Since $K$ must have finite $d$-diameter, it follows that $d(g_nK,1K){\rightarrow}\infty$ and so, since $\partial$ is maximal on $G/K$ and $d_H$ is a compatible metric, we have $\partial(g_nK, 1K){\rightarrow}\infty$. I.e., the action is metrically proper and $g\in G\mapsto gK\in G/K$ is a quasi-isometry.
Conversely, suppose that $d$ is a maximal compatible left-invariant metric on $G$ and that $K{\trianglelefteq}G$ is a closed normal subgroup with property (OB) relative to $G$. Then we claim that $d_H$ is a maximal compatible left-invariant metric on $G/K$.
For this, let $\partial$ be any other compatible left-invariant metric on $G/K$. Then $D(g,f)=d(g,f)+\partial(gK,fK)$ defines a compatible left-invariant metric on $G$, whereby, using maximality of $d$, there is a constant $C$ so that $$\partial(gK,fK)\leqslant D(g,f)\leqslant C\cdot d(g,f)+C$$ for all $g,f\in G$. Without loss of generality, we make take $C\geqslant {\rm diam}_d(K)$, whereby $d_H(gK,fK)=\inf_{k\in K}d(gk,f)\geqslant d(g,f)-C$ and hence $$\partial(gK,fK)\leqslant C \cdot d(g,f)+C\leqslant C\cdot d_H(gK,fK)+2C,$$ for all $g,f\in G$. In other words, $d_H$ is maximal on $G/K$.
\[cameron-vershik\] P. J. Cameron and A. M. Vershik [@vershik] have shown that that there is an invariant metric $d$ on the group ${\mathbb Z}$ for which the metric space $({\mathbb Z},d)$ is isometric to the rational Urysohn metric space ${\mathbb Q}{\mathbb U}$. Since $d$ is two-sided invariant, the topology $\tau$ it induces on ${\mathbb Z}$ is necessarily a group topology, i.e., the group operations are continuous. Thus, $({\mathbb Z},\tau)$ is a metrisable topological group and we claim that $({\mathbb Z},\tau)$ has a well-defined quasi-isometry type, namely, ${\mathbb U}$ or, equivalently, ${\mathbb Q}{\mathbb U}$.
To see this, we first verify that $d$ is metrically proper on $({\mathbb Z}, \tau)$. For this, note that, since $({\mathbb Z}, \tau)$ is isometric to ${\mathbb Q}{\mathbb U}$, we have that, for all $n,m \in {\mathbb Z}$ and ${\epsilon}>0$, if $r=\lceil\frac{d(n,m)}{\epsilon}\rceil$, then there are $k_0=n, k_1, k_2, \ldots, k_r=m\in{\mathbb Z}$ so that $d(k_{i-1},k_i)\leqslant {\epsilon}$. Thus, as $r$ is a function only of ${\epsilon}$ and of the distance $d(n,m)$, we see that $d$ satisfies the criteria in Example \[geodesic proper\] and hence is metrically proper on $({\mathbb Z},\tau)$. Also, as ${\mathbb Q}{\mathbb U}$ is large scale geodesic, so is $({\mathbb Z},d)$. It follows that the shift action of the topological group $({\mathbb Z},\tau)$ on $({\mathbb Z},d)$ is a metrically proper transitive action on a large scale geodesic space. So, by Theorem \[milnor large scale geodesic\], the identity map is a quasi-isometry between the topological group $({\mathbb Z},\tau)$ and the metric space $({\mathbb Z},d)$. As the latter is quasi-isometric to ${\mathbb Q}{\mathbb U}$, so is $({\mathbb Z},\tau)$.
By taking the completion of $({\mathbb Z},\tau)$, this also provides us with monothetic Polish groups quasi-isometric to the Urysohn space ${\mathbb U}$.
Affine actions on Banach spaces
===============================
By the Mazur–Ulam Theorem, every surjective isometry $A$ of a Banach space $X$ is [*affine*]{}, that is, there are a unique invertible linear isometry $T\colon X{\rightarrow}X$ and a vector $\eta\in X$ so that $A$ is given by $A(\xi)=T(\xi)+\eta$ for all $\xi \in X$. It follows that, if $\alpha\colon G\curvearrowright X$ is an isometric action of a group $G$ on a Banach space $X$, there is an isometric linear representation $\pi\colon G\curvearrowright X$, called the [*linear part*]{} of $\alpha$, and a corresponding [*cocycle*]{} $b\colon G{\rightarrow}X$ so that $$\alpha(g)\xi=\pi(g)\xi +b(g)$$ for all $g\in G$ and $\xi \in X$. The cocycle $b$ then satisfies the [*cocycle equation*]{} $$b(gf)=\pi(g)b(f)+b(g)$$ for $g,f\in G$. In particular, for all $g,f\in G$, it follows that $\alpha(g)0=b(g)$ and $b(gg{^{-1}})=b(1)=0$, whereby $$\begin{split}
{\lVert b(f)-b(g)\rVert}&={\lVertb(f)+b(g g{^{-1}})-b(g)\rVert}\\
&={\lVertb(f)+\pi(g)b(g{^{-1}})\rVert}\\
&={\lVert\pi(g{^{-1}})b(f)+b(g{^{-1}})\rVert}\\
&={\lVertb(g{^{-1}}f)\rVert}\\
&={\lVert\alpha(g{^{-1}}f)0-0\rVert}.
\end{split}$$
Using this, one sees that, if $G$ is a metrisable group with a compatible left-invariant metric $d$ and the action $\alpha\colon G\curvearrowright X$ is continuous, then the cocycle $b\colon (G,d){\rightarrow}X$ is actually uniformly continuous. Moreover, if the metric $d$ is metrically proper, then, as $\alpha(B)0$ is norm bounded for every $d$-bounded set $B\subseteq G$, $b\colon (G,d){\rightarrow}X$ is also bornologous. Finally, as$ {\lVert b(f)-b(g)\rVert}={\lVertb(g{^{-1}}f)\rVert}$, the cocycle $b\colon (G,d){\rightarrow}X$ is a coarse embedding if and only if it is metrically proper, which, as ${\lVertb( f)\rVert}={\lVert\alpha(f)0-0\rVert}$, happens if and only if $\alpha$ is a metrically proper action.
\[cocycle\] Assume $G$ is a metrisable group with a compatible metrically proper left-invariant metric $d$. Suppose $\alpha\colon G\curvearrowright X$ is a a continuous affine isometric action on a Banach space $X$ with associated cocycle $b$. Then $b\colon G{\rightarrow}X$ is a coarse embedding if and only if $\alpha$, or equivalently $b$, is metrically proper.
We shall adopt the standard notation $Z^1(G,\pi)$ for the vector space of continuous cocycles $b\colon G{\rightarrow}X$ associated to $\pi$, i.e., the continuous mappings $b$ satisfying the above cocycle equation.
Our goal in the following is twofold. On the one hand, we aim to offer various converses of the above observation, namely, we aim to show that, if $G$ is a metrisable group with a compatible metrically proper left-invariant metric admitting a coarse embedding into a Banach space with some geometric property, then $G$ admits a metrically proper affine isometric action on a Banach space with this same property. On the other hand, we also wish to construct such actions without necessarily beginning from a coarse embedding, but rather from some other data on the group.
Our study naturally splits into four different cases depending on the required geometric properties of the Banach space upon which $G$ acts, namely, general, reflexive, super-reflexive and Hilbert spaces.
The Arens–Eells space {#arens-eells}
---------------------
Suppose $X$ is a non-empty set. The space ${\mathbb M}(X)$ of [*molecules*]{} over $X$ is the vector space of finitely supported real-valued functions $m\colon X{\rightarrow}{\mathbb R}$ with mean zero, i.e., so that $$\sum_{x\in X}m(x)=0.$$ By induction on the size of its support, we note that every molecule $m\in {\mathbb M}(X)$ can be written as a finite linear combination of [*atoms*]{}, i.e., the molecules of the form $$m_{x,y}=\delta_x-\delta_y,$$ where $x,y\in X$ and $\delta_x$ is the Dirac measure at $x$.
If, moreover, $(X,d)$ is a metric space, we can define the [*Arens–Eells norm*]{} on ${\mathbb M}(X)$, by the formula $${{\lVertm\rVert}}_{\AE}=\inf\Big( \sum_{i=1}^n |a_i| d(x_i,y_i) \;{ \; \Big| \;}\; m=\sum_{i=1}^na_im_{x_i,y_i}\Big).$$ A simple application of the triangle inequality for $d$ shows that, in the computation of the norm, the infimum is attained at some presentation $m=\sum_{i=1}^na_im_{x_i,y_i}$ where $x_i$ and $y_i$ all belong to the support of $m$. This also shows that ${\lVert\cdot\rVert}_{\AE}$ is strictly positive on non-zero molecules, which verifies that it is indeed a norm. Moreover, as is well-known (see, e.g., [@weaver]), the norm is equivalently computed by $${\lVertm\rVert}_{{\AE}}=\sup\big( \sum_{x\in X}m(x)f(x){ \; \big| \;}f\colon X{\rightarrow}{\mathbb R}\text{ is $1$-Lipschitz }\big),$$ and so, in particular, ${\lVertm_{x,y}\rVert}_{\AE}=d(x,y)$ for all $x,y\in X$. The completion of ${\mathbb M}(X)$ with respect to the norm ${\lVert\cdot\rVert}_{\AE}$ will be denoted by ${\AE}(X,d)$, which we call the [*Arens-Eells*]{} space of $(X,d)$. Since the set of molecules that are rational linear combinations of atoms with support in a given dense subset of $X$ is dense in ${\text{\AE}}(X,d)$, one immediately sees that ${\text{\AE}}(X,d)$ is a separable Banach space provided $(X,d)$ itself is separable. See the book by N. Weaver [@weaver] for further information on $\AE(X,d)$.
Now, if $G$ is a group acting by isometries on $(X,d)$, one immediately obtains an isometric linear action $\pi\colon G\curvearrowright \big({\mathbb M}(X), {\lVert\cdot\rVert}_{\AE}\big)$ via $$\pi(g) m=m(g{^{-1}}\;\cdot\;),$$ or, equivalently, $$\pi(g)\big( \sum_{i=1}^na_im_{x_i,y_i}\big)=\sum_{i=1}^na_im_{gx_i,gy_i}$$ for any molecule $m=\sum_{i=1}^na_im_{x_i,y_i}\in {\mathbb M}(X)$ and $g\in G$. Again, this action, being isometric, extends automatically and uniquely to the completion $\AE(X,d)$ of ${\mathbb M}(X)$.
The Arens–Eells space can be seen as a linearisation of the metric space $(X,d)$ by the following procedure. Fix any base point $e\in X$ and let $\phi_e\colon X{\rightarrow}{\mathbb M}(X)$ be the injection defined by $$\phi_e(x)=m_{x,e}.$$ Since $${\lVert\phi_e(x)-\phi_e(y)\rVert}_{\AE}={\lVertm_{x,e}-m_{y,e}\rVert}_{\AE}={\lVertm_{x,y}\rVert}_{\AE}=d(x,y)$$ for all $x,y\in X$, the map $\phi_e$ is an isometric embedding of $(X,d)$ into the Banach space $\AE(X,d)$.
Similarly, associated to the isometric linear representation $\pi\colon G\curvearrowright \AE(X,d)$, we can construct a cocycle $b_e\colon G{\rightarrow}\AE(X,d)$ via the formula $$b_e(g)=m_{ge,e}.$$ To verify that $b_e\in Z^1(G,\pi)$, i.e., that $b_e$ satisfies the cocycle equation, note that for $g,h\in G$ $$b_e(gh)=m_{ghe,e}=m_{ge,e}+m_{ghe,ge}=b_e(g)+\pi(g)\big(b_e(h)\big).$$ We let $\alpha_e\colon G\curvearrowright \AE(X)$ denote the corresponding affine isometric representation with linear part $\pi$ and cocycle $b_e$. We remark that in this case the following diagram commutes for all $g\in G$. $$\begin{CD}
X @>g>> X\\
@V\phi_eVV @VV\phi_eV\\
{\mathbb M}(X)@>>\alpha_e(g)> {\mathbb M}(X)
\end{CD}$$ Namely, for any $x\in X$, $$\begin{split}
\big(\alpha_e(g)\circ \phi_e\big)(x)
&=\alpha_e(g)(m_{x,e})\\
&=\pi(g)(m_{x,e})+b_e(g)\\
&=m_{gx,ge}+m_{ge,e}\\
&=m_{gx,e}\\
&=\big(\phi_e\circ g\big)(x).
\end{split}$$ Therefore, for any choice of base point $e\in X$, the map $\phi_e\colon (X,d){\rightarrow}\AE(X,d)$ is an equivariant isometry between the isometric action $G\curvearrowright (X,d)$ and the affine isometric action $\alpha_e\colon G\curvearrowright \AE(X,d)$.
\[thm arens-eells\] Suppose $d$ is a compatible left-invariant metric on a topological group $G$. Then the affine isometric action $\alpha\colon G\curvearrowright \AE(G,d)$ with linear part $\pi\colon G\curvearrowright \AE(G,d)$ and cocycle $b\in Z^1(G,\pi)$ given by $b(g)=\delta_g-\delta_1$ is continuous and satisfies $${\lVertb(g)\rVert}_{\AE}=d(g,1)$$ for all $g\in G$. In particular, if $d$ is metrically proper, then so is the action $\alpha$.
The only point not addressed by the preceding discussion is the continuity of the action, which separates into continuity of $b\colon G{\rightarrow}\AE(G,d)$ and strong continuity of $\pi$. For $b$, note that $${\lVertb(g)-b(f)\rVert}_{\AE}={\lVert\delta_g-\delta_f\rVert}_{\AE}=d(g,f).$$ For strong continuity of $\pi$, note that, since the linear span ${\mathbb M}(G)$ of the atoms $\delta_g-\delta_f$ is dense in $\AE(G,d)$, it suffices to verify that the map $$h\in G\mapsto \pi(h)\big(\delta_g-\delta_f\big)\in \AE(G,d)$$ is continuous for all $ g,f\in G$. But this follows again from $$\begin{split}
{\lVert\pi(h)\big(\delta_g-\delta_f\big)-\pi(k)\big(\delta_g-\delta_f\big)\rVert}_{\AE}
&\leqslant {\lVert\delta_{hg}-\delta_{kg}\rVert}_{\AE}+{\lVert\delta_{hf}-\delta_{kf}\rVert}_{\AE}\\
&=d(hg,kg)+d(hf,kf),
\end{split}$$ thus finishing the proof.
Kernels conditionally of negative type and Hilbert spaces
---------------------------------------------------------
As the Arens–Eells space in general has very bad geometric properties even starting from a fairly well-behaved metric spaces, we are interested in other constructions that preserve more of the initial metric properties of $(X,d)$. The most regular case is of course Hilbert spaces, for which we need some background material on kernels conditionally of negative type. The well-known construction presented here originates in work of E. H. Moore [@moore].
A [*kernel conditionally of negative type*]{} on a set $X$ is a function $\Psi\colon X\times X{\rightarrow}{\mathbb R}$ so that
1. $\Psi(x,x)=0$ and $\Psi(x,y)=\Psi(y,x)$ for all $x,y\in X$,
2. for all $x_1,\ldots, x_n\in X$ and $r_1,\ldots, r_n\in {\mathbb R}$ with $\sum_{i=1}^nr_i=0$, we have $$\sum_{i=1}^n\sum_{j=1}^n r_ir_j\Psi(x_i,x_j)\leqslant 0.$$
For example, if $\sigma\colon X{\rightarrow}{\mathcal}H$ is any mapping from $X$ into a Hilbert space ${\mathcal}H$, then a simple calculation shows that $$\sum_{i=1}^n\sum_{j=1}^n r_ir_j{\lVert\sigma(x_i)-\sigma(x_j)\rVert}^2=-2{\big\lVert\sum_{i=1}^nr_i\sigma(x_i)\big\rVert}\leqslant 0,$$ whenever $\sum_{i=1}^nr_i=0$, which implies that $\Psi(x,y)={\lVert\sigma(x)-\sigma(y)\rVert}^2$ is a kernel conditionally of negative type.
Now, if $\Psi$ is a kernel conditionally of negative type on a set $X$ and ${\mathbb M}(X)$, as before, denotes the vector space of finitely supported real valued functions $\xi$ on $X$ of mean $0$, i.e., $\sum_{x\in X}\xi(x)=0$, we can define a positive symmetric linear form $\langle\cdot{ \; \big| \;}\cdot\rangle_\Psi$ on ${\mathbb M}(X)$ by $$\Big\langle\sum_{i=1}^nr_i\delta_{x_i}{ \; \Big| \;}\sum_{j=1}^ks_j\delta_{y_i}\Big\rangle_\Psi=-\frac 12\sum_{i=1}^n\sum_{j=1}^k r_is_j\Psi(x_i,y_j).$$ Also, if $N_\Psi$ denotes the null-space $$N_\Psi=\{\xi\in {\mathbb M}(X){ \; \big| \;}\langle\xi{ \; \big| \;}\xi\rangle_\Psi=0\},$$ then $\langle\cdot{ \; \big| \;}\cdot\rangle_\Psi$ defines an inner product on the quotient ${\mathbb M}(X)/N_\Psi$ and we obtain a real Hilbert space ${\mathcal}K$ as the completion of ${\mathbb M}(X)/N_\Psi$ with respect to $\langle\cdot{ \; \big| \;}\cdot\rangle_\Psi$.
We remark that, if $\Psi$ is defined by a map $\sigma\colon X{\rightarrow}{\mathcal}H$ as above and $e\in X$ is any choice of base point, the map $\phi_e\colon X{\rightarrow}{\mathcal}K$ defined by $\phi_e(x)=\delta_x-\delta_e$ satisfies ${\lVert\phi_e(x)-\phi_e(y)\rVert}_{\mathcal}K={\lVert\sigma(x)-\sigma(y)\rVert}_{\mathcal}H$. Indeed, $$\begin{split}
{\lVert\phi_e(x)-\phi_e(y)\rVert}^2_{\mathcal}K
&=\langle \phi_e(x)-\phi_e(y){ \; \big| \;}\phi_e(x)-\phi_e(y)\rangle\\
&=\langle \delta_x-\delta_y{ \; \big| \;}\delta_x-\delta_y\rangle\\
&=-\frac 12\big(\Psi(x,x)+\Psi(y,y)-\Psi(x,y)-\Psi(y,x)\big)\\
&=\Psi(x,y)\\
&={\lVert\sigma(x)-\sigma(y)\rVert}_{\mathcal}H^2.
\end{split}$$
Also, if $G\curvearrowright X$ is an action of a group $G$ on $X$ and $\Psi$ is $G$-invariant, i.e., $\Psi(gx,gy)=\Psi(x,y)$, this action lifts to an action $\pi\colon G\curvearrowright{\mathbb M}(X)$ preserving the form $\langle\cdot{ \; \big| \;}\cdot\rangle_\Psi$ via $\pi(g)\xi=\xi(g{^{-1}}\,\cdot\,)$. It follows that $\pi$ factors through to an orthogonal (i.e., isometric linear) representation $G\curvearrowright{\mathcal}K$.
Theorem \[maurey\] below is originally due to I. Aharoni, B. Maurey and B. S. Mityagin [@maurey] for the case of abelian groups and is also worked out by Y. de Cornulier, R. Tessera and A. Valette for compactly generated locally compact groups in [@tessera]. Since more care is needed when dealing with general amenable groups as opposed to locally compact, we include a full proof.
\[maurey\] Suppose $d$ is a compatible left-invariant metric on an amenable topological group $G$ and $\sigma\colon (G,d){\rightarrow}{\mathcal}H$ is a uniformly continuous and bornologous map into a Hilbert space ${\mathcal}H$ with compression and expansion moduli $\kappa_1$ and $\kappa_2$. Then there is a continuous affine isometric action $\alpha\colon G\curvearrowright {\mathcal}K$ on a real Hilbert space with associated cocycle $b\colon G{\rightarrow}{\mathcal}K$ so that $$\kappa_1\big(d(g,1)\big)\leqslant {\lVertb(g)\rVert}\leqslant \kappa_2\big(d(g,1)\big),$$ for all $g\in G$.
For fixed $g,h\in G$, we define a function $\phi_{g,h}\colon G{\rightarrow}{\mathbb R}$ via $$\phi_{g,h}(f)={\lVert\sigma(fg)-\sigma(fh)\rVert}^2.$$ Note first that $\phi_{g,h}\in \ell^\infty(G)$. For $d(fg,fh)=d(g,h)$ for all $f\in G$ and so, since $\sigma$ is bornologous, we have $${\lVert\phi_{g,h}\rVert}_\infty=\sup_{f\in G}|\phi_{g,h}(f)|=\sup_{f\in G}{\lVert\sigma(fg)-\sigma(fh)\rVert}^2\leqslant \kappa_2\big(d(g,h)\big)^2<\infty.$$
Secondly, we claim that $\phi_{g,h}$ is [*left*]{}-uniformly continuous, i.e., that for all ${\epsilon}>0$ there is $W\ni 1$ open so that $|\phi_{g,h}(f)-\phi_{g,h}(fw)|<{\epsilon}$, whenever $f\in G$ and $w\in W$. To see this, take some $\eta>0$ so that $4\eta{\lVert\phi_{g,h}\rVert}_\infty+4\eta^2<{\epsilon}$ and find, by uniform continuity of $\sigma$, some open $V\ni1$ so that ${\lVert\sigma(f)-\sigma(fv)\rVert}<\eta$ for all $f\in G$ and $v\in V$. Pick also $W\ni 1$ open so that $Wg\subseteq gV$ and $Wh\subseteq hV$. Then, if $f\in G$ and $w\in W$, there are $v_1,v_2\in V$ so that $wg=gv_1$ and $wh=hv_2$, whence $$\begin{split}
\big|\phi_{g,h}(f)-\phi_{g,h}(fw)\big|
&=\Big| {\lVert\sigma(fg)-\sigma(fh)\rVert}^2- {\lVert\sigma(fwg)-\sigma(fwh)\rVert}^2\Big| \\
&=\Big| {\lVert\sigma(fg)-\sigma(fh)\rVert}^2- {\lVert\sigma(fgv_1)-\sigma(fhv_2)\rVert}^2\Big|\\
&<4\eta{\lVert\phi_{g,h}\rVert}_\infty+4\eta^2\\
&<{\epsilon}.
\end{split}$$ Thus, every $\phi_{g,h}$ belongs to the closed linear subspace ${\rm LUC}(G)\subseteq \ell^\infty(G)$ of left-uniformly continuous bounded real-valued functions on $G$ and a similar calculation shows that the map $(g,h)\in G\times G\mapsto \phi_{g,h}\in \ell^\infty(G)$ is continuous.
Now, since $G$ is amenable, there exists a mean $m$ on ${\rm LUC}(G)$ invariant under the [*right*]{}-regular representation $\rho\colon G\curvearrowright {\rm LUC}(G)$ given by $\rho(g)\big(\phi\big)=\phi(\,\cdot\, g)$. Using this, we can define a continuous kernel $\Psi\colon G\times G{\rightarrow}{\mathbb R}$ by $$\Psi(g,h)=m(\phi_{g,h})$$ and note that $\Psi(fg,fh)=m(\phi_{fg,fh})=m\big(\rho(f)\big(\phi_{g,h}\big)\big)=m(\phi_{g,h})=\Psi(g,h)$ for all $g,h,f\in G$.
We claim that $\Psi$ is a kernel conditionally of negative type. To verify this, let $g_1,\ldots, g_n\in G$ and $r_1,\ldots, r_n\in {\mathbb R}$ with $\sum_{i=1}^nr_i=0$. Then, for all $f\in G$, $$\sum_{i=1}^n\sum_{j=1}^n r_ir_j\phi_{g_i,g_j}(f)=\sum_{i=1}^n\sum_{j=1}^n r_ir_j{\lVert\sigma(fg_i)-\sigma(fg_j)\rVert}^2\leqslant 0,$$ since $(g,h)\mapsto {\lVert\sigma(fg)-\sigma(fh)\rVert}^2$ is a kernel conditionally of negative type. Since $m$ is positive, it follows that also $$\sum_{i=1}^n\sum_{j=1}^n r_ir_j\Psi(g_i,g_j)
= m\Big(\sum_{i=1}^n\sum_{j=1}^n r_ir_j\phi_{g_i,g_j}\Big)
\leqslant 0.$$
As above, we define a positive symmetric form $\langle\cdot{ \; \big| \;}\cdot\rangle_\Psi$ on ${\mathbb M}(G)$. Note that, since $\Psi$ is left-invariant, the form $\langle\cdot{ \; \big| \;}\cdot\rangle_\Psi$ is invariant under the left-regular representation $\lambda\colon G\curvearrowright {\mathbb M}(G)$ given by $\lambda(g)(\xi)=\xi(g{^{-1}}\,\cdot\,)$ and so $\lambda$ induces a strongly continuous orthogonal representation $\pi$ of $G$ on the Hilbert space completion ${\mathcal}K$ of ${\mathbb M}(G)/N_\Psi$.
Moreover, as is easily checked, the map $b\colon G{\rightarrow}{\mathcal}K$ given by $b(g)=(\delta_g-\delta_1)+N_\Psi$ is a cocyle for $\pi$. Now $${\lVertb(g)\rVert}^2=\langle \delta_g-\delta_1 { \; \big| \;}\delta_g-\delta_1\rangle_\Psi=\Psi(g,1)=m(\phi_{g,1}),$$ so, since $$\kappa_1\big(d(g,1)\big)^2\leqslant \phi_{g,1}\leqslant \kappa_2\big(d(g,1)\big)^2,$$ it follows from the positivity of $m$ that $$\kappa_1\big(d(g,1)\big)\leqslant {\lVertb(g)\rVert}\leqslant \kappa_2\big(d(g,1)\big),$$ which proves the theorem.
We can now extend the definition of the Haagerup property from locally compact groups to the full category of metrisable groups.
A metrisable group $G$ is said to have the [*Haagerup property*]{} if it admits a metrically proper continuous affine isometric action on a Hilbert space.
Thus, based on Theorem \[maurey\], we have the following reformulation of the Haagerup property for amenable metrisable groups.
\[haagerup equiv\] The following are equivalent for an amenable separable metrisable group with the local property (OB),
1. $G$ admits a uniformly continuous coarse embedding into a Hilbert space $\eta \colon G{\rightarrow}{\mathcal}H$,
2. $G$ has the Haagerup property.
(2)${\Rightarrow}$(1): If $\alpha\colon G\curvearrowright {\mathcal}H$ is a metrically proper continuous affine isometric action, with corresponding cocycle $b\colon G{\rightarrow}{\mathcal}H$, then, by Observation \[cocycle\], $b\colon G{\rightarrow}{\mathcal}H$ is a uniformly continuous coarse embedding.
(1)${\Rightarrow}$(2): Fix a metrically proper compatible left-invariant metric $d$ on $G$. By Theorem \[maurey\], there is a continuous affine isometric action $\alpha\colon G\curvearrowright {\mathcal}K$ on a Hilbert space ${\mathcal}K$ with associated cocycle $b\colon G{\rightarrow}{\mathcal}K$ so that, for all $g\in G$, $$\kappa_1\big(d(g,1)\big)\leqslant {\lVertb(g)\rVert}\leqslant\kappa_2\big(d(g,1)\big),$$ where $\kappa_1(t)=\inf_{d(g,f)\geqslant t}{\lVert\eta(g)-\eta(f)\rVert}$ and $\kappa_2(t)=\sup_{d(g,f)\leqslant t}{\lVert\eta(g)-\eta(f)\rVert}$. Since $\eta$ is a coarse embedding, the cocycle $b\colon G{\rightarrow}{\mathcal}K$ is metrically proper and so is the action $\alpha$.
In the setting of amenable non-Archimedean Polish groups, the condition of uniform continuity of $\eta$ may be dropped. Indeed, fix an open subgroup $V\leqslant G$ with property (OB) relative to $G$ and note that, since $\eta$ is bornologous, there is a constant $K>0$ so that ${\lVert\eta(g)-\eta(f)\rVert}\leqslant K$ whenever $f{^{-1}}g\in V$. Letting $X\subseteq G$ denote a set of left-coset representatives for $V$, we define $\sigma(g)=\eta(h)$, where $h\in X$ is the coset representative of $gV$. Then ${\lVert\eta(g)-\sigma(g)\rVert}\leqslant K$ for all $g\in G$, so $\sigma\colon G{\rightarrow}{\mathcal}H$ is bornologous and clearly constant on left-cosets of $V$, whence also uniformly continuous with respect to some chosen metrically proper compatible left-invariant metric $d$ on $G$. So $\sigma$ is a uniformly continuous coarse embedding.
U. Haagerup [@haagerup] initially showed that finitely generated free groups have the Haagerup property. It is also known that amenable locally compact groups [@BCV] (see also [@CCJJV]) have the Haagerup property. However, this is not the case for amenable metrisable groups, as, e.g., the isometry group of the Urysohn metric space ${\mathbb U}$ provides a counter-example. We shall verify this in Section \[superrefl\].
There is also a converse to this. Namely, E. Guentner and J. Kaminker [@GK] showed that, if a finitely generated discrete group $G$ admits a affine isometric action on a Hilbert space whose cocycle $b$ growths faster than the square root of the word length, then $G$ is amenable (see [@tessera] for the generalisation to the locally compact case).
Approximate compactness, super-reflexivity and Rademacher type {#superrefl}
--------------------------------------------------------------
Weakening the geometric restrictions on the phase space from euclidean to uniformly convex, we still have a result similar to Theorem \[maurey\]. However, in this case, we must assume that the group in question is approximately compact and not only amenable.
A topological group $G$ is said to be [*approximately compact*]{} if there is a countable chain $K_0\leqslant K_1\leqslant \ldots \leqslant G$ of compact subgroups whose union $\bigcup_nK_n$ is dense in $G$.
This turns out to be a fairly common phenomenon among non-locally compact metrisable groups. For example, the unitary group $U({\mathcal}H)$ of separable infinite-dimensional Hilbert space with the strong operator topology is approximately compact. Indeed, if ${\mathcal}H_1\subseteq {\mathcal}H_2\subseteq \ldots\subseteq {\mathcal}H$ is an increasing exhaustive sequence of finite-dimensional subspaces and $U(n)$ denotes the group of unitaries pointwise fixing the orthogonal complement ${\mathcal}H_n^\perp$, then each $U(n)$ is compact and the union $\bigcup_nU(n)$ is dense in $U({\mathcal}H)$.
More generally, as shown by P. de la Harpe [@harpe], if $M$ is an approximately finite-dimensional von Neumann algebra, i.e., there is an increasing sequence $A_1\subseteq A_2\subseteq\ldots\subseteq M$ of finite-dimensional matrix algebras whose union is dense in $M$ with respect the strong operator topology, then the unitary subgroup $U(M)$ is approximately compact with respect to the strong operator topology.
Similarly, if $G$ contains a locally finite dense subgroup, this will witness approximate compactness. Again this applies to, e.g., ${\rm Aut}([0,1],\lambda)$ with the weak topology, where the dyadic permutations are dense, and ${\rm Isom}({\mathbb U})$ with the pointwise convergence topology (this even holds for the dense subgroup ${\rm Isom}({\mathbb Q}{\mathbb U})$ by a result of S. Solecki; see [@RZ] for a proof).
Of particular interest to us is the case of non-Archimedean Polish groups. By general techniques, these may be represented as automorphism groups of countable locally finite (i.e., any finitely generated substructure is finite) ultrahomogeneous structures. And, in this setting, we have the following reformulation of approximate compactness.
Let $\bf M$ be a locally finite, countable, ultrahomogeneous structure. Then ${\rm Aut}(\bf M)$ is approximately compact if and only if, for every finite substructure $\bf A\subseteq \bf M$ and all [*partial*]{} automorphisms $\phi_1,\ldots,
\phi_n$ of $\bf
A$, there is a larger finite substructure $\bf B$ with $$\bf A\subseteq \bf B\subseteq \bf M$$ and [*full*]{} automorphisms $\psi_1,\ldots,\psi_n$ of $\bf B$ extending $\phi_1,\ldots,\phi_n$ respectively.
We recall that a Banach space $V$ is [*super-reflexive*]{} if every other space crudely finitely representable in $V$ is reflexive. That is, if $X$ is a Banach space so that, for some fixed $K\geqslant 1$, every finite-dimensional subspace $F\subseteq X$ is $K$-isomorphic to a subspace of $V$, then $X$ is reflexive. In particular, every super-reflexive space is reflexive. Moreover, super-reflexive spaces are exactly those all of whose ultrapowers are reflexive. By a result of P. Enflo [@enflo] (see also G. Pisier [@pisier] for an improved result or [@fabian] for a general treatment), the super-reflexive spaces can also be characterised as those admitting an equivalent uniformly convex renorming.
For the case of super-reflexive spaces, we have the following result, which is due to V. Pestov [@pestov] in the case of a locally finite discrete group $G$.
\[pestov\] Suppose $d$ is a compatible left-invariant metric on an approximately compact topological group $G$. Assume that $\sigma\colon (G,d){\rightarrow}E$ is a uniformly continuous and bornologous map into a super-reflexive Banach space $E$ with compression and expansion moduli $\kappa_1$ and $\kappa_2$. Then there is a continuous affine isometric action of $G$ on a super-reflexive Banach space $V$ with corresponding cocycle $b$ so that $$\kappa_1\big(d(f,1)\big)\leqslant {\lVertb(f)\rVert}_V\leqslant \kappa_2\big(d(f,1)\big)$$ for all $f\in G$.
By renorming $E$, we may suppose that $E$ is uniformly convex. For a compact subgroup $K\leqslant G$, let $\mu$ denote the Haar measure on $K$ and $L^2(K, E)$ denote the Banach space of square integrable $E$-valued functions $\phi$ on $K$ with norm $${\lVert\phi\rVert}_{L^2}=\Big(\int_K{\lVert\phi(g)\rVert}_E^2 \;d\mu(g)\Big)^{\frac 12}.$$ Since $E$ uniformly convex, so is $L^2(K,E)$. Moreover, for every other compact subgroup $C\leqslant G$, the probability spaces $K$ and $C$ are isomorphic, whence the Banach spaces $L^2(K,E)$ and $L^2(C,E)$ are isometric. Thus, the modulus of uniform convexity of $L^2(K,E)$ is independent of the choice of $K$. For every $f\in G$, we define an element $[f]_K\in L^2(K,E)$ by $[f]_K(g)=\sigma(gf)$ for all $g\in K$. Note that since $\sigma$ is continuous, so is $[f]_K$ and hence $[f]_K$ is automatically square integrable on the compact group $K$.
Let also $\rho\colon K\curvearrowright L^2(K,E)$ denote the right-regular representation given by $\big(\rho(g)\phi\big)(h)=\phi(hg)$ and note that, in particular, $$\big(\rho(g)[f]_K\big)(h)=[f]_K(hg)=\sigma(hgf)=[gf]_K(h),$$ i.e., $\rho(g)[f]_K=[gf]_K$ for all $g\in K$ and $f\in G$.
Now, fix a non-principal ultrafilter ${\mathcal}U$ on ${\mathbb N}$ and let $K_0\leqslant K_1\leqslant\ldots\leqslant G$ be a countable chain of compact subgroups whose union is dense in $G$. Consider the ultraproduct $$W=\prod_{\mathcal}UL^2(K_n,E).$$ That is, $W$ is the quotient of $\big(\bigoplus_n L^2(K_n,E)\big)_\infty$ by the subspace $$N_{\mathcal}U=\{(\phi_n)\in \big(\bigoplus_n L^2(K_n,E)\big)_\infty{ \; \big| \;}\lim_{\mathcal}U{\lVert\phi_n\rVert}_{L^2}=0\}.$$ For $(\phi_n)\in \big(\bigoplus_n L^2(K_n,E)\big)_\infty$, we denote its image in $W$ by $(\phi_n)_{\mathcal}U$. Since the spaces $L^2(K_n,E)$ are all uniformly convex with the same modulus of uniform convexity, the ultraproduct remains uniformly convex and hence super-reflexive.
Note that, for all $f,h\in G$ and $n\in {\mathbb N}$, $$\begin{split}
{\big\lVert[f]_{K_n}-[h]_{K_n}\big\rVert}_{L^2}
&=\Big(\int_{K_n}{\lVert[f]_{K_n}(g)-[h]_{K_n}(g)\rVert}_E^2 \;d\mu(g)\Big)^{\frac 12}\\
&=\Big(\int_{K_n}{\lVert\sigma(gf)-\sigma(gh)\rVert}_E^2 \;d\mu(g)\Big)^{\frac 12}\\
&\leqslant \Big(\int_{K_n}\kappa_2\big(d(gf,gh)\big)^2 \;d\mu(g)\Big)^{\frac 12}\\
&= \Big(\int_{K_n}\kappa_2\big(d(f,h)\big)^2 \;d\mu(g)\Big)^{\frac 12}\\
&=\kappa_2\big(d(f,h)\big),
\end{split}$$ so the sequence $\big([f]_{K_n}-[h]_{K_n}\big)_n$ is uniformly bounded in the $L^2$-norms and thus belongs to $\big(\bigoplus_n L^2(K_n,E)\big)_\infty$. By the same reasoning, we also note that ${\big\lVert[f]_{K_n}-[h]_{K_n}\big\rVert}_{L^2}\geqslant \kappa_1\big(d(f,h)\big)$. It follows that, for all $f,h\in G$, $$\label{eq super}
\kappa_1\big(d(f,h)\big)\leqslant {\big\lVert\big([f]_{K_n}-[h]_{K_n}\big)_{\mathcal}U\big\rVert}_{W}\leqslant \kappa_2\big(d(f,h)\big).$$
By the above, it follows that we can define a map $b\colon G{\rightarrow}W$ by setting $b(f)=([f]_{K_n}-[1]_{K_n})_{\mathcal}U$. Note that, since $\sigma$ is uniformly continuous, for all ${\epsilon}>0$ there is $\delta>0$ so that $\kappa_2(\delta)<{\epsilon}$. In particular, $${\big\lVertb(f)-b(h)\big\rVert}_W={\big\lVert\big([f]_{K_n}-[h]_{K_n}\big)_{\mathcal}U\big\rVert}_{W}\leqslant \kappa_2\big(d(f,h)\big)<{\epsilon},$$ whenever $d(f,h)<\delta$. Thus, $b$ is uniformly continuous.
Also, for $g\in \bigcup_nK_n$, the right-regular representation $\rho(g)$ defines a linear isometry of $L^2(K_n,E)$ for all but finitely many $n\in {\mathbb N}$. Therefore, as the ultrafilter ${\mathcal}U$ is non-principal, this means that we can define an isometric linear representation $$\tilde\rho \colon \bigcup_nK_n\curvearrowright W$$ by letting $$\tilde\rho(g)\big((\phi_n)_{\mathcal}U\big)=\big(\rho(f)\phi_n\big)_{\mathcal}U.$$
We claim that $b\in Z^1(\bigcup_nK_n,\tilde \rho)$, i.e., that $b$ satisfies the cocycle identity $b(fh)=\tilde\rho(f)b(h)+b(f)$ for $f,h\in \bigcup_nK_n$. To see this, note that $$\begin{split}
\tilde\rho(f)b(h)+b(f)
&=\tilde\rho(f)\big([h]_{K_n}-[1]_{K_n}\big)_{\mathcal}U+\big([f]_{K_n}-[1]_{K_n}\big)_{\mathcal}U\\
&= \big([fh]_{K_n}-[f]_{K_n}\big)_{\mathcal}U+\big([f]_{K_n}-[1]_{K_n}\big)_{\mathcal}U\\
&= \big([fh]_{K_n}-[1]_{K_n}\big)_{\mathcal}U\\
&=b(fh).
\end{split}$$ In particular, we see that the linear span of $b[\bigcup_nK_n]$ is $\tilde\rho[\bigcup_nK_n]$-invariant. Moreover, since every $\tilde\rho(f)$ is an isometry and $b$ is continuous, the same holds for the closed linear span $V\subseteq W$ of $b[G]$.
We claim that, for every $\xi\in V$, the map $f\in \bigcup_nK_n\mapsto \tilde\rho(f)\xi$ is uniformly continuous. Since linear span of $b[\bigcup_nK_n]$ is dense in $V$, it suffices to prove this for $\xi\in b[\bigcup_nK_n]$. So fix some $g\in \bigcup_nK_n$ and note that, for $f,h\in \bigcup_nK_n$, we have $$\begin{split}
{\big\lVert\tilde\rho(f)b(g)-\tilde\rho(h)b(g)\big\rVert}_W
&={\big\lVert\big([fg]_{K_n}-[f]_{K_n}\big)_{\mathcal}U-\big([hg]_{K_n}-[h]_{K_n}\big)_{\mathcal}U\big\rVert}_W\\
&\leqslant{\big\lVert\big([fg]_{K_n}-[hg]_{K_n}\big)_{\mathcal}U\big\rVert}_W+{\big\lVert\big([h]_{K_n}-[f]_{K_n}\big)_{\mathcal}U\big\rVert}_W\\
&\leqslant\kappa_2\big( d(fg,hg) \big)+\kappa_2\big( d(h,f) \big).
\end{split}$$ Now, given ${\epsilon}>0$, there is $\delta>0$ so that $\kappa_2(\delta)<\frac {\epsilon}2$ and an $\eta>0$ so that $d(f,h)<\eta$ implies that $d(fg,hg)<\delta$. It follows that, provided $d(f,h)<\min \{\eta,\delta\}$, we have ${\big\lVert\tilde\rho(f)b(g)-\tilde\rho(h)b(g)\big\rVert}_W<{\epsilon}$, hence verifying uniform continuity.
Using our claim and the density of $\bigcup_nK_n$ in $G$, we can now uniquely extend $\tilde\rho$ to a strongly continuous isometric linear presentation $\tilde\rho\colon G\curvearrowright V$ so that the cocycle identity $b(fh)=\tilde\rho(f)b(h)+b(f)$ holds for all $f,h\in G$. It follows that we can define a continuous affine isometric action $\alpha\colon G\curvearrowright V $ by setting $\alpha(f)\xi=\tilde\rho(f)\xi+b(f)$ for $\xi\in V$ and $f\in G$.
Finally, note that $$\kappa_1\big(d(f,1)\big)\leqslant {\lVertb(f)\rVert}_V\leqslant \kappa_2\big(d(f,1)\big)$$ for all $f\in G$.
Repeating the proof of Theorem \[haagerup equiv\] and noting that approximately compact metrisable groups are separable, we obtain the following equivalence.
\[super reflexive equiv\] The following are equivalent for an approximately compact metrisable group with the local property (OB),
1. $G$ admits a uniformly continuous coarse embedding into a super-reflexive Banach space $\eta \colon G{\rightarrow}E$,
2. $G$ admits a metrically proper continuous affine isometric action on a super-reflexive Banach space.
The proof of Theorem \[pestov\] is fairly flexible and allows for several variations preserving different local structure of the Banach space $E$. As a concrete example, we shall study the preservation of Rademacher type in the above construction. For that we fix a [*Rademacher sequence*]{}, i.e., a sequence $({\epsilon}_n)_{n=1}^\infty$ of mutually independent random variables ${\epsilon}_n\colon \Omega{\rightarrow}\{-1,1\}$, where $(\Omega, \mathbb P)$ is some probability space, so that $\mathbb P({\epsilon}_n=-1)=\mathbb P({\epsilon}_n=1)=\frac 12$. E.g., we could take $\Omega=\{-1,1\}^{\mathbb N}$ with the usual coin tossing measure and let ${\epsilon}_n(\omega)=\omega(n)$.
A Banach space $X$ is said to have [*type*]{} $p$ for some $1\leqslant p\leqslant 2$ if there is a constant $C$ so that $$\Big(\mathbb E{\Big\lVert\sum_{i=1}^n{\epsilon}_ix_i\Big\rVert}^p\Big)^\frac 1p\leqslant C\cdot\Big(\sum_{i=1}^n{\lVertx_i\rVert}^p\Big)^\frac 1p$$ for every finite sequence $x_1,\ldots, x_n\in X$.
Similarly, $X$ has [*cotype*]{} $q$ for some $2\leqslant q<\infty$ if there is a constant $K$ so that $$\Big(\sum_{i=1}^n{\lVertx_i\rVert}^q\Big)^\frac 1q\leqslant K\cdot \Big(\mathbb E{\Big\lVert\sum_{i=1}^n{\epsilon}_ix_i\Big\rVert}^q\Big)^\frac 1q$$ for every finite sequence $x_1,\ldots, x_n\in X$.
We note that, by the triangle inequality, every Banach space has type $1$. Similarly, by stipulation, every Banach space is said to have cotype $q=\infty$.
Whereas the $p$ in the formula $\Big(\sum_{i=1}^n{\lVertx_i\rVert}^p\Big)^\frac 1p$ is essential, this is not so with the $p$ in $\Big(\mathbb E{\Big\lVert\sum_{i=1}^n{\epsilon}_ix_i\Big\rVert}^p\Big)^\frac 1p$. Indeed, the Kahane–Khintchine inequality (see [@albiac]) states that, for all $1<p<\infty$, there is a constant $C_p$ so that, for every Banach space $X$ and $x_1,\ldots, x_n\in X$, we have $$\mathbb E{\Big\lVert\sum_{i=1}^n{\epsilon}_ix_i\Big\rVert}
\leqslant
\Big(\mathbb E{\Big\lVert\sum_{i=1}^n{\epsilon}_ix_i\Big\rVert}^p\Big)^\frac 1p
\leqslant
C_p\cdot\mathbb E{\Big\lVert\sum_{i=1}^n{\epsilon}_ix_i\Big\rVert}.$$ In particular, for any $p,q\in [1,\infty[$, the two expressions $\Big(\mathbb E{\Big\lVert\sum_{i=1}^n{\epsilon}_ix_i\Big\rVert}^p\Big)^\frac 1p$ and $\Big(\mathbb E{\Big\lVert\sum_{i=1}^n{\epsilon}_ix_i\Big\rVert}^q\Big)^\frac 1q$ differ at most by some fixed multiplicative constant independent of the space $X$ and the vectors $x_i\in X$. We refer the reader to [@albiac] for more information on Rademacher type and cotype.
\[type\] Suppose $G$ is an approximately compact separable metrisable group with the local property (OB) admitting a uniformly continuous coarse embedding into a Banach space $E$ with type $p$ and cotype $q$ for some $1\leqslant p\leqslant 2\leqslant q\leqslant \infty$. Then $G$ admits a metrically proper affine isometric action on a space with type $p$ and cotype $q$.
Suppose $\sigma\colon G{\rightarrow}E$ is a uniformly continuous coarse embedding into a space $E$ with type $p$ and cotype $q$. We then repeat the proof of Thorem \[pestov\] using the fact that, as $E$ has type $p$ and cotype $q$, so do the spaces $L^2(K_n,E)$. Since also $L^2(K_n,E)$ and $L^2(K_m,E)$ are isometric for all $n,m$, the ultraproduct $W$ of the proof is isometric to the ultrapower of a single space $L^2(K_1,E)$. However, the ultrapower $X^{\mathcal}U$ of a Banach space $X$ is [*finitely representable*]{} in $X$, meaning that every finite-dimensional subspace $F\subseteq X^{\mathcal}U$ almost isometrically embeds into $X$. Thus, in particular, $W$ is finitely representable in $L^2(K_1,E)$ and therefore has type $p$ and cotype $q$. This similarly holds for the subspace $V=\overline{\rm span}\big(b[G]\big)\subseteq W$, which finishes the proof.
We mention that, by results of W. Orlicz and G. Nordlander (see [@albiac]), the space $L^p$ has type $p$ and cotype $2$, whenever $1\leqslant p\leqslant 2$, and type $2$ and cotype $p$, whenever $2\leqslant p<\infty$. So Theorem \[type\] applies, in particular, when $G$ is a separable approximately compact metrisable group with the local property (OB) admitting a uniformly continuous coarse embedding into an $L^p$ space.
In the interval $1\leqslant p\leqslant 2$, Theorem \[maurey\] gives us a somewhat better result, since it follows from results of J. Bretagnolle, D. Dacunha-Castelle and J.-L. Krivine [@bretagnolle] that $L^p$ coarsely embeds into $L^2$ for all $p\in [1,2]$.
Stable metrics and reflexive spaces
-----------------------------------
The case of reflexive spaces trivialises for locally compact second countable groups, since N. Brown and E. Guentner [@BG] showed that every countable discrete group admits a proper affine isometric action on a reflexive Banach space and U. Haagerup and A. Przybyszewska [@haagerup-affine] generalised this to locally compact second countable groups.
However, for general metrisable groups, the situation is significantly more complicated. Indeed, results of A. Shtern [@shtern] and M. Megrelishvili [@megrelishvili2], show that a topological group $G$ admits a topologically faithful isometric linear representation on a reflexive Banach space, i.e., $G$ is isomorphic to a subgroup of the linear isometry group of a reflexive Banach space with the strong operator topology, if and only if the continuous weakly almost periodic functions on $G$ separate points and closed sets. Moreover, there are examples, such as the group ${\rm Homeo}_+[0,1]$ of increasing homeomorphisms of the unit interval [@megrelishvili1], that admit no non-trivial continuous linear actions on a reflexive space.
We recall that a bounded function $\phi\colon G{\rightarrow}{\mathbb R}$ is said to be [*weakly almost periodic*]{} provided that its orbit $\lambda(G)\phi=\{\phi(g{^{-1}}\,\cdot\, ){ \; \big| \;}g\in G\}$ under the left regular representation is a relatively weakly compact subset of $\ell^\infty(G)$. Also, by a result of A. Grothendieck [@grothendieck], the weakly almost periodic functions on $G$ may be characterised by the following double limit criterion.
A a bounded function $\phi\colon G{\rightarrow}{\mathbb R}$ on a group $G$ is weakly almost periodic if and only if, for all sequences $(g_n)$ and $(f_m)$ in $G$ and ultrafilters ${\mathcal}U$ and ${\mathcal}V$ on ${\mathbb N}$, we have $$\lim_{n{\rightarrow}{\mathcal}U}\lim_{m{\rightarrow}{\mathcal}V}\phi(g_nf_m)=\lim_{m{\rightarrow}{\mathcal}V}\lim_{n{\rightarrow}{\mathcal}U}\phi(g_nf_m).$$
With a bit of effort, one may verify that, equivalently, $\phi$ is weakly almost periodic if and only if, for all sequences $(g_n)$ and $(f_m)$ in $G$, we have $$\lim_{n{\rightarrow}\infty}\lim_{m{\rightarrow}\infty}\phi(g_nf_m)=\lim_{m{\rightarrow}\infty}\lim_{n{\rightarrow}\infty}\phi(g_nf_m).$$ whenever the two limits exist.
Motivated by the notion of stability in model theory, J.-L. Krivine and B. Maurey [@KM] isolated the concept of a [*stable norm*]{} on a Banach space, which equivalently can be defined in terms of stability of the metric.
A metric $d$ on a set $X$ is said to be [*stable*]{} if, for all $d$-bounded sequences $(x_n)$ and $(y_m)$ in $X$ and ultrafilters ${\mathcal}U$ and ${\mathcal}V$ on ${\mathbb N}$, we have $$\lim_{n{\rightarrow}{\mathcal}U}\lim_{m{\rightarrow}{\mathcal}V}d(x_n,y_m)=\lim_{m{\rightarrow}{\mathcal}V}\lim_{n{\rightarrow}{\mathcal}U}d(x_n,y_m).$$
We remark that a simple, but tedious, inspection shows that, if $d$ is stable, then so is the uniformly equivalent bounded metric $D(x,y)=\max\{d(x,y),1\}$.
Now, N. Kalton [@kalton] showed that every stable metric space may be coarsely embedded into a reflexive Banach space. Moreover, he also showed that, e.g., the Banach space $c_0$ does not admit a coarse embedding into a reflexive Banach space.
For the case of groups, I. Ben Yaacov, A. Berenstein and S. Ferri [@ben; @yaacov] showed that a metrisable group admits a compatible left-invariant and stable metric if and only if it admits a topologically faithful isometric linear representation on a reflexive space.
Our goal here is to provide a group theoretical counter-part of Kalton’s theorem, that is, we wish to construct metrically proper continuous affine isometric actions on Banach spaces of topological groups admitting metrically proper stable compatible left-invariant metrics.
\[stable metric refl\] Suppose a topological group $G$ carries a compatible left-invariant metrically proper stable metric. Then $G$ admits a metrically proper continuous affine isometric action on a reflexive Banach space.
We should mention that this theorem is far from establishing an equivalence. Indeed, the Tsirelson space $T$ [@tsirelson] is a separable reflexive Banach space not containing isomorphic copies of any $\ell^p$, $1\leqslant p<\infty$, nor of $c_0$. Now, as noted in Exampe \[banach2\], the norm-metric on the additive group $(X,+)$ of a Banach space $(X,{\lVert\cdot\rVert})$ is maximal. It follows that the translation action of $T$ on itself is a metrically proper affine action on a reflexive space. However, by a result of Y. Raynaud (Thm. 4.1 [@raynaud]), if the additive group of an infinite-dimensional Banach space $E$ admits a compatible invariant stable metric, then $E$ must contain an isomorphic copy of some $\ell^p$, $1\leqslant p<\infty$. In other words, $T$ has no equivalent invariant stable metric, but has a metrically proper affine isometric action on a reflexive space.
In fact, the proof of Theorem \[stable metric refl\] will also require something less than a stable metric, namely, the existence of a sufficiently separating family of continuous weakly almost periodic functions.
\[wap refl\] Suppose $d$ is a compatible left-invariant metric on a topological group $G$ and assume that, for all $\alpha>0$, there is a continuous weakly almost periodic function $\phi\in \ell^\infty(G)$ with $d$-bounded support so that $\phi\equiv 1$ on $D_\alpha=\{g\in G{ \; \big| \;}d(g,1)\leqslant \alpha\}$.
Then $G$ admits a continuous isometric action $\pi\colon G\curvearrowright X$ on a reflexive Banach space $X$ with a corresponding continuous and metrically proper cocycle $b\colon (G,d){\rightarrow}X$.
Under the given assumptions, we claim that, for every integer $n\geqslant 1$, there is a continuous weakly almost periodic function $0\leqslant \phi_n\leqslant 1$ on $G$ so that
1. ${\lVert\phi_n\rVert}_\infty=\phi(1)=1$,
2. ${\lVert\phi_n-\lambda(g)\phi_n\rVert}_\infty\leqslant \frac 1{4^n} \text{ for all } g\in D_n$ and
3. ${\rm supp}(\phi_n)$ is $d$-bounded.
To see this, we pick inductively sequences of continuous weakly almost periodic functions $(\psi_i)_{i=1}^{4^n}$ and radii $(r_i)_{i=0}^{4^n}$ so that
1. $0=r_0<2n<r_1<r_1+2n<r_2<r_2+2n<r_3<\ldots<r_{4^n}$,
2. $0\leqslant \psi_i\leqslant 1$,
3. ${\psi}_{i}\equiv 1$ on $D_{r_{i-1}+n}$,
4. ${\rm supp}(\psi_i)\subseteq D_{r_i}$.
Note first that, by the choice of $r_i$, the sequence $$D_{r_0+n}\setminus D_{r_0},\; D_{r_1}\setminus D_{r_0+n}, \;D_{r_1+n}\setminus D_{r_1},\; D_{r_2}\setminus D_{r_1+n}, \; \ldots\;, D_{r_{4^n}}\setminus D_{r_{4^n-1}+n}, \; G\setminus D_{r_{4^n}}$$ partitions $G$. Also, for all $1\leqslant i\leqslant 4^n$, $$\psi_1\equiv \ldots\equiv \psi_i\equiv 0, \text{ while } \psi_{i+1}\equiv \ldots\equiv \psi_{4^n}\equiv 1\text{ on }D_{r_i+n}\setminus D_{r_i}$$ and $$\psi_1\equiv \ldots\equiv \psi_{i-1}\equiv 0, \text{ while } \psi_{i+1}\equiv \ldots\equiv \psi_{4^n}\equiv 1\text{ on }D_{r_i}\setminus D_{r_{i-1}+n}.$$ Setting $\phi_n=\frac 1{4^n}\sum_{i=1}^{4^n}\psi_i$, we note that, for all $1\leqslant i\leqslant 4^n$, $$\phi_n\equiv \frac {4^n-i}{4^n} \quad \text{ on }D_{r_i+n}\setminus D_{r_i}$$ and $$\frac {4^n-i}{4^n}\leqslant \phi_n\leqslant \frac {4^n-i+1}{4^n} \quad \text{ on }D_{r_{i}}\setminus D_{r_{i-1}+n}.$$
Now, if $g\in D_n$ and $f\in G$, then $|d(g{^{-1}}f,1)-d(f,1)|=|d(f,g)-d(f,1)|\leqslant d(g,1)\leqslant n$. So, if $f$ belongs to some term in the above partition, then $g{^{-1}}f$ either belongs to the immediately preceding, the same or the immediately following term of the partition. By the above estimates on $\phi_n$, it follows that $|\phi_n(f)-\phi_n(g{^{-1}}f)|\leqslant \frac1{4^n}$. In other words, for $g\in D_n$, we have $$\begin{split}
{\lVert\phi_n-\lambda(g)\phi_n\rVert}_\infty= \sup_{f\in G}|\phi_n(f)-\phi_n(g{^{-1}}f)|\leqslant \frac 1{4^n},
\end{split}$$ which verifies condition (2). Conditions (1) and (3) easily follow from the construction.
Consider now a specific $\phi_n$ as above and define $$W_n={\overline}{\rm conv}\big(\lambda(G)\phi_n\cup -\lambda(G)\phi_n\big)\subseteq \ell^\infty(G)$$ and, for every $k\geqslant 1$, $$U_{n,k}=2^kW_n+2^{-k}B_{\ell^\infty},$$ where $B_{\ell^\infty}$ denotes the unit ball in ${\ell^\infty}(G)$. Let ${\lVert\cdot\rVert}_{n,k}$ denote the gauge on ${\ell^\infty}(G)$ defined by $U_{n,k}$, i.e., $${\lVert\psi\rVert}_{n,k}=\inf(\alpha>0{ \; \big| \;}\psi\in \alpha\cdot U_{n,k}).$$
If $g\in D_n$, then ${\lVert\phi_n-\lambda(g)\phi_n\rVert}_\infty\leqslant \frac 1{4^n}$ and so, for $k\leqslant n$, $$\phi_n-\lambda(g)\phi_n\in \frac 1{2^n}\cdot 2^{-k}B_{\ell^\infty}\subseteq \frac 1{2^n}\cdot U_{n,k}.$$ In particular, $$\label{a}
{\lVert\phi_n-\lambda(g)\phi_n\rVert}_{n,k}\leqslant \frac 1{2^n},\;\;\text{ for all } k\leqslant n \text{ and }g\in D_n.$$
On the other hand, for all $g\in G$ and $k$, we have $\phi_n-\lambda(g)\phi_n\in 2W_n\subseteq \frac1{2^{k-1}}U_{n,k}$. Therefore, $$\label{b}
{\lVert\phi_n-\lambda(g)\phi_n\rVert}_{n,k}\leqslant \frac 1{2^{k-1} },\;\;\text{ for all } k\text{ and }g.$$
Finally, if $g\notin ({\rm supp}\;\phi_n){^{-1}}$, then ${\lVert\phi_n-\lambda(g)\phi_n\rVert}_{n,1}\geqslant \frac 12{\lVert\phi_n-\lambda(g)\phi_n\rVert}_\infty\geqslant \frac 12$. So $$\label{c}
{\lVert\phi_n-\lambda(g)\phi_n\rVert}_{n,1}\geqslant \frac 12,\;\;\text{ for all } g\notin ({\rm supp}\;\phi_n){^{-1}}.$$
It follows from (\[a\]) and (\[b\]) that, for $g\in D_n$, we have $$\begin{split}
\sum_k{\lVert\phi_n-\lambda(g)\phi_n\rVert}^2_{n,k}
&\leqslant \underbrace{\Big(\frac 1{2^n}\Big)^2+\ldots+\Big(\frac 1{2^n}\Big)^2}_{n\text{ times}}
+ \Big(\frac 1{2^{(n+1)-1}}\Big)^2+ \Big(\frac 1{2^{(n+2)-1}}\Big)^2+\ldots\\
&\leqslant\frac 1{2^{n-1}},
\end{split}$$ while using (\[c\]) we have, for $g\notin ({\rm supp}\;\phi_n){^{-1}}$, $$\sum_k{\lVert\phi_n-\lambda(g)\phi_n\rVert}^2_{n,k}\geqslant \frac1{4}.$$
Define ${|\!|\!|\cdot|\!|\!|}_n$ on $\ell^\infty(G)$ by ${|\!|\!|\psi|\!|\!|}_n=\big(\sum_k{\lVert\psi\rVert}_{n,k}^2\big)^{\frac 12}$ and set $$X_n=\{\psi\in {\overline}{\rm span}(\lambda(G)\phi_n)\subseteq \ell^\infty(G){ \; \big| \;}{|\!|\!|\psi|\!|\!|}_n<\infty\}\subseteq \ell^\infty(G).$$ By the main result of W. J. Davis, T. Figiel, W. B. Johnson and A. Pełczyński [@dfjp], the interpolation space $(X_n,{|\!|\!|\cdot|\!|\!|}_n)$ is a reflexive Banach space. Moreover, since $W_n$ and $U_{n,k}$ are $\lambda(G)$-invariant subsets of $\ell^\infty(G)$, one sees that ${\lVert\cdot\rVert}_{n,k}$ and ${|\!|\!|\cdot|\!|\!|}_n$ are $\lambda(G)$-invariant and hence we have an isometric linear representation $\lambda\colon G\curvearrowright (X_n, {|\!|\!|\cdot|\!|\!|}_n)$.
Note that, since $\phi_n\in W_n$, we have $\phi_n\in X_n$ and can therefore define a cocycle $b_n\colon G{\rightarrow}X_n$ associated to $\lambda$ by $b_n(g)=\phi_n-\lambda(g)\phi_n$. By the estimates above, we have $${|\!|\!|b_n(g)|\!|\!|}_n={|\!|\!|\phi_n-\lambda(g)\phi_n|\!|\!|}_n\leqslant\big( \frac 1{{\sqrt 2}}\big)^{n-1}$$ for $g\in D_n$, while $${|\!|\!|b_n(g)|\!|\!|}_n={|\!|\!|\phi_n-\lambda(g)\phi_n|\!|\!|}_n\geqslant\frac1{2}$$ for $g\notin ({\rm supp}\;\phi_n){^{-1}}$.
Let now $Y=\big(\bigoplus_n(X_n,{|\!|\!|\cdot|\!|\!|}_n)\big)_{\ell^2}$ denote the $\ell^2$-sum of the spaces $(X_n,{|\!|\!|\cdot|\!|\!|}_n)$. Let also $\pi\colon G\curvearrowright Y$ be the diagonal action and $b=\bigoplus b_n$ the corresponding cocycle. To see that $b$ is well-defined, note that, for $g\in D_n$, we have $$\begin{split}
{\lVertb(g)\rVert}_Y
&=\Big(\sum_{m=1}^\infty{|\!|\!|b_m(g)|\!|\!|}_m^2\Big)^\frac 12\\
&=\Big(\text{finite}+\sum_{m=n}^\infty{|\!|\!|b_m(g)|\!|\!|}_m^2\Big)^\frac 12\\
&\leqslant\Big(\text{finite}+\sum_{m=n}^\infty\frac 1{2^{m-1}}\Big)^\frac 12\\
&<\infty,
\end{split}$$ so $b(g)\in Y$.
Remark that, whenever $g\notin ({\rm supp}\;\phi_n){^{-1}}$, we have $${\lVertb(g)\rVert}_Y\geqslant \Big( \underbrace{\big(\frac 12\big)^2+\ldots+\big(\frac 12\big)^2}_{n\text{ times}} \Big)^\frac 12=\frac {\sqrt n}2.$$ As $({\rm supp}\;\phi_n){^{-1}}$ is $d$-bounded, this shows that the cocycle $b\colon (G,d){\rightarrow}Y$ is metrically proper. We leave the verification that the action is continuous to the reader.
Let us now see how to deduce Theorem \[stable metric refl\] from Theorem \[wap refl\]. So fix a compatible metrically proper stable left-invariant stable metric $d$ on $G$ with corresponding balls $D_\alpha$. Then, for every $\alpha>0$, we can define a continuous bounded weakly almost periodic function $\phi_\alpha\colon G{\rightarrow}{\mathbb R}$ by $$\phi_\alpha(g)=2-\min\Big\{1, \max\big\{\frac{d(g,1)}\alpha, 2\big\}\Big\}.$$ We note that $\phi_\alpha$ has $d$-bounded support, while $\phi_\alpha\equiv 1$ on $D_\alpha$, thus verifying the conditions of Theorem \[wap refl\].
Since, by Example \[banach2\], the additive group $(X,+)$ of a Banach space $(X,{\lVert\cdot\rVert})$ is quasi-isometric to $(X,{\lVert\cdot\rVert})$ itself, these provide examples of metrisable groups admitting metrically proper affine isometric actions on Banach spaces with various types of geometry.
For example, $c_0$ does not admit a coarse embedding into a reflexive Banach space [@kalton] and thus cannot have a metrically proper affine isometric action on a reflexive Banach space.
Also, by results of M. Mendel and A. Naor [@naor], $L^q$ does not embed coarsely into $L^p$, whenever $\max \{2,p\}<q<\infty$. Thus, $L^q$ cannot have a metrically proper affine isometric action on $L^p$ either. But, being super-reflexive, its shift-action on itself is a metrically proper affine isometric action on a super-reflexive space.
Open problems
=============
Suppose $M$ is a compact surface of genus $g\geqslant 1$. Does the group of orientation preserving homeomorphisms, ${\rm Homeo}^+(M)$, have the local property (OB)? Does the identity component ${\rm Homeo}^+_0(M)$ have property (OB) relative to ${\rm Homeo}^+(M)$? We note that, if the latter holds, then ${\rm Homeo}^+(M)$ would be quasi-isometric to the mapping class group ${\mathcal}M(M)={\rm Homeo}^+(M)/{\rm Homeo}^+_0(M)$.
Do diffeomorphism groups of smooth manifolds have well-defined quasi-isometry type? Cf. the results of [@brandenburgsky], in which certain right-invariant metrics on groups of measure-preserving diffeomorphisms are shown to be unbounded.
Is there an analogue of the Guentner–Kaminker result [@GK] valid for general metrisable groups? I.e., if a metrisable group $G$ admits a maximal metric and an affine isometric action on a Hilbert space with a cocycle growing faster that the square root of the distance, does it follow that $G$ is amenable?
Let $M$ be an approximately finite-dimensional von Neumann algebra, whence its unitary subgroup $U(M)$ is approximately compact [@harpe]. Does $U(M)$ have property (OB) or the local property (OB)? If so, on what kinds of Banach spaces does $U(M)$ act metrically properly by affine isometric transformations?
Find necessary and sufficient conditions for a metrisable group to have a metrically proper continuous affine isometric action on a reflexive Banach space. In particular, if $G$ admits such an action, what can be said about the weakly almost periodic functions on $G$?
Does the isometry group ${\rm Isom}({\mathbb Q}{\mathbb U})$ of the rational Urysohn metric space have a continuous affine isometric action on a reflexive space without a fixed point?
More generally, does a non-Archimedean Polish group have property (OB) if and only if all of its affine isometric actions on reflexive spaces fix a point? If $G$ has property (OB), the existence of such a fixed point follows immediately from the fixed point theorem of C. Ryll-Nardzewski [@ryll].
Suppose $\bf M$ is the countable atomic model of an $\omega$-stable theory $T$. Does ${\rm Aut}(\bf M)$ have the local property (OB)? We note that, if this is the case, then ${\rm Aut}(\bf M)$ admits a metrically proper continuous affine isometric action on a reflexive Banach space.
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[^1]: This work was partially supported by a grant from the Simons Foundation (\#229959 to Christian Rosendal) and likewise by NSF grant DMS 1201295
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abstract: 'We present a transfer matrix approach that combines the Blonder-Tinkham-Klapwijk (BTK) formalism and self-consistent solutions to the Bogolibuov-de Gennes (BdG) equations and use it to study the tunneling conductance and spin transport in ferromagnet (${\rm F}$)-superconductor (${\rm S}$) trilayers (${\rm F_1F_2 S}$) as functions of bias voltage. The self-consistency ensures that the spin and charge conservation laws are properly satisfied. We consider forward and angularly averaged conductances over a broad range of the strength of the exchange fields and ${\rm F}$ thicknesses, as the relative in-plane magnetization angle, $\phi$, between the two ferromagnets varies. The $\phi$-dependence of the self-consistent conductance curves in the trilayers can differ substantially from that obtained via a non-self-consistent approach. The zero bias forward conductance peak exhibits, as $\phi$ varies, resonance effects intricately associated with particular combinations of the geometrical and material parameters. We find, when the magnetizations are non-collinear, signatures of the anomalous Andreev reflections in the subgap regions of the angularly averaged conductances. When ${\rm F_1}$ is half-metallic, the angularly averaged subgap conductance chiefly arises from anomalous Andreev reflection. The in-plane components of the spin current are strongly bias dependent, while the out-of-plane spin current component is only weakly dependent upon voltage. The components of the spin current aligned with the local exchange field of one of the F layers are conserved in that layer and in the S region, while they oscillate in the other layer. We compute the spin transfer torques, in connection with the oscillatory behavior of spin currents, and verify that the spin continuity equation is strictly obeyed in our method.'
author:
- 'Chien-Te Wu'
- 'Oriol T. Valls'
- Klaus Halterman
title: 'Tunneling Conductance and Spin Transport in Clean Ferromagnet-Ferromagnet-Superconductor Heterostructures'
---
Introduction {#intro}
============
Over the last two decades, significant progress in fabrication techniques has allowed the development of spintronics devices, such as spin valves,[@igor] that utilize both charge and spin degrees of freedom. Traditional spin valves consist of magnetic materials only. There is another important type of spintronics devices, involving ferromagnet (F)-superconductor (S) heterostructures. These heterostructures have also received much attention because of the fundamental physics related to the interplay between ferromagnetic and superconducting order. Their potential applications in spintronics include magnetic memory technology where information storage is accomplished via control of the magnetic moment bit. It is then crucial to have precise control over the magnetization direction. Spin transfer torque (STT) is one effect that affords such control. The generation of spin-polarized supercurrents may be used to obtain a superconducting STT acting on the magnetization of a ferromagnet. This effect may be utilized in high density nanotechnologies that require magnetic tunnel junctions. Thus, the dissipationless nature of the supercurrent flow offers a promising avenue in terms of low energy nanoscale manipulation of superconducting and magnetic orderings.
Although ferromagnetism and $s$-wave superconductivity seem incompatible because of the inherently opposite natures of their order parameter spin configurations, superconductivity can still be induced in the F layers of F-S layered structures by the superconducting proximity effects.[@Buzdin2005] In essence, the superconducting proximity effects describe the leakage of superconductivity into a non-superconducting normal (N) or magnetic metal, as well as its depletion in S near the interface. However, proximity effects in F-S systems are very different from those in N-S structures due to the inherent exchange field in the F materials. As a consequence of this exchange field, the Cooper pair acquires a non-zero center-of-mass momentum[@demler; @Buzdin2005; @Halterman2001; @Halterman2002] and the overall Cooper pair wavefunction oscillates spatially in the F regions. Owing to this oscillatory nature, many new physical phenomena emerge in F-S heterostructures such as oscillations of the superconducting transition temperature, $T_c$, with the thickness of the F layers. [@Buzdin1990; @Buzdin2005; @demler; @Halterman2004]
It is of fundamental importance that superconducting proximity effects are governed by Andreev reflection,[@Andreev] which is a process of electron-to-hole conversion at N-S or F-S interfaces, and it involves the creation or annihilation of a Cooper pair. Therefore, consideration of Andreev reflection is central when studying the transport properties of N-S[@btk; @tanaka] or F-S systems.[@beenakker; @zv1; @zv2] Of particular interest[@btk; @tanaka; @beenakker; @zv1; @zv2] is the behavior of the tunneling conductance in the subgap region, where hybrid systems can carry a supercurrent due to Andreev reflection. In conventional Andreev reflection, the reflected hole has opposite spin to the incident particle. Accordingly, the exchange field in the F materials that causes the splitting of spin bands has a significant effect on the tunneling conductance in the subgap region. Most important, the qualitative behavior of the conductance peak in the zero bias limit is strongly influenced by the degree of conduction electron spin polarization in the F materials.[@beenakker; @zv1; @zv2; @mazin] Experimentally, this concept has been applied to quantify the spin polarization. [@raychaudhuri; @upad; @chalsani; @soulen; @hacohen]
An intriguing phenomenon in F-S structures is the induction of triplet pairing correlations.[@berg86; @Bergeret2007; @Wang2010; @Hubler2012; @hv2p] These correlations are very important when studying transport phenomena such as those found in SFS Josephson junctions.[@Keizer2006; @robinson; @khaire] In contrast to the short proximity length[@Halterman2002] of singlet Cooper pair condensates into F materials, the $m=\pm1$ triplet pairing correlations are compatible with the exchange fields and hence largely immune to the pair breaking effect produced by the latter. However, for such correlations to be induced F-S structures must possess a spin-flip mechanism. Examples include a spin-dependent scattering potential at the F-S interface [@Halterman2009; @Eschrig2008] and the introduction of another magnetic layer with a misoriented magnetic moment such as ${\rm F_1SF_2}$ superconducting spin valves.[@Halterman2007] The pairing state of $m=\pm1$ induced triplet correlations is at variance with the effects of [*conventional*]{} Andreev reflection, responsible for the generation of singlet Cooper pairs. Thus, recent studies[@linder2009; @visani; @niu; @ji; @feng] on the tunneling conductance propose the existence of [*anomalous*]{} Andreev reflection, that is, a reflected hole with the same spin as the incident particle can be Andreev reflected under the same circumstances as the generation of $m=\pm1$ triplet pairing correlations becomes possible. In this view, triplet proximity effects are correlated with the process of this anomalous Andreev reflection. This will be confirmed and discussed in this work.
Another important geometry for a superconducting spin valve consists of a conventional spin valve with a superconductor layer on top: a ${\rm F_1F_2S}$ trilayer. By applying an external magnetic field, or switching via STT, one is able to control the relative orientation of the intrinsic magnetic moments and investigate the dependence[@alejandro; @leksin; @zdravkov] of physical properties such as $T_c$ on the misorientation angle $\phi$ between the two magnetic layers. Due to the proximity effects, $T_c$ is often found to be minimized when the magnetizations are approximately perpendicular to each other,[@cko] reflecting the presence of long range triplet correlations, induced in ${\rm F_1F_2S}$ trilayers. Their existence has been verified both theoretically[@cko] and experimentally.[@alejandro] The non-monotonic behavior of $T_c$ as a function of $\phi$ has also been shown to be quantitatively[@alejandro] related to the long range triplet correlations, with excellent agreement between theory and experiment.
Motivated by these important findings, we will investigate here, in a fully self-consistent manner, the $\phi$ dependence of the tunneling conductance and other transport quantities of these ${\rm F_1F_2S}$ trilayers. Non-self-consistent theoretical studies of tunneling conductance have been performed on ${\rm F_1F_2S}$ trilayers in previous work.[@ji; @cheng] However, as we shall see in Sec. \[methods\], only self-consistent methods guarantee that conservation laws are not violated and (see Sec. \[results\]) only then can one correctly predict the proximity effects on the angular dependence of transport properties. The spin-polarized tunneling conductance of F-S bilayers only, was studied in Refs. . Also, in traditional spin valves e.g. ${\rm F_1}$-${\rm F_2}$ layered structures, the spin-polarized current generated in the ${\rm F_1}$ layer can transfer angular momentum to the ${\rm F_2}$ layer when their magnetic moments are not parallel to each other[@igor] via the effect of STT.[@berger; @myers] As a result, the spin current is not a conserved quantity and one needs a general law that relates local spin current to local STT.[@linder2009] The transport properties of ${\rm F_1SF_2}$ structures, in particular the dependence on applied bias of the spin-transfer torque and the spin-polarized tunneling conductance have been previously studied.[@romeo; @bozovic; @linder2009] Here, we consider charge transport and both spin current and spin-transfer torque in ${\rm F_1F_2S}$ trilayers. In previous theoretical work, such as that mentioned above, when computing tunneling conductance of N-S and F-S structures, using methods based on the Blonder-Tinkham-Klapwijk (BTK) procedure[@btk; @tanaka; @zv1; @zv2; @bozovic; @linder2007; @linder2009] and quasi-classical approximations,[@Eschrig2010] the superconducting pair amplitude was assumed to be a step function: a constant in S, dropping abruptly to zero at the N-S or F-S interface and then vanishing in the non-superconducting region. This assumption neglects proximity effects. Only qualitative predictions on the behavior of the tunneling conductance can be reliably made. Still, results exhibit many interesting features especially in F-S systems.[@zv1; @zv2] However, to fully account for the proximity effects, in the transport properties, one must use a self-consistent pair potential. This is because that reveals realistic information regarding the leakage and depletion of superconductivity. Also, as we shall discuss below, self-consistent solutions guarantee that conservation laws are satisfied. In Ref. , the tunneling conductance of F-S bilayers was extracted via self-consistent solutions of Bogoliubov-de Gennes (BdG) equations.[@degennes] However, the numerical methods used there required awkward fitting procedures that led to appreciable uncertainties and precluded their application to trilayers. The findings indicated that the self-consistent tunneling conductances for the bilayer are quantitatively different from those computed in a non-self-consistent framework, thus demonstrating the importance of properly accounting for proximity effects in that situation. Here we report on a powerful [**self-consistent**]{} approach and use it to compute the tunneling conductance of ${\rm F_1F_2S}$ trilayers. It is based on the BTK method, incorporated into a transfer matrix procedure similar to that used[@strinati] in Josephson junction calculations and simple F-S junctions within a Hubbard model[@ting]. As we shall demonstrate, this approach not only has the advantage of being more numerically efficient but also can be used to compute spin transport quantities. Thus, we are able to address many important points regarding both charge and spin transport in ${\rm F_1F_2S}$ trilayers, including the spin currents and spin-transfer torque, the proximity effects on the tunneling conductance, and the correlation between the anomalous Andreev reflection and the triplet correlations.
This paper is organized as follows: we present our self-consistent approach, and its application to compute the tunneling conductance, the spin-transfer torques, the spin current, and the proper way to ensure that conservation laws are satisfied, in Sec. \[methods\]. In Sec. \[results\] we present the results. In Subsec. \[bilayers\], we briefly compare the results of F-S bilayers obtained in our self-consistent approach with non-self-consistent ones. The rest of Subsec. \[trilayers\] includes our results for trilayers, that is, the main results of this work. The dependence on the tunneling conductance of ${\rm F_1F_2S}$ trilayers on the angle $\phi$ is extensively discussed as a function of geometrical and material parameters. Results for the effect of the anomalous Andreev reflection, the spin-transfer torque, and the spin current are also presented. We conclude with a recapitulation in Sec. \[conclusions\].
![(Color online) Schematic of the F$_1$F$_2$S trilayer. The exchange field, $\bm h$, denoted by a black solid arrow, is along the $+z$ direction in the outer magnetic layer (F$_1$) while within the inner magnetic layer (F$_2$), $\bm h$ is oriented at an angle $\phi$ in the $x-z$ plane. The outer magnetic layer and the superconducting layer are connected to electrodes that are biased with a finite voltage $V$.[]{data-label="figure1"}](newfig1.eps){width="45.00000%"}
Methods
=======
Description of the system {#description}
-------------------------
The geometry of our system is depicted in Fig. \[figure1\]. We denote the outer ferromagnet as ${\rm F_1}$ and the middle layer as ${\rm F_2}$. We choose our coordinate system so that the interfaces are parallel to the $x-z$ plane, and infinite in extent, while the system has a finite width $d=d_{F_1}+d_{F_2}+d_S$ in the $y$ direction. The Hamiltonian appropriate to our system is, $$\begin{aligned}
\label{ham}
%\begin{aligned}
{\cal H}_{eff}&=&\int d^3r \left\{ \sum_{\alpha}
\psi_{\alpha}^{\dagger}\left(\mathbf{r}\right){\cal H}_0
\psi_{\alpha}\left(\mathbf{r}\right)\right.\nonumber \\
&+&\left.\frac{1}{2}\left[\sum_{\alpha,\:\beta}\left(i\sigma_y\right)_{\alpha\beta}
\Delta\left(\mathbf{r}\right)\psi_{\alpha}^{\dagger}
\left(\mathbf{r}\right)\psi_{\beta}^{\dagger}
\left(\mathbf{r}\right)+H.c.\right]\right.\nonumber \\
&-&\left.\sum_{\alpha,\:\beta}\psi_{\alpha}^{\dagger}
\left(\mathbf{r}\right)\left(\mathbf{h}\cdot\bm{\sigma}
\right)_{\alpha\beta}\psi_{\beta}\left(\mathbf{r}\right)\right\},
%\end{aligned}\end{aligned}$$ where ${\cal H}_0$ is the single-particle Hamiltonian, $\mathbf{h}$ is a Stoner exchange field that characterizes the magnetism, and $\bm{\sigma}$ are Pauli matrices. The superconducting pair potential $\Delta(\mathbf{r})\equiv %ctw4
g\left(\mathbf{r}\right)\left\langle\psi_{\uparrow}
\left(\mathbf{r}\right)\psi_{\downarrow}\left(\mathbf{r}\right) %ctw4
\right\rangle$ is the product of pairing constant, $g\left(\mathbf{r}\right)$, in the singlet channel, and the pair amplitude. We begin by writing down the BdG equations, which we will solve self-consistently for our F$_1$F$_2$S trilayers. By performing the generalized Bogoliubov transformation[@degennes], $\psi_{\sigma}=\sum_n\left(u_{n\sigma}\gamma_n %kh moved
+\eta_{\sigma}v_{n\sigma}^{\ast}\gamma_n^{\dagger}\right)$, where $\sigma= (\uparrow, \downarrow)$ and $\eta_{\sigma}\equiv1(-1)$ for spin-down (up), the Hamiltonian \[Eq. (\[ham\])\] can be diagonalized. We can then for our geometry rewrite[@cko] Eq. (\[ham\]) as a quasi-one-dimensional eigensystem: $$\begin{aligned}
&\begin{pmatrix}
{\cal H}_0 -h_z&-h_x&0&\Delta \\
-h_x&{\cal H}_0 +h_z&\Delta&0 \\
0&\Delta&-({\cal H}_0 -h_z)&-h_x \\
\Delta&0&-h_x&-({\cal H}_0+h_z) \\
\end{pmatrix}
\begin{pmatrix}
u_{n\uparrow}\\u_{n\downarrow}\\v_{n\uparrow}\\v_{n\downarrow}
\end{pmatrix} \nonumber \\
&=\epsilon_n
\begin{pmatrix}
u_{n\uparrow}\\u_{n\downarrow}\\v_{n\uparrow}\\v_{n\downarrow}
\end{pmatrix}\label{bogo},\end{aligned}$$ where the $u_{n\sigma}$ and $v_{n\sigma}$ are respectively the quasiparticle and quasihole amplitudes with spin $\sigma$. The exchange field vanishes in the S region, while in ${\rm F_1}$ it is directed along $z$, $\mathbf{h}=h_1\hat{\mathbf{z}}\equiv \mathbf{h_1}$, and in ${\rm F_2}$ it can rotate in the $x$$-$$z$ plane, $\mathbf{h}=h_2\left(\sin\phi\hat{\mathbf{x}}+\cos\phi\hat{\mathbf{z}}\right)
\equiv \mathbf{h_2}$. The single-particle Hamiltonian now reads[@cko] ${\cal H}_0=-({1}/{2m})({d^2}/{dy^2})+{\epsilon_{\perp}}-E_F(y)$, where $\epsilon_{\perp}\equiv k_\perp^2/2m$ denotes the transverse kinetic energy in the $x-z$ plane. Also, $E_F(y)=E_{FS}\equiv{k_{FS}^2}/{2m}$ in the superconducting region and $E_F(y)=E_{FM}\equiv{k_{FM}^2}/{2m}$ in the ferromagnetic layers. Throughout this paper, we assume natural units $\hbar=k_B=1$ and measure all energies in units of $E_{FS}$. To take into account the more realistic situation where the F materials can in general have different bandwidths than the S layer, we define (as in Ref. ) a mismatch parameter $\Lambda$ via $E_{FM}\equiv\Lambda E_{FS}$. We are aiming here to solve the problem in a fully self consistent manner. The self-consistent pair potential $\Delta(y)$ can be expressed in terms of the quasi-particle and quasi-hole wavefunctions. Accordingly, $$\label{del}
\Delta(y) = \frac{g(y)}{2}{\sum_n}^\prime
\bigl[u_{n\uparrow}(y)v_{n\downarrow}^{\ast}(y)+
u_{n\downarrow}(y)v_{n\uparrow}^{\ast}(y)\bigr]\tanh\left(\frac{\epsilon_n}{2T}\right), \,$$ where the primed sum is over all eigenstates with energies $\epsilon_n$ smaller than a characteristic Debye energy, and $g(y)$ is the superconducting coupling constant in the $S$ region and vanishes elsewhere. We obtain the self-consistent pair potential by solving Eqs. (\[bogo\]) and (\[del\]) following the iterative numerical procedures discussed in previous work.[@hv2p; @cko]
Application of the BTK method {#BTK}
-----------------------------
The BTK formalism is a procedure to extract the transmitted and reflected amplitudes, and hence the conductance, from solutions to the BdG equations. This is accomplished by writing down the appropriate eigenfunctions in different regions. In this subsection, we review the relevant aspects of the formalism[@btk] for the non-self-consistent case (a step function pair potential) with the objective of establishing notation and methodology to describe, in the next subsection, the procedure to be used in the self-consistent case.
Consider first a spin-up quasi-particle with energy $\epsilon$, incident into the left side labeled “${\rm F}_1$", in Fig. \[figure1\]). Since the exchange fields in the ${\rm F_1}$ and ${\rm F_2}$ layers can be non-collinear, it follows from Eq. (\[bogo\]) that the spin-up (-down) quasi-particle wavefunction is not just coupled to the spin-down (-up) quasi-hole wavefunction, as is the case of F-S bilayers. Indeed, the wavefunction in the ${\rm F_1}$ layer is a linear combination of the original incident spin-up quasi-particle wavefunctions and various types of reflected wavefunctions, namely reflected spin-up and spin-down quasi-particle and quasi-hole wavefunctions (via both ordinary and Andreev reflections). We use a single column vector notation to represent these combinations, $$\label{f1waveup}
\Psi_{F1,\uparrow}\equiv\begin{pmatrix}e^{ik^+_{\uparrow1}y}+b_{\uparrow}e^{-ik^+_{\uparrow1}y}
\\b_{\downarrow}e^{-ik^+_{\downarrow1}y}
\\a_{\uparrow}e^{ik^-_{\uparrow1}y}
\\a_{\downarrow}e^{ik^-_{\downarrow1}y}\end{pmatrix}\enspace\enspace.
%\sigma=\uparrow.$$ If the incident particle has spin down, the corresponding wavefunction in ${\rm F_1}$ is $$\label{f1wavedown}
\Psi_{F1,\downarrow}\equiv\begin{pmatrix}b_{\uparrow}e^{-ik^+_{\uparrow1}y}
\\e^{ik^+_{\downarrow1}y}+b_{\downarrow}e^{-ik^+_{\downarrow1}y}
\\a_{\uparrow}e^{ik^-_{\uparrow1}y}
\\a_{\downarrow}e^{ik^-_{\downarrow1}y}\end{pmatrix}\enspace\enspace.% \sigma=\downarrow.$$ In these expressions $k^{\pm}_{\sigma 1}$ are quasi-particle $(+)$ and quasi-hole $(-)$ wavevectors in the longitudinal direction $y$, and satisfy the relation, $$\label{wavevector}
k^{\pm}_{\sigma m}=\left[\Lambda(1-\eta_{\sigma}{h}_m)\pm{\epsilon}-{k_\perp^2}\right]^{1/2},
%\end{align}$$ where $m=1$ (as used above) or $m=2$, used later. As mentioned above, all energies are in units of $E_{FS}$ and, in addition, we measure all momenta in units of $k_{FS}$. In this simple case, one can easily distinguish the physical meaning of each individual wavefunction. For instance in Eq. (\[f1waveup\]), $a_{\downarrow}\left(0,0,0,1\right)^{T}e^{ik^-_{\downarrow1}y}$ is the reflected spin-down quasi-hole wavefunction. The quasi-hole wavefunctions are the time reversed solutions of the BdG equations and carry a positive sign in the exponent for a left-going wavefunction. The relevant angles can be easily found in terms of wavevector components. Thus, e.g., the incident angle $\theta_i$ (for spin-up) at the ${\rm F_1-F_2}$ interface is $\theta_{i}=\tan^{-1}\left({k_{\perp}}/{k_{\uparrow 1}^{+}}\right)$, and the Andreev reflected angle $\theta_{r\downarrow}^-$ for reflected spin-down quasi-hole wavefunction is $\theta_{r\downarrow}^-=\tan^{-1}\left({k_{\perp}}/{k_{\downarrow 1}^-}\right)$. The conservation of transverse momentum leads to many important features[@bv; @zv1] when one evaluates the angularly averaged tunneling conductance, as we will see below. For the intermediate layer ${\rm F_2}$, the eigenfunction in general contains both left- and right-moving plane waves, that is, $$\label{f2wave}
\Psi_{F2}\equiv %otv6
\begin{pmatrix}c_1f^+_{\uparrow}e^{ik^+_{\uparrow2}y}+c_2f^+_{\uparrow}e^{-ik^+_{\uparrow2}y}
+c_3g^+_{\uparrow}e^{ik^+_{\downarrow2}y}+c_4g^+_{\uparrow}e^{-ik^+_{\downarrow2}y}\\
c_1f^+_{\downarrow}e^{ik^+_{\uparrow2}y}+c_2f^+_{\downarrow}e^{-ik^+_{\uparrow2}y}
+c_3g^+_{\downarrow}e^{ik^+_{\downarrow2}y}+c_4g^+_{\downarrow}e^{-ik^+_{\downarrow2}y}\\
c_5f^-_{\uparrow}e^{ik^-_{\uparrow2}y}+c_6f^-_{\uparrow}e^{-ik^-_{\uparrow2}y}
+c_7g^-_{\uparrow}e^{ik^-_{\downarrow2}y}+c_8g^-_{\uparrow}e^{-ik^-_{\downarrow2}y}\\
c_5f^-_{\downarrow}e^{ik^-_{\uparrow2}y}+c_6f^-_{\downarrow}e^{-ik^-_{\uparrow2}y}
+c_7g^-_{\downarrow}e^{ik^-_{\downarrow2}y}+c_8g^-_{\downarrow}e^{-ik^-_{\downarrow2}y}
\end{pmatrix},$$ where $k^{\pm}_{\uparrow 2}$ and $k^{\pm}_{\downarrow 2}$ are defined in Eq. (\[wavevector\]). The $\pm$ indices are defined as previously, and the up and down arrows refer to ${\rm F_1}$. The eigenspinors $f$ and $g$ that correspond to spin parallel or antiparallel to ${\bf h_2}$ respectively, are given, for $0\leq\phi\leq{\pi}/{2}$, by the expression, $$\label{f2spinor1}
\begin{pmatrix}f^+_{\uparrow}\\f^+_{\downarrow}\end{pmatrix}=
\frac{1}{\cal N}\begin{pmatrix}1\\\frac{1-\cos\phi}{\sin\phi}\end{pmatrix}
=\begin{pmatrix}f^-_{\uparrow}\\-f^-_{\downarrow}\end{pmatrix};\enspace
\begin{pmatrix}g^+_{\uparrow}\\g^+_{\downarrow}\end{pmatrix}=
\frac{1}{\cal N}\begin{pmatrix}-\frac{\sin\phi}{1+\cos\phi}\\1\end{pmatrix}
=\begin{pmatrix}-g^-_{\uparrow}\\g^-_{\downarrow}\end{pmatrix}$$ with the normalization constant ${\cal N}=\sqrt{{2}/{1+\cos\phi}}$. These spinors reduce to those for pure spin-up and spin-down quasi-particles and holes when $\phi=0$, corresponding to a uniform magnetization along $z$. One can also easily see that the particular wavefunction of Eq. (\[f2wave\]), $c_1\left(f_{\uparrow}^+,f_{\downarrow}^+,0,0\right)^{T}e^{ik_{\uparrow 2}^+y}$ denotes a quasi-particle with spin parallel to the exchange field in ${\rm F_2}$. When ${\pi}/{2}<\phi\leq\pi$, these eigenspinors read $$\label{f2spinor2}
\begin{pmatrix}f^+_{\uparrow}\\f^+_{\downarrow}\end{pmatrix}=
\frac{1}{\cal N}\begin{pmatrix}\frac{\sin\phi}{1-\cos\phi}\\1\end{pmatrix}
=\begin{pmatrix}-f^-_{\uparrow}\\f^-_{\downarrow}\end{pmatrix};\enspace
\begin{pmatrix}g^+_{\uparrow}\\g^+_{\downarrow}\end{pmatrix}=
\frac{1}{\cal N}\begin{pmatrix}1\\-\frac{1+\cos\phi}{\sin\phi}\end{pmatrix}
=\begin{pmatrix}g^-_{\uparrow}\\-g^-_{\downarrow}\end{pmatrix}$$ with ${\cal N}=\sqrt{{2}/{1-\cos\phi}}$. In this subsection where we are still assuming a non-self-consistent stepwise potential equal to $\Delta_0$ throughout the S region and to zero elsewhere, we have the superconducting coherence factors, $\sqrt{2}u_0=\left[\left(\epsilon+{\sqrt{\epsilon^2-\Delta_0^2}}\right)/{\epsilon}\right]^{1/2}$ and $\sqrt{2}v_0=\left[\left(\epsilon-{\sqrt{\epsilon^2-\Delta_0^2}}\right)/{\epsilon}\right]^{1/2}$. In this case the right-going eigenfunctions on the S side can be written as, $$\Psi_S\equiv %kh
\begin{pmatrix}t_1u_0e^{ik^+y}+t_4v_0e^{-ik^-y}\\
t_2u_0e^{ik^+y}+t_3v_0e^{-ik^-y}\\
t_2v_0e^{ik^+y}+t_3u_0e^{-ik^-y}\\
t_1v_0e^{ik^+y}+t_4u_0e^{-ik^-y}
\end{pmatrix},
\label{transmitted}$$ where, $k^{\pm}=\left[{1\pm\sqrt{{\epsilon}^2-{\Delta}_0^2}-k_\perp^2}\right]^{1/2}$ are quasi-particle (+) and quasi-hole (-) wavevectors in the S region. By using continuity of the four-component wavefunctions and their first derivatives at both interfaces, one can obtain all sixteen unknown coefficients in the above expressions for the wavefunctions by solving a set of linear equations of the form ${\cal M}_{F1}x_{F1,\sigma}={\cal M}_{F2}x_{F2}$ at the ${\rm F_1-F_2}$ interface and $\tilde{{\cal M}}_{F2}x_{F2}={\cal M}_Sx_S$ at the ${\rm F_2-S}$ interface simultaneously, where
\[interface\] $$\begin{aligned}
&x_{F1,\uparrow}^T=\left(1,b_{\uparrow},0,b_{\downarrow},0,a_{\uparrow},0,a_{\downarrow}\right)\\
&x_{F1,\downarrow}^T=\left(0,b_{\uparrow},1,b_{\downarrow},0,a_{\uparrow},0,a_{\downarrow}\right)\\ %ctw4 added the case for spin-down
&x_{F2}^T=\left(c_1,c_2,c_3,c_4,c_5,c_6,c_7,c_8\right)\\
&x_{S}^T=\left(t_1,0,t_2,0,t_3,0,t_4,0\right),\end{aligned}$$
and ${\cal M}_{F1}$, ${\cal M}_{F2}$, $\tilde{{\cal M}}_{F2}$, and ${\cal M}_S$ are appropriate $8\times8$ matrices, which are straightforward to write down. Use of these coefficients gives us all the reflected and transmitted amplitudes $a_\sigma$ and $b_\sigma$ which are used to compute the conductance, as discussed in the next two subsections.
Transfer matrix self consistent method {#transfer}
--------------------------------------
The non-self-consistent step potential assumption is largely unrealistic. Proximity effects lead to a complicated oscillatory behavior of the superconducting order parameter in the F layers and to the generation[@Buzdin2005; @Keizer2006; @Bergeret2007; @Wang2010; @visani; @Hubler2012; @Halterman2007; @hv2p] of triplet pairs as discussed in Sec. \[intro\]. The concomitant depletion of the pair amplitudes near the F-S interface means that unless the superconductor is thick enough, the pair amplitude does not saturate to its bulk value even deep inside the S regions. Furthermore, as we shall emphasize below, lack of self consistency may lead to violation of charge conservation: hence, while non-self-consistent approximations might be sometimes adequate for equilibrium calculations, their use must be eschewed for transport. Therefore, one should generally use a self-consistent pair potential that is allowed to spatially vary, as required by Eq. (\[del\]), and hence results in a minimum in the free energy of the system.
We begin by extending the BTK formalism to the spatially varying self-consistent pair potential obtained as explained below Eq. (\[del\]). Although the self-consistent solutions of the BdG equations reveal that the pair amplitudes are non-zero in the non-superconducting regions due to the proximity effects, the pair potential vanishes in these regions since $g(y)\equiv0$ there. Therefore, one can still use Eqs. (\[f1waveup\]) and (\[f1wavedown\]), with (\[f2wave\]), for the wavefunctions in the ${\rm F_1}$ and ${\rm F_2}$ regions. To deal with the spatially varying pair potential on the S side, we divide it into many very thin layers with microscopic thicknesses of order $k_{FS}^{-1}$. We treat each layer as a very thin superconductor with a constant pair potential, $\Delta_i$, as obtained from the self-consistent procedure. We are then able to write the eigenfunctions of each superconducting layer corresponding to that value of the pair potential. For example, in the $i$-th layer, the eigenfunction should contain all left and right going solutions, and it reads: $$\label{sstate}
\Psi_{Si}\equiv %kh
\begin{pmatrix}
t_{1i}u_ie^{ik_i^+y}+\bar{t}_{1i}u_ie^{-ik_i^+y}
+t_{4i}v_ie^{-ik_i^-y}+\bar{t}_{4i}v_ie^{ik_i^-y}\\
t_{2i}u_ie^{ik_i^+y}+\bar{t}_{2i}u_ie^{-ik_i^+y}
+t_{3i}v_ie^{-ik_i^-y}+\bar{t}_{3i}v_ie^{ik_i^-y}\\
t_{2i}v_ie^{ik_i^+y}+\bar{t}_{2i}v_ie^{-ik_i^+y}
+t_{3i}u_ie^{-ik_i^-y}+\bar{t}_{3i}u_ie^{ik_i^-y}\\
t_{1i}v_ie^{ik_i^+y}+\bar{t}_{1i}v_ie^{-ik_i^+y}
+t_{4i}u_ie^{-ik_i^-y}+\bar{t}_{4i}v_ie^{ik_i^-y}
\end{pmatrix},$$ where, $k_i^{\pm}=\left[1\pm\sqrt{{\epsilon}^2-{\Delta}_i^2}-k_{\perp}^2\right]^{1/2}$, and ${\Delta}_i$ represents the strength of the normalized self consistent pair potential in the $i$-th superconducting layer. The superconducting coherence factors $u_i$ and $v_i$ depend on $\Delta_i$ in the standard way. All the coefficients in Eq. (\[sstate\]) are unknown, and remain to be determined. However, in the outermost S layer (rightmost in our convention) the eigenfunctions are of a form identical to Eq. (\[transmitted\]) but with different locally constant pair potential. We see then that the price one has to pay for including the proximity effects is the need to compute a very large number of coefficients. To do so, we adopt here a transfer matrix method to solve for these unknowns.[@strinati] If one considers the interface between the $i$-th and the $(i+1)$-th layer, we have the linear relation $\tilde{{\cal M}}_ix_i={\cal M}_{i+1}x_{i+1}$, where, for a generic $i$, $$%\begin{align}
x_i^T=\left(t_{1i},\bar{t}_{1i},t_{2i},\bar{t}_{2i},
t_{3i},\bar{t}_{3i},t_{4i},\bar{t}_{4i}\right),
%&x_{i+1}^T=\left(t_{1i+1},\bar{t}_{1i+1},t_{2i+1},\bar{t}_{2i+1},
%t_{3i+1},\bar{t}_{3i+1},t_{4i+1},\bar{t}_{4i+1}\right)
%\end{align}$$ and the matrices, $\tilde{{\cal M}}_i$ and ${\cal M}_{i+1}$, can be written as discussed in connection with Eq. (\[interface\]). The coefficients in the $(i+1)$-th layer can be obtained in terms of those in the $i$-th layer as $x_{i+1}={\cal M}_{i+1}^{-1}\tilde{{\cal M}}_ix_i$. In the same way, for the interface between the $(i-1)$-th layer and the $i$-th layer, we can write $x_i={\cal M}_i^{-1}\tilde{{\cal M}}_{i-1}x_{i-1}$. From the above relations, one can write down the relation between $x_{i+1}$ and $x_{i-1}$, i.e. $x_{i+1}={\cal M}_{i+1}^{-1}\tilde{{\cal M}}_i
{\cal M}_i^{-1}\tilde{{\cal M}}_{i-1}x_{i-1}$. By iteration of this procedure, one can “transfer” the coefficients layer by layer and eventually relate the coefficients of the rightmost layer, $x_n$, to those of the leftmost layer in S and then on to the inner ferromagnetic layer ${\rm F_2}$: $$%kh should we write in terms of multiplication \Pi operator to be more compact?
\label{transferring}
x_n={\cal M}_n^{-1}\tilde{{\cal M}}_{n-1}{\cal M}_{n-1}^{-1}
\cdots\tilde{{\cal M}}_1{\cal M}_1^{-1}
\tilde{{\cal M}}_{F2}x_{F2}$$ By solving Eq. (\[transferring\]) together with ${\cal M}_{F1}x_{F1}={\cal M}_{F2}x_{F2}$, we obtain all the coefficients in the ${\rm F_1}$ region, where the wavefunction is [*formally*]{} still described by the expressions given in Eqs. (\[f1waveup\]) and (\[f1wavedown\]). Of course, all coefficients involved, including the energy dependent $a_\sigma$ and $b_\sigma$ values from which (see below) the conductance is extracted, are quite different from those in a non-self-consistent calculation. These differences will be reflected in our results. One can also prove that, when the pair potential in S is a constant (non-self-consistent), then ${\cal M}_{i+1}=\tilde{\cal M}_i$ and therefore Eq. (\[transferring\]) becomes $x_n=x_1={\cal M}_1^{-1}\tilde{{\cal M}}_{F2}x_{F2}$. This is formally identical to that we have seen in our discussion of the non-self-consistent formalism.
This efficient technique, besides allowing us to determine all the reflected and transmitted amplitudes in the outermost layers, permits us to perform a consistency check by recomputing the self-consistent solutions to the BdG equations (the eigenfunctions). Once we have determined the amplitudes $x_{F1}$, $x_{F2}$, and $x_n$, we can use them to find the amplitudes in any intermediate layer by “transferring” back the solutions. For example, the coefficients $x_{n-1}$ can be found by using $x_n={\cal M}_n^{-1}\tilde{\cal M}_{n-1}x_{n-1}$ if we know the coefficient $x_n$ for the rightmost layer. Knowledge of these coefficients in every region yields again the self-consistent wavefunctions of the system. These of course should be the same as the eigenfunctions found in the original procedure. Although the numerical computations involved in this consistency check are rather intensive, it is worthwhile to perform them: we have verified that, by plugging these solutions into Eq. (\[del\]) and considering all possible solutions with all possible incident angles to the BdG equations, the output pair potential obtained from the transport calculation is the same as the input pair potential obtained by direct diagonalization. This would obviously not have been the case if the initial pair potential had not been fully self consistent to begin with. The reflected and transmitted amplitudes calculated from the self-consistent solutions are in general very different from the non-self-consistent ones and lead to different quantitative behavior of the tunneling conductance, as we shall discuss in section \[results\].
Charge conservation {#conservation}
-------------------
We discuss now the important issue of the charge conservation laws. In transport calculations, it is fundamental to assure that they are not violated [@baym]. From the Heisenberg equation $$\frac{\partial}{\partial t}\left\langle\rho({\mathbf r})\right\rangle
=i\left\langle\left[{\cal H}_{eff},\rho({\mathbf r})\right]\right\rangle.$$ By computing the above commutator, we arrive at the following continuity condition $$\frac{\partial}{\partial t}\left\langle\rho(\mathbf r)\right\rangle
+\nabla\cdot{\mathbf j}=
-4e{\rm Im}\left[\Delta({\mathbf r})\left\langle %ctw4 negative sign
\psi_{\uparrow}^{\dagger}({\mathbf r})
\psi_{\downarrow}^{\dagger}({\mathbf r})\right\rangle\right].
\label{current}$$ In the steady state, which is all that we are considering here, the first term on the left is omitted. Eqn. (\[current\]) is then simply an expression for the divergence of the current. In our quasi one-dimensional system, and in terms of our wavefunctions, the conservation law can be rewritten as: $$\frac{\partial j_y(y)}{\partial y}= 2e {\rm Im}\left\{\Delta(y)\sum_n\left[u_{n \uparrow}^* %ctw2 missing a number 4 %ctw4
v_{n \downarrow}+u_{n\downarrow}^*v_{n\uparrow}\right]\tanh\left(\frac{\epsilon_n}{2T}\right)\right\} %ctw4
\label{currentuv}$$ When the system is in equilibrium the self-consistency condition on the pair potential causes the right hand side of Eqs. (\[current\]) or (\[currentuv\]) to vanish. This would [**not**]{} necessarily be the case if a non-self-consistent[@imaginary] solution were used.[@bagwell] It was shown that charge conservation is only guaranteed when self consistency is adhered to in microscopic Josephson junctions.[@sols2] Current-voltage calculations for N-S heterostructures show that self-consistency is crucial to properly account for all of the Andreev scattering channels arising when the current is constant throughout the system.[@sols] While non-self-consistent solutions are less computationally demanding, their validity when calculating transport quantities in the nonequilibrium regime is always suspect. In the problem we are considering, there exists a finite voltage bias $V$ between the two leads of the system (see Fig. \[figure1\]). This finite bias leads to a non-equilibrium quasi-particle distribution and results of course in a net current. Still, charge conservation must hold. To see how this works in this non-equilibrium case we first write down the net quasi-particle charge density in the $T\rightarrow 0$ limit (the case we consider here) by considering the excited state $|\bf{k_1k_2}\cdots\rangle$ caused by the finite bias $V$. Thus, this excited state contains all single particle states $|\mathbf{k_j}\rangle$ ($j=1,2,\cdots$) with energies less than $eV$. For simplicity, let us first consider the contribution by a single-particle state. We use $|\mathbf{k}\rangle$ to characterize this single particle state with an incident wavevector $\mathbf{k}=\mathbf{k_{\perp}}+k\hat{\mathbf{y}}$ and energy $\epsilon_\mathbf{k}$. The charge density associated with it is written as $$\begin{aligned}
\rho&=-e\sum_{\sigma}\left\langle \mathbf{k} \left| \psi_{\sigma}^{\dagger}\psi_{\sigma} \right| \mathbf{k} \right\rangle \\\nonumber
&=-e\sum_{n\sigma}\left(|u_{n\sigma}|^2
\left\langle \mathbf{k} \left| \gamma_n^{\dagger}\gamma_n \right| \mathbf{k} \right\rangle
+|v_{n\sigma}|^2\left\langle \mathbf{k} \left| \gamma_n\gamma_n^{\dagger} \right| \mathbf{k} \right\rangle\right)\\\nonumber
&=-e\sum_{n\sigma}\left(|u_{n\sigma}|^2
\left\langle \mathbf{k} \left| \gamma_n^{\dagger}\gamma_n \right| \mathbf{k} \right\rangle
+|v_{n\sigma}|^2\left\langle \mathbf{k} \left| 1-\gamma_n^{\dagger}\gamma_n \right| \mathbf{k} \right\rangle\right)\\\nonumber
&=-e\sum_{n\sigma}|v_{n\sigma}|^2
-e\sum_{n\sigma}\left(|u_{n\sigma}|^2-|v_{n\sigma}|^2\right)\delta_{n\mathbf{k}}\\\nonumber
&=-e\sum_{n\sigma}|v_{n\sigma}|^2
-e\sum_{\sigma}\left(|u_{\mathbf{k}\sigma}|^2-|v_{\mathbf{k}\sigma}|^2\right)\end{aligned}$$ The first term represents the ground state charge density. For a generic excited state, $|\bf{k_1k_2}\cdots\rangle$ , that can contain many single-particle states, one need to sum over all single-particle states for the charge density such that $$\rho=-e\sum_{n\sigma}|v_{n\sigma}|^2
-e\sum_{\epsilon_\mathbf{k}<eV}\sum_{\sigma}\left(|u_{\mathbf{k}\sigma}|^2-|v_{\mathbf{k}\sigma}|^2\right).$$ The quasi-particle current density from this generic excited state can also be computed, $$\begin{aligned}
\label{current2}
j_y&=-\frac{e}{2m}\sum_{\epsilon_\mathbf{k}<eV}\sum_{\sigma} %ctw4
\left\langle-i\psi_{\sigma}^{\dagger}\frac{\partial}{\partial y}\psi_\sigma
+i\left(\frac{\partial}{\partial y}\psi_{\sigma}^{\dagger}\right)\psi_{\sigma}\right\rangle_{\mathbf{k}}\\\nonumber
&=-\frac{e}{m}{\rm Im}\left[\sum_{n\sigma}v_{n\sigma}\frac{\partial v_{n\sigma}^{\ast}}{\partial y}
+\sum_{\epsilon_\mathbf{k}<eV}\sum_{\sigma}\left(u_{\mathbf{k}\sigma}^{\ast}\frac{\partial u_{\mathbf{k}\sigma}}{\partial y}
+v_{\mathbf{k}\sigma}^{\ast}\frac{\partial v_{\mathbf{k}\sigma}}{\partial y}\right)\right]\\\nonumber
&=-\frac{e}{m}{\rm Im}\left[
\sum_{\epsilon_\mathbf{k}<eV}\sum_{\sigma}\left(u_{\mathbf{k}\sigma}^{\ast}\frac{\partial u_{\mathbf{k}\sigma}}{\partial y}
+v_{\mathbf{k}\sigma}^{\ast}\frac{\partial v_{\mathbf{k}\sigma}}{\partial y}\right)\right],\end{aligned}$$ where $\langle...\rangle_\mathbf{k}$ is a shorthand notation of $\langle\mathbf{k}\left|...\right|\mathbf{k}\rangle$. The first term in the second line vanishes because it represents the net current for the system in the ground state with a real pair potential. The right hand side of the continuity equation, Eq. (\[currentuv\]), becomes $-4e{\rm Im}\left[\Delta\sum_{\epsilon_\mathbf{k}<eV}\left(u_{\mathbf{k}\uparrow}^{\ast}
v_{\mathbf{k}\downarrow}+v_{\mathbf{k}\uparrow}u_{\mathbf{k}\downarrow}^{\ast}\right)\right]$ and is responsible for the interchange between the quasi-particle current density and the supercurrent density[@btk]. We have numerically verified that by properly including these terms, all of our numerical results for the current density are constant throughout the whole system.
Extraction of the conductance {#extraction}
-----------------------------
We are now in a position to compute the differential tunneling conductances. We begin by discussing the extraction of the conductance from the BTK theory. As we mentioned in the previous subsection, the finite bias $V$ and the resulting non-equilibrium distribution leads to an electric current flowing in the junction. In the BTK theory, this current can be evaluated from the following[@btk] expression, $$\label{totalcurrent}
I(V)=\int G(\epsilon)\left[f\left(\epsilon-eV\right)
-f\left(\epsilon\right)\right]d\epsilon,$$ where $f$ is the Fermi function. The energy dependent tunneling conductance, $G(\epsilon)=\partial I/{\partial V}|_{V=\epsilon}$ in the low-$T$ limit, is given as: $$\begin{aligned}
\label{conductance}
&G(\epsilon,\theta_i)=\sum_\sigma P_\sigma G_{\sigma }(\epsilon,\theta_i) %wv+ %wv one deleted
%G_{\downarrow }({\epsilon},\theta_i)
\\\nonumber
&=\sum_{\sigma}P_{\sigma}\left(1+\frac{k^-_{\uparrow 1}}{k^+_{\sigma 1}}|a_{\uparrow }|^2
+\frac{k^-_{\downarrow 1}}{k^+_{\sigma 1}}|a_{\downarrow }|^2
-\frac{k^+_{\uparrow 1}}{k^+_{\sigma 1}}|b_{\uparrow }|^2
-\frac{k^+_{\downarrow 1}}{k^+_{\sigma 1}}|b_{\downarrow }|^2\right), %wv \end{aligned}$$ where we have used, as is customary, natural units of conductance $(e^2/h)$. In the above expression the different $k$ components are as explained in subsection \[BTK\] (see e.g. Eq. (\[wavevector\])) and the $a_\sigma$ and $b_\sigma$ are as defined in Eqns. (\[f1waveup\]) and (\[f1wavedown\]). These coefficients, which are of course energy dependent, are calculated using the self-consistent transfer matrix technique of subsection \[transfer\]. Therefore, even though Eq. (\[conductance\]) is formally the same in the self-consistent and non-self-consistent cases, the results for the reflection amplitudes or probabilities involved, $|a_{\uparrow}|^2$, $|a_{\downarrow}|^2$, $|b_{\uparrow}|^2$, and $|b_{\downarrow}|^2$ are different in these two schemes. The angle $\theta_i$ is the incident angle, discussed in terms of ${\bf k}$ components below Eq. (\[wavevector\]). The weight factor $P_{\sigma}\equiv\left(1-h_1\eta_{\sigma}\right)/2$ accounts for the number of available states for spin-up and spin-down bands in the outer electrode. The tunneling conductance can also be interpreted as the transmission coefficient for electrical current. The method enables us also to compute the current density directly from the wavefunctions, Eqs. (\[f1waveup\]) and (\[f1wavedown\]), in the ${\rm F_1}$ layer by using Eq. (\[current2\]) and we have been able to verify that the resulting current density is identical to the terms inside the bracket in the expression of $G(\epsilon)$, Eq. (\[conductance\]). In other words, in the low-$T$ limit the continuum-limit version of Eq. (\[current2\]) is equivalent to Eq. (\[totalcurrent\]). The conductance results Eq. (\[conductance\]) also depend on the incident angle of electrons, $\theta_i$. Experimentally, one can measure the forward conductance, $\theta_i=0$, via point contacts or, in most other experimental conditions, an angular average. Consequently, it is worthwhile to compute the angularly averaged conductance by using the following definitions, $$\label{aaG}
\langle G_{\sigma}({\epsilon})
\rangle=\frac{\int_0^{\theta_{c\sigma}} d\theta_i \cos\theta_i G_\sigma(\epsilon,\theta_i)}
{\int_0^{\theta_{c\sigma}} d\theta_i \cos\theta_i},$$ and $$\langle G \rangle=\sum_{\sigma} P_{\sigma}\langle G_{\sigma}\rangle,$$ where the critical angle $\theta_{c\sigma}$ is in general different for spin-up and spin-down bands. This critical angle arises from the conservation of transverse momentum and the corresponding Snell law: $$\label{snell}
\begin{aligned}
&\sqrt{\left({k_{\sigma 1}^+}^2+k_{\perp}^2\right)}\sin\theta_i
=\sqrt{\left({k_{\sigma' 1}^+}^2+k_{\perp}^2\right)}\sin\theta_{r\sigma'}^+\\
&=\sqrt{\left({k_{\sigma' 1}^-}^2+k_{\perp}^2\right)}\sin\theta_{r\sigma'}^-
=\sin\theta_S,
\end{aligned}$$ where we continue to measure wavevectors in units of $k_{FS}$. The angles $\theta_{r\sigma}^\pm$ satisfy $\tan^{-1}\left({k_{\perp}}/{k_{\sigma 1}^{\pm}}\right)$, and the $\sigma$ and $\sigma'$ are each $\uparrow$ or $\downarrow$. The last equality in Eq. (\[snell\]) represents the case of the transmitted wave in S, and $\theta_S$ is the transmitted angle. Although the self-consistent pair potential varies in S and so do the quasi-particle (hole) wavevectors, we here need only consider the transmitted angle $\theta_S$ in the rightmost layer: this follows in the same way as the usual Snell’s law in a layered system, as given in elementary textbooks. From Eq. (\[snell\]), one can determine the critical angles for different channels. Consider, e.g., a spin-up electron incident from ${\rm F_1}$ without any Fermi wavevector mismatch, i.e. $\Lambda=1$. Since we are only concerned with the case that the bias of tunneling junctions is of the order of superconducting gap and therefore much smaller than the Fermi energy, the approximate magnitude of the incident wavevector is $\sqrt{1+{h}_1}$, the Andreev approximation. We substitute this and similar expressions into Eq. (\[snell\]) and, with the help of Eq. (\[wavevector\]), we obtain $$\label{andreevr}
\sqrt{1+{h}_1}\sin\theta_i=\sqrt{1-{h}_1}\sin\theta_{r\downarrow}^-
=\sin\theta_S.$$ One can straightforwardly verify that, when the relation $\theta_i>\sin^{-1}[((1-{h}_1)/(1+{h}_1))^{1/2}]$ is satisfied for the incident angle, the conventional Andreev reflection becomes an evanescent wave[@zv2]. In this case, the conventional Andreev reflection does not contribute to the angular averaging. On the other hand, if the energy $\epsilon$ of the incident electron is less than the saturated value of the superconducting pair amplitude in S, all the contribution to the conductance from the transmitted waves in S also vanishes because $k^{\pm}$ acquires an imaginary part. However, even the condition that $\epsilon$ is greater than the saturated superconducting amplitude does not guarantee that the contribution from the transmitted waves to the conductance is nonvanishing. One still needs to consider the transmitted critical angle $\sin^{-1}[1/(1+h_1)^{1/2}]$. We define the critical angle $\theta_{c\sigma}$ to be the largest one among all the reflected and transmitted critical angles. It is obvious that the critical angles $\theta_{c\sigma}$ are different for spin-up and spin-down bands when $h_1\neq0$.
Spin transport {#spintran}
--------------
We consider now the spin-transfer torque and the spin current. As the charge carriers that flow through our system, along the $y$ direction in our convention, are spin polarized, the STT provides an additional probe of the spin degree of freedom. Unlike the charge current, that must be a constant throughout the system, the spin current density is generally not a conserved quantity in the ferromagnet regions as we will demonstrate below. The discussion in Sec. \[conservation\] on how the BTK formalism deals with the charge current can be extended to compute these spin dependent transport quantities. We need here the continuity equation for the local magnetization $\mathbf{m}\equiv-\mu_B\sum_{\sigma}\left\langle\psi^{\dagger}_{\sigma} %ctw4: typo
\bm{\sigma}\psi_{\sigma}\right\rangle$, where $\mu_B$ is the Bohr magneton. By using the Heisenberg equation $\frac{\partial}{\partial t}\left\langle\mathbf{m}({\mathbf r})\right\rangle
=i\left\langle\left[{\cal H}_{eff},\mathbf{m}({\mathbf r})\right]\right\rangle$ we obtain the relation: $$\label{spinconserve}
\frac{\partial}{\partial t}\langle m_i \rangle+ \frac{\partial}{\partial y} S_i= \tau_i,
\enspace\enspace i=x,y,z %ctw$$ where $\bm{\tau}$ is the spin-transfer torque, $\bm{\tau}\equiv 2\mathbf{m}\times\mathbf{h}$, and the spin current density $S_i$ is given by $$S_i\equiv\frac{i\mu_B}{2m}\sum_\sigma\left\langle \psi_\sigma^{\dagger}\sigma_i\frac{\partial \psi_\sigma}{\partial y}
-\frac{\partial \psi_\sigma^{\dagger}}{\partial y}\sigma_i\psi_\sigma\right\rangle.$$ The spin current density reduces from a tensor form to a vector because of the quasi-one-dimensional nature of our geometry. From Eq. (\[spinconserve\]), we can see that $\mathbf{S}$ is a local physical quantity and $\bm{\tau}$ is responsible for the change of local magnetizations due to the flow of spin-polarized currents. As we shall see in Sec. \[results\], the conservation law (with the source torque term) for the spin density is fundamental and one has to check it is not violated when studying these transport quantities. In the low-$T$ limit and with the presence of a finite bias, the non-equilibrium local magnetizations $m_i\equiv\sum_{\epsilon_\mathbf{k}<eV}\sum_{\sigma}-\mu_B\langle\psi_\sigma^\dagger\sigma_i\psi_\sigma\rangle_\mathbf{k}$ in Eq. (\[spinconserve\]) reads
\[mag\] $$\begin{aligned}
m_x=&-\mu_B\left[\sum_n\left(-v_{n\uparrow}v_{n\downarrow}^{\ast}-v_{n\downarrow}v_{n\uparrow}^{\ast}\right)\right.\\\nonumber
&\left.+\sum_{\epsilon_\mathbf{k}<eV}\left(u_{\mathbf{k}\uparrow}^{\ast}u_{\mathbf{k}\downarrow}
+v_{\mathbf{k}\uparrow}v_{\mathbf{k}\downarrow}^{\ast}
+u_{\mathbf{k}\downarrow}^{\ast}u_{\mathbf{k}\uparrow}
+v_{\mathbf{k}\downarrow}v_{\mathbf{k}\uparrow}^{\ast}\right)\right]\\
m_y=&-\mu_B\left[i\sum_n\left(v_{n\uparrow}v_{n\downarrow}^{\ast}-v_{n\downarrow}v_{n\uparrow}^{\ast}\right)\right.\\\nonumber %ctw4 : '-' sign missing
&\left.-i\sum_{\epsilon_\mathbf{k}<eV}\left(u_{\mathbf{k}\uparrow}^{\ast}u_{\mathbf{k}\downarrow}
+v_{\mathbf{k}\uparrow}v_{\mathbf{k}\downarrow}^{\ast}
-u_{\mathbf{k}\downarrow}^{\ast}u_{\mathbf{k}\uparrow}
-v_{\mathbf{k}\downarrow}v_{\mathbf{k}\uparrow}^{\ast}\right)\right]\\
m_z=&-\mu_B\left[\sum_n\left(|v_{n\uparrow}|^2-|v_{n\downarrow}|^2\right)\right.\\\nonumber
&\left.+\sum_{\epsilon_\mathbf{k}<eV}\left(|u_{\mathbf{k}\uparrow}|^2
-|v_{\mathbf{k}\uparrow}|^2
-|u_{\mathbf{k}\downarrow}|^2
+|v_{\mathbf{k}\downarrow}|^2\right)\right],\end{aligned}$$
where the first summations in the expressions for $m_i$ denote the ground state local magnetizations. The second summations appear as a consequence of the finite bias between electrodes. The expressions for the corresponding spin currents, $$S_i\equiv\frac{i\mu_B}{2m}\sum_{\epsilon_\mathbf{k}<eV}\sum_{\sigma}\left\langle \psi_\sigma^{\dagger}\sigma_i\frac{\partial \psi_\sigma}{\partial y}
-\frac{\partial \psi_\sigma^{\dagger}}{\partial y}\sigma_i\psi_\sigma\right\rangle_\mathbf{k},$$ becomes
\[spincur\] $$\begin{aligned}
S_x=&\frac{-\mu_B}{m}{\rm Im}\left[\sum_n\left
(-v_{n\uparrow}\frac{\partial v_{n\downarrow}^{\ast}}{\partial y}
-v_{n\downarrow}\frac{\partial v_{n\uparrow}^{\ast}}{\partial y}\right)\right.\\\nonumber
&\left.+\sum_{\epsilon_\mathbf{k}<eV}\left(u_{\mathbf{k}\uparrow}^{\ast}\frac{\partial u_{\mathbf{k}\downarrow}}{\partial y}
+v_{\mathbf{k}\uparrow}\frac{\partial v_{\mathbf{k}\downarrow}^{\ast}}{\partial y}
+u_{\mathbf{k}\downarrow}^{\ast}\frac{\partial u_{\mathbf{k}\uparrow}}{\partial y}
+v_{\mathbf{k}\downarrow}\frac{\partial v_{\mathbf{k}\uparrow}^{\ast}}{\partial y}\right)\right]\\
S_y=&\frac{\mu_B}{m}{\rm Re}\left[\sum_n\left(-v_{n\uparrow}
\frac{\partial v_{n\downarrow}^{\ast}}{\partial y}
+v_{n\downarrow}\frac{\partial v_{n\uparrow}^{\ast}}{\partial y}\right)\right.\\\nonumber
&\left.+\sum_{\epsilon_\mathbf{k}<eV}\left(u_{\mathbf{k}\uparrow}^{\ast}\frac{\partial u_{\mathbf{k}\downarrow}}{\partial y}
+v_{\mathbf{k}\uparrow}\frac{\partial v_{\mathbf{k}\downarrow}^{\ast}}{\partial y}
-u_{\mathbf{k}\downarrow}^{\ast}\frac{\partial u_{\mathbf{k}\uparrow}}{\partial y}
-v_{\mathbf{k}\downarrow}\frac{\partial v_{\mathbf{k}\uparrow}^{\ast}}{\partial y}\right)\right]\\
S_z=&\frac{-\mu_B}{m}{\rm Im}\left[\sum_n\left(v_{n\uparrow}\frac{\partial v_{n\uparrow}^{\ast}}{\partial y}
-v_{n\downarrow}\frac{\partial v_{n\downarrow}^{\ast}}{\partial y}\right)\right.\\\nonumber
&\left.+\sum_{\epsilon_\mathbf{k}<eV}\left(u_{\mathbf{k}\uparrow}^{\ast}\frac{\partial u_{\mathbf{k}\uparrow}}{\partial y}
-v_{\mathbf{k}\uparrow}\frac{\partial v_{\mathbf{k}\uparrow}^{\ast}}{\partial y}
-u_{\mathbf{k}\downarrow}^{\ast}\frac{\partial u_{\mathbf{k}\downarrow}}{\partial y}
+v_{\mathbf{k}\downarrow}\frac{\partial
v_{\mathbf{k}\downarrow}^{\ast}}{\partial y}\right)\right]. %ctw\end{aligned}$$
The first summations in Eq. (\[spincur\]) represent the static spin current densities when there is no bias. The static spin current does not need to vanish, since a static spin-transfer torque may exist near the boundary of two magnets with misaligned exchange fields. The finite bias leads to a non-equilibrium quasi-particle distribution for the system and results in non-static spin current densities that are represented by the second summation in Eq. \[spincur\]. Obviously, the spin-transfer torque has to vanish in the superconductor where the exchange field is zero. It is conventional to normalize $\mathbf{m}$ to[@Halterman2007] $-\mu_B(N_\uparrow+N_\downarrow)$, where the number densities $N_\uparrow=k_{FS}^3(1+h_m)^{3/2}/(6\pi^2)$ and $N_\downarrow=k_{FS}^3(1-h_m)^{3/2}/(6\pi^2)$. Following this convention, we normalize $\bm{\tau}$ to $-\mu_B(N_\uparrow+N_\downarrow)E_{FS}$ and $\mathbf{S}$ to $-\mu_B(N_\uparrow+N_\downarrow)E_{FS}/k_{FS}$.
Results
=======
![(Color online) Bias dependence of the results for the forward conductance, $G$, in thick F-S bilayers (see text). The values of $h$ are indicated. In both main panels the solid and dashed curves show $G$, in units of $e^2/h$ for non-self-consistent and self-consistent results, respectively. The bias $E$ is in units of the S bulk gap $\Delta_0$. In the top panel the (red) lower curves are for a mismatch parameter $\Lambda=0.25$, (green) the middle curves for $\Lambda=0.5$, and the (blue) higher curves for $\Lambda=1$. In the bottom panel, the (purple) top curves are for $\Lambda=1.41$, the (blue) curves are as in the top panel, and the (black) lower ones for $\Lambda=0.71$. The inset (see text) shows $G(E=1)$ vs. $\Lambda$ in the self consistent calculation (dots) and the non-self-consistent result (line). []{data-label="figure2"}](newfig2.eps){width="45.00000%"}
The forward scattering conductances $G$ are computed by considering a particle incident with an angle $\theta_i\cong0$ (normal incidence). Angular averaging has been discussed in the text above Eq. (\[aaG\]). The bias energy $E\equiv eV$ is in units of the zero temperature gap, $\Delta_0$, in bulk S material and $e^2/h$ is used as the natural unit of conductance. When the ${\rm F_1}$ and ${\rm F_2}$ regions are made of same F material, i.e., $h_1=h_2$ and $k_{\uparrow1,(\downarrow1)}^{\pm}=k_{\uparrow2,(\downarrow)2}^{\pm}$, we will use $h$ (not to be confused with Planck’s constant) and $k_{\uparrow,(\downarrow)}^{\pm}$ to denote their exchange fields and wavevectors. This is the case we will mostly study. All results are for the low-$T$ limit. All of the lengths are measured in unit of $k_{FS}^{-1}$ and denoted by capital letters, e.g. $D_S$ denotes $k_{FS} d_S$.
Bilayers
--------
We begin with a brief discussion of self-consistent results for the tunneling conductance in F-S bilayers, contrasting them with non-self-consistent results. We assume that the S layer is very thick so that the pair amplitude saturates to its bulk value deep inside the S region. In this subsection, the dimensionless superconducting coherence length $\Xi_0$ is taken to be $50$ and the thicknesses $D_F$ and $D_S$ of the F and S layers are both $15\Xi_0$. By computing the pair amplitudes via the direct diagonalization method,[@hv2p] we have verified that they indeed saturate to their bulk value with this large ratio of $D_S$ to $\Xi_0$.
As discussed in Sec. \[intro\], the replacement of non-magnetic metals with ferromagnets in a bilayer leads to strong suppression of the Andreev reflection in the subgap region. The decrease of the zero bias conductance (ZBC) strongly depends on the magnitude of the exchange field in F. This dependence is used to measure the degree of spin-polarization of magnetic materials experimentally.[@soulen; @upad] However, in early theoretical work,[@zv1; @zv2] it was shown that to accurately determine the degree of spin-polarization, one has to consider the Fermi wavevector mismatch (FWM), $\Lambda$, as well as the interfacial barriers. The ZBC peak is very sensitive to both spin-polarization and FWM and the dependence cannot be characterized by a single parameter.
We display in Fig. \[figure2\] forward conductance vs. bias results for both the self-consistent and non-self-consistent calculations, at two different values of the exchange fields and several FWM values. One sees at once that the self-consistent results approach the non-self-consistent ones in the zero bias limit, while deviating the most for energies near the superconducting gap. The ZBC decreases with increasing $h$ and with decreasing $\Lambda$. Also, larger $h$ indeed leads to a conspicuous reduction in the subgap conductance and so does the introduction of FWM. One can conclude that the behavior of the ZBC can not be characterized by only one parameter, either $h$ or $\Lambda$. Instead, one should expand the fitting parameter space to determine the degree of spin polarization.
In the non-self-consistent framework, the conductance at the superconducting gap ($E=1$ in our units) is independent of $\Lambda$ at a given $h$. However, earlier work[@bv] predicted that this conclusion is invalid in self-consistent approach, and that the conductance at the superconducting gap varies monotonically with increasing $\Lambda$. Here we verify this via our self-consistent transfer matrix method. The inset in the bottom panel of Fig. \[figure2\] clearly shows this dependence on $\Lambda$. Figure \[figure2\] also shows that the self-consistent results (dashed curves) on subgap conductances are in general lower than those obtained in the non-self-consistent framework (solid curves) for a strong exchange field. On the other hand, in the high bias limit, the self-consistent results become similar to the non-self-consistent ones. This is simply because the particle does not experience much of a difference between a step-like pair potential and a smooth pair potential when it is incident with high enough energy. Finally, clear cusps appear at the superconducting gap value in some cases, e.g., the forward scattering conductance curve at $h=0.866$ and $\Lambda=1$. This is consistent with what is found in previous work[@bv] for thick bilayers.
![(Color online) Comparison between the self-consistent and non-self-consistent forward scattering conductances of ${\rm F_1F_2S}$ trilayers. The solid and the dashed lines are for non-self-consistent and self-consistent results respectively. The (red) curves, highest at the critical bias (CB) are for $\phi=0^{\circ}$. The (blue) curves, lowest at CB, are for $\phi=180^{\circ}$. We have $D_{F1}=10$, $D_{F2}=12$, and $D_S=180$ (see text).[]{data-label="figure3"}](fig3a.eps){width="45.00000%"}
Trilayers
---------
We now discuss our results for ${\rm F_1F_2S}$ trilayers of finite widths. First, we discuss the dependence of the tunneling conductances on the angle $\phi$ between ${\mathbf h_1}$ and ${\mathbf h_2}$ (see below Eq. (\[bogo\]) and Fig. \[figure1\]). An important reason for considering trilayers with finite widths is the strong dependence of the superconducting transition temperatures $T_c$ on the angle $\phi$ due to proximity effects[@cko] and induced long-range triplet correlations.[@gol] Field induced switching effects[@oh] also make these structures attractive candidates for memory elements. The non-monotonic behavior of $T_c(\phi)$ with its minimum being near $\phi=90^{\circ}$, was extensively discussed in Ref. . This angular dependence has been shown to be related to the induced triplet pairing correlations[@alejandro]. The superconducting transition temperatures are also predicted to be positively correlated with the singlet pair amplitudes deep inside the S regions[@cko]. Therefore, it is of particular importance to consider systems of finite size to take into view the whole picture of proximity effects on the angular dependence of the tunneling conductance. For the results shown in this subsection, we assume the absence of FWM ($\Lambda=1$).
{width="90.00000%"}
### Forward Scattering
As a typical example of our results, we show in Fig. \[figure3\] results for the $\phi$ dependence of the forward scattering conductances. The exchange field we use here for both F layers is $h=0.3$, and the thicknesses of the F$_1$ and F$_2$ layers correspond to $D_{F1}=10$ and $D_{F2}=12$ respectively, while the S layer has width $D_S=180=1.5 \Xi_0$. Results obtained via the non-self-consistent approach are plotted for comparison. In the non-self-consistent framework where the single parameter $\Delta_0$ describes the stepwise pair potential, one sees in Fig. \[figure3\] that for all values of the angle $\phi$ the conductance curves drop when the bias is at $\Delta_0$, corresponding to $E=1$ in our units. In contrast, for the self-consistent results, one can clearly see in Fig. \[figure3\], that the drop in the conductance curves occurs at different bias values for different angles. We also see that this critical bias (which we will denote by CB) depends on $\phi$ non-monotonically, with $\phi=180^{\circ}$ corresponding to the largest and $\phi=90^{\circ}$ to the smallest bias values. Since the CB depends on the strength of the superconducting gap deep inside the S regions, the non-monotonicity of the CB in Fig. \[figure3\] is correlated with the non-monotonicity of $T_c$. The CB never reaches unity, in these trilayers, due to their finite size. Accordingly, this feature of the correct self-consistent results implies that one cannot adequately determine the angular dependence of the forward conductance in the non-self-consistent framework. This feature also provides experimentalists with another way to measure the strength of the superconducting gap for different angles in these trilayers by determining the CB in a set of conductance curves. The remaining results shown in this section are all computed self-consistently.
In Fig. \[figure45\], we present more results for the dependence of the forward scattering conductances on $\phi$. In the top panels the thicknesses of each layer and the coherence length are the same as Fig. \[figure3\]. In the bottom panels we increase the thickness of the inner magnetic layer to $D_{F2}=18$ while $D_{F1}$, $D_{S}$, and $\Xi_0$ remain unchanged. For each row of Fig. \[figure45\], results for three different exchange fields are plotted. In the top left panel ($h=0.5$) we see that the angular dependence of the CB (or the magnitude of the saturated pair amplitudes) is monotonic with $\phi$. Although this monotonicity is not common, we have verified that it is consistent with the theoretical results for $T_c(\phi)$ for the same particular case. The more usual non-monotonic dependence is found in all other panels, as discussed in the previous paragraph. In every case, we have also checked that the magnitude of the CB reflects the magnitude of the self-consistent pair amplitudes deep inside the superconductor.
For the ZBC, we see that the degree of its angular dependence is very sensitive to $h$. In the top left panel, with $h=0.5$, the ZBC is nearly independent on $\phi$. On the other hand, the ZBC in the top right panel, $h=0.6$, drops by almost a factor of two as $\phi$ varies from the relative parallel (P) orientation, $\phi=0^{\circ}$, to the antiparallel (AP) orientation, $\phi=180^{\circ}$. This is a consequence of interference between the spin-up and spin-down wavefunctions under the influence of the rotated exchange field in the middle layer. In the top left panel, we see that the conductance at CB decreases with increasing angle. In other words, the zero bias conductance peak (ZBCP) becomes more prominent as $\phi$ is increased. However, for the top middle panel, $h=0.45$, the development of the ZBCP is less noticeable when the angle $\phi$ is increased. In the top right panel, $h=0.6$, the ZBCP evolves into a zero bias conductance dip (ZBCD) as $\phi$ varies from $\phi=0^{\circ}$ to $\phi=180^{\circ}$, with a clear finite bias conductance peak (FBCP) appearing just below the CB. This behavior is reminiscent[@zv2] of that which occurs when a barrier, or mismatch, are present. In the bottom panels of this figure, corresponding to a larger value of $D_{F2}$ one can observe similar features. For example, a slight change from $h=0.35$ to $h=0.4$ causes by itself a very large change in the behavior of the ZBC. Moreover, the evolution of the ZBCP to a ZBCD accompanies the occurrence of a FBCP when $\phi>90^{\circ}$. The location of the FBCP also moves closer to the CB value when $\phi$ increases. That these features of the ZBC depend on both the strength of exchange field (reflected in $k_{\uparrow}^{\pm}$ and $k_{\downarrow}^{\pm}$) and the thickness of the ${\rm F_2}$ layer indicates that the ZBC shows the characteristics of a resonance scattering phenomenon as in an elementary quantum mechanical barrier. The main difference is that the scattering problem here involves the intricate interference between quasi-particle and quasi-hole spinors.
![(Color online) Resonance effects in the forward scattering conductance at zero bias for trilayers at $\phi=180^{\circ}$. In the top panel, the trilayers have same thicknesses as in the top panels of Fig. \[figure45\], and in the bottom panel, they are as in the bottom panels of Fig. \[figure45\].The (blue) dots are the results from our computations and the (red) curves from Eq. (\[resonance\]).[]{data-label="figure6"}](fig5f.eps){width="45.00000%"}
When the bias is high enough, the tunneling conductance approaches its normal state value. Thus, one can extract the magnetoresistance from the conductance at $E=2$. We only discuss here the magnetoresistance’s qualitative behavior. One can define a measure of the magnetoresistance as, $$\begin{aligned}
%M_R(E,\phi)\equiv[G(E,\phi=0^{\circ})-G(E,\phi)]/
%{G(E,\phi=0^{\circ})}.
M_G(E,\phi)\equiv\frac{G(E,\phi=0^{\circ})-G(E,\phi)}{ %wv
{G(E,\phi=0^{\circ})}}.\end{aligned}$$ For all results shown in the panels of Fig. \[figure45\], the conductance at $E=2$ decreases with increasing $\phi$, i.e., it is a monotonic function of $\phi$, the standard behavior for conventional, non-superconducting, spin-valves. Furthermore, one can also see that $M_G(E=2,\phi=180^{\circ})$ increases with exchange field. Therefore, the behavior of the magnetoresistance at large bias is as one would expect in the present self-consistent BTK framework. However, the behavior of $M_G(E=0,\phi=180^\circ)$ that is associated with the behavior of the ZBC is generally a non-monotonic function of $h$. We next investigate the high sensitivity of the ZBC to $h$ by examining its resonances for two different F widths arranged in an AP magnetic configuration ($\phi=180^{\circ}$). To do so, we performed an analytic calculation of the ZBC in the non-self-consistent framework in situations where (as discussed in connection with Fig. \[figure3\]) the results nearly coincide with those of self-consistent calculations. We find that the ZBC at $\phi=180^{\circ}$, $G(E=0,\phi=180^\circ)\equiv G_{ZB}$, for a given $h$ and $D_{F2}$ is: $$\label{resonance}
G_{ZB}=\frac{32 k_{\uparrow}^3k_{\downarrow}^3}{A+2\left(h^4-2h^2-2h^2k_{\uparrow}k_{\downarrow}\right)\cos\left[2\left(k_\uparrow-k_\downarrow\right)D_{F2}\right]}.
%&\frac{32 k_{\uparrow}^3k_{\downarrow}^3}{A[1-\cos(2(k_{\uparrow}+k_{\downarrow})D_{F2})]
%+B\cos(2(k_{\uparrow}-k_{\downarrow})D_{F2})+C[\cos(2k_{\uparrow}D_{F2})-\cos(2k_{\downarrow}D_{F2})]+D}$$ The expression for $A$ in Eq. (\[resonance\]) is: $$a_1\sin^2\left[\left(k_{\uparrow}+k_{\downarrow}\right)D_{F2}\right]
+a_2\left[\cos\left(2k_{\uparrow}D_{F2}\right)-\cos\left(2k_{\downarrow}D_{F2}\right)\right]+a_3,$$ where $a_1=4h^2(1-k_{\uparrow}k_{\downarrow})^2$, $a_2=4h^3$, and $a_3=h^4+(-2+h^2-2k_{\uparrow}k_{\downarrow})^2$. Here we have omitted the $\pm$ indices for the quasi-particle and quasi-hole wavevectors, since we are in the zero bias limit. In Fig. \[figure6\], we plot Eq. (\[resonance\]) as a function of $h$ for $D_{F2}=12$ (top panel) and $18$ (bottom panel). In this zero bias limit, the (blue) circles (self-consistent numerical results) are on top of the (red) curves (analytic results). As the thickness of the intermediate layer increases, the number of resonance maxima and minima increases. Therefore, the resonance behavior of the ZBC is more sensitive to $h$ for larger $D_{F2}$, as we have seen in Fig. \[figure45\]. For a given $D_{F2}$, the ZBC drops considerably as $\phi$ varies from $\phi=0^{\circ}$ to $\phi=180^{\circ}$ when $h$ is near the minimum of the resonance curve (rightmost panels of Fig. \[figure45\]). On the other hand, when $h$ is near the resonance maximum (leftmost panels of Fig. \[figure45\]), the ZBC is a very weak function of $\phi$ provided that $h$ is not too strong. By examining the denominator of Eq. (\[resonance\]), we find that the terms involved in $A$ are less important than the last term. This is because the wavelength $(k_{\uparrow}-k_{\downarrow})^{-1}$ associated with that term is the dominant characteristic wavelength in the theory of proximity effects in F-S structures.[@demler; @Halterman2002] In both panels of Fig. \[figure6\], we see that the ZBC for $\phi=180^{\circ}$ vanishes in the half-metallic limit. To show this analytically, one can use the conservation of probability currents and write down the relation, valid when the bias is smaller than the superconducting gap: $$\label{probabilitylaw} %wv fixed labels
\frac{k^-_{\uparrow 1}}{k^+_{\sigma 1}}|a_{\uparrow }|^2
+\frac{k^-_{\downarrow 1}}{k^+_{\sigma 1}}|a_{\downarrow }|^2
+\frac{k^+_{\uparrow 1}}{k^+_{\sigma 1}}|b_{\uparrow }|^2
+\frac{k^+_{\downarrow 1}}{k^+_{\sigma 1}}|b_{\downarrow }|^2=1. %ctw3 typo
%wv \enspace\enspace \sigma=\uparrow,\downarrow.$$ By combining Eq. (\[probabilitylaw\]) with Eq. (\[conductance\]), it becomes clear that the subgap conductances arise largely from Andreev reflection. In the half-metallic limit, conventional Andreev reflection is forbidden due to the absence of an opposite-spin band: this leads to zero ZBC at $\phi=180^{\circ}$. Same-spin Andreev reflection (see discussion in the paragraph above Eq. (\[f1waveup\])) is not allowed in collinear magnetic configurations. Equation (\[probabilitylaw\]) also reflects another important feature of the ZBC: the contributions to $G$ at zero bias from the spin-up and down channels are identical except for the weight factor $P_{\sigma}$: one can prove analytically that the sum of first two terms (related to Andreev reflection) in Eq. (\[probabilitylaw\]) is spin-independent. As a result, the sum of last two terms, related to ordinary reflection, is also spin-independent, and so is the ZBC.
![(Color online) Forward scattering conductance of a ${\rm F_1F_2S}$ trilayer with differing magnetic materials corresponding to exchange fields of $h_1=0.6$ and $h_2=0.1$. Various magnetic orientations, $\phi$, are considered as shown. Geometry and other parameters are as in the top panels of Fig. \[figure45\].[]{data-label="figure11a"}](fig6f.eps){width="45.00000%"}
{width="90.00000%"}
We briefly consider here one example where the two F materials in the trilayers have different field strengths. In this example all the thicknesses and the coherence length are as in the top panels of Fig. \[figure45\]. In Fig. \[figure11a\], we plot the forward scattering conductance for several $\phi$ at $h_1=0.6$ and $h_2=0.1$. One can quickly identify that the ZBC here is a non-monotonic function of $\phi$ with it maximum at the orthogonal relative magnetization angle, $\phi=90^{\circ}$. In contrast, results at equal exchange field strengths usually demonstrate monotonic behavior, as previously shown. However, many features are still the same, such as the formation of a FBCP when $\phi>90^{\circ}$. For $\phi=0^{\circ}$ and $\phi=30^{\circ}$, the conductance curves are not monotonically decreasing, as was the case at $h_1=h_2$. There, when $h_1=h_2$ and $\phi<90^{\circ}$, we always see monotonically decreasing behavior because the scattering effect due to misoriented magnetizations is not as great as at $\phi>90^{\circ}$. Also, when $h_1 \neq h_2$, we have to include in our considerations another scattering effect that comes from the mismatch between $k_{\uparrow1,(\downarrow 1)}^{\pm}$ and $k_{\uparrow 2,(\downarrow 2)}^{\pm}$. Specifically, when $\phi=0^{\circ}$, the only important scattering effect is that due to mismatch from $h_1\neq h_2$ and it leads to suppression of the ZBC at $\phi=0^{\circ}$. However, we see that the scattering due to the misoriented magnetic configuration ($\phi\neq 0^{\circ}$) compensates the effect of mismatch from $h_1 \neq h_2$ and ZBC is maximized when $\phi=90^{\circ}$. Qualitatively, one can examine Eqs. (\[f2spinor1\]) and (\[f2spinor2\]) and verify that the spinor at $\phi=90^{\circ}$ is composed of both pure spin-up and spin-down spinors with equal weight, apart from phase factors. As a result, the scattering effect due to mismatch from $k_{\uparrow1,(\downarrow 1)}^{\pm}$ and $k_{\uparrow 2,(\downarrow 2)}^{\pm}$ is reduced. We also verified that, when the strength of $h_2$ is increased towards $h_1$, the locations for the maximum of the ZBC($\phi$) curves gradually move from $\phi=90^{\circ}$ at $h_2=0.1$ to $\phi=0^{\circ}$ at $h_2=0.6$.
### Angularly averaged conductance
We now present results for the angularly averaged conductance, $\langle G \rangle$ as defined in Eq. (\[aaG\]). The details of the angular averaging are explained under Eq. (\[snell\]). The angularly averaged conductance is relevant to a much wider range of experimental results than the forward conductance, which is relevant strictly only for some point contact experiments. This is particularly true if one recalls that the critical angle $\theta_{c\sigma}$ and the weight factor for angular averaging in Eq. (\[aaG\]) used in this work can be modified based on a real experimental set-up or on the geometry of the junction.
In Fig. \[figure7\], we present results for $\langle G \rangle$ at $D_{F2}=12$ (left panels) and $D_{F2}=18$ (right panels). All curves are obtained with $D_{F1}=10$ and $D_S=180=1.5 \Xi_0$ at the values of $h$ indicated in each panel. Results are plotted over the entire range of $\phi$ values. The CB values obtained for $\langle G \rangle$ are again non-monotonic functions of $\phi$ and the non-monotonicity matches that of the saturated pair amplitudes, for the reasons previously given. The CB values for $\langle G \rangle$ in these cases are the same as those for the forward scattering conductance. One can also see that the resonance phenomenon is washed out in the angularly averaged conductance. For example, the resonance curve in the top panel of Fig. \[figure6\] tells us that $h\approx 0.3$ and $h\approx 0.6$ correspond respectively to a resonance maximum and minimum of the ZBC in the forward scattering $G$. However, in the top left panel of Fig. \[figure7\], the ZBC is no longer a weak function of $\phi$ and it gradually decreases when $\phi$ is increased. Near the resonance minimum, $h=0.6$, bottom left panel of Fig. \[figure7\], we can see a trace of the appearance of the FBCP when $\phi$ is above $90^{\circ}$. This FBCP in $\langle G \rangle$ is not as prominent as that in the forward scattering $G$, due to the averaging.
The magnetoresistance measure $M_G(E=2,\phi)$ is larger for $\langle G \rangle$ than for the forward scattering conductance. For example, $M_G (E=2,\phi=180^{\circ})$ in the forward scattering conductance for $h=0.6$ and $D_{F2}=12$ is half of that in $\langle G \rangle$. As for the zero bias magnetoresistance $M_G(E=0,\phi=180^{\circ})$ in $\langle G \rangle$, it is of about the same order as $M_G(E=2,\phi=180^{\circ})$ and it does not depend on where it is located in the resonance curve, Fig. \[figure6\] (recall that $M_G(E=0,\phi=180^{\circ})$ for the forward scattering conductance almost vanishes at the resonance maximum). In the right panels of Fig. \[figure7\], we plot results for a larger $D_{F2}$ with values of $h=0.35$ (near a resonance maximum) and $h=0.725$ (near a resonance minimum). They share very similar features with the thinner $D_{F2}$ case in the left panels. However, for $h=0.725$, we see that the ZBC values at different $\phi$ shrink to almost or less than unity and they are just barely higher than the conductance at $E=2$ because the contributions from Andreev reflection are strongly suppressed in such a high exchange field.
{width="90.00000%"}
Another important feature in the angularly averaged results for higher exchange fields (bottom panels in Fig. \[figure7\]) is the existence of cusps at the CB. To understand the formation of these cusps, we analyze $\langle G \rangle$ by dividing the contribution from all angles into two ranges: the range above and the range below the conventional Andreev critical angles $\theta_c^A$ \[see discussion below Eq. (\[andreevr\])\]. Consider e.g., the case of spin-up incident quasi-particles. When $\theta_c^A\equiv\sin^{-1}[\sqrt{(1-h)/(1+h)}]<\theta_i<
\sin^{-1}[\sqrt{{1}/(1+h)}]$, the conventional Andreev reflected waves become evanescent while the transmitted waves are still traveling waves above the CB. When $\theta_i>\sin^{-1}[\sqrt{{1}/(1+h)}]$, both the conventional Andreev reflected waves and the transmitted waves become evanescent. Here, $\theta_{c\uparrow}=\sin^{-1}[\sqrt{{1}/(1+h)}]$ is the upper limit in Eq. (\[aaG\]).
The case of spin-down incident quasi-particles is trivial, because the dimensionless incident momentum is $\sqrt{1-h}$ which is less than both the conventional Andreev reflected wavevector, $\sqrt{1+h}$, and the transmitted wavevector, (unity in our conventions). Therefore, all the reflected and transmitted waves above the CB are traveling waves. As a result, we should consider all possible incident angles and the upper limit of Eq. (\[aaG\]) is $\pi/2$. Let us therefore focus on the nontrivial spin-up component of $\langle G \rangle$. In Fig. \[figure8\] we separately plot the contributions to $\langle G_{\uparrow}\rangle$ from angles in the range above $\theta_c^A$ (top panels) and below (bottom panels) for the field values and geometry in the left panels of Fig. \[figure7\], in particular $D_{F2}=12$. These contributions we will denote as $\langle G_{\uparrow}(E)\rangle_{above}$ and $\langle G_{\uparrow}(E)\rangle_{below}$ respectively. The $\langle G_{\uparrow}(E)\rangle_{below}$ contributions, in the bottom panels of Fig. \[figure8\] are, for both $h=0.3$ and $h=0.6$, similar to the result for their total forward scattering counterpart (see Fig. \[figure3\] and the top right panel of Fig. \[figure45\]). Of course, the angular averaging leads to a smearing of the pronounced features originally in the forward scattering $G$. Qualitatively, the similarity comes from the propagating nature of all possible waves except the transmitted waves below the CB when $\theta_i<\theta_c^A$. Therefore, the forward scattering $G$ is just a special example with the incident angle perpendicular to the interface.
In the subgap region, the contribution to $\langle G_{\uparrow}(E)\rangle_{above}$ is vanishingly small although small humps appear when the exchange fields in the two F layers are non-collinear, i.e., $\phi\neq0,\pi$. These small humps are generated by the process of anomalous, equal-spin Andreev reflection. This process is possible in trilayers because, in a non-collinear magnetic configuration, a spin up quasiparticle can Andreev reflect as a spin-up hole. This can be seen from the matrix form of the BdG equations, Eq. (\[bogo\]). The occurrence of anomalous Andreev reflection leads to some important physics which we shall discuss in the next sub-subsection. One can see from Fig. \[figure8\], that when the exchange fields are strictly parallel or anti-parallel to each other, anomalous Andreev reflection does not arise.
Above $\theta_c^A$, the conventional Andreev-reflected wave is evanescent and it does not contribute to $\langle G_\uparrow \rangle$. When the bias is above the saturated pair amplitude, contributions to $\langle G_\uparrow \rangle$ from the upper range are provided by both the transmitted waves and by anomalous Andreev reflected waves. Recall that ordinary transmitted waves are propagating when $E$ is greater than the saturated pair amplitudes. We also see that $\langle G_\uparrow \rangle_{above}$ decreases with increasing $\phi$. At $\phi=180^{\circ}$, $\langle G_\uparrow \rangle$ is vanishingly small due to the effect of a large mismatch from the anti-parallel exchange field. Note also that the contribution from above $\theta_c^A$ is less in the $h=0.3$ case than at $h=0.6$. This is mainly due to a smaller fraction of states at $h=0.3$ with incident angles larger than $\theta_c^A$. On the other hand, the contribution from below $\theta_c^A$ is larger in the $h=0.3$ case. The increase of $\langle G_\uparrow \rangle_{above}$ and the decrease of $\langle G_\uparrow \rangle_{below}$ from $h=0.3$ to $h=0.6$ gives rise to the cusp at the CB, when adding these two contributions together.
### Anomalous Andreev reflection
As we have seen, equal-spin (anomalous) Andreev reflection (ESAR) can be generated when the magnetic configuration is non-collinear. We have previously shown that conventional Andreev reflection is forbidden when $\theta_i>\theta_c^A=
\sin^{-1}(\sqrt{(1-h_1)/(1+h_1)})$. Thus, $\theta_c^A$ vanishes in the half-metallic limit. In that case, conventional Andreev reflection is not allowed for any incident angle $\theta_i$ and the subgap $\langle G_\uparrow \rangle$ arises only from ESAR. For this reason, in this sub-subsection we present results for a trilayer structure that consists of one half-metal ($h_1=1$) and a much weaker ($h_2=0.1$) ferromagnet. The weaker ferromagnet serves the purpose of generating ESAR. A somewhat similar example that has been extensively discussed in the literature is that of half metal-superconductor bilayers with spin-flip interface.[@eschrig2003; @visani; @linder2010; @niu] There the spin-flip interface plays the same role as the weaker ferromagnet here. Another interesting phenomenon also related to ESAR is the induction of triplet pairing correlations in F-S structures.[@Halterman2007; @hv2p; @ji; @cko] To induce this type of triplet pairing, F-S systems must be in a non-collinear magnetic configuration such as ${\rm F_1F_2S}$ or ${\rm F_1SF_2}$ trilayers with $\phi\neq 0,\pi$. Hence, the mechanism behind induced triplet pairing correlations is also responsible for ESAR and these two phenomena are closely related.
![(Color online) The angularly averaged conductance of ${\rm F_1F_2S}$ trilayers with exchange field $h_1=1$ and $h_2=0.1$ for several values of $\phi$. See text for discussion.[]{data-label="figure9"}](fig9f.eps){width="45.00000%"}
![(Color online) Contributions to $G(E,\phi=150^\circ)$, computed for the parameter values used in Fig. \[figure9\], from the spin-up quasiparticle and spin-up quasihole ESAR (see text for discussion). The total $G$ is also shown. []{data-label="figure10"}](newfig10.eps){width="45.00000%"}
In Fig. \[figure9\], we plot the $\langle G \rangle$ of this particular system for several $\phi$. The geometrical parameters are again $D_{F1}=10$, $D_{F2}=12$, and $D_{S}=180$. We have $\langle G \rangle=\langle G_{\uparrow} \rangle$ because the weight factor $P_{\downarrow}=0$ in this half metallic case. For $\phi=0^{\circ}$ and $\phi=180^{\circ}$ the CB value is about 0.65 and, below the CB (in the subgap region), $\langle G \rangle$ vanishes because the conventional Andreev reflection is completely suppressed and ESAR is not allowed in the collinear cases. For $\phi=30^{\circ}$ and $\phi=150^{\circ}$, the CB is near 0.4 and 0.5 respectively and all of the subgap $\langle G \rangle$ is due to ESAR. The CB values for $\phi=60^{\circ}$, $90^{\circ}$, and $120^{\circ}$ are 0.15, 0.12, and 0.15. For these three angles, a FBCP clearly forms, arising from the ESAR in the subgap region.
To examine the conductance in the subgap region, which is in this case due only to ESAR, we choose the $\phi=150^{\circ}$ angle and plot, in Fig. \[figure10\], the contributions to $G$ (for this case $G$ and $\langle G \rangle$ are very similar) from the reflected spin-up particle and the reflected spin-up hole wavefunctions. The spin-down particle and spin-down hole wavefunctions are evanescent and do not contribute to the conductance. Thus, Eq. (\[conductance\]) reads $G=
1+{(k^-_{\uparrow 1}}/{k^+_{\uparrow 1})}|a_{\uparrow }|^2
-%wv2 {(k^+_{\uparrow 1}}/{k^+_{\sigma 1})}
|b_{\uparrow }|^2$. The quantities plotted are the second ((green) curve) and third ((red) curve, highest at the origin) terms in this expression. The value of $G$ is also plotted. One sees that the reflected ESAR amplitudes decay very quickly above the CB. However, these reflected amplitudes are quite appreciable in the subgap region. In other words, the supercurrent in the subgap region contains signatures of the triplet correlations. This confirms the simple picture[@btk] that above the CB the current flowing throughout the junction is governed by the transport of quasiparticles. However, below the CB it is dominated by ESAR.
![(Color online) The components of the spin current density, $S_x$, $S_y$, and $S_z$, calculated from Eq. (\[spincur\]) are plotted vs. $Y\equiv k_Fy$ for several values of the bias $E \equiv eV$ (main panels). We have $\phi=90^\circ$, $h=0.1$, $D_{F1}=250$, $D_{F2}=30$, $D_S=250=5\Xi_0$. The F$_2$-S interface is at $Y=0$ and the F$_1$-F$_2$ interface at $Y=-30$. Vertical lines at these interfaces in the top and bottom panels help locate the different regions. Only the central portion of the $Y$ range is included (see text). The ranges included depend on the component. The insets show the change in each component of the local magnetization, $\delta \mathbf{m}(E)\equiv \mathbf{m}(E)-\mathbf{m}(0)$, also as a function of $Y$. The values of $E$ are as in the main plot, the ranges included may be different. []{data-label="figure11"}](newfig11f.eps){width="45.00000%"}
### Spin current densities and spin-transfer torques
Finally, we now report on spin-dependent transport quantities, including the spin current, the spin-transfer torques, and their connections to the local magnetization at finite bias. An objective here is to demonstrate that the conservation law Eq. (\[spinconserve\]) which in the steady state is simply: $$\label{spinsteady}
\frac{\partial}{\partial y} S_i= \tau_i,
\enspace\enspace i=x,y,z, %ctw$$ is satisfied in our self consistent calculations for F$_1$F$_2$S trilayers. We consider these spin dependent quantities in a trilayer with $h=0.1$ and a non-collinear orthogonal magnetic configuration, $\phi=90^\circ$. Thus, the internal field in the outer electrode F$_1$ is along the $z$ axis, while that in F$_2$ is along $x$. The thicknesses are $D_{F1}=250$, $D_{F2}=30$, and $D_S=250=5\Xi_0$.
A set of results is shown in Fig. \[figure11\]. There, in the three main plots, we display the three components of the spin current density, computed from Eqs. (\[spincur\]) and normalized as explained below that equation. They are plotted as functions of the dimensionless position $Y\equiv k_F y$ for several values of the bias $V$, $E \equiv eV$. The F$_2$-S interface and the F$_1$-F$_2$ interface are located at $Y=0$ and $Y=-30$, respectively. For clarity, only the range of $Y$ corresponding to the ”central” region near the interfaces is included in these plots: the shape of the curves deeper into S or F$_1$ can be easily inferred by extrapolation. From these main panels, one sees that the current is spin-polarized in the $x$-direction (the direction of the exchange field in F$_2$) to the right of F$_1$-F$_2$ interface, including the S region. Furthermore, $S_x$ is found to be a constant except in the F$_1$ region, where it exhibits oscillatory behavior. This indicates the existence of a non-vanishing, oscillating spin-transfer torque in the F$_1$ layer, as we will verify below. We also see that $S_x$ vanishes when the bias is less than the superconducting gap in bulk S ($E<1$ in our notation). In fact, the behavior of $S_x$ with $V$ is similar to that of the ordinary charge current in an N-S tunneling junction with a very strong barrier where there is no current until $V>\Delta_0$. This phenomenon is very different from what occurs in ordinary spin valves (F$_1$-F$_2$), where the spin current is not blocked below any finite characteristic bias. The $S_y$ component, along the normal to the layers, is shown in the middle main panel of Fig. \[figure11\]. It depends extremely weakly on the bias $E$. It is very small except near the interface between the two ferromagnets but there it is about an order of magnitude larger than the other two components. Hence only a somewhat smaller $Y$ range is shown. Unlike the $S_x$ and $S_z$ components, $S_y$ does not vanish even when there is no bias applied to the trilayer (the (red) curve in this panel). From these observations, one can infer that $S_y$ is largely derived from its static part with only a very small contribution from the effect of finite bias. The emergence of a static spin current is due to the leakage of the local magnetization $m_z$ into the F$_2$ layer and of $m_x$ into the F$_1$ layer. This explains why the static spin current $S_y$ is mostly localized near the F$_1$-F$_2$ interface. The $S_z$ component (lower panel) is constant in the F$_1$ region, as one would expect. It oscillates in the F$_2$ region, and vanishes in the S layer. As opposed to the $S_x$ component, $S_z$ is non-vanishing, although very small, when $E<1$ It increases rapidly with bias when $E>1$. The oscillatory behavior of $S_z$, again, is related to the local spin-transfer torque as we will verify below.
![(Color online) The components of the spin-transfer torque $\bm{\tau}\equiv 2\mathbf{m}\times\mathbf{h}$ plotted vs. $Y$ for several bias values. All parameters and geometry are as in Fig. \[figure11\]. Vertical lines, denoting interfaces, are in the top and bottom panels. The insets show (for bias $E=1.6$) the torque ((blue) dashed line) and the derivative of the component of spin current density ((blue) circles). The lines and circles agree, proving that Eq. (\[spinsteady\]) holds.[]{data-label="figure12"}](newfig12f.eps){width="45.00000%"}
We can summarize the behavior of the spin current vector, in this $\phi=90^\circ$ configuration, as follows: when $E>1$, the spin current, which is initially (at the left side) spin-polarized in the $+z$ direction, is twisted to the $x$ direction under the action of the spin torques discussed below, as it passes through the second magnet, which therefore acts as a spin filter. The current remains then with its spin polarization in the $+x$ direction as it flows through the superconductor. Thus in this range of $E$ the trilayer switches the polarization of the spin current. On the other hand, when $E<1$, the small $z$-direction spin-polarized current tunneling into the superconductor is gradually converted into supercurrent and becomes spin-unpolarized.
In the insets of the three panels of Fig. \[figure11\], we illustrate the behavior with bias of the corresponding component of the local magnetization as it is carried into S. Specifically, we plot the components of the vector difference between the local magnetization with and without bias, $\delta\mathbf{m}(E)\equiv \mathbf{m}(E)-\mathbf{m}(E=0)$, as a function of $Y$. The range of $Y$ is chosen to display the salient aspects of the behavior of this quantity, and it is not the same as in the main plots, nor is it the same for each component. The bias values are the same as in the main plots, however. The magnetizations are computed from Eqs. (\[mag\]) and normalized in the usual way, as discussed below Eqs. (\[spincur\]). In these units, and at $h=0.1$ the value of the dominant component of ${\bf m}$ in the magnetic layers is about 0.15. This scale should be kept in mind.
The behavior of the $x$ component is nontrivial in the F$_2$ and S regions, and the corresponding $Y$ range is included in the top panel inset. When the applied bias is below the bulk S gap value, $\delta m_x(E)$ penetrates into the S layer with a decay length $\sim\Xi_0=50$. This decay length is much longer than that found for the static magnetization, $\mathbf{m}(0).$[@cko] When the bias is above the gap, $\delta m_x(E)$ penetrates even more deeply into the S layer, with a clearly very different behavior than for $E<1$. This long-range propagation is of course consistent with the behavior of $S_x$, as $S_x$, the spin current polarized in the $x$ direction, appears only when $E>1$. The magnitude of $\delta m_y$ is much smaller than that of $\delta m_x$ or $\delta m_z$. It peaks near the F$_1$-F$_2$ interface and that range of $Y$ is emphasized in the middle inset. Its overall scale monotonically increases with increasing bias. It damps away from the interface in an oscillatory manner. As to $\delta m_z$, which can conveniently be plotted in the same $Y$ range, it decays with a very short decay length and oscillates in F$_2$. The overall damped oscillatory behavior of $\delta m_y$ and $\delta m_z$ in the F$_2$ region reflects the precession, as a function of position, of the spin density around the local exchange field that points toward the $+x$ direction. This phenomenon is well known in spin-valves.[@ralph] The oscillation periods for $\delta m_y$, $\delta m_z$, $S_x$, and $S_z$ are very similar and of the order of $1/(hk_{FS})$.
Next, we investigate the spin-transfer torque, $\bm{\tau}\equiv 2\mathbf{m}\times\mathbf{h}$. This quantity, computed from the normalized values of $h$ and ${\bf m}$, is plotted as a function of position in Fig. \[figure12\] for the same system as in Fig. \[figure11\]. Results are shown for each of its three components in the main panels of the figure. One sees that at zero bias, $E=0$, both $\tau_x$ and $\tau_z$ vanish identically. In the F$_1$ layer, $\tau_x$ increases in magnitude with increasing $E$. It vanishes in F$_2$ and in S. The behavior of $\tau_z$ is, as one would expect, the converse: it vanishes in F$_1$ and S, and its magnitude increases in F$_2$. The oscillatory behavior of $\tau_x$ and $\tau_z$ is consistent, as we shall see below, with the results for $S_x$ and $S_y$. The component normal to the layers, $\tau_y$, is nonvanishing only near the ${\rm F_1}$-${\rm F_2}$ interface, although its peak there attains a rather high value, nearly two orders of magnitude larger than the peak value of the other components. It is independent of bias, consistent with the behavior of $S_y$.
In the insets, we verify, for each component, that Eq. (\[spinsteady\]) is satisfied, that is, that our self-consistent methods strictly preserve the conservation laws in this nontrivial case. (We have already mentioned that we have verified that the charge or particle current are independent of $y$). We specifically consider the bias value $E=1.6$ as an illustration. Consider first the top panel inset. There we plot both the $x$ component of the spin-transfer torque, $\tau_x$ (blue dashes), taken from the corresponding main plot, and the derivative of the spin current, $\partial S_x/\partial Y$ (blue circles), obtained by numerically differentiating the corresponding result in the top panel of Fig. \[figure11\]. Clearly, the curves are in perfect agreement. (One can easily check that with the normalizations and units chosen there should be no numerical factor between the two quantities). In the second panel, the same procedure is performed for the $y$ component, although in this case, because of the very weak dependence of both $S_y$ and $\tau_y$ on bias, the value of the latter is hardly relevant. Nevertheless, despite the evident difficulty in computing the numerical derivative of the very sharply peaked $S_y$, the agreement is excellent. For $\tau_z$, its vanishing in the F$_1$ region is in agreement with the constant spin current in that layer. The conservation law Eq. (\[spinsteady\]) is verified in the inset for this component, again at bias $E=1.6$. Just as for the $x$ component, the dots and the line are on top of each other. Thus the conservation law for each component is shown to be perfectly obeyed.
The results of this sub-subsection can be summarized as follows: the finite bias leads to spin currents. As opposed to the ordinary charge currents, these spin currents are generally not conserved locally because of the presence of the spin-transfer torques which act as source terms and are responsible for the change of spin-density. But a self-consistent calculation [*must*]{} still contain exactly the correct amount of non-conservation, that is, Eq.(\[spinsteady\]) must be satisfied. It is therefore of fundamental importance to verify that it is, as we have.
Conclusions
===========
In summary, we have investigated important transport properties of F$_1$F$_2$S trilayers, including tunneling conductances and spin transport. To properly take into account the proximity effects that lead to a spatially varying pair potential, we have incorporated a transfer matrix method into the BTK formalism. This allows us to use self-consistent solutions of the BdG equations. This technique also enables us to compute spin transport quantities including spin transfer torque and spin currents. We have shown that in F-S bilayers the self-consistent calculations lead to conductances at the superconducting gap that increase with the Fermi wavevector mismatch whereas non-self-consistent ones predict they are insensitive to this parameter. In F$_1$F$_2$S trilayers, we have found that the critical bias CB (where tunneling conductance curves drop) for different relative magnetization angles, $\phi$, depends on the strength of the superconducting order parameter near the interface. The angular dependence of the critical bias reflects that of the transition temperatures $T_c$, which are usually nonmonotonic functions of $\phi$. For forward scattering in these F$_1$F$_2$S trilayers, we found that the dependence of the zero bias conductance peak (ZBCP) on $\phi$ is related to both the strength of the exchange fields and the thickness of the $F_2$ layers. This remarkable behavior can be explained via quantum interference effects. At the resonance minimum, the ZBCP drops significantly and monotonically from $\phi=0^\circ$ to $\phi=180^\circ$. On the other hand, the $\phi$ dependence of the ZBCP is very weak when it is at its resonance maximum. For asymmetric cases where $h_1\neq h_2$, we found that the ZBCP is a nonmonotonic function of $\phi$ with its value at $\phi=\pi/2$ being the maximum. We have also investigated the angularly averaged tunneling conductances, $\langle G \rangle$, and found that features of resonance effects are then somewhat washed out due to the averaging. However, by studying $\langle G \rangle$ in the subgap regions, we found that anomalous (equal spin) Andreev reflection (ESAR) arises when $\phi$ corresponds to noncollinear orientations. The emergence of ESAR is correlated with the well-known induced triplet pairing correlations in proximity coupled F-S structures. When the outer magnet is a half metal, the $\langle G \rangle$ signatures arise chiefly from the process of ESAR. We have also studied the bias dependence of the spin currents and spin transfer torques and their general behavior in F$_1$F$_2$S trilayers with $\phi=90^\circ$ (the exchange fields in F$_1$ and F$_2$ point toward the $z$ and $x$ directions, respectively). The spin current components are in general non-conserved quantities. The $S_z$ component, parallel to the local exchange field in the F$_1$ layer, does not change in the F$_1$ region but shows damped oscillatory behavior in the F$_2$ layer and eventually vanishes in the S region. However, $S_x$ is a constant throughout the F$_2$ and S regions and oscillates in F$_1$ layers. We found that $S_y$ (the component normal to the layers) depends very weakly on the bias, and thus its spatial dependence arises largely from a static effect. The bias dependence of $S_x$ in the S region is very similar to that of the tunneling charge current in normal/superconductor systems with high barriers: $S_x$ vanishes in the subgap regions and arises right above the gap. The behavior of $\bf{m}$ is consistent with that of $\bf{S}$. We found that $m_x$, parallel to the local exchange fields in F$_2$, spreads out over the S regions when the bias is larger than the superconducting gap. We have also investigated the bias dependence of the spin transfer torques, and we have carefully verified that the appropriate continuity equation for the spin current is strictly obeyed in our self-consistent approach. Our method can be extended to include the effects of interfacial scattering and wavevector mismatch. It can also be used for further study of the intricate phenomena associated with spin transport in these systems.
Portions of this work were supported by IARPA grant No. N66001-12-1-2023. CTW thanks the University of Minnesota for a Dissertation Fellowship. The authors thank I. Krivorotov (Irvine) for helpful discussions.
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abstract: |
We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition $\mathbf{P}(S_{n}>0)\rightarrow \rho ,\
n\rightarrow \infty $, which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case $\rho \in (0,1)$, we investigate the case where the offspring distribution is in the domain of attraction of a stable law with parameter $1$, which implies that $\rho =0$ or $1$. We find the asymptotic behaviour of the survival probability of the population in these two cases.\
**AMS 2000 subject classifications.** Primary 60J80; Secondary 60G50.\
**Keywords.** Branching process, random environment, random walk, conditioned random walk, Spitzer’s condition
address:
- 'Congzao Dong, School of Mathematics and Statistics, Xidian University, 710126 Xian, P.R. China'
- 'Charline Smadi, Univ. Grenoble Alpes, IRSTEA, LESSEM, 38000 Grenoble, France and Univ. Grenoble Alpes, CNRS, Institut Fourier, 38000 Grenoble, France'
- 'Vladimir A. Vatutin, Department of Discrete Mathematics, Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkin Street, 117 966 Moscow GSP-1, Russia'
author:
- 'C. Dong'
- 'C. Smadi'
- 'V. A. Vatutin'
title: Critical branching processes in random environment and Cauchy domain of attraction
---
Introduction and main results
=============================
Branching processes have been introduced by Galton and Watson in the 19th century in order to study the extinction of family names [watson1875probability]{}. Since then they have been widely used to model the dynamics of populations or the spread of infections for instance [haccou2005branching,allen2010introduction]{}. Branching processes in random environment have been first introduced and studied by Smith and Wilkinson and Athreya and Karlin in the early seventies [smith1969branching,athreya1971branching,athreya19712branching]{}. By introducing such processes, their aim was to better understand the effect of the environmental stochasticity on the population dynamics. Initially restricted to environments satisfying strong assumptions or to particular offspring distributions, they have been later generalised. Their study has known a renewed interest during the last two decades, with the development of new techniques to investigate them, in particular by linking events on the trajectory of the population process until a certain generation $n$ with an other event of its associated random walk until the same time $n$ (see, for instance, [afanasyev2005criticality,afanasyev2012limit,bansaye2013lower,vatutin2013evolution]{} for more detail).
A branching process in an independent identically distributed (i.i.d.) random environment is specified by a sequence of i.i.d. random offspring generating functions $$f_{n}(s):=\sum_{k=0}^{\infty }f_{n}\left[ k\right] s^{k},\quad n\in
\{1,2,...\}=:\mathbb{N},\quad 0\leq s\leq 1. \label{DefF}$$Denoting by $Z_{n}$ the number of individuals in the process at time $n$, we assume that there is initially one individual in the population ($Z_{0}=1$) and we define its evolution by the relations $$\mathbf{E}[s^{Z_{n}}|f_{1},\ldots ,f_{n};Z_{0},Z_{1},\ldots
,Z_{n-1}]:=(f_{n}(s))^{Z_{n-1}},\quad n\in \mathbb{N}.$$
Let $$X_{k}:=\log f_{k}^{\prime }(1)=\log \mathbf{E}[Z_{k}|f_{k},Z_{k-1}=1],\quad
k\in \mathbb{N},$$and denote $$S_{0}:=0,\quad S_{n}:=X_{1}+X_{2}+\ldots +X_{n}$$the auxiliary random walk associated with the quenched expectation of offspring number. The long time behaviour of the process $\mathcal{Z}:=\left\{ Z_{n},\,n\geq 0\right\} $ is intimately related to the properties of the random walk $\mathcal{S}:=\left\{ S_{n},n\geq 0\right\} $ (see [geiger2001survival,geiger2003limit,afanasyev2005criticality]{} for instance). According to fluctuation theory of random walks (see [@fe]), three different cases are possible: either $\mathcal{S}$ drifts to $\infty $, or $\mathcal{S}$ drifts to $-\infty $, or the random walk oscillates: $$\limsup_{n\rightarrow \infty }S_{n}=+\infty \ \text{ and }\
\liminf_{n\rightarrow \infty }S_{n}=-\infty$$with probability $1$. Accordingly, the branching process is called *supercritical*, *subcritical*, or *critical* [@afanasyev2005criticality]. We consider the last possibility. In this case the stopping time $$T^{-}:=\min \{k\geq 1:S_{k}<0\}$$is finite with probability $1$ and, as a result (see [afanasyev2005criticality]{}), the extinction time $$T:=\min \{k\geq 1:Z_{k}=0\}$$of the process $\mathcal{Z}$ is finite with probability $1$.
In this work we will be interested in the asymptotic behaviour of the survival probability $\mathbf{P}(Z_{n}>0)$ of the population at large time. It is a natural question when dealing with populations, and it has been answered under various assumptions in the case of branching processes in random environment (see, for instance, [kozlov1977asymptotic,geiger2001survival,afanasyev2005criticality,afanasyev2012limit]{}).
We assume that the random walk $\mathcal{S}$ satisfies the Doney-Spitzer condition, which is a classical condition in fluctuation theory, and writes $$\lim_{n\rightarrow \infty } \frac{1}{n} \sum_{m=1}^n \mathbf{P}\left(
S_{m}>0\right) =:\rho . \label{Spit}$$ According to Bertoin and Doney [@bertoin1997spitzer], this condition is equivalent to $$\lim_{n\rightarrow \infty }\mathbf{P}\left( S_{n}>0\right) =:\rho .$$ The case $\rho \in (0,1)$ has been studied by Afanasyev and his coauthors in [@afanasyev2005criticality]. Under some mild additional assumptions they proved the following equivalent for the survival probability of the population at large times $n$, $$\mathbf{P}(Z_{n}>0)\sim \frac{l(n)}{n^{1-\rho }}, \label{tails}$$where $l(.)$ is a slowly varying function.
The aim of the present paper is to complement (\[tails\]) by considering the asymptotic behaviour of $\mathbf{P}(Z_{n}>0)$ as $n\rightarrow \infty $ in the cases $\rho =0$ and $\rho =1$.
Before stating our main results, we need to introduce some notation and a set of assumptions on the law of the random walk $\mathcal{S}$. The main assumption is that $\mathcal{S}$ is in the domain of attraction of a stable law with parameter $1$. It means that there exist a slowly varying function $L(\cdot )$, and two nonnegative numbers $p$ and $q,\ p+q=1,$ such that $$\mathbf{P}\left( X_{1}>x\right) \sim p\frac{L(x)}{x}\quad \text{and}\quad
\mathbf{P}\left( X_{1}<-x\right) \sim q\frac{L(x)}{x},\quad x\rightarrow
\infty . \label{BasicCond}$$
As we will see (Remark \[rho0p>q\]), $\mathcal{S}$ will satisfy the Doney-Spitzer condition with $\rho =0$ (resp. $\rho =1$) in the case $p>q$ (resp. $p<q$). To show that we introduce two scaling sequences which play the main role in the asymptotic behaviour of various quantities related to the random walk $\mathcal{S}$. The first sequence, $\left\{ a_{n},n\in \mathbb{N}\right\} $, satisfies, as $n\rightarrow \infty $ the relation $$\frac{L(a_{n})}{a_{n}}\sim \frac{1}{n}. \label{defan}$$Note that the sequence is regularly varying with parameter 1 as $n\rightarrow \infty $ (see [@Sen76]). We can thus rewrite it as $$a_{n}=nL_{4}(n) \label{AsympAn}$$where $L_{4}(.)$ is a slowly varying function as $n\rightarrow \infty $. The second sequence, $\left\{ h_{n},n\in \mathbb{N}\right\} $, is specified by $$h_{n}:=n\mu \left( a_{n}\right) \text{ where }\mu (x)=\mathbf{E}\left[ X_{1}\mathbf{1}_{\left\{ \left\vert X_{1}\right\vert \leq x\right\}} \right] ,
\label{defhn}$$ where $\mathbf{1}$ is the indicator function.
In addition, we suppose that$$\mu :=\mathbf{E}\left[ X_{1}\right] =0. \label{Def_mu}$$
Let $$l^{\ast }(z):=\int_{z}^{\infty }\frac{L\left( y\right) }{y}dy.
\label{deflstar}$$The relation between $p$ and $q$ and the value of $\rho $ in derive from the following properties:
\[rho0p>q\] (Lemma 7.3 in [@Berg17] and Proposition 1.5.9 in [bingham1989regular]{}) If the conditions (\[BasicCond\]) and (\[Def\_mu\]) hold and $p>q$, then, as $n\rightarrow \infty $ $$\mathbf{P}\left( S_{n}>0\right) \sim \frac{p}{p-q}\frac{L\left( \left\vert
h_{n}\right\vert \right) }{l^{\ast }\left( \left\vert h_{n}\right\vert
\right) }\quad \label{AsymS_up}$$and $$\sum_{k=1}^{n}\frac{1}{k}\mathbf{P}\left( S_{k}>0\right) \sim -\frac{p}{p-q}\log l^{\ast }\left( \left\vert h_{n}\right\vert \right) .$$Notice that, as $n\rightarrow \infty $ , $l^{\ast }(n)\rightarrow 0$, $l^{\ast }(\cdot )$ is slowly varying and $l^{\ast }(n)/L(n)\rightarrow
\infty $. The case $p<q$ is symmetric.
Thus, the situation $p>q$ corresponds to $\rho =0$ while $p<q$ corresponds to the case $\rho =1.$
As in [@afanasyev2005criticality], we need to impose restrictions on the standardized truncated second moment of the environment, namely: $$\zeta _{k}(a):=\sum_{y=a}^{\infty }y^{2}f_{k}[y]/\left( \sum_{y=0}^{\infty
}yf_{k}[y]\right) ^{2}, \label{defzetaa}$$for $a,k\in \mathbb{N}$. The moment condition depends on the value of $\rho $ in the Doney-Spitzer condition .
**Condition A**. ($\rho =0\leftrightarrow p>q$) There exist $a\in
\mathbb{N}$ and $\beta >0$ such that $$\mathbf{E}[\zeta _{1}^{\beta }(a)]<\infty \quad \text{and}\quad \mathbf{E}[U(X_{1})\zeta _{1}^{\beta }(a)]<\infty ,$$where $U$ is the renewal function associated to the strict descending ladder epochs of $\mathcal{S}$, $$\gamma _{0}:=0,\quad \gamma _{j+1}:=\min \left( n>\gamma
_{j}:S_{n}<S_{\gamma _{j}}\right) ,\quad j\in \mathbb{N}_{0}:=\mathbb{N\cup }\left\{ 0\right\} , \label{defstrictdesc}$$and is defined by $$U(x):=\sum_{j=0}^{\infty }\mathbf{P}(S_{\gamma _{j}}\geq -x),\quad x>0,\quad
U\left( 0\right) =1,\quad U\left( x\right) =0,\quad x<0. \label{Def_U}$$
**Condition B**. ($\rho =1\leftrightarrow p<q$) There exist $a\in
\mathbb{N}$ and $\beta >0$ such that $$\mathbf{E}\left[ \left( \log ^{+}\zeta _{1}(a)\right) ^{1+\beta }\right]
<\infty \quad \text{and}\quad \mathbf{E}\left[ U(X_{1})\left( \log ^{+}\zeta
_{1}(a)\right) ^{1+\beta }\right] <\infty .$$
Observe that the moment condition in [@afanasyev2005criticality] under the Doney-Spitzer condition with $\rho \in (0,1)$ was the existence of $\beta >0$ and $a\in \mathbb{N}$ such that: $$\mathbf{E}\left[ \left( \log ^{+}\zeta _{1}(a)\right) ^{1/\rho +\beta }\right] <\infty \quad \text{and}\quad \mathbf{E}\left[ U(X_{1})\left( \log
^{+}\zeta _{1}(a)\right) ^{1+\beta }\right] <\infty .$$Our **Condition B** is thus a natural extension of the moment condition to the case $\rho =1$. In contrast, such a natural extension for $\rho =0$ would have provided an infinite exponent for the logarithm and we could not obtain a moment condition on the logarithm only. Notice however that we can take $\beta $ as small as we want in **Condition A.** Thus**,** our moment condition is not very strong.
Last, for technical reasons, we need to add an assumption which will be used for the case $p>q$ only.
**Condition C**. There exists an integer-valued function $g(j)=e^{o(j)},\ j\rightarrow \infty ,$ such that $$\sum_{j=1}^{\infty }1/\Lambda(g(j))<\infty ,$$where $\Lambda$ is a slowly varying function (see the proof of Proposition 12 in [@kortchemski2019condensation]) defined by $$\label{def_Lambda}
\Lambda \left(\frac{1}{1-s}\right)= \exp \left( \sum_{k=1}^\infty \frac{\mathbf{P}(S_k \geq 0)}{k}s^k \right), \quad s \in [0,1).$$
We will provide in Example \[ex1\] an illustration of a slowly varying function $L(\cdot )$ meeting this condition.
As previously observed under different assumptions on the random environment (see, for instance, [@vatutin2013evolution] for a comprehensive review on the critical and subcritical cases (before 2013) or the recent monograph [@GV2017]) the survival of a branching process in random environment is essentially determined by its survival until the moment when the associated random walk $\mathcal{S}$ attains its infimum. The idea is that if we divide the trajectory of the process on the interval $[0,n]$ into two parts, one before the running infimum of the random environment $\mathcal{S}$, and one after this running infimum, the process will live in a favorable environment after the running infimum of the random environment, and will thus survive with a nonnegligible probability until time $n$, provided it survived until the time of the running infimum. This is essentially, in words, the idea of the proof of our main result (see Theorem \[T\_main\]). To state things more rigorously, we introduce the running infimum of the random walk $\mathcal{S}$: $$L_{n}:=\min \left\{ S_{0},S_{1},...,S_{n}\right\} ,\ n\in \mathbb{N}_{0}.
\label{defLn}$$
Depending on the relative positions of $p$ and $q$ (defined in ) or equivalently on the value of $\rho $ ($0$ or $1$) we have the two following possible asymptotics for the survival probability of the process $\mathcal{Z}$:
\[T\_main\]Assume that Conditions (\[BasicCond\]) and (\[Def\_mu\]) hold.
- If $p>q$, and Conditions A and C hold then there exists a constant $K_{1}\in \left( 0,\infty \right) $ such that, as $n\rightarrow \infty $$$\mathbf{P}\left( Z_{n}>0\right) \sim K_{1}\mathbf{P}\left( L_{n}\geq
0\right) \sim K_{1}\frac{L_{22}(n)}{n}, \label{survivP>Q}$$where $L_{22}(.)$ is a function slowly varying at infinity.
- If $p<q$ and Condition B holds then there exists a constant $K_{2}\in
\left( 0,\infty \right) $ such that, as $n\rightarrow \infty $$$\mathbf{P}\left( Z_{n}>0\right) \sim K_{2}\mathbf{P}\left( L_{n}\geq
0\right) \sim K_{2}L_{33}(n), \label{survivP<Q}$$where $L_{33}(.)$ is a function slowly varying at infinity.
Hence, despite the irregular behaviour of the associated random walk $S$ (a null expectation but a probability converging to $1$ to be positive (resp. negative)), the asymptotic behaviour of the survival probability is, except for the slowly varying function, the limit of the one obtained in [afanasyev2005criticality]{} by taking $\rho=0$ or $1$ instead of $\rho \in
(0,1)$.
The rest of the paper is structured as follows. Section \[sec\_supr\] is dedicated to the study of the running extrema of the random walk $S$. In Section \[sec\_chgtprob\], we perform a change of measure, obtained as a Doob-h transform, where the renewal function $U(\cdot )$ of $\mathcal{S}$ and the indicator of the event $\{L_{n}\geq 0\}$ are involved. Finally, the proof of the main result, Theorem \[T\_main\], is completed in Section [sec\_proof]{}.
Estimates for the suprema of the associated random walk {#sec_supr}
=======================================================
The aim of this section is to provide some bounds for the probabilities of the events related to the running infimum and maximum of the random walk $\mathcal{S}$. We recall the definition of the running infimum in , and introduce the running maximum via $$M_{n}:=\max \left\{ S_{1},...,S_{n}\right\} ,\ n\in \mathbb{N}.$$
We first list a number of known results which will be needed in our arguments. Recall definitions and . The following results have been first derived in Theorem 3.4 in [@Berg17], and then under weaker conditions in [@kortchemski2019condensation] (see Proposition 12 and Remark 13).
\[T\_Berg3\_4\] Assume that Conditions (\[BasicCond\]) and (\[Def\_mu\]) hold. Then when $n$ goes to infinity,
1\) if $p>q$ then $h_{n}\sim -\left( p-q\right) nl^{\ast }\left( a_{n}\right)
\rightarrow -\infty $ and $$\mathbf{P}\left( L_{n}\geq 0\right) \sim \frac{L\left( \left\vert
h_{n}\right\vert \right) }{\left\vert
h_{n}\right\vert}\Lambda(n) =:\frac{L_{22}(n)}{n}, \label{Ber1}$$ for some slowly varying functions $L_{22}$ (recall that the sowly varying function $\Lambda$ has been defined in ).
2\) if $p<q$ then $h_{n}\sim \left( q-p\right) nl^{\ast }\left( a_{n}\right)
\rightarrow +\infty $ and $$\mathbf{P}\left( L_{n}\geq 0\right) \sim \frac{1}{\tilde{\Lambda}(n)}=:L_{33}(n),
\label{Ber2}$$ for some slowly varying functions $L_{33}$. The slowly varying function $\tilde{\Lambda}$ is defined as $\Lambda$ but with $-\mathcal{S}$ in place of $\mathcal{S}$.
3\) if $p>q$ then$$\mathbf{P}\left( M_{n}<0\right) \sim \frac{1}{\Lambda(n)} =:L_{44}(n)
\label{Ber3}$$
4\) if $p<q$ then $$\mathbf{P}\left( M_{n}<0\right) \sim\frac{L\left( \left\vert
h_{n}\right\vert \right) }{\left\vert
h_{n}\right\vert}\tilde{\Lambda}(n) =:\frac{L_{55}(n)}{n}.
\label{Ber4}$$
Recall the definitions of the strict descending ladder epochs $\left\{
\gamma _{j},j\in \mathbb{N}_{0}\right\} $ of $\mathcal{S}$ and of their associated renewal function $U(\cdot )$ in and ([Def\_U]{}), respectively, and introduce the strict ascending ladder epochs $\left\{ \Gamma _{j},j\in \mathbb{N}_{0}\right\} $ of $\mathcal{S}$ and their associated renewal function $V(\cdot )$ via $$\Gamma _{0}:=0,\quad \Gamma _{j+1}:=\min (n>\Gamma _{j}:S_{n}>S_{\Gamma
_{j}}),\quad j\in \mathbb{N}_{0}, \label{defstrictasc}$$and $$V(x):=1+\sum_{j=1}^{\infty }\mathbf{P}(S_{\Gamma _{j}}<x),\quad x>0,\quad
V\left( 0\right) =1,\quad V(x)=0,\quad x<0.$$
For a slowly varying function $L_{ii}(\cdot )$ let $$\hat{l}_{ii}(n):=\int_{1}^{n}\frac{L_{ii}(x)}{x}dx.$$The next lemma provides bounds on the probabilities for the running extrema to be in a certain interval.
\[L\_roughEstimates\]Assume that Conditions (\[BasicCond\]) and ([Def\_mu]{}) hold. Then there exists a constant $C\in \left( 0,\infty
\right) $ such that, for every $x\geq 0$ and $n\in \mathbb{N}$, $$\mathbf{P}\left( L_{n}\geq -x\right) \leq \left\{
\begin{array}{ccc}
CU(x)n^{-1}\hat{l}_{22}\left( n\right) & \text{if} & p>q, \\
& & \\
CU(x)L_{33}\left( n\right) & \text{if} & p<q,\end{array}\right. \label{EstimMin2}$$and $$\mathbf{P}\left( M_{n}<x\right) \leq \left\{
\begin{array}{ccc}
CV(x)L_{44}\left( n\right) & \text{if} & p>q, \\
& & \\
CV(x)n^{-1}\hat{l}_{55}\left( n\right) & \text{if} & p<q.\end{array}\right. \label{EstimMax2}$$
We know by a Spitzer identity that, for any $\lambda \geq 0$$$\begin{aligned}
\sum_{n=0}^{\infty }s^{n}\mathbf{E}\left[ e^{\lambda L_{n}}\right] &=&\exp
\left\{ \sum_{n=1}^{\infty }\frac{s^{n}}{n}\mathbf{E}\left[ e^{\lambda \min
(0,S_{n})}\right] \right\} \\
&=&\exp \left\{ \sum_{n=1}^{\infty }\frac{s^{n}}{n}\mathbf{E}\left[
e^{\lambda S_{n}};S_{n}<0\right] \right\} \exp \left\{ \sum_{n=1}^{\infty }\frac{s^{n}}{n}\mathbf{P}\left( S_{n}\geq 0\right) \right\} .\end{aligned}$$A Sparre-Anderson identity (see, for instance, Theorem 4.3 in [@GV2017]) allows us to rewrite the first term at the right hand side as $$\begin{aligned}
\exp \left\{ \sum_{n=1}^{\infty }\frac{s^{n}}{n}\mathbf{E}\left[ e^{\lambda
S_{n}};S_{n}<0\right] \right\} &=&1+\sum_{n=1}^{\infty }s^{n}\mathbf{E}\left[ e^{\lambda S_{n}};\Gamma ^{\prime }>n\right] \\
&=&\int_{0}^{+\infty }e^{-\lambda x}U_{s}(dx),\end{aligned}$$where $$\Gamma ^{\prime }:=\min \left( n\in \mathbb{N},S_{n}\geq 0\right)$$and $$U_{s}(x)=\sum_{n=0}^{\infty }s^{n}\mathbf{P}\left( S_{n}\geq -x;\Gamma
^{\prime }>n\right) ,\ x\geq 0.$$Therefore, $$\begin{aligned}
\sum_{n=0}^{\infty }s^{n}\mathbf{P}\left( L_{n}\geq -x\right)
&=&U_{s}(x)\exp \left\{ \sum_{n=1}^{\infty }\frac{s^{n}}{n}\mathbf{P}\left(
S_{n}\geq 0\right) \right\} \notag \\
&=&U_{s}(x)\sum_{n=1}^{\infty }s^{n}\mathbf{P}\left( L_{n}\geq 0\right)
\label{Ln0}\end{aligned}$$for $x\geq 0$. Note that by the duality principle for random walks (see, for instance, [@GV2017] p. 63), $$\begin{aligned}
\lim_{s\uparrow 1}U_{s}(x) &=&\sum_{n=0}^{\infty }\mathbf{P}\left( S_{n}\geq
-x;\Gamma ^{\prime }>n\right) \notag \label{duality} \\
&=&1+\sum_{n=1}^{\infty }\mathbf{P}\left( S_{n}\geq
-x;S_{i}<0,i=1,...,n\right) \notag \\
&=&1+\sum_{n=1}^{\infty }\mathbf{P}\left( S_{n}\geq
-x;S_{n}<S_{j},j=0,1,...,n-1\right) \notag \\
&=&1+\sum_{n=1}^{\infty }\sum_{r=1}^{n}\mathbf{P}\left( S_{n}\geq -x;\gamma
_{r}=n\right) \\
&=&1+\sum_{r=1}^{\infty }\sum_{n=r}^{\infty }\mathbf{P}\left( S_{n}\geq
-x;\gamma _{r}=n\right) =1+\sum_{r=1}^{\infty }\mathbf{P}\left( S_{\gamma
_{r}}\geq -x\right) =U(x). \notag\end{aligned}$$
On the other hand, if $s\uparrow 1$ then (\[Ber1\]) and an application of Corollary 1.7.3 in [@bingham1989regular] with $\rho =0$ give for $p>q$, $$\sum_{n=0}^{\infty }s^{n}\mathbf{P}\left( L_{n}\geq 0\right) \sim
\sum_{n=0}^{\infty }s^{n}\frac{L_{22}(n)}{n}\sim \hat{l}_{22}\left( \frac{1}{1-s}\right) ,$$while (\[Ber2\]) and again an application of Corollary 1.7.3 in [bingham1989regular]{} but now with $\rho =1$ justify, for $p<q$ the asymptotics $$\sum_{n=0}^{\infty }s^{n}\mathbf{P}\left( L_{n}\geq 0\right) \sim
\sum_{n=0}^{\infty }s^{n}L_{33}(n)\sim \frac{L_{33}(1/\left( 1-s\right) )}{1-s}.$$
Thus if $p>q$ then, as $s\uparrow 1$$$\sum_{n=0}^{\infty }s^{n}\mathbf{P}\left( L_{n}\geq -x\right) \sim U(x)\hat{l}_{22}\left( \frac{1}{1-s}\right) , \label{Slowly1}$$and if $p<q$ then, as $s\uparrow 1$$$\sum_{n=0}^{\infty }s^{n}\mathbf{P}\left( L_{n}\geq -x\right) \sim U(x)\frac{L_{33}(1/\left( 1-s\right) )}{1-s}.$$
Using , and the monotonicity of $U_{s}(x)$ in $s$ we get, for $p>q$$$\sum_{n=0}^{\infty }s^{n}\mathbf{P}\left( L_{n}\geq -x\right) \leq
U(x)\sum_{n=0}^{\infty }s^{n}\mathbf{P}\left( L_{n}\geq 0\right) \sim U(x)\hat{l}_{22}\left( \frac{1}{1-s}\right) .$$
Since $\mathbf{P}\left( L_{n}\geq -x\right) $ is nonincreasing with $n,$ we have for $p>q$$$\begin{aligned}
\frac{n}{2}\left( 1-\frac{1}{n}\right) ^{n}\mathbf{P}\left( L_{n}\geq
-x\right) &\leq &\sum_{n/2\leq m\leq n}\left( 1-\frac{1}{n}\right) ^{m}\mathbf{P}\left( L_{m}\geq -x\right) \\
&\leq &CU(x)\hat{l}_{22}\left( n\right) ,\end{aligned}$$and, similarly, for $p<q$$$\frac{n}{2}\left( 1-\frac{1}{n}\right) ^{n}\mathbf{P}\left( L_{n}\geq
-x\right) \leq CU(x)nL_{33}\left( n\right) .$$As a result$$\mathbf{P}\left( L_{n}\geq -x\right) \leq \left\{
\begin{array}{ccc}
CU(x)n^{-1}\hat{l}_{22}\left( n\right) & \text{if} & p>q, \\
& & \\
CU(x)L_{33}\left( n\right) & \text{if} & p<q.\end{array}\right.$$By the same arguments and (\[Ber3\]) we have as $s\uparrow 1$ $$\sum_{n=1}^{\infty }s^{n}\mathbf{P}\left( M_{n}<0\right) \sim
\sum_{n=1}^{\infty }s^{n}L_{44}(n)\sim \frac{L_{44}(1/\left( 1-s\right) )}{1-s} \label{Regularity2}$$for $p>q$, and by (\[Ber4\]) $$\sum_{n=1}^{\infty }s^{n}\mathbf{P}\left( M_{n}<0\right) \sim
\sum_{n=1}^{\infty }s^{n}\frac{L_{55}(n)}{n}\sim \hat{l}_{55}\left( \frac{1}{1-s}\right) \label{Slowly2}$$for $p<q$. Thus $$\mathbf{P}\left( M_{n}<x\right) \leq \left\{
\begin{array}{ccc}
CV(x)L_{44}\left( n\right) & \text{if} & p>q, \\
& & \\
CU(x)n^{-1}\hat{l}_{55}\left( n\right) & \text{if} & p<q.\end{array}\right.$$This ends the proof.
Observe that$$\begin{gathered}
\sum_{n=1}^{\infty }s^{n}\mathbf{P}\left( L_{n}\geq 0\right)
\sum_{n=1}^{\infty }s^{n}\mathbf{P}\left( M_{n}<0\right) \\
=\exp \left\{ \sum_{n=1}^{\infty }\frac{s^{n}}{n}\mathbf{P}\left( S_{n}\geq
0\right) \right\} \times \exp \left\{ \sum_{n=1}^{\infty }\frac{s^{n}}{n}\mathbf{P}\left( S_{n}<0\right) \right\} =\frac{1}{1-s}.\end{gathered}$$Thus, as $n\rightarrow \infty $ $$\hat{l}_{22}\left( n\right) L_{44}(n)\sim 1,\;L_{33}(n)\hat{l}_{55}\left(
n\right) \sim 1. \label{Equival}$$
Set $$b_{n}:=\left( na_{n}\right) ^{-1},\quad n\in \mathbb{N}.$$The next statement describes some properties of the running extrema of $\mathcal{S}$.
\[P\_AGKV\](compare with Proposition 2.3 in [@afanasyev2012limit]) Assume that Conditions (\[BasicCond\]) and (\[Def\_mu\]) hold. Then there exists a constant $c$ such that, uniformly for all $x,y\geq 0$ and all $n\in \mathbb{N}$ $$\mathbf{P}_{x}\left( L_{n}\geq 0,y-1\leq S_{n}<y\right) \ \leq \
c\,b_{n}\,U(x)V(y)\ , \label{LocalMin}$$and $$\mathbf{P}_{-x}\left( M_{n}<0,-y\leq S_{n}<-y+1\right) \ \leq \
c\,b_{n}\,V(x)U(y)\ . \label{LocalMax}$$
We prove the latter statement only. Since the density of any $\alpha $-stable law is bounded, it follows from the Gnedenko [@GK54] and Stone [@Sto65] local limit theorems that there exists a finite constant $C$ such that for all $n\in \mathbb{N}$ and all $z,\Delta \geq 0,$ $$\mathbf{P}\left( S_{n}\in \lbrack -z,-z+\Delta )\right) \leq \frac{C\Delta }{a_{n}}. \label{EstS1}$$Let $x,y\geq 0$, $\mathcal{S}^{\prime }$ be the dual random walk $$S_{i}^{\prime }=S_{n}-S_{n-i}$$and $L_{i}^{\prime }$, $i\leq n$, the corresponding minima. Denote $$\begin{aligned}
A_{n}\ :=\ \{& M_{\lfloor n/3\rfloor }<x\} \\
A_{n}^{\prime }\ :=\ \{& L_{\lfloor n/3\rfloor }^{\prime }\geq -y\}\ , \\
A_{n}^{\prime \prime }\ :=\ \{& x-y\leq S_{n}<x-y+1\} \\
=\ \{& x-y-T_{n}\leq S_{\lfloor 2n/3\rfloor }-S_{\lfloor n/3\rfloor
}<x-y-T_{n}+1\}\ ,\end{aligned}$$with $$T_{n}:=S_{\lfloor n/3\rfloor }+S_{n}-S_{\lfloor 2n/3\rfloor }.$$Let $\mathcal{A}_{n}$ be the $\sigma $–field generated by $X_{1},\ldots
,X_{\lfloor n/3\rfloor }$ and $X_{\lfloor 2n/3\rfloor +1},\ldots ,X_{n}$. Then $T_{n}$ is $\mathcal{A}_{n}$–measurable, whereas $S_{\lfloor
2n/3\rfloor }-S_{\lfloor n/3\rfloor }$ is independent of $\mathcal{A}_{n}$. Consequently from (\[EstS1\]) and the fact that $\left\{ a_{n},n\in
\mathbb{N}\right\} $ is regularly varying there is a $c>0$ such that $$\mathbf{P}\left( A_{n}^{\prime \prime }\,|\,\mathcal{A}_{n}\right) \ \leq \
ca_{n}^{-1}\ .$$Since $A_{n},A_{n}^{\prime }$ are $\mathcal{A}_{n}$-measurable and independent, it follows that $$\mathbf{P}\left( A_{n}\cap A_{n}^{\prime }\cap A_{n}^{\prime \prime }\right)
\ \leq \ ca_{n}^{-1}\mathbf{P}\left( A_{n}\right) \mathbf{P}\left(
A_{n}^{\prime }\right) \ .$$Moreover, according to Lemma \[L\_roughEstimates\] $$\mathbf{P}\left( L_{\lfloor n/3\rfloor }^{\prime }\geq -y\right) \ \leq \
c_{1}U(y)n^{-1}\hat{l}_{22}\left( n\right) \ ,\quad \mathbf{P}(M_{\lfloor
n/3\rfloor }<x)\ \leq \ c_{2}V(x)L_{44}(n),$$if $p>q$ and $$\mathbf{P}\left( L_{\lfloor n/3\rfloor }^{\prime }\geq -y\right) \ \leq \
c_{1}U(y)L_{33}\left( n\right) \ ,\quad \mathbf{P}(M_{\lfloor n/3\rfloor
}<x)\ \leq \ c_{2}V(x)n^{-1}\hat{l}_{55}(n),$$if $p<q$. This and (\[Equival\]) give the uniform estimate $$\mathbf{P}\left( A_{n}\cap A_{n}^{\prime }\cap A_{n}^{\prime \prime }\right)
\ \leq \ cV(x)U(y)\,b_{n}$$for $c$ sufficiently large. Now notice that $$\{M_{n}<x,x-y\leq S_{n}<x-y+1\}\ \subset \ A_{n}\cap A_{n}^{\prime }\cap
A_{n}^{\prime \prime }\ .$$The fact that the event on the left hand side is included in $A_{n}\cap
A_{n}^{\prime \prime }$ is straightforward. It is also included in $A_{n}^{\prime }$ due to the following series of inequalities, which hold for any $0\leq i\leq n$ on the event $\{x-y\leq S_n,M_{n}<x\}$: $$x-y\leq S_{n}-S_{i}+S_{i}\leq S_{n}-S_{i}+M_{n}\leq
S_{n}-S_{i}+x=S_{n-i}^{\prime }+x.$$This ends the proof.
We have now all the tools needed to prove the following statement.
\[L\_minim\] Assume that Conditions (\[BasicCond\]) and (\[Def\_mu\]) hold. Then for every $x\geq 0$ as $n\rightarrow \infty $
1\) if $p>q$ then $$\begin{aligned}
\mathbf{P}\left( L_{n}\geq -x\right) &\sim &U(x)\mathbf{P}\left( L_{n}\geq
0\right) \sim U(x)\frac{L_{22}(n)}{n},\quad \label{ExectMin3} \\
\mathbf{P}\left( M_{n}<x\right) &\sim &V(x)\mathbf{P}\left( M_{n}<0\right)
\sim V(x)L_{44}(n);\quad \notag\end{aligned}$$
2\) if $p<q$ then$$\begin{aligned}
\mathbf{P}\left( L_{n}\geq -x\right) &\sim &U(x)\mathbf{P}\left( L_{n}\geq
0\right) \sim U(x)L_{33}(n),\quad \label{ExectMin4} \\
\mathbf{P}\left( M_{n}<x\right) &\sim &V(x)\mathbf{P}\left( M_{n}<0\right)
\sim V(x)\frac{L_{55}(n)}{n}. \notag\end{aligned}$$
As the derivations of the four equivalents are similar, we only check the first one. Let $$\tau _{n}:=\min \left\{ j\leq n:S_{j}=L_{n}\right\} .$$We have $$\begin{aligned}
\mathbf{P}\left( L_{n}\geq -x\right) &=&\sum_{j=0}^{n}\mathbf{P}\left(
L_{n-j}\geq 0\right) \mathbf{P}\left( S_{j}\geq -x;\tau _{j}=j\right) \\
&=&\sum_{j=0}^{n}\mathbf{P}\left( L_{n-j}\geq 0\right) \mathbf{P}\left(
S_{j}\geq -x;M_{j}<0\right) ,\end{aligned}$$where we used the duality principle as in . In view of ([Ber1]{}), for any $\varepsilon \in \left( 0,1\right) $ and $j\leq \varepsilon
n$, $$\mathbf{P}\left( L_{n-j}\geq 0\right) \sim \frac{n}{n-j}\mathbf{P}\left(
L_{n}\geq 0\right) ,\quad n\rightarrow \infty .$$Moreover, from , we have $$\sum_{j=0}^{n\varepsilon }\mathbf{P}\left( S_{j}\geq -x;M_{j}<0\right)
=\sum_{j=0}^{n\varepsilon }\mathbf{P}\left( S_{j}\geq -x;\Gamma ^{\prime
}>j\right) \sim U(x),\quad n\rightarrow \infty . \label{Main_term}$$We deduce that $$\sum_{j=0}^{n\varepsilon }\mathbf{P}\left( L_{n-j}\geq 0\right) \mathbf{P}\left( S_{j}\geq -x;M_{j}<0\right) -\mathbf{P}\left( L_{n}\geq 0\right)
U(x)=O(\varepsilon )\mathbf{P}\left( L_{n}\geq 0\right) U(x)$$when $n$ is large enough. Further, by (\[LocalMin\]), (\[Ber1\]) and , for any $\delta >0$ $$\begin{aligned}
\sum_{j=n\varepsilon }^{n}\mathbf{P}\left( L_{n-j}\geq 0\right) \mathbf{P}\left( S_{j}\geq -x;M_{j}<0\right) &\leq &Cb_{n}xU(x)\sum_{j=n\varepsilon
}^{n}\mathbf{P}\left( L_{n-j}\geq 0\right) \notag \\
&\leq &\frac{CxU(x)}{na_{n}}L_{22}(n) \notag \\
&=&\frac{CxU(x)}{n^{2}}\frac{L_{22}(n)}{L_{4}(n)}=o\left( \frac{1}{n^{2-\delta }}\right) \notag \\
&=&o\left( \mathbf{P}\left( L_{n}\geq 0\right) \right) ,\quad n\rightarrow
\infty , \label{Neglterm}\end{aligned}$$since $(L_{22}(n)/L_{4}(n))n^{-\delta }\rightarrow 0$ as $n\rightarrow
\infty $ for any $\delta >0.$ Combining (\[Main\_term\]) and (\[Neglterm\]) and letting $\varepsilon \rightarrow 0$ give (\[ExectMin3\]).
The last result of this section is a technical statement which will be needed in the proof of Theorem \[T\_main\]. As Lemma \[L\_minim\], it is a consequence of Lemma \[P\_AGKV\] and can be proven in the same way as Corollary 2.4 in [@afanasyev2012limit].
\[C\_exponential\] Assume that Conditions (\[BasicCond\]) and (\[Def\_mu\]) hold. For any $\theta >0$ there exists a finite $c$ (depending on $\theta $) such that for all $x,y\geq 0$ $$\mathbf{E}_{x}\big[e^{-\theta S_{n}};L_{n}\geq 0,S_{n}\geq y\big ]\leq c\
b_{n}V(x)U(y)\ e^{-\theta y}$$and $$\mathbf{E}_{-x}\big[e^{\theta S_{n}};M_{n}<0,S_{n}<-y\big ]\leq c\
b_{n}V(y)U(x)\ e^{\theta y}\ .$$
Change of measure {#sec_chgtprob}
=================
Recall the definition of the renewal function $U$ in (\[Def\_U\]). One of its fundamental properties is the identity (see, for instance, [kozlov1977asymptotic,bertoin1994conditioning]{}) $$\mathbf{E}\left[ U(x+X);X+x\geq 0\right] =U(x),\,x\geq 0. \label{DefV}$$This property has often been used to construct a change of probability measure (see for instance [@geiger2001survival]), and we will use such a construction in our proof.
Denote by ${\mathcal{F}}$ the filtration consisting of the $\sigma -$algebras ${\mathcal{F}}_{n}$ generated by the random variables $S_{0},...,S_{n}$ and $Z_{0},...,Z_{n}$. Taking into account $U(0)=1$ we may introduce probability measures $\mathbf{P}_{n}^{+}$ on the $\sigma $-fields $\mathcal{F}_{n}$ by means of the densities $$d\mathbf{P}_{n}^{+}\ :=\ U(S_{n})I_{\{L_{n}\geq 0\}}\,d\mathbf{P}\ .$$Because of the martingale property the measures are consistent, i.e., $\mathbf{P}_{n+1}^{+}|\mathcal{F}_{n}=\mathbf{P}_{n}^{+}$. Therefore (choosing a suitable underlying probability space), there exists a probability measure $\mathbf{P}^{+}$ on the $\sigma $-field $\mathcal{F}_{\infty }:=\bigvee_{n}\mathcal{F}_{n}$ such that $$\mathbf{P}^{+}|\mathcal{F}_{n}\ =\ \mathbf{P}_{n}^{+}\,,\quad n\geq 0\,.
\label{ppp}$$We note that (\[ppp\]) can be rewritten as $$\mathbf{E}^{+}\,\left[ Y_{n}\right] \ =\ \mathbf{E}[Y_{n}U(S_{n});L_{n}\geq
0] \label{measurechange}$$for every $\mathcal{F}_{n}$–measurable nonnegative random variable $Y_{n}$. This change of measure is the well-known Doob $h$-transform from the theory of Markov processes. In particular, under $\mathbf{P}^{+}$ the process $S$ becomes a Markov chain with state space $\mathbb{R}_{0}^{+}$ and transition kernel $$P^{+}(x;dy)\ :=\ \frac{1}{U(x)}\mathbf{P}\left( x+X\in dy\right) U(y)\
,\quad x\geq 0\ .$$In our context, we can show that $\mathbf{P}^{+}$ can be realised as the limit of the probability of the process conditioned to live in a nonnegative environment (in the sense that the running infimum is null). It is the content of the next lemma, and will allow us to link the survival probability of the population process to the probability for the running infimum to be null, in order to prove Theorem \[T\_main\].
\[conditioning\] (compare with Lemma 2.5 in [afanasyev2005criticality]{}) Assume that Conditions (\[BasicCond\]) and ([Def\_mu]{}) hold. For $k\in \mathbb{N}$ let $Y_{k}$ be a bounded real-valued $\mathcal{F}_{k}$–measurable random variable. Then, as $n\rightarrow \infty $, $$\mathbf{E}[Y_{k}\;|\;L_{n}\geq 0]\ \rightarrow \ \mathbf{E^{+}}\,Y_{k}\ .
\label{claim1_L_conditioning}$$More generally, let $Y_{1},Y_{2},\ldots $ be a uniformly bounded sequence of real-valued random variables adapted to the filtration $\mathcal{F}$, which converges $\mathbf{P}^{+}$–$a.s.$ to some random variable $Y_{\infty }$. Then, as $n\rightarrow \infty $, $$\mathbf{E}[Y_{n}\;|\;L_{n}\geq 0]\ \rightarrow \ \mathbf{E}^{+}\;Y_{\infty
}\ .$$
The proof of this lemma in the case $p>q$ coincides with the proof of Lemma 2.5 in [@afanasyev2005criticality] when taking $\rho =0$ and we omit it. In the case $p<q$ some modifications are needed to check the second claim of the lemma. Namely, writing $$m_{l}(x):=\mathbf{P}(L_{l}\geq -x)\quad \text{for}\quad x\geq 0,\ l\in
\mathbb{N}$$and using , and we deduce for $\lambda >1$, $k\leq n$ and $n$ large enough, the existence of a finite $C$ such that $$\begin{aligned}
\left\vert \mathbf{E}[Y_{n}-Y_{k}|I\left\{ L_{\lfloor \lambda n\rfloor }\geq
0\right\} ]\right\vert & \leq \mathbf{E}\left[ \left\vert
Y_{n}-Y_{k}\right\vert \frac{m_{\lfloor (\lambda -1)n\rfloor }(S_{n})}{m_{\lfloor \lambda n\rfloor }(0)}I\left\{ L_{n}\geq 0\right\} \right] \\
& \leq C\mathbf{E}\left[ \left\vert Y_{n}-Y_{k}\right\vert U(S_{n})I\left\{
L_{n}\geq 0\right\} \right] \\
& =C\mathbf{E}^{+}\left[ \left\vert Y_{n}-Y_{k}\right\vert \right] .\end{aligned}$$Letting sequentially $n$ and $k$ go to infinity and applying the dominated convergence theorem, we obtain that the right hand side of the previous series of inequalities vanishes. Applying now the first claim of the lemma and using the fact that $n\mapsto \mathbf{P}(L_{n}\geq 0)$ is slowly varying we obtain $$\mathbf{E}[Y_{n};L_{\lfloor \lambda n\rfloor }\geq 0]=\left( \mathbf{E}^{+}[Y_{\infty }]+o(1)\right) \mathbf{P}(L_{\lfloor \lambda n\rfloor }\geq
0)=\left( \mathbf{E}^{+}[Y_{\infty }]+o(1)\right) \mathbf{P}(L_{n}\geq 0)$$and $$\mathbf{E}[Y_{n};L_{n}\geq 0]-\mathbf{E}[Y_{n};L_{\lfloor \lambda n\rfloor
}\geq 0]=o\left( \mathbf{P}(L_{n}\geq 0)\right) .$$This ends the proof.
Let $\nu \geq 1$ be the time of the first *prospective minimal value* of $\mathcal{S}$, i.e., a minimal value with respect to the future development of the walk, $$\nu \ :=\ \min \{m\in \mathbb{N}\;:\;S_{m+i}\geq S_{m}\text{ for all }i\geq
0\}.$$Moreover, let $\iota \in \mathbb{N}$ be the first weak ascending ladder epoch of $S$, $$\iota \ :=\ \min \{m\in \mathbb{N}\;:\;S_{m}\geq 0\}\ .$$We denote $$\widetilde{f}_{n}\ :=\ f_{\nu +n}\ \mbox{ and }\ \widetilde{S}_{n}\ :=\
S_{\nu +n}-S_{\nu },\quad n\in \mathbb{N}.$$
The previous result allows us to rigorously express what we mean by *living in a good environment* for the population process. The next lemma and its proof are the same as Lemma 2.6 in [@afanasyev2005criticality] and its proof. We thus do not provide it and refer the reader to [afanasyev2005criticality]{}.
\[L\_tanaka\] (see Lemma 2.6 in [@afanasyev2005criticality]) Suppose that $\iota <\infty $ $\mathbf{P}$–$a.s.$ Then $\nu <\infty $ $\,\mathbf{P}^{+}$–$a.s.$ and
1. $(f_{1},f_{2},\ldots )$ and $(\widetilde{f}_{1},\widetilde{f}_{2},\ldots )$ are identically distributed with respect to $\mathbf{P}^{+}$;
2. $(\nu ,f_{1},\ldots ,f_{\nu })$ and $(\widetilde{f}_{1},\widetilde{f}_{2},\ldots )$ are independent with respect to $\mathbf{P}^{+}$;
3. $\mathbf{P}^+ \{\nu = k, S_{\nu} \in dx \} = \mathbf{P} \{\iota = k,
S_{\iota} \in dx \}$ for all $k \geq 1$.
Proof of Theorem \[T\_main\] {#sec_proof}
============================
Thanks to the results we have collected in the previous sections, we are now able to prove our main result. We have already demonstrated (Lemmas [conditioning]{} and \[L\_tanaka\]) that we can divide the survival probability of $\mathcal{Z}$ until time $n$ into two parts: the probability for the process to survive until the time when the running infimum $L_{n}$ is attained for the first time, and the probability that the process $\mathcal{Z}$ survives in a “good” environment, i.e., in an environment with a running infimum of $L$ null. We still have to prove that the population indeed has a nonnegligible probability to survive in this good environment, for large $n$. It is the content of the next result.
Let $$\eta _{k}\ :=\ \sum_{y=0}^{\infty }y(y-1)\;f_{k}\left[ y\right] \ \Big/\ \Big(\sum_{y=0}^{\infty }y\;f_{k}\left[ y\right] \Big)^{2}\,,\quad k\in
\mathbb{N}.$$
\[L\_borel\] Assume that Conditions (\[BasicCond\]) and (\[Def\_mu\]) hold. If $p>q$, and Conditions A and C hold or if $p<q$ and Condition B holds, then $$\sum_{k=0}^{\infty }\eta _{k+1}e^{-S_{k}}\ <\ \infty \quad \mathbf{P}^{+}\text{--}a.s.$$
Let us first assume that $p>q$, and Conditions A and C hold. Recall the definition of the standardized truncated second moment of the environment in . Following [@afanasyev2005criticality] Equation (2.24) we have the following bound, for any $a \in \mathbb{N}$, $$\begin{aligned}
\sum_{k=0}^{\infty }\eta _{k+1}e^{-S_{k}} &\leq a \sum_{k=0}^{\infty
}e^{-S_{k}} +\sum_{k=0}^{\infty } \zeta_{k+1}(a)e^{-S_{k}} \notag \\
& =: \mathcal{A}_a+\mathcal{B}_a. \label{defAandA}\end{aligned}$$
The first step of the proof consists in bounding the two sums by using the times $0:=\nu (0)<\nu (1)<\cdots $ of prospective minima of $\mathcal{S}$, defined by $$\nu (j)\ :=\ \min \{m>\nu ({j-1})\;:\;S_{m+i}\geq S_{m}\text{ for all }i\geq
0\}\,,\quad j\in \mathbb{N}. \label{renewal}$$
By definition, $$S_{k}\geq S_{\nu (j)},\ \mbox{ if }\,k\geq \nu (j). \label{simple}$$Thus, we get $$\mathcal{A}_a \leq a \sum_{j=0}^{\infty } (\nu(j+1)-\nu(j)) e^{-S_{v(j)}},$$and $$\mathcal{B}_a \leq \sum_{j=0}^{\infty } \left(\sum_{k=\nu(j)+1}^{\nu(j+1)}
\zeta_k(a) \right) e^{-S_{v(j)}}.$$
Now we aim at bounding the variables $\nu (j)$. For the sake of readability, let us introduce $$\nu _{j}=\nu (j)-\nu (j-1),\quad j\in \mathbb{N}.$$By Lemma \[L\_tanaka\].$($*1)* and *(2)*, $\nu (j)$ is the sum of $j$ nonnegative i.i.d. random variables, each having the distribution of $\nu =\nu (1)=\nu _{1}$. Lemma \[L\_tanaka\].$($*3)* and (\[Ber3\]) imply for large $k$ $$\begin{aligned}
\mathbf{P}^{+}\left( \nu >k\right) \ &=&\ \mathbf{P}\{\iota >k\}\ =\
\mathbf{P}\{M_{k}<0\}\ \\
&\leq &2L_{44}(k)= 2/ \Lambda(k).\end{aligned}$$ These estimates and Condition** C** imply $$\sum_{j=1}^{\infty }\mathbf{P}^{+}\left( \nu _{j}>g(j)\right) \leq
2\sum_{j=1}^{\infty } 1/\Lambda(g(j))<\infty .$$ Hence, by the Borel-Cantelli lemma there will be $\mathbf{P}^{+}$–$a.s.$ only a finite number of cases when $\nu _{j}>g(j)$. And as $g(i)=e^{o(i)},\
i\rightarrow \infty $, for any $\gamma >0$, $$\sum_{i=0}^{j}g(i)=o\left( e^{\gamma j}\right) ,\quad j\rightarrow \infty .$$Thus, there will be $\mathbf{P}^{+}$–$a.s.$ only a finite number of cases when $\nu (j)>e^{\gamma j}$.
Now we would like to bound the term $$\sum_{k=\nu (j)+1}^{\nu (j+1)}\zeta _{k}(a)$$in order to show that the random variable $\mathcal{B}_{a}$ is almost surely finite. The first step to obtain this bound is to use the inequality (2.25) in [@afanasyev2005criticality], that we now recall: for any $x\geq 0$, $$\mathbf{P}^{+}(\zeta _{k}(a)>x)\leq \mathbf{P}(\zeta _{1}(a)>x)+\mathbf{E}[U(X_{1});\zeta _{1}(a)>x]\mathbf{P}(L_{k-1}\geq 0).$$Applying it with $x=k^{\alpha /\gamma }$ (with $\alpha >0$ to be precised later on) and using the Markov inequality as well as Condition** A** yield for any $k\in \mathbb{N}$, $$\mathbf{P}^{+}(\zeta _{k}(a)>k^{\alpha /\gamma })\leq \frac{c}{k^{\alpha
\beta /\gamma }}+\frac{c}{k^{\alpha \beta /\gamma }}\mathbf{P}(L_{k-1}\geq
0)\leq \frac{c}{k^{\alpha \beta /\gamma }}+\frac{c}{k^{\alpha \beta /\gamma }}\frac{\hat{l}_{22}(k)}{k},$$where we applied and the value of $c$ can change from line to line. The constants $\alpha $ and $\beta $ are fixed. However, we know that $\gamma $ can be chosen as small as we want. In particular, we may select it in such a way that $\alpha \beta /\gamma =2$. Applying again the Borel-Cantelli lemma we deduce that there is $\mathbf{P}^{+}$–$a.s.$ only a finite number of cases when $\zeta _{k}(a)>k^{\alpha /\gamma }$.
Combining this fact with the previous results we obtain that for $j$ large enough and $k\in \lbrack \nu (j-1)+1,\nu (j)]$, $\mathbf{P}^{+}$–$a.s.$, $$\zeta _{k}(a)\leq k^{\alpha /\gamma }\leq \left( e^{\gamma j}\right)
^{\alpha /\gamma }=e^{\alpha j}\quad \text{and}\quad \nu _{j}\leq g(j).$$Hence for $j$ large enough, $\mathbf{P}^{+}$–$a.s.$, $$\sum_{k=\nu (j-1)+1}^{\nu (j)}\zeta _{k}(a)\leq e^{\alpha j}\nu _{j}\leq
e^{\alpha j}g(j).$$
The last part of the proof consists in estimating the $S_{\nu (j)}$ from below. According to Lemma \[L\_tanaka\] *(1)* and *(2)*, the random variable $S_{\nu (j)}$ is the sum of $j$ non-negative i.i.d. random variables with positive mean. Thus, there exists a $\lambda >0$ such that $$S_{\nu (j)}\ \geq \ \lambda j\qquad \text{eventually}\quad \mathbf{P}^{+}\text{--}a.s. \label{lln}$$
Choosing $\alpha <\lambda $ in the previous inequalities, we obtain
$$\begin{aligned}
\sum_{k=0}^{\infty }\eta _{k+1}e^{-S_{k}}\leq \mathcal{A}_{a}+\mathcal{B}_{a}& \leq c\sum_{j=0}^{\infty }(a+e^{\alpha (j+1)})\nu _{j+1}e^{-S_{v(j)}} \\
& \leq c\sum_{j=0}^{\infty }e^{\alpha (j+1)}g(j+1)e^{-\lambda j}<\infty \qquad
\mathbf{P}^{+}\text{--}a.s.,\end{aligned}$$
where the value of $c$ can change from line to line. It ends the proof for the case $p>q$.
The proof for the case $p<q$ is the same as the proof of Lemma 2.7 in [afanasyev2005criticality]{}. Indeed, even if the authors of the mentioned paper assume $\rho \in (0,1)$, their proof remains valid when we take $\rho
=1$ as it is the case when $p<q$.
In the following example, we illustrate the fact that **Condition C** is not too strong and that we can find slowly varying functions $L(\cdot )$ satisfying this assumption. To this aim we choose $g(j)=e^{j^{1-\theta }}$ for a $\theta \in (0,1)$.
\[ex1\] Let us consider a slowly varying function $L(\cdot )$ satisfying $$L(x)\sim \frac{c}{\log ^{m+1}x},\quad m>(p-q)/p,$$where $c$ is a positive constant. Then, as $x\rightarrow \infty $$$l^{\ast }(x)=\int_{x}^{\infty }\frac{L(u)}{u}du\sim \frac{c}{m\log ^{m}x}$$ and, by $$a_{n}\sim \frac{n}{\log ^{m+1}n}, \quad n\rightarrow \infty .$$Therefore, $$\left\vert h_{n}\right\vert \sim \left\vert \left( p-q\right) nl^{\ast
}\left( a_{n}\right) \right\vert \sim c_{2}\frac{n}{\log ^{m}n},\quad
n\rightarrow \infty ,$$for a positive $c_{2}.$
Now from Lemma 7.3 in [@Berg17], we know that $$\mathbf{P}(S_n\geq 0) \sim \frac{p}{p-q} \frac{L(|h_n|)}{l^\ast(|h_n|)}$$ Hence, using the previous calculations we obtain $$\mathbf{P}(S_n\geq 0) \sim \frac{p m}{(p-q)\ln n}.$$ As $$\frac{p m}{(p-q)} \sum_{k=2}^n \frac{1}{k\ln k} \sim \frac{p m}{(p-q)}\ln \ln n, \quad n \to \infty,$$ an application of Corollary 1.7.3 in [@bingham1989regular] yields, as $n\to\infty$ $$\sum_{k=1}^\infty \frac{\mathbf{P}(S_k \geq 0)}{k}\left(1-\frac{1}{n}\right)^k \sim \frac{p m}{(p-q)} \ln \ln n ,$$ and, in particular, for $\theta\in (0,1)$ as $j\to\infty$ $$\sum_{k=1}^\infty \frac{\mathbf{P}(S_k \geq 0)}{k}\left(1-\frac{1}{e^{j^{1-\theta }}}\right)^k \sim \frac{p m}{(p-q)} \ln \ln e^{j^{1-\theta }}
=\frac{p m}{(p-q)}(1-\theta)\ln j.$$ As a consequence, for any $\varepsilon>0$, there exist $j(\varepsilon)$ such that for $j \geq j(\varepsilon)$, $$\Lambda (e^{j^{1-\theta }}) \geq j^{(1-\varepsilon)(1-\theta)p m/(p-q)}.$$ As $p m/(p-q)>1$, we just have to choose $\varepsilon > 0$ and $\theta\in (0,1)$ such that $(1-\varepsilon)(1-\theta)p m/(p-q)>1$ to conclude that **Condition C** holds for $p>q.$
Introduce iterations of probability generating functions $\ \
f_{1}(.),f_{2}(.),...$ by setting $$f_{k,n}(s):=f_{k+1}(f_{k+2}(\ldots (f_{n}(s))\ldots ))$$for $0\leq k\leq n-1$, $0 \leq s \leq 1$, and letting $f_{n,n}(s):=s.$ By definition,$$\mathbf{P}\left( Z_{n}>0|\ f_{k+1},\ldots ,f_{n};Z_{k}=1\right) =1-f_{k,n}(0)$$and we have (see, for instance, formula (3.4) in [afanasyev2005criticality]{})$$1-f_{0,n}(0)\geq \left( e^{-S_{n}}+\sum_{k=1}^{n-1}\eta
_{k+1}e^{-S_{k}}\right) ^{-1}$$implying by Lemma \[L\_borel\] $$1-f_{0,\infty }(0):=\lim_{n\rightarrow \infty }\left( 1-f_{0,n}(0)\right)
\geq \left( \sum_{k=1}^{\infty }\eta _{k+1}e^{-S_{k}}\right) ^{-1}\text{ }\mathbf{P}^{+}-\text{a.s.} \label{Est_f2}$$
We finally provide the proof of our main result.
Let us begin with the case $p>q$. We write $$\begin{aligned}
\mathbf{P}\left( Z_{n}>0\right) &=&\sum_{k=0}^{n}\mathbf{P}\left(
Z_{n}>0;\tau _{n}=k\right) \notag \label{majprop3} \\
&=&\sum_{k=0}^{N}\mathbf{E}\left[ 1-f_{0,n}(0);\tau _{n}=k\right]
+\sum_{k=N+1}^{n\varepsilon }\mathbf{E}\left[ 1-f_{0,n}(0);\tau _{n}=k\right]
\notag \\
&&+\sum_{k=n\varepsilon +1}^{n}\mathbf{E}\left[ 1-f_{0,n}(0);\tau _{n}=k\right] ,\end{aligned}$$for some $N\in \mathbb{N}$ to be precised later on, and a small positive $\varepsilon $. Let us first bound the second term in the right hand side of $$\begin{aligned}
\sum_{k=N+1}^{n\varepsilon }\mathbf{E}\left[ 1-f_{0,n}(0);\tau _{n}=k\right]
&\leq &\sum_{k=N+1}^{n\varepsilon }\mathbf{E}\left[ 1-f_{0,k}(0);\tau _{n}=k\right] \\
&=&\sum_{k=N+1}^{n\varepsilon }\mathbf{E}\left[ 1-f_{0,k}(0);\tau _{k}=k\right] \mathbf{P}\left( L_{n-k}\geq 0\right) \\
&\leq &\sum_{k=N+1}^{n\varepsilon }\mathbf{E}\left[ e^{S_{k}};\tau _{k}=k\right] \mathbf{P}\left( L_{n-k}\geq 0\right) .\end{aligned}$$By the duality principle for random walks and Lemma \[C\_exponential\], with $x=y=0$, we have $$\mathbf{E}\left[ e^{S_{k}};\tau _{k}=k\right] =\mathbf{E}\left[
e^{S_{k}};M_{k}<0\right] \leq c\ b_{k}.$$This estimate, the equivalence $$b_{k}=\frac{1}{ka_{k}}\sim \frac{1}{k^{2}L_{4}(k)}$$and give$$\begin{aligned}
\sum_{k=N+1}^{n\varepsilon }\mathbf{E}\left[ 1-f_{0,n}(0);\tau _{n}=k\right]
&\leq &\mathbf{P}\left( L_{n(1-\varepsilon )}\geq 0\right)
\sum_{k=N+1}^{\infty }\frac{1}{k^{2}L_{4}(k)} \\
&\leq &C\frac{\mathbf{P}\left( L_{n(1-\varepsilon )}\geq 0\right) }{a_{N}}.\end{aligned}$$Now we focus on the third part of the right hand side of . Similarly as for the second part, we have the following series of inequalities, where the value of the finite constant $C$ may change from line to line and may depend on $\varepsilon $: $$\begin{aligned}
\sum_{k=n\varepsilon +1}^{n}\mathbf{E}\left[ 1-f_{0,n}(0);\tau _{n}=k\right]
&\leq &C\sum_{k=n\varepsilon +1}^{n}\frac{1}{k^{2}L_{4}(k)}\mathbf{P}\left(
L_{n-k}\geq 0\right) \notag \label{maj_part3} \\
&\leq &C\frac{L_{22}(n)}{n^{2}L_{4}(n)}=o\left( \frac{1}{n^{3/2}}\right) .\end{aligned}$$Finally, $$\sum_{k=0}^{N}\mathbf{E}\left[ 1-f_{0,n}(0);\tau _{n}=k\right]
=\sum_{k=0}^{N}\mathbf{E}\left[ 1-f_{k,n}^{Z_{k}}(0);\tau _{k}=k,L_{k,n}\geq
0\right]$$where $$L_{k,n}=\min_{k\leq j\leq n}\left( S_{j}-S_{k}\right) .$$Recalling Lemma \[conditioning\] and using the independency and homogeneity of the environmental components we conclude that for $k\leq N$, $$\begin{aligned}
\mathbf{E}\left[ 1-f_{k,n}^{Z_{k}}(0);\tau _{k}=k,L_{k,n}\geq 0\right]
&=&\sum_{j=1}^{\infty }\mathbf{P}\left( Z_{k}=j,\tau _{k}=k\right) \mathbf{P}\left( L_{n-k}\geq 0\right) \mathbf{E}\left[ 1-f_{0,n-k}^{j}(0)|L_{n-k}\geq 0\right] \\
&\sim &\mathbf{P}\left( L_{n}\geq 0\right) \sum_{j=1}^{\infty }\mathbf{P}\left( Z_{k}=j,\tau _{k}=k\right) \mathbf{E}^{+}\left[ 1-f_{0,\infty }^{j}(0)\right]\end{aligned}$$as $n\rightarrow \infty $. Note that by Lemma \[L\_borel\] and ([Est\_f2]{}) $$\mathbf{E}^{+}\left[ 1-f_{0,\infty }^{j}(0)\right] \geq \mathbf{E}^{+}\left[
1-f_{0,\infty }(0)\right] \geq \mathbf{E}^{+}\left[ \left(
\sum_{k=0}^{\infty }\eta _{k+1}e^{-S_{k}}\right) ^{-1}\right] >0.$$Thus, letting first $n$ to infinity, then $\varepsilon $ to zero and, finally, $N$ to infinity we prove (\[survivP>Q\]).
The proof for the case $q>p$ is very similar. The only difference is when looking for an equivalent of Equation . Applying yields $$\begin{aligned}
\sum_{k=n\varepsilon +1}^{n}\mathbf{E}\left[ 1-f_{0,n}(0);\tau _{n}=k\right]
&\leq &C\sum_{k=n\varepsilon +1}^{n}\frac{1}{k^{2}L_{4}(k)}\mathbf{P}\left(
L_{n-k}\geq 0\right) \\
&\leq &C_{\varepsilon }\frac{L_{33}(n)}{nL_{4}(n)}=o\left( \frac{1}{n^{1/2}}\right) =o\left( \mathbf{P}\left( L_{n}\geq 0\right) \right) .\end{aligned}$$We end the proof as for .
The authors thank I. Kortchemski for attracting their attention to reference [@kortchemski2019condensation] which allowed them to weaken the conditions for their main result to hold. C.S. thanks the CNRS for its financial support through its competitive funding programs on interdisciplinary research, the ANR ABIM 16-CE40-0001 as well as the Chair “Modélisation Mathématique et Biodiversité” of VEOLIA-Ecole Polytechnique-MNHN-F.X. She also thanks the French Embassy in Russia for its financial support via the André Mazon program.
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title: 'Measurement of t-tbar production with additional jet activity, including b quark jets, in the dilepton decay channel using pp collisions at sqrt(s) = 8 TeV'
---
=1
$Revision: 360507 $ $HeadURL: svn+ssh://svn.cern.ch/reps/tdr2/papers/TOP-12-041/trunk/TOP-12-041.tex $ $Id: TOP-12-041.tex 360507 2016-07-28 11:41:46Z cdiez $
\[1\][D[,]{}[ ]{}[\#1]{}]{}
Introduction
============
Precise measurements of production and decay properties provide crucial information for testing the expectations of the standard model (SM) and specifically of calculations in the framework of perturbative quantum chromodynamics (QCD) at high-energy scales. At the energies of the CERN LHC, about half of the events contain jets with transverse momentum () larger than 30that do not come from the weak decay of the system [@bib:TOP-12-018]. In this paper, these jets will be referred to as “additional jets” and the events as “+jets”. The additional jets typically arise from initial-state QCD radiation, and their study provides an essential test of the validity and completeness of higher-order QCD calculations describing the processes leading to multijet events.
A correct description of these events is also relevant because +jets processes constitute important backgrounds in the searches for new physics. These processes also constitute a challenging background in the attempt to observe the production of a Higgs boson in association with a pair ([$\PQt\PAQt\PH$]{}), where the Higgs boson decays to a bottom () quark pair (), because of the much larger cross section compared to the [$\PQt\PAQt\PH$]{}signal. Such a process has an irreducible nonresonant background from pair production in association with a pair from gluon splitting. Therefore, measurements of +jets and [$\PQt\PAQt\PQb\PAQb$]{}production can give important information about the main background in the search for the [$\PQt\PAQt\PH$]{}process and provide a good test of next-to-leading-order (NLO) QCD calculations.
Here, we present a detailed study of the production of events with additional jets and quark jets in the final state from pp collisions at $\sqrt{s} = 8\TeV$ using the data recorded in 2012 with the CMS detector, corresponding to an integrated luminosity of 19.7 . The pairs are reconstructed in the dilepton decay channel with two oppositely charged isolated leptons (electrons or muons) and at least two jets. The analysis follows, to a large extent, the strategy used in the measurement of normalized differential cross sections in the same decay channel described in Ref. [@bib:TOP-12-028].
The measurements of the absolute and normalized differential cross sections are performed as a function of the jet multiplicity for different thresholds for the jets, in order to probe the momentum dependence of the hard-gluon emission. The results are presented in a visible phase space in which all selected final-state objects are produced within the detector acceptance and are thus measurable experimentally. The study extends the previous measurement at $\sqrt{s} = 7\TeV$ [@bib:TOP-12-018], where only normalized differential cross sections were presented.
The absolute and normalized +jets production cross sections are also measured as a function of the and pseudorapidity ($\eta$) [@bib:CMS] of the leading additional jets, ordered by . The CMS experiment has previously published a measurement of the inclusive [$\PQt\PAQt\PQb\PAQb$]{}production cross section [@bib:ttbb_ratio:2014]. In the present analysis, the [$\PQt\PAQt\PQb\PAQb$]{}and [$\PQt\PAQt\PQb$]{}(referred to as “[$\PQt\PAQt\PQb\PAQb$]{}([$\PQt\PAQt\PQb$]{})” in the following) cross sections are measured for the first time differentially as a function of the properties of the additional jets associated with quarks, which will hereafter be called jets. The [$\PQt\PAQt\PQb\PAQb$]{}process corresponds to events where two additional jets are generated in the visible phase space, while [$\PQt\PAQt\PQb$]{}represents the same physical process, where only one additional jet is within the acceptance requirements. In cases with at least two additional jets or two jets, the cross section is also measured as a function of the angular distance between the two jets and their dijet invariant mass. The results are reported both in the visible phase space and extrapolated to the full phase space of the system to facilitate the comparison with theoretical calculations.
Finally, the fraction of events that do not contain additional jets (gap fraction) is determined as a function of the threshold on the leading and subleading additional-jet , and the scalar sum of all additional-jet . This was first measured in Refs. [@bib:TOP-12-018] and [@bib:atlas2].
The results are compared at particle level to theoretical predictions obtained with four different event generators: [@bib:madgraph], [@bib:mcatnlo], [@Frixione:2007vw], and [<span style="font-variant:small-caps;">MG5\_aMC@NLO</span>]{} [@MG5_amcnlo], interfaced with either [@Sjostrand:2006za] or [@bib:herwig], and in the case of with both. Additionally, the measurements as a function of the jet quantities are compared to the predictions from the event generator [[<span style="font-variant:small-caps;">PowHel</span>]{}]{} [@Garzelli:2014aba].
This paper is structured as follows. A brief description of the CMS detector is provided in Section \[sec:CMS\]. Details of the event simulation generators and their theoretical predictions are given in Section \[sec:theory\]. The event selection and the method used to identify the additional radiation in the event for both +jets and [$\PQt\PAQt\PQb\PAQb$]{}([$\PQt\PAQt\PQb$]{}) studies are presented in Sections \[sec:selection\] and \[sec:tt\_addjets\]. The cross section measurement and the systematic uncertainties are described in Sections \[sec:syst\] and \[sec:diffxsec\]. The results as a function of the jet multiplicity and the kinematic properties of the additional jets and jets are presented in Sections \[sec:diffxsecNJets\]–\[sec:diffxsecAddbJets\]. The definition of the gap fraction and the results are described in Section \[sec:gap\]. Finally, a summary is given in Section \[sec:summary\].
The CMS detector {#sec:CMS}
================
The central feature of the CMS apparatus is a superconducting solenoid of 6 internal diameter, providing a magnetic field of 3.8. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter, each composed of a barrel and two endcap sections. Extensive forward calorimetry complements the coverage provided by the barrel and endcap detectors. Muons are measured in gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [@bib:CMS].
Event simulation and theoretical predictions {#sec:theory}
============================================
Experimental effects coming from event reconstruction, selection criteria, and detector resolution are modelled using Monte Carlo (MC) event generators interfaced with a detailed simulation of the CMS detector response using (v. 9.4) [@Agostinelli:2002hh].
The (v. 5.1.5.11) [@bib:madgraph] generator calculates the matrix elements at tree level up to a given order in $\alpha_s$. In particular, the simulated sample used in this analysis is generated with up to three additional partons. The [[<span style="font-variant:small-caps;">MadSpin</span>]{}]{} [@bib:madspin] package is used to incorporate spin correlations of the top quark decay products. The value of the top quark mass is chosen to be $m_{\PQt} = 172.5\GeV$, and the proton structure is described by the CTEQ6L1 [@bib:cteq] set of parton distribution functions (PDF). The generated events are subsequently processed with (v. 6.426) [@Sjostrand:2006za] for fragmentation and hadronization, using the MLM prescription for the matching of higher-multiplicity matrix element calculations with parton showers [@Mangano:2006rw]. The parameters for the underlying event, parton shower, and hadronization are set according to the Z2\* tune, which is derived from the Z1 tune [@Field:2010bc]. The Z1 tune uses the CTEQ5L PDFs, whereas Z2\* adopts CTEQ6L.
In addition to the nominal sample, dedicated samples are generated by varying the central value of the renormalization ($\mu_\mathrm{R}$) and factorization ($\mu_\mathrm{F}$) scales and the matrix element/parton showering matching scale (jet-parton matching scale). These samples are produced to determine the systematic uncertainties in the measurement owing to the theoretical assumptions on the modelling of events, as well as for comparisons with the measured distributions. The nominal values of $\mu_\mathrm{R}$ and $\mu_\mathrm{F}$ are defined by the $Q^2$ scale in the event: $\mu_\mathrm{R}^2 =\mu_\mathrm{F}^2 = Q^2 = m_{\PQt}^2 + \sum{\pt^2(\text{jet})}$, where the sum runs over all the additional jets in the event not coming from the decay. The samples with the varied scales use $\mu_\mathrm{R}^2 =\mu_\mathrm{F}^2 = 4Q^2$ and $Q^2/4$, respectively. For the nominal sample, a jet-parton matching scale of 40is chosen, while for the varied samples, values of 60 and 30are employed, respectively. These scales correspond to jet-parton matching thresholds of 20for the nominal sample, and 40 and 10for the varied ones.
The (v. 1.0 r1380) and (v. 3.41) generators, along with the CT10 [@bib:CT10] and CTEQ6M [@bib:cteq] PDFs, are used, respectively, for comparisons with the data. The generator simulates calculations of production to full NLO accuracy, and is matched with two parton shower MC generators: the (v. 6.426) Z2\* tune (designated as in the following), and the [@bib:herwig] (v. 6.520) AUET2 tune [@bib:auet2tune] (referred to as in the following). The parton showering in is based on a transverse-momentum ordering of parton showers, whereas uses angular ordering. The generator implements the hard matrix element to full NLO accuracy, matched with (v. 6.520) for the initial- and final-state parton showers using the default tune. These two generators, and , are formally equivalent up to the NLO accuracy, but they differ in the techniques used to avoid double counting of radiative corrections that may arise from interfacing with the parton showering generators.
=1200 The cross section as a function of jet multiplicity and the gap fraction measurements are compared to the NLO predictions of the (v2) [@Frixione:2007vw] and [<span style="font-variant:small-caps;">MG5\_aMC@NLO</span>]{} [@MG5_amcnlo] generators. The (v2) generator is matched to the (v. 8.205) CUETP8M1 tune [@Astalos:2015ivw] (referred to as ), , and . In these samples the [<span style="font-variant:small-caps;">hdamp</span>]{}parameter of <span style="font-variant:small-caps;">powhegbox</span>, which controls the matrix element and parton shower matching and effectively regulates the high-radiation, is set to $m_{\PQt}= 172.5\GeV$. The [<span style="font-variant:small-caps;">MG5\_aMC@NLO</span>]{}generator simulates events with up to two additional partons at NLO, and is matched to the parton shower simulation using the <span style="font-variant:small-caps;">FxFx</span> merging prescription [@Frederix:2012ps]. The top quark mass value used in all these simulations is also 172.5and the PDF set is NNPDF3.0 [@Ball:2014uwa]. In addition, a sample matched to for the parton showering and hadronization is used for comparisons with the data.
=1200 The [$\PQt\PAQt\PQb\PAQb$]{}production cross sections are also compared with the predictions by the generator [[<span style="font-variant:small-caps;">PowHel</span>]{}]{} [@Garzelli:2014aba] ([<span style="font-variant:small-caps;">H</span>ELAC-NLO]{} [@Bevilacqua:2011xh] + <span style="font-variant:small-caps;">powhegbox</span> [@Alioli:2010xd]), which implements the full [$\PQt\PAQt\PQb\PAQb$]{}process at NLO QCD accuracy, with parton shower matching based on the NLO matching algorithm [@Nason:2004rx; @Frixione:2007vw]. The events are further hadronized by means of (v. 6.428), using parameters of the Perugia 2011 C tune [@Skands:2010ak]. In the generation of the events, the renormalization and factorization scales are fixed to $\mu_\mathrm{R} = \mu_\mathrm{F} = \HT/4$, where $\HT$ is the sum of the transverse energies of the final-state partons (, , , ) from the underlying tree-level process, and the CT10 PDFs are used.
The SM background samples are simulated with , , or , depending on the process. The generator is used to simulate $\PZ / \Pgg^*$ production (referred to as Drell–Yan, DY, in the following), production in association with an additional boson (referred to as $\ttbar$+$\PZ$, $\ttbar$+$\PW$, and $\ttbar$+$\Pgg$), and boson production with additional jets ($\PW$+jets in the following). Single top quark events ($\PQt\PW$ channel) are simulated using . Diboson ($\PW\PW$, $\PW\PZ$, and $\PZ\PZ$) and QCD multijet events are simulated using . For the [$\PQt\PAQt\PQb$]{}and [$\PQt\PAQt\PQb\PAQb$]{}measurements, the expected contribution from SM [$\PQt\PAQt\PH$]{}processes, simulated with , is also considered, although the final state has not yet been observed.
For comparison with the measured distributions, the events in the simulated samples are normalized to an integrated luminosity of $19.7\fbinv$ according to their predicted cross sections. These are taken from next-to-next-to-leading-order (NNLO) ($\PW$+jets [@Melnikov:2006di] and DY [@Melnikov:2006kv]), NLO + next-to-next-to-leading logarithmic (NNLL) (single top quark $\PQt\PW$ channel [@bib:twchan]), NLO (diboson [@bib:mcfm:diboson], $\ttbar$+$\PZ$ [@bib:ttV], $\ttbar$+$\PW$ [@bib:ttV], and $\ttbar$+$\PH$ [@Heinemeyer:1559921]), and leading-order (LO) (QCD multijet [@Sjostrand:2006za]) calculations. The contribution of QCD multijet events is found to be negligible.The predicted cross section for the $\ttbar$+$\Pgg$ sample is obtained by scaling the LO cross section obtained with the [<span style="font-variant:small-caps;">Whizard</span>]{} event generator [@bib:whizard] by an NLO/LO $K$-factor correction [@bib:ttgamma]. The simulated sample is normalized to the total cross section $\sigma_{\ttbar} = {\ensuremath{252.9\, \pm \,\xspace^{6.4}_{8.6} \text{(scale)} \pm 11.7 (\mathrm{PDF}+\alpha_s)\unit{pb}}\xspace}$, calculated with the <span style="font-variant:small-caps;">Top++2.0</span> program to NNLO in perturbative QCD, including soft-gluon resummation to NNLL order [@Czakon:2011xx], and assuming $m_{\PQt} = 172.5\GeV$. The first uncertainty comes from the independent variation of the factorization and renormalization scales, $\mu_\mathrm{R}$ and $\mu_\mathrm{F}$, while the second one is associated with variations in the PDF and $\alpha_s$, following the PDF4LHC prescription with the MSTW2008 68% confidence level () NNLO, CT10 NNLO, and NNPDF2.3 5f FFN PDF sets (see Refs. and references therein and Refs. ).
A number of additional pp simulated hadronic interactions (“pileup") are added to each simulated event to reproduce the multiple interactions in each bunch crossing from the luminosity conditions in the real data taking. Correction factors for detector effects (described in Sections \[sec:selection\] and \[sec:syst\]) are applied, when needed, to improve the description of the data by the simulation.
Event reconstruction and selection {#sec:selection}
==================================
The event selection is based on the decay topology of the events, where each top quark decays into a boson and a quark. Only the cases in which both bosons decayed to a charged lepton and a neutrino are considered. These signatures imply the presence of isolated leptons, missing transverse momentum owing to the neutrinos from boson decays, and highly energetic jets. The heavy-quark content of the jets is identified through tagging techniques. The same requirements are applied to select the events for the different measurements, with the exception of the requirements on the jets, which have been optimized independently for the +jets and [$\PQt\PAQt\PQb\PAQb$]{}([$\PQt\PAQt\PQb$]{}) cases. The description of the event reconstruction and selection is detailed in the following.
Events are reconstructed using a particle-flow (PF) algorithm, in which signals from all subdetectors are combined [@bib:pf2010; @CMS-PAS-PFT-09-001]. Charged particles are required to originate from the primary collision vertex [@Chatrchyan:2014fea], defined as the vertex with the highest sum of $\pt^2$ of all reconstructed tracks associated with it. Therefore, charged-hadron candidates from pileup events, originating from additional pp interactions within the same bunch crossing, are removed before jet clustering on an event-by-event basis. Subsequently, the remaining neutral-particle component from pileup events is accounted for through jet energy corrections [@Cacciari:2008gn].
Muon candidates are reconstructed from tracks that can be linked between the silicon tracker and the muon system [@Chatrchyan:2012xi]. The muons are required to have $\pt>20\GeV$, be within $|\eta|<2.4$, and have a relative isolation $I_{\text{rel}}<0.15$. The parameter $I_{\text{rel}}$ is defined as the sum of the of all neutral and charged reconstructed PF candidates, except the muon itself, inside a cone of $\Delta R\equiv\sqrt{\smash[b]{(\Delta\eta)^2+(\Delta\phi)^2}} < 0.3$ around the muon direction, divided by the muon , where $\Delta\eta$ and $\Delta\phi$ are the difference in pseudorapidity and azimuthal angle between the directions of the candidate and the muon, respectively. Electron candidates are identified by combining information from charged-track trajectories and energy deposition measurements in the ECAL [@Khachatryan:2015hwa], and are required to be within $|\eta|<2.4$, have a transverse energy of at least 20, and fulfill $I_{\text{rel}} < 0.15$ inside a cone of $\Delta R < 0.3$. Electrons from identified photon conversions are rejected. The lepton identification and isolation efficiencies are determined via a tag-and-probe method using boson events.
Jets are reconstructed by clustering the PF candidates, using the anti-$\kt$ clustering algorithm [@Cacciari:2008gp; @Cacciari:2011ma] with a distance parameter of $0.5$. The jet momentum is determined as the vectorial sum of all particle momenta in the jet, and is found in the simulation to be within 5 to 10% of the true momentum over the entire range and detector acceptance. Jet energy corrections are derived from the simulation, and are confirmed with in situ measurements with the energy balance of dijet and photon+jet events [@Chatrchyan:2011ds]. The jet energy resolution amounts typically to 15% at 10and 8% at 100. Muons and electrons passing less stringent requirements compared to the ones mentioned above are identified and excluded from the clustering process. Jets are selected in the interval $|\eta|<2.4$ and with $\pt >20\GeV$. Additionally, the jets identified as part of the decay products of the system (cf. Section \[sec:tt\_addjets\]) must fulfill $\pt >30\GeV$. Jets originating from the hadronization of quarks are identified using a combined secondary vertex algorithm (CSV) [@bib:btag004], which provides a tagging discriminant by combining identified secondary vertices and track-based lifetime information.
The missing transverse energy () is defined as the magnitude of the projection on the plane perpendicular to the beams of the negative vector sum of the momenta of all reconstructed particles in an event [@bib:MET]. To mitigate the effect of contributions from pileup on the resolution, we use a multivariate correction where the measured momentum is separated into components that originate from the primary and the other collision vertices [@Khachatryan:2014gga]. This correction improves the resolution by ${\approx}5\%$.
Events are triggered by requiring combinations of two leptons ($\ell$ = e or $\mu$), where one fulfills a threshold of 17and the other of 8, irrespective of the flavour of the leptons. The dilepton trigger efficiencies are measured using samples selected with triggers that require a minimum or number of jets in the event, and are only weakly correlated to the dilepton triggers used in the analysis.
Events are selected if there are at least two isolated leptons of opposite charge. Events with a lepton pair invariant mass less than 20are removed to suppress events from heavy-flavour resonance decays, QCD multijet, and DY production. In the $\mu\mu$ and ee channels, the dilepton invariant mass is required to be outside a boson mass window of $91\pm15\GeV$, and is required to be larger than 40.
For the +jets selection, a minimum of two jets is required, of which at least one must be tagged as a jet. A loose CSV discriminator value is chosen such that the efficiency for tagging jets from () quarks is ${\approx}85\%$ (40%), while the probability of tagging jets originating from light quarks ($\PQu$, $\PQd$, or $\PQs$) or gluons is around 10%. Efficiency corrections, depending on jet and $\eta$, are applied to account for differences in the performance of the tagging algorithm between data and simulation.
For the [$\PQt\PAQt\PQb\PAQb$]{}([$\PQt\PAQt\PQb$]{}) selection, at least three -tagged jets are required (without further requirements on the minimum number of jets). In this case, a tighter discriminator value [@bib:btag004] is chosen to increase the purity of the sample. The efficiency of this working point is approximately 70% (20%) for jets originating from a () quark, while the misidentification rate for light-quark and gluon jets is around 1%. The shape of the CSV discriminant distribution in simulation is corrected to better describe the efficiency observed in the data. This correction is derived separately for light-flavour and jets from a tag-and-probe approach using control samples enriched in events with a boson and exactly two jets, and events in the $\Pe\mu$ channel with no additional jets [@bib:HIG-13-029].
Identification of additional radiation in the event {#sec:tt_addjets}
===================================================
To study additional jet activity in the data, the identification of jets arising from the decay of the system is crucial. In particular, we need to identify correctly the two jets from the top quark decays in events with more than two jets. This is achieved by following two independent but complementary approaches: a kinematic reconstruction [@bib:Abbott:1997fv] and a multivariate analysis, optimized for the two cases under study, +jets and [$\PQt\PAQt\PQb\PAQb$]{}([$\PQt\PAQt\PQb$]{}), respectively. The purpose of the kinematic reconstruction is to completely reconstruct the system based on and the information on identified jets and leptons, taking into account detector resolution effects. This method is optimized for the case where the jets in the event only arise from the decay of the top quark pair. The multivariate approach is optimized for events with more jets than just those from the system. This method identifies the two jets that most likely originated from the top quark decays, and the additional jets, but does not perform a full reconstruction of the system. Both methods are described in the following sections.
Kinematic reconstruction in t-tbar+jets events {#sec:ttjetsreco}
----------------------------------------------
The kinematic reconstruction method was developed and used for the first time in the analysis from Ref. [@bib:TOP-12-028]. In this method the following constraints are imposed: is assumed to originate solely from the two neutrinos; the boson invariant mass is fixed to $80.4\GeV$ [@PDG2014]; and the top quark and antiquark masses are fixed to a value of $172.5\GeV$. Each pair of jets and lepton-jet combination fulfilling the selection criteria is considered in the kinematic reconstruction. Effects of detector resolution are accounted for by randomly smearing the measured energies and directions of the reconstructed lepton and jet candidates by their resolutions. These are determined from the simulation of signal events by comparing the reconstructed jets and leptons matched to the generated quarks and leptons from top quark decays. For a given smearing, the solution of the equations for the neutrino momenta yielding the smallest invariant mass of the system is chosen. For each solution, a weight is calculated based on the expected invariant mass spectrum of the lepton and jet from the top quark decays at the parton level. The weights are summed over 100 randomly smeared reconstruction attempts, and the kinematics of the top quark and antiquark are calculated as a weighted average. Finally, the two jets and lepton-jet combinations that yield the maximum sum of weights are chosen for further analysis. Combinations with two -tagged jets are chosen over those with a single -tagged jet. The efficiency of the kinematic reconstruction, defined as the number of events with a solution divided by the total number of selected +jets events, is approximately 94%. The efficiency in simulation is similar to the one in data for all jet multiplicities. Events with no valid solution for the neutrino momenta are excluded from further analysis. In events with additional jets, the algorithm correctly identifies the two jets coming from the decay in about 70% of the cases.
After the full event selection is applied, the dominant background in the $\Pe\mu$ channel originates from other decay channels and is estimated using simulation. This contribution corresponds mostly to leptonic $\tau$ decays, which are considered background in the +jets measurements. In the $\Pe\Pe$ and $\Pgm\Pgm$ channels, the dominant background contribution arises from $\PZ / \Pgg^*$+jets production. The normalization of this background contribution is derived from data using the events rejected by the boson veto, scaled by the ratio of events failing and passing this selection, estimated from simulation [@bib:TOP-11-002_paper]. The remaining backgrounds, including the single top quark $\PQt\PW$ channel, $\PW$+jets, diboson, and QCD multijet events, are estimated from simulation for all the channels.
In Fig.\[fig:ctrl:jetmult\], the multiplicity distributions of the selected jets per event are shown for different jet thresholds and compared to SM predictions. In this figure and the following ones, the sample is simulated using +, where only events with two leptons ($\Pe$ or $\mu$) from the boson decay are considered as signal. All other events, specifically those originating from decays via $\tau$ leptons, which are the dominant contribution, are considered as background. In the following figures, “Electroweak” corresponds to DY, $\PW$+jets, and diboson processes, and “bkg.” includes the $\ttbar$+$\Pgg/\PW/\PZ$ events. The data are well described by the simulation, both for the low jet threshold of 30and the higher thresholds of 60 and 100. The hatched regions in Figs.\[fig:ctrl:jetmult\]–\[fig:leadjet12\] correspond to the uncertainties affecting the shape of the simulated signal and background events (cf. Section \[sec:syst\]), and are dominated by modelling uncertainties in the former.
\
Additional jets in the event are defined as those jets within the phase space described in the event selection (cf. Section \[sec:selection\]) that are not identified by the kinematic reconstruction to be part of the system. The $\eta$ and distributions of the additional jets with the largest and second largest in the event (referred to as the leading and subleading additional jets in the following) are shown in Fig. \[fig:leadjet\]. Three additional event variables are considered: the scalar sum of the of all additional jets, $\HT$, the invariant mass of the leading and subleading additional jets, [$m_{\mathrm{jj}}$]{}, and their angular separation, ${\ensuremath{\Delta R_{\mathrm{jj}}}\xspace}=\sqrt{\smash[b]{(\Delta\eta)^2+(\Delta\phi)^2}}$, where $\Delta\eta$ and $\Delta\phi$ are the pseudorapidity and azimuthal differences between the directions of the two jets. These distributions are shown in Fig. \[fig:leadjet12\]. The predictions from the simulation, also shown in the figures, describe the data within the uncertainties.
\
Identification of t-tbar jets and additional jets in t-tbar-b-bbar events {#sec:ttbbreco}
-------------------------------------------------------------------------
The multivariate approach uses a boosted decision tree (BDT) to distinguish the jets stemming from the system from those arising from additional radiation for final states with more than two jets. This method is optimized for [$\PQt\PAQt\PQb\PAQb$]{}topologies in the dilepton final state of the system. The BDT is set up using the TMVA package [@Hocker:2007ht]. To avoid any dependence on the kinematics of the additional jets, and especially on the invariant mass of the two additional jets, the method identifies the jets stemming from the system by making use of properties of the system that are expected to be mostly insensitive to the additional radiation. The variables combine information from the two final-state leptons, the jets, and . All possible pairs of reconstructed jets in an event are considered. For each pair, one jet is assigned to the jet and the other to the jet. This assignment is needed to define the variables used in the BDT and is based on the measurement of the charge of each jet, which is calculated from the charge and the momenta of the PF constituents used in the jet clustering. The jet in the pair with the largest charge is assigned to the , while the other jet is assigned to the . The efficiency of this jet charge pairing is defined as the fraction of events where the assigned and are correctly matched to the corresponding generated b and jets, and amounts to 68%.
A total of twelve variables are included in the BDT. Some examples of the variables used are: the sum and difference of the invariant mass of the $\PQb\ell^+$ and $\PAQb\ell^-$ systems, $m^{\PQb\ell^+}\pm m^{\PAQb\ell^-}$; the absolute difference in the azimuthal angle between them, $ \lvert \Delta\phi^{ \PQb\ell^+,\PAQb\ell^- } \rvert $; the of the $\PQb\ell^+$ and $\PAQb\ell^-$ systems, $\pt^{\PQb\ell^+}$ and $\pt^{\PAQb\ell^-}$; and the difference between the invariant mass of the two jets and two leptons and the invariant mass of the pair, $m^{\PQb \PAQb \ell^+\ell^-}-m^{\bbbar}$. The complete list of variables can be found in Appendix \[ap:mvaVariables\]. The main challenge with this method is the large number of possible jet assignments, given four genuine jets and potential extra jets from additional radiation in each event. The basic methodology is to use the BDT discriminant value of each dijet combination as a measure of the probability that the combination stems from the system. The jets from the system are then identified as the pair with the highest BDT discriminant. From the remaining jets, those -tagged jets with the highest are selected as being the leading additional ones.
The BDT training is performed on a large and statistically independent sample of simulated [$\PQt\PAQt\PH$]{}events with the Higgs boson mass varied over the range 110–140. The [$\PQt\PAQt\PQb\PAQb$]{}events are not included in the training to avoid the risk of overtraining owing to the limited number of events in the available simulated samples. The simulated [$\PQt\PAQt\PH\,(\PQb\PAQb)$]{}sample is suited for this purpose since the four jets from the decay of the system and the Higgs boson have similar kinematic distributions. Since it is significantly harder to identify the jets from the system in [$\PQt\PAQt\PH$]{}events than in [$\PQt\PAQt\PQb\PAQb$]{}events, where the additional jets arise from initial- or final-state radiation, a good BDT performance with [$\PQt\PAQt\PH$]{}events implies also a good identification in [$\PQt\PAQt\PQb\PAQb$]{}events. The distributions of the BDT discriminant in data and simulation are shown in Fig. \[fig:mvaTopJets:cpWeights\] for all dijet combinations in an event, and for the combination with the highest weight that is assigned to the system. The subset “Minor bkg." includes all non-processes and $\ttbar$+$\PZ/\PW/\Pgg$ events. There is good agreement between the data and simulation distributions within the statistical uncertainties.
{width="40.00000%"}{width="40.00000%"}
The number of simulated events with correct assignments for the additional jets in [$\PQt\PAQt\PH$]{}events relative to the total number of events where those jets are selected and matched to the corresponding generator jets, is approximately 34%. In [$\PQt\PAQt\PQb\PAQb$]{}events, this fraction is about 40%. This efficiency is high enough to allow the measurement of the cross section as a function of the kinematic variables of the additional jets (the probability of selecting the correct assignments by choosing random combinations of jets is 17% in events with four jets and 10% in events with five jets). The relative increase in efficiency with respect to the use of the kinematic reconstruction for [$\PQt\PAQt\PQb\PAQb$]{}is about 15%. Additionally, the BDT approach improves the correlation between the generated and reconstructed variables, especially for the distribution of the invariant mass of the two leading additional jets [$m_{\PQb\PQb}$]{}and their angular separation $\Delta R_{\PQb\PQb} = \sqrt{\smash[b]{(\Delta\eta)^2+(\Delta\phi)^2}}$, where $\Delta\eta$ and $\Delta\phi$ are the pseudorapidity and azimuthal differences between the directions of the two jets.
The expected fraction of events with additional jets is not properly modelled in the simulation, in agreement with the observation of a previous CMS measurement [@bib:ttbb_ratio:2014]. This discrepancy between the +simulation and data can be seen in the jet multiplicity distribution, as shown in Fig. \[fig:control\_btagMult\].
{width="40.00000%"}{width="40.00000%"}
To improve the description of the data by the simulation, a template fit to the -tagged jet multiplicity distribution is performed using three different templates obtained from simulation. One template corresponds to the [$\PQt\PAQt\PQb$]{}and [$\PQt\PAQt\PQb\PAQb$]{}processes, defined at the generator level as the events where one or two additional jets are generated within the acceptance requirements, $\pt>20\GeV$ and ${\ensuremath{|\eta|}\xspace}<2.4$, (referred to as “+HF"). The [$\PQt\PAQt\PQb\PAQb$]{}and [$\PQt\PAQt\PQb$]{}processes are combined into a single template because they only differ by the kinematic properties of the second additional jet. Details about the definition of the jets and the acceptance are given in Section \[sec:diffxsec\]. The second template includes the background contribution coming from [$\PQt\PAQt\PQc\PAQc$]{}and +light-jets events (referred to as “ other”), where [$\PQt\PAQt\PQc\PAQc$]{}events are defined as those that have at least one jet within the acceptance and no additional jets. This contribution is not large enough to be constrained by data, therefore it is combined with the +light-jets process in a single template. The third template contains the remaining background processes, including [$\PQt\PAQt 2 \PQb$]{}, which corresponds to events with two additional hadrons that are close enough in direction to produce a single jet. This process, produced by collinear $\Pg \to \PQb \PAQb$ splitting, is treated separately owing to the large theoretical uncertainty in its cross section and insufficient statistical precision to constrain it with data. The normalizations of the first two templates are free parameters in the fit. The third is fixed to the corresponding cross section described in Section \[sec:theory\], except for the cross section for the [$\PQt\PAQt 2 \PQb$]{}process, which is corrected by a factor of $1.74{}_{-0.74}^{+0.69}$ [@bib:Zbb-xsec]. The normalization factors obtained for the template fit correspond to $1.66 \pm 0.43$ (+HF) and $1.00 \pm 0.01$ ( other). Details about the uncertainties in those factors are presented in Section \[sec:syst\_ttbb\]. The improved description of the jet multiplicity can be seen in Fig. \[fig:control\_btagMult\] (right).
Figure \[fig:cp\_bjets\] (top) shows the and [$|\eta|$]{}distributions of the leading additional jet, measured in events with at least three -tagged jets (using the tighter discriminator value described in Section \[sec:selection\]), after the full selection and including all corrections. The distributions of the and [$|\eta|$]{}of the second additional jet in events with exactly four -tagged jets, $\Delta R_{\PQb\PQb}$, and [$m_{\PQb\PQb}$]{}are also presented. The dominant contribution arises from the [$\PQt\PAQt\PQb\PAQb$]{}process. The decays into $\tau$ leptons decaying leptonically are included as signal to increase the number of [$\PQt\PAQt\PQb$]{}and [$\PQt\PAQt\PQb\PAQb$]{}events both in data and simulation. It has been checked that the distribution of the variables of relevance for this analysis do not differ between the leptons directly produced from $\PW$ boson decays and the leptons from $\tau$ decays within the statistical uncertainties in the selected [$\PQt\PAQt\PQb$]{}and [$\PQt\PAQt\PQb\PAQb$]{}events. In general, the variables presented are well described by the simulation, after correcting for the heavy-flavour content measured in data, although the simulation tends to predict smaller values of $\Delta R_{\PQb\PQb}$ than the data. After the full selection, the dominant background contribution arises from dilepton events with additional light-quark, gluon, and jets, corresponding to about 50% and 20% of the total expected yields for the [$\PQt\PAQt\PQb$]{}and [$\PQt\PAQt\PQb\PAQb$]{}cases, respectively. Smaller background contributions come from single top quark production, in association with or bosons, and events in the lepton+jets decay channels. The contribution from [$\PQt\PAQt\PH\,(\PQb\PAQb)$]{}is also small, amounting to 0.9% and 3% of the total expected events for the [$\PQt\PAQt\PQb$]{}and [$\PQt\PAQt\PQb\PAQb$]{}distributions. The contribution from background sources other than top quark production processes such as DY, diboson, or QCD multijet is negligible.
{width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"}
Systematic uncertainties {#sec:syst}
========================
Different sources of systematic uncertainties are considered arising from detector effects, as well as theoretical uncertainties. Each systematic uncertainty is determined individually in each bin of the measurement by varying the corresponding efficiency, resolution, or model parameter within its uncertainty, in a similar way as in the CMS previous measurement of the differential cross sections [@bib:TOP-12-028]. For each variation, the measured differential cross section is recalculated and the difference with respect to the nominal result is taken as the systematic uncertainty. The overall uncertainty in the measurement is then derived by adding all contributions in quadrature, assuming the sources of systematic uncertainty to be fully uncorrelated.
Experimental uncertainties
--------------------------
The experimental sources of systematic uncertainty considered are the jet energy scale (JES), jet energy resolution (JER), background normalization, lepton trigger and identification efficiencies, tagging efficiency, integrated luminosity, pileup modelling, and kinematic reconstruction efficiency.
The experimental uncertainty from the JES is determined by varying the energy scale of the reconstructed jets as a function of their and $\eta$ by its uncertainty [@Chatrchyan:2011ds]. The uncertainty from the JER is estimated by varying the simulated JER by its $\eta$-dependent uncertainty [@Chatrchyan:2011ds].
The uncertainty from the normalization of the backgrounds that are taken from simulation is determined by varying the cross section used to normalize the sample, see Section \[sec:theory\], by ${\pm}30\%$. This variation takes into account the uncertainty in the predicted cross section and all other sources of systematic uncertainty [@bib:TOP-11-005_paper; @bib:TOP-12-018; @bib:TOP-12-028]. In the case of the tW background, the variation of ${\pm}30\%$ covers the theoretical uncertainty in the absolute rate, including uncertainties owing to the PDFs. The contribution from the DY process, as determined from data, is varied in the normalization by ${\pm}30\%$ .
The trigger and lepton identification efficiencies in simulation are corrected by lepton $\pt$ and $\eta$ multiplicative data-to-simulation scale factors. The systematic uncertainties are estimated by varying the factors by their uncertainties, which are in the range 1–2%.
For the +jets measurements, the tagging efficiency in simulation is also corrected by scale factors depending on the and $\eta$ of the jet. The shape uncertainty in the tagging efficiency is then determined by taking the maximum change in the shape of the and [$|\eta|$]{}distributions of the jet, obtained by changing the scale factors. This is achieved by dividing the jet distributions in and [$|\eta|$]{}into two bins at the median of the respective distributions. The tagging scale factors for jets in the first bin are scaled up by half the uncertainties quoted in Ref. [@bib:btag004], while those in the second bin are scaled down, and vice versa, so that a maximum variation is assumed and the difference between the scale factors in the two bins reflects the full uncertainty. The changes are made separately in the and [$|\eta|$]{}distributions, and independently for heavy-flavour (b and c) and light-flavour (s, u, d, and gluon) jets, assuming that they are all uncorrelated. A normalization uncertainty is obtained by varying the scale factors up and down by half the uncertainties. The total uncertainty is obtained by summing in quadrature the independent variations.
=500 The uncertainty in the integrated luminosity is 2.6% [@bib:lumiPAS2013]. The effect of the uncertainty in the level of pileup is estimated by varying the inelastic pp cross section in simulation by ${\pm}5\%$.
The uncertainty coming from the kinematic reconstruction method is determined from the uncertainty in the correction factor applied to account for the small difference in efficiency between the simulation and data, defined as the ratio between the events with a solution and the total number of selected events.
### Specific systematic uncertainties associated with the t-tbar-b-bbar (t-tbar-b) measurements {#sec:syst_ttbb}
In the [$\PQt\PAQt\PQb\PAQb$]{}([$\PQt\PAQt\PQb$]{}) measurements, an additional uncertainty associated with the template fit to the -tagged jet multiplicity distribution is considered. Since the input templates are known to finite precision, both the statistical and systematic uncertainties in the templates are taken into account. The considered systematic uncertainties that affect the shapes of the templates are those of the JES, the CSV discriminant scale factors following the method described in [@bib:HIG-13-029], the cross section of the [$\PQt\PAQt\PQc\PAQc$]{}process, which is varied by ${\pm}50\%$ [@bib:HIG-13-029], and the uncertainty in the [$\PQt\PAQt 2 \PQb$]{}cross section. This is taken as the maximum between the largest uncertainty from the measurement described in Ref. [@bib:Zbb-xsec] and the difference between the corrected cross section and the prediction by the nominal simulation used in this analysis. This results in a variation of the cross section of about ${\pm}40\%$. This uncertainty is included as a systematic uncertainty in the shape of the background template.
Model uncertainties {#sec:modelsyst}
-------------------
The impact of theoretical assumptions on the measurement is determined by repeating the analysis, replacing the standard signal simulation by alternative simulation samples. The uncertainty in the modelling of the hard-production process is assessed by varying the common renormalization and factorization scale in the signal samples up and down by a factor of two with respect to its nominal value of the $Q$ in the event (cf. Section \[sec:theory\]). Furthermore, the effect of additional jet production in is studied by varying up and down by a factor of two the threshold between jet production at the matrix element level and via parton showering. The uncertainties from ambiguities in modelling colour reconnection (CR) effects are estimated by comparing simulations of an underlying-event (UE) tune including colour reconnection to a tune without it (Perugia 2011 and Perugia 2011 noCR tunes, described in Ref. [@Skands:2010ak]). The modelling of the UE is evaluated by comparing two different Perugia 11 (P11) tunes, mpiHi and TeV, to the standard P11 tune. The dependency of the measurement on the top quark mass is obtained using dedicated samples in which the mass is varied by ${\pm}1\GeV$ with respect to the default value used in the simulation. The uncertainty from parton shower modelling is determined by comparing two samples simulated with and , using either or for the simulation of the parton shower, underlying event, and hadronization. The effect of the uncertainty in the PDFs on the measurement is assessed by reweighting the sample of simulated signal events according to the 52 CT10 error PDF sets, at the 90% [@bib:CT10].
Since the total uncertainty in the [$\PQt\PAQt\PQb$]{}and [$\PQt\PAQt\PQb\PAQb$]{}production cross sections is largely dominated by the statistical uncertainty in the data, a simpler approach than for the +jets measurements is chosen to conservatively estimate the systematic uncertainties: instead of repeating the measurement, the uncertainty from each source is taken as the difference between the nominal +sample and the dedicated simulated sample at generator level. In the case of the uncertainty coming from the renormalization and factorization scales, the uncertainty estimated in the previous inclusive cross section measurement [@bib:ttbb_ratio:2014] is assigned.
Summary of the typical systematic uncertainties
-----------------------------------------------
Typical values of the systematic uncertainties in the absolute differential cross sections are summarized in Table \[tab:TypicalValSysUncertainties\] for illustrative purposes. They are the median values of the distribution of uncertainties over all bins of the measured variables. Details on the impact of the different uncertainties in the results are given in Sections \[sec:diffxsecNJets\] to \[sec:gap\].
In general, for the +jets case, the dominant systematic uncertainties arise from the uncertainty in the JES, as well as from model uncertainties such as the renormalization, factorization, and jet-parton matching scales and the hadronization uncertainties. For the [$\PQt\PAQt\PQb$]{}and [$\PQt\PAQt\PQb\PAQb$]{}cross sections, the total uncertainty, including all systematic uncertainties, is only about 10% larger than the statistical uncertainty. The experimental uncertainties with an impact on the normalization of the expected number of signal events, such as lepton and trigger efficiencies, have a negligible effect on the final cross section determination, since the normalization of the different processes is effectively constrained by the template fit.
\[tab:TypicalValSysUncertainties\]
[l|c|c]{}\
Source & +jets & [$\PQt\PAQt\PQb\PAQb$]{}([$\PQt\PAQt\PQb$]{})\
\
Trigger efficiency & 1.3 & 0.1\
Lepton selection & 2.2 & 0.1\
Jet energy scale & 6.8 & 11\
Jet energy resolution & 0.3 & 2.5\
Background estimate & 2.1 & 5.6\
$\PQb$ tagging & 0.5 & 12\
Kinematic reconstruction & 0.3 &\
Pileup & 0.3 & 1.7\
\
Fact./renorm. scale & 2.7 & 8.0\
Jet-parton matching scale & 1.3 & 3.0\
Hadronization & 4.5 & 5.2\
Top quark mass & 1.4 & 2.0\
PDF choice & 0.3 & 0.9\
Underlying event & 1.0 & 2.9\
Colour reconnection & 1.3 & 1.9\
Differential t-tbar cross section {#sec:diffxsec}
=================================
The absolute differential cross section is defined as: $$\frac{\rd\sigma_{\ttbar}}{\rd x_i}=\frac{\sum_j A_{ij}^{-1} (N^j_{\text{data}}-N^j_{\text{bkg}})}{\Delta_x^i \mathcal{L}},
\label{eq:xsec}$$ where $j$ represents the bin index of the reconstructed variable $x$, $i$ is the index of the corresponding generator-level bin, $N^j_{\text{data}}$ is the number of data events in bin $j$, $N^j_{\text{bkg}}$ is the number of estimated background events, $\mathcal{L}$ is the integrated luminosity, and $\Delta_x^i$ is the bin width. Effects from detector efficiency and resolution in each bin $i$ of the measurement are corrected by the use of a regularized inversion of the response matrix (symbolized by $A_{ij}^{-1}$) described in this section.
For the measurements of +jets, the estimated number of background events from processes other than production ($N_{\text{non \ttbar bkg}}$) is subtracted from the number of events in data ($N$). The contribution from other decay modes is taken into account by correcting the difference $N$–$N_{\text{non \ttbar bkg}} $ by the signal fraction, defined as the ratio of the number of selected signal events to the total number of selected events, as determined from simulation. This avoids the dependence on the inclusive cross section used for normalization. For the [$\PQt\PAQt\PQb$]{}and [$\PQt\PAQt\PQb\PAQb$]{}production cross sections, where the different contributions are fitted to the data, the expected contribution from all background sources is directly subtracted from the number of data events.
The normalized differential cross section is derived by dividing the absolute result, Eq. (\[eq:xsec\]), by the total cross section, obtained by integrating over all bins for each observable. Because of the normalization, the systematic uncertainties that are correlated across all bins of the measurement, the uncertainty in the integrated luminosity, cancel out.
Effects from the trigger and reconstruction efficiencies and resolutions, leading to migrations of events across bin boundaries and statistical correlations among neighbouring bins, are corrected using a regularized unfolding method [@bib:svd; @bib:blobel; @bib:TOP-12-028]. The response matrix $A_{ij}$ that corrects for migrations and efficiencies is calculated from simulated events using . The generalized inverse of the response matrix is used to obtain the unfolded distribution from the measured distribution by applying a $\chi^2$ technique. To avoid nonphysical fluctuations, a smoothing prescription (regularization) is applied. The regularization level is determined individually for each distribution using the averaged global correlation method [@bib:james]. To keep the bin-to-bin migrations small, the width of bins in the measurements are chosen according to their purity and stability. The purity is the number of events generated and correctly reconstructed in a certain bin divided by the total number of reconstructed events in the same bin. The stability is the ratio of the number of events generated and reconstructed in a bin to the total number of events generated in that bin. The purity and stability of the bins are typically larger than 40–50%, which ensures that the bin-to-bin migrations are small enough to perform the measurement. The performance of the unfolding procedure is tested for possible biases from the choice of the input model (the simulation). It has been verified that by reweighting the simulation the unfolding procedure based on the nominal response matrix reproduces the altered shapes within the statistical uncertainties. In addition, samples simulated with and are employed to obtain the response matrices used in the unfolding for the determination of systematic uncertainties of the model (Section \[sec:modelsyst\]). Therefore, possible effects from the unfolding procedure are already taken into account in the systematic uncertainties.
The differential cross section is reported at the particle level, where objects are defined as follows. Leptons from boson decays are defined after final-state radiation, and jets are defined at the particle level by applying the anti-$\kt$ clustering algorithm with a distance parameter of 0.5 [@Cacciari:2008gp] to all stable particles, excluding the decay products from boson decays into $\Pe\nu$, $\mu\nu$, and leptonic $\tau$ final states. A jet is defined as a jet if it has at least one hadron associated with it. To perform the matching between hadrons and jets, the hadron momentum is scaled down to a negligible value and included in the jet clustering (so-called ghost matching [@Cacciari:2008gn]). The jets from the decay are identified by matching the hadrons to the corresponding original quarks. The measurements are presented for two different phase-space regions, defined by the kinematic and geometric attributes of the decay products and the additional jets. The visible phase space is defined by the following kinematic requirements:
- Leptons: $\pt>20\GeV$, $|\eta|<2.4$,
- jets arising from top quarks: $\pt>30\GeV$, $|\eta|<2.4$,
- Additional jets and jets: $\pt>20\GeV$, $|\eta|<2.4$.
The full phase space is defined by requiring only the additional jets or jets be within the above-mentioned kinematic range, without additional requirements on the decay products of the system, and including the correction for the corresponding dileptonic branching fraction, calculated using the leptonic branching fraction of the boson [@PDG2014].
In the following sections, the differential cross section measured as a function of the jet multiplicity in the visible phase space and the results as a function of the kinematic variables of the additional jets in the event, measured in the visible and the full phase-space regions, are discussed. The absolute cross sections are presented as figures and compared to different predictions. The full results are given in tables in Appendix \[sec:summarytables\], along with the normalized differential cross sections measurements.
Differential t-tbar cross sections as a function of jet multiplicity
====================================================================
\[sec:diffxsecNJets\]
In Fig. \[fig:xsecjet\], the absolute differential cross section is shown for three different jet thresholds: $\pt >30$, $60$, and $100\GeV$. The results are presented for a nominal top quark mass of $172.5\GeV$. The lower part of each figure shows the ratio of the predictions from simulation to the data. The light and dark bands in the ratio indicate the statistical and total uncertainties in the data for each bin, which reflect the uncertainties for a ratio of 1.0. All predictions are normalized to the measured cross section in the range shown in the histogram, which is evaluated by integrating over all bins for each observable. The results are summarized in Table \[tab:dilepton:SummaryResultsJetMult\], together with the normalized cross sections. In general, the generator interfaced with , and interfaced both with and , provide reasonable descriptions of the data. The generator interfaced with does not generate sufficiently large jet multiplicities, especially for the lowest jet threshold. The sensitivity of to scale variations is investigated through the comparison of different renormalization, factorization, and jet-parton matching scales with respect to the nominal simulation. Variations in the jet-parton matching threshold do not yield large effects in the cross section, while the shape and normalization are more affected by the variations in the renormalization and factorization scales, which lead to a slightly worse description of the data up to high jet multiplicities, compared to their nominal values.
=400 In Fig. \[fig:xsecjetNewMC\], the results are compared to the predictions from and [<span style="font-variant:small-caps;">MG5\_aMC@NLO</span>]{}interfaced with , and the generator with the [<span style="font-variant:small-caps;">hdamp</span>]{}parameter set to $m_{\PQt}=172.5\GeV$ (labelled (h$_{\text{damp}}=m_{\PQt}$) in the legend), interfaced with , , and . The and [<span style="font-variant:small-caps;">MG5\_aMC@NLO</span>]{}simulations interfaced with predict larger jet multiplicities than measured in the data for all the considered thresholds. In general, no large deviations between data and the different predictions are observed.
The total systematic uncertainty in the absolute differential cross section ranges between 6 to 30%, while for the normalized cross section it varies from 2% up to 20% for the bins corresponding to the highest number of jets. In both cases, the dominant experimental systematic uncertainty arises from the JES, having a maximum value of 16% for the absolute cross section bin with at least six jets and $\pt> 30\GeV$. Typical systematic uncertainty values range between 0.5 and 8%, while the uncertainty in the normalized cross section is 0.5–4%. Regarding the modelling uncertainties, the most relevant ones are the uncertainty in the renormalization and factorization scales and the parton shower modelling, up to 6% and 10%, respectively. The uncertainties from the assumed top quark mass used in the simulation and the jet-parton matching threshold amount to 1–2%. Other modelling uncertainties such as PDF, CR, and UE have slightly smaller impact. These uncertainties cancel to a large extent in the normalized results, with typical contributions below 0.5%. The total contribution from the integrated luminosity, lepton identification, and trigger efficiency, which only affect the normalization, is 3.5%. This contribution is below 0.1% for every bin in the normalized results. The uncertainty from the estimate of the background contribution is around 2% for the absolute cross sections and typically below 0.5% for the normalized results.
Differential t-tbar cross sections as a function of the kinematic variables of the additional jets
==================================================================================================
\[sec:diffxsecJets\] The absolute and normalized differential cross sections are measured as a function of the kinematic variables of the additional jets in the visible phase space defined in Section \[sec:diffxsec\]. The results are compared to predictions from four different generators: interfaced with and , +, and + with varied renormalization, factorization, and jet-parton matching scales. All predictions are normalized to the measured cross section over the range of the observable shown in the histogram in the corresponding figures.
The absolute differential cross sections as a function of the of the leading and subleading additional jets and $\HT$, the scalar sum of the of all additional jets in the event, are shown in Fig. \[fig:inclusivept\]. The total uncertainties in the absolute cross sections range from 8–14% for the leading additional jet and $\HT$, and up to 40% for the subleading additional jet , while the systematic uncertainties in the normalized cross sections for the bins with the larger number of events are about 3–4%. The dominant sources of systematic uncertainties arise in both cases from model uncertainties, in particular the renormalization and factorization scales, and the parton shower modelling (up to 10% for the absolute cross sections), and JES (3–6% for the absolute cross sections). The typical contribution of other uncertainties such as the assumed top quark mass in the simulation, background contribution, etc., amounts to 1–3% and 0.5–1.5%, for the absolute and normalized cross sections, respectively.
In general, the simulation predictions describe the behaviour of the data for the leading additional jet momenta and $\HT$, although some predictions, in particular , favour a harder spectrum for the leading jet. The + prediction yields the largest discrepancies. The varied samples provide similar descriptions of the shape of the data, except for with the lower $\mu_\mathrm{R} = \mu_\mathrm{F}$ scale, which worsens the agreement.
The results as a function of [$|\eta|$]{}are presented in Fig. \[fig:inclusiveeta\]. The typical total systematic uncertainties in the absolute cross sections vary from 6.5–19% for the leading additional jet and about 11–20% for the subleading one. The uncertainty in the normalized cross section ranges from 1.5–9% and 5–14%, respectively. The shape of the [$|\eta|$]{}distribution is well modelled by +. The distributions from and yield a similar description of the data, being slightly more central than . Variations of the parameters have little impact on these distributions.
The differential cross section is also measured as a function of the dijet angular separation [$\Delta R_{\mathrm{jj}}$]{}and invariant mass [$m_{\mathrm{jj}}$]{}for the leading and subleading additional jets (Fig. \[fig:DeltaRmassjj\]). In general, all simulations provide a reasonable description of the distributions for both variables. All results are reported in Tables \[tab:dilepton:SummaryResultsJet1\]–\[tab:dilepton:SummaryResultsJet12\] in Appendix \[sec:summarytables\]. Representative examples of the migration matrices are presented in Fig. \[fig:migration\] in Appendix \[sec:migrationmatrix\].
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The absolute and normalized differential cross sections are also measured as a function of the kinematic variables of the additional jets and jets in the event for the full phase space of the system to facilitate comparison with theoretical calculations. In this case, the phase space is defined only by the kinematic requirements on the additional jets.
Figures \[fig:inclusiveptFull\] and \[fig:inclusiveetaFull\] show the absolute cross sections as a function of the and [$|\eta|$]{}of the leading and subleading additional jets and $\HT$, while the results as a function of [$\Delta R_{\mathrm{jj}}$]{}and [$m_{\mathrm{jj}}$]{}are presented in Fig. \[fig:DeltaRmassjjFull\].
The total uncertainties range between 8–12% for the leading jet and $\HT$, 10% at lower and 40% in the tails of distribution of the subleading jet . The uncertainties for $|\eta|$ are 6–16% and 10–30% for the leading and subleading additional jets, respectively. The typical uncertainties in the cross section as a function of [$\Delta R_{\mathrm{jj}}$]{}and [$m_{\mathrm{jj}}$]{}are on the order of 10–20%. The uncertainties are dominated by the JES, scale uncertainties, and shower modelling.
The numerical values are given in Tables \[tab:dilepton:SummaryResultsJet1Full\]–\[tab:dilepton:SummaryResultsJet12Full\] of Appendix \[sec:summarytables\], together with the normalized results. In the latter, the uncertainties are on average 2–3 times smaller than for the absolute cross sections, owing to the cancellation of uncertainties such as the integrated luminosity, lepton identification, and trigger efficiency, as well as a large fraction of the JES and model uncertainties, as discussed in Section \[sec:diffxsecNJets\]. The dominant systematic uncertainties are still the model uncertainties, although they are typically smaller than for the absolute cross sections.
The shapes of the distributions measured in the full and visible phase-space regions of the system are similar, while the absolute differential cross sections are a factor of 2.2 larger than those in the visible phase space of the system (excluding the factor due to the leptonic branching fraction correction $(4.54 \pm 0.10)\%$ [@PDG2014]).
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Differential t-tbar-b-bbar (t-tbar-b) cross sections as a function of the kinematic variables of the additional b jets
======================================================================================================================
\[sec:diffxsecAddbJets\]
Figure \[fig:xsec\_bjets\] shows the absolute differential cross sections in the visible phase space of the system and the additional jets as a function of the and [$|\eta|$]{}of the leading and subleading additional jets, and $\Delta R_{\PQb\PQb}$ and [$m_{\PQb\PQb}$]{}of the two jets. The uncertainties in the measured cross sections as a function of the jet kinematic variables are dominated by the statistical uncertainties, with values varying from 20–100%. The results are quantified in Tables \[tab:dilepton:SummaryResultsBJet\] and \[tab:dilepton:SummaryResultsBJet12\] in Appendix \[sec:summarytables\], together with the normalized results. The corresponding migration matrices between the reconstructed and particle levels for the kinematic properties of the additional jets are presented in Fig. \[fig:migrationttbb\] in Appendix \[sec:migrationmatrix\] for illustration purposes.
The dominant systematic uncertainties are the tagging efficiency and JES, up to 20% and 15%, respectively. Other uncertainties have typical values on the order of or below 5%. The experimental sources of systematic uncertainties affecting only the normalization, which are constrained in the fit, have a negligible impact. The largest model uncertainty corresponds to that from the renormalization and factorization scales of 8%. The effect of the assumed top quark mass and the PDF uncertainties have typical values of 1–2%. On average, the inclusion of all the systematic uncertainties increases the total uncertainties by 10%.
=400 The measured distributions are compared with the + prediction, normalized to the corresponding measured inclusive cross section in the same phase space. The measurements are also compared to the predictions from interfaced with and from with and . The normalization factors applied to the and predictions are found to be about 1.3 for results related to the leading additional jet. The predictions from both generators underestimate the [$\PQt\PAQt\PQb\PAQb$]{}cross sections by a factor 1.8, in agreement with the results from Ref. [@bib:ttbb_ratio:2014]. The normalization factors applied to are approximately 2 and 4 for the leading and subleading additional jet quantities, respectively, reflecting the observation that the generator does not simulate sufficiently large jet multiplicities. All the predictions have slightly harder spectra for the leading additional jet than the data, while they describe the behaviour of the [$|\eta|$]{}and [$m_{\PQb\PQb}$]{}distributions within the current precision. The predictions favour smaller $\Delta R_{\PQb\PQb}$ values than the measurement, although the differences are in general within two standard deviations of the total uncertainty.
{width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"}
=1200 The [$\PQt\PAQt\PQb\PAQb$]{}production cross sections are compared to the NLO calculation by [[<span style="font-variant:small-caps;">PowHel</span>]{}]{}+ in Fig. \[fig:xsec\_bjetsNLO\]. In the figure, the prediction is normalized to the absolute cross section given by the calculation of $20.8 \pm 0.6 \stat {}^{+7.9}_{-5.4} \text{(scale)}\unit{fb}$. The prediction describes well the shape of the different distributions, while the predicted absolute [$\PQt\PAQt\PQb\PAQb$]{}cross section is about 30% lower than the measured one, but compatible within the uncertainties.
{width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"}
The absolute differential cross sections measured in the visible phase space of the additional jets and the full phase space of the system are presented in Fig. \[fig:xsec\_bjetsFull\] and given in Tables \[tab:dilepton:SummaryResultsBJetFullPS\]–\[tab:dilepton:SummaryResultsBJet12FullPS\] of Appendix \[sec:summarytables\]. The results are corrected for acceptance and dileptonic branching fractions including $\tau$ leptonic decays ($6.43 \pm 0.14$)% [@PDG2014]. The results are compared to the same predictions as in Fig. \[fig:xsec\_bjets\], which are scaled to the measured cross section, obtained by integrating all the bins of the corresponding distribution. The normalization factor applied to the simulations is similar to the previous one for the results in the visible phase space of the system. The description of the data by the simulations is similar as well. The total measured $\sigma_{{\ensuremath{\PQt\PAQt\PQb\PAQb}\xspace}}$, as well as the agreement between the data and the simulation, is in agreement with the result obtained in Ref. [@bib:ttbb_ratio:2014]. In the full phase space, the inclusive [$\PQt\PAQt\PQb\PAQb$]{}cross section at NLO given by [[<span style="font-variant:small-caps;">PowHel</span>]{}]{}+ corresponds to $62 \pm 1 \stat {}^{+23}_{-17} \text{(scale)}\unit{fb}$ (excluding the dileptonic branching fraction correction). The comparison of the differential [$\PQt\PAQt\PQb\PAQb$]{}cross section with the NLO calculation is presented in Fig. \[fig:xsec\_bjetsNLOFull\].
{width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"} {width="40.00000%"}
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Differences between the kinematic properties of the additional jets and b jets are expected owing to the different production mechanisms [@Bevilacqua:2014qfa] of both processes. The dominant production mechanism of $\Pp\Pp\to{\ensuremath{\PQt\PAQt\PQb\PAQb}\xspace}$ is gluon-gluon (gg) scattering, while in the case of $\Pp\Pp\to{\ensuremath{\PQt\PAQt\mathrm{jj}}\xspace}$, the quark-gluon (qg) channel is equally relevant. The [$|\eta|$]{}distributions of the additional b jets seem to be more central than the corresponding distributions of the additional jets, see Figs. \[fig:inclusiveeta\] and \[fig:inclusiveetaFull\]. This difference can be attributed mainly to the contribution of the production via the qg channel, which favours the emission of jets at larger ${\ensuremath{|\eta|}\xspace}$. The distributions of the differential cross section as a function of [$m_{\PQb\PQb}$]{}peak at smaller invariant masses than those as a function of [$m_{\mathrm{jj}}$]{}, presented in Figs. \[fig:DeltaRmassjj\] and \[fig:DeltaRmassjjFull\], because of the larger contribution of the gg channel. Given the large uncertainties in the [$\PQt\PAQt\PQb\PAQb$]{}measurements, no statistically significant differences can be observed in the shape of the distributions of the additional b jets compared to the additional jets, shown in Figs. \[fig:inclusivept\] and \[fig:inclusiveptFull\].
Additional jet gap fraction {#sec:gap}
===========================
An alternative way to investigate the jet activity arising from quark and gluon radiation is to determine the fraction of events that do not contain additional jets above a given threshold [@bib:TOP-12-018; @bib:atlas2]. A threshold observable, referred to as the gap fraction, is defined as: $$f(\pt^j)=\frac{N(\pt^j)}{N_{\text{total}}},$$ where $N_{\text{total}}$ is the total number of selected events and $N(\pt^j)$ is the number of events that do not contain at least $j$ additional jets (apart from the two jets from the solution hypothesis) above a threshold, with $j$ corresponding to one or two jets. The measurements are presented as a function of the of the leading and subleading additional jets, respectively.
A modified gap fraction can be defined as: $$f(\HT)=\frac{N(\HT)}{N_{\text{total}}},$$ where $N(\HT)$ is the number of events in which the sum of the scalar of the additional jets $(\HT)$ is less than a certain threshold. In both cases, detector effects are unfolded using the simulation to obtain the results at the particle level. The additional jets at the generator level are defined as all jets within the kinematic acceptance, excluding the two jets originating from the quarks from top quark decay (see Section \[sec:diffxsec\]). For each value of the and $\HT$ thresholds the gap fraction at the generator level is evaluated, along with the equivalent distributions after the detector simulation and analysis requirements. Given the high purity of the selected events, above 70% for any bin for the leading additional jet and $\HT$, and above 85% for any bin for the subleading additonal jets, a correction for detector effects is applied by following a simpler approach than the unfolding method used for other measurements presented here. The data are corrected to the particle level by applying the ratio of the generated distributions at particle level to the simulated ones at the reconstruction level, using the nominal simulation.
The measured gap fraction distributions are compared to predictions from interfaced with , interfaced with and , interfaced with , and to the predictions with varied renormalization, factorization, and jet-parton matching scales. Figure \[fig:gap\] displays the gap fraction distribution as a function of the of the leading and subleading additional jets, and $\HT$. The lower part of the figures shows the ratio of the predictions to the data. The light band indicates the total uncertainty in the data in each bin. The threshold, defined at the value where the data point is shown, is varied from 25(lower value compared to previous measurements [@bib:TOP-12-018]) to 190. In general, interfaced with agrees with the data distributions of the three variables, while interfaced with and also provide a good description of the data, though they tend to predict a lower gap fraction than the measured ones. The generator interfaced with describes the data well as a function of the leading additional jet . However, it predicts higher values of the gap fraction as a function of the subleading jet and $\HT$. Modifying the renormalization and factorization scales in worsens the agreement with data, while variations of the jet-parton matching threshold provide similar predictions as the nominal simulation, in agreement with the results shown before.
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The results are also compared in Fig. \[fig:gapNewMC\] with the recently available simulations, described in Section \[sec:theory\], matched to different versions of the parton showering models. The and [<span style="font-variant:small-caps;">MG5\_aMC@NLO</span>]{}generators interfaced with predict up to 10% lower values of the gap fraction for all the variables, which reflects the fact that those simulations generate larger jet multiplicities, as discussed in Section \[sec:diffxsecNJets\]. Within the uncertainties, the predictions of the + simulation agree well with data, while the generator (with ${\textsc{hdamp}\xspace}= m_{\PQt}$) interfaced with and tends to overestimate and underestimate the measured values, respectively.
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The gap fraction is also measured in different [$|\eta|$]{}regions of the additional jets, with the results presented in Figs. \[fig:Gap1eta\]– \[fig:GapHTeta\] as a function of the leading additional jet , subleading additional jet , and $\HT$, respectively. In general, the gap fraction values predicted by the simulations describe the data better in the higher [$|\eta|$]{}ranges. The values given by and interfaced with are slightly below the measured ones in the central region for the leading jet and $\HT$, while + yields higher values of the gap fraction. In the case of the subleading jet , all predictions agree with the data within the uncertainties, except for + in the more central regions. Variations of the jet-parton matching threshold do not have a noticeable impact on the gap fraction, while with the varied renormalization and factorization scales provides a poorer description of the data.
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The total systematic uncertainty in the gap fraction distributions is about 5% for low values of the threshold (or $\HT$) and decreases to ${<}0.5\%$ for the highest values. The measurement of the gap fraction as a function of $\HT$ has larger uncertainties because of the impact of the lower-momentum jets that have a significantly larger uncertainty, as discussed in Section \[sec:diffxsecJets\]. The uncertainty in JES is the dominant source of systematic uncertainty, corresponding to approximately 4% for the smallest and $\HT$ values. Other sources with a smaller impact on the total uncertainty are the tagging efficiency, JER, pileup, and the simulated sample used to correct the data to the particle level.
Summary {#sec:summary}
=======
Measurements of the absolute and normalized differential top quark pair production cross sections have been presented using pp collisions at a centre-of-mass energy of 8, corresponding to an integrated luminosity of 19.7, in the dilepton decay channel as a function of the number of jets in the event, for three different jet thresholds, and as a function of the kinematic variables of the leading and subleading additional jets. The results have been compared to the predictions from interfaced with , interfaced with both and , interfaced with , and samples with varied renormalization, factorization, and jet-parton matching scales. In general, all these generators are found to give a reasonable description of the data.
The and generators interfaced with describe the data well for all measured jet multiplicities; while interfaced with generates lower multiplicities than observed for the lower-thresholds. The prediction from with varied renormalization and factorization scales does not provide an improved description of the data compared to the nominal simulation. These results are also compared to the predictions from with the [<span style="font-variant:small-caps;">hdamp</span>]{}parameter set to the top quark mass interfaced with , , and , which provide a reasonable description of the data within the uncertainties, and the predictions from and [<span style="font-variant:small-caps;">MG5\_aMC@NLO</span>]{}interfaced with , which generate higher jet multiplicities for all the thresholds.
The measured kinematic variables of the leading and subleading additional jets are consistent with the various predictions. The simulations also describe well the data distributions of the leading additional jet and $\HT$, although they tend to predict higher values and more central values in $\eta$. with varied parameters yields similar predictions, except for varying the renormalization and factorization scales, which tends to give higher $\HT$ values. The generator predicts lower yields than observed for the subleading additional jet .
The uncertainties in the measured [$\PQt\PAQt\PQb\PAQb$]{}([$\PQt\PAQt\PQb$]{}) absolute and normalized differential cross sections as a function of the jet kinematic variables are dominated by the statistical uncertainties. In general, the predictions describe well the shape of the measured cross sections as a function of the variables studied, except for $\Delta R_{\PQb\PQb}$, where they favour smaller values than the measurement. The predictions underestimate the total [$\PQt\PAQt\PQb\PAQb$]{}cross section by approximately a factor of 2, in agreement with previous measurements [@bib:ttbb_ratio:2014]. The calculation by [[<span style="font-variant:small-caps;">PowHel</span>]{}]{} [@Garzelli:2014aba] describes well the shape of the distributions, while the predicted absolute cross section is about 30% lower, but compatible with the measurements within the uncertainties.
The gap fraction has been measured as a function of the of the leading and subleading additional jets and $\HT$ of the additional jets in different $\eta$ ranges. For a given threshold value, the gap fraction as a function of $\HT$ is lower than the gap fraction as a function of the of the leading additional jet, showing that the measurement is probing multiple quark and gluon emission. Within the uncertainties, all predictions describe the gap fraction well as a function of the momentum of the first additional jet, while interfaced with fails to describe the gap fraction as a function of the subleading additional jet and $\HT$. In general, with decreased renormalization and factorization scales more poorly describes the observed gap fraction, while varying the jet-parton matching threshold provides a similar description of the data. The and [<span style="font-variant:small-caps;">MG5\_aMC@NLO</span>]{}generators interfaced with predict lower values than measured. The simulation with ${\textsc{hdamp}\xspace}= m_{\PQt}$ interfaced with is consistent with the data, while the simulation interfaced with and tends to worsen the comparison with the measurement.
In general, the different measurements presented are in agreement with the SM predictions as formulated by the various event generators, within their uncertainties. The correct description of +jets production is important since it constitutes a major background in searches for new particles in several supersymmetric models and in [$\PQt\PAQt\PH$]{}processes, where the Higgs boson decays into $\bbbar$. The [$\PQt\PAQt\PQb\PAQb$]{}([$\PQt\PAQt\PQb$]{}) differential cross sections, measured here for the first time, also provide important information about the main irreducible background in the search for [$\PQt\PAQt\PH\,(\PQb\PAQb)$]{}.
Acknowledgements {#acknowledgements .unnumbered}
================
=500 We thank M. V. Garzelli for providing the theoretical predictions from [[<span style="font-variant:small-caps;">PowHel</span>]{}]{}+. We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centres and personnel of the Worldwide LHC Computing Grid for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC and the CMS detector provided by the following funding agencies: the Austrian Federal Ministry of Science, Research and Economy and the Austrian Science Fund; the Belgian Fonds de la Recherche Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of Education and Science; CERN; the Chinese Academy of Sciences, Ministry of Science and Technology, and National Natural Science Foundation of China; the Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science, Education and Sport, and the Croatian Science Foundation; the Research Promotion Foundation, Cyprus; the Ministry of Education and Research, Estonian Research Council via IUT23-4 and IUT23-6 and European Regional Development Fund, Estonia; the Academy of Finland, Finnish Ministry of Education and Culture, and Helsinki Institute of Physics; the Institut National de Physique Nucléaire et de Physique des Particules / CNRS, and Commissariat à l’Énergie Atomique et aux Énergies Alternatives / CEA, France; the Bundesministerium für Bildung und Forschung, Deutsche Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher Forschungszentren, Germany; the General Secretariat for Research and Technology, Greece; the National Scientific Research Foundation, and National Innovation Office, Hungary; the Department of Atomic Energy and the Department of Science and Technology, India; the Institute for Studies in Theoretical Physics and Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di Fisica Nucleare, Italy; the Ministry of Science, ICT and Future Planning, and National Research Foundation (NRF), Republic of Korea; the Lithuanian Academy of Sciences; the Ministry of Education, and University of Malaya (Malaysia); the Mexican Funding Agencies (CINVESTAV, CONACYT, SEP, and UASLP-FAI); the Ministry of Business, Innovation and Employment, New Zealand; the Pakistan Atomic Energy Commission; the Ministry of Science and Higher Education and the National Science Centre, Poland; the Fundação para a Ciência e a Tecnologia, Portugal; JINR, Dubna; the Ministry of Education and Science of the Russian Federation, the Federal Agency of Atomic Energy of the Russian Federation, Russian Academy of Sciences, and the Russian Foundation for Basic Research; the Ministry of Education, Science and Technological Development of Serbia; the Secretaría de Estado de Investigación, Desarrollo e Innovación and Programa Consolider-Ingenio 2010, Spain; the Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the Ministry of Science and Technology, Taipei; the Thailand Center of Excellence in Physics, the Institute for the Promotion of Teaching Science and Technology of Thailand, Special Task Force for Activating Research and the National Science and Technology Development Agency of Thailand; the Scientific and Technical Research Council of Turkey, and Turkish Atomic Energy Authority; the National Academy of Sciences of Ukraine, and State Fund for Fundamental Researches, Ukraine; the Science and Technology Facilities Council, UK; the US Department of Energy, and the US National Science Foundation.
Individuals have received support from the Marie-Curie programme and the European Research Council and EPLANET (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Council of Science and Industrial Research, India; the HOMING PLUS programme of the Foundation for Polish Science, cofinanced from European Union, Regional Development Fund; the OPUS programme of the National Science Center (Poland); the Compagnia di San Paolo (Torino); the Consorzio per la Fisica (Trieste); MIUR project 20108T4XTM (Italy); the Thalis and Aristeia programmes cofinanced by EU-ESF and the Greek NSRF; the National Priorities Research Program by Qatar National Research Fund; the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University (Thailand); and the Welch Foundation, contract C-1845.
BDT variables {#ap:mvaVariables}
=============
The variables used for the BDT are listed below. The candidate jet is denoted with the superscript in the following equations, while the candidate anti-jet is denoted as $\PAQb$. Combinations of particles that are treated as a system by adding their four-momentum vectors are denoted without a comma, $\PQb\ell^+$ represents the jet and the antilepton system. The angular separation $\Delta R = \sqrt{\smash[b]{(\Delta \eta)^2 + (\Delta \phi)^2}}$ and the azimuthal angular difference $\Delta \phi$ between the directions of two particles is designated using the two particle abbreviations in a superscript, separated by a comma.
One variable is the difference in the jet charges, $c_\text{rel}$, of the and jets:
- $c_\text{rel}^{\bar{\text{b}}} - c_\text{rel}^{\text{b}}$
It is the only variable not directly related to the kinematical properties of the decay and the additional radiation. The values are by definition positive, as the jet with the highest charge is always assigned as the anti-jet.
There are three angular variables:
- $0.5\, \left( \lvert \Delta\phi^{\mathrm{b},{\ensuremath{{\vec p}_{\mathrm{T}}\hspace{-0.78em}/\kern0.45em}\xspace}} \rvert + \lvert \Delta\phi^{\mathrm{\bar{b}},{\ensuremath{{\vec p}_{\mathrm{T}}\hspace{-0.78em}/\kern0.45em}\xspace}} \rvert \right)$
- $ \lvert \Delta\phi^{\mathrm{b\ell}^{+},\mathrm{\bar{b}\ell^{-}}} \rvert $
- $\Delta{R}^{\mathrm{b,\ell^{+}}}$ and $\Delta{R}^{\mathrm{\bar{b},\ell^{-}}}$
Here, [${\vec p}_{\mathrm{T}}\hspace{-0.78em}/\kern0.45em$]{}denotes the missing transverse momentum in an event. The angles are defined such that $-\pi\leq\Delta\phi\leq\pi$, and consequently the absolute values are within $[0, \pi]$.
Two variables are the of the jet (jet) and charged antilepton (lepton) systems:
- $\pt^{\PQb\ell^+}$ and $\pt^{\PAQb\ell^-}$
The remaining variables are based on the invariant or transverse masses of several particle combinations:
- $m^{\PQb\ell^+} + m^{\PAQb\ell^-}$
- $m^{\PQb\ell^+} - m^{\PAQb\ell^-}$
- $m^{\PQb\PAQb\ell^+\ell^-} - m^{\bbbar}$
- $m^\mathrm{jets}_{\text{recoil}} - m^{\bbbar}$
- $0.5\, \left( m^{\PQb{\ensuremath{{\vec p}_{\mathrm{T}}\hspace{-0.78em}/\kern0.45em}\xspace}}_{\text{T}} + m^{\PAQb{\ensuremath{{\vec p}_{\mathrm{T}}\hspace{-0.78em}/\kern0.45em}\xspace}}_{\text{T}} \right) $
For any pair of jets, the variable $m^\mathrm{jets}_{\text{recoil}}$ is the invariant mass of all the other selected jets recoiling against this pair, all selected jets except these two.
Summary tables of absolute and normalized cross section measurements {#sec:summarytables}
====================================================================
\[tab:dilepton:SummaryResultsJetMult\]
\[tab:dilepton:SummaryResultsJet1\]
\[tab:dilepton:SummaryResultsJet2\]
\[tab:dilepton:SummaryResultsJet12\]
\[tab:dilepton:SummaryResultsJet1Full\]
\[tab:dilepton:SummaryResultsJet2Full\]
\[tab:dilepton:SummaryResultsJet12Full\]
\[tab:dilepton:SummaryResultsBJet\]
\[tab:dilepton:SummaryResultsBJet12\]
\[tab:dilepton:SummaryResultsBJetFullPS\]
\[tab:dilepton:SummaryResultsBJet12FullPS\]
Migration matrices {#sec:migrationmatrix}
==================
The migration matrices relating the kinematic properties of the additional jets and jets at the reconstruction level and particle level in the visible phase space of the decay products and the additional jets are presented in Figs. \[fig:migration\] and \[fig:migrationttbb\], respectively.
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{width="38.50000%"} {width="38.50000%"}
{width="38.50000%"} {width="38.50000%"}\
{width="38.50000%"} {width="38.50000%"}\
{width="38.50000%"} {width="38.50000%"}\
The CMS Collaboration \[app:collab\]
====================================
=5000=500=5000
|
---
abstract: 'The interplay between magnetism and superconductivity in the newly discovered heavy-fermion superconductor CePt$_3$Si has been investigated using the zero-field $\mu$SR technique. The $\mu$SR data indicate that the whole muon ensemble senses spontaneous internal fields in the magnetic phase, demonstrating that magnetism occurs in the whole sample volume. This points to a microscopic coexistence between magnetism and heavy-fermion superconductivity.'
author:
- 'A. Amato'
- 'E. Bauer'
- 'C. Baines'
bibliography:
- 'amato\_general.bib'
title: 'On the Coexistence Magnetism/Superconductivity in the Heavy-Fermion Superconductor CePt$_3$Si'
---
Introduction
============
In recent years, strongly correlated electron systems have played a leading role in solid state physics. The importance of this research field is illustrated by the discovery of novel phases in metals, intermetallics and oxides at low temperatures. One of the most relevant example is the discovery of unconventional superconductivity in heavy-fermion systems.
Unconventional superconductivity seems to result from the nature of the mechanism providing the attractive force necessary for the Cooper-pair formation. In conventional superconductors, the electrons are paired in a spin-singlet zero-angular-momentum state ($L=0$), which results from the fact that their binding is described in terms of the emission and absorption of phonons. This leads to the formation of an isotropic superconducting gap in the electronic excitations over the whole Fermi surface. On the other hand, heavy-fermion superconductivity is observed to show a close interplay with magnetic fluctuations. This seems to indicate that the attractive effective interaction between the strongly renormalized heavy quasiparticles in the superconducting heavy-fermion systems is not provided by the electron-phonon interaction as in ordinary superconductors, but rather is mediated by electronic spin fluctuations. This non-conventional (i.e. non-BCS) mechanism is believed to lead to an unconventional configuration of the heavy-fermion superconducting state, which may involve anisotropic, nonzero-angular-momentum states ($L \ne 0$, see Ref. for a review and references therein).
An additional feature in a number of systems is the observation of an apparent coexistence of heavy-fermion superconductivity and static magnetism. However, at ambient pressure, such a coexistence was up to recently solely confirmed on uranium-based heavy-fermion systems, and was discarded on cerium-based systems. Such conclusions were deduced from microscopic studies, in particular from the sensitive $\mu$SR technique [@amato_RMP_1997]. In this frame, the example of the first discovered heavy-fermion superconductor CeCu$_2$Si$_2$ is examplary as it exhibits a competition between both ground states, i.e. magnetism and superconductivity do not coexist, but appear as two different, mutually exclusive ground states of the same subset of electrons. Such competition was first discovered by $\mu$SR [@luke_PRL_1994; @feyerherm_PHYSREVB_1997] and only recently confirmed by neutron studies [@stockert_TBP].
Recently another Ce-based heavy-fermion system, namely CePt$_3$Si, was found [@bauer_PRL_2004] showing antiferromagnetism and superconductivity ($T_{\text N} = 2.2$ K and $T_c = 0.75$ K) at ambient pressure. This material crystallizes in a tetragonal structure (space group $P4mm$) lacking a center of inversion symmetry. This feature is attracting presently much interest since unconventional superconductivity with spin-triplet was to date thought to require such inversion symmetry to obtain the necessary degenerated electron states [@anderson_PRB_1984; @frigeri_PRL_2004]. In this article, we present $\mu$SR studies aiming to test - at the microscopic level - the coexistence between magnetic and superconducting state.
Experiment
==========
CePt$_3$Si was prepared by high frequency melting and subsequently heat treated at 880$^{\circ}$C for one week. Phase purity was evidenced by x-ray diffraction. The $\mu$SR experiments were carried out at the Swiss Muon Source of the Paul Scherrer Institute (Villigen, Switzerland). Measurements were performed on the GPS and LTF instruments of the $\pi$M3 beamline, using a He-flow cryostat (base temperature 1.7 K) and a $^3$He-$^4$He dilution refrigerator (base temperature 0.03 K), respectively. In order to avoid a depolarizing background $\mu$SR signal, the sample was glued onto a high-purity silver holder. Measurements on both instruments were performed on the same sample and showed a very good agreement in the overlapping temperature range. Measurements were performed in zero applied field (ZF), with an external-field compensation of the order of $\pm 20$ mOe.
Results and discussion
======================
ZF $\mu$SR is a local probe measurement of the magnetic field at the muon stopping site(s) in the sample. If the implanted polarized muons are subject to magnetic interactions, their polarization becomes time dependent, ${\mathbf P}_{\mu}(t)$. By measuring the asymmetric distribution of positrons emitted when the muons decay as a function of time, the time evolution of $P_{\mu}(t)$ can be deduced. The function $P_{\mu}(t)$ is defined as the projection of ${\mathbf P}_{\mu}(t)$ along the direction of the initial polarization: $P_{\mu}(t) ={\mathbf P}_{\mu}(t)\cdot {\mathbf P}_{\mu}(0)/P_{\mu}(0)=G(t)P_{\mu}(0)$. Hence, the depolarization function $G(t)$ reflects the normalized muon-spin autocorrelation function $G(t)=\langle{\mathbf S}(t)\cdot{\mathbf S}(0)\rangle/S(0)^2$, which depends on the average value, distribution, and time evolution of the internal fields, and therefore contains all the physics of the magnetic interactions of the muon inside the sample [@blundell_CONTEMP_1997].
Above $T_{\text N}$, the time evolution of the muon polarization is best described by the well known Kubo-Toyabe function [@kubo_MAGNRES_1967]: $$\label{equation_cept3si_kt}
G^{\text {para}}(t) = \frac{1}{3} + \frac{2}{3}(1-\Delta^2 t^2)\exp(-\frac{\Delta^2 t^2}{2})~,$$ where $\Delta^2/\gamma_{\mu}^2$ represents the second moment of the local field distribution at the muon site ($\gamma_{\mu}$ is the gyromagnetic ration of the muon). Such a depolarization function is characteristic of a paramagnetic state where the muon depolarization is solely due to the dipolar fields of the nuclear moments ($^{29}$Si and $^{195}$Pt). In the paramagnetic state, the electronic magnetic moments are often not observable by $\mu$SR due to their fast fluctuation rates. Alternatively, the nuclear magnetic moments appear static within the $\mu$SR time window and create a Gaussian field distribution at the muon stopping site, leading to the Kobo-Toyabe depolarization function reported in Eq. (\[equation\_cept3si\_kt\]). Note that this function posses an initial Gaussian character \[$\simeq \exp(-\Delta^2t^2)$ for $t \ll \Delta^{-1}$\] as observed in the data reported on Fig. \[figure\_cept3si\_raw\]. Fitting Eq. (\[equation\_cept3si\_kt\]) to the data provides a depolarization rate $\Delta = 0.06~\text{MHz}$ corresponding to field distribution width of $\sim0.7$ G at the muon site, in line with theoretical values computed for several possible stopping sites.
![\[figure\_cept3si\_raw\] Example of ZF $\mu$SR signals obtained in polycrystalline CePt$_3$Si in the paramagnetic phase (10 K), the magnetic phase (1 K) and below the superconducting transition (0.1 K). The lines represent fits obtained using Eq. (\[equation\_cept3si\_kt\]) and (\[equation\_cept3si\_afm\]). Note that for clarity, the fit for the data obtained at 1 K is ommitted.](figure_cept3si_raw){width="8cm"}
Below $T_{\text N}$, clear spontaneous oscillations are detected in the $\mu$SR signal indicating the occurrence of static finite magnetic fields sensed by the muons and arising from static electronic magnetic moments. In the antiferromagnetic state, the $\mu$SR signal is best described by the sum of two components, i.e.: $$\begin{aligned}
\label{equation_cept3si_afm}
G^{\text {AF}}(t) &= &A_1\big[\tfrac{1}{3} \exp(-\lambda_1 t) +\nonumber\\
& & \quad\,\,\,\,\tfrac{2}{3} \exp(-\lambda_1' t) \cos (2 \pi \nu_1 t + \phi_1)\big] +\nonumber\\
& & A_2\big[\tfrac{1}{3} \exp(-\lambda_2 t) + \nonumber\\
& & \quad\,\,\,\,\tfrac{2}{3} \exp(-\lambda_2' t) \cos (2 \pi \nu_2 t + \phi_2)\big]~.\end{aligned}$$ These components indicate the presence of two magnetically inequivalent muon stopping sites sensing internal fields $\lvert {\mathbf B}_{\mu}^i\rvert=2\pi\nu_i/\gamma_{\mu}$. As expected for a polycrystalline sample, the “$1/3$-term” of each component represents the fraction of the muons possessing an initial polarization along the same direction of the internal field. Therefore, the depolarization rates related to these fractions ($\lambda_i$) reflect solely the internal spins dynamics, whereas the depolarization rates $\lambda_i'$ are ascribed by both dynamical and static effects. The temperature evolution of the spontaneous frequencies $\nu_i$ are reported on Fig. \[figure\_cept3si\_frequencies\]. The values of the frequencies at $T \rightarrow 0$ correspond to internal field values of $\sim$ 160 G and 10 G, respectively.
![\[figure\_cept3si\_frequencies\] Temperature dependence of the spontaneous $\mu$SR frequencies $\nu_1$ and $\nu_2$ obtained by fitting Eq. (\[equation\_cept3si\_afm\]) to the $\mu$SR data. The measurements were performed in a polycrystalline CePt$_3$Si sample.](figure_cept3si_freq){width="8cm"}
Very recently, neutron scattering experiments determined the magnetic structure of CePt$_3$Si [@metoki_JPCM_2004]. Magnetic Bragg reflections observed at wave vector values ${\mathbf Q} = (0,0,1/2)$ and $(1,0,1/2)$, indicate that magnetic moments align ferromagnetically in the basal plane and stack antiferromagnetically along the $c$ axis with a strongly reduced value of $0.16~\mu_{\text B}$. By considering this magnetic structure, the values of the spontaneous $\mu^+$-frequencies provide information for a tentative determination of the muon stopping sites in the tetragonal structure. By assuming the magnetic moment direction pointing along the $a$ or $b$ axis, the most probable muon sites are located at two different 1(b) Wyckoff-positions, i.e. $(1/2,1/2,0.65)$ for the low-frequency component and $(1/2,1/2,0.82)$ for the high-frequency component. These sites are respectively located in the center of the Pt-plane formed by the Pt(1) ions and between the planes formed by Pt(1) and Pt(2) ions (see Fig. \[figure\_cept3si\_structure\], notation from Ref. ). In addition, for these sites, the calculated field distributions due to nuclear dipole moments are found in reasonable agreement with the observed depolarization rate $\Delta$ observed in the paramagnetic regime \[see Eq. (\[equation\_cept3si\_kt\])\]. Note also that both sites have the same multiplicity, which is in line with the observation that $A_1 \simeq A_2$ as shown on Fig. \[figure\_cept3si\_asymmetry\].
![\[figure\_cept3si\_structure\] Crystal structure of CePt$_3$Si. The smallest spheres represent the muon stopping sites discussed in the text.](figure_cept3si_structure){width="7cm"}
![\[figure\_cept3si\_asymmetry\] Temperature dependence of the amplitudes $A_1$ and $A_2$ of the spontaneous $\mu$SR frequencies in CePt$_3$Si \[see Eq. (\[equation\_cept3si\_afm\])\].](figure_cept3si_asymmetry){width="8cm"}
The first relevant observation is that $A_1 + A_2 = 1$ for all temperatures below $T_{\text N}$. This means that the whole muon ensemble is sensing the magnetic state, which in turn unambiguously demonstrates that the *whole* sample volume is involved in the magnetic phase below $T_{\text N}$. Together with thermodynamical studies, demonstrating that superconductivity has a bulk character, the present observation indicates a microscopic coexistence between magnetism and superconductivity. A similar conclusion was very recently drawn from NMR studies performed at different frequencies [@yogi_PRL_2004]. Note that the conclusion obtained by $\mu$SR is independent of the exact knowledge of the muon stopping sites. The behavior observed here in CePt$_3$Si is opposite to the one reported for CeCu$_2$Si$_2$ (see above), where the magnetic state is expelled from the sample upon cooling the below $T_c$. The observed coexistence in CePt$_3$Si is reminiscent of the situation observed in U-based heavy-fermion systems, as UPd$_2$Al$_3$ (Ref. ) or UNi$_2$Al$_3$ (Ref. ), where a model of two independent elctronic subsets, localized and itinerant (responsible for magnetism and superconductivity, respectively), was proposed in view of similar microscopic studies [@feyerherm_PRL_1994] and thermodynamics measurements [@caspary_PRL_1993].
![\[figure\_cept3si\_freq\_norm\] Temperature dependence of the spontaneous $\mu$SR frequencies normalized to their values at $T_c = 0.75$ K. Note the very slight change below $T_c$. The symbols correspond to those of Fig. \[figure\_cept3si\_frequencies\].](figure_cept3si_freq_norm){width="8cm"}
Upon cooling the system into the superconducting state, the $\mu$SR data suggest a slight change of the absolute spontaneous internal fields at the muon sites. As shown on Fig. \[figure\_cept3si\_freq\_norm\], one observes, for $T < T_c$, a slight reduction and increase of the low and high frequency signals, respectively. Note that such changes are at the limit of the measurement accuracy. In any case, two possible explanations could be invoked for these changes. The first one would be to consider a coupling between the superconducting and magnetic order parameters, reminiscent to the situation observed in UPt$_3$ [@aeppli_PRL_1989]. Alternatively, the frequency changes could have a more simple origin, as the muon is sensing interstitial fields and therefore only indirectly probes the strength of the magnetic order parameter. Hence, in addition to the dipolar interaction, the static 4$f$ magnetic moments will change the spin polarization of the conduction electrons at the muon site [@amato_RMP_1997], which results in an increased hyperfine field action on the muon. Such a contribution is a function of the density of normal electron states and will therefore be affected upon cooling the sample into the superconducting state. Below $T_c$, one expects a decrease in absolute value due to the opening of the superconducting gap. Depending on the muon stopping site and due to the oscillatory character of the RKKY interaction between the static 4$f$ moments and the conduction electrons, a decrease of the hyperfine field contribution can actually lead to either an increase or a decrease of the total internal fields at the muon site, as possibly observed in the present $\mu$SR data.
Conclusion
==========
Our zero-field $\mu$SR data have demonstrated the bulk character of the antiferromagnetic state in the heavy-fermion superconductor CePt$_3$Si, suggesting therefore a microscopic coexistence between magnetism and superconductivity. In addition, a slight change of the $\mu$SR response upon cooling the sample below $T_c$ can be ascribed to a coupling of the superconducting and magnetic order parameters and/or to the decrease of the hyperfine contact contribution acting on the muon. \
The $\mu$SR measurements reported here were performed at the Swiss Muon Source, Paul Scherrer Institute, Switzerland. Parts of the work were supported by the Austrian FWF (Fonds zur Förderung der wissenschaftlichen Forschung) project P16370.
|
---
author:
- 'A. Bremner[^1]'
- 'N. Tzanakis[^2]'
title: Lucas sequences whose 12th or 9th term is a square
---
Introduction
============
Let $P$ and $Q$ be non-zero relatively prime integers. The Lucas sequence $\{U_n(P,Q)\}$ is defined by $$\label{Lucas}
U_0=0, \quad U_1=1, \quad U_n= P U_{n-1}-Q U_{n-2} \quad (n \geq 2).$$ The sequence $\{U_n(1,-1)\}$ is the familiar Fibonacci sequence, and it was proved by Cohn [@Co1] in 1964 that the only perfect square greater than $1$ in this sequence is $U_{12}=144$. The question arises, for which parameters $P$, $Q$, can $U_n(P,Q)$ be a perfect square? This has been studied by several authors: see for example Cohn [@Co2] [@Co3] [@Co4], Ljunggren [@Lj], and Robbins [@Rob]. Using Baker’s method on linear forms in logarithms, work of Shorey & Tijdeman [@ST] implies that there can only be finitely many squares in the sequence $\{U_n(P,Q)\}$. Ribenboim and McDaniel [@RM1] with only elementary methods show that when $P$ and $Q$ are [*odd*]{}, and $P^2-4Q>0$, then $U_n$ can be square only for $n=0,1,2,3,6$ or $12$; and that there are at most two indices greater than 1 for which $U_n$ can be square. They characterize fully the instances when $U_n=\Box$, for $n=2,3,6$, and observe that $U_{12}=\Box$ if and only if there is a solution to the Diophantine system $$\label{Paulocurve}
P=\Box, P^2-Q = 2\Box, P^2-2Q=3\Box, P^2-3Q=\Box, (P^2-2Q)^2-3Q^2=6\Box.$$ When $P$ is [even]{}, a later paper of Ribenboim and McDaniel [@RM2] proves that if $Q \equiv 1 \pmod 4$, then $U_n(P,Q)=\Box$ for $n>0$ only if $n$ is a square or twice a square, and all prime factors of $n$ divide $P^2-4Q$. Further, if $p^{2t}|n$ for a prime $p$, then $U_{p^{2u}}$ is square for $u=1,\ldots,t$. In addition, if $n$ is even, then $U_n = \Box$ only if $P$ is a square or twice a square. A remark is made that no example is known of an integer pair $P$, $Q$, and an odd prime $p$, such that $U_{p^2}=\Box$ (note none can exist for $P$, $Q$ odd, $P^2-4Q>0$).
In this paper, we complete the results of Ribenboim and MacDaniel [@RM1] by determining all Lucas sequences $\{U_n(P,Q)\}$ with $U_{12}=\Box$ (in fact, the result is extended, because we do not need the restrictions that $P$, $Q$ be odd, and $P^2-4Q>0$): it turns out that the Fibonacci sequence provides the only example. Moreover, we also determine all Lucas sequences $\{U_n(P,Q)\}$ with $U_9=\Box$, subject only to the restriction that $(P,Q)=1$. Throughout this paper the symbol $\Box$ means square of a [*non-zero*]{} rational number.
\[Th1\] Let $(P,Q)$ be any pair of relatively prime non-zero integers. Then,
- $U_{12}(P,Q)=\Box$ iff $(P,Q)=(1,-1)$ (corresponding to the Fibonacci sequence).
- $U_9(P,Q)=\Box$ iff $(P,Q)=(\pm 2,1)$ (corresponding to the sequences $U_n=n$ and $U_n=(-1)^{n+1}n$).
The remainder of the paper is devoted mainly to the proof of this theorem. Theorems 3 and 6 of [@RM1] combined with the first statement of Theorem \[Th1\] imply the following.
\[Th2\] Let $P,Q$ be relatively prime odd integers, such that $P^2-4Q>0$. Then the $n$th term, $n>1$, of the Lucas sequence $U_n=U_n(P,Q)$ can be a square only if $n=2,3,6$ or $12$. More precisely[^3]:
- $U_2=\Box$ iff $P=a^2$.
- $U_3=\Box$ iff $P=a$, $Q=a^2-b^2$.
- $U_6=\Box$ iff $\displaystyle{P=3a^2b^2\,,\,Q=\frac{-a^8+12a^4b^4-9b^8}{2}}$.
- $U_{12}=\Box$ iff $(P,Q)=(1,-1)$. Moreover, this result is also valid even if we remove all restrictions on $P,Q$ except for $\gcd(P,Q)=1$.
The proof of Theorem \[Th1\] hinges, in both cases, upon finding all rational points on a curve of genus $2$. When the rank of the Jacobian of such a curve is less than $2$, then methods of Chabauty [@Ch], as expounded subsequently by Coleman [@Col], Cassels and Flynn [@Ca-Fl] and Flynn [@F1] may be used to determine the (finitely many) rational points on the curve. When the rank of the Jacobian is at least $2$, as is the case here, a direct application of these methods fails. In order to deal with such situations, very interesting methods have been developed recently by a number of authors; see chapter 1 of Wetherell’s Ph.D. thesis [@W], Bruin [@Br1; @Br2; @Br3], Bruin and Flynn [@Br-Fl1; @Br-Fl2], Flynn [@F2], and Flynn & Wetherell [@FW1; @FW2]. For the purpose of this paper, the method of [@F2] or [@FW1] is sufficient.
The Diophantine equations {#DioEqns}
=========================
The case $U_{12}$ {#u12genus2}
-----------------
For $U_{12}(P,Q)$ to be square, we have from (\[Lucas\]) $$\label{u12eqn}
U_{12}(P,Q)=P (P^2-3Q)(P^2-2Q)(P^2-Q)(P^4-4 P^2 Q+Q^2) = \Box.$$ Now $(P(P^2-3Q)(P^2-Q), (P^2-Q)(P^4-4 P^2 Q+Q^2))$ divides $2$, so that $U_{12}=\Box$ implies $$P(P^2-3Q)(P^2-Q) = \delta \Box, \quad (P^2-2Q)(P^4-4 P^2 Q+Q^2) = \delta \Box,$$ where $\delta = \pm 1, \pm 2$. With $x = Q/P^2$, we deduce $$(1-2 x)(1-4 x+x^2) = \delta \Box,$$ and of these four elliptic curves, only the curve with $\delta=2$ has positive rational rank. Torsion points on the three other curves do not provide any solutions for $P$, $Q$. We are thus reduced to considering the equations $$P(P^2-3Q)(P^2-Q) = 2 \Box, \quad (P^2-2Q)(P^4-4 P^2 Q+Q^2) = 2 \Box.$$ From the first equation, $$P(P^2-3Q) = \pm 2 \Box, P^2-Q = \pm \Box, \quad \mbox{or} \quad P(P^2-3Q) = \pm \Box,
P^2-Q = \pm 2 \Box.$$ The former case implies one of $$\begin{aligned}
\label{array1}
P=\delta \Box & P^2-3Q=2\delta \Box & P^2-Q=\Box \nonumber \\
P=\delta \Box & P^2-3Q=-2\delta \Box & P^2-Q=-\Box \nonumber \\
P=2\delta \Box & P^2-3Q=\delta \Box & P^2-Q=\Box \nonumber \\
P=2\delta \Box & P^2-3Q=-\delta \Box & P^2-Q=-\Box\end{aligned}$$ where $\delta=\pm 1, \pm 3$.\
The latter case implies one of $$\begin{aligned}
\label{array2}
P=\delta \Box & P^2-3Q=\delta \Box & P^2-Q=2\Box \nonumber \\
P=\delta \Box & P^2-3Q=-\delta \Box & P^2-Q=-2\Box\end{aligned}$$ where $\delta=\pm 1, \pm 3$.\
Solvability in $\R$ or elementary congruences shows impossibility of the above equations (\[array1\]), (\[array2\]), except in the following instances: $$\begin{aligned}
\label{array3}
P=-\Box, & P^2-3Q=-2\Box, & P^2-Q=\Box \nonumber \\
P=-3\Box, & P^2-3Q=-6\Box, & P^2-Q=\Box \nonumber \\
P=6\Box, & P^2-3Q=3\Box, & P^2-Q=\Box \nonumber \\
P=\Box, & P^2-3Q=-2\Box, & P^2-Q=-\Box \nonumber \\
P=6\Box, & P^2-3Q=-3\Box, & P^2-Q=-\Box \nonumber \\
P=\Box, & P^2-3Q=\Box, & P^2-Q=2\Box \nonumber \\
P=-3\Box, & P^2-3Q=-3\Box, & P^2-Q=2\Box.\end{aligned}$$ Recall now that $$(P^2-2Q)(P^4-4 P^2 Q+Q^2) = 2 \Box,$$ from which $$P^2-2Q = \eta \Box, P^4-4 P^2 Q+Q^2 = 2\eta \Box \quad \mbox{or} \quad P^2-2Q
= 2\eta \Box, P^4-4 P^2 Q+Q^2 = \eta \Box,$$ where $\eta=\pm 1, \pm 3$. The only locally solvable equations are $$\begin{aligned}
\label{array4}
P^2-2Q = -\Box, & P^4-4 P^2 Q+Q^2 = -2\Box \nonumber \\
P^2-2Q = 3\Box, & P^4-4 P^2 Q+Q^2 = 6\Box \nonumber \\
P^2-2Q = 2\Box, & P^4-4 P^2 Q+Q^2 = \Box \nonumber \\
$$ It is straightforward by elementary congruences to deduce from (\[array3\]), (\[array4\]), that we must have one of the following:
$P$ $P^2-3Q$ $P^2-Q$ $P^2-2Q$ $P^4-4P^2Q+Q^2$
--------- ---------- --------- ---------- -----------------
$-\Box$ $-2\Box$ $\Box$ $-\Box$ $-2\Box$
$6\Box$ $3\Box$ $\Box$ $2\Box$ $\Box$
$\Box$ $-2\Box$ $-\Box$ $-\Box$ $-2\Box$
$\Box$ $\Box$ $2\Box$ $3\Box$ $6\Box$
Now the rational ranks of the following elliptic curves are 0: $$(-x+1)(x^2-4x+1)=-2\Box, \quad (-3x+1)(x^2-4x+1)=3\Box, \quad (-x+1)(x^2-4x+1)=2\Box,$$ and consequently the rational points on the curves corresponding to the first three rows of the above table are straightforward to determine: they are $(P,Q)=(-1,1)$, $(0,-1)$, and $(1,1)$ respectively. These lead to degenerate Lucas sequences with $U_{12}=0$.\
It remains only to find all rational points on the following curve: $$P=\Box, P^2-3Q=\Box, P^2-Q=2\Box, P^2-2Q=3\Box, P^4-4 P^2 Q+Q^2 =6\Box\,,$$ satisfying $(P,Q)=1$. Note that this is the curve (\[Paulocurve\]), though we have removed the restriction that $P$ and $Q$ be odd, and $P^2-4Q>0$.\
Put $Q/P^2 = 1-2 u^2$, so that $$\label{genus9}
3u^2-1 = 2\Box, \qquad 4u^2-1 = 3\Box, \qquad 2u^4+2u^2-1 = 3\Box.$$ The equations (\[genus9\]) define a curve of genus $9$, with certainly only finitely many points. We restrict attention to the curve of genus $2$ defined by $$4u^2-1 = 3\Box, \qquad 2u^4+2u^2-1 = 3\Box.$$ Define $K=\Q(\sqrt{3})$, with ring of integers $\mathcal{O}_K=\Z[\sqrt{3}]$, and fundamental unit $2+\sqrt{3}$. Observe that $(u^2-1)^2-3 u^4 = - 3\Box$ implies $$u^2 -1 + u^2 \sqrt{3} = \epsilon \sqrt{3} \gamma^2,$$ for $\epsilon$ a unit of $\mathcal{O}_K$ of norm $+1$, and $\gamma \in \mathcal{O}_K$. If $\epsilon=2+\sqrt{3}$, the resulting equation is locally unsolvable above $3$, and so without loss of generality, $\epsilon=1$. Consider now $$u^2 (4u^2-1) (u^2(1+\sqrt{3})-1 ) = 3 \sqrt{3} V^2, \quad V \in K.$$ In consequence, $(x,y)=((12+4\sqrt{3})u^2, (36+12\sqrt{3})V)$ is a point defined over $K$ on the elliptic curve $$\label{E_1}
E_1: y^2 = x (x-(3+\sqrt{3})) (x-4\sqrt{3})$$ satisfying $\frac{(3-\sqrt{3})}{24} x \in \Q^2$. We shall see that the $K$-rank of $E_1$ is equal to $1$, with generator of infinite order $P=(\sqrt{3},3\sqrt{3})$.
The case $U_9$
--------------
For $U_9(P,Q)$ to be square, we have from (\[Lucas\]) $$\label{u9eqn}
U_9(P,Q) = (P^2-Q)(P^6-6 P^4 Q+9 P^2 Q^2-Q^3) = \Box.$$ So $$P^2-Q = \delta \Box, \qquad P^6-6 P^4 Q+9 P^2 Q^2-Q^3 = \delta \Box,$$ where $\pm \delta = 1, 3$. Put $Q=P^2-\delta R^2$. Then $$\label{u9eq}
\frac{3}{\delta} P^6-9 P^4 R^2+6 \delta P^2 R^4+\delta^2 R^6 = \Box,$$ with covering elliptic curves $$\label{J1}
\frac{3}{\delta}-9 x +6 \delta x^2+\delta^2 x^3 = \Box, \qquad
\frac{3}{\delta} x^3-9 x^2+6\delta x+\delta^2 = \Box.$$ For $\delta=\pm 1$, the first curve has rational rank $0$, and torsion points do not lead to non-zero solutions for $P$, $Q$. For $\delta=\pm 3$, both curves at (\[J1\]) have rational rank $1$, so that the rank of the Jacobian of (\[u9eq\]) equals $2$. To solve the equation $U_9(P,Q)=\Box$, it is necessary to determine all integer points on the two curves $$\label{u91}
P^6-9 P^4 R^2+18 P^2 R^4+9 R^6 = \Box,$$ and $$\label{u92}
-P^6-9 P^4 R^2 -18 P^2 R^4+9 R^6 = \Box.$$ To this end, let $L=\Q(\al)$ be the number field defined by $\al^3-3\al-1=0$. Gal($L/\Q$) is cyclic of order $3$, generated by $\sigma$, say, where $\al^\sigma=-1/(1+\al)=-2-\al+\al^2$. The ring of integers $\mathcal{O}_L$ has basis $\{1,\al,\al^2\}$, and class number $1$. Generators for the group of units in $\mathcal{O}_L$ are $\epsilon_1 =\al, \epsilon_2=1+\al$, with norms $\mbox{Norm}(\epsilon_1)=1$, $\mbox{Norm}(\epsilon_2)=-1$. The discriminant of $L/Q$ is 81, and the ideal $(3)$ factors in $\mathcal{O}_L$ as $(-1+\al)^3$.
### {#u9genus2}
Equation (\[u91\]) may be written in the form $$\mbox{Norm}_{L/\Q} (P^2 +(-5+\al+\al^2) R^2) = S^2, \mbox{ say},$$ and it follows that $$\label{normeq1}
P^2 +(-5+\al+\al^2) R^2 = \lambda U^2,$$ with $\lambda \in \mathcal{O}_L$ squarefree and of norm $+1$ modulo $L^{*^2}$.\
Applying $\sigma$, $$\label{normeq2}
P^2 +(-5-2\al+\al^2)R^2 = \lambda^\sigma V^2.$$ Suppose $\mathcal{P}$ is a first degree prime ideal of $\mathcal{O}_L$ dividing $(\lambda)$. Then for the norm of $\lambda$ to be a square, $\lambda$ must also be divisible by one of the conjugate prime ideals of $\mathcal{P}$. It follows that $\mathcal{P}$, or one of its conjugates, divides both $\lambda$ and $\lambda^\sigma$. Then this prime will divide $((-5+\al+\al^2)-(-5-2\al+\al^2)) = (3 \al) = (1-\al)^3$. So $\mathcal{P}$ has to be $(1-\al)$, with $(1-\al)^2$ dividing $\lambda$, contradicting $\lambda$ squarefree. If the residual degree of $\mathcal{P}$ is 3, then the norm of $\lambda$ cannot be square. Finally, the residual degree of $\mathcal{P}$ cannot be 2, otherwise $\theta = 5-\al-\al^2 \equiv m \pmod{\mathcal{P}}$, for some rational integer $m$, so that $\al = 1-2\theta+\frac{1}{3}\theta^2$ is congruent to a rational integer modulo $\mathcal{P}$, impossible. In consequence, $\lambda$ is forced to be a unit, of norm $+1$. Without loss of generality, the only possibilities are $\lambda=1, \epsilon_1, -\epsilon_2, -\epsilon_1 \epsilon_2.$ However, specializing the left hand side of (\[normeq1\]) at the root $\al_0=1.8793852415...$ of $x^3-3 x-1 = 0$ shows that $P^2+ 0.4114..R^2 = \lambda(\al_0) U(\al_0)^2$, so that $\lambda(\al_0) > 0$, giving unsolvability of (\[normeq1\]) for $\lambda = -\epsilon_2$, $-\epsilon_1 \epsilon_2$. There remain the two cases $\lambda=1$, with solution $(P,R,U)=(1,0,1)$, and $\lambda=\epsilon_1$, with solution $(P,R,U) = (0,1, 4-\al^2)$. From (\[normeq1\]) and (\[normeq2\]) we now have $$P^2 (P^2+(-5+\al+\al^2)R^2) (P^2 +(-5-2\al+\al^2)R^2) = \mu W^2,$$ with $\mu = \lambda \lambda^\sigma = 1$ or $1+\al-\al^2$. Accordingly, $X=P^2/R^2$ gives a point on the elliptic curve: $$\label{E1}
X (X+(-5+\al+\al^2)) (X+(-5-2\al+\al^2)) = \mu Y^2.$$ Now when $\mu=1$, a relatively straightforward $2$-descent argument shows that the $\Q(\al)$-rank of (\[E1\]) is equal to $0$ (we also checked this result using the Pari-GP software of Denis Simon [@Sim]). The torsion group is of order $4$, and no non-zero $P$, $Q$ arise.\
When $\mu=1+\al-\al^2$, then $(x,y)=(\frac{\mu}{(1-\al)^2} \frac{P^2}{R^2}, \frac{\mu^2}{(1-\al)^3} \frac{W}{R^3})$ is a point on the elliptic curve $E_2$ over $\Q(\al)$, where $$\label{E_2}
E_2 : y^2 = x(x+(-2-\al+\al^2)) (x+(-1+\al+\al^2)),$$ satisfying $\frac{(1-\al)^2}{\mu}x = (4+\al-2\al^2) x \in \Q^2$.
We shall see that the $\Q(\al)$-rank of $E_2$ is $1$, with generator of infinite order equal to $(1,\al)$.
### {#section}
Equation (\[u92\]) may be written in the form $$\mbox{Norm}_{L/{\Q}} (-P^2 +(-5+\al+\al^2) R^2) = S^2, \mbox{ say},$$ so that $$\label{normeq3}
-P^2 +(-5+\al+\al^2) R^2 = \lambda U^2,$$ with $\lambda \in \mathcal{O}_L$ squarefree and of norm $+1$ modulo $L^{*^2}$. Arguing as in the previous case, $\lambda$ must be a squarefree unit of norm $+1$, so without loss of generality equal to $1$, $\epsilon_1$, $-\epsilon_2$, $-\epsilon_1 \epsilon_2$. Only when $\lambda=\epsilon_1$ is (\[normeq3\]) solvable at all the infinite places. Thus $$-P^2 +(-5+\al+\al^2) R^2 = \al U^2, \quad -P^2 +(-5-2\al+\al^2) R^2 = (-2-\al+\al^2) V^2,$$ and $x=\frac{(1+\al-\al^2)}{(1-\al)^2} \frac{P^2}{R^2}$ is the $x$-coordinate of a point on the elliptic curve $$\label{E2'}
y^2 = x (x + (2+\al-\al^2)) (x + (1-\al-\al^2)).$$ satisfying $\frac{(1-\al)^2}{(1+\al-\al^2)}x=(4+\al-2\al^2)x \in \Q^2$. However, a straightforward calculation shows that the $\Q(\al)$-rank of $(\ref{E2'})$ is equal to $0$, with torsion group the obvious group of order $4$. There are no corresponding solutions for $P$, $Q$.
The Mordell-Weil basis {#MWbasis}
======================
Here we justify our assertions about the elliptic curves $E_1$ at (\[E\_1\]) and $E_2$ at (\[E\_2\]). These curves are defined over fields $F$ (where $F$=$K$ or $L$, respectively) with unique factorization, and have $F$-rational two-torsion. So standard two-descents over $F$ work analogously to the standard two-descent over $\Q$ for an elliptic curve with rational two-torsion; see for example Silverman [@Sil], Chapter 10.4. It is thus straightforward to determine generators for $E_i(F)/2E_i(F)$, $i=1,2$ (and in fact software packages such as that of Simon [@Sim] written for Pari-GP, and ALGAE [@BrA] of Bruin written for KASH, with m-ALGAE [@BrmA] for MAGMA, also perform this calculation effortlessly). Such generators are the classes of $P_1=(\sqrt{3}, 3\sqrt{3})$ for the curve $E_1$, and $P_2=(1,\alpha)$ for the curve $E_2$. In fact $P_1$ and $P_2$ are generators for the Mordell-Weil groups $E_1(\Q(\sqrt{3}))$ and $E_2(\Q(\alpha))$ respectively. To show this necessitates detailed height calculations over the appropriate number field, with careful estimates for the difference $\hat{h}(Q) - \frac{1}{2} h(Q)$ where $\hat{h}(Q)$ is the canonical height of the point $Q$, and $h(Q)$ the logarithmic height. The KASH/TECC package of Kida [@Ki] was useful here in confirming calculations. The standard Silverman bounds [@Sil2] are numerically too crude for our purposes, so recourse was made to the refinements of Siksek [@Sik]. Full details of the argument are given in Section 3 of [@BT]. Actually, determination of the full $F$-rational Mordell-Weil groups of $E_1$ and $E_2$ may be redundant; it is likely that the subsequent local computations can be performed subject only to a simple condition on the index in $E(F)$ of a set of generators for $E(F)/2E(F)$. The reader is referred to Bruin [@Br3] or Flynn & Wetherell [@FW2] for details and examples. This latter technique must be used of course when the height computations are simply too time consuming to be practical.
Finding all points on (\[E\_1\]) and (\[E\_2\]) under the rationality conditions {#the solutions}
================================================================================
General description of the method {#formalgroup}
---------------------------------
The problems to which we were led in section \[DioEqns\] are of the following shape.
[**Problem:**]{} Let $$\label{gen_xy_elliptic}
\cE\,:\, y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x +a_6 \;,$$ be an elliptic curve defined over $\Q(\alpha)$, where $\alpha$ is a root of a polynomial $f(X)\in\Z[X]$, irreducible over $\Q$, of degree $d\geq 2$ and $\beta\in\Q(\alpha)$ an algebraic integer. Find all points $(x,y)\in\cE(\Q(\alpha))$ for which $\beta x$ is the square of a [*rational number*]{}.
For the solution of this type of problem we adopt the technique described and applied in Flynn and Wetherell [@FW1][^4]. Several problems of this type have already been solved with a similar technique (besides [@FW1], see also [@Br3],[@Br-Fl2],[@F2],[@FW2]); therefore we content ourselves with a rather rough description of the employed method and refer the interested reader to section 4 of [@BT] for a detailed exposition.
We assume the existence of a rational prime $p$ with the following properties:
\(i) $f(X)$ is irreducible in $\Q_p[X]$.
\(ii) The coefficients of (\[gen\_xy\_elliptic\]) are in $\Z_p[\alpha]$.
\(iii) Equation (\[gen\_xy\_elliptic\]) is a minimal Weierstrass equation for $\cE/\Q_p(\alpha)$ at the unique discrete valuation $v$ defined on $\Q(\alpha)$ with $v(p)=1$.
\(iv) $\beta\in\Q_p(\alpha)$ is a $p$-adic unit.
We work with both (\[gen\_xy\_elliptic\]) and the associated model $w=z^3+a_1zw+a_2z^2w+a_3w^2+a_4zw^2+a_6w^3 $, which are related by means of the birational transformation\
$ (x,y) \mapsto (z,w)=(-x/y,-1/y) \;,\; (z,w) \mapsto (x,y)=(z/w,-1/w) .$
We also need the [*formal group law*]{} which is defined by means of two $p$-adically convergent power series $\cF(z_1,z_2)\in\Z[\alpha][[z_1,z_2]]$ (“sum") and $\iota(z)\in\Z[\alpha][[z]]$ (“inverse"), satisfying certain properties (see §2,chapter IV of [@Sil]). These series can be explicitly calculated up to any presision, and the operations $(z_1,z_2) \mapsto \cF(z_1,z_2)$, $z\mapsto \iota(z)$ make $p\Z_p[\alpha]$ a group $\hat{\cE}$ (or, more precisely, $\hat{\cE}/\Z_p[\alpha]$), which is the [*formal group*]{} associated to $\cE/\Q_p(\alpha)$.
There is a group isomorphism between $\hat{\cE}$ and the subgroup of ${\cal E}(\Q_p(\alpha))$ consisting of those points $Q$ whose reduction $\bmod{p}$ is the zero point of the reduced $\bmod{p}$ curve, defined by\
$z \mapsto Q$, where $Q=(z/w(z)\,,\,-1/w(z))$ if $z\neq 0$ and $Q={\cal O}$ if $z=0$,\
with $w(z)$ a $p$-adically convergent power series, that can be explicitly calculated up to any $p$-adic precision. The inverse map is given by $z({\cal O})=0$ and for $Q\in{\cal E}(\Q_p(\alpha))$ different from $\cal O$ whose reduction $\bmod{p}$ is zero, $z(Q)=-\frac{x(Q)}{y(Q)}$.
The remarkable property relating the functions $z$ and $\cF$ is that, for any points $Q_1,Q_2$ as $Q$ above, $$\label{z_sum}
z(Q_1+Q_2) = \cF(z(Q_1),z(Q_2))\:.$$ With respect to $\cal E$, a [*logarithmic*]{} function $\log$ is defined on $p\Z_p[\alpha]$ and an [*exponential*]{} function $\exp$ is defined on $p^r\Z_p[\alpha]$, where $r=1$ if $p>2$ and $r=2$ if $p=2$. These functions are mutually inverse and can be explicitly calculated as $p$-adic power series up to any precision. Moreover, if $r$ is as above and $z_1,z_2 \in p^r\Z_p[\alpha]$, then $$\label{z_linform}
\mbox{$\log\cF(z_1,z_2)=\log z_1 +\log z_2$ and $\exp(z_1+z_2)=\cF(\exp z_1,\exp z_2)$.}$$ Suppose now we know a point such that $z(Q)\in p^r\Z_p[\alpha]$ and assume further that, for a certain specifically known point $P\in\Q(\alpha)\cap \cE(\Z_p[\alpha])$, or for $P={\cal O}$, we want to find all $n\in\Z$ for which $\beta x(P+nQ)$ is a rational number (or, more particularly, a square of a rational). According to whether $P$ is a finite point or $P={\cal O}$, we express $\beta x(P+nQ)$ or $1/\beta x(nQ)$ first as an element of $\Z_p[\alpha][[z(nQ)]]$ and then, using properties (\[z\_sum\]) and (\[z\_linform\]), as a sum $ \theta_0(n)+\theta_1(n)\alpha+\cdots +\theta_{d-1}(n)\alpha^{d-1}$, where each series $\theta_i(n)$ is a $p$-adically convergent power series in $n$ with coefficients in $\Z_p$, which can be explicitly calculated up to any desired $p$-adic precision. In order that this sum be a rational number we must have $\theta_i(n)=0$ for $i=1,\ldots, d-1$. At this point we use Strassman’s Theorem[^5], which restricts the number of $p$-adic integer solutions $n$. If the maximum number of solutions implied by this theorem is equal to the number of solutions that we [*actually*]{} know, then we have explicitly all solutions. Sometimes, as in some instances of the sections below, even Strassman’s Theorem is not necessary.
In the following two sections we apply the above method to equations (\[E\_1\]) and (\[E\_2\]). At the suggestion of the referee, we give only a few computational details; most of our computational results, including the explicit form of the functions $\cF(z_1,z_2), w(z),\log$ and $\exp$ with the required precision, can be found in section 5.1 of [@BT].
Equation (\[E\_1\]) {#solve_u12}
-------------------
For this section, let $\al=\sqrt{3}$. We write (\[E\_1\]) as $\cE: y^2 = x^3-(3+5\alpha)x^2+12(1+\alpha)x $ and according to the discussion in section \[u12genus2\] we must find all points$(x,y)$ on this curve, such that $\beta x = u^2 \in\Q^2$, where $\beta=(3-\alpha)/24$.\
We work $p$-adically with $p=7$. According to section \[MWbasis\], any point on $\cE(\Q(\alpha))$ is of the form $n_1P_1+T$, where $P_1=(\alpha, 3\alpha)$ and $T\in\{{\cal O},(0, 0),(4\alpha, 0),(3+\alpha,0) \}$. For $Q=11P_1$ we have $z(Q)\in 7\Z_7[\alpha]$, and any point of $\cE(\Q(\alpha))$ can be written in the form $n_1P_1+T=(11n+r)P_1+T=nQ+P$, with $P=rP_1+T\,,\: -5\leq r \leq 5$ and $T$ a torsion point as above. There are 44 possibilities for $P$, one of which is $P={\cal O}$.
\(i) Consider first the case when $P$ is one out of the 43 possible finite points. As noted in section \[formalgroup\], we are led to a relation $\beta x(P+nQ)=\theta_0(n)+\theta_1(n)\alpha$, where the $7$-adically convergent series $\theta_i(n), i=0,1$ depend on $P$. In 35 out of the 43 cases, it turns out that $\theta_1(0)\not\equiv 0\pmod{7}$, which, in particular, implies that $\beta x(P+nQ)=\theta_0(n)+\theta_1(n)\alpha \equiv \theta_0(0)+\theta_1(0)\alpha \pmod{7}$ cannot be rational. The only cases that are not excluded in this way occur when $P$ is one of the following points: $\pm 4P_1+(0,0),\pm 3P_1+(0,0), \pm P_1+(0,0), (0,0),(3+\alpha,0)$.
We deal with these cases as follows: If $P=\pm 4P_1+(0,0)$, then $\theta_0(0)=5$, a quadratic non-residue of $7$; therefore, whatever $n$ may be, $\beta x(P+nQ)=\theta_0(n)+\theta_1(n)\alpha$ cannot be the square of a rational number. In a completely analogous manner we exclude $P=\pm 3P_1+(0,0)$, since, in this case, $\theta_0(0)=6$.
Next, consider $P=\pm P_1+(0,0)=(12+4\alpha,\pm(36+12\alpha))$. With the plus sign we compute $\theta_1(n)=7\cdot 94 n+7^2\cdot 40 n^2 +7^3\cdot 6 n^3 +\cdots $, and with the minus sign, $\theta_1(n)=7\cdot 249 n+7^2\cdot 40 n^2 +7^3 n^3 +\cdots $. In both cases, if $n\neq 0$, then dividing out by $7n$ we are led to an impossible relation $\bmod{7}$; hence $n=0$ and $x=x(P+nQ)=x(P)=12+4\alpha$, which gives $u^2=\beta (12+4\alpha)=1$ and $u=1$.
If $P=(0,0)$, then we compute $\theta_1(n)=7^3\cdot 6889 n^2 +7^4\cdot 1733 n^4 +7^7\cdot 2n^6+7^7\cdot 5n^8+\cdots $ and if $n\neq 0$ we divide out by $7^3n^2$ and we are led to an impossible relation $\bmod{7}$. Thus, $n=0$, which leads to $x=0$ and $u=0$. Finally, if $P=(3+\alpha,0)$, then $\theta_1(n)=7^2\cdot 288 n^2 +7^4n^4+\cdots $, forcing again $n=0$. Thus, $x=3+\alpha$ and $u^2=\beta (3+\alpha)=1/4$, hence $u=1/2$.
\(ii) Assume now that $P={\cal O}$. Then we have $1/\beta x(nQ)=\theta_0(n)+\theta_1(n)\alpha$, where $\theta_1(n)=7^2\cdot 244 n^2+7^4\cdot 2n^4+\cdots $. Since we are interested in finite points $(x,y)=nQ$, $n$ must be non-zero. Dividing out $\theta_1(n)=0$ by $7^2n^2$ we obtain an impossible equality.
[**Conclusion.**]{} The only points on (\[E\_1\]) satisfying $\beta x=u^2\in\Q^2$ are those with $x=12+4\alpha, 3+\alpha,0$, corresponding to $u=1,\frac{1}{2},0$. Only the first leads to a solution of (\[genus9\]) and this leads to the solution $(P,Q)=(1,-1)$ of (\[u12eqn\]) with $U_{12}(1,-1)=12^2$.
Equation (\[E\_2\]) {#solve_u9}
-------------------
We write (\[E\_2\]) as $ y^2 = x^3+(-3+2\alpha^2)x^2+(2-\alpha^2)x $. According to the discussion in section \[u9genus2\], it suffices to find all points $(x,y)$ on this curve such that $\beta x\in\Q^2$, where $\beta=4+\alpha-2\alpha^2$. We work $p$-adically with $p=2$. According to section \[MWbasis\], any point on $\cE(\Q(\alpha))$ is of the form $n_1P_1+T$, where $P_1=(1, \alpha)$ and $T\in\{{\cal O}\,,\,(1-\alpha-\alpha^2, 0)\,,\,(0, 0)\,,\,(2+\alpha-\alpha^2,0) \}$. For the point $Q=4P_1$ we have $z(Q)\in 4\Z_2[\alpha]$ and we write any point on $\cE(\Q(\alpha))$ in the form $ n_1P_1+T=(4n+r)P_1+T=nQ+P$, with $P=rP_1+T\,,\: r\in\{-1,0,1,2\}$ and $T$ a torsion point as above. Therefore, there are 16 possibilities for $P$, one of which is $P={\cal O}$.
Working as in section \[solve\_u12\] we check that the only solutions $(x,y)$ such that $\beta x\in\Q^2$ are $(x,y)=(0,0),2P_1+(0,0),-2P_1+(0,0)$. To give an idea of how we apply Strassman’s Theorem, let us consider the instance when $P=2P_1+T$ with $T=(0,0)$. We compute $\theta_1(n) = 2^6\cdot 7 n+2^6\cdot 3 n^2+0\cdot n^3+0\cdot n^4+0\cdot n^5+2^7n^6(\cdots)$. By Strassman’s Theorem, $\theta_1(n)=0$ can have at most two solutions in $2$-adic integers $n$. On the other hand, a straightforward computation shows that $\beta x(P+0\cdot Q)=4=\beta x(P-Q)$, which implies, in particular, that $\theta_1(0)=0=\theta_1(-1)$. Hence, $n=0,-1$ are the only solutions obtained for the above specific value of $P$, leading to the points $(x,y)=P+0\,Q=2P_1+(0,0)$ and $(x,y)=P+(-1)Q=-2P_1+(0,0)$, both having $x=\frac{4}{3}(1-\alpha^2)$ and $\beta x=4$.
[**Conclusion.**]{} The only points on (\[E\_2\]) satisfying $\beta x\in\Q^2$ are those with $x=0$ (leading to $P=0$), and $x=\frac{4}{3}(1-\alpha^2)$ giving successively (in the notation of section \[u9genus2\]) $\frac{P^2}{R^2}=4$ and $(P,Q)=(\pm 2,1)$, corresponding to degenerate Lucas sequences.
Acknowledgement {#acknowledgement .unnumbered}
===============
We are grateful to the referee for suggestions which have greatly improved the presentation of this paper.
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[^1]: Department of Mathematics, Arizona State University, Tempe AZ, USA, e-mail: bremner@asu.edu , http://math.la.asu.edu/\~andrew/bremner.html
[^2]: Department of Mathematics, University of Crete, Iraklion, Greece, e-mail: tzanakis@math.uoc.gr , http://www.math.uoc.gr/\~tzanakis
[^3]: Below it is understood that parameters $a,b$ are in every case chosen so that $P,Q$ are odd, relatively prime and $P^2-4Q>0$.
[^4]: But see also the references at the end of section \[introduction\].
[^5]: Theorem 4.1, in [@Ca].
|
---
abstract: 'The [*Herschel*]{} Orion Protostar Survey obtained well-sampled 1.2 – 870 spectral energy distributions (SEDs) of over 300 protostars in the Orion molecular clouds, home to most of the young stellar objects (YSOs) in the nearest 500 pc. We plot the bolometric luminosities and temperatures for 330 Orion YSOs, 315 of which have bolometric temperatures characteristic of protostars. The histogram of bolometric temperature is roughly flat; 29% of the protostars are in Class 0. The median luminosity decreases by a factor of four with increasing bolometric temperature; consequently, the Class 0 protostars are systematically brighter than the Class I protostars, with a median luminosity of 2.3 $L_\sun$ as opposed to 0.87 $L_\sun$. At a given bolometric temperature, the scatter in luminosities is three orders of magnitude. Using fits to the SEDs, we analyze how the luminosities corrected for inclination and foreground reddening relate to the mass in the inner 2500 AU of the best-fit model envelopes. The histogram of envelope mass is roughly flat, while the median corrected luminosity peaks at 15 $L_\sun$ for young envelopes and falls to 1.7 $L_\sun$ for late-stage protostars with remnant envelopes. The spread in luminosity at each envelope mass is three orders of magnitude. Envelope masses that decline exponentially with time explain the flat mass histogram and the decrease in luminosity, while the formation of a range of stellar masses explains the dispersion in luminosity.'
author:
- |
William J. Fischer$^{1,2}$, S. Thomas Megeath$^3$, Elise Furlan$^4$, Babar Ali$^5$, Amelia M. Stutz$^{6,7}$, John J. Tobin$^{8,9}$,\
Mayra Osorio$^{10}$, Thomas Stanke$^{11}$, P. Manoj$^{12}$, Charles A. Poteet$^1$, Joseph J. Booker$^3$, Lee Hartmann$^{13}$,\
Thomas L. Wilson$^{14}$, Philip C. Myers$^{15}$, and Dan M. Watson$^{16}$
title: 'The *Herschel* Orion Protostar Survey: Luminosity and Envelope Evolution'
---
INTRODUCTION {#s.intro}
============
For roughly the first 500,000 years in the formation of a young star [@eva09; @dun14], a rotating, infalling envelope feeds a circumstellar disk, which in turn accretes onto a hydrostatically supported central object. Young stellar objects (YSOs) with such envelopes are known as protostars. With observations over the last decade by the [*Spitzer*]{} [@wer04] and [*Herschel*]{} [@pil10] space telescopes, more than 1000 protostars and more than 4000 young stars that have lost their envelopes but retain disks have been identified in the nearest 0.5 kpc [@reb10; @dun15; @meg16]. The Orion molecular clouds are home to 504 [*Spitzer*]{}-identified candidate protostars [@meg16] and 16 additional [*Herschel*]{}-identified candidates [@stu13; @tob15], easily making it the largest single collection of protostars in this volume.
In the [*Herschel*]{} Orion Protostar Survey (HOPS), a key program of the [*Herschel*]{} Space Observatory, we obtained infrared (IR) imaging and photometry of over 300 of the Orion protostars at 70 and 160 with the Photoconductor Array Camera and Spectrometer (PACS) instrument [@pog10] aboard [*Herschel*]{}. We supplemented our [*Herschel*]{} observations with archival and newly obtained imaging, photometry, and spectra from 1.2 to 870 , allowing modeling of the protostellar spectral energy distributions (SEDs) and images. Details of the [*Herschel*]{} photometry are presented in B. Ali et al. (in preparation), while the 1.2 to 870 SEDs of the protostars are presented in @fur16.
With a sample of hundreds of protostars observed over three orders of magnitude in wavelength, we are able to reliably measure the bolometric properties of each source, constrain their underlying physical properties via modeling [@fur16], and perform a statistical study of the evolution of protostellar envelopes. Since the SEDs are strongly modified by the absorption and reprocessing of radiation from the central stars by dusty disks and infalling envelopes, the shape of an SED is expected to evolve as the protostar evolves (e.g., @ada87). To capture this evolution, YSOs were initially divided into classes based on the slopes $\alpha$ of their near-to-mid-IR SEDs from roughly 2 to 20 [@lad87; @gre94], where $\alpha=\left(d\log\lambda S_\lambda\right)/\left(d\log\lambda\right)$, $\lambda$ is the wavelength, and $S_\lambda$ is the flux density at $\lambda$. Class I sources have $\alpha\ge0.3$, flat-spectrum sources have $-0.3\le\alpha<0.3$, Class II sources have $-1.6\le\alpha<-0.3$, and Class III sources have $\alpha<-1.6$.
The discovery of Class 0 objects [@and93], which were difficult to detect in the mid-IR until the launch of [*Spitzer*]{}, motivated additional criteria not based on the slope of the SED. The bolometric temperature $T_{\rm bol}$, the effective temperature of a blackbody with the same mean frequency as the protostellar SED [@mye93], was adopted to distinguish between Class 0 and Class I sources. Class 0 objects have $T_{\rm bol}<70~{\rm K}$, Class I objects have $70~{\rm K}<T_{\rm bol}<650~{\rm K}$, and Class II objects have $650~{\rm K}<T_{\rm bol}<2800~{\rm K}$ [@che95]. Flat-spectrum sources in the $\alpha$-based system are not explicitly included in this scheme, although @eva09 suggest a range of 350 to 950 K, straddling the Class I/II boundary.
To classify the HOPS sample, @fur16 adopted a hybrid approach, using $T_{\rm bol}$ to distinguish Class 0 objects from more evolved sources and using $\alpha$ (measured between 4.5 and 24 ) to classify these more evolved sources as Class I, flat-spectrum, or Class II objects. They consider Class 0, I, and flat-spectrum objects to be protostars, while Class II objects are post-protostellar, when the envelope has dissipated and only a circumstellar disk remains. (See Section 7.2.3 of @fur16 for a small number of exceptions to this distinction.) While @hei15 found that only half of their flat-spectrum sources, which were selected based on the extinction-corrected 2 to 24 spectral index, have envelopes detected in HCO$^+$, @fur16 found that nearly all of the HOPS flat-spectrum sources have SEDs best fit with envelopes that are generally less massive than those of Class 0 and Class I protostars.
With model fits, @fur16 found a systematic decrease in envelope density from Class 0 to Class I to flat-spectrum protostars, with an overall decrease of a factor of 50. This decrease is consistent with the interpretation that SED classes describe an evolutionary progression driven by the gradual dissipation of the envelope. The classification, however, is affected by additional factors. Inclination can affect the SED, where a Class I protostar viewed through an edge-on disk can have a lower $T_{\rm bol}$ than a Class 0 protostar viewed from an intermediate inclination angle. Foreground reddening is a further complication, in that a more evolved object that lies behind extensive foreground dust may appear to have a more massive envelope (and therefore lower $T_{\rm bol}$) than it really does.
To disentangle observational degeneracies in probing the evolution of envelopes, radiative transfer models have been employed to constrain physical parameters such as envelope density and mass. Based on fits of models to SEDs, @rob06 proposed the use of stages, where the stage refers to the underlying physical state probed by observations. For protostars, Stage 0 refers to the period when the envelope mass $M_{\rm env}$ still exceeds the mass of the central object $M_*$, and Stage I refers to the period when $0<M_{\rm env}<M_*$. The physical stages correspond only roughly to the observational classes [@dun14]. Fitting models to the HOPS SEDs, @fur16 tabulated the properties of the best-fit models. They also analyzed uncertainties in the model fits, showing that, although models provide good fits to the data, the solutions are not necessarily unique, and degeneracies in model fit parameters can lead to large uncertainties. For this reason, the use of model fits provides an alternative means of examining the evolution of protostars, but it does not fully replace the use of observational criteria such as SED class.
The bolometric luminosity and temperature (BLT) plot is a common evolutionary diagram for protostars first presented by @mye93, analogous to the Hertzsprung-Russell diagram for stars. Data from the [*Spitzer*]{} program “From Molecular Cores to Planet-Forming Disks” (c2d) were used to derive the BLT diagram for 1024 YSOs in five molecular clouds that are closer than Orion [@eva09]. With the relative numbers of YSOs in each class, the c2d team estimated median lifetimes of 0.16 Myr for Class 0, 0.38 Myr for Class I, and 0.40 Myr for the flat-spectrum phase, with small revisions downward after correcting for interstellar extinction. The luminosities at each bolometric temperature were found to be spread over several orders of magnitude.
@eva09 compared these findings to the models of @you05, which feature a constant envelope infall rate and are an extension of the @shu77 inside-out collapse model. These models predict a small range of luminosities due to the formation of a range of stellar masses, and these luminosities are large compared to those typically observed. For Class I protostars, the model luminosities are of order 10 $L_\sun$, while the observed ones are generally $<3$ $L_\sun$. This is consistent with the classic luminosity problem first noted with *Infrared Astronomical Satellite* data by @ken90.
@dun10 explored the ability of various modifications to the @you05 model to reproduce the broad luminosity spread in the c2d BLT diagram. As suggested in the paper that originally established the luminosity problem, @dun10 found that the most successful modification was to add episodic accretion, where the infalling matter from the envelope accumulates in the disk. The growing mass in the disk contributes little to the observed luminosity until it abruptly accretes onto the star, yielding a luminosity outburst. Explanations for this phenomenon typically invoke disk instabilities, either thermal instabilities (e.g., @bel94), the magnetorotational and gravitational instabilities acting in concert [@zhu09; @zhu10], or the accretion of clumps formed when the accumulation of envelope material causes the disk to fragment [@vor05; @vor15]. The luminosity in this scenario is thus usually smaller than predicted by @you05, but it agrees when averaged over both the quiescent and outburst modes.
@off11 compared the broad spread in protostellar luminosities, also noted for Orion and other clouds in the nearest 1 kpc by @kry12, to the predictions of various star-formation models. They found that models with a roughly constant accretion time, not a constant accretion rate, better reproduced the observed luminosity distributions. They also found that tapered models, where the mass infall rate diminishes at late times, were able to produce a distribution where the typical Class 0 luminosity is equal to or greater than the typical Class I luminosity. These contrasting approaches to resolving the luminosity problem, episodic (stochastic) accretion on one hand and slow (secular) variations of the accretion rate on the other, were discussed in detail by @dun14 and are difficult to disentangle observationally.
Here we present the BLT diagram of the Orion protostars, showing the distribution for the largest number to date of completely sampled SEDs at a common distance. We then use the radiative transfer modeling by @fur16 to plot the inner envelope masses of the protostars, investigate luminosity evolution across the protostellar phase, and interpret these findings with simple models of star formation. Section 2 describes the sample selection and observations, Section 3 presents BLT diagrams for the entire HOPS sample as well as for regions within Orion, Section 4 introduces model-based diagnostics that trace luminosity and envelope evolution, Section 5 interprets the evolutionary diagrams, and Section 6 contains our conclusions.
SAMPLE DEFINITION AND OBSERVATIONS
==================================
For our analysis we adopt the same sample of 330 YSOs as @fur16, who have tabulated their coordinates, photometry, properties, and model fits. These are candidate protostars that were targeted by our [*Herschel*]{} observations and detected in the PACS 70 images. They are spread over the Orion A and B molecular clouds from declinations of $-8^\circ50'$ to $1^\circ54'$ and from right ascensions of $5^{\rm h}33^{\rm m}$ to $5^{\rm h}55^{\rm m}$. The Orion Nebula itself is excluded due to saturation in the [*Spitzer*]{} maps used for sample selection.
We used photometry and spectra from several archival and new surveys to construct the SEDs of sources in the sample, which are plotted in @fur16. Near-IR photometry from the Two Micron All Sky Survey (2MASS; @skr06) and mid-IR photometry from [*Spitzer*]{} appear in @meg12. Mid-IR spectra from the [*Spitzer*]{} Infrared Spectrograph (IRS) are plotted in @fur16. The [*Herschel*]{} photometry, including 70 and 160 photometry from HOPS and 100 photometry from the public archive, and photometry at 350 and 870 from the Atacama Pathfinder Experiment (APEX) appear in @fur16. The [*Herschel*]{} and APEX surveys will be discussed in greater detail by B. Ali et al. (in preparation) and T. Stanke et al. (in preparation), respectively.
Using $T_{\rm bol}$, the 4.5 to 24 spectral slope, and qualitative assessment of the SEDs, @fur16 found 92 Class 0 protostars, 125 Class I protostars, 102 flat-spectrum protostars, and 11 Class II objects among the 330 sources. In the fitting of their SEDs, six of the 330 sources were found to lack envelope emission.
[lcccccc]{} All & $(-8.9,+1.9)$ & 330 & 315 & 91 & 224 & $0.29\pm0.03$\
L 1641 & $(-8.9,-6.1)$ & 173 & 160 & 32 & 128 & $0.20\pm0.03$\
ONC & $(-6.1,-4.6)$ & 79 & 77 & 27 & 50 & $0.35\pm0.05$\
Orion B & $(-2.5,+1.9)$ & 78 & 78 & 32 & 46 & $0.41\pm0.06$
BOLOMETRIC LUMINOSITIES AND TEMPERATURES
========================================
With far-IR photometry, we sample the peaks of the protostellar SEDs and thus derive more accurate bolometric properties than otherwise possible. In a BLT diagram, the bolometric luminosity $L_{\rm bol}$ is the luminosity integrated over the observed SED. It can differ from the true luminosity of the protostar due to inclination along the line of sight, where a protostar viewed through its edge-on disk will appear less luminous than the same protostar viewed along its axis of rotation, or due to extinction. The bolometric temperature is $$T_{\rm bol} = 1.25\times10^{-11}\int_0^\infty \nu S_\nu\, d\nu \bigg/ \int_0^\infty S_\nu\, d\nu~{\rm K~Hz^{-1}},$$ where $\nu$ is the frequency and $S_\nu$ is the flux density at that frequency [@mye93]. It is as low as 20 K for the most embedded protostars [@stu13] and increases as the envelope and disk accrete onto the star, reaching the effective temperature of the central star when circumstellar material is negligible. For a given protostar, $T_{\rm bol}$ also depends on the inclination.
We obtained bolometric luminosities and temperatures by trapezoidal integration under the available photometry and IRS spectra using `tsum.pro` from the IDL Astronomy Users’ Library.[^1] Upper limits are ignored, and the IRS spectra are rebinned to 16 fluxes. For the luminosities, we assume a distance of 420 pc to Orion based on high-precision parallax measurements of non-thermal sources in the Orion Nebula region [@san07; @men07; @kim08; @kou17].
The BLT Diagram
---------------
The BLT diagram for the 330 HOPS targets treated in this paper appears in Figure \[f.blt\], and classification statistics appear in Table \[t.target\]. There are 91 Class 0 sources, 224 Class I sources, and 15 Class II sources. Of the 315 protostars (Class 0 and Class I objects), 29% are in Class 0. Because we consider only $T_{\rm bol}$, these counts differ slightly from the results of @fur16, reviewed in Section 2.
While the standard classification scheme by $T_{\rm bol}$ does not contain a flat-spectrum category, the sources classified as such by @fur16 have $T_{\rm bol}$ ranging from 83 to 1200 K with the middle 80% falling between 190 and 640 K; the mean is 431 K. This distribution features lower temperatures than that of @eva09, who found that the middle 79% of their flat sources have $T_{\rm bol}$ between 350 and 950 K with a mean of 649 K. After correcting for extinction, the middle 77% of their flat sources have $T_{\rm bol}^\prime$ between 500 and 1450 K with a mean of 844 K. Compared to the results from @fur16, their larger temperatures before extinction correction are likely due to different definitions of the class, where @fur16 use the spectral index between 4.5 and 24 and @eva09 use the index between 2 and 24 . The first definition allows sources that have rising SEDs from 2 to 4.5 (a sign of extinction, either intrinsic to the source or foreground) and thus have relatively lower $T_{\rm bol}$ to be classified as flat. In Section \[s.diagnostics\] we show how $T_{\rm bol}$ is dependent on foreground reddening, particularly for sources with low envelope densities. Differences among authors in the definition of spectral slope and the means of correction for foreground reddening, if any, add uncertainty in the claimed range of $T_{\rm bol}$ for flat-spectrum sources.
In Figure \[f.blt\] we also display the histogram of $L_{\rm bol}$, which is the protostellar luminosity function of the sample, and the histogram of $T_{\rm bol}$. As seen in Table \[t.lum\], the bolometric luminosities of the HOPS protostars range over nearly five orders of magnitude, from 0.017 to 1500 $L_\sun$, with a mean of 13 $L_\sun$ and a median of 1.1 $L_\sun$. The luminosity shows a clear peak near 1 $L_\sun$, a width at half-maximum in $\log (L/L_\sun)$ of 2, and a tail extending beyond 100 $L_\sun$. The overall shape is similar to that determined by the extrapolation of [*Spitzer*]{} photometry by @kry12 for the Orion molecular clouds as well as for other giant molecular clouds forming massive stars, such as Cep OB3 and Mon R2. The protostellar luminosity function derived from the [*Spitzer*]{} c2d and Gould Belt surveys by @dun13 peaks at a higher luminosity. That diagram uses luminosities corrected for extinction, but it shows a similar width to the Orion luminosity function.
In contrast, the histogram of $T_{\rm bol}$ is quite flat. Each of the bins between 30 and 600 K contains 15 to 20% of the sample. Note that the drop-off for Class II sources is a selection effect due to the focus of HOPS on protostars. Across Orion, the number of Class II sources exceeds the number of protostars by a factor of three [@meg16].
To examine how luminosity depends on evolutionary state, we divide the sample into five bins of equal spacing in $\log\ T_{\rm bol}$. Table \[t.bltreg\] shows the five bins, the number of sources in each bin, and the median and interquartile range of their luminosities. (The interquartile range is the difference between the third and first quartiles of the distribution.) These results also appear as the large red diamonds in Figure \[f.blt\], with the interquartile ranges plotted as vertical red bars. They show a monotonic decrease in the median $L_{\rm bol}$ with increasing $T_{\rm bol}$ across the full range of protostars. They also show a wide range of luminosities in each bin, a spread of three orders of magnitude. The monotonic decline in median luminosities and broad spread in luminosities are the two most salient properties of the HOPS BLT diagram.
This decrease in luminosity can also be shown by dividing the sample into Class 0 and Class I protostars. The Class 0 luminosities are larger, ranging from 0.027 to 1500 $L_\sun$ with a mean of 30 $L_\sun$ and a median of 2.3 $L_\sun$. The Class I luminosities range from 0.017 to 360 $L_\sun$ with a mean of 6.5 $L_\sun$ and a median of 0.87 $L_\sun$. A two-sample Kolmogorov-Smirnov (KS) test reveals a probability of only $5.5\times10^{-4}$ that the Class 0 and Class I luminosity histograms were drawn from the same distribution. Figure \[f.histo\] shows the histograms of the two classes, plotted both as the number per bin and as the fraction of each class per bin. As we discuss in Section \[s.bias\], the difference in luminosity is unlikely to be due to the effects of incompleteness and extinction on the BLT diagram.
![Bolometric luminosities and temperatures of all 330 YSOs in the sample. Dashed lines show the traditional divisions into Class 0, Class I, and Class II. Large diamonds show the median luminosities in each of five bins that are equally spaced in $\log\ T_{\rm bol}$, and the solid vertical lines show the interquartile luminosity ranges. The histograms show the marginal distributions for luminosity and temperature. The blue line connects the pre- and post-outburst positions of HOPS 223; the symbol that happens to lie near its midpoint represents a different protostar. Pink boxes mark the post-outburst locations of the other three luminosity outbursts in the sample: HOPS 376 is the more luminous of the two Class I outbursts, HOPS 388 is the other, and HOPS 383 is the Class 0 outburst.\[f.blt\]](fig_blt_all.eps){width="\hsize"}
[lcccc]{} All & 0.017 & 1500 & 1.1 & 13\
L 1641 & 0.017 & 220 & 0.70 & 5.0\
ONC & 0.046 & 360 & 2.4 & 12\
Orion B & 0.027 & 1500 & 1.5 & 30\
All & 0.027 & 1500 & 2.3 & 30\
L 1641 & 0.027 & 140 & 1.6 & 12\
ONC & 0.25 & 38 & 4.2 & 9.1\
Orion B & 0.062 & 1500 & 2.9 & 65\
All & 0.017 & 360 & 0.87 & 6.5\
L 1641 & 0.017 & 220 & 0.69 & 3.7\
ONC & 0.046 & 360 & 1.9 & 14\
Orion B & 0.027 & 33 & 0.89 & 5.7
![Histograms of bolometric luminosity for the 91 Class 0 and 224 Class I protostars. The left panel shows the number per bin, and the right panel shows the fraction of each class per bin to facilitate comparison.\[f.histo\]](lc0c1hist.eps){width="\hsize"}
Dependence of the BLT Diagram on Region
---------------------------------------
With 330 sources, we can divide Orion into regions and retain enough protostars in each to examine BLT trends as a function of location or environment. Due to the roughly north-south alignment of the Orion molecular clouds, we define the regions simply as declination ranges. Figure \[f.orion\] shows how the 330 sources, color-coded by $T_{\rm bol}$ class, are divided into regions. This division into groups is beneficial, because we can compare BLT diagrams for two separate molecular clouds within the Orion OB association: Orion A and B.
[lcccccccc]{} (20, 46) & 49 & 2.9 (5.8) & 13 & 1.2 (3.4) & 17 & 6.6 (7.2) & 19 & 2.1 (5.3)\
(46, 110) & 90 & 1.5 (3.4) & 44 & 0.94 (2.0) & 20 & 3.3 (4.4) & 26 & 2.2 (4.3)\
(110, 240) & 67 & 1.1 (2.1) & 42 & 0.71 (1.1) & 14 & 1.9 (5.4) & 11 & 0.89 (19.)\
(240, 550) & 86 & 0.72 (3.0) & 47 & 0.63 (1.7) & 20 & 1.1 (4.8) & 19 & 0.54 (6.4)\
(550, 1300) & 38 & 0.63 (2.5) & 27 & 0.62 (1.5) & 8 & 2.1 (4.4) & 3 & 2.8 (15.)
![Locations of the 330 sources within Orion and the dividing lines that separate them into regions. Sources are coded by $T_{\rm bol}$ class as shown. Names of the regions used for statistics are printed in black, while names of the Orion B subregions are printed in gray italics.\[f.orion\]](regions.eps){width="\hsize"}
The HOPS sources north of $-2.5^\circ$ are part of the Orion B molecular cloud [e.g., @wil05; @bal09]. This consists of three distinct fields: the Lynds 1622 field, the field containing the NGC 2068/2071 nebulae, and the field containing the NGC 2024/2023 nebulae [@meg12]. These fields contain two clusters, a number of groups, and relatively isolated stars [@meg16]. While there is some disagreement as to whether these are parts of a single coherent cloud, they have similar distance and velocity, so we combine all 78 sources in Orion B for this work.
Orion A contains HOPS sources south of $-4.6^\circ$. (Due to the gap between Orion A and B, there are no HOPS sources between $-2.5^\circ$ and $-4.6^\circ$.) We divide Orion A into two regions, setting the boundary at $-6.1^\circ$. The northern region is the Orion Nebula Cluster (ONC). While the Orion Nebula itself contains no HOPS sources due to saturation in the 24 [*Spitzer*]{} band used to identify them, the outer regions of the ONC are rich in HOPS protostars. It contains 79 sources. Our ONC field, while larger than some definitions of the ONC and encompassing Orion Molecular Cloud (OMC) 2, 3, and 4, approximates the boundaries in @car00 and @meg16. The southern region of Orion A is Lynds 1641 (L 1641); it contains 173 sources, including multiple clusters, groups, and isolated protostars. Dividing the Orion A cloud thus gives us the opportunity to compare the BLT diagram of a rich cluster to that of a cloud dominated by smaller groups, clusters, and relatively isolated stars.
Table \[t.target\] lists the regions and the number of sources, number of protostars of each class, and fraction of Class 0 protostars for each. Tables \[t.lum\] and \[t.bltreg\] give the luminosity statistics for each region, and Figures \[f.blt\_l1641\] through \[f.blt\_orionb\] show the BLT diagrams for each region.
The division of protostars between Class 0 and Class I is similar among the three regions and the whole sample, but there are important differences. The fraction of protostars in Class 0 increases from south to north, going from $0.20\pm0.03$ in L 1641 to $0.35\pm0.05$ in the ONC to $0.41\pm0.06$ in Orion B. @stu15 found a similar increase from south to north within L 1641 and the ONC. The larger Class 0 fraction in Orion B meshes with the finding of @stu13 that the fraction of sources that are PACS Bright Red Souces (PBRS, a class of extremely young protostars) is higher in Orion B (0.17) than in Orion A (0.01).
![Bolometric luminosities and temperatures of the 173 sources in L 1641 (between declinations $-8.9^\circ$ and $-6.1^\circ$). Temperature bins are the same as in Figure \[f.blt\].\[f.blt\_l1641\]](fig_blt_l1641.eps){width="\hsize"}
![Bolometric luminosities and temperatures of the 79 sources in the ONC (between declinations $-6.1^\circ$ and $-4.6^\circ$). Temperature bins are the same as in Figure \[f.blt\].\[f.blt\_onc\]](fig_blt_onc.eps){width="\hsize"}
![Bolometric luminosities and temperatures of the 78 sources in Orion B (between declinations $-2.5^\circ$ and $1.9^\circ$). Temperature bins are the same as in Figure \[f.blt\].\[f.blt\_orionb\]](fig_blt_orionb.eps){width="\hsize"}
The typical bolometric luminosities of the protostars are largest in the ONC and smallest in L 1641, with the median luminosity declining from 2.4 $L_\sun$ in the ONC to 1.5 $L_\sun$ in Orion B to 0.70 $L_\sun$ in L 1641. In each region, the median Class 0 source is more luminous than the median Class I source by a factor ranging from 2.2 in the ONC to 3.3 in Orion B. In Orion B and L 1641, the [*mean*]{} bolometric luminosity is also larger in Class 0 than in Class I. This is not the case in the ONC; there, the high mean luminosity for Class I protostars is mainly due to HOPS 370 (OMC 2 FIR 3; @mez90; @ada12). It has $L_{\rm bol}=361~L_\sun$ and $T_{\rm bol}=71.5$ K, near the Class 0/I boundary. Without this source, the mean luminosity for Class I ONC sources is 7.3 $L_\sun$, less than that of the Class 0 protostars in the region.
We also show the luminosities in the five $T_{\rm bol}$ bins discussed above. In each region, the bolometric luminosity decreases with increasing bolometric temperature, except for the bins of highest $T_{\rm bol}$ in the ONC and in Orion B, which contain very few sources, and between the two bins of lowest $T_{\rm bol}$ in Orion B. The interquartile ranges vary between 1 and 8 $L_\sun$ for most bins, although the lightly populated bins for $T_{\rm bol} > 110$ K in Orion B have ranges up to 19 $L_\sun$.
Luminosity Outbursts
--------------------
Five Orion protostars have been identified as outbursting sources. (See @aud14 for a recent review of the outburst phenomenon in YSOs.) Reipurth 50 [@str93] lacks a HOPS identifier; it was saturated in the 4.5 [*Spitzer*]{} band used to find protostars when establishing the HOPS target catalog and is not part of the @fur16 sample. V883 Ori (HOPS 376; @str93) and V1647 Ori (McNeil’s Nebula; HOPS 388; @mcn04) began their outbursts before they were imaged with [*Spitzer*]{}. The pre-outburst SED of HOPS 383 [@saf15] was faint and poorly sampled, and a firm estimate of its pre-outburst bolometric properties is impossible. For HOPS 376, 383, and 388, the @fur16 properties used here are based on only their post-outburst SEDs. They are shown with pink boxes in Figure \[f.blt\].
The fifth outburst, V2775 Ori (HOPS 223; @car11 [@fis12]), has a well-sampled SED both before and after its outburst. @fur16 tabulated its BLT properties based on its combined pre- and post-outburst SEDs, acknowledging that this gives unreliable numbers but aiming for a uniform treatment of the large sample. We find a pre-outburst bolometric luminosity and temperature of 1.93 $L_\sun$ and 348 K, and we find post-outburst BLT properties of 18.0 $L_\sun$ and 414 K. (Pre-outburst data are from Table 1 of @fis12, while post-outburst data combine photometry from Table 2 of that paper with photometry derived from the 2011 IRTF spectrum presented therein.) While the pre-outburst properties are less reliable due to a lack of photometry beyond 70 , HOPS 223 is a member of Class I at both epochs. The pre- and post-outburst positions of HOPS 223 in BLT space are connected with a blue line in Figure \[f.blt\]. They are not used in the calculations of statistics; for this we retain the bolometric properties tabulated by @fur16, which place the object in the cluster of three points near 20 $L_\sun$ and 250 K.
Effect of Incompleteness and Extinction\[s.bias\]
-------------------------------------------------
When comparing the luminosities of the Class I and Class 0 protostars, potential biases due to incompleteness and extinction must be considered. In the [*Spitzer*]{} data, detection schemes can miss very deeply embedded Class 0 protostars with weak fluxes at wavelengths $\le24$ . To mitigate this source of incompleteness, @stu13 augmented the HOPS sample with 70 images acquired by [*Herschel*]{}/PACS to find new protostars not identified with [*Spitzer*]{}. They found that the original [*Spitzer*]{}-based detection [@meg12; @meg16] was not significantly incomplete, as there were only 15 likely protostars detected at 70 that were missed in the [*Spitzer*]{} sample of more than 300. @tob15 subsequently found one more. The majority of the newly detected protostars (14/16) are located in L 1641 or Orion B, not in regions of high nebulosity like the ONC, indicating that these sources were not previously detected due to their unusually faint 24 fluxes and not due to incompleteness in the [*Spitzer*]{} data resulting from confusion with nebulosity.
Another concern is that the far-IR nebulosity may hinder the detection of faint protostars in the 70 band. However, the decrease in luminosity between the Class 0 and Class I sources persists across various regions within Orion, including the high-background ONC and the low-background L 1641 (Figures \[f.blt\_l1641\] through \[f.blt\_orionb\]). This suggests that the difference in the luminosities is not the result of incompleteness to faint Class 0 protostars.
A final potential bias in the data is that foreground extinction may lead to the misclassification of protostars. @stu15 studied the effects of extinction-driven misclassification of Class I and Class 0 protostars, both by foreground material and “self-extinction” due to inclination. They find that when far-IR data are included in the SED analysis, as is the case here, the extinction-driven misclassification probability is negligible over statistical sample sizes such as ours. Specifically, they find that for protostars with measured $T_{\rm bol}=70$ K (that is, borderline Class 0 YSOs), the probability of misclassification is $< 15\%$ with foreground extinction levels of $A_V=30$ mag and steeply decreases with lower extinction levels. Furthermore, they find median extinction levels for HOPS protostars of $A_V=23.3$ mag in the ONC and $\sim 12$ mag in L 1641, indicating that misclassification of this type is not a concern when far-IR data are included in protostellar SED analysis.
A related concern is the potential misclassification of reddened Class II objects as flat-spectrum or Class I protostars. @fur16 classified the 330 sources with $T_{\rm bol}$, the 4.5 to 24 slope, and qualitative assessment of the SEDs, finding that 319 are Class 0, I, or flat-spectrum. Since $A_{[4.5]}$ is about 0.5 $A_{K_s}$ [@fla07], the slope from 4.5 to 24 is less influenced by foreground reddening than slopes that include data from shorter wavelengths. Additionally, far-IR photometry exists for the entire sample, and far-IR emission is affected very little by extinction. If an envelope exists, the far-IR emission will be stronger than if there is just a disk, so envelope- and disk-dominated sources are more easily distinguishable with such data. When modeling the sources, @fur16 found that for 324 of the 330, the far-IR emission is best fit with a model that includes an envelope. Our assessment, using only $T_{\rm bol}$, finds 315 Class 0, I, or flat-spectrum sources. (Five sources that are Class I by $T_{\rm bol}$ alone are Class II in the multi-pronged analysis by @fur16, and nine sources that are Class II by $T_{\rm bol}$ alone are Class I or flat-spectrum in their analysis.) While there is minor disagreement between an analysis limited to $T_{\rm bol}$ and one that uses additional information, multiple lines of evidence suggest that nearly all of our sources have protostellar envelopes.
UNDERSTANDING PROTOSTELLAR EVOLUTION VIA SED MODELING {#s.model}
=====================================================
Modeling of the 330 SEDs is described in detail by @fur16. Here we review the most important points. The HOPS team created a grid of 3040 SED models, each viewed from ten inclinations, with the code of @whi03. This code performs Monte Carlo simulations of radiative transfer through a dusty circumstellar environment. It uses an axisymmetric geometry and includes a central luminosity source, a flared disk with power-law scale height and radial density profiles, an envelope defined by the rotating spherical collapse model of @ulr76, and a bipolar envelope cavity with walls described by a polynomial expression.
The models sample parameters of interest in the study of protostars: 19 mass infall rates that scale the envelope density profile (including the case of no envelope), four disk radii, and five cavity opening (half-)angles. The system luminosity can take on values between 0.05 and 600 $L_\sun$. Other parameters, including the dust properties, are held constant, as described in @fur16. The quality of the model fits is evaluated with the parameter $R$. This is a measure of the average, weighted, logarithmic deviation between the observed and model SEDs; the model with the minimum value of $R$ is the best-fit model. @fur16 found that most protostars are well fit by models from the grid, although there are some degeneracies among model parameters, and the quality of the best-fit model for each protostar depends in part on how well-constrained the SED is. They estimate the reliability of each model fit by examining the modes of parameter values of models within a certain range of the best-fit $R$. We refer the reader to that paper for plots showing the quality of the fit to each object.
Among other results, @fur16 report the modeled envelope mass inside 2500 AU for each source, which is a function of other model parameters as shown below. In this section we show the utility of this mass in diagnosing envelope evolution. We then show how differences between the total luminosities of protostars and their observed luminosities may be accounted for via SED modeling. Finally, we examine the relationship between the evolutionary states and total luminosities of the HOPS sources using results from the fitting.
Model-based Masses as an Envelope Diagnostic\[s.diagnostics\]
-------------------------------------------------------------
Since a primary goal of studies of protostellar evolution is to track the flow of mass from the molecular cloud onto the central forming star, the envelope mass $M_{\rm env}$ remaining inside some radius $r$ is a useful diagnostic of envelope evolution. The youngest protostars have massive envelopes, while Class II objects have little to no remnant envelope. Further, the envelope mass is an easily understood quantity that changes in a straightforward way with the inclusion of outflow cavities and is independent of inclination angle. While we expect the envelope mass within 2500 AU to be correlated with both the ultimate main-sequence mass of the star and the age of the protostar, the envelope masses we model extend over four orders of magnitude, and the stars formed will mostly have masses that extend over about two orders of magnitude. Thus the envelope mass is mainly sensitive to age and is expected to be the intrinsic property that best traces age.
We set $r$, the radius inside which we consider the envelope mass, equal to 2500 AU. This corresponds to the 6 half-width at half maximum of the 160 PACS beam at the distance of Orion. This is the largest spatial scale probed by the HOPS point-source photometry near the expected peaks of the SEDs in the sample. The analysis in Section 5.2 assumes that envelope material inside 2500 AU is participating in free-fall toward the star, which is expected to be the case for all but the youngest sources.
The models we use assume axisymmetry, with deviations from spherical symmetry due to rotational flattening of the envelope and the presence of outflow cavities. These are characterized, respectively, by the centrifugal radius $R_C$ and the cavity opening angle $\theta_{\rm cav}$. The centrifugal radius gives the outer radius at which the infalling envelope material accumulates onto the central Keplerian disk. It may initially be equal to the outer radius of the disk, and this is assumed to be the case in our grid of models, although viscous spreading will cause the disk to expand outward. The cavity opening angle is the angle from the pole to the cavity edge at a height above the disk plane equal to the envelope radius. (See Figure 6 of @fur16 for a schematic illustration.)
The masses inside 2500 AU are easily scaled to other radii $r^\prime$, as seen in the top panel of Figure \[f.massenv\]. To a close approximation, the masses can be multiplied by $(r^\prime/2500~{\rm AU})^{1.5}$. Points of the same color and increasing mass show the effect of increasing $R_C$ from 5 to 500 AU. Points of different colors show the effect of changing $\theta_{\rm cav}$ from 5$^\circ$ to 45$^\circ$. The largest discrepancies between the actual masses within 5000 or 10,000 AU and those extrapolated from 2500 AU occur for large $R_C$ and $\theta_{\rm cav}$.
In the case of spherical symmetry, we can relate the mass to the infall rate, which is often used to parameterize models [@whi03]. The relationship is $$\begin{split}
M_{\rm env}\left(<r\right)=0.105\:M_\sun\bigg(\frac{\dot{M}_{\rm env}}{10^{-6}\:M_\sun\,{\rm yr}^{-1}}\bigg)\!\bigg(\frac{M_*}{0.5\:M_\sun}\bigg)^{-1/2}\\\times\bigg(\frac{r}{10^4\:{\rm AU}}\bigg)^{3/2}\label{e.mdot},
\end{split}$$ where $\dot{M}_{\rm env}$ is the rate at which matter from the envelope accumulates onto the disk and $M_*$ is the mass of the central star. Note that this assumes a constant, spherical infall, with the dominant mass being the central protostar.
Another common model parameter is $\rho_1$, the envelope density at 1 AU in the limit of no rotation [@ken93]. The relationship between envelope mass and $\rho_1$ is $$M_{\rm env}\left(<r\right)=0.139~M_\sun \bigg(\frac{\rho_1}{10^{-14}~{\rm g~cm}^{-3}}\bigg)\bigg(\frac{r}{10^4~{\rm AU}}\bigg)^{3/2}.$$ Like the envelope infall rate, this quantity does not account for changes in the cavity opening angle. For envelopes with $R_C\gg1~{\rm AU}$, it also gives densities much larger than actually exist in the envelope [@fur16].
We explore the effect of deviations from spherical symmetry on envelope mass in the second panel of Figure \[f.massenv\], which shows the effect of centrifugal radius $R_C$ and cavity opening angle $\theta_{\rm cav}$ on the mass inside 2500 AU. For a rotating envelope ($R_C>0$), the mass depends weakly on $R_C$ for $R_C\ll r$, inducing a small vertical spread in points of different colors. The mass depends more strongly on the cavity opening angle $\theta_{\rm cav}$, since large fractions of the envelope are removed with increasing $\theta_{\rm cav}$. The mass is reduced by up to 45% for the largest cavity opening angle. The top two panels of Figure \[f.massenv\] show results for the models in our SED grid with $\dot{M}_{\rm env}=10^{-6}~M_\sun~{\rm yr}^{-1}$ and with $M_*=0.5~M_\sun$, but the behavior is the same for other $\dot{M}_{\rm env}$ and $M_*$. This demonstrates the value of an envelope diagnostic that includes the effects of different cavity opening angles; reporting only the envelope infall rate or a representative density can be misleading.
![[*Top panel:*]{} Ratio of envelope mass inside 5000 or 10,000 AU to that at 2500 AU, plotted against envelope mass inside 2500 AU for models with $\dot{M}_{\rm env}=10^{-6}~M_\sun~{\rm yr}^{-1}$. Points of the same color but increasing mass correspond to increasing $R_C$. Points of differing color correspond to different $\theta_{\rm cav}$. The envelope mass is not dependent on inclination angle. The dashed lines show the ratios expected for a strict $r^{1.5}$ mass dependence. [*Second panel:*]{} Comparison of envelope masses from the grid to results for the angle-averaged solution with no cavity. The grid masses depend mildly on $R_C$ and dramatically on cavity angle. [*Third panel:*]{} Bolometric temperature versus envelope mass inside 2500 AU for a selection of models with the indicated $\log \dot{M}_{\rm env}$ (in $M_\sun$ yr$^{-1}$) and cavity opening angles. The spread in $T_{\rm bol}$ at each envelope mass is due to varying inclination angle, from low $T_{\rm bol}$ near edge-on to high $T_{\rm bol}$ near face-on. Dashed lines mark the traditional boundaries between SED classes. [*Bottom panel:*]{} Same as above, except the results are shown only for an inclination angle of $63^\circ$ as the SED is subjected to foreground extinction ranging from $A_V=0$ to 19.0 mag.\[f.massenv\]](massenv.eps){width="\hsize"}
In the rest of Figure \[f.massenv\], we show how these masses compare to $T_{\rm bol}$. While it has the advantage of being directly measurable from observed SEDs, $T_{\rm bol}$ depends strongly on source inclination and foreground reddening. In the third panel of Figure \[f.massenv\], we compare $T_{\rm bol}$ to $M_{\rm env}\left(<2500~{\rm AU}\right)$ for selected models with $\dot{M}_{\rm env}$ of $10^{-7}$, $10^{-6}$, $10^{-5}$, and $10^{-4}~M_\sun~{\rm yr}^{-1}$ and cavity opening angles of $15^\circ$ and $35^\circ$. For all models, there is a large spread in $T_{\rm bol}$ as the inclination runs from $18^\circ$ (highest $T_{\rm bol}$) to $87^\circ$ (lowest $T_{\rm bol}$), in some cases crossing the traditional boundaries between SED classes.
In the bottom panel of Figure \[f.massenv\], we compare $T_{\rm bol}$ to $M_{\rm env}\left(<2500~{\rm AU}\right)$ for the same models, except the inclination angle is held constant at $63^\circ$ and the foreground reddening is varied. The largest $T_{\rm bol}$ in each case is for $A_V=0$ mag, and the bolometric temperature decreases as $A_V$ increases. We show results for $A_V$ at the first, second, and third quartiles of the distribution used to model the HOPS sources, or $A_V=2.5$, 9.0, and 19.0 mag. Varying $A_V$ over this range has less of an effect than varying the inclination angle over its entire range, but the spread is still several hundred K for the least massive envelopes.
The envelope mass inside a particular radius is dependent on the assumed density distribution within the disk and envelope. The SED models presented by @fur16 fit the observations well, suggesting that the density distributions are plausible if not necessarily unique. Compared to $T_{\rm bol}$, this mass is an alternative diagnostic of envelope evolution that is insensitive to inclination angle and foreground reddening. In Figure \[f.menvtbol\] we plot the envelope mass within 2500 AU for each best-fit model against the bolometric temperature of the observed SED. There is a weak anticorrelation between the two, with substantial scatter due to the dependence of $T_{\rm bol}$ on not only envelope mass, but also on source inclination and foreground reddening.
![Model-derived envelope mass within 2500 AU versus observed bolometric temperature for the 330 YSOs. The dashed lines mark the divisions into SED classes.\[f.menvtbol\]](menvvtbol.eps){width="\hsize"}
Model-based Total Luminosities
------------------------------
![Total (model-derived) luminosity versus bolometric (observed) luminosity for the 330 YSOs. The dashed line marks equality, and the dotted line marks the case where the total luminosity is ten times the bolometric luminosity. Cases with $L_{\rm tot} \gg L_{\rm bol}$ are addressed in Section 5.2.\[f.ltotbol\]](ltotvlbol.eps){width="\hsize"}
The total luminosity of a protostar generally differs from its bolometric luminosity due to foreground extinction and inclination. Foreground extinction reduces the flux at all wavelengths (although trivially at far-IR wavelengths and longer), so correcting for this always increases the luminosity from its observed value. Inclination can affect the luminosity in either direction. Converting an observed flux to a luminosity involves multiplying by $4\pi$ sr, which assumes that the source is isotropic. Due to high extinction by a circumstellar disk in the (approximate) equatorial plane of the star and low extinction through the cavity aligned with the rotation axis, protostars are brighter when viewed along their rotational axes. Thus, multiplying by $4\pi$ sr overestimates the luminosity of a face-on protostar and underestimates the luminosity of an edge-on protostar. (See Figure 7 of @fur16 for an example of how a protostellar SED changes with inclination angle.)
We correct for these effects with SED fitting. In short, the colors of a protostar shortward of 70 are sensitive to inclination, while the colors at longer wavelengths are sensitive to envelope mass, and fitting attempts to break the degeneracy between the two by simultaneously accounting for both wavelength regimes [@ali10]. We subsequently analyze the total luminosity of the best-fit model from @fur16 rather than the observed luminosity.
Figure \[f.ltotbol\] compares the modeled total luminosity $L_{\rm tot}$ to the observed bolometric luminosity $L_{\rm bol}$ for the 330 YSOs in the sample. The quantity $\log (L_{\rm tot}/L_{\rm bol})$ has a mean of 0.47 and a standard deviation of 0.39, consistent with the ratios generally being greater than unity. In almost all cases, the total luminosity is larger than the bolometric luminosity, because foreground extinction always reduces the bolometric luminosity, while inclination effects can either inflate or reduce the bolometric luminosity. To consider the effect of inclination alone, we define $L_{\rm mod}$ as the integrated luminosity of the best-fit SED at the best-fit inclination, with the modeled foreground extinction removed. The quantity $\log (L_{\rm tot}/L_{\rm mod})$ has a mean of 0.19 and a standard deviation of 0.36. This shows that inclination tends to reduce the bolometric luminosity from the total luminosity; configurations that are sufficiently edge-on to reduce it are more likely than configurations that are sufficiently face-on to increase it.
Total Luminosity versus Envelope Mass
-------------------------------------
![Total luminosity versus envelope mass (TLM) inside 2500 AU for 324 protostars across the entire HOPS survey region. (Six sources with $M_{\rm env} = 0$ in their best-fit models are excluded.) The histograms show the marginal distributions for luminosity and mass. Large diamonds show the median luminosities in each of the mass bins indicated by dotted vertical lines, and the solid vertical lines show the interquartile luminosity ranges.\[f.tlm\]](fig_tlm_all.eps){width="\hsize"}
In Figure \[f.tlm\], we plot the total luminosity $L_{\rm tot}$ and envelope mass inside 2500 AU of the best-fit SED model assigned to each source, creating a total luminosity versus mass (TLM) diagram. Models in the grid have luminosities of 0.1, 0.3, 1, 3, 10, 30, 100, or 300 $L_\sun$, and the luminosity is adjusted by a factor between 0.5 and 2 to improve the SED fit [@fur16]. Therefore, the possible luminosities extend continuously from 0.05 to 600 $L_\sun$. The mass inside 2500 AU is set by the envelope infall rate, the centrifugal radius, and the cavity opening angle. The possible nonzero masses extend from $3.6\times10^{-4}$ to 10 $M_\sun$; there are 898 unique masses over this range. The fractional change from one mass to the next largest ranges from 10$^{-4}$ to 0.36 with a median of $3\times10^{-4}$.
Of the 330 sources, six are fit with models that contain no envelope. These are excluded from the analysis of how luminosity changes with envelope mass. The 324 remaining sources have total luminosities ranging from 0.06 to 600 $L_\sun$, roughly the same as the allowed range, and they have envelope masses ranging from $3.6\times10^{-4}$ $M_\sun$ (the minimum nonzero mass possible) to 7.3 $M_\sun$. The median total luminosity of all 324 sources is 3.0 $L_\sun$, larger than the median bolometric luminosity of 1.1 $L_\sun$, and the median envelope mass inside 2500 AU is 0.03 $M_\sun$.
We divide the sources into bins by envelope mass, with the edges of the bins at $10^{-4}$, $10^{-3}$, 0.01, 0.1, 1, and 10 $M_\sun$. While there is substantial scatter in the luminosities at each envelope mass, the median luminosities in each bin show a clear trend. They rise from 5.1 $L_\sun$ for the most massive envelopes to a peak of 15 $L_\sun$ in the bin extending from 0.1 to 1 $M_\sun$, and then they diminish from 2.6 to 1.7 $L_\sun$ over the three remaining bins. The number of sources in each bin and their median luminosities and interquartile ranges are reported in Table \[t.medlum\].
With medians and interquartile ranges, it can be difficult to assess whether the progression in luminosity with envelope mass is statistically significant. Therefore we also ran two-sample KS tests on the luminosity distributions in each pair of mass bins to assess the likelihood that these luminosities were drawn from the same underlying distribution. From least massive to most massive, the KS probabilities for the first and second, second and third, and first and third bins were, respectively, 92%, 81% and 39%, consistent with our claim that there is little evolution in luminosity over the least massive envelopes. The probabilities that the fourth bin was drawn from the same distribution as any of the first three bins are all between 10$^{-7}$ and 10$^{-6}$, indicative of a statistically significant decline from the fourth to the third bin. The KS probabilities for the fifth (most massive) bin compared to the first through fourth bins, are, respectively, 8%, 20%, 52%, and 9%.
Figures \[f.tlm\_l1641\] through \[f.tlm\_orionb\] show TLM diagrams for each of our three defined regions, and Table \[t.medlum\] summarizes the median total luminosities as a function of envelope mass by region. While the median total luminosity of all 324 sources with envelopes is 3.0 $L_\sun$, this quantity varies from region to region. It is largest in the ONC, at 6.2 $L_\sun$, and it falls to 3.3 $L_\sun$ in Orion B and 2.0 $L_\sun$ in L 1641. In all three regions, the median luminosity is largest for envelopes between 0.1 and 1 $M_\sun$, again with much scatter. The median luminosity falls for the next most massive bin and then tapers or remains roughly constant over the two least massive bins.
The total luminosity and envelope mass determine different properties of the SEDs. The former determines the overall flux level, while the latter roughly sets the amount of emission at mid- to far-IR wavelengths. To examine whether any degeneracies in the model fits may drive the reported trend of luminosity with envelope mass, for each source we consider the spread in total luminosities and inner envelope masses of all models that have $R<R_{\rm best}+2$, where the subscript refers to the best-fit model. The number of fits that satisfy this criterion varies from source to source, but on average it allows the 448 best fits per source. When considering all models that satisfy this criterion for all 330 sources, the standard deviation of $L/L_{\rm best}$ is 0.31 orders of magnitude, and the standard deviation of $M/M_{\rm best}$ is 1.29 orders of magnitude. While large, these ratios are not correlated. Instead, they are slightly anti-correlated, with a correlation coefficient of $-0.23$. Therefore, uncertainty in the model fitting is unlikely to be the source of the trends discussed in this and subsequent sections.
[lcccccccc]{} ($10^{0}$,$10^{1}$) & 22 & 5.1 (31.) & 4 & 5.1 (4.4) & 10 & 5.7 (58.) & 8 & 6.3 (9.5)\
($10^{-1}$,$10^{0}$) & 67 & 15. (45.) & 22 & 14. (31.) & 22 & 32. (54.) & 23 & 9.9 (45.)\
($10^{-2}$,$10^{-1}$) & 103 & 2.6 (5.5) & 53 & 2.0 (5.0) & 23 & 5.1 (7.8) & 27 & 3.0 (5.5)\
($10^{-3}$,$10^{-2}$) & 82 & 2.0 (4.8) & 59 & 2.0 (4.4) & 13 & 5.8 (17.) & 10 & 0.91 (1.2)\
($10^{-4}$,$10^{-3}$) & 50 & 1.7 (5.5) & 30 & 1.7 (2.9) & 10 & 6.1 (5.7) & 10 & 0.82 (5.8)\
------------------------------------------------------------------------
All & 324 & 3.0 (9.8) & 168 & 2.0 (4.7) & 78 & 6.2 (31.) & 78 & 3.3 (15.)
![Total luminosity versus envelope mass inside 2500 AU for 168 protostars in L 1641.\[f.tlm\_l1641\]](fig_tlm_l1641.eps){width="\hsize"}
![Total luminosity versus envelope mass inside 2500 AU for 78 protostars in the ONC.\[f.tlm\_onc\]](fig_tlm_onc.eps){width="\hsize"}
![Total luminosity versus envelope mass inside 2500 AU for 78 protostars in Orion B.\[f.tlm\_orionb\]](fig_tlm_orionb.eps){width="\hsize"}
ANALYSIS OF LUMINOSITY AND ENVELOPE EVOLUTION
=============================================
In this section, we discuss three trends apparent in the BLT diagram that persist when we switch from observed parameters to intrinsic properties estimated via SED modeling. These are
- the relatively flat distributions of bolometric temperature and envelope mass when considering the entire sample,
- the decrease in luminosity with decreasing envelope mass or increasing bolometric temperature, and
- the broad scatter in luminosity at each envelope mass or bolometric temperature.
The Flat Distribution of Envelope Mass
--------------------------------------
The flat histogram of bolometric temperature noted in Section 3 persists when we transition to the envelope mass in Figure \[f.tlm\]. The fraction of the objects in each bin varies between 10 and 20% for masses between $3 \times 10^{-4}$ and 0.3 $M_\sun$. At larger masses the histogram declines; there are fewer objects with envelopes of $\sim1$ $M_\sun$ or greater inside 2500 AU. These presumably form higher-mass stars; our consideration of the initial mass function in Section 5.3 suggests that such massive stars and envelopes should be rare.
The relatively flat histogram suggests that $dN/d(\log M_{\rm env})$ is constant. Expanding this expression, $$\frac{dN}{d(\log M_{\rm env})} = \frac{dN}{dt} \frac{dt}{d(\log M_{\rm env})}$$ is constant. The first term on the right is the star-formation rate; if this is constant, then $d(\log M_{\rm env})/dt$ is also constant. This implies an exponential decline in the envelope mass with time, which also suggests a roughly exponential decline in $\dot{M}$, the envelope infall rate.
This form for the infall rate is motivated by the work of @bon96, @mye98, @sch04, and @vor10. It is a consequence of the rate being roughly proportional to the remaining envelope mass, as expected if the rate equals the mass divided by some characteristic time, for example, the free-fall time for the mass within a given radius. In contrast, @oso99 found that the formation of the most massive stars (early B types and hotter) is best modeled with infall rates that increase with time. Some of the most massive envelopes in the HOPS sample may exhibit this feature; however, our focus is on the lower-mass objects that dominate the sample.
Decreasing Luminosities with Evolution\[s.evo\]
-----------------------------------------------
{width="\hsize"}
In the TLM diagram, the steep drop in median luminosity from the 0.1 – 1 $M_\sun$ mass bin to the next less massive one, followed by a more gradual decline, suggests an exponentially declining form for the dependence of the protostellar luminosity on envelope mass. This is also seen in the BLT diagram, where there is a slow decrease of $L_{\rm bol}$ with increasing $T_{\rm bol}$. Here we show how this feature of the BLT and TLM diagrams is a consequence of the exponentially declining envelope masses hypothesized above.
For consistency with results from the radiative transfer model, we consider the envelope mass within a radius of 2500 AU (dropping the $<2500$ AU notation for simplicity). The time dependence of the envelope mass is defined as $$M_{\rm env}(t)=M_{\rm env,0}\times e^{-t\ln 2/t_H},$$ where $t$ is the time elapsed, $M_{\rm env,0}$ is the initial mass, and $t_H$ is the time it takes for the mass to fall to half its initial value (the half-life).
The infall rate is $\dot{M}=M_{\rm env}/t_{\rm ff}$, where $t_{\rm ff}$ is the free-fall time within that radius, $$t_{\rm ff} = \frac{\pi\,(2500\ {\rm AU})^{3/2}}{2\sqrt{2G(M_{\rm env}+M_*)}}.$$ This can be expressed as $t_{\rm ff}=q\,(M_{\rm env}+M_*)^{-1/2}$, where $q=2.2\times10^4$ yr $\sqrt{M_\sun}$. Then $\dot{M}=M_{\rm env}\,(M_{\rm env}+M_*)^{1/2}/q$. At each time step, the stellar mass $M_*$ is $\int_0^t \dot{M}(t)\,dt$.
For comparison with the TLM diagram, we also calculate the luminosity as a function of time. This is the sum of the stellar luminosity $L_*$ and the accretion luminosity $L_{\rm acc}$. For the stellar luminosity we use a fit to the model tracks of @sie00 for stars of mass 0.1 to 3 $M_\sun$ at a model age of $5\times10^5$ yr, $$L_*=3.1\left(\frac{M_*}{0.9\,M_\sun}\right)^{1.34}.\label{e.lum}$$
The accretion luminosity is $\eta\,GM_*\dot{M}/R_*$, where $G$ is the gravitational constant and $\eta$ is a factor of order unity that depends on the details of the accretion process and characterizes how much of the accretion energy is radiated away. We do not explicitly include a circumstellar disk in this model, although the disk may act as a mass reservoir such that the accretion rate onto the star is not instantaneously equal to the envelope infall rate. Here we set $\eta=0.8$, which is typical of accreting young stars [@mey97]. The stellar radius is again a fit to the @sie00 models, $$R_*=3.2\left(\frac{M_*}{0.9\,M_\sun}\right)^{0.34}.$$
The parameters of the central star are a source of uncertainty in this effort. There are few observational constraints on stellar masses and radii for deeply embedded protostars, age is an ambiguous quantity at early times, and the accretion history of a given protostar is expected to have an important influence on its properties [@bar17]. For simplicity, we adopt the @sie00 models at a stated age of $5\times10^5$ yr. These authors’ models and the “hybrid” accretion case of @bar17 are similar at an age of 1 Myr, the earliest time at which the latter are tabulated.
In Figure \[f.model\], we explore the relationship between total luminosity and envelope mass under these assumptions for three cases that produce stars of differing final masses $M_{*,f}$. We arrange the simulations to yield final stellar masses such that the final stellar luminosities bracket the median total luminosity in the bin with the lowest envelope masses. These luminosities are at the tenth, fiftieth, and ninetieth percentiles of the distribution, or 0.21, 1.7, and 14 $L_\sun$. According to Equation (\[e.lum\]), the final stellar masses are then 0.12, 0.58, and 2.8 $M_\sun$.
The free parameters are the initial envelope mass inside 2500 AU and the half-life for the envelope mass. These are chosen to yield the final stellar masses of interest and to reach the lowest envelope masses in the 0.5 Myr expected lifetime for protostars [@eva09]. The inital masses are 0.085, 0.24, and 0.69 $M_\sun$. (The initial envelope mass inside 2500 AU can be less than the final stellar mass due to the infall of material from beyond 2500 AU.) The half-lives are 0.07, 0.06, and 0.05 Myr, respectively. The characteristics of all three models are shown in Table \[t.modelprop\]. The left panels of Figure \[f.model\] show the infall rate, stellar mass, envelope mass, stellar luminosity, accretion luminosity, and total luminosity for the case where $M_{\rm env,0}=0.24$ $M_\sun$ and $M_{*,f}=0.58$ $M_\sun$. The right panel shows the path of this model through TLM space as well as those of the models with smaller and larger final stellar masses.
[lccc]{} $M_{\rm env,0}$ ($M_\sun$) & 0.085 & 0.24 & 0.69\
$\dot{M}_0$ (10$^{-6}$ $M_\sun$ yr$^{-1}$) & 1.1 & 5.3 & 26\
$t_H$ (Myr) & 0.07 & 0.06 & 0.05\
$M_{*,f}$ ($M_\sun$) & 0.12 & 0.58 & 2.8
The luminosity is dominated by accretion over most of each track. Since $R_*\propto M_*^{0.34}$, the accretion luminosity is proportional to $M_*^{0.66}\dot{M}$. It initially rises quickly due to the increasing mass of the star and then falls off slowly due to the decline in the infall rate, leveling out at the luminosity of a star with the resulting final mass. These curves are qualitatively similar to those shown in Figure 6b of @and00.
Uncertainties in the properties of the central star affect the plotted curves in a straightforward way. For example, if the radius of the central star is actually 10% larger than assumed, the total luminosity at early times will be 10% smaller than assumed, when it is dominated by accretion and is inversely proportional to the stellar radius. The total luminosity at late times will be 20% larger than assumed, when it is dominated by the star and is approximately proportional to the stellar radius squared. Both such effects are small compared to the range of the logarithmic luminosity axis in Figure \[f.model\], and the qualitative shapes and positions of the curves will not change appreciably for discrepancies of this magnitude.
In the right panel of Figure \[f.model\], none of the models pass through the median total luminosity in each mass bin, as one would expect if the median total luminosities represented a typical protostar as it moved through the various stages of envelope evolution. By bracketing the majority of the data points, the models instead show how exponentially declining infall rates that produce a range of stellar masses can account for luminosities that, on average, decrease with evolution but are widely scattered.
The asterisks in Figure \[f.model\], which show the positions of the models at 1, 2, 3, 4, and $5 \times 10^5$ yr, are equally spaced in $\log M_{\rm env}$ for each model. Every 100,000 years, the envelope mass drops by 63%, 69%, or 75% for the low-, medium-, and high-mass models, respectively. This is consistent with the roughly flat distributions in $\log T_{\rm bol}$ (Figure \[f.blt\]) and $\log M_{\rm env}$ (Figure \[f.tlm\]).
Some outliers warrant additional attention. First we look at those with high masses and low luminosities. The dashed curve in the figure shows the predicted luminosities for spherical starless cores that range in mass from 0.1 to 10 $M_\sun$ and have uniform temperature 15 K, radius 2500 AU, and power-law density profile with exponent $-2$. Objects much less luminous and more massive than this curve are inconsistent with centrally illuminated sources of the given mass. Visual inspection of the [*Herschel*]{} images indicates that in the extreme cases and many of those near the curve, the envelope masses are likely overestimated due to the inclusion of mass in the aperture that is not part of the protostellar envelope. Another issue may be limitations to the grid of radiative transfer models and degeneracies that could yield unphysical parameters. We do not attempt to explain these cases with our model, although we note that, without them, the median luminosity at earlier times is even higher, supporting the scenario of an early period of rapid accretion. The circled protostars in Figure \[f.model\] are PBRS, the extremely young protostars discovered by @stu13. For PBRS near the curve, visual inspection of the [*Herschel*]{} and APEX images available in @stu13 reveals bright point sources consistent with protostars that have large envelope masses and are truly at a young evolutionary stage.
Objects in the opposite corner of the TLM diagram, with total luminosities greater than 30 $L_\sun$ and envelopes less massive than 0.01 $M_\sun$, are also far from the regime covered by the model tracks. Such a population of luminous, late-stage protostars does not appear in the observational BLT diagram (Figure \[f.blt\]). These ten sources typically have model-derived luminosities much larger than their bolometric luminosities; eight of them have $L_{\rm tot}/L_{\rm bol} > 10$. These large ratios are due either to nearly edge-on orientations or to very large model-derived extinctions $A_V$ along the line of sight to the source. In four cases, $A_V$ exceeds 50 mag. @fur16 judge the quality of the fit for each parameter by comparing the best-fit value to the mode of all fits within some range of acceptability. For all of these sources, $A_V$ is not well constrained by this measure.
The large extinctions may in some cases be due to the outer regions of a nearby protostellar envelope. One of the sources in this group, HOPS 165, was modeled as such in @fis10 due to its being only 5500 projected AU from HOPS 203. Two others are also within 6000 projected AU of another HOPS source.
In their analysis of scattered-light images of the HOPS protostars, J. J. Booker et al. (in preparation) found that protostars that are undetected at 1.6 , which usually correspond to Class 0 sources, have larger bolometric luminosities than protostars that are point sources at 1.6 , which usually correspond to flat-spectrum sources. This is additional evidence for a decrease in luminosity with evolution uncovered through a different means of classifying sources.
### The Stage 0 Lifetime\[s.stage0\]
The distribution of protostars with respect to class can be used to estimate the lifetime of Class 0. Assuming continuous star formation and a Class II half-life of 2 Myr [@eva09], @dun14 estimated a combined protostellar (Class 0 and I) lifetime of 0.5 Myr. @dun15 accounted for additional sources of uncertainty in lifetime calculations. In particular, they argued that the Class II half-life of 2 Myr that is the basis for such calculations may be better estimated as 3 Myr, and that Class III objects that retain disks should be added to the Class II count. With these additional effects, plausible lifetimes for the protostellar phase extend from 0.46 to 0.72 Myr. With 30% of the HOPS protostars in Class 0, the implied Class 0 lifetime is 30% of 0.5 Myr, or 0.15 Myr. This extends from 0.14 to 0.22 Myr if the @dun15 uncertainties are included.
As discussed in the Introduction, the SED class is not a perfect evolutionary indicator. The physical [*stage*]{} of a YSO describes its actual evolutionary condition, which is only suggested by the observed [*class*]{} [@rob06; @dun14]. Although it is difficult to determine, it is worthwhile to investigate the lifetime of Stage 0, when the envelope mass is greater than the mass of the central star. A Stage 0 lifetime that is short relative to the envelope lifetime would point to an early period of rapid mass accretion for protostars, suggesting that Stage I is a relatively long period of lower-level accretion punctuated by episodic bursts. In this case, the true Stage 0 population might feature some very luminous but heavily extinguished young protostars.
In each model plotted in Figure \[f.model\], we can determine when the central star reaches half its final mass, corresponding to the transition from Stage 0 to I. For the models shown, these are 0.075, 0.070, and 0.063 Myr for the low-, , and high-mass models, respectively. These are less than the Class 0 duration of 0.14 to 0.22 Myr. If we instead choose models in which half the stellar mass is assembled in about 0.15 Myr, matching the observationally derived Class 0 duration and requiring larger $t_H$, then the envelope masses inside 2500 AU are still of order $10^{-2}$ $M_\sun$ at 0.5 Myr, and many of the HOPS protostars correspond to models at times near 1 Myr, which is inconsistent with published estimates of the envelope lifetime. This roughly 0.07 Myr Stage 0 lifetime is only about a factor of three greater than the 0.025 Myr lifetime estimated for the PBRS by @stu13 based on the fraction of protostars in that category, assuming PBRS represent a distinct phase of star formation.
If the Stage 0 lifetime is shorter than the Class 0 lifetime, then some of the objects that are young according to observational diagnostics are really more evolved; i.e., some of the Class 0 sources are actually Stage I sources viewed at inclinations near edge-on. The time it takes to assemble half the star is then shorter than estimated from SED analysis; i.e., several times $10^4$ yr instead of more than $10^5$ yr. In this case, most of the envelope infall period of star formation would be characterized by a state of slowly declining, low-level accretion.
Scatter in Luminosities: Episodic Accretion
-------------------------------------------
Exponentially declining infall rates that form a reasonable distribution of stellar masses can explain most of the scatter in the BLT and TLM diagrams. While episodic accretion is not a predominant factor in this scenario, it clearly occurs and is likely responsible for some of the spread. The luminosity changes of V2775 Ori (HOPS 223), HOPS 383, and V1647 Ori (HOPS 388) were factors of 9.3, 35, and 9.7 [@fis12; @saf15; @and04], while the ratio of the third quartile total luminosity to the first quartile total luminosity in each mass bin ranges from 6.5 to 17. (See the blue line for HOPS 223 in Figure \[f.blt\].)
It has been argued from radiative-transfer and hydrodynamical modeling [@dun12; @vor15] and from the detection of CO$_2$ ice features as evidence of past high temperature [@kim12] that protostellar accretion outbursts must be frequent. E. J. Safron et al. (in preparation) searched for direct evidence of these outbursts by comparing IR photometry of the HOPS protostars at two epochs. They find statistical evidence that protostars undergo hundreds of low-amplitude ($\sim10\times$) bursts during their formation periods. These outbursts would lead to scatter away from model tracks that lack episodic accretion.
For a more complete investigation of the influence of episodic accretion on the TLM diagram, we generated luminosity histograms from exponentially declining model tracks. For each of the envelope mass bins in Table \[t.medlum\] except the most massive one, we randomly chose 1000 stars from an initial mass function. We used a function $dN/dM \propto M^{-\alpha}$, where $\alpha=0$ for $M_*<0.07$ $M_\sun$ [@all05], $\alpha=1.05$ for $0.07\ M_\sun <M_*<0.5\ M_\sun$ [@kro02], and $\alpha=2.35$ for $M>0.5$ $M_\sun$ [@kro02].[^2] We then calculated the tracks through TLM space needed to yield stars of these masses and checked the luminosity along each track at a randomly selected mass within the bin. The half-life is assumed to be 0.06 Myr; varying this by 0.01 Myr in either direction has no appreciable effect on the results.
The model and data-derived histograms have about the same widths in all bins: the third quartile luminosity is about a factor of 10 greater than the first quartile luminosity. The median luminosities differ, however. The SED-derived luminosities (Table \[t.medlum\]) are 15.1, 2.56, 2.01, and 1.74 $L_\sun$, while the modeled ones are 3.63, 0.79, 0.35, and 0.33 $L_\sun$. The former values are factors of 3 to 6 greater than the latter ones, although they follow the same trend of sharply decreasing then leveling out as the envelope mass diminishes.
Besides episodic accretion, two other factors could play an additional role in creating these discrepancies. First, the mass and radius of the central source are poorly understood at early times. If our assumed ratios of $M_*/R_*$ are too small, then the predicted accretion luminosities will be too small. Second, the model distributions are influenced by low-luminosity protostars ($<0.1~L_\sun$) that are not detected in our observations. We demonstrated earlier that any incompleteness is not dependent on SED class, but protostars of sufficiently low luminosity at all stages may be missed.
Identifying cases of episodic accretion in the absence of a historical outburst is non-trivial. It would not be evident from SED fitting, which cannot cleanly distinguish outbursts from truly massive and luminous objects. A promising avenue for determining the outburst history of an object is to look instead for unusually extended C$^{18}$O emission as a sign of past heating [@jor15]. Additional signs of recent outbursts may include a small disk mass or radius due to depletion by rapid accretion onto the star or a large cavity opening angle due to clearing by enhanced mass loss. The @fur16 fit to the HOPS 383 outburst gives both a small disk (5 AU in radius) and a large cavity opening angle (45$^\circ$), consistent with this scenario. With spectroscopy of the accretion region in the inner disk, accretion rates may be determined, providing direct evidence for episodic accretion [e.g., @fis12], and spectra indicative of an optically thick inner disk also point to outburst conditions [@con10].
Our data and approach for Orion, a high-mass cloud, can be contrasted with that of @dun10. In their explanation of the BLT distribution of YSOs in five nearby low-mass molecular clouds, they postulated constant infall rates with the variation in luminosities mainly due to episodic accretion. We instead postulate declining infall rates with a reduced but nonzero role for episodic accretion. Our finding recalls the theoretical work of @off11, where models with constant accretion times and larger infall rates for larger final masses produce a broad protostellar luminosity function. Continued analysis of the occurrence rate, magnitudes, and decay times of protostellar luminosity outbursts as a function of both cloud mass and environment will be crucial for understanding the importance of episodic accretion in explaining protostellar evolutionary diagrams.
Variation as a Function of Environment:\
Changing Star-Formation Rates?
----------------------------------------
The histogram of $T_{\rm bol}$ for the HOPS sample is remarkably flat. The model-derived histogram of envelope mass is also flat for masses below 0.3 $M_\sun$. Assuming a constant star-formation rate over the lifetime of protostars, we used this to argue for exponentially decreasing envelope densities. The HOPS program deliberately selected out YSOs thought to be of Class II, and the number of sources in our sample with $T_{\rm bol}$ consistent with Class II is highly incomplete. Typically in Orion, Class II sources outnumber protostars by a factor of three [@meg16].
Although the assumption of a constant star-formation rate may be valid for the full sample, the $T_{\rm bol}$ and $M_{\rm env}$ histograms of the different regions within the Orion complex suggest that the star-formation rate may not be constant within each region. The larger fraction of Class 0 protostars in Orion B, 41% as opposed to 35% in the ONC and 20% in L 1641, is consistent with the finding of @stu13 that the fraction of deeply embedded PBRS is largest in Orion B. The larger fraction of Class 0 protostars in the ONC than in L 1641 is consistent with the finding of @stu15 that the fraction of Class 0 protostars is larger in the north of Orion A than in the south; these authors found that this fraction is correlated with the column density distribution shape (N-PDF) variation across Orion A. Combined with the statistically larger luminosities for Class 0 protostars, this indicates that the protostellar luminosity may be rooted in the molecular cloud N-PDF structure, which in turn is rooted in the density structure of the star-forming material (see, e.g., @stu16). That is, regions with more local (or centrally concentrated and potentially filamentary) mass may produce higher-luminosity protostars.
These variations are evident in the $T_{\rm bol}$ and $M_{\rm env}$ histograms. The Orion B cloud shows a distinct decrease in the number of protostars with increasing evolutionary state, progressing from the young Class 0 or Stage 0 protostars, with low $T_{\rm bol}$ and high $M_{\rm env}$, to more evolved Class I or Stage I protostars. This decline may be the result of an increasing star-formation rate, with the rate increasing by roughly a factor of three over the protostellar lifetime. Although the low star-formation efficiency of Orion B [@meg16] may suggest that it is younger than Orion A, observations of pre-main-sequence stars show that star formation has occurred over 2 Myr in this cloud [@fla08]. Thus, the increase in the star-formation rate may be recent. The higher fraction of Class 0 protostars in the ONC may also be the result of a recent increase in the star-formation rate. In contrast, L 1641 shows a hint of a decrease in the star-formation rate. In general, these diagrams suggest that although the star-formation rate over the last 0.5 Myr is relatively stable when considered over the entire cloud, it may not be as constant when considering smaller regions within it.
CONCLUSIONS
===========
We have determined the bolometric luminosities and temperatures for 330 YSOs, 315 of which have $T_{\rm bol}$ consistent with protostars, in the Orion molecular clouds that were targets of HOPS, the [*Herschel*]{} Orion Protostar Survey. The $L_{\rm bol}$ histogram is broad, ranging over nearly five orders of magnitude, with a peak near 1 $L_\sun$, a width at half-maximum of two orders of magnitude, and a tail extending beyond 100 $L_\sun$. The $T_{\rm bol}$ histogram is flat, with logarithmic bins between 30 and 600 K each containing approximately equal numbers of protostars. The BLT diagram features a broad spread in luminosities at each bolometric temperature, with 29% of the sources having $T_{\rm bol}<70$ K, the dividing line for Class 0 and Class I protostars.
The BLT diagram shows a systematic decline in the median luminosity with increasing bolometric temperature. This decline is reflected in different luminosities for the Class 0 and Class I protostars. The median Class 0 luminosity is 2.3 $L_\sun$ compared to 0.87 $L_\sun$ for Class I, indicating that more deeply embedded protostars are more luminous. The Class 0 luminosity histogram has less than a 0.1% probability of being drawn from the same underlying distribution as the Class I luminosity histogram.
We divided the sample into regions; from south to north, these are L 1641, the ONC, and Orion B. The fraction of protostars in Class 0 increases from south to north, from 20% in L 1641 to 41% in Orion B. Within each region, trends seen in the entire sample persist. The Class 0 protostars are statistically brighter than the Class I protostars, and $L_{\rm bol}$ declines with $T_{\rm bol}$ while having a large dispersion. We argued that these trends are unlikely to be driven by incompleteness or an inaccurate accounting for foreground extinction. In the ONC and Orion B, there is a decrease in the number of protostars at progressively larger $T_{\rm bol}$, suggesting a relatively recent increase in the star-formation rate.
These findings are further confirmed via our SED-fitting analysis. When we fit SED models to the protostars and Class II objects to estimate their total luminosities and envelope masses inside 2500 AU, the median total luminosity increases to 3.0 $L_\sun$. The mass inside 2500 AU is not the entire mass of the envelope; it is only the portion that is warm enough to be traced by our far-IR measurements. In all but the earliest stages, this gas is falling toward the central protostar and is continually replenished by gas from outside 2500 AU.
Trends in total luminosity and envelope mass are similar to those in $L_{\rm bol}$ and $T_{\rm bol}$. The histogram of envelope masses is quite flat, luminosities are largest for envelope masses between 0.1 and 1 $M_\sun$ and fall as the envelopes become less massive, and there is a spread in luminosity of about three orders of magnitude in each mass bin. The flat histogram of envelope mass and the decrease in luminosity can largely be explained by an overall exponential decrease in the envelope infall rate with time as postulated by @bon96, @mye98, @sch04, and @vor10. We show that simple models invoking an exponentially decreasing envelope mass can approximately reproduce most aspects of the observed distribution of sources in an $L_{\rm tot}$ versus $M_{\rm env}$ diagram. In these models, we find that the time to assemble half the star, which corresponds to the time of the physical transition from Stage 0 to Stage I, is about half the observationally derived Class 0 lifetime.
The initial mass of the envelope and the half-life of the envelope mass set the final mass of the star. When the exponentially declining models are applied to an ensemble of cases that yield a typical initial mass function of main-sequence stars, the luminosities in each mass bin have a similar spread to those derived from the data but are systematically lower. In this model, the distribution of luminosities at each envelope mass is largely due to the expected distribution in the final masses of the forming stars.
Support for this work was provided by the National Aeronautics and Space Administration (NASA) through awards issued by the Jet Propulsion Laboratory/California Institute of Technology (JPL/Caltech). This work is based on observations made with the [*Spitzer*]{} Space Telescope, which is operated by JPL/Caltech under a contract with NASA; it is also based on observations made with the [*Herschel*]{} Space Observatory, a European Space Agency Cornerstone Mission with significant participation by NASA. We include data from the Atacama Pathfinder Experiment, a collaboration between the Max-Planck Institut für Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory. Finally, this publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/Caltech, funded by NASA and the National Science Foundation.
This work was supported by NASA Origins of Solar Systems grant 13-OSS13-0094. The work of WJF was supported in part by an appointment to the NASA Postdoctoral Program at Goddard Space Flight Center, administered by the Universities Space Research Association through a contract with NASA. JJT acknowledges past support from grant 639.041.439 from the Netherlands Organisation for Scientific Research (NWO) and from NASA through Hubble Fellowship grant \#HST-HF-51300.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. AS is thankful for funding from the “Concurso Proyectos Internacionales de Investigación, Convocatoria 2015” (project code PII20150171) and the BASAL Centro de Astrofísica y Tecnologías Afines (CATA) PFB-06/2007. MO acknowledges support from MINECO (Spain) grant AYA2014-57369-C3 (co-funded wth FEDER funds).
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[^1]: See <http://idlastro.gsfc.nasa.gov/>.
[^2]: This mass function is implemented in an IDL routine `cnb_imf.pro` by C. Beaumont, available at <http://www.ifa.hawaii.edu/users/beaumont/code/cnb_imf-code.html>.
|
---
abstract: 'Two measures are defined to evaluate the coupling strength of smeared interpolating operators to hadronic states at a variety of momenta. Of particular interest is the extent to which strong overlap can be obtained with individual high-momentum states. This is vital to exploring hadronic structure at high momentum transfers on the lattice and addressing interesting phenomena observed experimentally. We consider a novel idea of altering the shape of the smeared operator to match the Lorentz contraction of the probability distribution of the high-momentum state, and show a reduction in the relative error of the two-point function by employing this technique. Our most important finding is that the overlap of the states becomes very sharp in the smearing parameters at high momenta and fine tuning is required to ensure strong overlap with these states.'
author:
- 'Dale S. Roberts'
- Waseem Kamleh
- 'Derek B. Leinweber'
- 'M. S. Mahbub'
- 'Benjamin J. Menadue'
title: Accessing High Momentum States In Lattice QCD
---
Introduction
============
Lattice QCD has enjoyed great success as a tool for first-principles hadron-structure calculations. Early pion electromagnetic form factor calculations [@Martinelli:1987bh; @Draper:1988bp] and nucleon form factor calculations [@Martinelli:1988rr; @Draper:1989pi; @Leinweber:1990dv] established the formalism and presented first results establishing the challenges ahead for obtaining precision form factors to confront experimental data. Nucleon form factors continue to be an active area of research [@Leinweber:2004tc; @Boinepalli:2006xd; @Alexandrou:2010uk; @Alexandrou:2010hf; @Alexandrou:2010cm; @Aoki:2010xg; @Yamazaki:2009zq; @Syritsyn:2009mx] and a comprehensive review of recent form factor calculations can be found in [@Zanotti:2008zm] and references therein.
In practice, current lattice calculations were limited to a momentum transfer of approximately $Q^2=3\,\mathrm{GeV}^2$ due to a challenge of increasing statistical errors. Recently, calculations of the nucleon and pion form factors at $Q^2=6\,\mathrm{GeV}^2$ have been performed using variational techniques [@Lin:2010fv]. In this paper we explore very high momentum states and propose that, with sufficient optimisation of the smearing parameters alone, momentum transfers of the order $Q^2=10\,\mathrm{GeV}^2$ can be accomplished in lattice hadron structure calculations.
Smearing techniques have seen wide spread use in many applications in lattice QCD since first being applied to fermion operators [@Gusken:1989qx]. The most notable impacts can be found in spectroscopy calculations using variational methods [@Michael:1985ne; @Luscher:1990ck; @McNeile:2000xx; @Leinweber:2004it; @Mahbub:2009aa; @Mahbub:2010rm]. In spite of these successes, there has been little in the way of the optimisation of smearing parameters for high-momenta states. For low-momenta states there is no real need for optimization as the overlap of states is typically slowly varying with the smearing parameters. In the following we reveal that this is not the case for high-momenta states and finely tuned optimization is very beneficial in accessing these states on the lattice.
Isolation of the ground state at high-momentum is essential to removing otherwise large and problematic excited state contaminations. However, suppression of excited states through Euclidean evolution alone encounters a rapid onset of statistical noise. We introduce two different measures to quantify the coupling of a smeared operator to the ground state of a proton relative to the near-by excited states, and show how these measures determine the optimal smeared operator for ground state isolation early in Euclidean time.
We also introduce anisotropy into the smeared operators in the direction of momentum in an effort to improve the coupling to these Lorentz-contracted high-momentum states. Our results are complementary to the variational techniques of Ref. [@Lin:2010fv] in that the optimal set of smearings for accessing a variety of momenta can be combined to create a correlation matrix providing an effective basis for eigenstate isolation.
Two-Point Functions
===================
The two-point function of a baryon on the lattice in momentum space is given by where $\chi_i$ and $\bar\chi_i$ annihilate and create the baryon respectively at the sink point $x$ and source point $0$ and the index $i$ admits various spin-flavor structures for the interpolators. In the case of the proton, the annihilation operator is $$\chi_1 = \epsilon^{abc}(u_a^TC\gamma_5d_b)u_c,$$ where $u$ and $d$ represent the spinors for the up and down quarks respectively and $C$ is the charge conjugation matrix. It can be shown that, for positive parity states, where the sum over $B$ represents the ground and excited states of the baryon. It is common to average the $(1,1)$ and $(2,2)$ elements of the Dirac matrix where the signal for positive parity states is large. At zero momentum, the Dirac matrix contribution is then 1. The coefficient $\lambda_B$ provides a measure of the total overlap of $\bar\chi_i$ at the source and $\chi_i$ at the sink with the state $B$. It is the product of the source and sink overlaps which may be different if different smearings are used at the source and the sink. In this investigation the source will be fixed to a point source such that variation in $\lambda_B$ is proportional to the variation in the overlap of $\chi_i$ which will encounter a wide range of different sink smearings.
Each state decays at a rate proportional to the exponential of its energy. By evolving forward in Euclidean time, excited state contributions die away allowing the ground state to be isolated. This is less than ideal for the calculation of three-point functions that require effective ground state isolation close to the source to avoid large Euclidean time evolution and an associated loss of signal. It is for this reason that various techniques have been implemented for earlier Euclidean-time isolation of the ground state.
When calculating the two-point function, it is possible to choose the momentum of the baryon. On the finite lattice, momentum is quantised where $N_L$ is the spatial extent of the lattice, $a$ is the lattice spacing and $p_x$, $p_y$, $p_z$ are integers restricted to the range Due to the construction of the discrete fermion propagator, momentum input into the two-point function becomes proportional to $\mathrm{sin}(\vec{p})$, therefore, it is only reasonable to consider momentum states where such that the dispersion relation is approximately satisfied.
Gaussian Smearing
=================
Gaussian smearing is an iterative procedure applied to the source or sink of the two-point function in order to improve the relative coupling to the ground state of the particle. Consider with [@Gusken:1989qx] $$\begin{aligned}
F(x,y) &=& (1-\alpha)\, \delta_{xy} \\
&&+ \frac{\alpha}{6} \sum_{\mu=1}^3 \left (
U^\dagger_\mu(x-a\hat\mu)\, \delta_{x-\hat\mu,y} +
U_\mu(x)\, \delta_{x+\hat\mu,y} \right ) \, , \nonumber \end{aligned}$$ where $\alpha$ is a constant, which we set to $0.7$. We can introduce anisotropy to the smearing by introducing a new constant $\alpha_x$, which will act only in the $x$ direction, the expression for the smearing then becomes,
where $\alpha_o=0.7$ and $\alpha$ and $\alpha_x$ are normalised such that
Measures
========
Gusken [@Gusken:1989qx] introduced the measure for quantifying the ground state isolation of a hadron. By taking a point, $t'$, sufficiently late in time such that the excited state contributions become negligible, the ground state can be evolved back to the source via $e^{+m_0\, t'}$ to evaluate the fraction of $G_2(0)$ it holds. However, with sufficient smearing, states can contribute negatively to the two-point function, allowing this ratio to exceed $1$ and making it difficult to interpret the results.
The first measure we introduce follows from this idea by determining the deviation of $G_2(t)$ from the ideal two-point function of a single ground state. It is similar in principle to Gusken’s measure, however, it is capable of taking into account the presence of states with negative coupling to the operator. The measure, $M_1$ is defined as, where $\tilde G(t) = G(t)/G(t_0)$. The factor $-1$ makes this measure maximal when $G(t)$ is a pure exponential of the ground state. The energy $E_0$ is determined from a $4\times 4$ source-sink-smeared variational analysis [@Mahbub:2009nr] of the zero momentum state with the correct dispersion relation applied for finite-momentum states.
Another common method of extracting coupling effectiveness is to perform a four parameter, two exponential fit on a region close to the source of the two-point function, [*i.e.*]{} However, this method tends to prove unreliable with the parameters varying with the fit window. The method is limited by the fact that it can not take into account any states with higher energy than the two considered.
The second measure we introduce works similar to this. However, the parameters of the exponentials are predetermined by a variational analysis [@Mahbub:2009nr]. This leads to a simple linear fit of known exponentials, [*i.e.*]{} We can then find the proportion of the $i$-th state in the two-point function with the measure
Lattice Details
===============
Our calculations are performed on configurations of size $32^3\times
64$ with a lattice spacing of $0.0907\,\mathrm{fm}$ provided by the PACS-CS collaboration [@Aoki:2008sm]. These lattices have $2+1$ sea quark flavours generated with the Iwasaki gauge action [@Iwasaki:1983ck] and the non-perturbatively improved Clover fermion action [@Sheikholeslami:1985ij] with the $\kappa$ values for the light quarks and the strange quark given by $0.13754$ and $0.13640$ respectively, and $C_{SW}=1.715$. This gives a pion mass of $m_\pi=389\,\mathrm{MeV}$.
In order to eliminate any bias caused by smearing in the source, we use a single set of propagators generated with a point source. All of the smearing is then applied to the sink, making the two-point functions smearing dependent. All momentum will be in the $x$ direction, [*i.e.*]{} $p_y=0$ and $p_z=0$ in Eq. (\[quantP\]).
We use a $4\times 4$ correlation matrix to extract our excited state masses, constructed from the $\chi_1$ operator with 16, 35, 100 and 200 sweeps of smearing. We choose to use the larger basis in order to ensure that the first three eigenstate energies are accurately determined.
We have verified that no multi-particle states are present in the variational analysis by applying the single-particle dispersion relation to the zero momentum effective state masses to successfully predict the effective masses of the same states with non-zero momentum.
Our error analysis is performed with the second-order single-elimination jackknife method. Linear fits are performed using the normal equations with exact matrix inversion where possible and singular value decomposition otherwise.
Results
=======
Isotropic Smearing
------------------
![The measure from Eq. (\[M1\]) at $p_x=0$ in Eq. (\[quantP\]). Deviation from the ideal two-point function increases by a factor of 10 less than $30$ sweeps from the ideal smearing level, as shown in the inset graph []{data-label="diffMeasip00"}](diffMeasip00inset.ps){width="0.7\linewidth"}
We first calculate the measure from Eq. (\[M1\]) where the two-point functions have been normalised $1$ time slice after the source, with $t_i=1$ and $t_f=6$. The two-point function is calculated at every sweep of sink smearing between $1$ and $480$, up to an rms radius of $13.68$ in lattice units. For this particular ensemble, the two-point function that shows the highest proportion of ground state has $136$ sweeps of smearing at the sink, or an rms radius of $6.92$ lattice units as seen in Fig. \[diffMeasip00\]. Also apparent is that the effectiveness of the smearing at isolating the ground state is significantly reduced fairly close to the optimal amount of smearing. At only $30$ sweeps away from the ideal number of sweeps, the deviation from the ideal two-point function has increased by a factor of $10$.
{width="0.32\linewidth"} {width="0.32\linewidth"}
. \[diffMeasip01\]
When we move to $p_x=1$ in Eq. (\[quantP\]), which gives momentum in the $x$ direction of $427\,\mathrm{MeV}$, the ideal number of smearing sweeps reduces by just one sweep to $135$ (rms radius $6.90$ lattice units), as shown in Fig. \[diffMeasip01\]. This can be explained by considering the relativistic $\gamma$ factor, which is given by the ratio of the relativistic energy momentum relation and the ground state mass. The fitted ground state mass for the proton is $M_P=1.273(21)\,\mathrm{GeV}$, giving a relativistic energy of $E_P\vert_{p=1}=1.343(23)\,\mathrm{GeV}$ and $\gamma=1.05$. Given that all of the excited states are more massive, and therefore exhibit less Lorentz contraction than the ground state, it is feasible that there is very little difference in the probability distribution between this state and the zero momentum state, thus the ideal amount of smearing should be very similar to the zero momentum state.
{width="0.32\linewidth"} {width="0.32\linewidth"}
At $p_x=3$ in Fig. \[diffMeasip01\], the optimal number of smearing sweeps has decreased to $98$. The maximum value of the measure has also decreased relative to the lower momentum states, indicating relatively more excited state contamination, though still achieving good isolation. The ratio of the rms radius of the optimal smearing for this state to the optimal smearing for the ground state is $0.85$, compared to the relativistic $\gamma^{-1}$ factor of $0.72$. At $p_x=5$, corresponding to a momentum transfer of approximately $4.55\,\mathrm{GeV}^2$, shown in Fig. \[diffMeasip05\], the optimal number of sweeps is $52$ (rms radius $4.27$ lattice units). However, the maximum value of the measure is close to the maximum value for the $p_x=3$ case, indicating that very efficient isolation is possible, even at larger momentum transfers.
Moving to $p_x=7$, equivalent to a momentum transfer of $8.93\,\mathrm{GeV}^2$, there is significant noise far from the source in the two-point function, even for highly optimised smearing values. Hence we consider $t_f=5$ in the measure from Eq. (\[M1\]) at this value of momentum. The ideal number of sweeps decreases to $27$ sweeps, or $3.08$ lattice units rms radius, seen in Fig. \[diffMeasip05\]. Notably, the deviation from the ideal two-point function increases by a factor of $10$ only $5$ sweeps from this optimal value, corresponding to a change in rms radius of less than $0.3$ lattice units.
![Ground state proportion from the three exponential fit at $p_x=0$ in Eq. (\[quantP\]). There is insufficient information on the second excited state close to the optimal amount of smearing, thus requiring use of the two exponential fit to determine the optimal amount of smearing with this measure.[]{data-label="ThreeExpGSPip00"}](ThreeExpGSPip00.ps){width="0.7\linewidth"}
Using the measure described in Eq. (\[M2\]), we first consider the three exponential fit between time slices $1$ and $6$ after the source with masses $1.273(21)\,\mathrm{GeV}$, $2.301(28)\,\mathrm{GeV}$ and $2.786(95)\,\mathrm{GeV}$ as determined in our correlation matrix analysis. From the results in Fig. \[ThreeExpGSPip00\], we can see that, in the region where the first measure predicts ideal smearing levels, there is a sharp change in the structure of the graph. In order to determine the cause of this, we compare with the fits containing only the ground and first excited states. Fig. \[GSPip00\] shows that the optimal number of smearing sweeps lies close to the value predicted by the first measure. The overlap at the optimal number of sweeps, $138$ in this case, is $99.31(8)\%$, indicating that, in the three exponential fit, we are attempting to fit two quickly decaying exponentials using only $0.69\%$ of the signal available. This leads us to believe that, in the regions of ground state dominance where we are most interested, the coefficient from the quickly decaying third state cannot be determined accurately, therefore dominates well beyond where it should be allowed to contribute at all. For this reason, we will only consider fits using the ground and first excited states.
![Ground state proportion at $p_x=0$ in Eq. (\[quantP\]). Contamination due to excited states increases rapidly away from the optimal smearing level. There is good agreement between the two exponential fit here and the three exponential fit in Fig. \[ThreeExpGSPip00\] away from the optimum smearing levels.[]{data-label="GSPip00"}](GSPip00.ps){width="0.7\linewidth"}
The contamination due to excited states in the two exponential fit at zero momentum increases rapidly away from the optimum smearing level. Of the smearing sweeps used to extract the masses from the variational analysis, the one that shows the most overlap with the ground state is $200$ sweeps, or an rms radius of $8.55$ lattice units, with $77.69(7)\%$, or $32$ times more excited state contamination than the optimal smearing level.
![Ground state proportion at $p_x=3$ in Eq. (\[quantP\]). As momentum increases, the contamination due to excited states increases more rapidly away from the ideal smearing level.[]{data-label="GSPip03"}](GSPip03.ps){width="0.7\linewidth"}
At the first non-zero momentum state, the results present similarly to the first measure, the optimal amount of smearing is $1$ sweep less than that of the non-zero momentum ground state, and $2$ sweeps more than the optimal amount determined by the first measure. At $p_x=3$ in Eq. (\[quantP\]) shown in Fig. \[GSPip03\], the overlap is maximised at $101$ sweeps of smearing, or an rms radius of $5.95$ lattice units, once again agreeing within only a few sweeps of the optimum level suggested by the first measure. Remarkably, considering the use of a point source, the proportion of ground state present at this optimal amount of smearing is $98.87(12)\%$.
{width="0.32\linewidth"} {width="0.32\linewidth"}
At $p_x=5$ and $p_x=7$ in Fig. \[GSPip05\] there is again good agreement between the two measures, with the optimal smearing level being $53$ and $26$ sweeps respectively. Even at a momentum transfer of $8.93\,\mathrm{GeV}^2$, $97.20(20)\%$ overlap is achieved with the ground state, and once again, very few sweeps from the optimum level, the overlap drops dramatically. At $p_x=7$, far from the optimal number of smearing sweeps, it is unlikely that any highly Lorentz contracted state would couple to such a large sink. The second peak in Fig. \[GSPip05\] can therefore be considered to signify a limit to the domain of validity of the measure.
Anisotropic Smearing
--------------------
As anisotropy is introduced to the smearing as described in Eq. (\[anissmear\]), we consider the first measure from Eq. (\[M1\]) at the first non-zero momentum state and find that there is no improvement to the ground state isolation, as shown in Fig. \[M1Surfip01\]. There is, however, an ideal number of sweeps that increases for decreasing $\alpha_x$ that shows approximately equal ground-state proportion relative to the isotropic smearing case.
At $p_x=3$ in Eq. (\[quantP\]), in spite of the clear difference in the smearing sweeps required to maximise overlap with the source, Fig. \[M1Surfip03\] shows that introducing anisotropy to the smearing does not result in improved isolation of the ground state. The structure of the curve is similar to that of the $p_x=1$ state, where there is an optimal number of sweeps for every value of $\alpha_x$ which increases with decreasing $\alpha_x$.
Once again, there is no improvement in the ability of anisotropic smearing to isolate the ground state at the momentum of $p_x=5$, as shown in Fig. \[M1Surfip05\]. The structure revealed in the lower momentum states persists for this state and for the $p_x=7$ state in Fig. \[M1Surfip07\]. From these results, optimisation of the number of smearing sweeps alone is sufficient to achieve good isolation of the ground state of the two-point function at a range of momenta.
We now investigate how anisotropic smearing affects the signal-to-noise ratio or quality of the two-point function at high momenta. Since we have ensured that the ground state is isolated as close to the source as possible, we now determine the quality of the signal a few time slices away from the source. We consider the relative error of the two-point function four times slices after the source at the optimal number of smearing sweeps for each value of our anisotropy parameter, $\alpha_x$.
For $p_x=3$, Fig. \[relerrip03\] shows the two-point function at $t=4$. The smallest relative error occurs when the smearing is isotropic. Increasing the momentum to $p_x = 5$ lattice units shows that there is only a small improvement to the relative error for values of $\alpha_x \sim 0.48$. It is worth noting that the first of the minima visible in Fig. \[relerrip05\] at $\alpha_x = 0.36$ corresponds to the anisotropy expected due to Lorentz contraction as $\alpha_x/\alpha = 0.51$ equals $\gamma^{-1} = 0.51$.
The banding structure visible in Fig. \[relerrip05\] is a result of the optimal number of smearing sweeps increasing for decreasing values of $\alpha_x$. Each discontinuity in the graph for $\alpha_x > 0.36$ is the result of the optimal number of smearing sweeps decreasing by $1$. It is an artifact resulting from the density of the points in $\alpha_x$ being much finer than the density of the points in the number of smearing sweeps.
Moving to $p_x=7$ in Fig. \[relerrip05\] we see a distinct improvement in the correlation-function relative error when anisotropy is introduced. Both $\alpha_x=0.26$ and $0.32$ provide a $10\%$ reduction in the error relative to that observed at the isotropic value of 0.7. The values of $\alpha_x \simeq 0.26$ to 0.32 provide $\alpha_x/\alpha = 0. 37$ to 0.46, in accord with the value of $\gamma^{-1} = 0.39$ predicted by Lorentz contraction.
Conclusion
==========
We have presented two new measures of the effectiveness of smeared operators in isolating the ground state of a hadron in the two-point function. Both measures show good agreement with each other. We have performed a detailed analysis of ground state isolation with each measure and have shown that optimisation of the smearing can lead to remarkable improvement to the ground state isolation. Furthermore, the ability to isolate the ground state decreases dramatically a few sweeps from the optimal number of smearing sweeps for the higher momentum states. In selecting a basis for a correlation matrix analysis, these optimal smearing parameters are preferred.
On the introduction of anisotropy to the smearing, we found that there was no appreciable improvement to the overlap with the ground state. The relative proportion of the ground state for an isotropic source is already high. Optimising the number of sweeps of isotropic smearing alone is sufficient to ensure maximal isolation of high-momentum ground states. The introduction of anisotropy does provide a small improvement to the correlation function of high-momentum states a few Euclidean time slices after the source.
Our results indicate that future studies of high-momentum states should adopt this relatively cheap program of tuning the smearing parameters to optimize isolation and overlap with the states of interest. We anticipate this approach will be of significant benefit in future form factor studies.
Acknowledgments
===============
This research was undertaken on the NCI National Facility in Canberra, Australia, which is supported by the Australian Commonwealth Government. This research is supported by the Australian Research Council.
{width="0.49\linewidth"} {width="0.49\linewidth"}
{width="0.49\linewidth"} {width="0.49\linewidth"}
{width="0.49\linewidth"} {width="0.49\linewidth"}
{width="0.49\linewidth"} {width="0.49\linewidth"}
![Relative error in the two-point function measured four time slices after the source for $p_x=3$ as in Eq. (\[quantP\]). At this momentum, isotropic smearing provides the best relative error. []{data-label="relerrip03"}](relErrip03.ps){width="0.7\linewidth"}
{width="0.32\linewidth"} {width="0.32\linewidth"}
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|
---
abstract: 'A unification of left-right ${\mathrm{SU}(3)_{\mathrm{L}}}\times {\mathrm{SU}(3)_{\mathrm{R}}}$, colour ${\mathrm{SU}(3)_{\mathrm{C}}}$ and family ${\mathrm{SU}(3)_{\mathrm{F}}}$ symmetries in a maximal rank-8 subgroup of ${\rm{E}}_8$ is proposed as a landmark for future explorations beyond the Standard Model (SM). We discuss the implications of this scheme in a supersymmetric (SUSY) model based on the trinification gauge $\left[{\mathrm{SU}(3)_{\mathrm{}}}\right]^3$ and global ${\mathrm{SU}(3)_{\mathrm{F}}}$ family symmetries. Among the key properties of this model are the unification of SM Higgs and lepton sectors, a common Yukawa coupling for chiral fermions, the absence of the $\mu$-problem, gauge couplings unification and proton stability to all orders in perturbation theory. The minimal field content consistent with a SM-like effective theory at low energies is composed of one $\mathrm{E}_6$ $\bm{27}$-plet per generation as well as three gauge and one family ${\mathrm{SU}(3)_{\mathrm{}}}$ octets inspired by the fundamental sector of ${\rm{E}}_8$. The details of the corresponding (SUSY and gauge) symmetry breaking scheme, multi-scale gauge couplings’ evolution, and resulting effective low-energy scenarios are discussed.'
author:
- 'José E. Camargo-Molina'
- 'António P. Morais'
- Astrid Ordell
- Roman Pasechnik
- Jonas Wessén
bibliography:
- 'bib.bib'
title: |
Scale hierarchies, symmetry breaking and particle spectra\
in SU(3)-family extended SUSY trinification
---
LU TP 17-37
Introduction
============
Finding successful candidate theories unifying the strong and electroweak interactions, leading to a detailed understanding of the SM origin, with all its parameters, hierarchies, symmetries and particle content remain a big challenge for the theoretical physics community. Some of the most popular SM extensions are based on supersymmetric (SUSY) GUTs where the SM gauge interactions are unified under symmetry groups such as ${\mathrm{SU}(5)_{\mathrm{}}}$ and $\mbox{SO}(10)$ [@Georgi:1974sy; @Fritzsch:1974nn; @Chanowitz:1977ye; @Georgi:1978fu; @Georgi:1979dq; @Georgi:1979ga; @Georgi:1982jb] as well as $\mathrm{E}_6$[^1] and $\mathrm{E}_7$ [@Gursey:1976dn]. A particularly appealing scenario proposed by Glashow in 1984 [@original] is based upon the rank-6 trinification symmetry $[{\mathrm{SU}(3)_{\mathrm{}}}]^3 \equiv {\mathrm{SU}(3)_{\mathrm{L}}} \times
{\mathrm{SU}(3)_{\mathrm{R}}} \times {\mathrm{SU}(3)_{\mathrm{C}}} \rtimes \mathbb{Z}_3 \subset \mathrm{E}_6$ (T-GUT, in what follows) where all matter fields are embedded in bi-triplet representations and due to the cyclic permutation symmetry $\mathbb{Z}_3$, the corresponding gauge couplings unify at the T-GUT Spontaneous Symmetry Breaking (SSB) scale, or GUT scale in what follows.
There have been many phenomenological and theoretical studies of T-GUTs, in both SUSY and non-SUSY formulations, motivated by their unique features (see e.g. Refs. [@Dias:2010vt; @Reig:2016tuk; @Babu:1985gi; @He:1986cs; @Greene:1986jb; @Wang:1992hu; @Lazarides:1993uw; @Dvali:1994wj; @Maekawa:2002qv; @Kim:2003cha; @Kim:2004pe; @Carone:2004rp; @Carone:2005ha; @Sayre:2006ma; @DiNapoli:2006kq; @Cauet:2010ng; @Stech:2012zr; @Stech:2014tla; @Hetzel:2015bla; @Hetzel:2015cca; @Pelaggi:2015kna; @Pelaggi:2015knk; @Rodriguez:2016cgr; @Camargo-Molina:2016bwm]). For example, due to the fact that quarks and leptons belong to different gauge representations in T-GUT scenarios, the baryon number is naturally conserved by the gauge sector [@Babu:1985gi], only allowing for proton decay via Yukawa and scalar interactions, if at all present. As was shown for a particular T-GUT realisation in Ref. [@Sayre:2006ma], the proton decay rates were consistent with experimental limits in the case of low-scale SUSY, or completely unobservable in the case of split SUSY. Many T-GUTs can also accommodate any quark and lepton masses and mixing angles [@Babu:1985gi; @Stech:2014tla] whereas neutrino masses are generated by a see-saw mechanism [@Kim:2004pe] of radiative [@Sayre:2006ma] or inverse [@Cauet:2010ng] type.
Despite a notable progress in exploring gauge coupling unification, neutrino masses, Dark Matter candidates, TeV-scale Higgs partners, collider and other phenomenological implications of GUTs, there are several yet unresolved problems. One of problems emerging in the case of SUSY T-GUT model building is the longstanding issue of avoiding GUT scale masses for the would-be SM leptons. To circumvent this, the usual solution is to add several $\bm{27}$-plets of ${\rm E_6}$ with scalar components responsible for SSB of gauge trinification [@Babu:1985gi; @Wang:1992hu; @Dvali:1994wj; @Maekawa:2002qv; @Willenbrock:2003ca; @Carone:2005ha; @Sayre:2006ma; @Cauet:2010ng; @Stech:2012zr; @Stech:2014tla; @Hetzel:2015bla; @Pelaggi:2015kna], or to simply add higher dimensional operators [@Nath:1988xn; @Dvali:1994wj; @Maekawa:2002qv; @Carone:2005ha; @Cauet:2010ng]. These approaches typically require a significant fine-tuning in high-scale parameter space (especially, in the Yukawa sector) [@Sayre:2006ma]. Otherwise, they exhibit phenomenological issues with proton stability [@Babu:1985gi; @Maekawa:2002qv; @Sayre:2006ma] and with a large amount of unobserved light states [@original; @Nath:1988xn; @Dvali:1994wj; @Stech:2014tla; @Hetzel:2015bla; @Pelaggi:2015knk]. Despite continuous progress, the SM-like EFTs originating from T-GUTs still remain underdeveloped in comparison to other GUT models such as ${\mathrm{SU}(5)_{\mathrm{}}}$, ${\mathrm{SO}(10)_{\mathrm{}}}$ or even ${\rm E_6}$ (see e.g. Ref. [@Hetzel:2015cca] and references therein).
In this paper, we explore in detail the SUSY T-GUT model proposed in [@Camargo-Molina:2016yqm] with a global ${\mathrm{SU}(3)_{\mathrm{F}}}$ family symmetry inspired by the embedding of ${\rm E_6} \times {\mathrm{SU}(3)_{\mathrm{}}}$ into ${\rm E_8}$. We will refer to this model as the SUSY Higgs-Unified Trinification (SHUT) model (for alternative ways of extending the SM by means of an ${\mathrm{SU}(3)_{\mathrm{F}}}$ symmetry see e.g. Refs. [@Berezhiani:1990wn; @Berezhiani:1989fp; @Berezhiani:1990jj; @Sakharov:1994pr]). As we will see, the SHUT model offers solutions to some of the problems faced by previous T-GUTs. As the light Higgs and lepton sectors are unified, the model can be embedded into a single ${\rm E_8}$ representation. Furthermore, the embedding suggests the introduction of adjoint scalars and a family ${\mathrm{SU}(3)_{\mathrm{F}}}$, where the former protects a sufficient amount of fermionic states from acquiring masses before EWSB to be in agreement with the SM. The interplay of the family ${\mathrm{SU}(3)_{\mathrm{F}}}$ also provides a unification of the high-scale Yukawa sector into a single coupling. This is in contrast to well-known $\mathrm{SO}(10)$ and Pati-Salam models where the Yukawa unification is constrained to the third family only (see e.g. Refs. [@Blazek:2001sb; @Baer:2001yy; @Anandakrishnan:2013cwa; @Blazek:2002ta; @Tobe:2003bc; @Baer:2009ie; @Badziak:2011wm; @Anandakrishnan:2012tj; @Joshipura:2012sr; @Anandakrishnan:2013nca; @Anandakrishnan:2014nea; @Badziak:2013eda; @Ajaib:2013zha]).
The Yukawa and gauge couplings unification in the SHUT model largely reduces its parameter space, making a complete analysis of its low-energy EFT scenarios technically feasible. The model also has a particular feature in that no further spontaneous breaking of the symmetry towards the SM gauge group is provided by the SUSY conserving part of the model, and that the energy scales at which the symmetry is further broken are instead associated with the soft SUSY-breaking operators. As such, both the electro-weak scale and the scales of intermediate symmetry breaking are naturally suppressed relative to the GUT scale.
In Sect. \[Sec:LRCF\] we briefly discuss the key features of the SHUT model and its SSB scheme, and in Sect. \[sec:SUSYHS\] the high-scale SHUT model is introduced in its minimal setup in detail. In particular, we discuss its features and the details on how it solves the longstanding problems of previous T-GUT realizations and how the GUT scale SSB in this model leads to a Left-Right (LR) symmetric SUSY theory. In Sect. \[sec:BreakingSUSY\] we discuss the inclusion of soft SUSY-breaking interactions and how they lead to a breaking of the remaining gauge symmetries down to the SM gauge group, and in Sect \[sec:masses2\] we present a short overview of the low-energy limits of the SHUT model. Finally, Sect. \[sec:EFTs\] contains an analysis of RG evolution of gauge couplings at one loop and extraction of characteristic values of the GUT and soft scales, before concluding in Sect. \[sec:conclusions\].
A short note on notation {#a-short-note-on-notation .unnumbered}
------------------------
In this article we adopt the following notations:
- Supermultiplets are always written in bold (e.g. ${\bf \Delta}$). As usual, the scalar components of chiral supermultiplets and fermionic components of vector supermultiplets carry a tilde (e.g. $\widetilde{\Delta}$), except for the Higgs-Higgsino sector where the tilde serves to identify the fermion ${\mathrm{SU}(2)_{\mathrm{L}}} \times {\mathrm{SU}(2)_{\mathrm{R}}}$ bi-doublets (e.g. $\widetilde{H}$).
- Fundamental representations carry superscript indices while anti-fundamental representations carry subscript indices.
- ${\mathrm{SU}(3)_{\mathrm{K}}}$ and ${\mathrm{SU}(2)_{\mathrm{K}}}$ (anti-)fundamental indices are denoted by $k,k',k_1,k_2 \dots$ for $K=L,R$, respectively, while colour indices are denoted by $x,x',x_1,x_2 \dots$.
- Indices belonging to (anti-)fundamental representations of ${\mathrm{SU}(3)_{\mathrm{F}}}$ are denoted by $i,j,k \dots$.
- If a field transforms both under gauge and global symmetry groups, the index corresponding to the global one is placed within the parenthesis around the field, while the indices corresponding to the gauge symmetries are placed outside.
- Global symmetry groups will be indicated by $\{\dots\}$.
Left-Right-Color-Family unification {#Sec:LRCF}
===================================
In Glashow’s formulation of the trinified $[{\mathrm{SU}(3)_{\mathrm{L}}}\times {\mathrm{SU}(3)_{\mathrm{R}}}\times {\mathrm{SU}(3)_{\mathrm{C}}}]\rtimes \mathbb{Z}_3 \subset \mathrm{E}_6$ (LRC-symmetric) gauge theory [@original], three families of the fermion fields from the SM are arranged over three $\bm{27}$-plet copies of the $\mathrm{E}_6$ group, namely, $$\begin{aligned}
\bm{27}^i &\to& {\big(\bm{L}^{ i} \big)^{ l }{}_{ r }}\oplus{\big(\bm{Q}_{\mathrm{L}}^{ i} \big)^{ x }{}_{ l }}\oplus{\big(\bm{Q}_{\mathrm{R}}^{ i} \big)^{ r }{}_{ x }} \\
&\equiv& (\bm{3}^l,\bm{\bar{3}}_r,\bm{1})^i\oplus (\bm{\bar{3}}_l,\bm{1},
\bm{3}^x)^i\oplus (\bm{1},\bm{3}^r,\bm{\bar{3}}_x)^i \,,\end{aligned}$$ while the Higgs fields responsible for a high-scale SSB are typically introduced via e.g. an additional $\bm{27}$-plet. Here and below, the left, right, and color ${\mathrm{SU}(3)_{\mathrm{}}}$ indices are denoted by $l,\,r,$ and $x$, respectively, while the fermion families are labelled by an index $i=1,2,3$.
The SHUT model first presented in Ref. [@Camargo-Molina:2016yqm], in contrast to the Glashow’s trinification, introduces the global family symmetry ${\mathrm{SU}(3)_{\mathrm{F}}}$ which acts in the generation-space. In this case, the light Higgs and lepton superfields, as well as quarks and colored scalars, all are unified into a single $(\bm{27},\bm{3})$-plet under $\mathrm{E}_6\times {\mathrm{SU}(3)_{\mathrm{F}}}$ symmetry, i.e. $$\begin{aligned}
(\bm{27},\bm{3}) &\to& {\big(\bm{L}^{ i} \big)^{ l }{}_{ r }}\oplus{\big(\bm{Q}_{\mathrm{L}}^{ i} \big)^{ x }{}_{ l }}\oplus{\big(\bm{Q}_{\mathrm{R}}^{ i} \big)^{ r }{}_{ x }} \\
&\equiv& (\bm{3}^l,\bm{\bar{3}}_r,\bm{1},\bm{3}^i)\oplus (\bm{\bar{3}}_l,\bm{1},
\bm{3}^x,\bm{3}^i)\oplus (\bm{1},\bm{3}^r,\bm{\bar{3}}_x,\bm{3}^i) \,.\end{aligned}$$ The leptonic tri-triplet superfield ${\big(\bm{L}^{ i} \big)^{ l }{}_{ r }}$ that unifies the SM left- and right-handed leptons and SM Higgs doublets can be conveniently represented as $$\begin{aligned}
&& {\scalebox{0.95}{${\big(\bm{L}^{ i} \big)^{ l }{}_{ r }} =\begin{pmatrix}
\bm{H}_{11} & \bm{H}_{12} & \bm{e}_{\mathrm{L}}\\
\bm{H}_{21} & \bm{H}_{22} & \bm{\nu}_{\mathrm{L}}\\
\bm{e}_{\mathrm{R}}^{c} & \bm{\nu}_{\mathrm{R}}^{c} & \bm{\phi}
\end{pmatrix}^{i}\,,$}}
\label{eq:L-tri-triplet}\end{aligned}$$ Besides, the left-quark ${\big(\bm{Q}_{\mathrm{L}}^{ i} \big)^{ x }{}_{ l }}$ and right-quark ${\big(\bm{Q}_{\mathrm{R}}^{ i} \big)^{ r }{}_{ x }}$ tri-triplets are $$\begin{aligned}
\begin{aligned}
&{\scalebox{0.95}{${\big(\bm{Q}_{\mathrm{L}}^{ i} \big)^{ x }{}_{ l }}=\begin{pmatrix}\bm{u}_{\mathrm{L}}^x & \bm{d}_{\mathrm{L}}^x & \bm{D}_{\mathrm{L}}^x
\end{pmatrix}^{i}$}}, \label{eq:Q-tri-triplets}
\\
&{\scalebox{0.95}{${\big(\bm{Q}_{\mathrm{R}}^{ i} \big)^{ r }{}_{ x }}=\begin{pmatrix}\bm{u}_{\mathrm{R}x}^c & \bm{d}_{\mathrm{R}x}^c & \bm{D}_{\mathrm{R}x}^c
\end{pmatrix}^{\top\;\;i}$}}\,.
\end{aligned}\end{aligned}$$ In addition, the SHUT model also incorporates the adjoint (namely, ${\mathrm{SU}(3)_{\mathrm{L,R,C,F}}}$ octet) superfields $\bm{\Delta}_{\mathrm{L,R,C,F}}$. The first SSB step in the SHUT model ${\mathrm{SU}(3)_{\mathrm{L,R,F}}}\to {\mathrm{SU}(2)_{\mathrm{L,R,F}}}\times {\mathrm{U}(1)_{\mathrm{L,R,F}}}$ is triggered at the GUT scale by the SUSY-preserving vacuum expectation values (VEVs) in the scalar components of the corresponding octet superfields while all the subsequent low-scale SSB steps are triggered by VEVs in the leptonic tri-triplet ${\big(\bm{L}^{ i} \big)^{ l }{}_{ r }}$ through the soft SUSY-breaking operators.
Along this work, we will be focused on the symmetry breaking scheme shown in Fig. \[fig:1112abc\]. There it can be seen that an accidental global ${\mathrm{U}(1)_{\mathrm{B}}} \times {\mathrm{U}(1)_{\mathrm{W}}}$ symmetry (which is marked in red and will be discussed in detail in the next section) appears in the high-scale theory. As we will see, although alternative breaking schemes are possible, this is the one leading to the low energy SM-like scenarios we find most interesting. As we shall see in Sec. \[sec:masses2\], dimension-3 operators that softly break ${\mathrm{U}(1)_{\mathrm{W}}}$, and consequently its low-energy descendants (that will be denoted below as ${\mathrm{U}(1)_{\mathrm{S',T'}}}$), are needed for a phenomenologically viable low-scale fermion spectrum. Such interactions do not have a perturbative origin from the high-scale theory and are added to the effective theory that emerges once the heavy degrees of freedom of the SHUT model are integrated out.
Supersymmetric trinification with global ${\mathrm{SU}(3)_{\mathrm{F}}}$ {#sec:SUSYHS}
========================================================================
This section contains a review of the SHUT model before and after the T-GUT symmetry is broken spontaneously by adjoint field VEVs. We here present the symmetries, particle content and interactions of the model at both stages, in addition to showing how it addresses the shortcomings of previous T-GUTs.
Tri-triplet sector {#sec:tritri}
------------------
In the following, we consider the SHUT model – a SUSY T-GUT theory based on the trinification gauge group with an accompanying global ${\mathrm{SU}(3)_{\mathrm{F}}}$ family symmetry, i.e. $$\begin{aligned}
\nonumber
G_{333\{3\}} &\equiv& [{\mathrm{SU}(3)_{\mathrm{L}}} \times {\mathrm{SU}(3)_{\mathrm{R}}} \times {\mathrm{SU}(3)_{\mathrm{C}}} ] \\
&\rtimes& \mathbb{Z}_3^{(\mathrm{LRC})} \times \{ {\mathrm{SU}(3)_{\mathrm{F}}} \} \,.
\label{eq:3333}\end{aligned}$$ Here and below, curly brackets indicate global (non-gauge) symmetries. The minimal chiral superfield content (shown in Tab. \[table:ChiralSuperE6\]) that can accommodate the SM (Higgs and fermion) fields, is comprised of three tri-triplet representations of $G_{333\{3\}}$ which we label as $\bm{L}$, $\bm{Q}_{\mathrm{L}}$ and $\bm{Q}_{\mathrm{R}}$ respectively (for their explicit relation to the SM field content up to a possible mixing, see Eqs. (\[eq:L-tri-triplet\]) and (\[eq:Q-tri-triplets\])).
The $\mathbb{Z}_3^{(\mathrm{LRC})}$ in Eq. is realized on the chiral and vector superfields as the simultaneous cyclic permutation within $\{ \bm{L}, \bm{Q}_\mathrm{L}, \bm{Q}_\mathrm{R} \}$ and $\{\bm{V}_\mathrm{L},\bm{V}_\mathrm{C},\bm{V}_\mathrm{R}\}$ sets, respectively, where $\bm{V}_\mathrm{L,R,C}$ are the vector (super)fields for the respective gauge ${\mathrm{SU}(3)_{\mathrm{L,R,C}}}$ groups. The $\mathbb{Z}_3^{(\mathrm{LRC})}$ symmetry enforces the gauge couplings of the ${\mathrm{SU}(3)_{\mathrm{L,R,C}}}$ groups to unify, i.e. $g_\mathrm{L}=g_\mathrm{R}=g_\mathrm{C}\equiv g_\mathrm{U}$.
As mentioned previously, all fields in Tab. \[table:ChiralSuperE6\] can be contained in a $(\bm{27},\bm{3})$ representation of $\mathrm{E}_6 \times {\mathrm{SU}(3)_{\mathrm{F}}}$. In turn, the group $\mathrm{E}_6 \times {\mathrm{SU}(3)_{\mathrm{F}}}$ is a maximal subgroup of $\mathrm{E}_8$, $${\rm E_8} \supset { \rm E_6} \times {\mathrm{SU}(3)_{\mathrm{F}}} \,,
\label{eq:E8maximal}$$ where the $(\bm{27},\bm{3})$ fits neatly into the $\bm{248}$ irrep of $\mathrm{E}_8$ whose branching rule is given by $$\bm{248} = \left(\bm{1}, \bm{8} \right) \oplus \left( \bm{78},\bm{1} \right) \oplus \left( \bm{27},\bm{3} \right) \oplus \left(\bm{\overline{27}},
\bm{\overline{3}} \right)\,.
\label{eq:248}$$ Note, for clarity, that we are only considering representations of the subgroup ${\left[}{\mathrm{SU}(3)_{\mathrm{}}}{\right]}^4$ of $\mathrm{E}_8$, which are chiral rather than vector-like, in agreement with the chiral fermion content of the SM. In this work, we treat ${\mathrm{SU}(3)_{\mathrm{F}}}$ as a global symmetry. While considerably simpler, the trinification model with global ${\mathrm{SU}(3)_{\mathrm{F}}}$ can be viewed as the principal part of the fully gauged version in the limit of a vanishingly small family-gauge coupling $g_{\mathrm{F}} \ll g_{\mathrm{U}}$. In that case, Goldstone bosons would become the longitudinal d.o.f of massive ${\mathrm{SU}(3)_{\mathrm{F}}}$ gauge bosons instead of remaining as massless scalars. Such a restricted model can thus be a first step towards the fully gauged $\mathrm{E}_8$-inspired version.
Considering only renormalizable interactions, the symmetry group $G_{333\{3\}}$ allows for just a single term in the superpotential with the tri-triplet superfields, $$\label{eq:WE6}
W = \lambda_{\bm{27}}\, \varepsilon_{ijk} {\big(\bm{L}^{ i} \big)^{ l }{}_{ r }} {\big(\bm{Q}_{\mathrm{L}}^{ j} \big)^{ x }{}_{ l }} {\big(\bm{Q}_{\mathrm{R}}^{ k} \big)^{ r }{}_{ x }} \,.$$ where $\lambda_{\bm{27}}$ can be taken to be real without any loss of generality, as any phase can be absorbed with a field redefinition. As the light Higgs and lepton sectors are fully contained in the single tri-triplet $\bm{L}$, this construction provides an exact unification of Yukawa interactions of the fundamental superchiral sector and the corresponding scalar quartic couplings to a common origin, $\lambda_{\bm{27}}$.
The superpotential in Eq. has an accidental ${\mathrm{U}(1)_{\mathrm{W}}} \times {\mathrm{U}(1)_{\mathrm{B}}} $ symmetry as we can perform independent phase rotations on two of the tri-triplets as long as we do a compensating phase rotation on the third. We can arrange the charges of the tri-triplets under ${\mathrm{U}(1)_{\mathrm{W}}} \times {\mathrm{U}(1)_{\mathrm{B}}} $ as shown in Tab. \[tab:AccSym\], such that ${\mathrm{U}(1)_{\mathrm{B}}}$ is identified as the symmetry responsible for baryon number conservation. With this, we have proton stability to all orders in perturbation theory.
${\mathrm{U}(1)_{\mathrm{W}}}$ ${\mathrm{U}(1)_{\mathrm{B}}}$
--------------------- -------------------------------- --------------------------------
$\bm{L}$ $+1$ $0$
$\bm{Q}_\mathrm{L}$ $-1/2$ $+1/3$
$\bm{Q}_\mathrm{R}$ $-1/2$ $-1/3$
: *Charge assignment of the tri-triplets under the accidental symmetries.*[]{data-label="tab:AccSym"}
The model with the superpotential in Eq. also exhibits an accidental symmetry under LR-parity $\mathds{P}$. This is realized at the superspace level as $$\label{eq:SUSYParity}
\begin{aligned}
& {\big(\bm{L}^{ i} \big)^{ s }{}_{ t }} {\overset{\mathds{P}}{\rightarrow}}-{\big(\bm{L}^*_{ i} \big)_{ t }{}^{ s }} \quad , \, \left( \bm{Q}_{\mathrm{L,R}}^i \right)^x{}_s {\overset{\mathds{P}}{\rightarrow}}\left( \bm{Q}^*_{\mathrm{R,L}}{}_i \right)_s{}^x \, \\
& \qquad \qquad \bm{V}_{\mathrm{L,R,C}}^a {\overset{\mathds{P}}{\rightarrow}}-\bm{V}_{\mathrm{R,L,C}}^a \quad \, ,
\end{aligned}$$ accompanied by $$x^\mu {\overset{\mathds{P}}{\rightarrow}}x_\mu \quad , \quad \theta_\alpha {\overset{\mathds{P}}{\rightarrow}}{\mathrm i}\theta^{\dagger\;\alpha} \, .$$ Here, $\alpha$ is the spinor index on the Grassman valued superspace coordinate $\theta$. Note that $s$ and $t$ in Eq. label both ${\mathrm{SU}(3)_{\mathrm{L,R}}}$ indices as such representations are swapped under LR-parity. At the Lagrangian level, the LR-parity transformation rules become $$\label{eq:LagrParity}
\begin{aligned}
{\big(\tilde{L}^{ i} \big)^{ s }{}_{ t }} {\overset{\mathds{P}}{\rightarrow}}-{\big(\tilde{L}^*_{ i} \big)_ { t }{}^{ s }}\, , & \quad \left[{\big(L^{ i} \big)^{ s }{}_{ t }}\right]_\alpha {\overset{\mathds{P}}{\rightarrow}}- {\mathrm i}\left[{\big(L^{\dagger}_{ i} \big)_{ t }{}^{ s }}\right]^\alpha \, , \\
{\big(\tilde{Q}_{\mathrm{L}}^{ i} \big)^{ x }{}_{ s }} {\overset{\mathds{P}}{\rightarrow}}{\big(\tilde{Q}_{\mathrm{R} i}^* \big)_{ s }{}^{ x }} \, , & \quad \left[ \left( Q^i_{\mathrm{L,R}} \right)^x{}_s \right]_\alpha {\overset{\mathds{P}}{\rightarrow}}{\mathrm i}\left[ \left( Q_{\mathrm{R,L}}^{\dagger}{}_i \right)_s{}^x\right]^\alpha \, , \\
G_{\mathrm{L,R,C}}^a{}_\mu {\overset{\mathds{P}}{\rightarrow}}G_{\mathrm{R,L,C}}^{a}{}^\mu \, , &
\quad \left[ \widetilde{\lambda}^a_{\mathrm{L,R,C}} \right]_\alpha {\overset{\mathds{P}}{\rightarrow}}-{\mathrm i}\left[ \widetilde{\lambda}^a_{\mathrm{R,L,C}} {}^\dagger \right]^{\alpha} \, ,
\end{aligned}$$ which can be verified by expanding out the components of the superfields in Eq. . In this model, LR-parity exists already at the ${\mathrm{SU}(3)_{\mathrm{}}}$ level, unlike common ${\mathrm{SU}(2)_{\mathrm{L}}}\times{\mathrm{SU}(2)_{\mathrm{R}}}$ LR-symmetric realisations. Note also that there exist the corresponding accidental Right-Colour and Colour-Left parity symmetries due to the $\mathbb{Z}_3^{(\mathrm{LRC})}$ permutation symmetry imposed in the SHUT model.
As mentioned in the introduction, one of the main drawbacks of a SUSY T-GUT (as well as any SUSY GUT with very few free parameters) is the difficulty for spontaneous breaking of high-scale symmetries. For example, while the non-SUSY T-GUT in Ref. [@Camargo-Molina:2016bwm] has no problem with SSB down to a LR-symmetric theory, when including SUSY the additional relations between potential and gauge couplings make it so that there is no minimum of the potential allowing for that breaking. Moreover, even when relaxing the family symmetry, any VEV in e.g. $\widetilde{L}^i$ induces mass terms that mix the $L^i$ fermions with the gauginos $\widetilde{\lambda}^a_{\rm L,R}$ through $\mathcal{D}$-term interactions of the type $$\label{eq:FermiD}
\mathcal{L}_{\mathcal{D}} = -\sqrt{2} g_{\rm U} {\big(\tilde{L}^*_{ i} \big)_ { l_1 }{}^{ r }} \left(T_a\right)^{l_1}{}_{l_2}{\big(L^{ i} \big)^{ l_2 }{}_{ r }} \widetilde{\lambda}^a_{\mathrm{L}} \,.$$ This is a common problem in the previous T-GUT realizations as the number of light fields would not be enough to accommodate the particle content of the SM at low energies. While it is possible to get around this issue by adding extra Higgs multiplets to the theory and making them responsible for the SSB, this significantly increases the amount of light exotic fields that might be present at low energies but are unobserved. Such theories typically contain a very large number of free parameters and a fair amount of fine tuning which significantly reduces their predictive power.
In the SHUT model, this issue is instead solved by the inclusion of adjoint ${\mathrm{SU}(3)_{\mathrm{L,R,C,F}}}$ chiral supermultiplets, $\bm{\Delta}_{\mathrm{L,R,C,F}}$. By triggering the first SSB, while preserving SUSY, VEVs in scalar components of $\bm{\Delta}_{\mathrm{L,R,F}}$ do not lead to heavy would-be SM lepton fields. In addition, the scalar and fermion components of $\bm{\Delta}_{\mathrm{L,R,C}}$ are all automatically heavy after the breaking and thus do not remain in the low-energy theory.
${\mathrm{SU}(3)_{\mathrm{}}}$ adjoint superfields {#sec:SUSYGaugeAdjoints}
--------------------------------------------------
The addition of gauge adjoint superfields is the main feature preventing SM-like leptons from getting a GUT scale mass. As was briefly mentioned above, the gauge and family ${\mathrm{SU}(3)_{\mathrm{}}}$ adjoints are motivated by the $(\bm{78},\bm{1})$ and $(\bm{1},\bm{8})$ representations of $\mathrm{E}_6 \times {\mathrm{SU}(3)_{\mathrm{F}}}$ (which can be inspired by the branching rule of the $\bm{248}$-rep in its embedding into $\mathrm{E}_8$ as shown in Eq. ). Indeed, the $\bm{78}$-rep, in turn, branches as $$\label{eq:78}
{\scalebox{0.9}{$\bm{78} = \left( \bm{8}, \bm{1}, \bm{1} \right) \oplus \left( \bm{1}, \bm{8}, \bm{1} \right) \oplus \left( \bm{1}, \bm{1}, \bm{8} \right) \oplus
\left( \bm{3}, \bm{3}, \bm{\overline{3}} \right) \oplus \left( \bm{\overline{3}}, \bm{\overline{3}}, \bm{3} \right) $}} ,$$ under $\mathrm{E}_6 \supset [{\mathrm{SU}(3)_{\mathrm{}}}]^3$. We include three gauge-adjoint chiral superfields $\bm{\Delta}_{\mathrm{L,R,C}}$ corresponding to $\left( \bm{8}, \bm{1}, \bm{1} \right)$, $\left( \bm{1}, \bm{8}, \bm{1} \right)$ and $\left( \bm{1}, \bm{1}, \bm{8} \right)$ in Eq. , respectively, as well as the family ${\mathrm{SU}(3)_{\mathrm{F}}}$ adjoint, $\bm{\Delta}_{\mathrm{F}}$ (all listed in Table \[table:ChiralSuperAdj\]). The transformation rule for the $\mathbb{Z}_3^{(\mathrm{LRC})}$ symmetry in $G_{333\{3\}}$ of Eq. is now accompanied by the cyclic permutation of $\{ \bm{\Delta}_\mathrm{L}, \bm{\Delta}_\mathrm{C}, \bm{\Delta}_\mathrm{R} \}$ fields.
In order to keep the minimal setup, in this work we will not consider the fields that correspond to $\left( \bm{3}, \bm{3}, \bm{\overline{3}} \right)$ and $ \left( \bm{\overline{3}}, \bm{\overline{3}}, \bm{3} \right)$ from Eq. . In practice, they can be made very heavy and only couple to the tri-triplets via gauge interactions.
By introducing the adjoint chiral superfields, we have to add the following terms $$\begin{aligned}
\nonumber
W \supset &\sum_{A=\mathrm{L,R,C}} \left[\frac{1}{2} \mu_\mathbf{78} \,
\bm{\Delta}_A^a \bm{\Delta}_A^a +
\frac{1}{3!} \lambda_\mathbf{78} \, d_{abc} \bm{\Delta}_A^a \bm{\Delta}_A^b
\bm{\Delta}_A^c \right] \\
& + \frac{1}{2} \mu_{\bm{1}} {\bm{\Delta}_{\mathrm{F}}^{ a}} {\bm{\Delta}_{\mathrm{F}}^{ a}}
+ \frac{1}{3!} \lambda_{\bm{1}} d_{abc} {\bm{\Delta}_{\mathrm{F}}^{ a}} {\bm{\Delta}_{\mathrm{F}}^{ b}} {\bm{\Delta}_{\mathrm{F}}^{ c}} \,,
\label{eq:SuperPotentialGaugeAdj}\end{aligned}$$ to the superpotential in Eq. . Here, $d_{abc} = 2 \mathrm{Tr}[\lbrace T_a, T_b \rbrace T_c]$ are the totally symmetric ${\mathrm{SU}(3)_{\mathrm{}}}$ coefficients.
Note that bilinear terms are only present for the adjoint superfields and not for the fundamental ones, as they are forbidden by the T-GUT symmetry. Since the VEVs of the adjoint scalars set the first scale where the T-GUT symmetry is spontaneously broken, while all subsequent breaking steps occur at scales given by the soft parameters. In other words, the model is free of the so-called $\mu$-problem.
We can pick the phase of $\bm{\Delta}_{\mathrm{L,R,C,F}}$ to make $\mu_{\bm{78}}$ and $\mu_{\bm{1}}$ real, which makes $\lambda_{\bm{78}}$ and $\lambda_{\bm{1}}$ complex, in general. Notice that the superpotential provides no renormalisable interaction terms between the adjoint superfields and the tri-triplets. The accidental ${\mathrm{U}(1)_{\mathrm{W}}} \times {\mathrm{U}(1)_{\mathrm{B}}} $ symmetry of the tri-triplet sector is not affected by $\bm{\Delta}_{\mathrm{L,R,C,F}}$ as we can take these fields simply to not transform under this symmetry. The gauge interactions are parity-invariant with the following definitions for the transformation rules, $$\tilde{\Delta}_{\mathrm{L,R,C,F}}^a {\overset{\mathds{P}}{\rightarrow}}\tilde{\Delta}_{\mathrm{R,L,C,F}}^{*a},
\quad \left[ \Delta_{\mathrm{L,R,C,F}}^a \right]_{\alpha} {\overset{\mathds{P}}{\rightarrow}}{\mathrm i}\left[ \Delta_{\mathrm{R,L,C,F}}^{\dagger a} \right]^{\alpha},
\label{eq:Delta-parities}$$ or, equivalently, $\bm{\Delta}^a_{\mathrm{L,R,C,F}} {\overset{\mathds{P}}{\rightarrow}}\bm{\Delta}^{*a}_{\mathrm{R,L,C,F}}$ at the superfield level. However, LR-parity is not generally respected by the $\mathcal{F}$-term interactions unless $\lambda_{\bm{78}}$ and $\lambda_{\bm{1}}$ are real. In what follows, we assume a real $\lambda_{\bm{78}}$, whereas the accidental LR-parity can be explicitly broken by the soft SUSY-breaking sector of the theory, at or below the GUT scale.
Now, for illustration, let us discuss briefly the first symmetry breaking step which determines the GUT scale in the SHUT model (see Fig. \[fig:1112abc\]). Eq. leads to a scalar potential containing several SUSY-preserving minima with VEVs that can be rotated to the eighth component of $\tilde{\Delta}^8_{\mathrm{L,R,F}}$. In particular, there is an ${\mathrm{SU}(3)_{\mathrm{C}}}$ and LR-parity preserving minimum with $$\langle \tilde{\Delta}_{\mathrm{L,R}}^a \rangle = \frac{v_{\mathrm{L,R}}}{\sqrt{2}} \delta^a_8 \quad \mbox{with} \quad v_{\mathrm{L,R}} = v \equiv2 \sqrt{6} \,
\frac{ \mu_\mathbf{78}}{\lambda_\mathbf{78}}, \quad v_{\mathrm{C}}=0\,,$$ for the gauge-adjoints, and $$\langle \tilde{\Delta}_\mathrm{F}^a \rangle = \frac{v_\mathrm{F}}{\sqrt{2}} \delta^a_8 \quad \mbox{with} \quad v_\mathrm{F} =
2 \sqrt{6} \, \frac{ \mu_\mathbf{1}}{\lambda_\mathbf{1}} \, \label{vF-VEV}\,,$$ for the family-adjoint, setting the GUT scale $v\sim v_{\mathrm{F}}$. The vacuum structure $\langle \tilde{\Delta}_{\mathrm{L,R,F}}^8 \rangle \neq 0$ leads to the spontaneous breaking (see Appendix \[sec:Symmetries\] for the corresponding generators and ${\mathrm{U}(1)_{\mathrm{}}}$ charges), resulting in the unbroken group $$\begin{aligned}
\label{eq:322113}
G_{32211\lbrace 21 \rbrace} &\equiv& {\mathrm{SU}(3)_{\mathrm{C}}} \times [ {\mathrm{SU}(2)_{\mathrm{L}}} \times {\mathrm{SU}(2)_{\mathrm{R}}}
\\ &\times&
{\mathrm{U}(1)_{\mathrm{L}}} \times {\mathrm{U}(1)_{\mathrm{R}}} ] \times \lbrace {\mathrm{SU}(2)_{\mathrm{F}}} \times{\mathrm{U}(1)_{\mathrm{F}}} \rbrace \, .
\nonumber \end{aligned}$$ LR-parity also remains unbroken since $v_{\mathrm{L}} = v_{\mathrm{R}}^*$, which is true as long as $\lambda_{\bm{78}}$ is taken to be real.
By making the shift $$\label{eq:GaugeAdjShift}
\bm{\Delta}_{\mathrm{L,R}}^a \rightarrow \bm{\Delta}_{\mathrm{L,R}}^a + \frac{v}{\sqrt{2}} \delta^a_8,\;\;
\bm{\Delta}_{\mathrm{F}}^a \rightarrow \bm{\Delta}_{\mathrm{F}}^a + \frac{v_\mathrm{F}}{\sqrt{2}} \delta^a_8$$ and substituting $\mu_\mathbf{78} =\frac{\lambda_\mathbf{78} \, v}{2 \sqrt{6}}$, $\mu_\mathbf{1} =\frac{\lambda_\mathbf{1} \, v_{\mathrm{F}}}{2 \sqrt{6}}$ in the superpotential, we obtain
$$\begin{aligned}
W&\supset \sum_{B=\mathrm{L,R}} \left[ \frac{ \lambda_\mathbf{78} \, v}{2\sqrt{2}} \left( d_{aa8} +
\frac{1}{2\sqrt{3}} \right) \bm{\Delta}_B^a \bm{\Delta}_B^a + \frac{1}{3!} \lambda_\mathbf{78} \,
d_{abc} \bm{\Delta}_B^a \bm{\Delta}_B^b \bm{\Delta}_B^c \right] + \frac{ \lambda_\mathbf{1} \, v_\mathrm{F}}{2\sqrt{2}} \left( d_{aa8} +
\frac{1}{2\sqrt{3}} \right) \bm{\Delta}_\mathrm{F}^a \bm{\Delta}_\mathrm{F}^a \\
&+\frac{1}{3!} \lambda_\mathbf{78} \,
d_{abc} \bm{\Delta}_\mathrm{F}^a \bm{\Delta}_\mathrm{F}^b \bm{\Delta}_\mathrm{F}^c +
\frac{ \lambda_\mathbf{78} \, v }{4 \sqrt{6}} \, \bm{\Delta}_{\mathrm{C}}^a
\bm{\Delta}_{\mathrm{C}}^a + \frac{1}{3!} \lambda_\mathbf{78} \,
d_{abc} \bm{\Delta}_{\mathrm{C}}^a \bm{\Delta}_{\mathrm{C}}^b
\bm{\Delta}_{\mathrm{C}}^c + \mbox{const.}
\end{aligned}$$
The quadratic terms in the superpotential vanish for $\bm{\Delta}_{\mathrm{L,R,F}}^{4,5,6,7}$, since $d_{aa8}=-1/(2 \sqrt{3})$ for $a=4,5,6,7$, meaning that these fields receive no $\mathcal{F}$-term contribution to their masses (contrary to the other components of $\bm{\Delta}_\mathrm{L,R}$ and $\bm{\Delta}_\mathrm{F}$ which receive GUT scale masses $m_{{\scalebox{0.7}{$\Delta$}}}^2 \sim \lambda_{\bm{78}}^2 v^2$ and $\lambda_{\bm{1}}^2 v_{\mathrm{F}}^2$, respectively). While the global Goldstone bosons $\mathrm{Re} [\tilde{\Delta}_{\mathrm{F}}^{4,5,6,7}]$ are present in the physical spectrum, the gauge ones become the longitudinal polarisation states of the heavy gauge bosons related to the breaking $G_{333} \rightarrow G_{32211}$.
The presence of massless scalar degrees of freedom can only be avoided in the extended model with the gauged family symmetry. It is clear, however, that even in the case of an approximately global ${\mathrm{SU}(3)_{\mathrm{F}}}$ with $g_{\rm F}\ll g_{\rm U}$ there are no massless Goldstones in the spectrum (provided that the accidental symmetries are softly broken at low energies) but a set of relatively light family gauge bosons very weakly interacting with the rest of the spectrum.
By performing the shifts in Eq. in the $\mathcal{D}$-terms, we obtain $$\begin{aligned}
\mathcal{D}^a_B \supset -{\mathrm i}f^{abc} \tilde{\Delta}^b_B {}^\dagger \tilde{\Delta}^c_B &\rightarrow& -
{\mathrm i}\frac{v}{\sqrt{2}} f^{a8b} \left( \tilde{\Delta}^b_B - \tilde{\Delta}^b_B {}^\dagger \right) \\
&-& {\mathrm i}f^{abc} \tilde{\Delta}^b_B {}^\dagger \tilde{\Delta}^c_B \, ,\end{aligned}$$ for $B=\mathrm{L,R}$, leading to the universal GUT scale mass term $m^2 = 3 g_{\rm U}^2 v^2/4$ for the gauge-adjoints $\mathrm{Im}[\tilde{\Delta}^{4,5,6,7}_\mathrm{L,R} ]$, while $\tilde{\Delta}_{\mathrm{F}}^{4,5,6,7}$ have no $\mathcal{D}$-term contributions (or a small one in the case of approximately global ${\mathrm{SU}(3)_{\mathrm{F}}}$ with $g_{\rm F}\ll g_{\rm U}$). Hence, all components of the gauge adjoints and $\tilde{\Delta}^{1,2,3,8}_\mathrm{F}$ receive masses of order GUT scale and are integrated out in the low-energy EFT. The remaining $\tilde{\Delta}_{\mathrm{F}}^{4,5,6,7}$, on the other hand, receive a much smaller mass from the soft SUSY-breaking sector (and strongly suppressed $\mathcal{D}$-terms) and stay in the physical spectrum of the EFT. In what follows, we shall denote by $\bm{\mathcal{H}}^i_F$ the superfields containing $\mathrm{Im}[\tilde{\Delta}^{4,5,6,7}_\mathrm{L,R} ]$, and by $\bm{\mathcal{G}}^i_F$ the superfields containing $\mathrm{Re}[\tilde{\Delta}^{4,5,6,7}_\mathrm{L,R} ]$.
LR-symmetric SUSY theory {#sec:SUSYLR}
------------------------
In this section we describe the details of the supersymmetric theory left after the adjoint fields acquire VEVs. As shown in the previous section, all components of the gauge adjoint chiral superfields receive masses of the order of the GUT scale ($\mathcal{O}(v)$) in the vacuum given by Eq. . This means that to study the low-energy predictions of the theory, we need to integrate out $\bm{\Delta}_\mathrm{L,R,C}$, as well as components 1, 2, 3 and 8 of $\bm{\Delta}_\mathrm{F}$.
For the gauge sector of the SHUT model, $\langle \tilde{\Delta}_{\mathrm{L,R}} \rangle$ naturally triggers a ${\mathrm{SU}(3)_{\mathrm{\mathrm{L,R}}}}\rightarrow{\mathrm{SU}(2)_{\mathrm{\mathrm{L,R}}}}\times{\mathrm{U}(1)_{\mathrm{\mathrm{L,R}}}}$ breaking also for the tri-triplets (whose interactions with $\tilde{\Delta}_{\mathrm{L,R}}$ are mediated via $V_{\mathrm{L,R}}^a$ gauge bosons). For the global ${\mathrm{SU}(3)_{\mathrm{F}}}$ sector, there is no coupling of $\tilde{\Delta}_{\mathrm{F}}$ to the tri-triplets and, thus, the ${\mathrm{SU}(3)_{\mathrm{F}}}$ symmetry remains intact (or approximate in the case of $g_{\rm F}\ll g_{\rm U}$) in the tri-triplet sector, resulting in $G_{32211\{3\}}$ rather than $G_{32211\{21\}}$. Integrating out $\bm{\Delta}_\mathrm{L,R,C}$, and components 1, 2, 3 and 8 of $\bm{\Delta}_\mathrm{F}$, therefore leaves us with a supersymmetric theory based on the symmetry group $G_{32211\{3\}}$, with a chiral superfield content given by $\bm{\Delta}^{4-7}_\mathrm{F}$ and by the branching of $\bm{L}$, $\bm{Q}_\mathrm{L}$ and $\bm{Q}_\mathrm{R}$.
Writing the trinification tri-triplets in terms of $G_{32211\{3\}}$ representations one gets, $$\begin{aligned}
\label{eq:tri-triplets}
\renewcommand\arraystretch{1.3}
&&{\big(\bm{L}^{ i} \big)^{ l }{}_{ r }} =
\mleft(
\begin{array}{cc|c}
\bm{H}_{11} & \bm{H}_{12} & \bm{e}_{\mathrm{L}}\\
\bm{H}_{21} & \bm{H}_{22} & \bm{\nu}_{\mathrm{L}}\\
\hline
\bm{e}_{\mathrm{R}}^{c} & \bm{\nu}_{\mathrm{R}}^{c} & \bm{\phi}
\end{array}
\mright)^{i}\;, \\
&&
\begin{aligned}
\renewcommand\arraystretch{1.3}
&{\big(\bm{Q}_{\mathrm{L}}^{ i} \big)^{ x }{}_{ l }}=
\mleft(
\begin{array}{cc|c}
\bm{u}_{\mathrm{L}}^x & \bm{d}_{\mathrm{L}}^x & \bm{D}_{\mathrm{L}}^x
\end{array}
\mright)^{i}\;,
\\
\renewcommand\arraystretch{1.3}
&{\big(\bm{Q}_{\mathrm{R}}^{ i} \big)^{ r }{}_{ x }}=
\mleft(
\begin{array}{cc|c}
\bm{u}_{\mathrm{R}x}^c & \bm{d}_{\mathrm{R}x}^c & \bm{D}_{\mathrm{R}x}^c
\end{array}
\mright)^{\top\;\;i}\;,
\end{aligned}\end{aligned}$$ where the vertical and horizontal lines denote the separation of the original tri-triplets into ${\mathrm{SU}(2)_{\mathrm{}}}$-doublets and singlets after the first SSB step. We will refer to the lepton and quark ${\mathrm{SU}(2)_{\mathrm{L,R}}}$ doublets as $\bm{E}_{\mathrm{L,R}}$ and $\bm{q}_{\mathrm{L,R}}$. With this, we find that the most general superpotential consistent with $G_{32211\{3\}}$ is $$\label{eq:WLR}
\begin{aligned}
W &= \varepsilon_{ijk} \left\lbrace y_1 \bm{\phi}^i {\bm{D}_\mathrm{L}}^j {\bm{D}_\mathrm{R}}^k +
y_2 (\bm{H}^i)^L{}_R ({\bm{q}_\mathrm{L}}^j)_L ({\bm{q}_\mathrm{R}}^k)^R \right. \\
& \left. + y_3 ({\bm{E}_\mathrm{L}}^i)^L ({\bm{q}_\mathrm{L}}^j)_L {\bm{D}_\mathrm{R}}^k + y_4
({\bm{E}_\mathrm{R}}^i)_R {\bm{D}_\mathrm{L}}^j ({\bm{q}_\mathrm{R}}^k)^R \right\rbrace \,.
\end{aligned}$$ Note, in this effective SUSY LR theory one could naively add a mass term like $\varepsilon_{i j} \tilde{\mu} \bm{\mathcal{H}}^i_F \bm{\mathcal{G}}^i_F$ (that is symmetric under ${\mathrm{SU}(2)_{\mathrm{F}}}\times {\mathrm{U}(1)_{\mathrm{F}}}$ but not under full ${\mathrm{SU}(3)_{\mathrm{F}}}$) between the massless components of the family-adjoint superfield, $\bm{\mathcal{H}}^i_F$, and the massless superfield $\bm{\mathcal{G}}^i_F$ containing the Goldstone bosons. Such an effective $\mu$-term is matched to zero at tree level at the GUT scale. Due to SUSY non-renormalisation theorems [@Grisaru:1979wc], in the exact SUSY limit this term cannot be regenerated radiatively at low energies so $\tilde{\mu}$ is identically zero and was not included in the superpotential given by Eq. (\[eq:WLR\]). So, the resulting superpotential contains only fundamental superfields coming from $\bm{L}$, $\bm{Q}_\mathrm{L}$ and $\bm{Q}_\mathrm{R}$ and is indeed invariant under ${\mathrm{SU}(3)_{\mathrm{F}}}$.
In the GUT scale theory, a complex $\lambda_{\bm{78}}$ would be the only source of LR-parity violation. In the low energy theory this should lead to $y_3 \neq y_4^*$. Otherwise, $y_3 = y_4^*$ and after the matching is performed we can always make any $y_{1,2,3,4}$ real by field redefinitions. The same argument applies for the equality of the corresponding LR gauge couplings for ${\mathrm{SU}(2)_{\mathrm{L,R}}} \times {\mathrm{U}(1)_{\mathrm{L,R}}}$ symmetries.
Since we now have an effective LR-symmetric SUSY model with a ${\mathrm{U}(1)_{\mathrm{L,R}}}$ symmetry, there is a possibility of having gauge kinetic mixing. The ${\mathrm{U}(1)_{\mathrm{L,R}}}$ $D$-term contribution to the Lagrangian is given by $$\mathcal{L} \supset \frac{1}{2} (\chi\, \mathcal{D}_{\mathrm{L}}\mathcal{D}_{\mathrm{R}} + \mathcal{D}_{\mathrm{L}}^2 +
\mathcal{D}_{\mathrm{R}}^2) - \kappa (\mathcal{D}_{\mathrm{L}} - \mathcal{D}_{\mathrm{R}}) +
X_{\mathrm{L}} \mathcal{D}_{\mathrm{L}} + X_{\mathrm{R}} \mathcal{D}_{\mathrm{R}} \,,$$ where the terms proportional to $\kappa$ are the Fayet-Iliopoulos terms, while the $D$-terms and the expressions for $X_{\mathrm{L,R}}$ are shown in Appendix \[sec:ADterms\].
The values of the parameters $\{ y_{1,2,3,4}, {g_{{\scalebox{0.5}{$\mathrm{C}$}}}}, {g_{{\scalebox{0.5}{$\mathrm{L,R}$}}}}, {g_{{\scalebox{0.5}{$\mathrm{L,R}$}}}}^{\prime}, \chi, \kappa \}$ in the LR-symmetric SUSY theory are determined by the values of the parameters $\{ \lambda_{\bm{27}}, \lambda_{\bm{78}}, g_{\mathrm{U}}, v \}$ in the high-scale trinification theory at the GUT scale boundary through a matching procedure[^2]. Regarding the RG evolution of the couplings, we note that the only dimensionful parameter in the effective theory is the Fayet-Iliopoulos parameter $\kappa$. This means that $\beta_\kappa \propto \kappa$ so that if $\kappa=0$ at the matching scale (which is true, at least, at tree level), then $\kappa$ will remain zero throughout the RG flow yielding no spontaneous SUSY-breaking. Thus, we stick to the concept of soft SUSY-breaking in what follows.\
Softly broken SUSY {#sec:BreakingSUSY}
==================
In this section we describe the details of adding soft SUSY-breaking terms before the SHUT symmetry is broken spontaneously by adjoint field VEVs. One of the most important results is treated in Sec. \[sec:SU3breakingSoft2\], where it is shown that the symmetry breakings below the GUT scale are triggered solely by the soft SUSY-breaking sector. This in turn allows for a strong hierarchy between the GUT scale and the scale of the following VEVs.\
The soft SUSY-breaking Lagrangian {#sec:SSBSU3}
---------------------------------
The soft SUSY-breaking scalar potential terms respecting the imposed $G_{333\{3\}}$ symmetry, are bilinear and trilinear interactions given by
$$\begin{aligned}
\label{eq:Vsg}
\begin{aligned}
V_{{\rm{soft}}}^{\rm G} = &~\Bigg\{
~m^2_{\bm{27}}\, {\big(\tilde{L}^{ i} \big)^{ l }{}_{ r }} {\big(\tilde{L}^*_{ i} \big)_ { l }{}^{ r }} + m^2_{\bm{78}} {\tilde{\Delta}_{\mathrm{L}}^{\ast a}} {\tilde{\Delta}_{\mathrm{L}}^{ a}}
+ \left[ \frac{1}{2} b_{\bm{78}} {\tilde{\Delta}_{\mathrm{L}}^{ a}} {\tilde{\Delta}_{\mathrm{L}}^{ a}}
+ \mathrm{c.c} \right] \\
&
+ d_{a b c}\, \left[ \frac{1}{3!} A_{\bm{78}} {\tilde{\Delta}_{\mathrm{L}}^{ a}} {\tilde{\Delta}_{\mathrm{L}}^{ b}} {\tilde{\Delta}_{\mathrm{L}}^{ c}}
+ \frac{1}{2} C_{\bm{78}} {\tilde{\Delta}_{\mathrm{L}}^{\ast a}} {\tilde{\Delta}_{\mathrm{L}}^{ b}} {\tilde{\Delta}_{\mathrm{L}}^{ c}} + \mathrm{c.c.} \right]
\\
&
+ \left[ A_{\mathrm{G}} \, {\tilde{\Delta}_{\mathrm{L}}^{ a}} \left( T_a \right)_{\phantom{l}l_2}^{l_1} {\big(\tilde{L}^*_{ i} \big)_ { l_1 }{}^{ r }} {\big(\tilde{L}^{ i} \big)^{ l_2 }{}_{ r }}
+ A_{\bar{\mathrm{G}}} \, {\tilde{\Delta}_{\mathrm{R}}^{ a}} \left( T_a \right)_{\phantom{l} r_1}^{r_2} {\big(\tilde{L}^*_{ i} \big)_ { l }{}^{ r_1 }} {\big(\tilde{L}^{ i} \big)^{ l }{}_{ r_2 }} +
\mathrm{c.c.} \right] \\
&
+ (\mathbb{Z}_3^{(\mathrm{LRC})} \mbox{ permutations}) \Bigg\} +\, \left[ A_{\bm{27}} \,
\varepsilon_{ijk}{\big(\tilde{Q}_{\mathrm{L}}^{ i} \big)^{ x }{}_{ l }} {\big(\tilde{Q}_{\mathrm{R}}^{ j} \big)^{ r }{}_{ x }} {\big(\tilde{L}^{ k} \big)^{ l }{}_{ r }} +\mathrm{c.c.} \right]\,,
\end{aligned}\end{aligned}$$
for the gauge-adjoints and pure tri-triplet terms, and
$$\label{eq:SUSYbreakingNoDeltaHS}
\begin{aligned}
V_{{\rm{soft}}}^{\rm F} = & ~ m^2_{\bm{1}} {\tilde{\Delta}_{\mathrm{F}}^{\ast a}} {\tilde{\Delta}_{\mathrm{F}}^{ a}} + \left[ \frac{1}{2} b_{\bm{1}} {\tilde{\Delta}_{\mathrm{F}}^{ a}} {\tilde{\Delta}_{\mathrm{F}}^{ a}} +
\mathrm{c.c} \right] + \,d_{a b c} \left[ \frac{1}{3!} A_{\bm{1}} {\tilde{\Delta}_{\mathrm{F}}^{ a}} {\tilde{\Delta}_{\mathrm{F}}^{ b}} {\tilde{\Delta}_{\mathrm{F}}^{ c}} +
\frac{1}{2} C_{\bm{1}} {\tilde{\Delta}_{\mathrm{F}}^{\ast a}} {\tilde{\Delta}_{\mathrm{F}}^{ b}} {\tilde{\Delta}_{\mathrm{F}}^{ c}} + \mathrm{c.c.} \right] \\
& +\left[A_{\mathrm{F}} {\tilde{\Delta}_{\mathrm{F}}^{ a}} \left( T_a \right)^i {}_j {\big(\tilde{L}^*_{ i} \big)_ { l }{}^{ r }} {\big(\tilde{L}^{ j} \big)^{ l }{}_{ r }} + \mathrm{c.c.} +
(\mathbb{Z}_3^{(\mathrm{LRC})} \mbox{ permutations}) \right] \,.
\end{aligned}$$
for the family adjoint. All parameters here are assumed to be real for simplicity. We note that although trilinear terms with the gauge singlets (such as ${\tilde{\Delta}_{\mathrm{F}}^{\ast }} {\tilde{\Delta}_{\mathrm{F}}^{ }} {\tilde{\Delta}_{\mathrm{F}}^{ }} $ above) are not in general soft, due to the family symmetry and the fact that $\sum_a d_{aab} = 0 $, the dangerous tadpole diagrams do indeed cancel and do not lead to quadratic divergences.
The terms in Eq. (\[eq:Vsg\]) and (\[eq:SUSYbreakingNoDeltaHS\]), which account for the most general soft SUSY-breaking scalar potential consistent with $G_{333\{3\}}$ and real parameters, also respect the accidental ${\mathrm{U}(1)_{\mathrm{W}}} \times {\mathrm{U}(1)_{\mathrm{B}}}$ symmetry of the original SUSY theory. However, accidental LR-parity is, in general, softly-broken as long as $A_{\mathrm{G}} \not= A_{\bar{\mathrm{G}}}$, and this breaking can then be transmitted to the other sectors of the effective theory radiatively (e.g. via RG evolution and radiative corrections at the matching scale).
The only dimensionful parameters entering in the tree-level tri-triplet masses come from soft SUSY-breaking parameters, such that the corresponding scalar fields receive masses of the order of the soft SUSY-breaking scale. The full expressions are given in Appendix \[sec:masses\], from which we notice that positive squared masses requires $$\begin{aligned}
\nonumber
{\left| A_{\mathrm{G},\bar{\mathrm{G}}} \right| } v &\sim & {\left| A_{\rm F} \right| } v_{\rm F} \sim {\left| A_{\bm 1} \right| }
v_{\rm F} \\ &\sim & m^2_{\rm soft} \Rightarrow
\begin{cases}
{\left| A_{\mathrm{G},\bar{\mathrm{G}},\mathrm{F}} \right| } & \lesssim \tfrac{m^2_{\bm{27}}}{v}\sim \frac{m_{\rm soft}^2}{v}\,,\\
{\left| A_{\bm 1} \right| } & \lesssim \tfrac{m^2_{\bm{1}}}{v_{\rm F}} \,.
\end{cases}
\label{eq:softCond}\end{aligned}$$ For more details, see Sect. \[Sec:VacStab\].
Note that the $A_{\mathrm{F}}$-term in the soft sector introduces small ${\mathrm{SU}(3)_{\mathrm{F}}}$ violating (but ${\mathrm{SU}(2)_{\mathrm{F}}} \times {\mathrm{U}(1)_{\mathrm{F}}}$ preserving) effects on the interactions in the effective theory once $\langle \tilde{\Delta}_{\rm F} \rangle \neq 0$. Consider, for example, effective quartic interactions between components of $\tilde{L}$ that come from two $A_{\mathrm{F}}$ tri-linear vertices connected by an internal $\tilde{\Delta}_{\mathrm{F}}^{1,2,3}$ or $\tilde{\Delta}_{\mathrm{F}}^{8}$ propagator. The value of this diagram is $\sim {\mathrm i}A_{\mathrm{F}}^2 / \lambda_{\bm{1}}^2 v_{\mathrm{F}}^2$ neglecting the external momentum in the propagator. Using Eq. , we see that this diagram behaves as $[m_{\rm soft}/v]^4$.
The possible fermion soft SUSY-breaking terms are the Majorana mass terms for the gauginos and the Dirac mass terms between the gauginos and the fermion components of $\bm{\Delta}_{\mathrm{L,R,C}}$, namely, $$\begin{aligned}
\label{eq:Lsoft}
\mathcal{L}^{\rm{fermion}}_{\rm{soft}} &=& \Big[ -\dfrac{1}{2} M_0 \widetilde{\lambda}^a_{\rm L}
\widetilde{\lambda}^a_{\rm L} - M^{\prime}_0 \widetilde{\lambda}^a_{\rm L} {\Delta_{\mathrm{L}}^{ a}} +
\mathrm{c.c.} \\
&+& (\mathbb{Z}_3^{(\mathrm{LRC})} \mbox{ permutations}) \Big] \,, \nonumber\end{aligned}$$ From the transformation rules in Eqs. and it follows that LR-parity is not respected by $\mathcal{L}^{\rm{fermion}}_{\rm{soft}}$ unless $M^{\prime}_0 = 0$.\
Vacuum in the presence of soft SUSY-breaking terms {#sec:SU3breakingSoft2}
--------------------------------------------------
Here we show how the scalar potential changes in the presence of soft SUSY-breaking interactions. In particular, how soft SUSY-breaking terms trigger a VEV in ${\big(\tilde{L}^{ 3} \big)^{ 3 }{}_{ 3 }} \equiv \widetilde{\phi}^3$ of the same order as the soft SUSY-breaking scale.
With $\langle \Delta_{\mathrm{L,R,F}}^8 \rangle \equiv \frac{1}{\sqrt{2}} v_{\mathrm{L,R,F}}$ and $\langle \widetilde{\phi}^3\rangle \equiv \frac{1}{\sqrt{2}} v_{\varphi}$ being the VEVs present, our potential evaluated in the vacuum is given by
$$\begin{aligned}
V_{\rm vac} = & \left[ \frac{1}{2} m_{\bm{27}}^2 - \frac{1}{\sqrt{6}} \left( A_\mathrm{G} v_{\mathrm{L}} + A_{\bar{\mathrm{G}}}
v_{\mathrm{R}}+A_\mathrm{F} v_\mathrm{F} \right) \right] v_\varphi^2 + \frac{1}{12} g_{\mathrm{U}}^2 v_{\varphi}^4 +
\Big\{ \frac{1}{2} \left( m_{\bm{78}}^2 + b_{\bm{78}} \right) v_{\mathrm{L}}^2
\\
& - \frac{1}{\sqrt{6}}
\left( \frac{1}{3!} A_{\bm{78}} + \frac{1}{2} C_{\bm{78}} \right) v_{\mathrm{L}}^3 + \frac{1}{2}
v_{\mathrm{L}}^2 \, \left( \frac{1}{2 \sqrt{6}} \lambda_{\bm{78}} v_{\mathrm{L}} -
\mu_{\bm{78}} \right)^2 +(v_{\mathrm{L}} \rightarrow v_{\mathrm{R}} ) \Big\}
\\
& +\frac{1}{2} \left( m_{\bm{1}}^2 + b_{\bm{1}} \right) v_{\mathrm{F}}^2 - \frac{1}{\sqrt{6}}
\left( \frac{1}{3!} A_{\bm{1}} + \frac{1}{2} C_{\bm{1}} \right) v_{\mathrm{F}}^3 \,.
\end{aligned}$$
As all other fields (that do not acquire VEVs) only enter in bi-linear combinations, it suffices to consider the above terms to solve the conditions for vanishing first derivatives of the scalar potential. We retain the notation $v=2 \sqrt{6} \mu_{\bm{78}}/\lambda_{\bm{78}}$ for the VEVs of $\tilde{\Delta}_{\mathrm{L,R}}^8$ in the absence of soft terms. Assuming that the soft terms are much smaller than the GUT scale, i.e. $m_{\rm soft}\ll v$, we can approximately solve the extremum conditions for $v_{\mathrm{L,R},\varphi}$ by Taylor expanding them to the leading order in soft terms. Doing so we find $$\begin{aligned}
v_\varphi^2 \approx & \, \frac{3}{g^2_{\mathrm{U}} } \left[ - m_{\bm{27}}^2 +
\sqrt{\frac{2}{3}} \left( A_\mathrm{G} + A_{\bar{\mathrm{G}}} \right) v +
\sqrt{\frac{2}{3}} A_\mathrm{F} v_\mathrm{F} \right] \, , \\
v_{\mathrm{L,R}} \approx & \, v + \frac{24}{\lambda_{\bm{78}}} \Big[ - \frac{ m_{\bm{78}}^2 + b_{\bm{78}} }{v} +
\sqrt{\frac{3}{2}} \left( \frac{1}{3!} A_{\bm{78}} + \frac{1}{2} C_{\bm{78}} \right) \\
+&
\frac{1}{\sqrt{6}} A_{\mathrm{G},\bar{\mathrm{G}}} \left(\frac{v_\varphi}{v} \right)^2 \Big],
\end{aligned}$$ where in the top equation we see that the $\widetilde{\phi}^3$ VEV is of the order of the soft SUSY-breaking scale. In other words, the $\widetilde{\phi}^3$ VEV cannot be triggered unless soft terms are introduced. As is described in Sec. \[sec:SSBSU3\], the soft tri-linear couplings $A_{{\rm G},\bar{\rm G}}$, $A_{\bm{78}}$ and $C_{\bm{78}}$ need to be $\lesssim m_{\bm{27}}^2 / v$ for having positive squared masses.
Adding the soft terms shifts the values of the VEVs $v_{\mathrm{L,R}}$ described in Sec. \[sec:SUSYGaugeAdjoints\] by a relative amount behaving as $$\begin{aligned}
\sim \Big[ \frac{m_{\rm soft}}{v} \Big]^2\,.\end{aligned}$$ Furthermore, we note that the presence of $v_{\varphi}$ slightly affects the equality of $v_{\mathrm{L,R}}$, $$v_{\mathrm{L}} - v_{\mathrm{R}} \approx \frac{4 \sqrt{6} }{\lambda_{\bm{78}}}
\left( \frac{v_\varphi}{v} \right)^2 \, \left( A_\mathrm{G} - A_{\bar{\mathrm{G}}} \right) \,,$$ as long as $A_\mathrm{G}\neq A_{\bar{\mathrm{G}}}$. The relative difference between $v_{\mathrm{L,R}}$, therefore, behaves as $$\begin{aligned}
\sim \Big[ \frac{m_{\rm soft}}{v} \Big]^4\,.\end{aligned}$$ That is, although the VEVs of $\tilde{\Delta}_{\mathrm{L,R}}$ are shifted by the soft terms, the effect is very small, if not negligible, for $m_{\rm soft}\ll v$.
With a non-zero $v_\varphi\sim m_{\rm soft}\ll v$, the symmetry is further broken as $$\begin{aligned}
\label{eq:SymmetriesAfterPhiVev}
&& {\mathrm{U}(1)_{\mathrm{L}}} \times {\mathrm{U}(1)_{\mathrm{R}}} \times \{ {\mathrm{U}(1)_{\mathrm{F}}} \times {\mathrm{U}(1)_{\mathrm{W}}} \} \\
&& \overset{\langle \widetilde{\phi}^3 \rangle}{\rightarrow} {\mathrm{U}(1)_{\mathrm{L+R}}} \times
\{ {\mathrm{U}(1)_{\mathrm{S}}} \times {\mathrm{U}(1)_{\mathrm{S^\prime}}} \} \,, \nonumber\end{aligned}$$ where ${\mathrm{U}(1)_{\mathrm{L+R}}}$ consists of simultaneous ${\mathrm{U}(1)_{\mathrm{L,R}}}$ phase rotations by the same phase. ${\mathrm{U}(1)_{\mathrm{S}}}$ and ${\mathrm{U}(1)_{\mathrm{S'}}}$ are also simultaneous ${\mathrm{U}(1)_{\mathrm{L,R}}}$ phase rotations, but with opposite phase, which is compensated by an appropriate ${\mathrm{U}(1)_{\mathrm{F}}}$ and ${\mathrm{U}(1)_{\mathrm{W}}}$ transformation, respectively. All generators are presented in Appendix \[sec:Symmetries\].
In the limit of vanishingly small $A_{\mathrm F}\to 0$ in Eq. , the model exhibits an exact global ${\mathrm{SU}(3)_{\mathrm{F'}}} \times {\mathrm{SU}(3)_{\mathrm{F''}}}$ symmetry as we could then perform independent ${\mathrm{SU}(3)_{\mathrm{}}}$ family rotations on $(\bm{L},\bm{Q}_{\mathrm{L,R}})$ and $\bm{\Delta}_{\mathrm{F}}$. With non-zero $v_\varphi$ and $v_{\mathrm{F}}$, we would in this case end up with Goldstone fields built up out of $\widetilde{\phi}^{1,2}$ and $\mathrm{Re}
[ \tilde{\Delta}_{\mathrm{F}}^{4,5,6,7}]$ from the spontaneous breaking of ${\mathrm{SU}(3)_{\mathrm{F'}}}$ and ${\mathrm{SU}(3)_{\mathrm{F''}}}$, respectively. With $A_\mathrm{F} \neq 0$ the ${\mathrm{SU}(3)_{\mathrm{F'}}} \times {\mathrm{SU}(3)_{\mathrm{F''}}}$ symmetry softly breaks to the familiar ${\mathrm{SU}(3)_{\mathrm{F}}}$. This causes $\widetilde{\phi}^{1,2}$ and $\mathrm{Re}[ \tilde{\Delta}_{\mathrm{F}}^{4,5,6,7}]$ to arrange themselves into one pure Goldstone and one pseudo-Goldstone ${\mathrm{SU}(2)_{\mathrm{F}}}$ doublet (the mass of the latter is proportional to $A_\mathrm{F}$). Since $v_\varphi \ll v_{\mathrm{F}}$, the pure Goldstone is mostly $\mathrm{Re} [ \tilde{\Delta}_{\mathrm{F}}^{4,5,6,7}]$ (it has a small $\mathcal{O}(v_\varphi/v)$ admixture of $\widetilde{\phi}^{1,2}$, while the pseudo-Goldstone mode is mostly $\widetilde{\phi}^{1,2}$ containing an $\mathcal{O}(v_\varphi/v)$ amount of $\mathrm{Re} [ \tilde{\Delta}_{\mathrm{F}}^{4,5,6,7}] $).
Masses in presence of soft SUSY-breaking terms {#sec:massespres}
----------------------------------------------
The inclusion of soft SUSY-breaking interactions results in non-zero masses for the fundamental scalars contained in the $\bm{L}$, $\bm{Q}_{\rm L}$ and $\bm{Q}_{\rm R}$ superfields as well as for the gauginos. By construction, the soft SUSY-breaking parameters are small in comparison to the GUT scale, i.e. $m_{\rm soft} \ll v$, which means that the heavy states in the SUSY theory discussed in Sect. \[sec:SUSYHS\] will remain heavy and only those that were massless will receive contributions whose size is relevant for the low-energy EFT.
The masses of the fundamental scalars are purely generated in the soft SUSY-breaking sector. Furthermore, for a vacuum where only adjoint scalars acquire VEVs as in Eq. , there is no mixing among the components of the fundamental scalars corresponding to the physical eigenstates at the first breaking stage shown in Fig. \[fig:1112abc\].
The Higgs-slepton masses (no summation over the indices is implied) read $$\begin{aligned}
\nonumber
m^2_{{\big(\tilde{L}^{ i} \big)^{ l }{}_{ r }}} &=& m^2_{\bm{27}} + 2 \Big[ A_{\rm G} v {\left(}{T_{\mathrm{}}^{8}}{\right)}^l_l \\
&+& A_{\bar{\rm G}}{\left(}{T_{\mathrm{}}^{8}}{\right)}^r_r + A_{\rm F} v_{\rm F} {\left(}{T_{\mathrm{}}^{8}}{\right)}^i_i\Big]\,,\end{aligned}$$ while the corresponding squark masses are given by $$\begin{aligned}
\begin{aligned}
m^2_{{\big(\tilde{Q}_{\mathrm{L}}^{ i} \big)^{ }{}_{ l }}} &= m^2_{\bm{27}} + 2 {\left[}A_{\rm G} v {\left(}{T_{\mathrm{}}^{8}}{\right)}^l_l + A_{\rm F} v_{\rm F} {\left(}{T_{\mathrm{}}^{8}}{\right)}^i_i{\right]}\,, \\
m^2_{{\big(\tilde{Q}_{\mathrm{R}}^{ i} \big)^{ r }{}_{ }}} &= m^2_{\bm{27}} + 2 {\left[}A_{\bar{\rm G}} v {\left(}{T_{\mathrm{}}^{8}}{\right)}^r_r + A_{\rm F} v_{\rm F} {\left(}{T_{\mathrm{}}^{8}}{\right)}^i_i{\right]}\,.
\end{aligned}\end{aligned}$$ In Tab. \[table:SpecFund\] of Appendix \[sec:masses\] we show the masses for each fundamental scalar component in the LR-parity symmetric limit corresponding to $ A_{\rm G} = A_{\bar{\rm G}}$, for simplicity. Moreover, the $\widetilde{\mathcal{H}}_{\rm F}$ mass is given by $$\begin{aligned}
m^2_{\widetilde{\mathcal{H}}_{\rm F}} \simeq 2m^2_{\bm{1}} + \mathcal{O}{\left(}m^4_\mathrm{soft}/v^2_\mathrm{F}{\right)}\,,\end{aligned}$$ The exact expressions for scalar fields’ squared masses can be found in Tab. \[table:SpecAdj\] of Appendix \[sec:masses\].
The massless superpartners of the gauge bosons associated with the unbroken symmetries also acquire soft-scale masses. In particular, they mix with the chiral adjoint fermions via Dirac-terms whose strength, $M_0^{\prime}$ in Eq. , is also of the order $m_{\rm soft}$. Typically, for minimal Dirac-gaugino models, the ad-hoc introduction of adjoint chiral superfields has the undesirable side effect of spoiling the gauge couplings’ unification. However, in the model studied in Refs. [@Benakli:2014cia; @Benakli:2016ybe], this problem is resolved by evoking trinification as the natural embedding for the required adjoint chiral scalars needed to form Dirac mass terms with gauginos. With this point in mind, we want to note that the SHUT model, with softly broken SUSY at the GUT scale, is on its own a Dirac-gaugino model and a possible high-scale framework for such a class of models.
The mass matrix for the adjoint fermions in the basis $\{\widetilde{\lambda}^{1,2,3}_{\rm L,R}, \Delta^{1,2,3}_{\rm L,R},
\widetilde{\lambda}^{8}_{\rm L,R}, \Delta^{8}_{\rm L,R}\}$ is then $$\begin{aligned}
\label{eq:Mgauginos}
\renewcommand\arraystretch{1.3}
\mathcal{M}_{{\scalebox{0.6}{$\widetilde{\lambda},\Delta$}}} = \mleft(
\begin{array}{ccccccc}
M_0 & & M_0^{\prime} & & 0 & & 0 \\
M_0^{\prime} & & \frac{v \lambda_{\bm{78}} }{\sqrt{6}} + \mu_{\bm{78}} & & 0 & & 0 \\
0 & & 0 & & M_0 & & M_0^{\prime} \\
0 & & 0 & & M_0^{\prime} & & \frac{v \lambda_{\bm{78}} }{\sqrt{6}} - \mu_{\bm{78}}
\end{array}
\mright)\;.\;\end{aligned}$$ We denote the resulting mass eigenstates as $\{\mathcal{T}_{\mathrm{L,R}},\mathcal{T}^\perp_{\mathrm{L,R}}, \mathcal{S}_{\mathrm{L,R}}, \mathcal{S}^\perp_{\mathrm{L,R}}\}$ where $\mathcal{S}_{\mathrm{L,R}}$ and $\mathcal{T}_{\mathrm{L,R}}$ are the light (soft-scale) adjoint fermions while $\mathcal{S}^\perp_{\mathrm{L,R}}$ and $\mathcal{T}^\perp_{\mathrm{L,R}}$ denote the heavy (GUT scale) ones. Note that, due to a small mixing, both the low- and high-scale gauginos are essentially Majorana-like. Indeed, the mass of the former ones are approximately given by $M_0$, while the high-scale adjoint fermions $\mathcal{T}^\perp_{\mathrm{L,R}}$ and $\mathcal{S}^\perp_{\mathrm{L,R}}$ get their masses from $\mathcal{F}$-terms being approximately equal to $\big(\mathcal{M}_{{\scalebox{0.6}{$\widetilde{\lambda},\widetilde{\Delta}$}}}\big)^2_2$ and $\big(\mathcal{M}_{{\scalebox{0.6}{$\widetilde{\lambda},\widetilde{\Delta}$}}}\big)^4_4$, respectively.
The same effect is observed for the gluinos $\tilde{g}^a$ whose masses, in the limit $M_0 \sim M_0^\prime \ll v \sim \mu_{78}$, are equal to $M_0$, for the light states, and $\mu_{\bm{78}}$, for the heavy states. There is also an ${\mathrm{SU}(2)_{\mathrm{F}}}$-doublet fermion $\mathcal{H}_{\rm F}$ that acquires a mass of the order of soft SUSY-breaking scale $m_{\rm soft}$. Note that $\mathcal{H}_{\rm F}$ as well as its superpartner $\widetilde{\mathcal{H}}_{\rm F}$ receive $\mathcal{D}$-term contributions if ${\mathrm{SU}(3)_{\mathrm{F}}}$ is gauged. Finally, the chiral fundamental fermions are massless at this stage.
Particle masses at lower scales – a qualitative analysis {#sec:masses2}
========================================================
In this section we give a short overview of the low-energy limits of the SHUT model, i.e. the spectrum after $\widetilde{\phi}^3$, $\widetilde{\phi}^2$ and $\widetilde{\nu}_\mathrm{R}^1$ acquire VEVs. In particular, we investigate whether the SM-extended symmetry, $G_{\rm SM} \times {\mathrm{U}(1)_{\mathrm{T}}} \times {\mathrm{U}(1)_{\mathrm{T'}}}$ as represented at the bottom of Fig. \[fig:1112abc\], leaves enough freedom to realise the SM particle spectrum. Note that $\mathrm{SU}(2)$ (anti-)fundamental indices are denoted with lowercase letters for the remainder of the text, rather than with uppercase letters.
Colour-neutral fermions
-----------------------
Once the ${\mathrm{SU}(2)_{\mathrm{R}}} \times {\mathrm{SU}(2)_{\mathrm{F}}}$ symmetries are broken, the tri-doublet $\widetilde{H}^{f\,l}_r$ and the bi-doublet $\widetilde{h}^l_r$ are split into three distinct generations of ${\mathrm{SU}(2)_{\mathrm{L}}}$ doublets. We will then rename them as $\widetilde{H}_{r=1}^{f\,l} \equiv \widetilde{H}_{\rm u}^{f\,l}$, $\widetilde{h}_{r=1}^{l} \equiv \widetilde{h}_{\rm u}^{l}$, $\widetilde{H}_{r=2}^{f\,l} \equiv \widetilde{H}_{\rm d}^{f\,l}$ and $\widetilde{h}_{r=2}^{l} \equiv \widetilde{h}_{\rm d}^{l}$, such that $$\begin{aligned}
\begin{aligned}
\widetilde{H}^{i}_{\rm u} &= {\left(}\begin{array}{c}
\widetilde{H}_\mathrm{u}^{i\,0} \\
\widetilde{H}_\mathrm{u}^{i\,+}
\end{array}
{\right)}\qquad
\widetilde{H}^{i}_{\rm d} = {\left(}\begin{array}{c}
\widetilde{H}_\mathrm{d}^{i\,-} \\
\widetilde{H}_\mathrm{d}^{i\,0}
\end{array}
{\right)}\qquad
E^{i}_{\rm L} = {\left(}\begin{array}{c}
e_\mathrm{L}^i \\
\nu_\mathrm{L}^i
\end{array}
{\right)}\\
\widetilde{h}_{\rm u} &= {\left(}\begin{array}{c}
\widetilde{H}_\mathrm{u}^{3\,0} \\
\widetilde{H}_\mathrm{u}^{3\,+}
\end{array}
{\right)}\qquad
\widetilde{h}_{\rm d} = {\left(}\begin{array}{c}
\widetilde{H}_\mathrm{d}^{3\,-} \\
\widetilde{H}_\mathrm{d}^{3\,0}
\end{array}
{\right)}\qquad
\mathcal{E}_{\rm L} = {\left(}\begin{array}{c}
e_\mathrm{L}^3 \\
\nu_\mathrm{L}^3
\end{array}
{\right)}\end{aligned}\end{aligned}$$ where $i=1,2$, and where their scalar counterparts follow the same notation but without and with tildes, respectively. From this we can build mass terms for the charged lepton and charged Higgsinos as $$\begin{aligned}
\label{eq:DiracM}
\mathcal{L}_{C} &=&
\begin{pmatrix} e_\mathrm{L}^1 & e_\mathrm{L}^2 & e_\mathrm{L}^3 & \widetilde{H}_{\rm d}^{1\,-} &
\widetilde{H}_{\rm d}^{2\,-} & \widetilde{H}_{\rm d}^{3\,-} \end{pmatrix}
\mathcal{M}^C \\
&\times & \begin{pmatrix} e_\mathrm{R}^1 & e_\mathrm{R}^2 & e_\mathrm{R}^3 & \widetilde{H}_{\rm u}^{1\,+} &
\widetilde{H}_{\rm u}^{2\,+} & \widetilde{H}_{\rm u}^{3\,+} \end{pmatrix}^{\mathrm{T}}
+{\rm c.c.}\,. \nonumber\end{aligned}$$ Let us start by classifying all possible EW Higgs doublet and complex-singlet bosons, whose VEVs may have a role in the SM-like fermion mass spectrum. There are three types of Higgs doublets distinguished in terms of their ${\mathrm{U}(1)_{\mathrm{Y}}} \times {\mathrm{U}(1)_{\mathrm{T}}}$ charges and one possibility for complex singlets (and their complex conjugates). In particular, we can have
1. $( 1,\,1 )$: $H_{\rm u}^{2}$, $h_{\rm u}$, $H_{\rm d}^{*1}$, $\widetilde{E}_\mathrm{L}^{*2}$, $\widetilde{\mathcal{E}}_{\rm L}^*$, with VEVs denoted as $\bullet$-type.
2. $( 1,\,5 )$: $H_{\rm u}^{1}$, $H_{\rm d}^{* 2}$, $h_{\rm d}^{*}$, with VEVs denoted as $\star$-type.
3. $( 1,\,-3 )$: $\widetilde{E}_\mathrm{L}^{* 1}$, with VEVs denoted as $\ast$-type.
4. $( 0,\,-4 )$: $\widetilde{S}_{1,2}$, with VEVs denoted as $\diamond$-type.
Note that the doublets in each line can mix, in particular, in the last line the two complex singlets emerge from the mixing $\big(\widetilde{\phi}^{*1},\,
\widetilde{\nu}_\mathrm{R}^2,\, \widetilde{\nu}_\mathrm{R}^3\big) \mapsto
\big( \widetilde{S}_{1}, \, \widetilde{S}_{2}, \, \mathcal{G}_s \big)$ induced by the third breaking step in Fig. \[fig:1112abc\], with $\mathcal{G}_s$ being a complex Goldstone boson[^3].
According to the quantum numbers shown in Tab. \[Table:EFT-content-snu1Phi\] of Appendix \[sec:Symmetries\], the matrix $\mathcal{M}^C$ has the structure $$\begin{aligned}
\label{eq:MLTp}
\mathcal{M}^{C} & \sim {\left(}\begin{array}{ccc|ccc}
0 \; & \; \star \; & \; \star \; & \; 0\; & \; 0\; &\; 0 \\
\star \; &\; \bullet\; &\; \bullet\; &\; 0\; &\; 0\; &\; 0 \\
\star \; &\; \bullet\; &\; \bullet\; &\; 0\; &\; 0\; &\; 0 \\
\hline
\star \; &\; \bullet\; &\; \bullet\; &\; 0\; &\; 0\; &\; 0 \\
\bullet \; &\; 0\; &\; 0\; &\; 0\; &\; 0\; &\; 0 \\
\bullet & \; 0\; & \; 0\; & \; 0\; & \; 0\; & \; 0
\end{array}
{\right)}\,,\end{aligned}$$ where the symbols denote the type of VEVs contributing to the entry. In this case, the rank of the matrix $\mathcal{M}^{C}$ is at most three, which means that while we may be able to identify the correct patterns for the masses of the charged leptons in the SM, there will be massless charged Higgsinos remaining in the spectrum after EWSB, which is in conflict with phenomenology. The mass terms are forbidden by the ${\mathrm{U}(1)_{\mathrm{T'}}}$ symmetry, which remains unbroken after EWSB, and the latter is independent on the number of Higgs doublets involved.
In order to get a particle content consistent with the SM, one needs to break the ${\mathrm{U}(1)_{\mathrm{W}}}$ symmetry, thus avoiding the remnant ${\mathrm{U}(1)_{\mathrm{T'}}}$ symmetry. The most general ${\mathrm{U}(1)_{\mathrm{W}}}$ violating terms after $\langle\Delta^8_\mathrm{L,R,F}\rangle$ (obeying all other symmetries) are $$\begin{aligned}
\begin{aligned}
V^{\mathrm{\slashed{{\scalebox{0.5}{$W$}}}}}_{\rm soft}= &
\varepsilon_{f f'} \varepsilon_{l l'} \varepsilon^{r r'}{\left(}A_{Hh\phi} H^{l\,f}_{r} h^{l'}_{r'} \widetilde{\phi}^{f'}
+ A_{hEE} h^{l}_{r} \widetilde{E}_{\rm L}^{f \,l'} \widetilde{E}_{{\rm R}\,r'}^{f'}
\right. \\
&
\left.
+ A_{hE\mathcal{E}} H^{f\,l}_{r} \widetilde{E}_{\rm L}^{f' l'} \widetilde{\mathcal{E}}_{{\rm R}\,r'}
+ \bar{A}_{hE\mathcal{E}} H^{f\,l}_{r} \widetilde{E}_{{\rm R}\,r'}^{f'} \widetilde{\mathcal{E}}^{l'}_{{\rm L}\,} {\right)}+ \mathrm{c.c.} \,,
\label{eq:accbrk}
\end{aligned}\end{aligned}$$ with $A_{ijk} \ll v$. The charged lepton mass matrix now reads $$\begin{aligned}
\label{eq:MLT}
\mathcal{M}^{C} & \sim {\left(}\begin{array}{ccc|ccc}
0 \; & \;\star\; &\; \star\; &\; 0\; &\; \diamond\; &\; \diamond \\
\star \; &\; \bullet\; &\; \bullet\; &\; \diamond\; &\; \blacklozenge\; &\; \blacklozenge \\
\star \; &\; \bullet\; &\; \bullet\; &\; \diamond\; &\; \blacklozenge\; &\; \blacklozenge \\
\hline
\star \; &\; \bullet\; &\; \bullet\; &\; \diamond\; &\; \blacklozenge\; &\; \blacklozenge \\
\bullet \; &\; \ast\; &\; \ast\; &\; \blacklozenge\; &\; \diamond\; &\; \diamond \\
\bullet \; &\; \ast\; &\; \ast\; &\; \blacklozenge\; &\; \diamond\; &\; \diamond \\
\end{array}
{\right)}\,,\end{aligned}$$ where $\blacklozenge$ labels entries related to the $\widetilde{\nu}_{\rm{R}}^1$ VEV and can thus be well above the EW scale. We now have a mass matrix of rank-6 which means that no charged leptons and Higgsinos are left massless after EWSB. Note that before the EW symmetry is broken there are three massless lepton doublets, as the matrix in with only $\blacklozenge$-type entires has rank 3, in accordance with the SM. Furthermore, due to large $\blacklozenge$-type entries, the structure of $\mathcal{M}^{C}$ allows for three exotic lepton eigenstates heavier than the EW scale. Similarly, in the neutrino sector, no massless states remain after EWSB.
We see from the structure of Eq. that, while the maximal amount of light ${\mathrm{SU}(2)_{\mathrm{L}}}$ Higgs doublets is nine, the minimal low-scale model needs at least two Higgs doublets, one of the $\star$-type and one of the $\bullet$-type, for the rank of the matrix to remain at 6. Note also that the low-scale remnant of the family symmetry, ${\mathrm{U}(1)_{\mathrm{T}}}$, is non-universal in the space of fermion generations. As such, the various generations of Higgs bosons couple differently to different families of the SM-like fermions, offering a starting point for a mechanism explaining the mass and mixing hierarchies among the charged leptons. In addition, with the only tree-level interaction among fundamental multiplets arising from the high scale term $\bm{L}^i\bm{Q}_{\rm L}^j\bm{Q}_{\rm R}^k\epsilon_{ijk}$, the masses for all leptons must be generated at loop-level, providing a possible explanation for the lightness of the charged leptons observed in nature.
To see this, we write the allowed lepton Yukawa terms (omitting the heavy vector-like lepton contributions) $$\begin{aligned}
-\mathcal{L}_\mathrm{Y} = \Pi^a_{ij} \overline{\ell_{\mathrm{L}i}} H_a e_{\mathrm{R}j} + \mathrm{c.c.}.\end{aligned}$$ Note that the equation above is written in terms of Dirac spinors rather than left-handed Weyl spinors (such that the charges for all right-handed spinors in Table VI should be conjugated). Also, to match conventional notation, the left-handed spinor $E_\mathrm{L}$ is here denoted as $\ell_\mathrm{L}$.
For the case of the three Higgs doublets being $H_u^1$, $H_u^2$ and $H_d^2*$ (which is one of the possible scenarios enabling the Cabbibo mixing at tree-level, as shown in the following subsection), the charged lepton mass form reads $$\begin{aligned}
{\scalebox{0.9}{$M_e = \dfrac{1}{\sqrt{2}}\mleft(
\begin{array}{ccc}
0 & v_1 \Pi^1_{12} + v_d \Pi^3_{12} & v_1 \Pi^1_{13} + v_d \Pi^3_{13} \\
v_1 \Pi^1_{21} + v_d \Pi^3_{21} & v_2 \Pi^2_{22} & v_2 \Pi^2_{23} \\
v_1 \Pi^1_{31} + v_d \Pi^3_{31} & v_2 \Pi^2_{32} & v_2 \Pi^2_{33}
\end{array}
\mright) $}} \,,\end{aligned}$$ where $v_1,\; v_2$ and $v_d$ is the VEV of $H_u^1,\;H_u^2$ and $H_d^2*$, respectively. The Yukawa couplings $\Pi^a_{ij}$ are generated radiatively, by a higher-order sequential matching of the EFT to the high-scale SHUT theory at each of the breaking steps (tree-level matching yields $\Pi^a_{ij}=0$).
With this form, and with $\Pi^a_{ij}$ as free parameters, there is enough freedom to reproduce the pattern of charged SM-like lepton masses. However, whether or not it can be derived in terms of the high-scale SHUT parameters remains to be seen after the RG evolution and the calculations of the radiative threshold corrections have been carried out.
Finally, consider the neutrino sector of the model composed of 15 neutral leptons emerging from the leptonic tri-triplet ${\big(\bm{L}^{ i} \big)^{ l }{}_{ r }}$ after the EWSB, $$\begin{aligned}
{\scalebox{0.85}{$\Psi_N = \{\phi^1\, \phi^2\, \phi^3\, \nu^1_\mathrm{R}\, \nu^2_\mathrm{R}\, \nu^3_\mathrm{R}\, \nu^1_\mathrm{L}\, \nu^2_\mathrm{L}\, \nu^3_\mathrm{L}\,
\widetilde{H}^{1\,0}_\mathrm{d}\, \widetilde{H}^{2\,0}_\mathrm{d}\, \widetilde{H}^{3\,0}_\mathrm{d}\, \widetilde{H}^{1\,0}_\mathrm{u}\,
\widetilde{H}^{2\,0}_\mathrm{u}\, \widetilde{H}^{3\,0}_\mathrm{u}\} $}} \,.\end{aligned}$$ Note, in this first consideration we ignore the adjoint (chiral superfields ${\bm{\Delta}_{\mathrm{L,R,F}}^{ a}}$ and neutral gaugino $\widetilde{\lambda}^a_{\rm L,R}$) sectors for the sake of simplicity, while they should be included in a complete analysis of the neutrino sector involving the RG running and the radiative threshold corrections at every symmetry breaking scale. The corresponding 15$\times$15 mass form with all the Dirac and Majorana terms allowed after the EWSB $$\begin{aligned}
\mathcal{L}_\mathrm{N} = \Psi_N \mathcal{M}^N \Psi_N^{\top} \,,\end{aligned}$$ has the following generic structure $$\begin{aligned}
{\scalebox{0.95}{$\mathcal{M}^{N} =
{\left(}\begin{array}{cccccccccccccccccc}
0 & 0 & 0 & 0 & \otimes & \otimes & 0 & 0 & 0 & 0 & \vee & \vee & 0 & \vee & \vee \\
0 & \times & \times & \times & 0 & 0 & 0 & \vee & \vee & \vee & \vee & \vee & \vee & \vee & \vee \\
0 & \times & \cup & \times & 0 & 0 & 0 & \vee & \vee & \vee & \vee & \vee & \vee & \vee & \vee \\
0 & \times & \times & \times & 0 & 0 & 0 & \vee & \vee & \vee & \vee & \vee & \vee & \vee & \vee \\
\otimes & 0 & 0 & 0 & 0 & 0 & \vee & \vee & \vee & \vee & 0 & 0 & \vee & 0 & 0 \\
\otimes & 0 & 0 & 0 & 0 & 0 & \vee & \vee & \vee & \vee & 0 & 0 & \vee & 0 & 0 \\
0 & 0 & 0 & 0 & \vee & \vee & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \vee & \vee & \vee & \vee & \vee & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \otimes &\otimes \\
0 & \vee & \vee & \vee & \vee & \vee & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\otimes & \otimes \\
0 & \vee & \vee & \vee & \vee & \vee & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\otimes & \otimes \\
\vee & \vee & \vee & \vee & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\otimes & 0 & 0 \\
\vee & \vee & \vee & \vee & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \otimes & 0 & 0 \\
0 & \vee & \vee & \vee & \vee & \vee & 0 & 0 & 0 & 0 & \otimes &\otimes & 0 & 0 & 0 \\
\vee & \vee & \vee & \vee & 0 & 0 & 0 & \otimes & \otimes & \otimes & 0 & 0 & 0 & 0 & 0 \\
\vee & \vee & \vee & \vee & 0 & 0 & 0 & \otimes & \otimes & \otimes & 0 & 0 & 0 & 0 & 0 \\
\end{array}
{\right)}$}}\end{aligned}$$ where the symbol $\cup$ denotes the only Majorana bilinear below the $\langle\widetilde{\phi}^3\rangle$ scale, $\times$ the Majorana bilinears below the $\langle\widetilde{\phi}^2\rangle$ and $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$ scales, $\otimes$ the Dirac bilinears below $\langle\widetilde{\phi}^2\rangle$ and $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$ scales, and $\vee$ the Dirac bilinears at the lowest EWSB scale. For mass terms receiving contributions from more than one symmetry breaking scale, only the highest scale is displayed in the matrix above. Note that all bilinears with both fields having zero charge under all U(1) groups are referred to as Majorana bilinears, and not just combinations consisting of a field with itself.
Despite of the absence of tree-level Yukawa interaction for the leptonic tri-triplet ${\big(\bm{L}^{ i} \big)^{ l }{}_{ r }}$ at the GUT scale, the Majorana mass terms in the upper-left 3$\times$3 block of the mass form are generated at tree-level at the intermediate matching $\langle\widetilde{\phi}^3\rangle$, $\langle\widetilde{\phi}^2\rangle$ and $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$ scales due to interactions with gauginos, while all other Majorana and Dirac terms are generated radiatively, either at one- or two-loop level. With this structure, and with the hierarchy of scales presented in Sec. \[sec:EFTs\], there are solutions with three sub-eV neutrino states. Two of these states are present for a wide range of parameter values, while a third light state, in the considered simplistic approach, typically requires a tuned suppression of one or more entries in the lower right 8$\times$8 block. Whether this can be obtained with less fine-tuning when including the full set of neutral states coming from the adjoint superfields, remains to be seen once the full RG evolution and matching has been carried out.
Quark sector {#quarkquark}
------------
In the absence of the accidental ${\mathrm{U}(1)_{\mathrm{T'}}}$ symmetry, the low-energy limit of the SHUT model also offers good candidates for SM quarks without massless states after EWSB. To see this we first note that once $\widetilde{\phi}^3$ develops a VEV at the second SSB stage shown in Fig. \[fig:1112abc\], two generations of $D$-quarks mix and acquire mass terms of the form $ m_D D_{\rm L}^f D_{\rm R}^{f'} \varepsilon_{f f'}$, with $m_D = \mathcal{O} \big(m_{\rm soft}\big) \gg M_{\rm EW}$. Then, at the third breaking stage, the $\widetilde{\nu}_{\rm R}^1$ and $\widetilde{\phi}^2$ VEVs trigger a mixing between the R-type quarks $D_{\rm R}^i$ and $d_{\rm R}^i$ $$\begin{aligned}
\begin{pmatrix}
\vspace{0.5mm}
{d}_{\rm R}^1
\\
\vspace{0.5mm}
{D}_{\rm R}^2
\\
\vspace{0.5mm}
{D}_{\rm R}^3
\\
\vspace{0.5mm}
{D}_{\rm R}^1
\\
\vspace{0.5mm}
{d}_{\rm R}^2
\\
\vspace{0.5mm}
{d}_{\rm R}^3
\\
\end{pmatrix}
=
\begin{pmatrix}
\vspace{0.5mm}
1 &0 & 0 & 0 & 0 & 0
\\
\vspace{0.5mm}
0 & a_1 & a_2 & 0 & 0 & 0
\\
\vspace{0.5mm}
0 & a_3 & a_4 & 0 & 0 & 0
\\
\vspace{0.5mm}
0 & 0 & 0 & a_5 & a_6 & a_7
\\
\vspace{0.5mm}
0 & 0 & 0 & a_8 & a_9 & a_{10}
\\
\vspace{0.5mm}
0 & 0 & 0 & a_{11} & a_{12} & a_{13}
\end{pmatrix}
\begin{pmatrix}
\vspace{0.5mm}
\mathpzc{d}_{\rm R}^1
\\
\vspace{0.5mm}
\mathpzc{d}_{\rm R}^2
\\
\vspace{0.5mm}
\mathpzc{D}_{\rm R}^1
\\
\vspace{0.5mm}
\mathpzc{d}_{\rm R}^3
\\
\vspace{0.5mm}
\mathpzc{D}_{\rm R}^2
\\
\vspace{0.5mm}
\mathpzc{D}_{\rm R}^3
\\
\end{pmatrix},
\label{eq:changeOfBasis}\end{aligned}$$ where the parameters $a_{1}$ through $a_{13}$ are not all independent as the matrix is unitary. At the classical level, and with $\langle\widetilde{\phi}^3\rangle=\langle\widetilde{\phi}^2\rangle=\langle\widetilde{\nu}_{\rm R}^1\rangle$, the parameters are given by $$\begin{aligned}
&& a_{1,3,4,12}=-a_{2,9}=\frac{1}{\sqrt{2}},\;\;a_{5,8,11}=\frac{1}{\sqrt{3}}\,, \\
&& a_6=0,\;\;a_7=-\sqrt{\frac{2}{3}},\;\;a_{10,13}=\frac{1}{\sqrt{6}},\end{aligned}$$ while the corresponding expressions for general $\langle\widetilde{\phi}^3\rangle$, $\langle\widetilde{\phi}^2\rangle$, $\langle\widetilde{\nu}_{\rm R}^1\rangle$ are too extensive to be presented here.
Defining the components of the ${\mathrm{SU}(2)_{\mathrm{L}}}$ quark doublets as $Q_\mathrm{L}^{1,2} \equiv \big(u_\mathrm{L}^{1,2}, d_\mathrm{L}^{1,2} \big)^{\mathrm{T}}$ and $q_\mathrm{L} \equiv \big(u_\mathrm{L}^3, d_\mathrm{L}^3 \big)^{\mathrm{T}}$, we can construct the Lagrangian for the SM-like quarks as $$\begin{aligned}
\nonumber
\mathcal{L}_{\rm quarks} &=&
\begin{pmatrix} {u_{\rm L}^1} & {u_{\rm L}^2} & {u_{\rm L}^3} \end{pmatrix}
\mathcal{M}^{\rm u}
\begin{pmatrix} u_{\rm R}^1 \\\\ u_{\rm R}^2 \\\\ u_{\rm R}^3 \end{pmatrix} \\
&+&
\begin{pmatrix} {d_{\rm L}^1} & {d_{\rm L}^2} & {d_{\rm L}^3} \end{pmatrix}
\mathcal{M}^{\rm d}
\begin{pmatrix} \mathpzc{d}_{\rm R}^1 \\\\ \mathpzc{d}_{\rm R}^2 \\\\ \mathpzc{d}_{\rm R}^3 \end{pmatrix}
+{\rm c.c.}
\label{eq:quarks}\end{aligned}$$ With the different possibilities found for the Higgs sector, the most generic structure for $\mathcal{M}^{\rm u}$ and $\mathcal{M}^{\rm d}$ matrices obey the following patterns: $$\mathcal{M}^{\rm u} \sim
\begin{pmatrix}
\ast \; &\; \bullet\; &\; \bullet \\
\bullet\; &\; \star\; &\; \star \\
\bullet\; &\; \star\; &\; \star \\
\end{pmatrix}
, \;\;\;\;
\mathcal{M}^{\rm d} \sim
\begin{pmatrix}
0\; &\; \bullet\; &\; \star \\
\star \; & \; \ast \; &\; \bullet \\
\star\; & \; \ast \; &\; \bullet
\end{pmatrix}\label{eq:Md}
\,.$$ In order for all quarks to gain a mass after EWSB, the matrices in Eq. must be of rank-3. As such, the low-scale limit of the SHUT model requires, at least, two Higgs doublets, where both $\bullet$-type and $\star$-type ones are present. In contrast to charged leptons, for which the contributions arise solely from effective Yukawa couplings, in Eq. there are allowed tree-level bilinears for the SM-like quarks.
Next, let us consider the possible flavour structure in the low-scale limit. At the classical level, we have Cabbibo mixing with a minimum of three Higgs doublets. For a realistic mass spectrum, it is also required to incorporate RG effects as well as loop-induced threshold corrections, which make the Yukawa couplings different from each other. Take for example the 3HDM with two up-type Higgs doublets, $H_{\rm u}^{1}$ and $H_{\rm u}^{2}$ and a down type Higgs doublet $H_\mathrm{d}^2$. In the classical limit of the theory, this corresponds to $$\begin{aligned}
\nonumber
\mathcal{M}^{\rm u} &= \dfrac{\lambda_{27}}{\sqrt{2}}{\left(}\begin{array}{ccc}
0 & 0 & -v_2 \\
0 & 0 & v_1 \\
v_2 & -v_1 & 0 \\
\end{array}
{\right)}\,,\, \\
\mathcal{M}^{\rm d} &= \dfrac{\lambda_{27}}{\sqrt{2}}{\left(}\begin{array}{ccc}
0\; &\; 0\; &\; -\frac{1}{\sqrt{3}}v_d \\
0 \; & \; 0 \; &\; 0 \\
v_d\; & \; 0 \; &\; 0
\end{array}\label{eq:MuMd}
{\right)}\,,\end{aligned}$$
where $v_{1,2,d}$ are the corresponding Higgs VEVs and where $\lambda_{27}$ is the high-scale Yukawa coupling. With this, the Cabbibo angle satisfies $\tan \theta_C=\frac{v_1}{v_2}$ and results in the quark mass spectrum $$\begin{aligned}
\begin{split}
\label{eq:degenerateQuarkMasses}
&m^2_{\rm{c,t}}=\frac{1}{2}\lambda_{27}^2({v_1}^2 + {v_2}^2),\\
&m^2_{\rm{b}}=3m^2_{\rm{s}}=\frac{1}{2}\lambda_{27}^2v_{\rm{d}}^2,\;\;\;\; m^2_{\rm{u,d}}=0,
\end{split}\end{aligned}$$ i.e. the lowest order contributions to the particle spectrum imply a degeneracy of charm and top quark masses, while strange and bottom quark masses squared are related with a factor three.
![Diagrams contributing to the one-loop matching conditions for Yukawa interactions with the upper diagram representing the dominant contributing to the top quark mass and the lower one a correction to the charm mass.[]{data-label="fig:loop-Yuk"}](diagrams.pdf)
When radiative corrections are considered as well, the mass forms become more involved. Indeed, for an effective quark Yukawa Lagrangian the allowed terms (omitting, for simplicity, the heavy vector-like quark Yukawa terms) $$\begin{aligned}
-\mathcal{L}^q_\mathrm{Y} = \Gamma^a_{ij} \overline{q_{\mathrm{L}i}} H_a d_{\mathrm{R}j}
+ \Delta^a_{ij} \overline{q_{\mathrm{L}i}} \widetilde{H}_a u_{\mathrm{R}j} + \mathrm{c.c.}
\end{aligned}$$ written again in terms of Dirac fermions rather than left-handed Weyl fermions, and where the tilde on the Higgs doublet refers to $\widetilde{H}^l=\varepsilon^{ll'}H^*_{l'}$ and not it being a Higgsino, as in the other parts of the paper. With the three Higgs doublets again being $H_u^{1,2}$ and $H_d^{2*}$, we have the mass forms $$\begin{aligned}
\nonumber
&&\mathcal{M}^{\rm u} \approx \dfrac{1}{\sqrt{2}}\mleft(
\begin{array}{ccc}
0 & v_2 \Delta^2_{12} & v_2 \Delta^2_{13} \\
v_2 \Delta^2_{21} & v_1 \Delta^1_{22} + v_d \Delta^3_{22} & v_1 \Delta^1_{23} + v_d \Delta^3_{23} \\
v_2 \Delta^2_{31} & v_1 \Delta^1_{32} + v_d \Delta^3_{32} & v_1 \Delta^1_{33} + v_d \Delta^3_{33}
\end{array}
\mright) , \\
&&\mathcal{M}^{\rm d} \approx \dfrac{1}{\sqrt{2}}\mleft(
\begin{array}{ccc}
0 & v_2 \Gamma^2_{12} & v_1 \Gamma^1_{13} + v_d \Gamma^3_{13} \\
v_1 \Gamma^1_{21} + v_d \Gamma^3_{21} & 0 & v_2 \Gamma^2_{23} \\
v_1 \Gamma^1_{31} + v_d \Gamma^3_{31} & 0 & v_2 \Gamma^2_{33}
\end{array}
\mright) . \label{eq:MuMd-full}\end{aligned}$$ where the zeros are put in as a good approximation since the corresponding Yukawa terms come from higher-loop contributions which are generated only at the ${\mathrm{U}(1)_{\mathrm{T}}}$ breaking scale.
Let us estimate whether the radiative corrections can be sufficiently large to correct for the degeneracy in Eq. . As a demonstration, we will consider the largest mass discrepancy, namely the degeneracy between the top and charm mass whose tree level value is proportional to $\lambda_{\bm{27}}$. The key idea here is that $\lambda_{\bm{27}} \sim \mathcal{O}{\left(}10^{-2}{\right)}$ which readily generates a viable charm mass but leaves the top quark two orders of magnitude lighter than its measured value. To lift such a degeneracy, one needs an order $\mathcal{O}{\left(}1{\right)}$ correction to $\Delta_{32}^1$ while leaving $\Delta_{13}^2 \lesssim \mathcal{O}{\left(}10^{-2}{\right)}$. To have an estimate for these radiative corrections, we can start with an instance of the mass forms ${\mathcal{M}}^{\mathrm{u},\mathrm{d}}$, with textures as in Eq. , that reproduce measured quark masses and mixing angles [@Bora:2012tx; @Tanabashi:2018oca], e.g. $$\begin{aligned}
&\mathcal{M}^{\rm u} = \mleft(
\begin{array}{ccc}
0 & -7.287 & 0.636 \\
-0.0013 & -0.159 - i 0.521 & -0.0016 - i 0.005 \\
0.124 & -171.944 & 0.00011
\end{array}
\mright) \mathrm{GeV}
\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\\
&\mathcal{M}^{\rm d} = \mleft(
\begin{array}{ccc}
0 & -0.013 & 0.055 \\
-0.0006 & 0 & 0.013 \\
2.814 & 0 & 0.188
\end{array}
\mright) \mathrm{GeV} \,.
\end{aligned}\label{massNum}$$
Keeping in mind that $v_1^2 + v_2^2 + v_d^2=\left(246\;\mathrm{GeV}\right)^2$, we then get an idea of what the values for $\Delta_{32}^1$ and $\Delta_{13}^2$ need to be. In particular, we see that the magnitude of $\Delta_{32}^1$ has to be larger than 0.7.
The one-loop dominating contributions[^4] for the Yukawa couplings $\Delta_{32}^1$ and $\Delta_{13}^2$ are illustrated in Fig. \[fig:loop-Yuk\]. When a propagator in the loop becomes heavier than the renormalization scale, thus integrated out, we generate a threshold correction. For illustration purposes we will choose this scale to be either the gluino or the squark mass.
At this scale, the squark (gluino) propagators are resummed such that the masses are given by their $\overline{\mathrm{MS}}$ values at the gluino (squark) mass scale, which should be some function of quartic couplings, soft parameters and VEVs. We also have that $y_{41,42}$ are approximately equal to $\sqrt{2}g_\mathrm{S}$, with $\alpha_\mathrm{S} \sim 0.03$ at the $\langle\widetilde{\phi}^3\rangle$ scale, such that the two diagrams only differ when it comes to one of the couplings, and possibly by a mass difference for the squarks in the loop. The analytic expression for both diagrams, in the zero external momentum limit, is given by $$\begin{aligned}
\begin{split}
\frac{ 2i\alpha_{\mathrm{S}} Z m_3}{ 3\pi\left(m_2^2-m_3^2\right) }\left( \frac{m_3^2\log\left(\frac{m_1^2}{m_3^2}\right)}{m_3^2-m_1^2} -
\frac{m_2^2\log\left(\frac{m_1^2}{m_2^2}\right)}{m_2^2-m_1^2} \right) \,,
\end{split}\label{loop}\end{aligned}$$ with $$Z=\lambda_{162}\langle\widetilde{\nu}_\mathrm{R}^1\rangle, \;\;\{m_1,m_2,m_3\}=\{m_{\widetilde{u}_{\mathrm{L}}^3},m_{\widetilde{u}_{\mathrm{R}}^2\widetilde{D}_{\mathrm{L}}^3},m_{\widetilde{g}}\}
\label{eq:pars-top}$$ for the top diagram in Fig. \[fig:loop-Yuk\], and with $$Z=\lambda_{70}\langle\widetilde{\nu}_\mathrm{R}^1\rangle, \;\;\{m_1,m_2,m_3\}=\{m_{\widetilde{u}_{\mathrm{L}}^1},m_{\widetilde{u}_{\mathrm{R}}^3\widetilde{D}_{\mathrm{L}}^1},m_{\widetilde{g}}\}
\label{eq:pars-charm}$$ for the bottom diagram. Note that the result is finite also in the limit of degenerate masses and has the form $$\dfrac{i \alpha_S Z}{3 \pi m_{\tilde{g}} }.
\label{loopDeg}$$ In what follows we will consider the case where the intermediate symmetries are simultaneously broken by the VEVs $$\langle\widetilde{\phi}^3\rangle\sim\langle\widetilde{\phi}^2\rangle\sim\langle\widetilde{\nu}_\mathrm{R}^1\rangle\sim 8.8\cdot 10^{10}\; \mathrm{GeV},$$ consistent with Sec. \[sec:EFTs\], and with couplings as specified in Appendix \[sec:FullEffectiveLs\].
The magnitude of the dominant contributions to the top and charm Yukawa couplings are shown for a selection of gluino and squark masses in tabs. \[tab:Vals\] and \[tab:Vals2\] respectively. Here we have, for example, a scenario with squark masses at the TeV scale, offering an interesting phenomenological probe to be studied in the context of LHC searches, or alternatively a scenario where both the gluino and squark masses in the top diagram are closely degenerate. Interestingly enough, we see that radiative corrections to the charm quark are sub-leading if at least one squark propagator is heavy enough and close to the $\langle \widetilde{\phi}^3 \rangle$ scale. With the examples provided we see that a hierarchy in the squark sector is reflected as a hierarchy in the radiative Yukawa couplings, necessary for the phenomenological viability of the model.
$\Delta^1_{32}$ $\lambda_{162}$ $m_{\tilde{g}}$ $m_{\tilde{u}_\mathrm{L}^3}$ $m_{\tilde{u}_\mathrm{R}^2,\,D_\mathrm{L}^3}$
----------------- ----------------- ----------------- ------------------------------ -----------------------------------------------
$1$ $10^{-2}$ $10^8$ $10^3$ $10^3$
$1$ $10^{-2}$ $10^6$ $10^6$ $ 10^6$
: *[Order of magnitude of the radiative correction to the top-quark Yukawa coupling (first column) and of the parameters contributing to the one-loop function (second to fifth columns). Masses are expressed in $\mathrm{GeV}$]{}*[]{data-label="tab:Vals"}
$\Delta^2_{13}$ $\lambda_{70}$ $m_{\tilde{g}}$ $m_{\tilde{u}_\mathrm{L}^1}$ $m_{\tilde{u}_\mathrm{R}^3,\,D_\mathrm{L}^2}$
----------------- ---------------- ----------------- ------------------------------ -----------------------------------------------
$10^{-5}$ $10^{-2}$ $10^8$ $10^{10}$ $10^3$
$10^{-6}$ $10^{-2}$ $10^6$ $10^{10}$ $ 10^6$
: *[Order of magnitude of the radiative correction to the charm-quark Yukawa coupling (first column) and of the parameters contributing to the one-loop function (second to fifth columns). Masses are expressed in $\mathrm{GeV}$]{}*[]{data-label="tab:Vals2"}
Note that for the degenerate scenario $\Delta^1_{32} = 2.8\times 10^8~\mathrm{GeV} {\left(}\lambda_{162}/m_{\tilde{g}}{\right)}$, which means that a viable correction to the top quark mass requires the ratio $\lambda_{162}/m_{\tilde{g}} \sim \mathcal{O}{\left(}10^{-8}~\mathrm{GeV^{-1}}{\right)}$. This means that, depending on the details of the renormalization procedure that may enhance or suppress the quartic coupling $\lambda_{162}$, an appropriate choice of the free gluino mass parameter will in principle make it possible to naturally lift the top-charm mass degeneracy in the right direction.
The required parameter values for compatible couplings at the EW scale remains unknown until the full RG evolution and sequential matching of all couplings in the model has been carried out, which is a subject of a further much more involved and dedicated study. What we can say at this point is that there do exist parameter space points with a potential of reproducing the correct hierarchy between the top and charm masses.
Estimating the scales of the theory {#sec:EFTs}
===================================
In this section we estimate the symmetry breaking scales of the model, i.e. the GUT scale $\langle\widetilde{\Delta}^8_\mathrm{L,R,F}\rangle\sim v$, and the intermediate scales $\langle\widetilde{\phi}^3\rangle$, $\langle\widetilde{\phi}^2\rangle$ and $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$, by forcing the unified gauge coupling at the GUT scale to evolve such that it reproduces the measured values of the $\mathrm{SU}(3)_\mathrm{C} \times \mathrm{SU}(2)_\mathrm{L} \times \mathrm{U}(1)_\mathrm{Y}$ gauge couplings at the EW scale. This is done through a matching and running procedure, where the gauge couplings are matched at tree-level accuracy and evolved with one-loop RGEs, as a first step before matching at one-loop in future work. At each breaking scale, fermions obtaining a mass from the associated VEV are integrated out, giving rise to four intermediate energy ranges of RG evolution with different $\beta$-functions. We will refer to these regions as $$\begin{aligned}
&\mathrm{Region\;{ \textup{\uppercase\expandafter{\romannumeral1}}}:}\;\mu\in{\left[}\langle\widetilde{\phi}^3\rangle,v{\right]}, \\
&\mathrm{Region\;{ \textup{\uppercase\expandafter{\romannumeral2}}}:}\;\mu\in{\left[}\langle\widetilde{\phi}^2\rangle,\langle\widetilde{\phi}^3\rangle{\right]},
\\
&\mathrm{Region\;{ \textup{\uppercase\expandafter{\romannumeral3}}}:}\;\mu\in{\left[}\langle\widetilde{\nu}_\mathrm{R}^1\rangle,\langle\widetilde{\phi}^2\rangle{\right]},
\\
&\mathrm{Region\;{ \textup{\uppercase\expandafter{\romannumeral4}}}:}\;\mu\in{\left[}m_{Z},\langle\widetilde{\nu}_\mathrm{R}^1\rangle{\right]}.
\end{aligned}$$ The symmetry alone does not dictate the structure of the scalar mass spectrum, and we will therefore have to make assumptions about what scalars are to be integrated out at each matching scale. However, by studying the extreme cases we will show that the soft SUSY-breaking scale (which we associate with the scale of the largest tri-triplet VEV, $\langle\widetilde{\phi}^3\rangle$) is bounded from below by roughly $10^{11}$ GeV, independently of the scalar content.
With the $\beta$-functions and matching conditions presented in Appendix \[sec:betabeta\], we may set up a system of equations with three known values, the SM couplings at the $Z$-mass scale, and five unknown quantities, $\alpha^{-1}_g(v)$, $\log(\langle\widetilde{\phi}^3\rangle/v)$, $\log(m_Z/\langle\widetilde{\phi}^2\rangle)$, $\log(\langle\widetilde{\phi}^2\rangle/\langle\widetilde{\phi}^3\rangle)$ and $\log(\langle\widetilde{\nu}_\mathrm{R}^1\rangle/\langle\widetilde{\phi}^2\rangle)$: $$\begin{aligned}
\begin{split}
\label{eq:evolutionequations}
{\alpha}^{-1}_{g_\mathrm{C}}(m_Z)&={\alpha}^{-1}_{g}(v) - \frac{b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{g_\mathrm{C}}}{2\pi} \log\left(\frac{\langle\widetilde{\phi}^2\rangle}{\langle\widetilde{\phi}^3\rangle}\right) \\
& - \frac{b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{C}}}{2\pi} \log\left(\frac{\langle\widetilde{\nu}_\mathrm{R}^1\rangle}{\langle\widetilde{\phi}^2\rangle}\right)
- \frac{b^{{ \textup{\uppercase\expandafter{\romannumeral4}}}}_{g_\mathrm{C}}}{2\pi} \log\left(\frac{m_{{Z}}}{\langle\widetilde{\nu}_\mathrm{R}^1\rangle}\right),
\end{split}\end{aligned}$$ $$\begin{aligned}
\begin{split}
\label{eq:evolutionequations2}
{\alpha}^{-1}_{g_\mathrm{L}}(m_{{Z}})&={\alpha}^{-1}_{g}(v) - \frac{b^{{ \textup{\uppercase\expandafter{\romannumeral1}}}}_{g_\mathrm{L,R}}}{2\pi}\log\left(\frac{\langle\widetilde{\phi}^3\rangle}{v}\right)
\\
& - \frac{b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{g_\mathrm{L}}}{2\pi}\log\left(\frac{\langle\widetilde{\phi}^2\rangle}{\langle\widetilde{\phi}^3\rangle}\right) - \frac{b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{L}}}{2\pi}\log\left(\frac{\langle\widetilde{\nu}_\mathrm{R}^1\rangle}{\langle\widetilde{\phi}^2\rangle}\right)
\\
& - \frac{b^{{ \textup{\uppercase\expandafter{\romannumeral4}}}}_{g_\mathrm{L}}}{2\pi}\log\left(\frac{m_{{Z}}}{\langle\widetilde{\nu}_\mathrm{R}^1\rangle}\right),
\end{split}\end{aligned}$$ $$\begin{aligned}
\begin{split}
\label{eq:evolutionequations3}
{\alpha}^{-1}_{\widetilde{g}_\mathrm{Y}}(m_{{Z}}) &= \frac{5}{3}{\alpha}^{-1}_{g}(v) + \frac{b^{{ \textup{\uppercase\expandafter{\romannumeral4}}}}_{\widetilde{g}_\mathrm{Y}}}{2\pi}\log\left(\frac{\langle\widetilde{\nu}_\mathrm{R}^1\rangle}{m_{{Z}}}\right)
\\
&-\frac{1}{2\pi} \log\left(\frac{\langle\widetilde{\phi}^3\rangle}{v}\right)\left[{b^{{ \textup{\uppercase\expandafter{\romannumeral1}}}}_{g_\mathrm{L,R}}} + \frac{2}{3}{b^{{ \textup{\uppercase\expandafter{\romannumeral1}}}}_{\widetilde{g}_\mathrm{L,R}}}\right]
\\
&- \frac{1}{2\pi}\log\left(\frac{\langle\widetilde{\phi}^2\rangle}{\langle\widetilde{\phi}^3\rangle} \right) \left[{b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{g_\mathrm{R}}} + \frac{1}{3}{b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{\widetilde{g}_\mathrm{L+R}}}\right]
\\
&
- \frac{1}{2\pi}\log\left(\frac{\langle\widetilde{\nu}_\mathrm{R}^1\rangle}{\langle\widetilde{\phi}^2\rangle} \right) \left[{b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{R}}} + \frac{1}{3}{b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{\widetilde{g}_\mathrm{L+R}}}\right],
\end{split}\end{aligned}$$ with the following known parameters at the $m_Z$ scale ($\sim 91.2$ GeV) [@Patrignani:2016xqp] $$\begin{aligned}
\begin{split}
\label{betti2}
\alpha_{g_\mathrm{C}}^{-1}(m_{{Z}})&\sim 8.5, \\
\alpha_{g_\mathrm{L}}^{-1}(m_{{Z}})&=\sin^2(\theta_W)\cdot 128\sim 29.6, \\
\alpha_{\widetilde{g}_\mathrm{Y}}^{-1}(m_{{Z}})&=\cos^2(\theta_W)\cdot 128\sim 98.4. \end{split}\end{aligned}$$ As we have more than three unknowns, the scales cannot be solved for uniquely, but are functions of $\log(\langle\widetilde{\phi}^2\rangle/\langle\widetilde{\phi}^3\rangle)$ and $\log(\langle\widetilde{\nu}_\mathrm{R}^1\rangle/\langle\widetilde{\phi}^2\rangle)$. If we take, for example, the scenario of having no hierarchies between these three scales, $$\label{scenario11}
\langle\widetilde{\phi}^3\rangle \sim \langle\widetilde{\phi}^2\rangle \sim \langle\widetilde{\nu}_\mathrm{R}^1\rangle \sim m_{\mathrm{soft}},$$ we end up with the following values $$\begin{aligned}
\begin{split}
\label{eq:finalscales}
m_{\mathrm{soft}} & \sim 8.8\cdot 10^{10}\;\mathrm{GeV}, \\
v & \sim 4.9\cdot 10^{17}\;\mathrm{GeV}, \\
\alpha^{-1}_{g}(v)& \sim 31.5, \end{split}\end{aligned}$$ where hence the unified gauge coupling satisfies the perturbativity constraint, the GUT scale is below $\mathrm{M}_\mathrm{Planck}$ and the soft scale is well separated from both the GUT scale and the EW scale. Note that while the hierarchy between the GUT scale and the soft SUSY-breaking scale is stable with respect to radiative corrections, the hierarchy between the EW scale and the soft SUSY-breaking scale needs to be finely tuned.
![ [**Left panel:**]{} *Figure showing that the $\langle\widetilde{\phi}^3\rangle$ scale is minimised when there is no hierarchy between the soft scales, i.e. where both lines meet in the lower right corner. The purple (solid) line corresponds to VEVs for which the gauge couplings run down to the measured standard model values. The grey (dashed) line corresponds to the case of no hierarchy between the VEVs. Hence, the optimal choice corresponds to $\langle\widetilde{\phi}^3\rangle\sim \langle\widetilde{\phi}^2\rangle\sim \langle\widetilde{\nu}_\mathrm{R}^1\rangle$ and as such the scalar content in the intermediate regions will not affect the running of the gauge couplings.*\
[**Right panel:**]{} *RG evolution of the gauge couplings for the scenario where there is no hierarchy between the three intermediate scales. To match the gauge couplings measured at the EW scale, the soft scale ends up at $8.8\cdot10^{10}$ GeV and the GUT scale at $4.9\cdot10^{17}$ GeV, i.e. we end up with a distinct hierarchy between all three scales.* []{data-label="fig:RGE"}](scales.pdf){width="1.05\linewidth"}
![ [**Left panel:**]{} *Figure showing that the $\langle\widetilde{\phi}^3\rangle$ scale is minimised when there is no hierarchy between the soft scales, i.e. where both lines meet in the lower right corner. The purple (solid) line corresponds to VEVs for which the gauge couplings run down to the measured standard model values. The grey (dashed) line corresponds to the case of no hierarchy between the VEVs. Hence, the optimal choice corresponds to $\langle\widetilde{\phi}^3\rangle\sim \langle\widetilde{\phi}^2\rangle\sim \langle\widetilde{\nu}_\mathrm{R}^1\rangle$ and as such the scalar content in the intermediate regions will not affect the running of the gauge couplings.*\
[**Right panel:**]{} *RG evolution of the gauge couplings for the scenario where there is no hierarchy between the three intermediate scales. To match the gauge couplings measured at the EW scale, the soft scale ends up at $8.8\cdot10^{10}$ GeV and the GUT scale at $4.9\cdot10^{17}$ GeV, i.e. we end up with a distinct hierarchy between all three scales.* []{data-label="fig:RGE"}](RGEplot2.pdf){width="1.2\linewidth"}
Let us investigate whether the introduction of a hierarchy between $\langle\widetilde{\phi}^3\rangle$, $\langle\widetilde{\phi}^2\rangle$ and $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$ can lower the soft scale $\langle\widetilde{\phi}^3\rangle$. By solving for $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$ in Eq. and inserting all known values, we have the equation $$\begin{aligned}
\nonumber
\langle\widetilde{\nu}_\mathrm{R}^1\rangle=&m_{Z}\exp\Bigg\{20.69 - \frac{1}{19}\log\left(\frac{\langle\widetilde{\phi}^3\rangle}{\langle\widetilde{\phi}^2\rangle}\right)\Big[4b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{g_\mathrm{C}} - 9b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{g_\mathrm{L}}
\\&
+ {3}{b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{g_\mathrm{R}}}+ {b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{\widetilde{g}_\mathrm{L+R}}} \Big]
- \frac{1}{19}\log\left(\frac{\langle\widetilde{\phi}^3\rangle}{\langle\widetilde{\phi}^2\rangle}\right)
\nonumber
\\&
\left[4b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{C}} - 9b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{L}} + {3}{b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{R}}}+ {b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{\widetilde{g}_\mathrm{L+R}}} \right]
\Bigg\}.
\label{equationForM3}\end{aligned}$$
The $b$-values will vary depending on the scalar field content with the extreme values presented in Appendix \[sec:betabeta\]. To minimise the argument of the exponential (and thereby minimising the value of $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$), we should maximise the values of $b^{{ \textup{\uppercase\expandafter{\romannumeral2}}},{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{C}}$, $b^{{ \textup{\uppercase\expandafter{\romannumeral2}}},{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{R}}$ and $b^{{ \textup{\uppercase\expandafter{\romannumeral2}}},{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{\widetilde{g}_\mathrm{L+R}}$, while minimising $b^{{ \textup{\uppercase\expandafter{\romannumeral2}}},{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{L}}$. This occurs when including all scalars apart from the left-handed doublets $Q_\mathrm{L}^{1,2,3}$, $E_\mathrm{L}^{1,2,3}$ and $H^3$, in both region ${ \textup{\uppercase\expandafter{\romannumeral2}}}$ and ${ \textup{\uppercase\expandafter{\romannumeral3}}}$. In that case the values are $$\begin{aligned}
\nonumber
b^{{ \textup{\uppercase\expandafter{\romannumeral2}}},{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{C}}=-\frac{13}{3},\;\;\;\;
b^{{ \textup{\uppercase\expandafter{\romannumeral2}}},{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{L}}=-\frac{2}{3},\\
b^{{ \textup{\uppercase\expandafter{\romannumeral2}}},{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{R}}=\frac{4}{3},\;\;\;\;
b^{{ \textup{\uppercase\expandafter{\romannumeral2}}},{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{\widetilde{g}_\mathrm{L+R}}=\frac{40}{3}.
\label{eq:bvalues}\end{aligned}$$ When ranging over various hierarchies using the $b$-values in , we see that the scale of $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$ decreases as the hierarchy between $\langle\widetilde{\phi}^2\rangle$ and $\langle\widetilde{\phi}^3\rangle$ increases. The soft scale $\langle\widetilde{\phi}^3\rangle$, on the other hand, is minimised when it is equal to $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$, i.e. when there are no hierarchies, as shown in Fig. \[fig:RGE\] (left), by which we conclude that Eq. is in fact the optimal scenario in the sense that it provides the strongest hierarchy between the GUT scale and the soft SUSY-breaking scale. In Fig. \[fig:RGE\] (right) we show the evolution of the gauge couplings for this scenario.
It is important to mention that these scales are obtained from gauge couplings evolved to one-loop accuracy but matched at tree-level, where one-loop matching conditions could introduce significant corrections, due to the many fields involved, as indicated in Ref. [@Braathen:2018]. As the resulting scales could be sensitive to potentially significant threshold corrections, we are careful not to draw any strong conclusions at this point.
Furthermore, there is a possibility for lowering the soft scale by relaxing the $\mathbb{Z}_3$ symmetry at the GUT scale, with gauge unification instead happening at the $\mathrm{E}_6$ level. In fact, as was demonstrated in [@Chakrabortty:2008zk], a non-universal gauge coupling at the GUT breaking scale may arise from corrections to the gauge kinetic terms induced by dimension 5 operators, emerging due to higher dimensional $\mathrm{E}_6$ representations. This would also open up the possibility for the emergence of new gauge bosons at, or at least close to, the TeV scale. We leave the question about a significance of such effects and its phenomenological implications for a further study.
Summary {#sec:conclusions}
=======
Here, we would like to summarise the basic features of the LRCF-symmetric SHUT theory considered in this paper:
- In contrast to previous GUT scale formulations based on gauge trinification, all three fermion generations are unified into a single $(\bm{27},\bm{3})$-plet of ${\mathrm{SU}(3)_{\mathrm{F}}}\times \mathrm{E}_6$, and no copies of any fundamental $\mathrm{E}_6$ reps are required for its consistent breaking down to the gauge symmetry of the SM. The considered ${\mathrm{SU}(3)_{\mathrm{F}}}\times \mathrm{E}_6$ symmetry can be embedded into $\mathrm{E}_8$, motivating the addition of $(\bm{1},\bm{8})$ and $(\bm{78},\bm{1})$ multiplets corresponding to four ${\mathrm{SU}(3)_{\mathrm{}}}$-octet reps. The gauge couplings are enforced to unify by means of a cyclic permutation symmetry $\mathbb{Z}_3$ acting on the trinification subgroup of the LRCF-symmetry in the same way as in the Glashow’s formulation.
- The chiral-adjoint sector $\bm{\Delta}_{\rm F}^a=(\bm{1},\bm{8})$ and $\bm{\Delta}_{\rm L,R,C}^a\subset(\bm{78},\bm{1})$ is necessary for a consistent breaking of the LRCF-symmetry down to the SM gauge symmetry in the softly-broken SUSY formulation of the theory while none of the adjoint fields remain at the EW scale. In our model, the fields developing VEVs at lower energies (the tri-triplets) happen to have the mass terms of $\mathcal{O}(\mathrm{m}_\mathrm{soft})$, while the fields whose VEVs spontaneously break the high-scale SHUT LRCF-symmetry (the adjoints) have their GUT scale mass term in the superpotential. Hence, our model does not exhibit an analogue of the $\mu$-problem in the MSSM.
- With the first symmetry breaking being triggered at the GUT scale by VEVs in the adjoint (octet) scalars, mass terms in the fundamential ($\bm{L},\,\bm{Q}_{\rm L},\,\bm{Q}_{\rm R}$ tri-triplet) sector are forbidden. This means that the SM-like quarks and leptons remain massless until EWSB.
- In the SHUT model, all possible tree-level masses for fermions come from a single term in the superpotential, $\bm{L}^i\bm{Q}_{\rm L}^j\bm{Q}_{\rm R}^k\epsilon_{ijk}$. As we have seen, only two generations of would-be SM quarks get such contributions to their masses. As such, the model offers a starting point for a mechanism explaining the mass hierarchies of the SM, where, for example, the charged leptons are all light as they have no allowed tree-level masses and instead attain their masses radiatively (i.e. via loop-induced threshold corrections). Also, with three Higgs doublets at low energies, the model has Cabbibo quark mixing at tree-level, while radiatively generated (and RG evolved) Yukawa interactions open the possibility of reproducing the complete structure.
- The symmetry breaking scales below the GUT scale (including the EW scale) are fully determined by the dynamics of the soft SUSY-breaking interactions and are thus naturally protected from the GUT scale radiative corrections. A particularly relevant multi-stage symmetry breaking scheme in the SHUT theory down to the SM-like gauge effective theory has been shown in Fig. \[fig:1112abc\].
- The LRCF-symmetric theory contains an accidential ${\mathrm{U}(1)_{\mathrm{B}}}$ baryon symmetry, by which the proton remains stable to all orders in perturbation theory. Other accidental ${\mathrm{U}(1)_{\mathrm{W}}}$ and LR-parity symmetries can be (softly) broken in the low-energy EFT ensuring there being no massless charged leptons below the EWSB scale, and allowing the breaking of ${\mathrm{SU}(2)_{\mathrm{R}}}$ and ${\mathrm{SU}(2)_{\mathrm{L}}}$ symmetries at different energy scales, respectively.
- The smallest possible hierarchy between the EW scale and the soft scale, and the largest possible hierarchy between the soft scale and the GUT scale, occurs as the VEVs of $\widetilde{\phi}^3$, $\widetilde{\phi}^2$ and $\widetilde{\nu}_\mathrm{R}^1$ are all put at the same scale. For this scenario, the soft scale ends up at $\sim9\cdot10^{10}$ GeV and the GUT scale at $\sim5\cdot10^{17}$ GeV. However, these numbers do not take into account potentially large one-loop threshold corrections.
- While our estimates have shown a potential agreement with the SM particle spectrum, and in particular the possibility to lift the top-charm mass degeneracy via quantum effects, it is not less true that the large $\langle \widetilde{\phi}^3 \rangle$, $\langle \widetilde{\phi}^2 \rangle$ and $\langle \widetilde{\nu}_\mathrm{R}^1 \rangle$ VEVs introduce fine-tuning in the scalar sector in order to satisfy the requirement of light Higgs doublets and possibly light squarks. We have pointed out that to solve this issue we need to relax the $\mathbb{Z}_3$ symmetry and transfer the unification of gauge interactions to the ${\mathrm{E}_{6}}$ level, which is left for a future work.
Given the above properties, the SHUT model offers interesting new possibilities for deriving the structure and parameters of the SM from the GUT scale physics. This is a good motivation for investigations of this model, its multi-scale symmetry breaking patterns, loop-level matching and RG flow. Among the first natural steps would be to uncover some of the features of the simplest SM-like low-energy EFTs in a symmetry-based study without invoking the full-fledged radiative analysis of the SHUT theory. The EFT scenarios studied in this work pave the ground for further phenomenological studies of trinification based GUTs and move beyond the most common issues of such theories in the past.
The authors would like to thank N.-E. Bomark, A. E. Cárcamo Hernández, C. Herdeiro, S. Kovalenko, W. Porod, J. Rathsman, H. Serodio and F. Staub for insightful discussions in the various stages of this work. J. E. C.-M., J.W. and R.P. thank Prof. C. Herdeiro for support of the project and hospitality during their visits at Aveiro University. J. E. C.-M. was partially supported by the Crafoord Foundation and Lund University. A.P.M. was initially funded by FCT grant SFRH/BPD/97126/2013. A.P.M. is supported by Fundação para a Ciência e a Tecnologia (FCT), within project UID/MAT/04106/2019 (CIDMA) and by national funds (OE), through FCT, I.P., in the scope of the framework contract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-Law 57/2016, of August 29, changed by Law 57/2017, of July 19. A.P.M. is also supported by the *Enabling Green E-science for the Square Kilometer Array Research Infrastructure* (ENGAGESKA), POCI-01-0145-FEDER-022217, and by the project *From Higgs Phenomenology to the Unification of Fundamental Interactions*, PTDC/FIS-PAR/31000/2017. R.P., A.O. and J.W. were partially supported by the Swedish Research Council, contract numbers 621-2013-428 and 2016-05996. R.P. was also partially supported by CONICYT grants PIA ACT1406 and MEC80170112. The work by R.P. was partially supported by the Ministry of Education, Youth and Sports of the Czech Republic project LT17018, as well as by the COST Action CA15213. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 668679). This work is also supported by the CIDMA project UID/MAT/04106/2013.
Symmetry breaking schemes and charges {#sec:Symmetries}
=====================================
In this appendix we provide a summary of the SSB scheme from the high-scale GUT symmetry down to that of the SM.
Breaking path and generators
----------------------------
The breaking path from the GUT symmetry down to a LR-symmetric effective theory reads
$$\begin{aligned}
&&\left[ {\mathrm{SU}(3)_{\mathrm{C}}} \times {\mathrm{SU}(3)_{\mathrm{L}}} \times {\mathrm{SU}(3)_{\mathrm{R}}}\right] \rtimes \mathbb{Z}^{\rm (LRC)}_3
\times \{{\mathrm{SU}(3)_{\mathrm{F}}} \times {\mathrm{U}(1)_{\mathrm{W}}} \times {\mathrm{U}(1)_{\mathrm{B}}} \} \nonumber \\
&& \overset{v,v_{\rm F}}{\to} \quad \; {\mathrm{SU}(3)_{\mathrm{C}}}\times \left[ {\mathrm{SU}(2)_{\mathrm{L}}} \times {\mathrm{SU}(2)_{\mathrm{R}}} \times
{\mathrm{U}(1)_{\mathrm{L}}} \times {\mathrm{U}(1)_{\mathrm{R}}} \right] \times \{{\mathrm{SU}(2)_{\mathrm{F}}} \times {\mathrm{U}(1)_{\mathrm{F}}} \times {\mathrm{U}(1)_{\mathrm{W}}} \times {\mathrm{U}(1)_{\mathrm{B}}} \} \nonumber \\
&& \overset{\langle \widetilde{\phi}^{3}\rangle}{\to} \quad {\mathrm{SU}(3)_{\mathrm{C}}} \times [{\mathrm{SU}(2)_{\mathrm{L}}} \times
{\mathrm{SU}(2)_{\mathrm{R}}}] \times \mathrm{U}(1)_{\mathrm{L}+\mathrm{R}} \times
\{ {\mathrm{SU}(2)_{\mathrm{F}}}\times\mathrm{U}(1)_{\mathrm{S}}
\;\times\; {\mathrm{U}(1)_{\mathrm{S'}}} \times {\mathrm{U}(1)_{\mathrm{B}}}\} \equiv G_{3221\{21\}}\,, \label{eq:brk2}\end{aligned}$$
where global symmetries (including the accidental ones) are indicated by $\{ \cdots \}$. The generators of the ${\mathrm{U}(1)_{\mathrm{}}}$ groups after the GUT SSB are $$\label{generators0}
\begin{aligned}
{\scalebox{0.95}{$
{T_{\mathrm{L}}^{8}}\,, \qquad {T_{\mathrm{R}}^{8}}\,, \qquad {T_{\mathrm{F}}^{8}}\,, \qquad {T_{\mathrm{W}}^{}} \,, \qquad {T_{\mathrm{B}}^{}}\,,
$}}
\end{aligned}$$ whereas after the $\langle \widetilde{\phi}^{3}\rangle$ VEV we have $$\label{generators1}
\begin{aligned}
{T_{\mathrm{L+R}}^{}} = {T_{\mathrm{L}}^{8}} &+ {T_{\mathrm{R}}^{8}} \,, \quad {T_{\mathrm{S}}^{}} = {T_{\mathrm{L}}^{8}} - {T_{\mathrm{R}}^{8}} - 2 {T_{\mathrm{F}}^{8}} \,, \\
&{T_{\mathrm{S'}}^{}} = {T_{\mathrm{L}}^{8}} - {T_{\mathrm{R}}^{8}} + \tfrac{2}{\sqrt{3}} {T_{\mathrm{W}}^{}} \,.
\end{aligned}$$ with normalization factors conveniently chosen to provide integer charges for leptons and scalar bosons.
Note that, according to the discussion in Sect. \[sec:SSBSU3\] the LR-parity can be explicitly broken in the soft SUSY-breaking sector and is therefore absent in the effective theory.
We may also place a VEV in $\widetilde{\phi}^{2}$ and $\widetilde{\nu}_{\mathrm{R}}^{1}$. In such a case the breaking scheme takes the form
$$\label{eq:brknuphi}
\begin{aligned}
G_{3221\{21\}}&\overset{\langle \widetilde{\nu}_{\mathrm{R}}^{1}\rangle\,, \langle \widetilde{\phi}^{2} \rangle}{\longrightarrow} \,
{\mathrm{SU}(3)_{\mathrm{C}}}\times {\mathrm{SU}(2)_{\mathrm{L}}} \times \mathrm{U}(1)_{\mathrm{Y}} \, \\
&\times\, \{\mathrm{U}(1)_{\mathrm{T}} \,\times\,
\mathrm{U}(1)_{\mathrm{T'}} \times {\mathrm{U}(1)_{\mathrm{B}}}\} \,,
\end{aligned}$$
where the generators of ${\mathrm{U}(1)_{\mathrm{Y}}}$, ${\mathrm{U}(1)_{\mathrm{T}}}$ and ${\mathrm{U}(1)_{\mathrm{T'}}}$ read $$\begin{aligned}
\nonumber
&&{T_{\mathrm{Y}}^{}} = -\tfrac{1}{\sqrt{3}}{\left(}{T_{\mathrm{L+R}}^{}} + \sqrt{3} {T_{\mathrm{R}}^{3}} {\right)}\,, \quad
{T_{\mathrm{T}}^{}} = {T_{\mathrm{R}}^{3}} + \tfrac{1}{3\sqrt{3}}{T_{\mathrm{S}}^{}} - \tfrac{2}{3}{T_{\mathrm{F}}^{3}} \,, \\
&&\qquad \qquad {T_{\mathrm{T'}}^{}} = {T_{\mathrm{S'}}^{}} + \tfrac{1}{3}{T_{\mathrm{S}}^{}} -\tfrac{2}{\sqrt{3}} {T_{\mathrm{F}}^{3}} \,.
\label{generators3}\end{aligned}$$
Quantum numbers
---------------
In this section we present the representations and charges of the light states after each breaking step. We consider as light states all fields that are decoupled from the GUT scale after the first SSB step.
In what follows, the Higgs bi-doublets are referred to as ${H}^{1,2,3}$, the singlet Higgs-lepton fields denoted as $\phi^{1,2,3}$ and the lepton doublets as $E_{\mathrm{L,R}}^{1,2,3}$, while the quark multiplets split up into $\mathcal{Q}_{\mathrm{L,R}}^{1,2,3}$ and $D_{\rm L,R}^{1,2,3}$, where $\mathcal{Q}$ are the $3 \times 2$ blocks and $D$ the $3 \times 1$ blocks. The superscript $1,2,3$ is the generation number. Whenever convenient we will adopt a simplifying notation according to $$\label{1}
\begin{aligned}
H^{3} \rightarrow h\,, \\
E_{\rm{L,R}}^{3} \rightarrow \mathcal{E}_{\rm{L,R}}\,, \\
\mathcal{Q}_{\rm{L,R}}^{3} \rightarrow q_{\rm{L,R}}\,,\\
\end{aligned} \qquad
\begin{aligned}
\phi^{3} \rightarrow \varphi\,, \\
D_{\rm{L,R}}^{3} \rightarrow \mathcal{B}_{\rm{L,R}}\,,\\
X^{1,2} \rightarrow X^f\,,
\end{aligned}$$ where $f$ is a family index running over the first two generations with $X$ representing any of such ${\mathrm{SU}(2)_{\mathrm{F}}}$ doublets.
The quantum numbers of the light eigenstates after the $v$ and $v_{\rm F}$ VEVs are given in Tab. \[Table:EFT-content\] while those of the model after $\widetilde{\phi}^{3}$ VEV are shown in Tab. \[Table:EFT-content-phi\]. In Tab. \[Table:EFT-content-snu1Phi\] we show the charges of the SM-like EFT after the $\widetilde{\nu}_{\rm R}^{1}$ and $\widetilde{\phi}^{2}$ VEVs which may either occur simultaneously or at separate scales. Note that the ${\left \langle \varphi \right \rangle }$ VEV enables mixing between the first and second generations of singlet (s)quarks. For example, it allows fermion mass terms of the form $ m_D D_{\rm L}^f D_{\rm R}^{f'} \varepsilon_{f f'}$.
Particle masses in the high-scale theory {#sec:masses}
========================================
Scalar spectra and minimisation conditions
------------------------------------------
The extremizing conditions obtained after taking the first derivatives of the scalar potential of the SHUT model can be solved, e.g. w.r.t. the soft parameters $m^2_{\bm{78}}$ and $m^2_{\bm{1}}$ from where we obtain $$\begin{aligned}
\nonumber
m^2_{\bm{78}} = & - b_{\bm{78}} + \tfrac{v}{12} \left( \sqrt{6} A_{\bm{78}} + 3 \sqrt{6} C_{\bm{78}} -
v \lambda^2_{\bm{78}} \right) \\ &+
\tfrac{\sqrt{6}}{4} v \lambda_{\bm{78}} \mu_{\bm{78}} - \mu_{\bm{78}}^2 \,, \label{eq:tadE8} \\
m^2_{\bm{1}} = & - b_{\bm{1}} + \tfrac{{v_{\mathrm{F}}}}{12} \left( \sqrt{6} A_{\bm{1}} - {v_{\mathrm{F}}} \lambda^2_{\bm{1}} \right) +
\tfrac{\sqrt{6}}{4} {v_{\mathrm{F}}} \lambda_{\bm{1}} \mu_{\bm{1}} - \mu_{\bm{1}}^2\,.\nonumber\end{aligned}$$ The minimisation conditions are then used in the Hessian matrix whose eigenvalues corresponding to the fundamental and adjoint scalar sectors are shown in Tabs. \[table:SpecFund\] and \[table:SpecAdj\], respectively. Note that, for simplicity, we use the LR-symmetric case with $A_{\bar{\mathrm{G}}} = A_\mathrm{G}$. The branching rule for a fundamental representation of ${\mathrm{SU}(3)_{\mathrm{A}}}$, ${\rm A = L,R,F}$ when it is broken down to reads $$\label{eq:3to21}
\bm{3} \rightarrow \bm{2}_1 \oplus \bm{1}_{-2}\,,$$ where, up to an overall normalization factor, the subscripts represent the ${\mathrm{U}(1)_{\mathrm{A}}}$ charge. Therefore, after the SSB, the eigenstates shown in Tab. \[table:SpecFund\] form representations of the $G_{32211\lbrace 21 \rbrace}$ symmetry given in Eq. and transform as singlets, doublets, bi-doublets and tri-doublets under the ${\mathrm{SU}(2)_{\mathrm{L,R,F}}}$ symmetries, as schematically represented by the blocks in Eq. [^5]. The LR-parity discussed in Sect. \[sec:tritri\] yields identical masses for the ${\mathrm{SU}(2)_{\mathrm{L}}}$ and ${\mathrm{SU}(2)_{\mathrm{R}}}$ eigenstates at the trinification SSB scale.
The adjoint scalars $\widetilde{\Delta}^a_{\rm{A=L,R,F}}$ are complex octets whose branching rule is given by $$\label{eq:8to3221}
\bm{8} \rightarrow \bm{3}_0 \oplus \bm{2}_1 \oplus \bm{2}_{-1} \oplus \bm{1}_0\,,$$ where the complex octet is a reducible representation while its real and imaginary parts are the irreducible representations. As such, we end up with two real triplets, two real singlets and two complex doublets and their complex conjugates after the SSB. Each broken symmetry provides four Goldstone degrees of freedom out of which eight correspond to breaking of the local symmetries whereas four of them – to the global ones. While the triplet mass eigenstates, $\bm{3}_0$, can be written as $$\begin{aligned}
\nonumber
\renewcommand{1.4}{1.3}
&\widetilde{\mathcal{T}}_{\rm A} \equiv \dfrac{1}{\sqrt{2}} \mleft(
\begin{array}{ccc}
\rm{Re}[\widetilde{\Delta}^1_{\rm A}] - {\mathrm i}\rm{Re}[\widetilde{\Delta}^2_{\rm A}]\\
\sqrt{2} \rm{Re}[\widetilde{\Delta}^3_{\rm A}]\\
\rm{Re}[\widetilde{\Delta}^1_{\rm A}] + {\mathrm i}\rm{Re}[\widetilde{\Delta}^2_{\rm A}]
\end{array}
\mright)\;,$$ $$\begin{aligned}
\renewcommand{1.4}{1.3}
&\widetilde{\mathcal{T}}_{\rm A}^{\prime} \equiv \dfrac{1}{\sqrt{2}} \mleft(
\begin{array}{ccc}
\rm{Im}[\widetilde{\Delta}^1_{\rm A}] - {\mathrm i}\rm{Im}[\widetilde{\Delta}^2_{\rm A}]\\
\sqrt{2} \rm{Im}[\widetilde{\Delta}^3_{\rm A}]\\
\rm{Im}[\widetilde{\Delta}^1_{\rm A}] + {\mathrm i}\rm{Im}[\widetilde{\Delta}^2_{\rm A}]
\end{array}
\mright)\;,\label{eq:Triplets}
$$ the two real singlets $\bm{1}_0$ read $$\begin{aligned}
\label{eq:Singlets}
\widetilde{\mathcal{S}}_{\rm A} \equiv \rm{Re}[\widetilde{\Delta}^8_{\rm A}]\;,\; \widetilde{\mathcal{S}}_{\rm A}^{\prime} \equiv
\rm{Im}[\widetilde{\Delta}^8_{\rm A}]\,.\end{aligned}$$
Finally, there are two complex doublets from the real part of $\widetilde{\Delta}^a_{\rm{L,R,F}}$, transforming as $\bm{2}_{-1}$ and $\bm{2}_{1}$ $$\begin{aligned}
\nonumber
\renewcommand{1.4}{1.3}
&\widetilde{G}_{\rm A}\equiv \dfrac{1}{\sqrt{2}} \mleft(
\begin{array}{ccc}
-\rm{Re}[\widetilde{\Delta}^6_{\rm A}] - {\mathrm i}\rm{Re}[\widetilde{\Delta}^7_{\rm A}]\\
\rm{Re}[\widetilde{\Delta}^4_{\rm A}] + {\mathrm i}\rm{Re}[\widetilde{\Delta}^5_{\rm A}]
\end{array}
\mright) \;,\; $$ $$\begin{aligned}
\renewcommand{1.4}{1.3}
&\widetilde{G}_{\rm A}^*= \dfrac{1}{\sqrt{2}} \mleft(
\begin{array}{ccc}
-\rm{Re}[\widetilde{\Delta}^6_{\rm A}] + {\mathrm i}\rm{Re}[\widetilde{\Delta}^7_{\rm A}]\\
\rm{Re}[\widetilde{\Delta}^4_{\rm A}] - {\mathrm i}\rm{Re}[\widetilde{\Delta}^5_{\rm A}]
\end{array}
\mright) \;,\label{eq:Doublets}
$$ and two complex doublets from the imaginary part of $\widetilde{\Delta}^a_{\rm{L,R,F}}$, transforming as $\bm{2}_{-1}$ and $\bm{2}_{1}$ $$\begin{aligned}
\nonumber
\renewcommand{1.4}{1.3}
&\mathcal{H}_{\rm A} \equiv \dfrac{1}{\sqrt{2}} \mleft(
\begin{array}{ccc}
-\rm{Im}[\widetilde{\Delta}^6_{\rm A}] - {\mathrm i}\rm{Im}[\widetilde{\Delta}^7_{\rm A}]\\
\rm{Im}[\widetilde{\Delta}^4_{\rm A}] + {\mathrm i}\rm{Im}[\widetilde{\Delta}^5_{\rm A}]
\end{array}
\mright) \;,\; \\
\renewcommand{1.4}{1.3}
&\mathcal{H}_{\rm A}^* = \dfrac{1}{\sqrt{2}} \mleft(
\begin{array}{ccc}
-\rm{Im}[\widetilde{\Delta}^6_{\rm A}] + {\mathrm i}\rm{Im}[\widetilde{\Delta}^7_{\rm A}]\\
\rm{Im}[\widetilde{\Delta}^4_{\rm A}] - {\mathrm i}\rm{Im}[\widetilde{\Delta}^5_{\rm A}]
\end{array}
\mright) \;,\label{eq:Doublets2}
$$ respectively, where the subscript $-1$ stands for the doublet with negative $\mathrm{T}^8$ eigenvalue.
### Scalar mass spectrum {#Sec:VacStab}
It is possible to write the minimisation conditions in a convenient way by recasting the scalar masses. In particular, the fundamental scalar masses can be collectively written as $$\begin{aligned}
\label{eq:27scalars}
m^2_{\widetilde{\varphi}_i} =m^2_{\bm{27}} + c_1^i A_{\rm G} v + c_2^i A_{\rm F} v_{\rm F} \,,\end{aligned}$$ where $c^i_{1,2}$ are constants with index $i$ running over all fundamental scalar eigenstates. For simplicity, the soft SUSY-breaking parameters and the family breaking VEV can be redefined in terms of a dimensionless parameter times a common scale $v$ as follows $$\begin{aligned}
\label{eq:RelateTov}
v_{\rm F} = \beta v\,,~
m_{\bm{27}}^{2} = \alpha_{\bm{27}} v^{2}\,, ~
A_{\rm G} = \sigma_{\rm G}v\,,~
A_{\rm F} = \sigma_{\rm F}v\,,\end{aligned}$$ where, in the limit of low-scale SUSY-breaking, $\alpha_{\bm{27}},\,\sigma_{\rm G},\,\sigma_{\rm F}\ll 1$ and $\beta \sim {\cal O}\left( 1 \right)$ such that both gauge and family SSBs occur simultaneously at the GUT scale. Eq. allows one to rewrite the scalar masses in terms of the common scale $v$ $$\begin{aligned}
\label{eq:27recast}
m^2_{\widetilde{\varphi}_i} = v^2 \left( \alpha_{\bm{27}} + c^i_1 \sigma_{\rm G} + c^i_2 \beta \sigma_{\rm F} \right)
\equiv v^2 \omega_{\widetilde{\varphi}_i} \,,\end{aligned}$$ such that $\omega_{\widetilde{\varphi}_i} \ll 1$. As the expression for the fundamental scalar masses contains three independent parameters, we may characterize the entire spectrum by the following three definitions $$\begin{aligned}
\label{eq:ScalarPars}
\omega_{\widetilde{H}^{(3)}} \equiv \xi\,,~ \omega_{\widetilde{E}_{\rm L,R}^{(1,2)}} \equiv \delta\,,~\omega_{\widetilde{H}^{(1,2)}} \equiv \kappa\,,\end{aligned}$$ where the dimensionless parameters $\xi$, $\delta$ and $\kappa$ can span the entire spectrum by laying in the interval of 0 to 1, as the common mass scale is chosen to be the largest scale in the model, i.e. the GUT scale $v$. With this, we can recast the scalar mass terms in the resulting EFT as $$\label{eq:RedefMasses}
\begin{aligned}
m_{\widetilde{H}^{(3)}}^{2} &= v^{2}\xi\,, \\
m_{\widetilde{E}_{\rm L,R}^{(3)}}^{2} &= v^{2}\left(\delta+\xi-\kappa\right)\,, \\
m_{\widetilde{\phi}^{(3)}}^{2} &= v^{2}\left(2\delta+\xi-2\kappa\right)\,, \\
m_{\widetilde{\mathcal{Q}}_{\rm L,R}^{(3)}}^{2} &= \tfrac{1}{3}v^{2}\left(\delta+3\xi-\kappa\right)\,, \\
m_{\widetilde{D}_{\rm L,R}^{(3)}}^{2} &= \tfrac{1}{3}v^{2}\left(4\delta+3\xi-4\kappa\right)\,, \\
\end{aligned} \qquad
\begin{aligned}
m_{\widetilde{H}^{(1,2)}}^{2} &= v^{2}\kappa\,, \\
m_{\widetilde{E}_{\rm L,R}^{(1,2)}}^{2} &= v^{2}\delta\,, \\
m_{\widetilde{\phi}^{(1,2)}}^{2} &= v^{2}\left(2\delta-\kappa\right)\,, \\
m_{\widetilde{\mathcal{Q}}_{\rm L,R}^{(1,2)}}^{2} &= \tfrac{1}{3}v^{2}\left(\delta+2\kappa\right)\,, \\
m_{\widetilde{D}_{\rm L,R}^{(1,2)}}^{2} &= \tfrac{1}{3}v^{2}\left(4\delta-\kappa\right)\,.
\end{aligned}$$ Using Eq. the general set of conditions necessary to set the positivity of the fundamental scalar mass spectrum reads $$\begin{aligned}
\nonumber
\kappa > 0\, &\wedge& \, \Big[ \left( \dfrac{\kappa}{2} \leq \delta \leq \kappa \, \wedge \, \xi > -2 \delta + 2 \kappa \right)\, \\
&\vee& \,
\left( \delta > \kappa \, \wedge \, \xi > 0 \right) \Big]\,.
\label{eq:LQQcond}\end{aligned}$$ Following the same procedure, we may redefine the parameters of the adjoint sector in terms of the GUT SSB scale $v$ as follows $$\label{eq:RedefAdj}
\begin{aligned}
b_{\bm{1}} &= \tau_{\bm{1}} v^{2}\,, \\
b_{\bm{78}} &= \tau_{\bm{78}} v^{2}\,, \\
\mu_{\bm{1}} &= \alpha_{\bm{1}} v\,, \\
\mu_{\bm{78}} &= \alpha_{\bm{78}} v\,, \\
\end{aligned} \qquad
\begin{aligned}
A_{\bm{1}} &= \sigma_{\bm{1}} v\,, \\
A_{\bm{78}} &= \sigma_{\bm{78}} v\,, \\
C_{\bm{78}} &= \theta_{\bm{78}} v\,.
\end{aligned}$$ Substituting Eqs. in Tab. \[table:SpecAdj\] and, similarly to Eq. , choosing $$\begin{aligned}
\nonumber
&\omega_{\tilde{\mathcal{T}}_{\rm F}} \equiv \eta_{\rm F} \,,~ \omega_{\mathcal{H}_{\rm F}} \equiv \rho_{\rm F} \,,
~\omega_{\tilde{\mathcal{T}}^{\prime}_{\rm F}} \equiv \eta^{\prime}_{\rm F}\,,~ \omega_{\tilde{\mathcal{T}}_{\rm L,R}} \equiv \eta\,, \\
&~~~~~ \omega_{\tilde{\mathcal{H}}_{\rm L,R}} \equiv \rho\,,~ \omega_{\tilde{\Delta}^{\prime}_{\rm C}} \equiv \vartheta\,,\label{eq:ScalarPars}\end{aligned}$$ where now $\omega_{\tilde{\varphi}_i \neq {{\scalebox{0.7}{$\mathcal{H}_{\rm F}$}}} } \sim O(1)$ since only $\mathcal{H}_{\rm F}$ does not contain large $\mathcal{F}$- and $\mathcal{D}$-term contributions. Solving the system of equations w.r.t $\sigma_{\bm{1}},\, \tau_{\bm{1}},\,
\alpha_{\bm{1}},\,\sigma_{\bm{78}},\, \tau_{\bm{78}},\, \alpha_{\bm{78}}$ we obtain $$\begin{aligned}
&m_{\tilde{\mathcal{T}}_{\rm F}}^{2} = \eta_{\rm F} v^{2}\,, \nonumber \\
&m_{\tilde{\mathcal{T}}^{\prime}_{\rm F}}^{2} = \eta^{\prime}_{\rm F} v^{2}\,, \nonumber \\
&m_{\tilde{\mathcal{S}}_{\rm F}}^{2} = \tfrac{1}{6} v^{2} \left(\beta^2 \lambda^2_{\bm{1}} - 2 \eta_{\rm F} \right)\,, \nonumber \\
&m_{\tilde{\mathcal{S}}^{\prime}_{\rm F}}^{2} = \tfrac{1}{6} v^{2} \left(\beta^2 \lambda^2_{\bm{1}} - 2 \eta^{\prime}_{\rm F} + 8 \rho_{\rm F} \right)\,, \nonumber \\
&m_{\mathcal{H}_{\rm F}}^{2} = \rho_{\rm F} v^{2}\,, \nonumber \\
&m_{\tilde{\Delta}_{\rm C}}^{2} = \tfrac{1}{12} v^{2} \left( 4 \eta - \lambda^2_{\bm{78}} \right)\,, \label{eq:RecastMassAdj} \\
&m_{\tilde{\mathcal{T}}_{\rm L,R}}^{2} = \eta v^{2}\,, \nonumber \\
&m_{\tilde{\mathcal{T}}^{\prime}_{\rm L,R}}^{2} = \tfrac{1}{4} v^2 \left( \lambda^2_{\bm{78}} + 6 g^2_{\rm U} + 12 \vartheta - 8 \rho \right)\,, \nonumber \\
&m_{\tilde{\mathcal{S}}_{\rm L,R}}^{2} = \tfrac{1}{6} v^2 \left( \lambda^2_{\bm{78}} - 2 \eta \right)\,, \nonumber \\
&m_{\tilde{\mathcal{S}}^{\prime}_{\rm L,R}}^{2} = \tfrac{1}{12} v^2 \left(\lambda^2_{\bm{78}} - 18 g^2_{\rm U} - 12 \vartheta + 24 \rho \right) \,, \nonumber \\
&m_{\mathcal{H}_{\rm L,R}}^{2} = \rho v^{2}\,, \nonumber \\
&m_{\tilde{\Delta}^{\prime}_{\rm C}}^{2} = \vartheta v^{2}\,. \nonumber \end{aligned}$$
The scalar field components of the gauge and family adjoint sectors are treated separately. Noting that $\rho_\mathrm{F} \ll 1$, the general stability condition for the masses of the family sector read $$\label{eq:Fcond}
\rho_{\rm F} \geq 0 \, \wedge \, \left( \eta^{\prime}_{\rm F} > 4 \rho_{\rm F} \, \wedge \, x > 2 \eta^{\prime}_{\rm F} - 8 \rho_{\rm F} \, \wedge \,
\eta_{\rm F} < \dfrac{x}{2} \right) \,,$$ where we have defined $\beta^2 \lambda^2_{\bm{1}} \equiv x > 0$. Finally, the positivity conditions for the gauge sector are $$\begin{aligned}
\nonumber
&& \eta > 0 \, \wedge \, 2 \eta < y < 4 \eta \, \wedge \, \vartheta > 0 \,\wedge \\
&& \dfrac{1}{24} \left( z - y + 12 \vartheta \right) < \rho < \dfrac{1}{8} \left( y + 6 z + 12 \vartheta \right)\,,\label{eq:LRcond}\end{aligned}$$ where we have defined $ \lambda^2_{\bm{78}} \equiv y > 0$ and $g^2_{\rm U} \equiv z > 0$. When conditions , and are simultaneously satisfied, the tree-level vacuum of the SHUT model is stable.
---- ------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------
8 $m^2_{\bm{27}} - \tfrac{1}{\sqrt{6}} \left( A_\mathrm{G} v + 2 A_\mathrm{F} {v_{\mathrm{F}}} \right) $ $\widetilde{\nu}_{\rm R}^{(3)}\,,
\widetilde{e}_{\rm R}^{(3)}\,, \widetilde{\nu}_{\rm L}^{(3)}\,, \widetilde{e}_{\rm L}^{(3)}$
2 $m^2_{\bm{27}} - \tfrac{1}{\sqrt{6}} \left( 4 A_\mathrm{G} v + 2 A_\mathrm{F} {v_{\mathrm{F}}} \right) $ $\widetilde{\phi}^{(3)}$
8 $m^2_{\bm{27}} + \tfrac{1}{\sqrt{6}} \left( 2 A_\mathrm{G} v - 2 A_\mathrm{F} {v_{\mathrm{F}}} \right) $ $H_{11}^{(3)}\,, H_{21}^{(3)} \, ,
H_{12}^{(3)}\,, H_{22}^{(3)}$
4 $m^2_{\bm{27}} - \tfrac{1}{\sqrt{6}} \left( 4 A_\mathrm{G} v - A_\mathrm{F} {v_{\mathrm{F}}} \right) $ $\widetilde{\phi}^{(1,2)}$
16 $m^2_{\bm{27}} - \tfrac{1}{\sqrt{6}} \left( A_\mathrm{G} v - A_\mathrm{F} {v_{\mathrm{F}}} \right) $ $\widetilde{\nu}_{\rm R}^{(1,2)}\,,
\widetilde{e}_{\rm R}^{(1,2)}\,, \widetilde{\nu}_{\rm L}^{(1,2)}\,, \widetilde{e}_{\rm L}^{(1,2)}$
16 $m^2_{\bm{27}} + \tfrac{1}{\sqrt{6}} \left( 2 A_\mathrm{G} v + A_\mathrm{F} {v_{\mathrm{F}}} \right) $ $H_{11}^{(1,2)}\,, H_{21}^{(1,2)} \, ,
H_{12}^{(1,2)}\,, H_{22}^{(1,2)}$
24 $m^2_{\bm{27}} + \tfrac{1}{\sqrt{6}} \left( A_\mathrm{G} v - 2 A_\mathrm{F} {v_{\mathrm{F}}} \right) $ $\widetilde{u}_{\rm L}^{(3)}\,,
\widetilde{d}_{\rm L}^{(3)}\,,\widetilde{u}_{\rm R}^{(3)}\,,\widetilde{d}_{\rm R}^{(3)}$
12 $m^2_{\bm{27}} - \tfrac{1}{\sqrt{6}}\left( 2 A_\mathrm{G} v + 2 A_\mathrm{F} {v_{\mathrm{F}}} \right) $ $\widetilde{D}_{\rm L}^{(3)}\,,
\widetilde{D}_{\rm R}^{(3)}$
48 $m^2_{\bm{27}} + \tfrac{1}{\sqrt{6}} \left( A_\mathrm{G} v + A_\mathrm{F}{v_{\mathrm{F}}} \right) $ $\widetilde{u}_{\rm L}^{(1,2)}\,,
\widetilde{d}_{\rm L}^{(1,2)}\,,\widetilde{u}_{\rm R}^{(1,2)}\,,\widetilde{d}_{\rm R}^{(1,2)}$
24 $m^2_{\bm{27}} - \tfrac{1}{\sqrt{6}} \left( 2 A_\mathrm{G} v - A_\mathrm{F} {v_{\mathrm{F}}} \right) $ $\widetilde{D}_{\rm L}^{(1,2)}\,,
\widetilde{D}_{\rm R}^{(1,2)}$
---- ------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------
: *Scalar masses squared in the SHUT model for fields in the fundamental (tri-triplet) representation of the $[{\mathrm{SU}(3)_{\mathrm{}}}]^3\times {\mathrm{SU}(3)_{\mathrm{F}}}$ symmetry.*[]{data-label="table:SpecFund"}
---- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------
12 $0$ $\widetilde{G}_{\rm L,R,F}$
3 $\sqrt{\tfrac{3}{2}} \tfrac{{v_{\mathrm{F}}}}{2} \left( 3 \lambda_{\bm{1}} \mu_{\bm{1}} + A_{\bm{1}} \right) $ $\widetilde{\mathcal{T}}_{\rm F}$
1 $ \tfrac{{v_{\mathrm{F}}}}{12} \left( 2 {v_{\mathrm{F}}} \lambda^2_{\bm{1}} - 3\sqrt{6}\lambda_{\bm{1}} \mu_{\bm{1}} - \sqrt{6} A_{\bm{1}} \right) $ $\widetilde{\mathcal{S}}_{\rm F}$
1 $ -2 b_{\bm{1}} + \tfrac{{v_{\mathrm{F}}}}{12} \left( \sqrt{6} \lambda_{\bm{1}} \mu_{\bm{1}} + 3 \sqrt{6} A_{\bm{1}} \right) $ $\widetilde{\mathcal{S}}_F^{\prime}$
4 $ -2 b_{\bm{1}} + \tfrac{{v_{\mathrm{F}}}}{12} \left( 2 \sqrt{6} \lambda_{\bm{1}} \mu_{\bm{1}} - {v_{\mathrm{F}}} \lambda^2_{\bm{1}} + 2 \sqrt{6} A_{\bm{1}}\right)$ $\mathcal{H}_{\rm F}$
3 $ -2 b_{\bm{1}} + \tfrac{{v_{\mathrm{F}}}}{12} \left( 5 \sqrt{6} \lambda_{\bm{1}} \mu_{\bm{1}} + 2 {v_{\mathrm{F}}} \lambda^2_{\bm{1}} - \sqrt{6} A_{\bm{1}}\right)$ $\widetilde{\mathcal{T}}_{\rm F}^{\prime}$
6 $ \sqrt{\tfrac{3}{2}} \tfrac{v}{2} \left( 3 \lambda_{\bm{78}} \mu_{\bm{78}} + A_{\bm{78}} + 3 C_{\bm{78}} \right) $ $\widetilde{\mathcal{T}}_{\rm L,R}$
8 $ \tfrac{v}{12} \left( - v \lambda^2_{\bm{78}} + 3 \sqrt{6} \lambda_{\bm{78}}\mu_{\bm{78}} + \sqrt{6} A_{\bm{78}} + 3 \sqrt{6} C_{\bm{78}} \right) $ $\rm{Re}[\widetilde{\Delta}^{1,\cdots,8}_{\rm C}]$
2 $\tfrac{v}{12} \left( 2 v \lambda^2_{\bm{78}} - 3 \sqrt{6} \lambda_{\bm{78}}\mu_{\bm{78}} - \sqrt{6} A_{\bm{78}} - 3 \sqrt{6} C_{\bm{78}} \right) $ $\widetilde{\mathcal{S}}_{\rm L,R}$
2 $ - 2 b_{\bm{78}} + \tfrac{\sqrt{6}}{12} v \left( \lambda_{\bm{78}} \mu_{\bm{78}} + 3 A_{\bm{78}} + C_{\bm{78}} \right) $ $\widetilde{\mathcal{S}}_{\rm L,R}^{\prime}$
8 $ - 2 b_{\bm{78}} + \tfrac{3}{4} g^2_{\rm U} v^2 + \tfrac{v^2}{12}\lambda^2_{\bm{78}} + \tfrac{\sqrt{6}}{6} v \left(\lambda_{\bm{78}} $\mathcal{H}_{\rm L,R}$
\mu_{\bm{78}} + A_{\bm{78}} + C_{\bm{78}} \right) $
8 $ - 2 b_{\bm{78}} - \tfrac{v^2}{12}\lambda^2_{\bm{78}} + \tfrac{\sqrt{6}}{12} v \left( 3 \lambda_{\bm{78}} \mu_{\bm{78}} + $ \rm{Im}[\widetilde{\Delta}^{1,\cdots,8}_{\rm C}] $
A_{\bm{78}} + 3 C_{\bm{78}} \right) $
6 $ - 2 b_{\bm{78}} + \tfrac{v^2}{6}\lambda^2_{\bm{78}} + \tfrac{\sqrt{6}}{12} v \left( 5 \lambda_{\bm{78}} \mu_{\bm{78}} - $\widetilde{\mathcal{T}}_{\rm L,R}^{\prime}$
A_{\bm{78}} + 5 C_{\bm{78}} \right) $
---- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------
: *Scalar masses squared in the SHUT model for fields in the adjoint representations of the ${\mathrm{SU}(3)_{\mathrm{L,R,C,F}}}$ symmetries.*[]{data-label="table:SpecAdj"}
Fermion Masses
--------------
The masses of the fermions that originate from the gauge-adjoint sector are somewhat more complicated. For the sake of simplicity, we use a shortened notation and show the exact expressions for the fermion masses squared in Tab. \[table:FermiSpect\]. In particular, we parametrize the octet masses by $X^{\bm{8}}_{\rm C}$, $Y^{\bm{8}}_{\rm C}$ and $Z^{\bm{8}}_{\rm C}$, where the number in the superscript denotes the representation under the symmetry labeled in the subscript. The explicit form of such parameters reads $$\begin{aligned}
X^{\bm{8}}_{\rm C} =& 4 M^2_0 +2 M^{\prime 2}_0 + \mu^2_{\bm{78}}\,, \label{eq:gluinos1} \\
Y^{\bm{8}}_{\rm C} =& 4 M^{\prime 2}_0 \left( 2 M_0 + \mu_{\bm{78}} \right)^2\,, \label{eq:gluinos2} \\
Z^{\bm{8}}_{\rm C} =& \left( \mu^2_{\bm{78}} - 4 M^2_0 \right)^2\,. \label{eq:gluinos3}\end{aligned}$$ The singlet and triplet fermion masses depend on the $X^{\bm{1,3}}_{\rm{L,R}}$, $Y^{\bm{1,3}}_{\rm{L,R}}$ and $Z^{\bm{1,3}}_{\rm{L,R}}$ parameters which are given by $$\begin{aligned}
X^{\bm{1,3}}_{\rm{L,R}} &=& \Big[ 2 v^2 \lambda^2_{\bm{78}} \mp 4 \sqrt{6} v \lambda_{\bm{78}} \mu_{\bm{78}} + 12 \Big( 4 M^2_0 \nonumber\\
&+& 2 M^{\prime 2}_0 + \mu^2_{\bm{78}} \Big)\Big]\,, \label{eq:singlinos1} $$ $$\begin{aligned}
Y^{\bm{1,3}}_{\rm{L,R}} &=& \Big[ \pm 2 \sqrt{6} v \lambda_{\bm{78}} \mu_{\bm{78}} - v^2 \lambda^2_{\bm{78}} - 6 \Big( 4 M_0^2 \nonumber\\
&+& 2 M_0^{\prime 2} + \mu^2_{\bm{78}} \Big) \Big]^2\,, \label{eq:singlinos2} $$ $$\begin{aligned}
Z^{\bm{1,3}}_{\rm{L,R}} &=& 192 \Big[ 3 M_0^{\prime 4} \pm 2 M_0 M_0^{\prime 2} \left( \sqrt{6} v \lambda_{\bm{78}} \mp 6 \mu_{\bm{78}} \right) \nonumber \\
&+& 2 M_0^2 \left( v^2 \lambda^2_{\bm{78}} \mp 2 \sqrt{6} v \lambda_{\bm{78}} \mu_{\bm{78}} +
6 \mu^2_{\bm{78}} \right) \Big]\,. \label{eq:singlinos3}\end{aligned}$$ For the new doublet fermions, the mass eigenstates are written in terms of $X^{\bm{2}}_{\rm{L,R}}$, $Y^{\bm{2}}_{\rm{L,R}}$ and $Z^{\bm{2}}_{\rm{L,R}}$ which read $$\begin{aligned}
X^{\bm{2}}_{\rm{L,R}} =& 96 M_0^2 + 48 M_0^{\prime 2} + 36 v^2 g^2_{\rm U} + v^2 \lambda^2_{\bm{78}} \nonumber \\
-& 4 \sqrt{6} v \lambda_{\bm{78}} \mu_{\bm{78}} + 24 \mu^2_{\bm{78}}\,, \label{eq:Higgsinos1} \end{aligned}$$ $$\begin{aligned}
Y^{\bm{2}}_{\rm{L,R}} =& v^4 \lambda^4_{\bm{78}} - 8 \sqrt{6} v^3 \lambda^3_{\bm{78}} \mu_{\bm{78}} + 24 v^2 \lambda^2_{\bm{78}} \Big( 4 M_0^{\prime 2} \nonumber \\
-& 8 M^2_0 + 3 v^2 g^2_{\rm U} + 6 \mu^2_{\bm{78}}\Big)\,, \label{eq:Higgsinos2} $$ $$\begin{aligned}
Z^{\bm{2}}_{\rm{L,R}} =& 96 \Big\{ 6 \left[ 4 M^{\prime 2}_0 + \left( \mu_{\bm{78}} - 2 M_0 \right)^2 \right] \Big[ 3 v^2 g^2_{\rm U} \nonumber \\
+& \left( \mu_{\bm{78}} + 2 M_0 \right)^2\Big] + \sqrt{6} v \lambda_{\bm{78}} \Big( 6 v^2 g^2_{\rm U} M_0 \nonumber \\
-& 8 M_0 M^{\prime 2}_0 +
8 M_0^2 \mu_{\bm{78}} - 4 M_0^{\prime 2} \mu_{\bm{78}} \nonumber \\
-& 3 v^2 g^2_{\rm U} \mu_{\bm{78}} - 2 \mu^3_{\bm{78}} \Big) \Big\}\,. \label{eq:Higgsinos3}\end{aligned}$$
Note that the doublets $\widetilde{\mathcal{H}}_{\rm A}$, which are the left-handed Weyl fermions defined to transform as $\bm{2}_{1}$, form mass terms of the form $m\widetilde{\mathcal{H}}_{\rm A} \widetilde{\mathcal{H}}^{\prime}_{\rm A}$ with $\widetilde{\mathcal{H}}^{\prime}_{\rm A}$ being also the left-handed Weyl fermions transforming as $\bm{2}_{-1}$.
Gauge boson masses
------------------
The gauge bosons of the ${\mathrm{SU}(3)_{\mathrm{C}}}$ group remain massless and are identified with the SM gluons whereas the massive gauge bosons are generated upon the SSB of the ${\mathrm{SU}(3)_{\mathrm{L,R}}}$ symmetries. The covariant derivative of the GUT symmetry reads $$\begin{aligned}
\bm{D}^{\mu} =& {\partial}^{\mu} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}} {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$R$}}}} {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$C$}}}} - {\mathrm i}g_{\rm U}
\sum^{8}_{a=1} \Big[ G^{\mu a}_{\rm L} \bm{T}_{\rm L}^a {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$R$}}}} {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$C$}}}} \nonumber \\
+ &
G^{\mu a}_{\rm R} \bm{T}_{\rm R}^a {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}} {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$C$}}}} +
G^{\mu a}_{\rm C} \bm{T}_{\rm C}^a {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}} {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$R$}}}}\Big] \,,
\label{eq:covDeriv}\end{aligned}$$ where $G^{\mu a}_{\rm L}$ are the gauge fields of the ${\mathrm{SU}(3)_{\mathrm{L}}}$ symmetry which cyclically transform into $G^{\mu a}_{\rm R}$ and $G^{\mu a}_{\rm C}$ by means of $\mathbb{Z}_3$-permutations. Considering the gauge-breaking VEVs , the relevant kinetic terms that couple the vector and scalar fields evaluated in the vacuum of the theory are given by $$\begin{aligned}
\label{eq:covDerivVEV2}
\left \lvert D^{\mu} \left \langle \widetilde{\Delta}^b_{\rm L,R} \right\rangle \right\rvert^2 = \dfrac{3}{4} g^2_{\rm U} v^2
\sum^{7}_{a=4} \eta_{\mu \nu} G^{\mu a}_{\rm L,R} G^{\nu a}_{\rm L,R}\,.\end{aligned}$$ Therefore, there are eight massive gauge bosons in the model which transform as complex $\bm{2}_{1}$ representations of ${\mathrm{SU}(2)_{\mathrm{L,R}}} \times {\mathrm{U}(1)_{\mathrm{L,R}}}$ whose charge eigenstates read $$\begin{aligned}
\label{eq:GaugeBosonDoublets}
\renewcommand{1.4}{1.3}
\mathcal{G}^{\mu}_{\rm L,R}\equiv \dfrac{1}{\sqrt{2}} \mleft(
\begin{array}{ccc}
G^{\mu 5}_{\rm L,R} + {\mathrm i}G^{\mu 4}_{\rm L,R}\\
G^{\mu 7}_{\rm L,R} + {\mathrm i}G^{\mu 6}_{\rm L,R}
\end{array}
\mright)\;,\end{aligned}$$ with mass $m^2_{\mathcal{G}} = \tfrac{3}{4} g^2_{\rm U}\, v^2 $. In addition to the unbroken colour sector, the remaining gauge bosons are also massless at the SHUT SSB scale.
Gauge couplings: $\beta$-functions and matching conditions {#sec:betabeta}
==========================================================
In general, the one-loop $\beta$-function for a gauge coupling is given by [@Luo:2002ti] $$\begin{aligned}
\label{bebe}
\beta(g_i)&=&-\frac{g_i^3}{(4\pi)^2}\Big(\frac{11}{3}C_2(G) - \frac{4}{3}\kappa S_2(F) \nonumber \\
&-& \frac{1}{3}S_2(S)\Big)\equiv\frac{b_ig_i^3}{(4\pi)^2},\end{aligned}$$
where $\kappa=1/2$ for Weyl fermions, $C_2(G)=N$ is the Casimir index, $S_2(F)$ is the Dynkin index for a fermion and $S_2(S)$ is the Dynkin index for a complex scalar. The one-loop $\beta$-function for the gauge coupling of a $\mathrm{U}(1)$ theory reads
$$\label{2}
\beta(\tilde{g}_i)=\frac{\tilde{g}_i^3}{12\pi^2}\left(\kappa\sum_f Q_f^2 + \frac{1}{4}\sum_s Q_s^2\right)\equiv\frac{b_i\tilde{g}_i^3}{(4\pi)^2}.$$
where again $\kappa$ is equal to $1/2$ for Weyl fermions, and where $Q_f$ and $Q_s$ are, respectively, the charges for all fermions and scalars in the theory.
Rewriting the gauge couplings in terms of the inverse of the structure constants, $\alpha^{-1}=4\pi/g^2$, the solutions of and reads
$$\label{eqeq}
\alpha_i^{-1}(\mu_2)= \alpha_i^{-1}(\mu_1) - \frac{b_i}{2\pi}\log\left(\frac{\mu_2}{\mu_1}\right),$$
where the $b_i$-coefficients are dependent on the number of particles and respective charges of a given EFT. Below, we specify such information for each of the four regions and provide the corresponding results for the one-loop $\beta$-functions.
Region ${ \textup{\uppercase\expandafter{\romannumeral1}}}$ {#subsecRegion1}
------------------------------------------------------------
As discussed in Sec. \[sec:massespres\], all components of the fundamental scalars and fermions remain in the spectrum after the breaking of the T-GUT symmetry. In this region, the fermion sector also contains two adjoint triplets, $\mathcal{T}_\mathrm{L,R}$, two adjoint singlets, $\mathcal{S}_\mathrm{L,R}$ and one adjoint octet in color $\widetilde{g}^a$. Here adjoint triplets/doublets/singlets refers to triplet/doublet/singlet representations coming from an $\mathrm{SU}(3)$ octet. Heavy states, with masses the size of the T-GUT scale, are marked with a symbol $\perp$ in Tab. \[table:FermiSpect\] of Appendix \[sec:masses\], and are integrated out. For the adjoint doublets, on the other hand, there is no distinct hierarchy between $\tilde{\mathcal{H}}_{\rm L,R},\tilde{\mathcal{H}}^{\dagger}_{\rm L,R}$ and their heavy counterparts, and can hence all be excluded from the spectrum.
With this, there is a total of 18 fermions and 18 scalars in the fundamental/anti-fundamental rep of $\mathrm{SU}(3)_\mathrm{C}$, 18 fermions and 18 scalars in the fundamental/anti-fundamental rep of $\mathrm{SU}(2)_\mathrm{L}$ and $\mathrm{SU}(2)_\mathrm{R}$, one fermion and no scalars in the adjoint rep of $\mathrm{SU}(3)_\mathrm{C}$ and one fermion and no scalars in the adjoint rep of $\mathrm{SU}(2)_\mathrm{L}$ and $\mathrm{SU}(2)_\mathrm{R}$, resulting in
$$\label{factor1}
b^{{ \textup{\uppercase\expandafter{\romannumeral1}}}}_{g_{\mathrm{C}}}=0 \hspace{2mm} \mathrm{and}\hspace{2mm} b^{{ \textup{\uppercase\expandafter{\romannumeral1}}}}_{g_{\mathrm{L,R}}}=3,$$
with $b_i$ defined as $\beta(g_i)\equiv\frac{b_ig_i^3}{(4\pi)^2}$. Here ${g_{\mathrm{C}}}$ is the gauge coupling for $\mathrm{SU}(3)_{\mathrm{C}}$ and $g_{\mathrm{L,R}}$ is the gauge coupling for $\mathrm{SU}(2)_\mathrm{L}\times\mathrm{SU}(2)_\mathrm{R}$.
For the $\mathrm{U}(1)_{\mathrm{L}}\times\mathrm{U}(1)_{\mathrm{R}}$ coupling, $\widetilde{g}_\mathrm{L,R}$, the $\beta$-function is calculated using the charges in Tab. \[Table:EFT-content\] of Appendix \[sec:Symmetries\]. With this we obtain
$$\label{factor}
b^{{ \textup{\uppercase\expandafter{\romannumeral1}}}}_{\widetilde{g}_\mathrm{L,R}}=9.$$
Region ${ \textup{\uppercase\expandafter{\romannumeral2}}}$ {#subsecRegion22}
------------------------------------------------------------
In region ${ \textup{\uppercase\expandafter{\romannumeral2}}}$, the adjoint scalars are integrated out, in addition to $D_\mathrm{L,R}$ in the second and third generation, which are the only fermions able to form a Dirac mass at this stage. When it comes to the fundamental scalars, there are no clear hierarchies in the spectrum, so here we will instead present the possible extreme values.
As apparent from Eq. and , the extreme values for each $b$ occur for the minimal- and maximal number of scalars, respectively. The maximal $b$-values are hence obtained when keeping all fundamental scalars, while the minimal $b$-values correspond to keeping only $H^f$, $\widetilde{E}_\mathrm{R}^f$ and $\widetilde{\phi}^f$. The latter scenario cannot be further reduced, as $H^f$ is required to remain as it contains the minimal amount of Higgs $\mathrm{SU}(2)_\mathrm{L}$-doublets required for Cabbibo mixing at tree-level ($H_u^{1,2}$ and $H_d^2$), while $\widetilde{E}_\mathrm{R}^f$ and $\widetilde{\phi}^f$ are required as they are involved in the breaking scheme down to the SM.
With this, the $b$-values lie in the following intervals
$$\begin{aligned}
\nonumber
&-\frac{19}{3}\leq b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{g_\mathrm{C}}\leq-\frac{10}{3},\;\;\;\; -\frac{2}{3} \leq b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{g_\mathrm{L}}\leq\frac{5}{3},\\
&-\frac{1}{3} \leq b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{g_\mathrm{R}}\leq\frac{5}{3},\;\;\;\;\frac{31}{3}\leq b^{{ \textup{\uppercase\expandafter{\romannumeral2}}}}_{\widetilde{g}_\mathrm{L+R}}\leq\frac{46}{3},
\label{equationRegion2}\end{aligned}$$
where hence the upper bound corresponds to the maximal field content and the lower bound to the minimal field content.
Region ${ \textup{\uppercase\expandafter{\romannumeral3}}}$ {#subsecRegion3}
------------------------------------------------------------
In region ${ \textup{\uppercase\expandafter{\romannumeral3}}}$, the fermion spectrum remains the same, while for the scalar sector we once again investigate the extreme values. The maximal field content is still to keep all fundamental scalars, while for the minimal field content we may now remove $\widetilde{E}_\mathrm{L}^2$, as $\mathrm{SU}(2)_\mathrm{F}$ is broken and only $\widetilde{E}_\mathrm{L}^1$ is involved in the breaking scheme down to the SM.
With this, all $b$-values are identical to those in region ${ \textup{\uppercase\expandafter{\romannumeral2}}}$, apart from the lower bound of $b_{\widetilde{g}_\mathrm{L+R}}$
$$\begin{aligned}
\nonumber
&-\frac{19}{3}\leq b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{C}}\leq-\frac{10}{3},\;\;\;\; -\frac{2}{3} \leq b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{L}}\leq\frac{5}{3},\\
&-\frac{1}{3} \leq b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{g_\mathrm{R}}\leq\frac{5}{3},\;\;\;\;\frac{59}{6}\leq b^{{ \textup{\uppercase\expandafter{\romannumeral3}}}}_{\widetilde{g}_\mathrm{L+R}}\leq\frac{46}{3}, \label{equationRegion3}\end{aligned}$$
where again the upper bound corresponds to the maximal field content and the lower bound to the minimal field content.
Region ${ \textup{\uppercase\expandafter{\romannumeral4}}}$ {#subsecRegion2}
------------------------------------------------------------
In region ${ \textup{\uppercase\expandafter{\romannumeral4}}}$, the minimal field content corresponds to integrating out all scalars apart from three Higgs doublets, e.g. $H_u^{1,2}$ and $H_d^2$ and the field responsible for breaking the $\mathrm{U}(1)_\mathrm{T}$ symmetry, e.g. $\widetilde{\phi}^1$. A minimum of two Higgs doublets are required to remain in order for all SM particles to gain a mass, while a third is needed for getting the appropriate Cabbibo mixing at tree level, as discussed in Sec. \[quarkquark\].
Among the fermions, ${D}_\mathrm{L}^{1,2,3}$ $\mathcal{D}_\mathrm{R}^{1,2,3}$, $\nu_\mathrm{R}^{1,2,3}$, $\phi^{1,2,3}$ and all Higgsinos are integrated out, as they can form massive states without the Higgs VEV. This can be seen from Tab. \[Table:EFT-content-snu1Phi\] of Appendix \[sec:Symmetries\] (with $\mathrm{U}(1)_\mathrm{T'}$ broken). The remainder of the fundamental fermions are kept in the spectrum. Regarding the adjoints, both the octets $\widetilde{g}^a$, and the triplets, $\mathcal{T}_\mathrm{L}^i,\mathcal{T}_\mathrm{R}^{\pm}$ are integrated out, resulting in
$$\label{mm1}
b^{{ \textup{\uppercase\expandafter{\romannumeral4}}}}_{g_\mathrm{C}}=-7 \hspace{2mm} \mathrm{and}\hspace{2mm} b^{{ \textup{\uppercase\expandafter{\romannumeral4}}}}_{g_\mathrm{L}}=-\frac{17}{6},$$
where $g_\mathrm{L}$ is the gauge coupling for $\mathrm{SU}(2)_\mathrm{L}$.
For $\mathrm{U}(1)_\mathrm{Y}$, the charges in Tab. \[Table:EFT-content-snu1Phi\] of Appendix \[sec:Symmetries\], results in
$$\label{mm2}
b^{{ \textup{\uppercase\expandafter{\romannumeral4}}}}_{\widetilde{g}_\mathrm{Y}}=\frac{43}{6}$$
where $\widetilde{g}_\mathrm{Y}$ is the gauge coupling for $\mathrm{U}(1)_\mathrm{Y}$.
Matching conditions {#subsecMatching}
-------------------
The gauge couplings unification condition at the GUT scale reads
$$\label{MGUT}
\alpha^{-1}_{\widetilde{g}_\mathrm{L,R}}(v)=\alpha^{-1}_{{g}_\mathrm{L,R}}(v)=\alpha^{-1}_{g}(v),$$
with the charges in Tab. \[Table:EFT-content\] of Appendix \[sec:Symmetries\].
At the soft scale, the gauge coupling matching conditions are obtained by finding the gauge boson mass eigenstates after the VEVs $\langle\widetilde{\phi}^2\rangle$, $\langle\widetilde{\phi}^3\rangle$ and $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$, respectively, by expanding our old basis in terms of the new one, e.g. $\{G_\mathrm{R}^3,B_\mathrm{L},B_\mathrm{R}\}$ in terms of $\{B_\mathrm{L+R}, ...\}$[^6]. With this we have
$${\alpha}^{-1}_{\widetilde{g}_\mathrm{L+R}}(\langle\widetilde{\phi}^3\rangle)={\alpha}^{-1}_{\widetilde{g}_\mathrm{L}}(\langle\widetilde{\phi}^3\rangle) + {\alpha}^{-1}_{\widetilde{g}_\mathrm{R}}(\langle\widetilde{\phi}^3\rangle),$$
at the $\langle\widetilde{\phi}^3\rangle$ scale, and
$${\alpha}^{-1}_{\widetilde{g}_\mathrm{Y}}(\langle\widetilde{\nu}_\mathrm{R}^1\rangle)={\alpha}^{-1}_{g_\mathrm{R}}(\langle\widetilde{\nu}_\mathrm{R}^1\rangle) + \frac{1}{3}{\alpha}^{-1}_{\widetilde{g}_\mathrm{L+R}}(\langle\widetilde{\nu}_\mathrm{R}^1\rangle),$$
at the $\langle\widetilde{\nu}_\mathrm{R}^1\rangle$ scale, while the matching at the $\langle\widetilde{\phi}^2\rangle$ scale is trivial, ${\alpha}^{-1}_{\widetilde{g}_\mathrm{L+R}}(\langle\widetilde{\phi}^2\rangle)={\alpha}^{-1}_{\widetilde{g}_\mathrm{L+R}}(\langle\widetilde{\phi}^2\rangle)$.
Finally, at the $Z$-boson mass scale, the matching conditions between the electromagnetic coupling, the hypercharge coupling and the $\mathrm{SU}(2)_\mathrm{L}$ coupling are already well-known
$$\label{betti}
{\alpha}^{-1}_{\widetilde{g}_\mathrm{Y}} = {\cos^2\theta_\mathrm{W}}{\alpha}^{-1}_{\mathrm{EM}}\;\;\;\;\mathrm{and}\;\;\;\; {\alpha}^{-1}_{\widetilde{g}_L} = {\sin^2\theta_\mathrm{W}}{\alpha}^{-1}_{\mathrm{EM}},$$
where $\theta_\mathrm{W}$ is the weak mixing angle, $\sin^2(\theta_\mathrm{W})\sim 0.2312$ [@Patrignani:2016xqp].
Lagrangian of the LR-symmetric effective theory {#sec:FullEffectiveLs}
===============================================
The field content of the EFT is derived from the mass spectrum after the T-GUT symmetry breaking. As a general rule, the light fields, i.e. those with a mass scale much smaller than the GUT scale $v$, are kept in the EFT spectrum whereas those with masses of the same order of magnitude as $v$ are integrated out.
The light field components and their group transformations under the LR-symmetry obtained after $v$ and $v_{\rm F}$ VEVs (see Eq. ) are shown in Tab. \[Table:EFT-content\], where we use the notation given in Eq. .
The scalar potential of the LR-symmetric effective model
--------------------------------------------------------
The scalar potential of the effective LR-symmetric theory generated after the T-GUT breaking can be summarized by $$\begin{aligned}
V_{\rm LR} = V_2 + V_3 + V_4\,,\end{aligned}$$ where $V_2$, $V_3$ and $V_4$ denote the quadratic, cubic and quartic scalar self-interactions, respectively. For simplicity, we will suppress colour indices in $V_{\rm LR}$ and, for all those terms that can be written from LR-parity transformations on the fields, we will show them within square brackets as $\widehat{\mathcal{P}}_{\rm LR} [\cdots ]$. Note that here we use this notation for both the cases of invariance or not under LR-parity. For instance, while for the LR-parity symmetric case we should preserve the couplings, for the LR-parity broken case we should also read $m\to\bar{m},\, A\to \bar{A},\, \lambda \to \bar{\lambda}$ whenever LR-parity transformation is applied.
We start by writing the scalar mass terms, $$\begin{aligned}
\begin{split}
V_2 = & {m_H^2}H^{\ast\,r}_{f\,l} H^{f\,l}_{r}
+m_h^2 h^{\ast\,r}_{l} h^l_{r}
+ m_{\phi}^2 \widetilde{\phi}^\ast_{f} \widetilde{\phi}^f
+ m_{\varphi}^2 \widetilde{\varphi}^\ast \widetilde{\varphi}
\\
+& m_{\Delta}^2 \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast}
\widetilde{\mathcal{H}}_{\rm F}^f
+ \widehat{\mathcal{P}}_{\rm LR} \left[ m_{E}^2\widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
+ m_{\mathcal{E}}^2\widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
\right . \\
+ &\left. m_{\mathcal{Q}}^2\widetilde{\mathcal{Q}}_{{\rm L}\,f}^{\ast\,l}
\widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f}
+ m_{q}^2\widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l}
+ m_{D}^2\widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f}
+ m_{\mathcal{B}}^2 \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} \right]
\end{split}\end{aligned}$$ whereas the trilinear interactions are expressed as $$\begin{aligned}
\begin{split}
V_3 = & \varepsilon_{f f'}\Big\{ \widehat{\mathcal{P}}_{\rm LR} \Big[
A_{1} \widetilde{\mathcal{Q}}_{{\rm R}}^{f\,r} h^{l}_{r} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'}
+A_{2} \widetilde{D}_{\rm R}^f \widetilde{\varphi} \widetilde{D}_{\rm L}^{f'} \Big]
\\&
+ \widehat{\mathcal{P}}_{\rm LR} \Big[A_{3} \widetilde{q}_{\rm R}^{r} H^{f\,l}_{r} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'}
+ A_4 \widetilde{\mathcal{B}}_{\rm R} \widetilde{\phi}^{f'} \widetilde{D}_{\rm L}^{f}
\\&
+ A_{5} \widetilde{\mathcal{B}}_{\rm R} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \widetilde{E}_{\rm L}^{f'\,l}
+ A_{6} \widetilde{D}_{\rm R}^{f} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'} \widetilde{\mathcal{E}}_{\rm L}^{\,l}
\\&
+A_{7}\widetilde{D}_{\rm R}^f \widetilde{q}_{{\rm L}\,l} \widetilde{E}_{\rm L}^{f'\,l} + {\rm c.c.}
\Big] \Big\}
\end{split}\end{aligned}$$
Due to a large number of possible contractions of four scalar fields in the effective LR-symmetric model, we will employ a condensed notation to express the scalar quartic self-interactions. We describe below the five possible types of terms.
For the first type, which we denote “sc1”, we consider terms with *one* reoccurring index, where we define the reoccurring index as an index possessed by all the four fields. For such a combination there are three possible contractions, out of which two of them are linearly independent. In particular, we have $$\begin{aligned}
V_{\rm sc1} \supset
&\lambda_{k_1}\widetilde{D}_{{\rm L}\,x\,f'}^\ast\widetilde{D}_{\rm L}^{x\,f'} H^{\ast\,r}_{f\,l} H^{f\,l}_{r}
+
\lambda_{k_2}\widetilde{D}_{{\rm L}\,x\,f'}^\ast\widetilde{D}_{\rm L}^{x\,f} H^{\ast\,r}_{f\,l} H^{f'\,l}_r
\nonumber \\
\equiv
&{\lambda_{{\scalebox{0.50}{$k_1 \-- k_2$}}}} \widetilde{D}_{{\rm L}\,f'}^\ast\widetilde{D}_{\rm L}^{f'} H^{\ast\,r}_{f\,l} H^{f\,l}_{r}\,,
\label{sc1}\end{aligned}$$ where colour indices are suppressed in the condensed form.
For terms with *two* reoccurring indices, denoted as “sc2”, no matter if they are ${\mathrm{SU}(2)_{\mathrm{}}}$ indices or ${\mathrm{SU}(3)_{\mathrm{}}}$ indices[^7], there are four linearly independent contractions that read $$\begin{aligned}
&V_{\rm sc2} \supset \nonumber \\
&\lambda_{n_1}\widetilde{E}_{{\rm L}\;l'\,f'}^\ast \widetilde{E}_{\rm L}^{l'\,f'} \widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,x\,f}^{\ast\,l}
\widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,l}^{x\,f}
+
\lambda_{n_2}\widetilde{E}_{{\rm L}\;l'\,f'}^\ast \widetilde{E}_{\rm L}^{l'\,f}\widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,x\,f}^{\ast\,l}
\widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,l}^{x\,f'}
\nonumber \\
&+
\lambda_{n_3}\widetilde{E}_{{\rm L}\;l'\,f'}^\ast \widetilde{E}_{\rm L}^{l\,f'}\widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,x\,f}^{\ast\,l'}
\widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,l}^{x\,f}
+
\lambda_{n_4}\widetilde{E}_{{\rm L}\;l'\,f'}^\ast \widetilde{E}_{\rm L}^{l\,f}\widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,x\,f}^{\ast\,l'}
\widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,l}^{x\,f'}
\nonumber \\
&\equiv
{\lambda_{{\scalebox{0.50}{$n_1 \-- n_4$}}}} \widetilde{E}_{{\rm L}\;l'\,f}^\ast \widetilde{E}_{\rm L}^{l'\,f} \widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,f'}^{\ast\,l}
\widetilde{\mathcal{\mathcal{Q}}}_{{\rm L}\,l}^{\,f'} \,.
\label{sc2}\end{aligned}$$ The third type involves terms with *two* reoccurring indices (either ${\mathrm{SU}(2)_{\mathrm{}}}$ or ${\mathrm{SU}(3)_{\mathrm{}}}$ indices) but *identical* fields. We denote this case as “sc3” and observe that there are only two linearly independent terms of the form $$\begin{aligned}
&V_{\rm sc3} \supset \nonumber \\
&\lambda_{j_1}\widetilde{D}_{{\rm L}\;x'\,f'}^\ast \widetilde{D}_{\rm L}^{x'\,f'} \widetilde{D}_{{\rm L}\;x\,f}^\ast \widetilde{D}_{\rm L}^{x\,f}
+
\lambda_{j_2}\widetilde{D}_{{\rm L}\;x'\,f'}^\ast \widetilde{D}_{\rm L}^{x\,f'} \widetilde{D}_{{\rm L}\;x\,f}^\ast \widetilde{D}_{\rm L}^{x'\,f}
\nonumber \\
& \equiv {\lambda_{{\scalebox{0.50}{$j_1 \-- j_2$}}}} \widetilde{D}_{{\rm L}\,f'}^\ast \widetilde{D}_{\rm L}^{f'} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f}\,,
\label{sc3}\end{aligned}$$ where colour contractions are once again implicit.
For terms with *three* reoccurring indices and identical fields, labeled as “sc4”, there are four linearly independent combinations that we write as $$\begin{aligned}
V_{\rm sc4} \supset
&\lambda_{m_1}H^{\ast\,r'}_{f'\,l'} H^{f'\,l'}_{r'} H^{\ast\,r}_{f\,l} H^{f\,l}_r
+
\lambda_{m_2}H^{\ast\,r'}_{f'\,l'} H^{f'\,l'}_r H^{\ast\,r}_{f\,l} H^{f\,l}_{r'}
\nonumber \\
+&
\lambda_{m_3}H^{\ast\,r'}_{f'\,l'} H^{f\,l'}_{r'} H^{\ast\,r}_{f\,l} H^{f'\,l}_{r}
+
\lambda_{m_4}H^{\ast\,r'}_{f'\,l'} H^{f'\,l}_{r'} H^{\ast\,r}_{f\,l} H^{f\,l'}_{r}
\nonumber \\
\equiv &{\lambda_{{\scalebox{0.50}{$m_1 \-- m_4$}}}} H^{\ast\,r'}_{f'\,l'} H^{f'\,l'}_{r'}H^{\ast\,r}_{f\,l} H^{f\,l}_r
\label{sc4}\end{aligned}$$ Note that the case with three reoccurring indices and different fields does not exist and the only case with one reoccurring index and identical fields is the one involving the gauge singlet $\phi^f$.
Finally, the fifth type (“sc5”) involves terms without reoccurring indices or terms with one reoccurring index but four identical fields such as $$\begin{aligned}
\label{sc5}
V_{\rm sc5} \supset {\lambda_{{\scalebox{0.5}{$i$}}}} h^{\ast\,r}_{l} h^l_{r} \widetilde{\phi}^\ast_{f} \widetilde{\phi}^f + {\lambda_{{\scalebox{0.5}{$j$}}}} \widetilde{\phi}^\ast_{f'}\widetilde{\phi}^{f'}
\widetilde{\phi}^\ast_{f}\widetilde{\phi}^f \,.\end{aligned}$$ Note that, for ease of notation, we assume that combinatorial factors were absorbed by various ${\lambda_{{\scalebox{0.5}{$i$}}}}$ and ${\lambda_{{\scalebox{0.50}{$i \-- j$}}}}$.
We will then consider five different scenarios organized according to the type of index contractions as described in detail in Eqs. , , , and : $$V_4 = V_{\rm sc1} + V_{\rm sc2}+ V_{\rm sc3} + V_{\rm sc4} + V_{\rm sc5}\,.$$ The first contribution reads
$$\begin{aligned}
\begin{split}
&V_{\rm sc1} =
{\lambda_{{\scalebox{0.50}{$1 \-- 2$}}}} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l} \widetilde{q}_{{\rm R}\,r}^{\ast} \widetilde{q}_{\rm R}^{r}
+ {\lambda_{{\scalebox{0.50}{$3 \-- 4$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{\mathcal{B}}_{\rm R}^\ast \widetilde{\mathcal{B}}_{\rm R}
+{\lambda_{{\scalebox{0.50}{$5 \-- 6$}}}} H^{\ast\,r}_{f'\,l} H^{f'\,l}_r \widetilde{\phi}^\ast_{f}\widetilde{\phi}^f
+{\lambda_{{\scalebox{0.50}{$7 \-- 8$}}}} \widetilde{E}_{{\rm L}\;f'\,l}^\ast \widetilde{E}_{\rm L}^{f'\,l} \widetilde{E}_{{\rm R}\;f}^{\ast\,r} \widetilde{E}_{{\rm R}\,r}^{f}
+\widehat{\mathcal{P}}_{\rm LR} \left[ {\lambda_{{\scalebox{0.50}{$9 \-- 10$}}}} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l}
\widetilde{\mathcal{Q}}_{{\rm R}\,f\,r}^{\ast} \widetilde{\mathcal{Q}}_{{\rm R}}^{f\,r}
\right. \\
&
\left.
+{\lambda_{{\scalebox{0.50}{$11 \-- 12$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{D}_{{\rm R}\;f}^\ast \widetilde{D}_{\rm R}^{f}
+{\lambda_{{\scalebox{0.50}{$13 \-- 14$}}}}\widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l}\widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f}
+{\lambda_{{\scalebox{0.50}{$15 \-- 16$}}}}\widetilde{\mathcal{Q}}_{{\rm L}\;f}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L}
+ {\lambda_{{\scalebox{0.50}{$17 \-- 18$}}}} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l}\widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L}
+{\lambda_{{\scalebox{0.50}{$19 \-- 20$}}}}\widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l} \widetilde{D}_{{\rm R}\;f}^\ast \widetilde{D}_{\rm R}^{f}
\right. \\
&
\left.
+{\lambda_{{\scalebox{0.50}{$21 \-- 22$}}}}\widetilde{\mathcal{Q}}_{{\rm L}\;f}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \widetilde{\mathcal{B}}_{\rm R}^\ast \widetilde{\mathcal{B}}_{\rm R}
+ {\lambda_{{\scalebox{0.50}{$23 \-- 24$}}}} \widetilde{\mathcal{Q}}_{{\rm L}\;f}^{\ast\,l'} \widetilde{\mathcal{Q}}_{{\rm L}\,l'}^{f} h^{\ast\,r}_{l} h^l_{r}
+{\lambda_{{\scalebox{0.50}{$25 \-- 26$}}}} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l}\widetilde{\mathcal{B}}_{\rm R}^\ast \widetilde{\mathcal{B}}_{\rm R}
+{\lambda_{{\scalebox{0.50}{$27 \-- 28$}}}} \widetilde{q}_{\rm L}^{\ast\,l'} \widetilde{q}_{{\rm L}\,l'} H^{\ast\,r}_{f\,l} H^{f\,l}_{r}
+ {\lambda_{{\scalebox{0.50}{$29 \-- 30$}}}} \widetilde{q}_{\rm L}^{\ast\,l'} \widetilde{q}_{{\rm L}\,l'} h^{\ast\,r}_{l} h^l_{r}
\right. \\
&
\left.
+ {\lambda_{{\scalebox{0.50}{$31 \-- 32$}}}} \widetilde{D}_{{\rm L}\,f'}^\ast \widetilde{D}_{\rm L}^{f'} H^{\ast\,r}_{f\,l} H^{f\,l}_{r}
+ {\lambda_{{\scalebox{0.50}{$33 \-- 34$}}}} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l} \widetilde{E}_{{\rm L}\;f\,l'}^\ast \widetilde{E}_{\rm L}^{f\,l'}
+ {\lambda_{{\scalebox{0.50}{$35 \-- 36$}}}} \widetilde{\mathcal{Q}}_{{\rm L}\,f}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \widetilde{\mathcal{E}}_{{\rm L}\;l'}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l'}
+{\lambda_{{\scalebox{0.50}{$37 \-- 38$}}}} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l} \widetilde{\mathcal{E}}_{{\rm L}\;l'}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l'}
\right. \\
&
\left.
+ {\lambda_{{\scalebox{0.50}{$39 \-- 40$}}}} \widetilde{\mathcal{Q}}_{{\rm R}\;f'}^{\ast\;r} \widetilde{\mathcal{Q}}_{{\rm R}\;\,r}^{f'} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
+ {\lambda_{{\scalebox{0.50}{$41 \-- 42 $}}}}\widetilde{D}_{{\rm L}\,f'}^\ast \widetilde{D}_{\rm L}^{f'} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
+{\lambda_{{\scalebox{0.50}{$43 \-- 44$}}}} \widetilde{D}_{{\rm R}\;f'}^\ast \widetilde{D}_{\rm R}^{f'} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
+ {\lambda_{{\scalebox{0.50}{$45 \-- 46$}}}} \widetilde{\mathcal{Q}}_{{\rm L}\,f'}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'} \widetilde{\phi}^\ast_{f}\widetilde{\phi}^f
\right. \\
&
\left.
+{\lambda_{{\scalebox{0.50}{$47 \-- 48$}}}} \widetilde{D}_{{\rm L}\,f'}^\ast \widetilde{D}_{\rm L}^{f'}\widetilde{\phi}^\ast_{f}\widetilde{\phi}^f
+ {\lambda_{{\scalebox{0.50}{$49 \-- 50$}}}} h^{\ast\,r}_{l'} h^{l'}_{r} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
+ {\lambda_{{\scalebox{0.50}{$51 \-- 52$}}}} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
+{\lambda_{{\scalebox{0.50}{$53 \-- 54$}}}} H^{\ast\,r}_{f\,l'} H^{f\,l'}_{r} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
\right. \\
&
\left.
+{\lambda_{{\scalebox{0.50}{$55 \-- 56$}}}} h^{\ast\,r}_{l'} h^{l'}_{r} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
+ {\lambda_{{\scalebox{0.50}{$57 \-- 58$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f}
+{\lambda_{{\scalebox{0.50}{$59 \-- 60$}}}} \widetilde{E}_{{\rm L}\;f'\,l}^\ast \widetilde{E}_{\rm L}^{f'\,l} \widetilde{\phi}^\ast_{f}\widetilde{\phi}^f
+ {\lambda_{{\scalebox{0.50}{$61 \-- 62$}}}} \widetilde{\phi}^f H^{f'\,l}_r \widetilde{E}_{{\rm L}\;f'\,l}^\ast \widetilde{E}_{{\rm R}\;f}^{\ast\,r}
\right. \\
&
\left.
+ \left( {\lambda_{{\scalebox{0.50}{$63 \-- 64$}}}} h^{\ast\,r}_{l'} H^{f\,l'}_{r} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
+ {\lambda_{{\scalebox{0.50}{$65 \-- 66$}}}} h^{\ast\,r}_{l'} H^{f\,l'}_{r} \widetilde{\mathcal{Q}}_{{\rm L}\,f}^{\ast\,l} \widetilde{q}_{{\rm L}\,l}
+ {\lambda_{{\scalebox{0.50}{$67 \-- 68$}}}} \widetilde{E}_{{\rm L}\;f\,l'}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l'} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f}
+ {\lambda_{{\scalebox{0.50}{$69 \-- 70$}}}} \widetilde{D}_{{\rm L}\,f'}^\ast \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'} \widetilde{E}_{{\rm R}\;f}^{\ast\,r} H^{f\,l}_{r}
\right. \right. \\
&
\left. \left.
+ {\lambda_{{\scalebox{0.50}{$71 \-- 72$}}}} \widetilde{D}_{{\rm L}\,f'}^\ast \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'} \widetilde{E}_{\rm L}^{f\,l} \widetilde{\phi}^\ast_{f}
+ {\lambda_{{\scalebox{0.50}{$73 \-- 74$}}}} \widetilde{\mathcal{B}}_{\rm L} \widetilde{\mathcal{Q}}_{{\rm R}\;\,r}^{f} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{q}_{{\rm R}\,r}^{\ast}
+ {\rm c.c.} \right) \right] + V_{\rm sc1}^{\rm gen}
\,,
\end{split}\end{aligned}$$
with $V_{\rm sc1}^{\rm gen}$ corresponding to the interactions generated only after the matching procedure, i.e. not directly obtained by expansion of the Lagrangian of the original theory, and given by $$\begin{aligned}
\begin{split}
&V_{\rm sc1}^{\rm gen} =
\widehat{\mathcal{P}}_{\rm LR} \Big[
{\delta_{{\scalebox{0.5}{$1 \-- 2$}}}} \widetilde{\mathcal{Q}}_{{\rm L}\,f'}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'} \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
+{\delta_{{\scalebox{0.5}{$3 \-- 4$}}}}\widetilde{D}_{{\rm L}\,f'}^\ast \widetilde{D}_{\rm L}^{f'}\widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
\\
&
+ {\delta_{{\scalebox{0.5}{$5 \-- 6$}}}} \widetilde{E}_{{\rm L}\;f'\,l}^\ast \widetilde{E}_{\rm L}^{f'\,l} \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
\Big]
+ {\delta_{{\scalebox{0.5}{$7 \-- 8$}}}} H^{\ast\,r}_{f'\,l} H^{f'\,l}_r \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f \,.
\end{split}\end{aligned}$$ The effective quartic interactions with two reoccurring indices are given by $$\begin{aligned}
\begin{split}
&V_{\rm sc2} =
{\lambda_{{\scalebox{0.50}{$75 \-- 78$}}}} \widetilde{\mathcal{Q}}_{{\rm L}\,f'}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'} \widetilde{\mathcal{Q}}_{{\rm R}\,f\,r}^{\ast}
\widetilde{\mathcal{Q}}_{{\rm R}}^{f\,r}
+{\lambda_{{\scalebox{0.50}{$79 \-- 82$}}}} \widetilde{D}_{{\rm L}\,f'}^\ast \widetilde{D}_{\rm L}^{f'} \widetilde{D}_{{\rm R}\;f}^\ast \widetilde{D}_{\rm R}^{f}
\\
&
+{\lambda_{{\scalebox{0.50}{$83 \-- 86$}}}} h^{\ast\,r'}_{l'} h^{l'}_{r'} H^{\ast\,r}_{f\,l} H^{f\,l}_{r}
+ \widehat{\mathcal{P}}_{\rm LR} \left[ {\lambda_{{\scalebox{0.50}{$75 \-- 78$}}}} \widetilde{\mathcal{Q}}_{{\rm L}\;f'}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'}
\widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f}
\right. \\
&
\left.
+ {\lambda_{{\scalebox{0.50}{$87 \-- 90$}}}}\widetilde{\mathcal{Q}}_{{\rm L}\;f'}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'} \widetilde{D}_{{\rm R}\;f}^\ast \widetilde{D}_{\rm R}^{f}
+ {\lambda_{{\scalebox{0.50}{$91 \-- 94$}}}} \widetilde{q}_{\rm L}^{\ast\,l'} \widetilde{q}_{{\rm L}\,l'} \widetilde{\mathcal{Q}}_{{\rm L}\,f}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f}
\right. \\
&
\left.
+{\lambda_{{\scalebox{0.50}{$95 \-- 98$}}}} \widetilde{\mathcal{Q}}_{{\rm L}\;f'}^{\ast\,l'} \widetilde{\mathcal{Q}}_{{\rm L}\,l'}^{f'} H^{\ast\,r}_{f\,l} H^{f\,l}_{r}
+ {\lambda_{{\scalebox{0.50}{$99 \-- 102$}}}}\widetilde{\mathcal{Q}}_{{\rm L}\,f'}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'} \widetilde{E}_{{\rm L}\;f\,l'}^\ast
\widetilde{E}_{\rm L}^{f\,l'}
\right. \\
&
\left.
+{\lambda_{{\scalebox{0.50}{$103 \-- 106$}}}} H^{\ast\,r}_{f'\,l'} H^{f'\,l'}_{r} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l} \right]\,.
\end{split}\end{aligned}$$ The third contribution, which accounts for identical multiplets and two reoccurring indices, has the form $$\begin{aligned}
\begin{split}
&V_{\rm sc3} =
{\lambda_{{\scalebox{0.50}{$107 \-- 108$}}}} h^{\ast\,r'}_{l'} h^{l'}_{r'} h^{\ast\,r}_{l} h^l_{r} + \widehat{\mathcal{P}}_{\rm LR} \left[ {\lambda_{{\scalebox{0.50}{$101 \-- 102$}}}}\widetilde{q}_{\rm L}^{\ast\,l'}
\widetilde{q}_{{\rm L}\,l'} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l} \right. \\
&
\left.
+ {\lambda_{{\scalebox{0.50}{$109 \-- 110$}}}} \widetilde{D}_{{\rm L}\,f'}^\ast \widetilde{D}_{\rm L}^{f'} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f}
\right. \\
&
\left.
+{\lambda_{{\scalebox{0.50}{$111 \-- 112$}}}}\widetilde{E}_{{\rm L}\;f'\,l'}^\ast \widetilde{E}_{\rm L}^{f'\,l'} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l} \right] \,,
\end{split}\end{aligned}$$ while the forth scenario, where identical fields with three reoccurring indices are considered, reads $$\begin{aligned}
\begin{split}
&V_{\rm sc4} =
{\lambda_{{\scalebox{0.50}{$113 \-- 116$}}}} H^{\ast\,r'}_{f'\,l'} H^{f'\,l'}_{r'} H^{\ast\,r}_{f\,l} H^{f\,l}_{r} \\
+ &
\widehat{\mathcal{P}}_{\rm LR} \left[ {\lambda_{{\scalebox{0.50}{$117 \-- 120$}}}}
\widetilde{\mathcal{Q}}_{{\rm L}\;f'}^{\ast\,l'} \widetilde{\mathcal{Q}}_{{\rm L}\,l'}^{f'} \widetilde{\mathcal{Q}}_{{\rm L}\,f}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \right]\,.
\end{split}\end{aligned}$$ Finally, for those terms that contain only one independent type of contraction we have
$$\begin{aligned}
\begin{split}
& V_{\rm sc5} =
{\lambda_{{\scalebox{0.5}{$121$}}}} h^{\ast\,r}_{l} h^l_{r} \widetilde{\phi}^\ast_{f}\widetilde{\phi}^f
+ {\lambda_{{\scalebox{0.5}{$122$}}}} H^{\ast\,r}_{f\,l} H^{f\,l}_{r} \widetilde{\varphi}^\ast \widetilde{\varphi}
+ {\lambda_{{\scalebox{0.5}{$123$}}}} h^{\ast\,r}_{l} h^l_{r} \widetilde{\varphi}^\ast \widetilde{\varphi}
+ {\lambda_{{\scalebox{0.5}{$124$}}}} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
\widetilde{\mathcal{E}}_{\rm R}^{\ast\,r} \widetilde{\mathcal{E}}_{{\rm R}\,r}
+{\lambda_{{\scalebox{0.5}{$125$}}}} \widetilde{\phi}^\ast_{f'}\widetilde{\phi}^{f'} \widetilde{\phi}^\ast_{f}\widetilde{\phi}^f
+ {\lambda_{{\scalebox{0.5}{$126$}}}} \widetilde{\phi}^\ast_{f}\widetilde{\phi}^f \widetilde{\varphi}^\ast \widetilde{\varphi}
\nonumber \\
&
+ {\lambda_{{\scalebox{0.5}{$127$}}}} \widetilde{\varphi}^\ast \widetilde{\varphi}\,\widetilde{\varphi}^\ast \widetilde{\varphi}
+ \widehat{\mathcal{P}}_{\rm LR} \left[
{\lambda_{{\scalebox{0.5}{$128$}}}} \widetilde{\varphi} h^l_{r} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm R}^{\ast\,r}
+ {\lambda_{{\scalebox{0.5}{$129$}}}} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{\mathcal{B}}_{\rm R}^\ast \widetilde{D}_{\rm R}^{f}
+ {\lambda_{{\scalebox{0.5}{$130$}}}} \widetilde{\mathcal{Q}}_{{\rm L}\,f}^{\ast\,l} \widetilde{q}_{{\rm L}\,l} \widetilde{q}_{{\rm R}\,r}^{\ast} \widetilde{\mathcal{Q}}_{{\rm R}}^{f\,r}
+ {\lambda_{{\scalebox{0.5}{$131$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L}
\right. \nonumber \\
&
\left.
+ {\lambda_{{\scalebox{0.5}{$132$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} H^{\ast\,r}_{f\,l} H^{f\,l}_{r}
+ {\lambda_{{\scalebox{0.5}{$133$}}}} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f} h^{\ast\,r}_{l} h^l_{r}
+ {\lambda_{{\scalebox{0.5}{$134$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} h^{\ast\,r}_{l} h^l_{r}
+ {\lambda_{{\scalebox{0.5}{$135$}}}} \widetilde{q}_{{\rm R}\,r}^{\ast} \widetilde{q}_{\rm R}^{r} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
+ {\lambda_{{\scalebox{0.5}{$136$}}}} \widetilde{\mathcal{Q}}_{{\rm R}\,f\,r}^{\ast} \widetilde{\mathcal{Q}}_{{\rm R}}^{f\,r} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
\right. \nonumber \\
&
\left.
+ {\lambda_{{\scalebox{0.5}{$137$}}}} \widetilde{q}_{{\rm R}\,r}^{\ast} \widetilde{q}_{\rm R}^{r} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
+ {\lambda_{{\scalebox{0.5}{$138$}}}} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
+ {\lambda_{{\scalebox{0.5}{$139$}}}} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
+ {\lambda_{{\scalebox{0.5}{$140$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
+ {\lambda_{{\scalebox{0.5}{$141$}}}} \widetilde{\mathcal{B}}_{\rm R}^\ast \widetilde{\mathcal{B}}_{\rm R} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
\right. \nonumber \\
&
\left.
+ {\lambda_{{\scalebox{0.5}{$142$}}}} \widetilde{D}_{{\rm R}\;f}^\ast \widetilde{D}_{\rm R}^{f} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
+ {\lambda_{{\scalebox{0.5}{$143$}}}} \widetilde{\mathcal{B}}_{\rm R}^\ast \widetilde{\mathcal{B}}_{\rm R} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
+ {\lambda_{{\scalebox{0.5}{$144$}}}} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l} \widetilde{\phi}^\ast_{f}\widetilde{\phi}^f
+ {\lambda_{{\scalebox{0.5}{$145$}}}} \widetilde{\mathcal{Q}}_{{\rm L}\,f}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \widetilde{\varphi}^\ast \widetilde{\varphi}
+ {\lambda_{{\scalebox{0.5}{$146$}}}} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l} \widetilde{\varphi}^\ast \widetilde{\varphi}
+ {\lambda_{{\scalebox{0.5}{$147$}}}}\widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{\phi}^\ast_{f}\widetilde{\phi}^f
\right. \nonumber \\
&
\left.
+ {\lambda_{{\scalebox{0.5}{$148$}}}} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{D}_{\rm L}^{f} \widetilde{\varphi}^\ast \widetilde{\varphi}
+ {\lambda_{{\scalebox{0.5}{$149$}}}}\widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{\varphi}^\ast \widetilde{\varphi}
+ {\lambda_{{\scalebox{0.5}{$150$}}}} \widetilde{\mathcal{E}}_{{\rm L}\;l'}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l'} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
+ {\lambda_{{\scalebox{0.5}{$151$}}}} \widetilde{\mathcal{E}}_{{\rm L}\;l'}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l'} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
+ {\lambda_{{\scalebox{0.5}{$152$}}}} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{E}_{{\rm R}\;f}^{\ast\,r} \widetilde{E}_{{\rm R}\,r}^{f}
\right. \nonumber \\
&
\left.
+{\lambda_{{\scalebox{0.5}{$153$}}}} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{\phi}^\ast_{f}\widetilde{\phi}^f
+{\lambda_{{\scalebox{0.5}{$154$}}}} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l} \widetilde{\varphi}^\ast \widetilde{\varphi}
+{\lambda_{{\scalebox{0.5}{$155$}}}} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{\varphi}^\ast \widetilde{\varphi}
+\left(
{\lambda_{{\scalebox{0.5}{$156$}}}} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{\varphi}^\ast \widetilde{\phi}^f
+ {\lambda_{{\scalebox{0.5}{$157$}}}} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{\phi}^f h^l_{r} \widetilde{\mathcal{E}}_{\rm R}^{\ast\,r}
\right. \right. \nonumber\\
&
\left. \left.
+ {\lambda_{{\scalebox{0.5}{$158$}}}} H^{f\,l}_{r} \widetilde{E}_{{\rm R}\;f}^{\ast\,r} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\varphi}
+ {\lambda_{{\scalebox{0.5}{$159$}}}} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{D}_{\rm L}^{f}
+ {\lambda_{{\scalebox{0.5}{$160$}}}} \widetilde{\varphi}^\ast \widetilde{\phi}^f \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{\mathcal{B}}_{\rm L}
+ {\lambda_{{\scalebox{0.5}{$161$}}}} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f}
+ {\lambda_{{\scalebox{0.5}{$162$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{q}_{{\rm L}\,l} \widetilde{E}_{{\rm R}\;f}^{\ast\,r} H^{f\,l}_{r}
\right. \right.\nonumber \\
&
\left. \left.
+ {\lambda_{{\scalebox{0.5}{$163$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \widetilde{E}_{{\rm R}\;f}^{\ast\,r} h^{l}_{r}
+ {\lambda_{{\scalebox{0.5}{$164$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{q}_{{\rm L}\,l} \widetilde{\mathcal{E}}_{\rm R}^{\ast\,r} h^{l}_{r}
+ {\lambda_{{\scalebox{0.5}{$165$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{q}_{{\rm L}\,l} \widetilde{E}_{\rm L}^{f\,l} \widetilde{\phi}^\ast_{f}
+ {\lambda_{{\scalebox{0.5}{$166$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{\phi}^\ast_{f}
+ {\lambda_{{\scalebox{0.5}{$167$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{q}_{{\rm L}\,l} \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{\varphi}^\ast
\right. \right. \nonumber\\
&
\left. \left.
+ {\lambda_{{\scalebox{0.5}{$168$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{D}_{\rm L}^{f} \widetilde{E}_{{\rm R}\;f}^{\ast\,r} \widetilde{\mathcal{E}}_{{\rm R}\,r}
+ {\lambda_{{\scalebox{0.5}{$169$}}}} \widetilde{D}_{{\rm L}\, f}^{\ast} \widetilde{q}_{{\rm L}\,l} \widetilde{\mathcal{E}}_{\rm R}^{\ast\,r} H^{f\,l}_{r}
+ {\lambda_{{\scalebox{0.5}{$170$}}}} \widetilde{D}_{{\rm L}\, f}^{\ast} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \widetilde{\mathcal{E}}_{\rm R}^{\ast\,r} h^{l}_{r}
+ {\lambda_{{\scalebox{0.5}{$171$}}}} \widetilde{D}_{{\rm L}\, f}^{\ast} \widetilde{q}_{{\rm L}\,l} \widetilde{E}_{\rm L}^{f\,l} \widetilde{\varphi}^\ast
\right. \right. \nonumber \\
&
\left. \left.
+{\lambda_{{\scalebox{0.5}{$172$}}}} \widetilde{D}_{{\rm L}\, f}^{\ast} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{\varphi}^\ast
+ {\rm c.c.} \right)\right]
+ V_{\rm sc5}^{\rm gen}\,.
\end{split}\end{aligned}$$
Here, the terms generated after the breaking are $$\begin{aligned}
\begin{split}
&V_{\rm sc5}^{\rm gen} =
{\lambda_{{\scalebox{0.5}{$173$}}}} h^{\ast\,r}_{l} H^{f\,l}_{r} \widetilde{\phi}^\ast_{f} \widetilde{\varphi}
+ {\lambda_{{\scalebox{0.5}{$174$}}}} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{\mathcal{E}}_{\rm R}^{\ast\,r} \widetilde{E}_{{\rm R}\,r}^{f}
\\
&
+ {\delta_{{\scalebox{0.5}{$9$}}}} h^{\ast\,r}_{l} h^l_{r} \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
+ {\delta_{{\scalebox{0.5}{$10$}}}} \widetilde{\mathcal{H}}_{{\rm F} f'}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^{f '} \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
\\
&
+ {\delta_{{\scalebox{0.5}{$11$}}}} \widetilde{\varphi}^\ast \widetilde{\varphi} \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
+ {\delta_{{\scalebox{0.5}{$12$}}}} \widetilde{\phi}^\ast_{f'}\widetilde{\phi}^{f'} \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
\\
&
+ \widehat{\mathcal{P}}_{\rm LR} \left[
{\lambda_{{\scalebox{0.5}{$175$}}}} h^{\ast\,r}_{l} H^{f\,l}_{r} \widetilde{D}_{{\rm L}\;f}^\ast \widetilde{\mathcal{B}}_{\rm L}
+ {\lambda_{{\scalebox{0.5}{$176$}}}} \widetilde{\varphi}^\ast \widetilde{\phi}^f \widetilde{\mathcal{Q}}_{{\rm L}\,f}^{\ast\,l} \widetilde{q}_{{\rm L}\,l}
\right. \\
&
\left.
+ {\lambda_{{\scalebox{0.5}{$177$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{D}_{\rm L}^{f} \widetilde{E}_{{\rm L}\,f\,l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l}
+ {\lambda_{{\scalebox{0.5}{$178$}}}} \widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{q}_{{\rm R}\,r}^{\ast} \widetilde{\mathcal{Q}}_{{\rm R}}^{f\,r}
\right. \\
&
\left.
+ {\delta_{{\scalebox{0.5}{$13$}}}} \widetilde{q}_{\rm L}^{\ast\,l} \widetilde{q}_{{\rm L}\,l} \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
+ {\delta_{{\scalebox{0.5}{$14$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \widetilde{\mathcal{B}}_{\rm L} \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
\right. \\
&
\left.
+ {\delta_{{\scalebox{0.5}{$15$}}}} \widetilde{\mathcal{E}}_{{\rm L}\;l}^\ast \widetilde{\mathcal{E}}_{\rm L}^{l} \widetilde{\mathcal{H}}_{{\rm F} f}^{\ast} \widetilde{\mathcal{H}}_{\rm F}^f
\right]\,.
\end{split}\end{aligned}$$
The fermion sector of the LR-symmetric EFT
------------------------------------------
The part of the Lagrangian of the effective LR-symmetric theory that involves purely quadratic fermion interactions as well as the Yukawa terms reads $$\begin{aligned}
\mathcal{L}_{\rm fermi} = \mathcal{L}_{\rm M} + \mathcal{L}_{\rm Yuk}\,.
\label{eq:Lfermi-LR}\end{aligned}$$ For the mass terms we have $$\begin{aligned}
\begin{aligned}
\label{LM}
&\mathcal{L}_{\rm M} =
\widehat{\mathcal{P}}_{\rm LR} {\left[}\tfrac{1}{2} m_{S_{\rm L}} S_{\rm L} S_{\rm L}
+ \tfrac{1}{2} m_{\mathcal{T}_{\rm L}} \mathcal{T}^i_{\rm L} \mathcal{T}^i_{\rm L} + {\rm c.c.} {\right]}\\
+& \widehat{\mathcal{P}}_{\rm LR} {\left[}\tfrac{1}{2} m_{\tilde{g}} \tilde{g}^a \tilde{g}^a + m_{{\rm LR}} S_{\rm L} S_{\rm R}
+ \tfrac{1}{2} m_{\mathcal{H}} \mathcal{H}_{{\rm F} f}^{\ast} \mathcal{H}_{\rm F}^f {\right]}\,,
\end{aligned}\end{aligned}$$ while for the Yukawa ones we write for convenience, $$\begin{aligned}
\mathcal{L}_{\rm Yuk} = \mathcal{L}_{\rm 3c} + \mathcal{L}_{\rm 2c} + \mathcal{L}_{\rm 1c} + \mathcal{L}_{\mathcal{S}} +
\mathcal{L}_{\mathcal{T}} + \mathcal{L}_{\rm \widetilde{g}}\,,\end{aligned}$$ where the first three terms, which involve only the fields from the fundamental representations of the trinification group, denote three, two and one ${\mathrm{SU}(2)_{\mathrm{}}}$ contractions, respectively, whereas the last ones describe the Yukawa interactions of the singlet $\mathcal{S}$, triplet $\mathcal{T}$ and octet $\widetilde{g}^a$ fermions.
The terms with three ${\mathrm{SU}(2)_{\mathrm{}}}$ contractions are given by $$\begin{aligned}
\begin{split}
\label{L3c}
&\mathcal{L}_{\rm 3c} =
\varepsilon_{f f'} \left(\widehat{\mathcal{P}}_{\rm LR} {\left[}{\mathrm{y}_{{\scalebox{0.5}{$1$}}}} {\mathcal{Q}}_{\rm R}^{f\,r} h^{l}_{r} \mathcal{Q}_{{\rm L}\,l}^{f'}{\right]}+ \widehat{\mathcal{P}}_{\rm LR} \left[{\mathrm{y}_{{\scalebox{0.5}{$2$}}}} \widetilde{q}_{\rm R}^{r} \widetilde{H}^{f\,l}_{r} \mathcal{Q}_{{\rm L}\,l}^{f'}
\right. \right. \\
&
\left. \left.
+ {\mathrm{y}_{{\scalebox{0.5}{$3$}}}} \widetilde{\mathcal{Q}}_{\rm R}^{f\,r}\, \widetilde{h}^{l}_{r} \mathcal{Q}_{{\rm L}\,l}^{f'}
+ {\mathrm{y}_{{\scalebox{0.5}{$4$}}}} q_{\rm R}^{\,r} H^{f\,l}_{r} \mathcal{Q}_{{\rm L}\,l}^{f'} + {\rm c.c.} \right] \right) \,,
\end{split}\end{aligned}$$ those with two ${\mathrm{SU}(2)_{\mathrm{}}}$ contractions are written as $$\begin{aligned}
\begin{split}
\label{L2c}
&\mathcal{L}_{\rm 2c} =
\varepsilon_{f f'} \widehat{\mathcal{P}}_{\rm LR} \left[{\mathrm{y}_{{\scalebox{0.5}{$5$}}}} \widetilde{\mathcal{B}}_{\rm R} \mathcal{Q}_{{\rm L}\,l}^{f} E_{\rm L}^{f'\,l}
+{\mathrm{y}_{{\scalebox{0.5}{$6$}}}} \widetilde{D}_{\rm R}^{f} \mathcal{Q}_{{\rm L}\,l}^{f'} \mathcal{E}_{\rm L}^{l}
\right. \\
&
\left.
+{\mathrm{y}_{{\scalebox{0.5}{$7$}}}} \widetilde{D}_{\rm R}^{f} q_{{\rm L}\,l} E_{\rm L}^{f'\,l}
+{\mathrm{y}_{{\scalebox{0.5}{$8$}}}}\mathcal{B}_{\rm R} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f} E_{\rm L}^{f'\,l}
\right. \\
&
\left.
+{\mathrm{y}_{{\scalebox{0.5}{$9$}}}} D_{\rm R}^{f} \widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f'} \mathcal{E}_{\rm L}^{l}
+{\mathrm{y}_{{\scalebox{0.5}{$10$}}}} D_{\rm R}^{f} \widetilde{q}_{{\rm L}\,l} E_{\rm L}^{f'\,l}
+{\mathrm{y}_{{\scalebox{0.5}{$11$}}}} \mathcal{B}_{\rm R} \mathcal{Q}_{{\rm L}\,l}^{f} \widetilde{E}_{\rm L}^{f'\,l}
\right. \\
&
\left.
+{\mathrm{y}_{{\scalebox{0.5}{$12$}}}} D_{\rm R}^{f} \mathcal{Q}_{{\rm L}\,l}^{f'} (\widetilde{\mathcal{E}}_{\rm L})^{l}
+{\mathrm{y}_{{\scalebox{0.5}{$13$}}}} D_{\rm R}^{f} q_{{\rm L}\,l} \widetilde{E}_{\rm L}^{f'\,l} + {\rm c.c.} \right]\,,
\end{split}\end{aligned}$$ and for those with one ${\mathrm{SU}(2)_{\mathrm{}}}$ contraction we have $$\begin{aligned}
\begin{split}
\label{L1c}
&\mathcal{L}_{\rm 1c} =
\varepsilon_{f f'} \left( \widehat{\mathcal{P}}_{\rm LR} {\left[}{\mathrm{y}_{{\scalebox{0.5}{$14$}}}} D_{\rm R}^f \widetilde{\varphi} D_{\rm L}^{f'}{\right]}+ \widehat{\mathcal{P}}_{\rm LR} \left[
{\mathrm{y}_{{\scalebox{0.5}{$15$}}}} \widetilde{\mathcal{B}}_{\rm R} \phi^f D_{\rm L}^{f'}
\right. \right. \\
&
\left. \left.
+{\mathrm{y}_{{\scalebox{0.5}{$16$}}}} \widetilde{D}_{\rm R}^{f} \phi^{f'} \mathcal{B}_{\rm L}
+{\mathrm{y}_{{\scalebox{0.5}{$17$}}}} \widetilde{D}_{\rm R}^{f} \varphi D_{\rm L}^{f'}
+{\mathrm{y}_{{\scalebox{0.5}{$18$}}}} \mathcal{B}_{\rm R} \widetilde{\phi}^f D_{\rm L}^{f'} + {\rm c.c.} \right] \right) \,.
\end{split}\end{aligned}$$ The part of the Lagrangian involving the singlets $\mathcal{S}_{\rm L,R}$ reads $$\begin{aligned}
\begin{split}
\label{LS}
&\mathcal{L}_{\mathcal{S}} =
\widehat{\mathcal{P}}_{\rm LR} \left[
{\mathrm{y}_{{\scalebox{0.5}{$19$}}}} \widetilde{\mathcal{Q}}^{\ast\,l}_{{\rm L}\,f}\mathcal{S}_{\rm L} {\mathcal{Q}}_{{\rm L}\,l}^{f}
+{\mathrm{y}_{{\scalebox{0.5}{$20$}}}} \widetilde{q}^{\ast\,l}_{\rm L}\mathcal{S}_{\rm L} {q}_{{\rm L}\,l}
+{\mathrm{y}_{{\scalebox{0.5}{$21$}}}} \widetilde{D}_{{\rm L}\,f}^\ast \mathcal{S}_{\rm L} {D}_{\rm L}^{f}
\right. \\
&
\left.
+{\mathrm{y}_{{\scalebox{0.5}{$22$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \mathcal{S}_{\rm L} {\mathcal{B}}_{\rm L}
+{\mathrm{y}_{{\scalebox{0.5}{$23$}}}} H^{\ast \,r}_{f\,l} \mathcal{S}_{\rm L} \widetilde{H}^{f\,l}_{r}
+{\mathrm{y}_{{\scalebox{0.5}{$24$}}}} h^{\ast \,r}_{l}\mathcal{S}_{\rm L} \widetilde{h}^{l}_{r}
\right. \\
&
\left.
+{\mathrm{y}_{{\scalebox{0.5}{$25$}}}} \widetilde{E}_{{\rm L}\,f\,l}^\ast \mathcal{S}_{\rm L} {E}_{\rm L}^{f\,l}
+{\mathrm{y}_{{\scalebox{0.5}{$26$}}}} \widetilde{\mathcal{E}}_{{\rm L}\,l}^\ast \mathcal{S}_{\rm L}{\mathcal{E}}_{\rm L}^{\,l}
+{\mathrm{y}_{{\scalebox{0.5}{$27$}}}}{\widetilde{\phi}^\ast_f} \mathcal{S}_{\rm L} {\phi}^{f}
\right. \\
&
\left.
+{\mathrm{y}_{{\scalebox{0.5}{$28$}}}}{\widetilde{\varphi}^\ast} \mathcal{S}_{\rm L} {\varphi}
+y \mathcal{H}_{{\rm F} f}^{\ast} \mathcal{S}_{\rm L} \mathcal{H}_{\rm F}^f
+{\mathrm{y}_{{\scalebox{0.5}{$29$}}}} \widetilde{E}_{{\rm L}\,f\,l}^\ast \mathcal{S}_{\rm R} {E}_{\rm L}^{f\,l}
\right. \\
&
\left.
+{\mathrm{y}_{{\scalebox{0.5}{$30$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast \mathcal{S}_{\rm R} {\mathcal{B}}_{\rm L}
+{\mathrm{y}_{{\scalebox{0.5}{$31$}}}} \widetilde{D}_{{\rm L}\,f}^\ast \mathcal{S}_{\rm R} {D}_{\rm L}^{f}
+{\mathrm{y}_{{\scalebox{0.5}{$32$}}}} \widetilde{\mathcal{Q}}^{\ast\,l}_{{\rm L}\,f}\mathcal{S}_{\rm R} {\mathcal{Q}}_{{\rm L}\,l}^{f}
\right. \\
&
\left.
+{\mathrm{y}_{{\scalebox{0.5}{$33$}}}} \widetilde{q}^{\ast\,l}_{\rm L}\mathcal{S}_{\rm R} {q}_{{\rm L}\,l}
+{\mathrm{y}_{{\scalebox{0.5}{$34$}}}} \widetilde{\mathcal{E}}_{{\rm L}\,l}^\ast \mathcal{S}_{\rm R}{\mathcal{E}}_{\rm L}^{\,l}
+ {\rm c.c.}
\right]\,,
\end{split}\end{aligned}$$ while those interactions that couple to $\mathcal{T}^i_{\rm L,R}$ read $$\begin{aligned}
\begin{split}
\label{LT}
&\mathcal{L}_{\mathcal{T}} =
\widehat{\mathcal{P}}_{\rm LR} \left[ \left({\sigma_i}\right)^{l}_{l'} \left(
{\mathrm{y}_{{\scalebox{0.5}{$35$}}}} \widetilde{\mathcal{Q}}^{\ast\,l'}_{{\rm L}\,f}\mathcal{T}^i_L {\mathcal{Q}}_{{\rm L}\,l}^{f}
+{\mathrm{y}_{{\scalebox{0.5}{$36$}}}} \widetilde{q}^{\ast \,l'}_{\rm L} \mathcal{T}^i_L{q}_{{\rm L}\,l}
\right. \right. \\
&
\left. \left.
+{\mathrm{y}_{{\scalebox{0.5}{$37$}}}} H^{\ast \,r}_{f\,l} \mathcal{T}^i_L \widetilde{H}^{f\,l'}_{r}
+{\mathrm{y}_{{\scalebox{0.5}{$38$}}}} h^{\ast \,r}_{l} \mathcal{T}^i_L \widetilde{h}^{l'}_{r}
\right. \right. \\
&
\left. \left.
+{\mathrm{y}_{{\scalebox{0.5}{$39$}}}} \widetilde{E}_{{\rm L}\,f\,l}^\ast \mathcal{T}^i_L {E}_{\rm L}^{f\,l'}
+{\mathrm{y}_{{\scalebox{0.5}{$40$}}}} \widetilde{\mathcal{E}}_{{\rm L}\,l}^\ast \mathcal{T}^i_L {\mathcal{E}}_{\rm L}^{l'}
+ {\rm c.c.}
\right) \right]\,.
\end{split}\end{aligned}$$ Finally, the Yukawa interactions involving gluinos are given by $$\begin{aligned}
\begin{split}
\label{Lglu}
&\mathcal{L}_{\widetilde{g}} =
\widehat{\mathcal{P}}_{\rm LR} \left[
{\mathrm{y}_{{\scalebox{0.5}{$41$}}}} \widetilde{\mathcal{Q}}^{\ast \,l}_{{\rm L}\,f}{\bm{T}^a}\widetilde{g}^a \mathcal{Q}_{{\rm L}\,l}^{f}
+{\mathrm{y}_{{\scalebox{0.5}{$42$}}}} \widetilde{q}^{\ast \,l}_{\rm L} {\bm{T}^a}\widetilde{g}^a{q}_{{\rm L}\,l}
\right. \\
&
\left.
+{\mathrm{y}_{{\scalebox{0.5}{$43$}}}} \widetilde{D}_{{\rm L}\,f}^\ast {\bm{T}^a}\widetilde{g}^a {D}_{\rm L}^{f}
+{\mathrm{y}_{{\scalebox{0.5}{$44$}}}} \widetilde{\mathcal{B}}_{\rm L}^\ast {\bm{T}^a}\widetilde{g}^a{\mathcal{B}}_{\rm L}
+ {\rm c.c.}
\right] \,.
\end{split}\end{aligned}$$
The gauge sector of the LR-symmetric EFT
----------------------------------------
In this section, we consider interactions involving the gauge bosons of the effective SHUT-LR model. For ease of reading, we separate those into the gauge-scalar (gs), gauge-fermion (gf) and pure-gauge (pg) interaction types, $$\begin{aligned}
\mathcal{L}_{\rm gauge} = \mathcal{L}_{\rm gs} + \mathcal{L}_{\rm gf} + \mathcal{L}_{\rm pg}\,,\end{aligned}$$ where Eqs. , and of appendix \[sec:Cov\] can be employed to write $$\begin{aligned}
\begin{split}
\label{Lg}
&\mathcal{L}_{\rm gs} =
{\left(}\bm{D}_{\mu} \widetilde{\varphi} {\right)}^{\ast} {\left(}\bm{D}^{\mu} \widetilde{\varphi} {\right)}+ {\left(}\bm{D}_{\mu} \widetilde{\phi} {\right)}^{\ast}_f {\left(}\bm{D}^{\mu} \widetilde{\phi} {\right)}^f
\\
&
+ {\left(}\bm{{D}}_{\mu} h {\right)}^{\dagger\,r}_{l} {\left(}\bm{{D}}^{\mu} h {\right)}^{l}_{r}
+ {\left(}\bm{{D}}_{\mu} H {\right)}^{\dagger\,r}_{f\,l} {\left(}\bm{{D}}^{\mu} H {\right)}^{f\,l}_{r}
\\
&
+ \eta_{\mu \nu} \widehat{\mathcal{P}}_{\rm LR} {\left[}{\left(}\bm{D}^{\nu} \widetilde{\mathcal{E}}_{\rm L} {\right)}^{\dagger}_{l} {\left(}\bm{D}^{\mu}\widetilde{\mathcal{E}}_{\rm L} {\right)}^{l}
+ {\left(}\bm{D}^{\nu} \widetilde{E}_{\rm L} {\right)}^{\dagger}_{f\,l} {\left(}\bm{D}^{\mu} \widetilde{E}_{\rm L} {\right)}^{f\,l}
\right.
\\
&
\left.
+ {\left(}\bm{D}^{\nu} \widetilde{q}_{\rm L} {\right)}^{\dagger\,l} {\left(}\bm{D}^{\mu}\widetilde{q}_{\rm L} {\right)}_{l}
+ {\left(}\bm{D}^{\nu} \widetilde{\mathcal{Q}}_{\rm L} {\right)}^{\dagger\,l}_{f} {\left(}\bm{D}^{\mu}\widetilde{\mathcal{Q}}_{\rm L} {\right)}^{f}_{l}
\right.
\\
&
\left.
+ {\left(}\bm{D}^{\nu} \widetilde{\mathcal{B}}_{\rm L} {\right)}^{\dagger} {\left(}\bm{D}^{\mu} \widetilde{\mathcal{B}}_{\rm L} {\right)}+ {\left(}\bm{D}^{\nu}\widetilde{D}_{\rm L} {\right)}^{\dagger}_{f} {\left(}\bm{D}^{\mu} \widetilde{D}_{\rm L} {\right)}^{f} {\right]}\\
&\mathcal{L}_{\rm gf} =
{\mathrm i}\varphi^{\dagger} \overline{\sigma}_{\mu} \bm{D}^{\mu} \varphi
+ {\mathrm i}\phi^{\dagger}_f \overline{\sigma}_{\mu} {\left(}\bm{D}^{\mu} \phi{\right)}^f
+ {\mathrm i}\widetilde{h}^{\dagger\,r}_{l} \overline{\sigma}_{\mu} {\left(}\bm{{D}}^{\mu} \widetilde{h} {\right)}^{l}_{r}
\\
&
+ {\mathrm i}\widetilde{H}^{\dagger\,r}_{f\,l} \overline{\sigma}_{\mu} {\left(}\bm{{D}}^{\mu} \widetilde{H} {\right)}^{f\,l}_{r}
+ \widehat{\mathcal{P}}_{\rm LR} {\left[}{\mathrm i}\mathcal{E}^{\dagger}_{{\rm L}\,l} \overline{\sigma}_{\mu} {\left(}\bm{D}^{\mu}\mathcal{E}_{\rm L} {\right)}^{l}
\right. \\
&
\left.
+ {\mathrm i}E^{\dagger}_{{\rm L}\,f\,l} \overline{\sigma}_{\mu} {\left(}\bm{D}^{\mu} E_{\rm L} {\right)}^{f\,l}
+ {\mathrm i}q^{\dagger\,l}_{\rm L} \overline{\sigma}_{\mu} {\left(}\bm{D}^{\mu} q_{\rm L} {\right)}_{l}
\right. \\
&
\left.
+ {\mathrm i}\mathcal{Q}^{\dagger\,l}_{{\rm L}\,f} \overline{\sigma}_{\mu} {\left(}\bm{D}^{\mu} \mathcal{Q}_{\rm L} {\right)}^{f}_{l}
+ {\mathrm i}\mathcal{B}^{\dagger}_{\rm L} \overline{\sigma}_{\mu} \bm{D}^{\mu} \mathcal{B}_{\rm L}
+ {\mathrm i}D^{\dagger}_{{\rm L}\,f} \overline{\sigma}_{\mu} {\left(}\bm{D}^{\mu} D_{\rm L} {\right)}^{f} {\right]}\\
&
+ \sum_{\rm A = L,R} {\left[}{\mathrm i}\mathcal{S}^\dagger_A\overline{\sigma}_\mu\partial^\mu\mathcal{S}_A
+ {\mathrm i}\mathcal{T}^{i\dagger}_A \overline{\sigma}_\mu {\left(}\bm{D}^{\mu} \mathcal{T}_A {\right)}^i {\right]}\\
&
+ {\mathrm i}\widetilde{g}^{a\,\dagger} \overline{\sigma}_\mu {\left(}\bm{D}^{\mu} \widetilde{g} {\right)}^a + {\rm c.c.}
\\
&
\mathcal{L}_{\rm pg} =
- \dfrac{1}{4} {\left[}\sum_{\rm A = L,R} {\left(}B^{\mu \nu}_{{{\scalebox{0.7}{$\rm A$}}}} B_{{{\scalebox{0.7}{$\rm A$}}} \mu \nu}
+ F^{\mu \nu\,i}_{{{\scalebox{0.7}{$\rm A$}}}} F^{i}_{{{\scalebox{0.7}{$\rm A$}}} \mu \nu} {\right)}\right. \\
&
\left.
+ G^{\mu \nu\,a} G^{a}_{\mu \nu}
+ B^{\mu \nu}_{{{\scalebox{0.7}{$\rm L$}}}} B_{{{\scalebox{0.7}{$\rm R$}}} \mu \nu} {\right]}\,.
\end{split}\end{aligned}$$
### Covariant derivatives and field strengths {#sec:Cov}
The covariant derivatives of the LR-symmetric effective model can be written in a compact matrix form as follows $$\label{CovDer-1}
\begin{aligned}
&\bm{D}^{\mu} \left(H,h\right) = \left( {\mathbb{1}_{{\scalebox{0.6}{$L$}}}} {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$R$}}}}{\partial}^{\mu} - {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{\rm L}$}}}} A^{\mu\, i}_{\rm L}\bm{\tau}^i {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$R$}}}} - {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{\rm R}$}}}} A^{\mu\,i}_{\rm R}\bm{ \tau}^i {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}}
\right.
\\
&
\left.
+ {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{L}$}}}}^{\prime} {Y_{{\scalebox{0.5}{$\mathrm{\rm L}$}}}} B^{\mu}_{\rm L} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}} {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$R$}}}}
+ \;{\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{R}$}}}}^{\prime} {Y_{{\scalebox{0.5}{$\mathrm{R}$}}}} B^{\mu}_{\rm R} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}} {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$R$}}}}\right) \left(H,h\right) \,,
\\
&
\widehat{\mathcal{P}}_{\rm LR} \left[\bm{D}^{\mu} \left(E_L,\mathcal{E}_L\right)\right]= \widehat{\mathcal{P}}_{\rm LR}
\left[\left( {\mathbb{1}_{{\scalebox{0.6}{$L$}}}}{\partial}^{\mu} - {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{\rm L}$}}}} A^{\mu\, i}_{\rm L}\bm{ \tau}^i
\right.\right.
\\
&
\left.\left.
+ {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{L}$}}}}^{\prime} {Y_{{\scalebox{0.5}{$\mathrm{\rm L}$}}}} B^{\mu}_{\rm L} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}} +
{\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{R}$}}}}^{\prime} {Y_{{\scalebox{0.5}{$\mathrm{R}$}}}} B^{\mu}_{\rm R}{\mathbb{1}_{{\scalebox{0.6}{$L$}}}} \right) \left(E_L,\mathcal{E}_L\right)\right]
\\
&\bm{D}^{\mu} \left(\phi,\varphi\right) =\left({\partial}^{\mu} + {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{L}$}}}}^{\prime} {Y_{{\scalebox{0.5}{$\mathrm{\rm L}$}}}} B^{\mu}_{\rm L} +
{\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{R}$}}}}^{\prime} {Y_{{\scalebox{0.5}{$\mathrm{R}$}}}} B^{\mu}_{\rm R} \right) \left(\phi,\varphi\right) \,,
\\
&
\widehat{\mathcal{P}}_{\rm LR} \left[\bm{D}^{\mu}\left(Q_L,q_L\right) \right]=\widehat{\mathcal{P}}_{\rm LR}
\left[\left( {\mathbb{1}_{{\scalebox{0.6}{$C$}}}}{{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}} {\partial}^{\mu} - {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{C}$}}}} G^{\mu \, a}_{\rm C} \bm{T}^a {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}}
\right. \right.
\\
&\left. \left.
- {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{\rm L}$}}}} A^{\mu\, i}_{\rm L}\bm{ \tau}^i {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$C$}}}}
+ {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{L}$}}}}^{\prime} {Y_{{\scalebox{0.5}{$\mathrm{\rm L}$}}}} B^{\mu}_{\rm L}{\mathbb{1}_{{\scalebox{0.6}{$C$}}}} {{{\scalebox{0.7}{$\otimes$}}}} {\mathbb{1}_{{\scalebox{0.6}{$L$}}}}\right) \left(Q_L,q_L\right) \right]\,,
\\
&\widehat{\mathcal{P}}_{\rm LR} \left[\bm{D}^{\mu}\left(D_L,\mathcal{B}_L\right)\right] =\widehat{\mathcal{P}}_{\rm LR}
\left[\left( {\mathbb{1}_{{\scalebox{0.6}{$C$}}}} {\partial}^{\mu} - {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{C}$}}}} G^{\mu \, a}_{\rm C} \bm{T}^{a}
\right. \right.
\\
&\left. \left.
+ {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{L}$}}}}^{\prime} {Y_{{\scalebox{0.5}{$\mathrm{\rm L}$}}}} B^{\mu}_{\rm L} {\mathbb{1}_{{\scalebox{0.6}{$C$}}}} \right) \left(D_L,\mathcal{B}_L\right)\right] \,,
\end{aligned}$$ $$\label{CovDer-2}
\begin{aligned}
&\bm{D}^{\mu}\;\mathcal{T}_A =\left( {\mathbb{1}^{{\scalebox{0.6}{$\rm adj$}}}_{{\scalebox{0.6}{$L,R$}}}} {\partial}^{\mu} - {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{\rm L,R}$}}}} A^{\mu\, i}_{\rm L,R}\bm{\tau}_{{\scalebox{0.6}{$\rm adj$}}}^i \right) \mathcal{T}_A
\\
&\bm{D}^{\mu}\;\widetilde{g} =\left( {\mathbb{1}^{{\scalebox{0.6}{$\rm adj$}}}_{{\scalebox{0.6}{$C$}}}} {\partial}^{\mu} - {\mathrm i}{g_{{\scalebox{0.5}{$\mathrm{\rm C}$}}}} G^{\mu\, a}_{\rm C}\bm{T}_{{\scalebox{0.6}{$\rm adj$}}}^a\right) \widetilde{g} \,,
\end{aligned}$$ where summation is assumed over each pair of repeated indices, ${Y_{{\scalebox{0.5}{$\mathrm{\rm A}$}}}}$ is the ${\mathrm{U}(1)_{\mathrm{A}}}$ hypercharge and ${\mathbb{1}_{{\scalebox{0.6}{$A$}}}}$ and ${\mathbb{1}^{{\scalebox{0.6}{$\rm adj$}}}_{{\scalebox{0.6}{$A$}}}}$ are the identity matrices with the same dimensions of the fundamental and adjoint representations, respectively. The field strength tensors of the ${\mathrm{U}(1)_{\mathrm{A}}}$, ${\mathrm{SU}(2)_{\mathrm{A}}}$ and ${\mathrm{SU}(3)_{\mathrm{C}}}$ gauge symmetries are given by $$\label{Fmn}
\begin{aligned}
B^{\mu \nu}_{{{\scalebox{0.7}{$\rm A$}}}} &= {\partial}^{\mu} B_{\rm A}^{\nu} - {\partial}^{\nu} B_{\rm A}^{\mu} \\
F^{\mu \nu\,i}_{{{\scalebox{0.7}{$\rm A$}}}} &= {\partial}^{\mu} A_{\rm A}^{\nu\,i} - {\partial}^{\nu} A_{\rm A}^{\mu\,i} +
{g_{{\scalebox{0.5}{$\mathrm{A}$}}}} \varepsilon^{ijk} A_{\rm A}^{\mu\,j} A_{\rm A}^{\nu\,k} \\
G^{\mu \nu\,a} &= {\partial}^{\mu} G_{\rm C}^{\nu\,a} - {\partial}^{\nu} G_{\rm C}^{\mu\,a} +
{g_{{\scalebox{0.5}{$\mathrm{C}$}}}} f^{abc} G_{\rm C}^{\mu\,b} G_{\rm C}^{\nu\,c} \,.
\end{aligned}$$
### Abelian $D$-terms {#sec:ADterms}
The ${\mathrm{U}(1)_{\mathrm{L,R}}}$ $D$-terms of the LR-symmetric theory read $$\begin{aligned}
\begin{split}
&\mathcal{D}_{\mathrm{L}} = \frac{1}{\left(1-\frac{\chi^2}{4}\right)}
\left[-\frac{1}{2}\chi\left( X_{\mathrm{R}} - \kappa\right) +
X_{\mathrm{L}} + \kappa \right] \,,
\\
&\mathcal{D}_{\mathrm{R}} = \frac{1}{\left(1-\frac{\chi^2}{4}\right)}
\left[-\frac{1}{2}\chi\left( X_{\mathrm{L}} + \kappa\right) +
X_{\mathrm{R}} - \kappa \right] \,,
\\
& X_{\mathrm{L}} = H^{\ast\,r}_{f\,l} H^{l\,f}_{r}
-2
\widetilde{\phi}_{f}^\ast\widetilde{\phi}^{f}
+
\widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
-2
\widetilde{E}_{{\rm R}\;f}^{\ast\,r} \widetilde{E}_{{\rm R}\,r}^{f}
\\
&\qquad \qquad \quad
-\,
\widetilde{\mathcal{Q}}_{{\rm L}\,f}^{\ast\,l}\widetilde{\mathcal{Q}}_{{\rm L}\,l}^{f}
+2
\widetilde{D}_{{\rm L}\;f}^\ast\widetilde{D}_{\rm L}^{f} \,,
\\
& X_{\mathrm{R}} =
-H^{\ast\,r}_{f\,l} H^{l,f}_{r}
+2
\widetilde{\phi}_{f}^\ast\widetilde{\phi}^{f}
+2
\widetilde{E}_{{\rm L}\;f\,l}^\ast \widetilde{E}_{\rm L}^{f\,l}
-
\widetilde{E}_{{\rm R}\;f}^{\ast\,r} \widetilde{E}_{{\rm R}\,r}^{f}
\\
&\qquad \qquad \quad
+\,
\widetilde{\mathcal{Q}}_{{\rm R}\,f\,r}^{\ast}\widetilde{\mathcal{Q}}_{{\rm R}}^{f\,r}
-2
\widetilde{D}_{{\rm R}\;f}^\ast\widetilde{D}_{\rm R}^{f} \,,
\end{split}\end{aligned}$$ with $f=1,2,3$.
[^1]: The $\mathrm{E}_6$-based models are typically motivated by heterotic string theories where massless sectors consistent with the chiral structure of the SM are naturally described by an ${\rm E_8} \times {\rm E^{\prime}_8}$ gauge theory. For more details we refer the reader to Refs. [@Gursey:1975ki; @Gursey:1981kf; @Achiman:1978vg]
[^2]: Before adding soft SUSY-breaking interactions, $\Delta_\mathrm{F}$ is completely decoupled from the fundamental sector when taking ${\mathrm{SU}(3)_{\mathrm{F}}}$ to be global, meaning that $\lambda_{\bm{1}}$ and $v_{\rm F}$ do not enter in the matching conditions.
[^3]: The breaking ${\mathrm{SU}(2)_{\mathrm{R}}} \times {\mathrm{SU}(2)_{\mathrm{F}}} \times {\mathrm{U}(1)_{\mathrm{L+R}}} \times
{\mathrm{U}(1)_{\mathrm{S}}} \to {\mathrm{U}(1)_{\mathrm{Y}}} \times {\mathrm{U}(1)_{\mathrm{T}}}$ gives rise to six Goldstone bosons, three gauge and three global ones, where the former are $\rm{Im}{\left[}\widetilde{\nu}_\mathrm{R}^1 {\right]},\, \rm{Re}{\left[}\widetilde{e}_\mathrm{R}^1 {\right]}$ and $\rm{Im}{\left[}\widetilde{e}_\mathrm{R}^1 {\right]}$ while the latter ones are $\rm{Im}{\left[}\widetilde{\phi}^2{\right]}$ and $\mathcal{G}_s$.
[^4]: Which diagrams that dominate depends on the specific parameter point and the details of the RG evolution. However, the gauge coupling for $\mathrm{SU}(3)_{\mathrm{C}}$ is larger than any other gauge coupling in the model at all scales, and as such the diagram with the gluino propagator dominates over diagrams with other gauginos, unless the gluino would be significantly heavier.
[^5]: The family ${\mathrm{SU}(3)_{\mathrm{F}}}$ triplets are also split up into ${\mathrm{SU}(2)_{\mathrm{F}}}$ doublets, containing the first and second generations, and singlets corresponding to the third generation.
[^6]: Here, $G_\mathrm{R}^3$ is the gauge boson corresponding to the third generator of $\mathrm{SU}(2)_\mathrm{R}$, $B_\mathrm{L,R}$ the gauge bosons for $\mathrm{U}(1)_\mathrm{L,R}$ and $B_\mathrm{L+R}$ the gauge boson for $\mathrm{U}(1)_\mathrm{L+R}$.
[^7]: The two types coincide since for ${\mathrm{SU}(2)_{\mathrm{}}}$ the three combinations reduce down to two, using that $\varepsilon_{ij}\varepsilon^{kl}=\delta_i^k\delta_j^l-\delta_i^l\delta_j^k$, while for ${\mathrm{SU}(3)_{\mathrm{}}}$ there are only two possible contractions to begin with, and no Levi-Civita tensor to impose a reduction.
|
---
abstract: 'In this paper, we establish several gradient estimates for the positive solution of Porous Medium Equations (PMEs) and Fast Diffusion Equations (FDEs). Our proof is probabilistic and uses martingale techniques and Forward and Backward Stochastic Differential Equations (FBSDEs).'
author:
- 'Ying Hu[^1], Zhongmin Qian[^2] and Zichen Zhang[^3]'
title: Gradient estimates for porous medium and fast diffusion equations via FBSDE approach
---
Introduction
============
The Porous Medium Equation (PME) on $\mathbb{R}^{n}$ is $$\partial _{t}u=\Delta u^{m}\text{, \ \ on \ }(0,\infty )\times \mathbb{R}^{n}\text{,} \label{eq:PME}$$where $m>1$. It is a nonlinear version of heat equation arising in modelling fluid flow or heat transfer in physics. Moreover, this equation is degenerate on $u=0$, hence having finite speed of propagation. On the other hand, when $m\in (0,1)$, (\[eq:PME\]) is called Fast Diffusion Equation (FDE). It appears as diffusion in plasma in physics. We refer to [MR2286292]{} for a comprehensive study.
An important aspect in the study of (\[eq:PME\]) in connection with the existence / uniqueness problem and the regularity problem is to establish a priori estimates, which is an interesting question by its own. There is a large body of knowledge in this respect for the similar partial differential equations in particular those linear equations, namely for linear heat equations or in general for parabolic equations. For porous medium equations, usually only nonnegative solutions are considered for both cases where $m<1$ or $m>1$, as this is natural from physical point of view.
To study positive solutions, mathematically a way of utilizing the non-negativity is to do the transformation: $v=\frac{m}{m-1}u^{m-1}$. It is an analogy of the Hopf transformation for the heat equation in the sense that $$\lim_{m\rightarrow 1}\left( v-\frac{m}{m-1}\right) =\log u\text{ .}$$It is easy to see that $v$ evolves according to the following non-linear parabolic equation: $$\frac{\partial v}{\partial t}=\left( m-1\right) v\Delta v+\left\vert \nabla
v\right\vert ^{2}\text{.} \label{v-eq1}$$It is this equation which can be used to derive various a priori estimates for the non-negative solutions to (\[eq:PME\]).
The fact that the gradient term on the right-hand side of (\[v-eq1\]) is quadratic is useful for many calculations. However, the diffusion coefficient in (\[v-eq1\]) is degenerate at $v=0$. A standard way to overcome this degeneracy is to consider strictly positive solutions which are bounded away from zero at first, then use an approximation argument such as on page 200 in [@MR2286292] to pass to the non-negative solutions. Therefore, our result for nonnegative initial data should be understood in distribution sense. From now on, however, in order to avoid such technical steps which have been treated thoroughly in [@MR2286292], by a positive solution to the porous medium equation we mean a solution which is bounded below by a positive constant, whose initial data is Lipschitz continuous. We on the other hand wish to emphasize that the a priori estimates we are going to establish in this article are independent of the positive lower bound of a positive solution.
In [@MR524760], Aronson and Benilan established that if $m>1-\frac{2}{n}$, then for any positive smooth solution $u$ of (\[eq:PME\]), $$\Delta v\geq -\frac{\alpha }{\left( m-1\right) t}$$with $\alpha =\frac{n\left( m-1\right) }{\left( m-1\right) n+2}$ and $v=\frac{m}{m-1}u^{m-1}$, or equivalently $$\left\vert \nabla v\right\vert ^{2}-\frac{\partial v}{\partial t}\leq \frac{\alpha v}{t}\text{.} \label{gr-e1}$$The proceeding *a priori* estimate plays a key role in the study of existence of initial value problem for PME. A lot of work have been done to improve this result. Let us recall here some results which are related to gradient estimates for positive solutions to (\[eq:PME\]). We apologize for any possible omissions due to authors’ limited knowledge.
In [@MR2487898] P. Lu, L. Ni, J. L. Vazquez and C. Villani established a local version of Aronson-Benilan’s estimate for (\[eq:PME\]) where $m>1-\frac{2}{n}$ on a complete Riemannian manifold with Ricci curvature bounded from below. Assume that Ric$\geq -\left( n-1\right) K^{2}$ on $B\left(
\mathcal{O},R\right) $ for some $K\geq 0$, and $u$ is a positive smooth solution $u$ to (\[eq:PME\]) on $B\left( \mathcal{O},R\right) \times \left[
0,T\right] $ ($B\left( \mathcal{O},R\right) $ denotes the ball of radius $R>0 $ with centre $\mathcal{O}$). If $m>1$ and $\beta >1$, they proved that $$\left\vert \nabla v\right\vert ^{2}-\beta v_{t}\leq \alpha \beta ^{2}\left(
\frac{1}{t}+C_{2}K^{2}v_{\max }^{R,T}\right) v+\alpha \beta ^{2}\frac{v_{\max }^{R,T}}{R^{2}}C_{1}v$$on $B\left( \mathcal{O},\frac{R}{2}\right) \times \left[ 0,T\right] $, with $$C_{1}=40\left( m-1\right) \left[ 3+\left( n-1\right) \left( 1+KR\right) \right] +\frac{200\alpha \beta ^{2}m^{2}}{\beta -1}$$and $C_{2}=\frac{\left( m-1\right) \left( n-1\right) }{\beta -1}$, where $v_{\max }^{R,T}=\max_{B\left( \mathcal{O},R\right) \times \left[ 0,T\right]
}v$. They also established that, if $m\in \left( 1-\frac{2}{n},1\right) $ and $\gamma \in \left( 0,1\right) $, $$\left\vert \nabla v\right\vert ^{2}-\gamma v_{t}\leq \frac{\alpha \gamma ^{2}}{C_{3}}\left( \frac{1}{t}+C_{4}\sqrt{C_{3}}K^{2}\bar{v}_{\max
}^{R,T}\right) v+\frac{\alpha \gamma ^{2}}{C_{3}}\frac{\bar{v}_{\max }^{R,T}}{R^{2}}C_{5}v$$on $B\left( \mathcal{O},\frac{R}{2}\right) \times \left[ 0,T\right] $, where $$\begin{aligned}
C_{3} &=&1-\alpha \left( 1-\gamma \right) -\left( 1-\gamma \right) \frac{\left( 1-\alpha +\epsilon _{1}\right) ^{2}}{\left( 1-\gamma \right) -\alpha
-\left( 1-\gamma \right) \epsilon _{2}^{2}}\text{,} \\
C_{4} &=&\frac{\left( m-1\right) \left( n-1\right) }{\left( \gamma -1\right)
\epsilon _{2}}\text{,} \\
C_{5} &=&1600m^{2}\frac{\alpha \gamma ^{2}}{2\epsilon _{1}\left( \gamma
-1\right) }+40\left( 1-m\right) \left[ 3+\left( n-1\right) \left(
1+KR\right) \right]\end{aligned}$$and $\bar{v}_{\max }^{R,T}=\max_{B\left( \mathcal{O},R\right) \times \left[
0,T\right] }\left( -v\right) $, and $\epsilon _{1}$ and $\epsilon _{2}$ are positive constants such that$$\begin{aligned}
1-\gamma -\alpha -\left( 1-\gamma \right) \epsilon _{2}^{2} &>&0\text{,} \\
1-\alpha \left( 1-\gamma \right) -\left( 1-\gamma \right) \frac{\left(
1-\alpha +\epsilon _{1}\right) ^{2}}{1-\gamma -\alpha -\left( 1-\gamma
\right) \epsilon _{2}^{2}} &>&0\text{.}\end{aligned}$$
On the other hand, some gradient bounds of Hamilton type have been established by various authors in recent years. This is kind of estimates on the space derivatives of the solution. In L. A. Caffarelli, J. L. Vazquez and N. I. Wolanski [@MR891781], the problem of (\[eq:PME\]) with a compactly supported initial data has been studied thoroughly. Among of other interesting results, they proved that for $m>1$, there exist a time $T=T\left( u_{0}\right) >0$ and a constant $c=c\left( m,n\right) >0$ such that for any $t>T$ and almost everywhere $x\in \mathbb{R}^{n}$ $$\left\vert \nabla v\left( x,t\right) \right\vert \leq c\left( \left( \frac{v}{t}\right) ^{\frac{1}{2}}+\frac{\left\vert x\right\vert }{t}\right)$$if $u_{0}\geq 0$ is integrable with a compact support. In a recent paper [@MR2853544] by X. Xu, positive solutions of (\[eq:PME\]) on a complete Riemannian manifold have been considered, where $m\in \left( 1-\frac{4}{n+3},\infty \right) $ and the Ricci curvature is bounded below: Ric$\geq -k$ for some $k\geq 0$. In particular, it was proved in [@MR2853544] that if $m>1$ and if there exists a constant $\delta \in \left( 0,\frac{4}{n-1}\right] $ such that $$1\leq \frac{v_{\max }^{R,T}}{v_{\min }^{R,T}}<\frac{1}{1+\delta }\left(
\frac{4m}{\left( n-1\right) \left( m-1\right) }+1\right)$$then $$\frac{\left\vert \nabla v\right\vert }{v_{\max }^{R,T}\left( 1+\delta
\right) -v}\leq C\left( n,m\right) \left( \frac{1+\delta }{\rho \delta R}+\frac{1}{\sqrt{\frac{m-1}{m}v_{\max }^{R,T}\delta \rho T}}+\sqrt{\frac{k}{\delta }}\right)$$on $B\left( x_{0},\frac{R}{2}\right) \times \left[ t_{0}-\frac{T}{2},t_{0}\right] $, where $$\rho =2m-\frac{\left( n-1\right) \left( m-1\right) }{2}\frac{v_{\max
}^{R,T}\left( 1+\delta \right) -v_{\min }^{R,T}}{v_{\min }^{R,T}},$$$v_{\max }^{R,T}=\sup_{B\left( x_{0},R\right) \times \left[ t_{0}-T,t_{0}\right] }v$ and $v_{\min }^{R,T}=\inf_{B\left( x_{0},R\right) \times \left[
t_{0}-T,t_{0}\right] }v$. In particular, if $n=1$ $$\frac{\left\vert \nabla v\right\vert }{v_{\max }^{R,T}\left( 1+\delta
\right) -v}\leq C\left( m\right) \left( \frac{1+\delta }{\delta R}+\frac{1}{\sqrt{\frac{m-1}{m}v_{\max }^{R,T}\delta T}}\right)$$on $B\left( x_{0},\frac{R}{2}\right) \times \left[ t_{0}-\frac{T}{2},t_{0}\right] $ for any $\delta >0$. On the other hand, for $m\in \left( 1-\frac{4}{n+3},1\right) $, it was also proved in [@MR2853544] that $$\frac{\left\vert \nabla v\right\vert }{-v}\leq C\left( m,n\right) \left(
\frac{1}{R}+\frac{1}{\sqrt{\frac{1-m}{m}\bar{v}_{\min }^{R,T}T}}+\sqrt{k}\right)$$on $B\left( x_{0},\frac{R}{2}\right) \times \left[ t_{0}-\frac{T}{2},t_{0}\right] $, where $\bar{v}_{\min }^{R,T}=\inf_{B\left( x_{0},R\right) \times \left[ t_{0}-T,t_{0}\right] }\left( -v\right) $. In X. Zhu [@MR2763753], the author proved a similar result for positive solution of (\[eq:PME\]) for $m\in \left( 1-\frac{2}{n},1\right) $ which reads as the following $$\frac{\left\vert \nabla v\right\vert }{\sqrt{-v}}\leq C\left( m,n,\bar{v}_{\max }^{R,T}\right) \sqrt{\bar{v}_{\max }^{R,T}}\left( \frac{1}{R}+\frac{1}{\sqrt{T}}+\sqrt{k}\right)$$on $B\left( x_{0},\frac{R}{2}\right) \times \left[ t_{0}-\frac{T}{2},t_{0}\right] $, where $\bar{v}_{\max }^{R,T}=\sup_{B\left( x_{0},R\right) \times \left[ t_{0}-T,t_{0}\right] }\left( -v\right) $.
In this paper, by using a connection of a solution and its gradient to a backward stochastic differential equation, we derive global estimates with explicit constants for the gradients of a positive solution to the porous medium equation. Our estimates, in contrast with the preceding estimates, involve only the supremum norm of the initial data rather than the norm over the solution over whole evolving time interval $[0,T]$, furthermore the constants appearing in our a priori estimate are explicit and are close to optimal. Our method uses some martingale methods and the link between PDE and BSDE borrowing some idea from the paper [@HQ].
\[mth1a\]Let $u>0$ be a positive solution to the porous medium equation (\[eq:PME\]).
1\) If $m\in (1,1+\frac{2}{n})$, then $$|\nabla u^{\frac{3}{2}(m-1)}(t,\cdot )|\leq \frac{3||u_{0}^{m-1}-1||_{\infty
}^{2}}{\sqrt{2m}t}\frac{1}{\sqrt{t}}\text{,\ \ \ \ \ \ \ }\forall t>0\text{ .} \label{e-6-5-1}$$
2\) If $n=1$ and $m>1$, then$$|\nabla u^{m-1}(t,\cdot )|\leq \frac{\sqrt{2(m-1)||u_{0}^{m-1}-1||_{\infty }}}{m}\frac{1}{\sqrt{t}}\text{, \ \ \ \ \ \ \ }\forall t>0\text{.}
\label{e-6-5-2}$$
3\) If $1-\frac{6}{n+8}\leq m<1$, then $$|\nabla u^{1-m}(t,\cdot )|\leq \frac{\sqrt{2}||u_{0}^{1-m}-1||_{\infty }\sqrt{||u_{0}^{1-m}-1||_{\infty }+1}}{\sqrt{m}}\frac{1}{\sqrt{t}}\text{, \ \
\ \ \ \ }\forall t>0\text{.} \label{e-6-5-3}$$
4\) If $m\in (\frac{n-1}{n+3},1)$, then$$|\nabla \log u(t,\cdot )|\leq \frac{2\sqrt{\frac{m}{1-m}||u_{0}^{1-m}-1||_{\infty }}}{m^{2}\sqrt{\left\vert 2m-4-\frac{\sqrt{2}}{m}\beta
_{2}\right\vert }}\frac{1}{\sqrt{t}}\text{, \ \ \ \ }\forall t>0\text{ }
\label{e-6-5-4}$$where $\beta _{2}$ is given in equation (\[beta2\]).
This paper is organized as follows: in the next section, we introduce the PME, SDE and BSDE and their link. In section 3, we use martingale method to state the main result on gradient estimate when $m>1$. The last section is devoted to the gradient estimate when $0<m<1$.
Fundamental equations and FBSDE
===============================
Consider a positive solution $u$ to the porous medium equation (\[eq:PME\]). We will use the following transformation: $$f=\frac{m}{m-1}(u^{m-1}-1)$$in the case that $m>1$. Since$$f=\frac{m}{m-1}(u^{m-1}-u^{0})\rightarrow \log u\text{ \ \ as }m\downarrow 1\text{,}$$the transformation coincides with the Hopf transformation for the linear heat equation, though in the case that $m\in (0,1)$ we will use a slightly different transformation, see (\[eq-5-29-0\]) below. Then $$(m-1)f+m=mu^{m-1}>0\text{ ,}$$a fact will be used throughout the paper.
It is easy to verify that $f$ satisfies equations:$$\nabla f=mu^{m-2}\nabla u\text{, \ }|\nabla f|^{2}=m^{2}u^{2m-4}|\nabla
u|^{2}\text{,} \label{5-6-1}$$$$\begin{aligned}
\Delta f &=&\frac{m}{m-1}\Delta (u^{m})^{\frac{m-1}{m}} \notag \\
&=&(u^{m})^{\frac{m-1}{m}-1}\Delta u^{m}-\frac{1}{m}m^{2}u^{-m-1}u^{2m-2}|\nabla u|^{2} \notag \\
&=&u^{-1}\Delta u^{m}-\frac{1}{m}u^{-m+1}|\nabla f|^{2} \label{5-6-2}\end{aligned}$$and$$\frac{\partial }{\partial t}f=mu^{m-2}\frac{\partial }{\partial t}u\text{ .}
\label{5-6-3}$$Hence$$\Delta f=u^{-m+1}\frac{1}{m}\partial _{t}f-\frac{1}{m}u^{-m+1}|\nabla f|^{2}$$that is$$\frac{\partial }{\partial t}f=((m-1)f+m)\Delta f+|\nabla f|^{2}\text{.}
\label{eq1}$$
If $m\in (0,1)$, then we use a slight different transformation $$f=\frac{m}{1-m}(u^{1-m}-1)\text{.} \label{eq-5-29-0}$$Then $f$ is a solution to $$\frac{\partial }{\partial t}f=\frac{m^{2}}{(1-m)f+m}\Delta f+\frac{m^{2}(2m-1)}{\left( (1-m)f+m\right) ^{2}}|\nabla f|^{2}\text{.}
\label{eqmless1}$$
The link between Partial Differential Equation (PDE) and Backward Stochastic Differential Equation (BSDE) is by now well known, see, e.g. [@MR1176785], [@MR1262970], [@MR2257138]. In this section, we introduce the SDE and BSDE associated with PME (\[eq:PME\]) and recall this link in our framework. Throughout this paper, we suppose that $u$ is a bounded smooth solution to (\[eq:PME\]).
Given $T>0$. We start with the PDE (\[eq1\]), and write down a Forward-Backward Stochastic Differential Equation (FBSDE) for this PDE. Let $(\Omega ,\mathcal{F},P)$ be a complete probability space, and $W$ be an $n$-dimensional Brownian motion on it. And let $\{\mathcal{F}_{t}\}$ be the natural augmented filtration of $W$.
Let $X=(X_{t})$ be the diffusion process defined to be the unique (strong) solution flow of the following stochastic differential equation (SDE)$$dX=\sqrt{2((m-1)f(T-t,X)+m)}\circ dW\text{, \ }X_{0}=x \label{sde-6-5-1}$$where $\circ dW$ denotes the Stratonovich differentiation, which may be written in terms of Itô integral:$$dX=\sqrt{2((m-1)f(T-t,X)+m)}dW+\frac{m-1}{2}\nabla f(T-t,X)dt\text{ .}
\label{sde-6-5-2}$$
Let $Y=f(T-\cdot ,X)$, $U=2((m-1)Y+m)$, $Z^{i}=\sqrt{U}\frac{\partial f}{\partial x^{i}}(T-\cdot ,X)$ and $Z=(Z^{i})=\sqrt{U}\nabla f(T-\cdot ,X)$. Then $$dX=\sqrt{U}dW+\frac{m-1}{2\sqrt{U}}Zdt\text{,}\quad X_{0}=x\text{ .}
\label{eq:FSDE}$$Itô’s formula applying to $f(T-t,X_{t})$ yields $$Y_{T}-Y_{t}=\int_{t}^{T}ZdW+\int_{t}^{T}\frac{m-3}{2}\frac{|Z|^{2}}{U}ds\text{,}$$that is$$dY=ZdW+\frac{m-3}{2}\frac{|Z|^{2}}{U}dt\text{,}\quad Y_{T}=f(0,X_{T})\text{.}
\label{eq:BSDE}$$The coupled system (\[eq:FSDE\]) and (\[eq:BSDE\]) form a system of FBSDEs.
Next, let us recall the representation of $Z$ by the stochastic flow of $Y$.
Let $J_{j}^{i}=\frac{\partial }{\partial x^{j}}X^{i}$ be the Jacobi matrix, its inverse $K_{j}^{i}$. Let $Y^{i}=\frac{\partial }{\partial x^{i}}Y$ and $Z_{j}^{i}=\frac{\partial }{\partial x^{j}}Z^{i}$. We will use the following relation$$Y^{i}=\frac{\partial }{\partial x^{i}}Y=\frac{\partial }{\partial X^{k}}f(T-t,X)J_{i}^{k}=\frac{1}{\sqrt{U}}Z^{k}J_{i}^{k}$$so that $Z^{i}=\sqrt{U}Y^{l}K_{i}^{l}$, here and thereafter the Einstein convention is enforced, that is, repeated indices are summed up within their range.
Finally, we are in a position to write down the stochastic differential for $Z$ from this representation. The following computation involving the tangent process is standard, see for detail K. Elworthy [@MR675100], N. Ikeda and S. Watanabe [@MR637061].
Differentiating the SDE for $X$ (forward equation) to obtain the SDE for $J$:$$\begin{aligned}
dJ_{j}^{i} &=&\frac{(m-1)Y^{j}}{\sqrt{U}}dW^{i}-\frac{(m-1)^{2}}{2U^{3/2}}Y^{j}Z^{i}dt \\
&&+\frac{m-1}{2\sqrt{U}}Z_{j}^{i}dt\text{ .}\end{aligned}$$We then derive the equation for $K$ by using $KJ=JK=I$, which is given by$$\begin{aligned}
dK_{j}^{i} &=&-\frac{m-1}{U}Z^{j}K_{k}^{i}dW^{k}-\frac{m-1}{2\sqrt{U}}K_{k}^{i}K_{j}^{l}Z_{l}^{k}dt \notag \\
&&+\frac{3(m-1)^{2}}{2U^{2}}K_{k}^{i}Z^{k}Z^{j}dt\text{ .} \label{e-6-5-a}\end{aligned}$$
Differentiating the backward equation we obtain a SDE for $Y^{i}$:$$dY^{i}=Z_{i}^{k}dW^{k}+\frac{(m-3)Z^{k}Z_{i}^{k}}{U}dt-\frac{(m-3)(m-1)}{U^{2}}|Z|^{2}Y^{i}dt\text{ .} \label{e-6-5-b}$$Thus, using (\[e-6-5-a\]) and (\[e-6-5-b\]), together with integration by parts finally in a position to write down the equation for $Z$, which takes the following form:$$\begin{aligned}
dZ^{i} &=&\sqrt{U}K_{i}^{l}Z_{l}^{k}\left( dW^{k}+\frac{(3m-7)}{2}\frac{Z^{k}}{U}dt\right) \notag \\
&&-\frac{(m-3)(m-1)}{2}\frac{Z^{i}|Z|^{2}}{U^{2}}dt \notag \\
&&-\frac{m-1}{\sqrt{U}}Z^{i}K_{k}^{l}Z_{l}^{k}dt\text{.} \label{e-6-5-c}\end{aligned}$$
Porous Medium Equations
=======================
In this section we develop explicit estimates for the derivative of a positive solution $u$ to the porous equation (\[eq:PME\]), where $m>1$, and $f=\frac{m}{m-1}(u^{m-1}-1)$.
Gradient Estimates
------------------
The first ingredient of our approach is a Girsanov transformation of probability. In order to have some flexibility, we introduce a family of change of probability depending on a parameter $\varepsilon$.
\[lema1\]$\frac{Z}{U}\cdot W$ is a BMO martingale.
Making change of probability and define$$d\tilde{W}^{k}=dW^{k}-\varepsilon \frac{Z^{k}}{U}dt\text{.} \label{6-5-10}$$Then (\[e-6-5-c\]) and (\[eq:BSDE\]) can be rewritten as$$\begin{aligned}
dZ^{i} &=&\sqrt{U}K_{i}^{l}Z_{l}^{k}d\tilde{W}^{k}-\frac{(m-3)(m-1)}{2}\frac{Z^{i}|Z|^{2}}{U^{2}}dt \notag \\
&&+\frac{(3m-7+2\varepsilon )}{2}K_{i}^{l}Z_{l}^{k}\frac{Z^{k}}{\sqrt{U}}dt-\frac{m-1}{\sqrt{U}}Z^{i}K_{k}^{l}Z_{l}^{k}dt \label{5-7-1}\end{aligned}$$and$$dY=Z^{k}d\tilde{W}^{k}+\frac{m-3+2\varepsilon }{2}\frac{|Z|^{2}}{U}dt\text{.}
\label{5-7-2}$$
According to [@HQ], we work out Doob-Meyer’s decomposition for $|Z|^{2}$ under the probability $Q$, and identify conditions under which $|Z|^{2}$ is a $Q$-submartingale.
\[lema2\]Let $\theta _{k}=\sum_{l}K_{k}^{l}Z_{l}^{k}$, and $\delta
=3m-7+2\varepsilon $. Then$$\begin{aligned}
d|Z|^{2} &=&2\sqrt{U}Z^{i}K_{i}^{l}Z_{l}^{k}d\tilde{W}^{k} \notag \\
&&+\left[ (n+3-(n+1)m)(m-1)+(m-1)\delta -\frac{\delta ^{2}}{4}\right] \frac{|Z|^{4}}{U^{2}}dt+A\text{,} \label{5-9-5}\end{aligned}$$where$$\begin{aligned}
A &=&U\sum_{i\neq k}\left\vert \sum_{l}K_{i}^{l}Z_{l}^{k}+\frac{\delta }{2U^{3/2}}Z^{i}Z^{k}\right\vert ^{2} \notag \\
&&+U\left\vert \theta _{k}-\frac{1}{2U^{3/2}}\left( \delta
|Z^{k}|^{2}-2(m-1)|Z|^{2}\right) \right\vert ^{2}. \label{5-9-5aa}\end{aligned}$$
By (\[5-7-1\]) and integration by parts we obtain$$\begin{aligned}
d|Z|^{2} &=&2\sqrt{U}Z^{i}K_{i}^{l}Z_{l}^{k}d\tilde{W}^{k}+(3-m)(m-1)\frac{|Z|^{4}}{U^{2}}dt \notag \\
&&+U\sum_{k=1}^{n}\left[ \frac{\theta _{k}}{U^{3/2}}\left( \delta
|Z^{k}|^{2}-2(m-1)|Z|^{2}\right) +\left\vert \theta _{k}\right\vert ^{2}\right] dt \notag \\
&&+U\sum_{i\neq k}\left( \left\vert \sum_{l}K_{i}^{l}Z_{l}^{k}\right\vert
^{2}+\frac{\delta }{U^{3/2}}Z^{i}Z^{k}K_{i}^{l}Z_{l}^{k}\right) dt\text{.}
\label{5-7-3}\end{aligned}$$While$$\begin{aligned}
&&\frac{\theta _{k}}{U^{3/2}}\left( \delta |Z^{k}|^{2}-2(m-1)|Z|^{2}\right)
+\left\vert \theta _{k}\right\vert ^{2} \\
&=&\left\vert \theta _{k}-\frac{1}{2U^{3/2}}\left( \delta
|Z^{k}|^{2}-2(m-1)|Z|^{2}\right) \right\vert ^{2} \\
&&-\frac{\delta ^{2}}{4U^{3}}|Z^{k}|^{4}+(m-1)\delta |Z^{k}|^{2}\frac{|Z|^{2}}{U^{3}}-\frac{|Z|^{4}}{U^{3}}(m-1)^{2}\end{aligned}$$and the last term on the right-hand side of (\[5-7-3\]) can be handled as the following$$\begin{aligned}
&&\sum_{i\neq k}\left[ \left\vert \sum_{l}K_{i}^{l}Z_{l}^{k}\right\vert ^{2}+\frac{\delta }{U^{3/2}}Z^{i}Z^{k}\sum_{l}K_{i}^{l}Z_{l}^{k}\right] \\
&=&\sum_{i\neq k}\left\vert \sum_{l}K_{i}^{l}Z_{l}^{k}+\frac{\delta }{2U^{3/2}}Z^{i}Z^{k}\right\vert ^{2}-\frac{\delta ^{2}}{4U^{3}}\sum_{i\neq
k}|Z^{i}|^{2}|Z^{k}|^{2}\end{aligned}$$so that (\[5-7-3\]) yields (\[5-9-5\]).
\[lema3\]Let us suppose $m\leq 1+\frac{2}{n}$ and choose $\delta =2(m-1)$. Then $|Z|^{2}$ becomes a $Q$-submartingale.
Choosing $\delta =2(m-1)$ in (\[5-9-5\]), we deduce that $$d|Z|^{2}\geq 2\sqrt{U}Z^{i}K_{i}^{l}Z_{l}^{k}d\tilde{W}^{k}+(n+2-nm)(m-1)\frac{|Z|^{4}}{U^{2}}dt\text{,} \label{5-9-4}$$from which we deduce the result.
Finally from the BSDE (\[5-7-2\]) satisfied by $(Y,Z)$ under $Q$ and Lemma 2.2 in [@DHB], $$E^{Q}[\int_{0}^{T}|Z|^{2}dt]\leq 4||f_{0}||_{\infty }^{2}.$$This together with the fact that $|Z|^{2}$ is a $Q$-submartingale gives us the main result of this section.
\[thm:main\] Suppose $m\leq 1+\frac{2}{n}$. Then $$((m-1)f(T,x)+m)|\nabla f(T,x)|^{2}\leq \frac{2||f_{0}||_{\infty }^{2}}{T}
\label{est-1}$$which is equivalent to (\[e-6-5-1\]).
Since $$E^{Q}[\int_{0}^{T}|Z|^{2}dt]\leq 4||f_{0}||_{\infty }^{2}$$and $|Z|^{2}$ is a $Q$-submartingale, we deduce that $$T|Z_{0}|^{2}\leq 4||f_{0}||_{\infty }^{2}$$which yields (\[est-1\]).
One-dimensional case
--------------------
In Theorem \[thm:main\], we have given an estimate on $((m-1)f+m)|\nabla
f|^{2}$. The objective of this subsection is to try to obtain an estimate on $|\nabla f|^{2}$. The idea comes from the observation that from BSDE ([5-7-2]{}), $$\left\vert \frac{m-3+2\varepsilon }{2}\right\vert E^{Q}\left[ \int_{0}^{T}\frac{|Z|^{2}}{U}dt\right] \leq 2||f_{0}||_{\infty }\text{ .}$$Hence it is sufficient to give some condition under which $M=\frac{|Z|^{2}}{U}$ is a $Q$-submartingale.
\[lema4\]Let $M=\frac{|Z|^{2}}{U}$ and $\beta =2\varepsilon -3-m$. Then$$\begin{aligned}
dM &=&\left( \frac{2}{\sqrt{U}}Z^{i}K_{i}^{l}Z_{l}^{k}-2(m-1)U^{-2}|Z|^{2}Z^{k}\right) d\tilde{W}^{k}
\notag \\
&&+\left( (m-1)^{2}(1-n)-\beta ^{2}\right) \frac{|Z|^{4}}{U^{3}}dt+B\text{,}
\label{5-9-11}\end{aligned}$$where $$\begin{aligned}
B &=&\sum_{k=1}^{n}\left\vert \theta _{k}+\frac{\beta
|Z^{k}|^{2}-2(m-1)|Z|^{2}}{2U^{3/2}}\right\vert ^{2} \notag \\
&&+\sum_{i\neq k}\left\vert \sum_{l}K_{i}^{l}Z_{l}^{k}+\frac{\beta }{2U^{3/2}}Z^{i}Z^{k}\right\vert ^{2}\text{.} \label{6-5-12}\end{aligned}$$
Recall that, by (\[5-7-3\]), $$\begin{aligned}
d|Z|^{2} &=&2\sqrt{U}Z^{i}K_{i}^{l}Z_{l}^{k}d\tilde{W}^{k}+(3-m)(m-1)\frac{|Z|^{4}}{U^{2}}dt \notag \\
&&+U\sum_{k=1}^{n}\left[ \frac{\theta _{k}}{U^{3/2}}\left( \left(
3m-7+2\varepsilon \right) |Z^{k}|^{2}-2(m-1)|Z|^{2}\right) +\left\vert
\theta _{k}\right\vert ^{2}\right] dt \notag \\
&&+U\sum_{i\neq k}\left( \left\vert \sum_{l}K_{i}^{l}Z_{l}^{k}\right\vert
^{2}+\frac{\left( 3m-7+2\varepsilon \right) }{U^{3/2}}Z^{i}Z^{k}K_{i}^{l}Z_{l}^{k}\right) dt\text{,} \label{5-9-7}\end{aligned}$$where $\theta _{k}=\sum_{l}K_{k}^{l}Z_{l}^{k}$, $U=2(m-1)Y+2m$, and$$dY=Z^{k}d\tilde{W}^{k}+\frac{m-3+2\varepsilon }{2}\frac{|Z|^{2}}{U}dt\text{.}
\label{5-9-8}$$According to Itô’s formula$$\begin{aligned}
d\frac{1}{U} &=&d\left[ 2(m-1)Y+2m\right] ^{-1} \notag \\
&=&-2(m-1)U^{-2}Z^{k}d\tilde{W}^{k}-\left( 1+2\varepsilon -3m\right) (m-1)\frac{|Z|^{2}}{U^{3}}dt\text{,} \notag\end{aligned}$$so that, integrating by parts we obtain$$\begin{aligned}
dM &=&\frac{2}{\sqrt{U}}Z^{i}K_{i}^{l}Z_{l}^{k}d\tilde{W}^{k}-2(m-1)U^{-2}|Z|^{2}Z^{k}d\tilde{W}^{k} \notag \\
&&+2(m+1-\varepsilon )(m-1)\frac{|Z|^{4}}{U^{3}}dt \notag \\
&&+\sum_{k=1}^{n}\left[ \left( \beta |Z^{k}|^{2}-2(m-1)|Z|^{2}\right) \frac{\theta _{k}}{U^{3/2}}+\left\vert \theta _{k}\right\vert ^{2}\right] dt
\notag \\
&&+\sum_{i\neq k}\left( \left\vert \sum_{l}K_{i}^{l}Z_{l}^{k}\right\vert
^{2}+\frac{\beta }{U^{3/2}}Z^{i}Z^{k}\sum_{l}K_{i}^{l}Z_{l}^{k}\right) dt\text{.} \label{5-9-9}\end{aligned}$$By using the elementary equalities$$\begin{aligned}
&&\sum_{i\neq k}\left( \left\vert \sum_{l}K_{i}^{l}Z_{l}^{k}\right\vert ^{2}+\frac{\beta }{U^{3/2}}Z^{i}Z^{k}\sum_{l}K_{i}^{l}Z_{l}^{k}\right) \\
&=&\sum_{i\neq k}\left\vert \sum_{l}K_{i}^{l}Z_{l}^{k}+\frac{\beta }{2U^{3/2}}Z^{i}Z^{k}\right\vert ^{2}-\frac{\beta ^{2}}{4}\frac{1}{U^{3}}\sum_{i\neq
k}|Z^{i}|^{2}|Z^{k}|^{2}\end{aligned}$$and$$\begin{aligned}
&&\sum_{k=1}^{n}\left[ \left( \beta |Z^{k}|^{2}-2(m-1)|Z|^{2}\right) \frac{\theta _{k}}{U^{3/2}}+\left\vert \theta _{k}\right\vert ^{2}\right] \\
&=&\sum_{k=1}^{n}\left\vert \theta _{k}+\frac{\beta |Z^{k}|^{2}-2(m-1)|Z|^{2}}{2U^{3/2}}\right\vert ^{2} \\
&&-\frac{\beta ^{2}}{4U^{3}}\sum_{k=1}^{n}|Z^{k}|^{4}+(m-1)\left( \beta
-nm+n\right) \frac{|Z|^{4}}{U^{3}}\text{.}\end{aligned}$$Thus (\[5-9-11\]) follows from (\[5-9-9\]).
It is trivial to see that if $n=1$ and $\beta=0$, then $M$ is a $Q$-submartingale.
\[lema5\]If $n=1$, and $\varepsilon =\frac{3+m}{2}$, then $M$ is a $Q$-submartingale.
\[thm3\]If $n=1$ and $m>1$, then$$|\nabla f(T,x)|^{2}\leq \frac{2||f_{0}||_{\infty }}{mT}$$which is equivalent to (\[e-6-5-2\]).
From BSDE (\[5-7-2\]) for $(Y,Z)$, we know that $${m}E^{Q}[\int_{0}^{T}M_{t}dt]\leq 2||f_{0}||_{\infty },$$from which we deduce $$|\nabla f(T,x)|^{2}\leq \frac{2||f_{0}||_{\infty }}{mT}.$$
Fast diffusion Equations
========================
Forward-Backward Systems
------------------------
If $0<m<1$ and $u$ is a positive solution to $\partial _{t}u=\Delta u^{m}$ we use a different change of variable $f=\frac{m}{1-m}(u^{1-m}-1)$. Note that $$f=\frac{m}{1-m}(u^{1-m}-u^{0})\rightarrow \log u\text{ \ \ as }m\uparrow 1\text{. }$$Then$$\partial _{t}f=\frac{m^{2}}{(1-m)f+m}\Delta f+\frac{m^{2}(2m-1)}{\left(
(1-m)f+m\right) ^{2}}|\nabla f|^{2}\text{.} \label{5-10-1}$$
As in Section 2, we first derive a coupled system of FBSDEs for this PDE. Run a diffusion $X$ with generator $L=\frac{m^{2}}{(1-m)f+m}\Delta $. Consider the following diffusion$$dX=m\sqrt{2}\left( (1-m)f(T-\cdot ,X)+m\right) ^{-\frac{1}{2}}\circ dW\text{, \ }X_{0}=x\text{. } \label{5-10-2}$$
Let $Y=f(T-\cdot ,X)$,$$U=(1-m)f(T-\cdot ,X)+m\text{ and }Z=\frac{m\sqrt{2}}{\sqrt{U}}\nabla
f(T-\cdot ,X)\text{ .} \label{5-10-4}$$Then, we may rewrite the SDE (\[5-10-2\]) as the following (in Itô’s integral)
$$dX=\frac{m\sqrt{2}}{\sqrt{U}}dW-\frac{\sqrt{2}m(1-m)}{4}\frac{Z}{U^{\frac{3}{2}}}dt\text{ .} \label{5-10-10}$$
For simplicity, let $J_{j}^{i}=\frac{\partial }{\partial x^{j}}X^{i}$, $K_{j}^{i}$ is the inverse of $J$. Let $Y_{j}=\frac{\partial }{\partial x^{j}}Y,U_{j}=\frac{\partial }{\partial x^{j}}U$, $Z_{j}^{\alpha }=\frac{\partial
}{\partial x^{j}}Z^{\alpha }$. Then$$U_{j}=(1-m)\sum_{i=1}^{n}J_{j}^{i}\frac{\partial f}{\partial x^{i}}(T-\cdot
,X)=(1-m)Y_{j} \label{5-10-5}$$and$$Y_{j}=\sum_{i=1}^{n}\frac{\partial }{\partial X^{i}}f(T-t,X)J_{j}^{i}=\frac{\sqrt{U}}{\sqrt{2}m}Z^{i}J_{j}^{i}\text{.} \label{5-10-6}$$Therefore$$Z^{i}=\sqrt{2}mU^{-\frac{1}{2}}Y_{j}K_{i}^{j}\text{ .} \label{5-10-7}$$As $Z=\frac{m\sqrt{2}}{\sqrt{U}}\nabla f(T-\cdot ,X),$we have $$|\nabla f|^{2}(T-\cdot ,X)=\frac{1}{2m^{2}}U|Z|^{2}.$$By Itô’s formula to $f(T-\cdot ,X)$, $$dY=Z.dW-\frac{(3m-1)}{4}\frac{|Z|^{2}}{U}dt\text{ .} \label{5-10-8}$$
(\[5-10-10\]) and (\[5-10-8\]) form a coupled system of FBSDEs. Now differentiating in the initial data we deduce that$$dY_{j}=Z_{j}^{i}dW^{i}-\frac{(3m-1)}{2}Z_{j}^{i}\frac{Z^{i}}{U}dt+\frac{(3m-1)(1-m)}{4}\frac{|Z|^{2}Y_{j}}{U^{2}}dt\text{ .} \label{5-10-9}$$On the other hand, differentiating $X$ in the initial data in the stochastic differential equation (\[5-10-10\]) we obtain$$dJ_{j}^{i}=-\frac{(1-m)}{2}\frac{Z^{a}}{U}J_{j}^{a}dW^{i}+\frac{3}{2}\frac{(1-m)^{2}}{4}\frac{Z^{i}Z^{a}}{U^{2}}J_{j}^{a}dt-\frac{\sqrt{2}m(1-m)}{4}\frac{Z_{j}^{i}}{U^{\frac{3}{2}}}dt\text{ .} \label{5-10-11}$$It follows that, by using $KJ=I$, to obtain$$dK_{j}^{i}=\frac{(1-m)}{2}\frac{K_{k}^{i}Z^{j}}{U}dW^{k}+\frac{\sqrt{2}m(1-m)}{4}\frac{K_{c}^{i}K_{j}^{k}Z_{k}^{c}}{U^{\frac{3}{2}}}dt-\frac{1}{2}\frac{(1-m)^{2}}{4}\frac{K_{k}^{i}Z^{k}Z^{j}}{U^{2}}dt\text{.} \label{5-10-12}$$We then can derive the stochastic differential equation for $Z^{i}$. Indeed by integrating by parts the following relation $Z^{i}=\sqrt{2}mU^{-\frac{1}{2}}Y_{j}K_{i}^{j}$ , we deduce $$\begin{aligned}
\frac{1}{\sqrt{2}m}dZ^{i} &=&\frac{\sum_{l}K_{i}^{l}Z_{l}^{k}}{\sqrt{U}}dW^{k}-\frac{3(3m-1)(m-1)}{8}\frac{Y_{j}K_{i}^{j}}{\sqrt{U}U}\frac{|Z^{k}|^{2}}{U}dt \notag \\
&&-\frac{5m-1}{4}\frac{K_{i}^{c}Z_{c}^{k}Z^{k}}{U\sqrt{U}}dt-\frac{m-1}{2}\frac{Z_{j}^{k}K_{k}^{j}Z^{i}}{U\sqrt{U}}dt\text{.} \label{6-5-13}\end{aligned}$$
Gradient Estimates
------------------
We are now in a position to establish the following
If $1-\frac{6}{n+8}\leq m\leq 1$, then $$|\nabla f(T,x)|\leq \frac{\sqrt{2}||f_{0}||_{\infty }\sqrt{(1-m)||f_{0}||_{\infty }+m}}{m\sqrt{T}} \label{e-6-7-1}$$which yields (\[e-6-5-3\]).
From (\[6-5-13\]), we have $$\begin{aligned}
d|Z|^{2} &=&2\sqrt{2}m\frac{K_{i}^{j}Z_{j}^{k}Z^{i}}{\sqrt{U}}dW^{k}+\frac{3}{4}(3m-1)(1-m)\frac{|Z|^{4}}{U^{2}}dt \notag \\
&&-\frac{\sqrt{2}(5m-1)m}{2}\frac{K_{i}^{j}Z_{j}^{k}Z^{i}Z^{k}}{U\sqrt{U}}dt+\sqrt{2}(1-m)m\frac{Z_{j}^{k}K_{k}^{j}|Z|^{2}}{U\sqrt{U}}dt \notag \\
&&+2m^{2}\sum_{k}\left\vert \frac{K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}\right\vert
^{2}dt\text{.} \label{5-11-3}\end{aligned}$$Let us return to the equation for $Y$, which may be written as $$dY=\sum_{k}Z^{k}\left( dW^{k}-\varepsilon \frac{Z^{k}}{U}dt\right) -(\frac{3m-1}{4}-\varepsilon )\frac{|Z|^{2}}{U}dt. \label{5-11-14}$$Hence we first define a new probability such that $$d\tilde{W}^{k}=dW^{k}-\varepsilon \frac{Z^{k}}{U}dt\text{ }$$is a Brownian motion. Then $$\begin{aligned}
d|Z|^{2} &=&2\sqrt{2}m\frac{K_{i}^{j}Z_{j}^{k}Z^{i}}{\sqrt{U}}d\tilde{W}^{k}+\frac{3}{4}(3m-1)(1-m)\frac{|Z|^{4}}{U^{2}}dt \notag \\
&&+\sum_{k}\left[ \left( \delta |Z^{k}|^{2}+\sqrt{2}(1-m)m|Z|^{2}\right)
\frac{1}{U}\frac{\theta _{k}}{\sqrt{U}}+2m^{2}\left\vert \frac{\theta _{k}}{\sqrt{U}}\right\vert ^{2}\right] dt \notag \\
&&+\sum_{k\neq i}\left[ 2m^{2}\left\vert \frac{\sum_{j}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}\right\vert ^{2}dt+\delta \frac{Z^{i}Z^{k}}{U}\frac{\sum_{j}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}\right] dt \label{5-11-7}\end{aligned}$$where $$\delta =2\sqrt{2}m\varepsilon -\frac{\sqrt{2}(5m-1)m}{2}\text{, \ \ }\theta
_{k}=\sum_{j}K_{k}^{j}Z_{j}^{k}\text{.}$$Since$$\begin{aligned}
&&\sum_{k}\left[ \left( \delta |Z^{k}|^{2}+\sqrt{2}(1-m)m|Z|^{2}\right)
\frac{1}{U}\frac{\theta _{k}}{\sqrt{U}}+2m^{2}\left\vert \frac{\theta _{k}}{\sqrt{U}}\right\vert ^{2}\right] \\
&=&2m^{2}\sum_{k}\left\vert \frac{\theta _{k}}{\sqrt{U}}+\left( \delta
|Z^{k}|^{2}+\sqrt{2}(1-m)m|Z|^{2}\right) \frac{1}{4m^{2}U}\right\vert ^{2} \\
&&-\left( \frac{n(1-m)^{2}}{4}+\frac{\sqrt{2}\delta (1-m)}{4m}\right) \frac{|Z|^{4}}{U^{2}}-\frac{\delta ^{2}}{8m^{2}U^{2}}\sum_{k}|Z^{k}|^{4}\end{aligned}$$and$$\begin{aligned}
&&\sum_{k\neq i}\left[ 2m^{2}\left\vert \frac{\sum_{j}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}\right\vert ^{2}dt+\delta \frac{Z^{i}Z^{k}}{U}\frac{\sum_{j}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}\right] \\
&=&2m^{2}\sum_{k\neq i}\left\vert \frac{\sum_{j}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}+\frac{\delta }{4m^{2}}\frac{Z^{i}Z^{k}}{U}\right\vert ^{2}-\frac{\delta ^{2}}{8m^{2}}\frac{1}{U^{2}}\sum_{k\neq i}|Z^{i}|^{2}|Z^{k}|^{2}\text{ .}\end{aligned}$$Hence$$d|Z|^{2}=2\sqrt{2}m\frac{Z^{i}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}d\tilde{W}^{k}+(A+B)dt+\frac{G(\delta )}{4}\frac{|Z|^{4}}{U^{2}}dt \label{5-11-9}$$where$$G(\delta )=3(3m-1)(1-m)-n(1-m)^{2}-\frac{\sqrt{2}(1-m)}{m}\delta -\frac{1}{2m^{2}}\delta ^{2}\text{,} \label{5-11-12}$$$$A=2m^{2}\sum_{k\neq i}\left\vert \frac{\sum_{j}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}+\frac{\delta }{4m^{2}}\frac{Z^{i}Z^{k}}{U}\right\vert ^{2}$$and$$B=2m^{2}\sum_{k}\left\vert \frac{\theta _{k}}{\sqrt{U}}+\left( \delta
|Z^{k}|^{2}+\sqrt{2}(1-m)m|Z|^{2}\right) \frac{1}{4m^{2}U}\right\vert ^{2}\text{.}$$
Setting$$\delta =-\sqrt{2}m(1-m)$$then$$G(\delta )=\left[ (n+8)m-2-n)\right] (1-m)\ge 0$$if and only if$$1-\frac{6}{n+8}\le m\le 1\text{.}$$
Using the same argument as that in Theorem \[thm:main\], we obtain ([e-6-7-1]{}).
Estimates on Logarithm of Gradient
----------------------------------
Let us prove the following
\[thm6\] If $1>m\geq \frac{n-1}{n+3}$, then$$|\nabla \log u(T,x)|\leq \frac{2\sqrt{||f_{0}||_{\infty }}}{m^{2}\sqrt{T\left\vert 2m-4-\frac{\sqrt{2}}{m}\beta _{2}\right\vert }}\text{,}
\label{6-5-14}$$where $\beta _{2}$ is given in (\[beta2\]), which yields (\[e-6-5-4\])
The idea of the proof for this theorem is to identify conditions under which $M=\frac{|Z|^{2}}{U}$is a $Q$-submartingale.
Since$$d\frac{1}{U}=-(1-m)U^{-2}\sum_{k}Z^{k}d\tilde{W}^{k}+(1-m)(\frac{3}{4}-\varepsilon -\frac{m}{4})U^{-2}\frac{|Z|^{2}}{U}dt, \notag$$ together with (\[5-11-7\]) we get $$\begin{aligned}
UdM &=&2\sqrt{2}m\frac{K_{i}^{j}Z_{j}^{k}Z^{i}}{\sqrt{U}}d\tilde{W}^{k}-(1-m)\frac{|Z|^{2}}{U}Z^{k}d\tilde{W}^{k} \\
&&+\sum_{k}\left[ \left( \beta |Z^{k}|^{2}+\sqrt{2}(1-m)m|Z|^{2}\right)
\frac{1}{U}\frac{\theta _{k}}{\sqrt{U}}+2m^{2}\left\vert \frac{\theta _{k}}{\sqrt{U}}\right\vert ^{2}\right] dt \\
&&+\sum_{k\neq i}\left[ 2m^{2}\left\vert \frac{\sum_{j}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}\right\vert ^{2}dt+\beta \frac{Z^{i}Z^{k}}{U}\frac{\sum_{j}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}\right] dt \\
&&+(1-m)(2m-\varepsilon )\frac{|Z|^{4}}{U^{2}}dt\end{aligned}$$where$$\beta =\delta -2\sqrt{2}m(1-m)\text{, \ }\delta =2\sqrt{2}m\varepsilon -\frac{\sqrt{2}(5m-1)m}{2}$$so that$$\varepsilon =\frac{\sqrt{2}\beta }{4m}+\frac{m+3}{4}\text{. }$$Finally, $$\begin{aligned}
UdM &=&2\sqrt{2}m\frac{K_{i}^{j}Z_{j}^{k}Z^{i}}{\sqrt{U}}d\tilde{W}^{k}-(1-m)\frac{|Z|^{2}}{U}Z^{k}d\tilde{W}^{k} \\
&&+H(\beta )\frac{|Z|^{4}}{4U^{2}}+Ldt,\end{aligned}$$where $$H(\beta )=(1-m)\left( (7+n)m-3-n\right) -\frac{2\sqrt{2}\beta (1-m)}{m}-\frac{\beta ^{2}}{2m^{2}},$$and$$\begin{aligned}
&&L=2m^{2}\sum_{k}\left\vert \frac{\theta _{k}}{\sqrt{U}}+\left( \beta
|Z^{k}|^{2}+\sqrt{2}(1-m)m|Z|^{2}\right) \frac{1}{4m^{2}U}\right\vert ^{2} \\
&&+2m^{2}\sum_{k\neq i}\left\vert \frac{\sum_{j}K_{i}^{j}Z_{j}^{k}}{\sqrt{U}}+\frac{\beta }{4m^{2}}\frac{Z^{i}Z^{k}}{U}\right\vert ^{2}.\end{aligned}$$
Hence, if $m\geq \frac{n-1}{n+3}$, then if $\beta \in \lbrack \beta
_{1},\beta _{2}]$, with $$\beta _{1}=-2\sqrt{2}m(1-m)-m\sqrt{2(1-m)((3+n)m+1-n)} \label{beta1}$$and$$\beta _{2}=-2\sqrt{2}m(1-m)+m\sqrt{2(1-m)((3+n)m+1-n)}\text{.} \label{beta2}$$
Recall that$$dY=\sum_{k}Z^{k}d\tilde{W}^{k}-\left( 2m-4-\frac{\sqrt{2}}{m}\beta \right)
\frac{|Z|^{2}}{4U}dt$$and$$d\tilde{W}^{k}=dW^{k}-\left( \frac{\sqrt{2}}{m}\beta +3+m\right) \frac{Z^{k}}{4U}dt$$
Hence, $$\left\vert 2m-4-\frac{\sqrt{2}}{m}\beta \right\vert \int_{0}^{T}\frac{|Z|^{2}}{4U}dt\leq 2||f_{0}||_{\infty }.$$The same argument as that in Section 3.2 leads (\[6-5-14\]).
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[^1]: IRMAR, Université Rennes 1, 35042 Rennes Cedex, France. Email: ying.hu@univ-rennes1.fr. This author is partially supported by the Marie Curie ITN Grant, “Controlled Systems”, GA no.213841/2008.
[^2]: Mathematical Institute, University of Oxford, Oxford OX1 3LB, England. Email: qianz@maths.ox.ac.uk.
[^3]: Mathematical Institute, University of Oxford, Oxford OX1 3LB, England. Email: zhangz@maths.ox.ac.uk.
|
---
abstract: 'Near field cosmology is practiced by studying the Local Group (LG) and its neighbourhood. The present paper describes a framework for simulating the “near field” on the computer. Assuming the model as a prior and applying the Bayesian tools of the Wiener filter (WF) and constrained realizations of Gaussian fields to the Cosmicflows-2 (CF2) survey of peculiar velocities, constrained simulations of our cosmic environment are performed. The aim of these simulations is to reproduce the LG and its local environment. Our main result is that the LG is likely a robust outcome of the scenario when subjected to the constraint derived from CF2 data, emerging in an environment akin to the observed one. Three levels of criteria are used to define the simulated LGs. At the base level, pairs of halos must obey specific isolation, mass and separation criteria. At the second level the orbital angular momentum and energy are constrained and on the third one the phase of the orbit is constrained. Out of the 300 constrained simulations 146 LGs obey the first set of criteria, 51 the second and 6 the third. The robustness of our LG ‘factory’ enables the construction of a large ensemble of simulated LGs. Suitable candidates for high resolution hydrodynamical simulations of the LG can be drawn from this ensemble, which can be used to perform comprehensive studies of the formation of the LG.'
author:
- 'Edoardo Carlesi,$^{1}$ [^1] Jenny G. Sorce,$^{2}$ Yehuda Hoffman,$^{1}$ Stefan Gottlöber,$^{2}$ Gustavo Yepes,$^{3,4}$'
- 'Noam I. Libeskind,$^{2}$ Sergey V. Pilipenko,$^{5,6}$ Alexander Knebe,$^{3,4}$ Hélène Courtois,$^{7}$'
- |
R. Brent Tully,$^{8}$ Matthias Steinmetz$^{2}$\
\
$^{1}$Racah Institute of Physics, 91040 Givat Ram, Jerusalem, Israel\
$^{2}$Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, D-144 Potsdam, Germany\
$^{3}$Grupo de Astrofísica, Departamento de Fisica Teorica, Modulo C-8, Universidad Autónoma de Madrid, Cantoblanco E-280049, Spain\
$^{4}$Astro-UAM, UAM, Unidad Asociada CSIC\
$^{5}$Moscow Institute of Physics and Technology, Institutskij per. 9, 141700 Dolgoprudnyj, Russia\
$^{6}$Astro Space centre of Lebedev Physical Institute of Russian Academy of Sciences, Profsojuznaja st. 84/32, 117997 Moscow, Russia\
$^{7}$University of Lyon, UCB Lyon 1/CNRS/IN2P3; IPN Lyon, France\
$^{8}$Institute for Astronomy (IFA), University of Hawaii, 2680 Woodlaum Drive, HI 96822, US
bibliography:
- 'biblio.bib'
date: Submitted XXXX January XXXX
title: 'Constrained Local UniversE Simulations: A Local Group Factory'
---
\[firstpage\]
methods:$N$-body simulations – galaxies: haloes – cosmology: theory – dark matter
Introduction {#sec:intro}
============
These are exciting times for cosmology. Observations of the anisotropies of the cosmic microwave background (CMB) radiation by the WMAP and Planck observatories have provided spectacular validation of the standard model of cosmology, the $\Lambda$CDM. Observations of distant objects, spanning a look-back time of over 12 Gyrs, provide further support for the $\Lambda$CDM predictions for the growth of structure in the universe. The basic tenets of the model consist of an early inflationary phase, a prolonged phase a homogeneous and isotropic expansion dominated by the dark matter (DM) and dark energy, and structure that emerges out of a primordial perturbation field via gravitational instability.
Cosmology is the science of the biggest possible generalization. It deals with the Universe as a whole. This leads to an inherent tension between the drive, on the hand, to study the general properties of everything that we observe and, on the other, the wish to study the particularities of our own patch of the Universe. It follows that cosmology can be practiced by observing the Universe on the largest possible scales. But it can be practiced also by observing the very ‘local’ universe, resulting in the so-called near field cosmology [@Bland-Hawtorn:2006; @Bland-Hawtorn:2014]. Near field cosmology can test some of the predictions of the $\Lambda$CDM model, and indeed possible conflicts have been uncovered. Locally observed dwarf and satellite galaxies seem to be at odd with predictions based on cosmological galaxy formation simulations [e.g. @Peebles:2010; @Oman:2015 and references therein]. It is this tension between cosmology, practiced at large, and the near-field cosmology which motivates the Constrained Local UniversE Simulations (CLUES) project[^2] in general and the present paper in particular. Our aim here is to present a numerical laboratory which enables the testing of the near field cosmology against cosmological simulations.
The modus operandi of standard cosmological simulations is that they are designed to represent a typical and random realization of the Universe within a given computational box. Indeed, cosmological simulations have been the leading research tool in cosmology and the formation of the large scale structure (LSS). Near field cosmology poses a challenge to the standard cosmological simulations - how to associate environs and objects from the random simulations with our own Local Group (LG) and its environment? Bayesian reconstruction methods and constrained simulations provide an alternative to standard cosmological simulations. The basic idea behind these methods is the use of a Bayesian inference methodology to construct constrained realizations of the local universe of either the present epoch or initial conditions for numerical simulations. These constrained realizations are designed to obey a set of observational data and an assumed theoretical prior model. Two main streams have been followed - one uses galaxy peculiar velocity surveys and the other galaxy redshift surveys. @Ganon:1993 were the first to generate constrained initial conditions from peculiar velocity surveys, and these were used by @Kolatt:1996 to run the first constrained simulations of the local universe. This early work has been extended by the CLUES project, within which many velocity constrained simulations have been performed [@Gottloeber:2010; @Yepes:2014 and references therein]. @Bistolas:1998 and later @Mathis:2002 ran the first redshift survey constrained simulations of the local universe, based on the IRAS survey [@Davis:1991]. The application of Bayesian quasi-linear Hamiltonian Markov Chain Monte Carlo sampling methods to galaxy redshift surveys has provided a new and interesting way to reconstruct the local universe [see e.g. @Kitaura:2010; @Jasche:2010; @Wang:2014; @Ata:2015; @Lavaux:2016]. Yet, these methods have a limited scope of resolution of a few megaparsecs and therefore are unable to resolve the LG itself. Presently the only available constrained simulations of the LG are the ones conducted by the CLUES project, to be described below.
In the case of the old CLUES simulations [i.e. the ones reported in @Yepes:2014] the underlying methodology gives raise to only a small number of “realistic“ (numerical) LGs - ones with mass, distance and relative velocities akin to the observed LG. In fact, roughly 200 constrained simulations yielded only 4 acceptable LG candidates. Though the smallness of the sample hindered a statistically systematic study of the LG within the cosmology, performing high resolution zoom simulations of two of them it has nonetheless been possible to address a large number of relevant cosmological and astrophysical issues, such as the universality of the DM halo profiles [@Mariposa:2013], properties of substructure [@Libeskind:2010], local implications of the Warm Dark Matter paradigm [@Libeskind:2013a] and peculiarities of the mass aggregation history of the LG [@Forero-Romero:2011].
One of the major aims of the CLUES project is to turn it into a ’factory’ that produces on demand LG-like simulated objects, allowing for a systematic study of the properties of the LG - within the framework of the model and the Cosmicflows-2 data base of peculiar velocities [CF2; @Tully:2013 see the discussion below]. The present paper shows how the incorporation of the CF2 new data and the improved methodology have turned the CLUES project into an efficient ’factory’ that produces LGs, essentially on demand.
The basic pillar of the CLUES approach rests on the fact that in the standard model of cosmology the primordial perturbation field constitutes a Gaussian random field. The Bayesian linear tools of the Wiener filter (WF) and the constrained realisations (CRs) of Gaussian fields enable the construction of ICs constrained both by a given observational data base and an assumed prior model [@Hoffman:1991; @Zaroubi:1995]. It follows that there are two attractors that ’pull’ the ICs - the prior (cosmological) model and the observational data. Where - either in configuration or in the resolution (k) space - the data is strong the ICs reproduce the constraints, and otherwise they correspond to random realizations of the prior model. To the extent that ’strong’ data is used as constraints the resulting ICs are likely to reproduce the observed local universe. The so-called constrained variance of the constrained realizations is significantly smaller than the cosmic variance of the prior model. Consequently, the reduction in the cosmic variance measures the ’strength’ of the data, for the given prior model.
The improvement in the CLUES constrained simulations has proceeded along two main streams. In the methodological one the original WF/CR algorithm has been amended by the application of the Reverse Zeldovich Approximation (RZA) [@Doumler:2013a; @Doumler:2013b; @Doumler:2013c; @Sorce:2014] technique, which accounts for the Zeldovich displacement of the data points. On the data stream, the CF2 [@Tully:2013] database of galaxy peculiar velocities, corrected for the Malmquist bias by the method described by [@Sorce:2015], is used to constrain the ICs. @Sorce:2016 recently presented this new generation CLUES simulations. The present paper extends it to the case of zoom simulations of the LG.
The work starts with [Section \[sec:cs\]]{}, that explains the basic ideas behind the Constrained Simulation (CS) method and its numerical implementation. [Section \[sec:ln\]]{} describes the Local Universe, listing the properties which will be used as a benchmark for the quality of the reconstruction, while [Section \[sec:simu\]]{} discusses the design of the simulations in relation to the aims of the present study. Then, [Section \[sec:llse\]]{} is devoted to the analysis of the reconstructed Local Neighbourhood: The Local Void, the local filament and the Virgo cluster. [Section \[sec:lg\]]{} contains a discussion on the identification and the properties of LG-like pairs, showing that in spite of the large role played by the random short-wave modes a substantial number of candidates can be identified using distinct classification criteria. In [Section \[sec:end\]]{} the results are summarized and the plan of the future applications of this method is sketched.
Constrained Simulations {#sec:cs}
=======================
At the very core of the constrained simulations lies the [[WF/RZA/CR]{}]{} method. Its main features will be briefly reviewed in the following subsections, addressing the interested reader to the original works for a comprehensive theoretical review.
Methods
-------
The WF is a powerful tool for reconstructing a continuous (Gaussian) field from a sparse dataset, assuming a prior model. Its use in cosmology has been pioneered by @Rybicki:1992 while @Zaroubi:1995 first applied the method to the reconstruction of fields from observational dataset. In practice, the WF reconstruction results in an estimate of the true underlying field which is dominated by the *data* in the region where the sampling is dense while reproducing the *priors* where the data is lacking or uncertain due to large observational errors. The recovered velocity field is then easily converted into the cosmic displacement field, which in turn is used to trace back the object to its progenitor’s position. The latter is called the RZA procedure, which shifts backwards the observed *radial* component of the velocity only. The RZA can be improved replacing the *radial* constraints at these positions by the full three dimensional WF estimation of the velocity field’s components [@Sorce:2014].
However, the recovered velocity field will tend to zero in the regions where data is lacking or dominated by the error, since the WF will tend towards the mean field, which is the assumed prior. This can be compensated by means of the Constrained Realisation algorithm (CR) [@Hoffman:1991], that basically fills the data-poor regions with a random Gaussian field while converging to the constraints’ values where these are present, also ensuring the overall compliance with the prior’s power spectrum. This means that the outcome of a simulation will be determined by the interplay between these random modes and the constraints. In particular, the additional random components will play a role in two separate regimes: 1) The very large scales, where the data is sparse and error dominated and 2) the very small scales, which are dominated by the intrinsic non-linearity of the processes involved and cannot be constrained by the RZA. In our case, understanding of the latter kind of randomness is crucial, since it is dominant on the scales that affect the formation of LG-like objects.
Numerical implementation
------------------------
The constrained white noise fields are generated using the techniques described by @Doumler:2013a [@Doumler:2013b; @Doumler:2013c; @Sorce:2014] on $256^3$ nodes grids, which represent the minimum scale on which the constraints are effective [@Sorce:2016]. Short-wave $k$-modes can be added to the input white noise field using the `Ginnungagap` [^3] numerical package.
This numerical setup, that combines two different codes, turns out to be extremely relevant to our study, since (as anticipated) one will have to deal with two sources of randomness, i.e. the one coming from the constrained white noise field, affecting the data-poor large scales, and the small scale one, introduced by `Ginnungagap` when increasing the mass resolution. This is why, among the other reasons that are explained later, we chose to design and study different sets of simulations that separately relied on the two codes, exploiting their different capabilities in different ways and thus allowing us to spot any numerics-induced effect on the results.
The Local Universe {#sec:ln}
==================
Our definition of the Local Universe encompasses two separate sets of objects; one is the Local Neighbourhood, which we define as the ensemble of the largest structures around the MW, and the other is the Local Group itself. Even though our main goal is the simultaneous reproduction of both of them, the differences in scales and methods to be used demands a step-by-step separate treatment. In what follows, a general description of these objects will be provided, with the aim of establishing a benchmark to gauge the quality of the reconstruction method at different scales and resolutions.
The Local Neighbourhood
-----------------------
$
\begin{array}{ccc}
\includegraphics[height=5.5cm]{wf_cf2_xy} &
\includegraphics[height=5.5cm]{wf_cf2_yz} &
\includegraphics[height=5.5cm]{wf_cf2_xz} \\
(a) & (b) & (c) \\
\end{array}
$
Around the supergalactic coordinates $(-3.1, 11.3, -0.58)$ is located Virgo, the closest cluster of galaxies, whose estimated virial mass exceeds $4\times10^{14}$ [@Tully:2009] and is part of the much larger Laniakea supercluster [@Tully:2014]. Virgo is the most massive object in our neighbourhood, and the effects of its gravitational pull can be observed even at the level of the Local Group, since MW and M31 both live in a filament [@Klypin:2003a; @Libeskind:2015a; @Forero-Romero:2015] that stretches from it to the Fornax cluster.
The main properties of this Local Filament are derived by analysing the eigenvectors of the (observational) velocity shear tensor around the LG position, which can be recovered from the CF2 dataset [@Tully:2013]. The CF2 is a catalog of more than 8000 galaxy distances and peculiar velocities; which have to be further grouped to smooth out the small scale non-linearities, reducing the final number of constraints to less than 5000 points, and treated in order to minimize the biases [@Sorce:2015],
[Fig. \[img:wf\_cf2\]]{} shows the WF reconstruction applied to the CF2 data, plotting three density slices of $\pm25$ side and $\pm5$ thickness centered around the LG position, showing both the velocity streams and overdensity contours; a similar reconstruction was obtained by [@Courtois:2013] using the CF1 dataset. This image clearly depicts the nature of the large scale flows in our neighbourhood; for instance, in the supergalactic $X-Y$ plane it can be seen the flow moving from the center of the box towards Virgo, along the local filament, which then points in the direction of the Great Attractor (not shown) towards the negative $X$. In the $X-Z$ and $Y-Z$ planes, instead, the effect of the local void can be seen in the $Z>0$ quadrant, where the velocity streams clearly indicate an outward movement from such a region.
The local void was first detected and described by @Tully:1987; however, its nature and extension are still uncertain so that estimations of its diameter range from $20$ Mpc [@Nasonova:2011] to $> 45$Mpc [@Tully:2008]. It is this large underdense region, situated in the proximity of the LG, which is responsible for the observed peculiar velocities of galaxies in its vicinity, including MW and M31 [@Tully:2008]. @Libeskind:2015a also showed that the center of the local void is aligned with $\hat{\mathbf{e}}_1$ (the eigenvector associated with the largest eigenvalue, its numerical value is shown in [Table \[tab:ev\_obs\]]{}) the direction that defines the strongest infall of matter, again showing the strong influence of this region on the dynamics in the neighbourhood of the MW.
SGX SGY SGZ $\lambda$
---------------------- ---------- ---------- ---------- -----------
$\hat{\mathbf{e}}_1$ $-0.331$ $-0.318$ $0.881$ $0.148$
$\hat{\mathbf{e}}_2$ $0.788$ $0.423$ $0.446$ $0.051$
$\hat{\mathbf{e}}_3$ $0.517$ $-0.848$ $-0.110$ $-0.160$
: Eigenvector coordinates and eigenvalues at the LG position for the velocity shear tensor, taken from @Libeskind:2015a. Since eigenvectors are non-directional lines, the $+/-$ directions are arbitrary. []{data-label="tab:ev_obs"}
The Local Group
---------------
--------------------------- ------------- ---------------------------
$r_{M31}$ kpc $770\pm40$
$\mathbf{r}_{M31}$ kpc $(-378.9, 612.7, -283.1)$
$\sigma_{\mathbf{r},M31}$ kpc $(18.9, 30.6, 14.5)$
$\mathbf{v}_{M31}$ km s$^{-1}$ $(66.1, -76.3, 45.1)$
$\sigma_{\mathbf{v},M31}$ km s$^{-1}$ $(26.7, 19.0, 26.5)$
$V_{M31, tan}$ km s$^{-1}$ $< 34.4$
$V_{M31, rad}$ km s$^{-1}$ $-109.3\pm4.4$
$M_{200, tot}$ $10^{12}$ $3.14\pm0.58$
$M_{200, MW}$ $10^{12}$ $1.6\pm0.5$
$M_{200, M31}$ $10^{12}$ $1.6\pm0.5$
--------------------------- ------------- ---------------------------
: Observational constraints for MW and M31 in the MW frame of reference. Kinematic properties are taken from @Marel:2012, distances and errors from with @Marel:2008, while MW and M31 masses are consistent with @Marel:2012 [@Boylan-Kolchin:2013]. []{data-label="tab:mc_lg"}
The term Local Group refers to the group of galaxies dominated by M31 and MW. There are more than 70 additional galaxies belonging to it, most of whom are of dwarf type and are co-rotating around thin planes centered around either of the two hosts [@Lynden-Bell:1976; @Pawlowski:2013; @Ibata:2013]. The MW and M31 and their satellites form a relatively isolated system, the largest closest object being Cen A, a group of galaxies located at a distance of around $2.7$. [Table \[tab:mc\_lg\]]{} shows the kinematic properties of the LG’s main players, from which it can be noticed a substantial uncertainty on the mass of the system and a stark difference between the tangential and radial components of the system’s velocity. In particular, the latter property makes the LG an outlier from the dynamical point of view, since a factor of three difference between these two components in halo pairs of this size is very rarely observed in cosmological simulations [@Fattahi:2015].
The simulations {#sec:simu}
===============
The simulation of the Local Universe is a complex task that passes throughout several phases, due to the variety of the techniques involved and to the different nature (large scale and small scale) of the objects to be simulated. The present section motivates the choice of the design of the simulations in relation to the different aims of the present paper, with a particular emphasis on the intrinsic limitations of the CS method.
Aims
----
The properties of the MW and M31 satellites, namely their number and anisotropic distribution, have deep implications for the model itself [@Klypin:1999b; @Kroupa:2005] and is hence one of the strongest motivations for the study of the nearby Universe. Even though these phenomena cannot be constrained directly, the CS framework provides a powerful and robust pipeline to reproduce the environment in which the LG forms, which is an element that might help explaining the peculiarities of MW and M31 [@Gonzalez:2014].
In the context of simulations, this means that one has to disentangle the randomness of the short-wave modes from the large scale constrains and to gauge the effectiveness of CSs and their variance.
Since the reconstruction of the environment is itself a non-trivial task, unaffected by what happens at LG scales, we decide to take a two steps approach. This way one can ensure that larger structures such as Virgo are well recovered without first bothering about what happens to the LG, which is way more problematic due to the predominance of random modes. If this is the case, i.e. the reconstructed local neighbourhood compares well to the observational benchmarks previously outlined, one can at least hope for a successful outcome at smaller scales if some deeper relation between the LG and its environment exists.
In practice, to do this one has to run two series of simulations, addressing different (but related) questions, namely:
- how stable are the properties of the local neighbourhood? Which role does the random component of the CR plays on these scales?
- how many LG-like objects can be obtained at the expected position? How can they be properly defined and characterized?
In the first case it is clear that there is no need for high resolution, since a sufficiently large number of simulations is enough in order to address the question of the stability of the results and to estimate the constrained variance of the CS method in the quasi-linear regime. Even though the CS produces stable results when using boxes of side 500, as shown by @Sorce:2016, smaller simulation volumes deserve a separate analysis. In fact, when dealing with sub-Megaparsec scales it is necessary to deal with the issues of mass resolution (which needs to be high in order to reliably identify MW and M31-sized haloes) and sample variance (which demands to perform a very large number of simulations). This trade-off can be minimized by the use of zoom-in techniques. In order to do this while answering to the questions above, we have designed three series of simulations, which will be described in the following subsection.
Settings
--------
Name $N_{simu}$ $L_{box}$ $R_{zoom}$ $m_{p}$ $z_{start}$
-------------------- ------------ ----------- ------------ -------------------- -------------
[[SimuLN256]{}]{} $100$ 100 NO $5.26\times10^{9}$ $60$
[[SimuLN512]{}]{} $12$ 100 NO $6.57\times10^{8}$ $80$
[[SimuLGzoom]{}]{} $300$ 100 12 $6.57\times10^{8}$ $80$
In [Table \[tab:simu\]]{} are shown the parameter settings for the Initial Conditions (ICs) of the three series of simulations. The size of the box has been chosen as a compromise between the need for resolution on sub-Mpc scales and the necessity to minimize the effects of periodic boundary conditions on Virgo and the local filament. All of the simulations are run using the following cosmological parameters: $\Omega_{m}=0.312$, $\Omega_{\Lambda}=0.688$, $h=0.677$ and $\sigma_8=0.807$ which are compatible with the Planck-I results [@Planck:2013]. To minimize boundary effects and maximize the power of the constraints, the observer is placed at the center of the box: This will be considered our expected LG position, around which candidate pairs will be searched for.
This means that the question about LG-like structures has to be addressed by a different type of simulations. The approach in this case has been that of adding a sufficiently large region around the box center with an equivalent resolution of $6.57\times10^8$, which allow to identify this kind of objects with $\approx 10^3$ Dark Matter (DM) particles. This is enough to ensure their stability in the process of increasing the resolution by adding further levels of mass refinement. The size of this first zoom-in region has been chosen as follows. For each [[SimuLN256]{}]{} run, all the particles within a sphere of $R=5$ around the box center (the expected LG location) at $z=0$ were taken and traced back to their original positions in the ICs. We then computed the center of mass of these particles and the radius $R_{zoom}$ of the smallest sphere enclosing all of them. It turns out that both these quantities are extremely stable so that a single sphere of $R_{zoom}=10$ placed around the coordinates $\mathbf{X}_{init}=(55, 43, 54)$ ( units) would encircle all the 100 spheres generated for each one of the simulations. Then, to generate the ICs for the LG-tailored simulations, the resolution was increased within a sphere of $R_{zoom}=12$ around $\mathbf{X}_{init}$. This pipeline has been tested using several runs with different levels of refinement, in order to ensure that this choice would prevent contamination coming from high mass (low resolution) particles within the region of interest for LG-like objects. This series of zoom-in simulations is labeled [[SimuLGzoom]{}]{}. The large scale white noise field is then drawn from 40 out of the 100 [[SimuLN256]{}]{} runs, while the short waves necessary to run the zoom-ins were generated using the same set of 10 random seeds for 20 simulations and 5 out of those 10 seeds for the remaining ones, making sure not to double count any of the original simulations.
In addition, we ran another series of full box simulations with $512^3$ particles, labeled [[SimuLN512]{}]{}, making use of twelve white noise fields (on a $256^3$ grid) from the [[SimuLN256]{}]{} runs and then increasing the resolution. With this, it was ensured that (1) the resolution of the [[SimuLN256]{}]{} simulations was not affecting the results, i.e. the local neighbourhood results largely overlap and (2) that the properties of the zoom in region of the [[SimuLGzoom]{}]{} runs are the same of those [[SimuLN256]{}]{} ran with the same pair of large scale and small scale random seeds. Once these issues have been settled, ensuring that resolution related issues are under control, no further use of the [[SimuLN512]{}]{}runs has been made.
All the simulations were executed using the publicly available Tree-PM $N$-Body code `GADGET2` [@Springel:2005] on the MareNostrum supercluster at the Barcelona Supercomputing Center, with individual simulations from the [[SimuLN256]{}]{} set running in $\approx60$ core-hours each and the [[SimuLGzoom]{}]{} ones taking approximately twice as much time.
Recovery of the local neighbourhood {#sec:llse}
===================================
\[img:llse\] $
\begin{array}{cc}
\includegraphics[height=7.0cm]{ln_xy} &
\includegraphics[height=7.0cm]{ln_yz}
\end{array}
$
{height="7.0cm"}
This section deals with the issue of the stability of the main large scale structures in the nearby Universe, focusing on the properties of the Virgo cluster and the local filament. The discussion of the local void and its properties will be marginal, since its reconstruction goes beyond the scope of this paper, due to the smallness of the box size. A visual example of the reconstruction of the local neighbourhood is presented in [Fig. \[img:llse\]]{}, where all of the aforementioned structures can be spotted in the dark matter particles distribution and large scale velocity streams.
Naively, one could think that the presence of these objects should be granted by the very nature of the linear WF reconstruction, which is effective on these scales as shown for the CF2 data reconstruction. However, the small box and therefore the effect of periodic boundary conditions effects make the issue of correct reproduction of the local neighbourhood highly non-trivial, deserving to be treated on its own. It is only ensuring that a proper environment stems from the application of the [[WF/RZA/CR]{}]{} that one can move forward and look for LG-candidates.
In what follows, the `AHF` halo finder [@Knollmann:2009] has been used to find Virgo and determine its mass and position. Different algorithms can be used to define a filament [see e.g. @Hahn:2007; @Forero-Romero:2009; @Sousbie:2011; @Falack:2012; @Tempel:2013; @Chen:2015], in this work we used the one described by @Hoffman:2012, which is based on the dimensionless velocity shear tensor
$$\label{eq:shear}
\Sigma_{\alpha\beta} = - \frac{1}{2 H_0} \left ( \frac{\partial v_{\alpha}}{\partial r_{\beta}} -
\frac{\partial v_{\beta}}{\partial r_{\alpha}} \right)$$
to classify a region in space given the number of positive $\Sigma_{\alpha\beta}$ eigenvalues $\lambda$ at that point. Namely, a *void* is defined as an area with no $\lambda>0$, a *sheet* has one $\lambda>0$, a *filament* two and a *knot* three. The stability of the results has been also checked computing the shear tensor over different grids ($128^3$ and $256^3$ nodes) and smoothing lengths (2.5 and 3 ), showing no particular dependence on the particular choice of their value [@Libeskind:2014].
In the case of the local void, we will refrain from providing a detailed quantitative analysis of our results, which is left for future studies. In fact, due to the limitations imposed by the boundary conditions, we know in advance that it is not possible to quantitatively reconstruct a structure of that size within the context of this reconstruction. It is nonetheless worth noticing that a visual inspection of density contours and velocity streamlines consistently reveals the presence of an extended underdense zone in the $Z>0$ area above the center of the box. Moreover, the shear tensor eigenvectors and eigenvalues show how the push coming from this region is compatible with the observational estimates of [@Libeskind:2015a], ensuring that our local void reconstruction, albeit incomplete, is however satisfactory from the point of view of the Local Group and the aims of the current paper.
Virgo
-----
To identify the simulated Virgo candidate, one has to look for the closest and largest structure around the expected observational position in the `AHF` catalogs. The properties of these simulated Virgos are listed in [Table \[tab:virgo\]]{}. The first remarkable result is that the variance of the reconstructed masses and positions is small (in agreement with the findings of @Sorce:2016) as can be deducted by looking at both the spread between the maximum and minimum values and the standard deviation of each of the properties shown.
median $\sigma$ $obs.$
----------- -------- ---------- --------
$SGX$ -2.50 1.06 -2.56
$SGY$ +10.3 0.83 10.9
$SGZ$ +1.87 1.12 -0.512
$M_{200}$ 2.09 0.69 $>4$
: Mean and standard deviation of the simulated Virgo mass ($M_{200}$, in $10^{14}$) and its position in super-galactic coordinates and ). The corresponding observational values are taken from the EDD, [@Tully:2009]. All the coordinates are rescaled by shifting the box center from $(50, 50, 50)$ to $(0, 0, 0)$.[]{data-label="tab:virgo"}
This is a proof of the power of this method on these scales, even with a simulation box of only $100$, where in principle the effect of the periodic boundary conditions could be strong enough to spoil the stability of the results. It is clear that the effect of the random modes is at best marginal and the whole reconstruction pipeline results in very stable cluster-sized Virgo look-alikes, with a median error in the reconstructed position of only $2.67$. As a side note, it can be noticed that the $M_{200}$ is just half of the observed value for the Virgo mass, which is $> 4\times10^{14}$. This is however an expected result, most likely related to the smallness of the box size. In fact, it can be shown [@Sorce:2016] that with a box of $500$ per side the median reconstructed Virgo mass is $\approx3.5\times10^{14}$, and is thus much closer to the observed value. Whereas this effect is known for standard $N$-body simulations [@Power:2006], a detailed investigation of the correlation between the box size and mass of the local supercluster in CSs is beyond the scope of this work and is left for a separate analysis. For the aim of this paper it is sufficient to prove that our local neighbourhood reconstruction includes a cluster-sized object, placed at the right position, whose properties are largely independent from the specific random realisation. A stable Virgo look-alike is hence the first result of the [[WF/RZA/CR]{}]{} pipeline, in agreement with the findings of @Sorce:2016 and a substantial improvement over the past outcomes of constrained simulations performed with observational data [@Gottloeber:2010], where reconstructed local superclusters had a position displacement $> 10$ with respect to the expected value.
The Local Filament
------------------
Filaments are identified using the same procedure described in [Section \[sec:llse\]]{}, based on the algorithm of @Hoffman:2012. It is important to remark that the use of a smoothing length of $2.5$ is dictated by the need to compare to observations, filtering out nonlinearities induced by the unconstrained short wave modes.
The local filament can be found by looking at structures around the center of the box. Since the reconstructed Virgo position turns out to be displaced by $\approx 2.5$ with respect to the observational value, one would first look for the local filament at a position shifted by an equal amount from the box center. However, it turns out that nearest grid point (NGP) of such a displaced position would have (in almost all realisations) only one positive eigenvalue, meaning that the environment should be classified as a sheet. Indeed, the center of the box is an excellent choice to look for a filament, since in 87 out of 100 cases the NGP is characterized by two positive eigenvalues. Moreover, even in the remaining 13 cases a filament can be always found within $\approx5$ from the center, meaning that this kind of structure is a constant feature of the expected LG position.
![Velocity shear tensor eigenvalues and their standard deviations at the center of the box. The red triangles are the averages and standard deviations obtained for the [[SimuLN256]{}]{} while the black circles represent the observational values computed by [@Libeskind:2015a]. The two estimates are in very good agreement, even though they are computed using different smoothing lengths. \[img:ev\_wf\]](ev_wf_simu.png){height="6.0cm"}
[cc]{}
\[tab:align\]
$\quad$ mean $\sigma$
----------------- --------- ---------- --
$\sin\theta _1$ $-0.07$ $0.23$
$\sin\theta _2$ $-0.19$ $0.21$
$\sin\theta _3$ $0.13$ $0.19$
&
\[tab:align2\]
$\quad$ $simu$ $obs$
------------- ------------------ ------------------
$\lambda_1$ $0.174\pm0.062$ $0.148\pm0.038$
$\lambda_2$ $0.052\pm0.075$ $0.051\pm0.039$
$\lambda_3$ $-0.270\pm0.074$ $-0.160\pm0.033$
\
(a) & (b)\
To quantify the goodness of our filament reconstruction we compare our results to @Libeskind:2015a, which were previously shown in [Table \[tab:ev\_obs\]]{}, In [Fig. \[img:ev\_wf\]]{} are plotted the $\lambda$s and their standard deviations, noticing that our estimated values are in very good agreement with their findings. This means that the strength of matter accretion (along $\mathbf{e}_1$ and $\mathbf{e}_2$) and expulsion (along $\mathbf{e}_3$) is very well reproduced by the simulated filament. Moreover, not only the intensity but also the directions of the eigenvectors are well reproduced, as shown in [Table \[tab:align\]]{}. In fact, the small values of $\sin\theta$ presented there (defining $\theta$ as the angle between reconstructed $\mathbf{e}_i$ and $\mathbf{e}^{aWF}_i$) indicate a good alignment between the two vectors.
Recovery of the Local Group {#sec:lg}
===========================
\[img:lg\] $
\begin{array}{cc}
\includegraphics[height=7.0cm]{lg_xy} &
\includegraphics[height=7.0cm]{lg_yz} \\
\end{array}
$
{height="7.0cm"}
Once it is ensured that the reconstruction methods employed here give rise to a configuration of the local neighbourhood which is both stable and realistic, one can start to look for possible LG candidates. As we discussed in [Section \[sec:simu\]]{}, resolution-related problems were bypassed designing a series of zoom-in simulations (labeled [[SimuLGzoom]{}]{}) with a sufficient number of smaller mass particles just around the expected LG position. The white noise field of these simulations has been generated in the following way. We selected $40$ white noise fields from different SimuLN runs, to have a sufficient number of different CR of the large scales to factor out possible specific seed-related effects. Then, for each of these fields were generated five to ten different random realisations of the additional layer of refinement in the Lagrangian region, using `Ginnungagap` giving rise to different sub-Mpc configurations. For some of these realisations, it has been checked that the results in the zoom-in regions overlap with those taken from the set of [[SimuLN512]{}]{} full box $512^3$ simulations.
Identification criteria
-----------------------
The first issue that is encountered in the identification of the LG candidates is related to the details of the definition of a LG-like object. Indeed, given both the observational uncertainties and uneven relevance of the many LG-defining properties to some general problems, the debate on what really constitutes a meaningful candidate is far from settled and several authors in the past have used different criteria [e.g. @Forero-Romero:2013; @Gonzalez:2014; @Sawala:2014; @Libeskind:2015a] to define LG-like pairs in cosmological simulations. Due to the predominance of the random component at sub-Mpc scales, it is not expected that these objects should be a feature of the simulations in the same way the local neighbourhood was, so that imposing restrictive identification criteria from the very start might conceal other interesting results. Moreover, there is a huge number of variables that could be checked and used as criteria to match a simulated LG to the real one. Therefore, we decided to split the identification issue into several steps, taking into account an increasingly larger subset of variables to define our candidates. Namely, we introduced (1) a first broad selection criterion based solely on the mass and position of isolated halo pairs, (2) a second one based on the requirement that angular momentum and energy fall within observational bounds and (3) additionally restricting our sample to pairs with radial and tangential velocities compatible to the LG values. In what follows, we will refer to [[LG-gen]{}]{} as the first LG-compatibility criterion, [[LG-dyn]{}]{} as the second one and [[LG-vel]{}]{} as the third one.
Specifically, a [[LG-gen]{}]{}-type LG candidate will satisfy the following requirements:
- the total mass of the halo pair must be smaller than $5\times10^{12}$
- the mass of the smallest candidate must be larger than $5\times10^{11}$
- the distance $d$ between the two haloes must satisfy $0.3 < d < 1.5$
- there is no other halo of mass greater or equal than the smallest candidate within a radius of $2.5$
- the center of mass of the halo pair must be located within 5 from the box center
This criterion allows us to obtain a first assessment of the success of our method in producing isolated halo pairs at the right position and with a mass of roughly a factor two within the observational constraints on the LG . These are general prerequisite characteristics of any realistic LG, and allow us to construct a first large halo sample which could be used to address a large number of open issues, such as the use of the timing argument [@Li:2008; @Partridge:2013; @Gonzalez:2014]
To identify [[LG-dyn]{}]{} halo pairs the approach of @Forero-Romero:2013 was followed, considering the (reduced) values of energy angular momentum of the [[LG-gen]{}]{} candidates to identify those whose *global* dynamic status is compatible with the actual observations. Defining
$$\label{eq:e}
e = \frac{1}{2}\mathbf{v}_{M31}^2 - \frac{GM}{| \mathbf{r}_{M31}|}$$
as the reduced energy and
$$\label{eq:l}
l = | \mathbf{r}_{M31} \times \mathbf{v}_{M31} |$$
as the reduced angular momentum, one can proceed generating contours in the $e-l$ plane through $10^{7}$ Monte Carlo iterations, realised drawing from the observational values listed in [Table \[tab:mc\_lg\]]{} and assuming Gaussian priors on the $2\sigma$ intervals. Then, for each [[LG-gen]{}]{} pair $e$ and $l$ are computed and all those pairs that fall within the $95\%$ contours are labeled as [[LG-dyn]{}]{}-type. [[LG-vel]{}]{} candidates are those whose radial and tangential velocities fall within $2\sigma$ of the observational values.
Local Group like objects
------------------------
Using the identification criteria outlined in the previous section, one starts looking for potential LG candidates in each of the 300 [[SimuLN256]{}]{} realisations. We proceed as follows. First, a sphere of 5 is taken around the box center and all the [[LG-gen]{}]{} halo pairs therein are listed. The environmental type is then determined associating each halo to its NGP on a $256^3$ grid, where the eigenvalues of the shear tensor were previously computed. In this way we can double-check the results of [Section \[sec:llse\]]{} and make sure that one can identify a filament also at the *actual* LG position. Finally, one can look at the dynamics of the [[LG-gen]{}]{} candidates. On the one hand, it is interesting to see whether their conserved properties are generally compatible with the observed ones in the sense of the [[LG-dyn]{}]{} criterion, singling out candidate pairs disregarding the specific transient dynamical state. On the other hand, however, one can be also interested in checking whether some of those pairs actually have radial and tangential velocities in agreement with observations (type [[LG-vel]{}]{}). [Table \[tab:lg\]]{} shows the number of candidates according to each selection criterion.
Type $N$ $N_f$
---------------- ----- -------
[[LG-gen]{}]{} 146 120
[[LG-dyn]{}]{} 51 42
[[LG-vel]{}]{} 6 6
: Total number of LG-like objects ($N$) and total number of objects located on a filament ($N_f$) in the [[SimuLGzoom]{}]{} runs satisfying the [[LG-gen]{}]{} [[LG-dyn]{}]{} and [[LG-vel]{}]{} criteria.[]{data-label="tab:lg"}
The first important thing to notice is that in approximately half of the simulations (146 out of 300) it is possible to identify a [[LG-gen]{}]{}-type of candidate within 5 of the expected position. These objects are then characterised by the NGP eigenvalues of the two haloes, allowing the pair to be composed of objects living in different environment types. Checking the numbers explicitly for each candidate pair, we find that 120 out of 146 candidates actually belong to a filament. This is an important results for the CSs: one can produce at a $40\%$ rate objects that can be broadly classified as LGs, within just 5 from the expected position and that live within a filament. [Fig. \[img:lg\]]{} offers a visual impression of one of such LG candidates for a single realisation, where the two main haloes can be clearly seen within a filamentary stream of dark matter particles. It is quite clear that such a high success rate has to bear a connection to the shown stability of the local neighbourhood. However, such an investigation is beyond the scope of the present paper and will be dealt with in a following work.
One can compare these rates to what has been found by other authors in the literature to quantify the improvement obtained over previous results. For instance, @Fattahi:2015 found 12 objects looking within a random cosmological simulation box of $100$ per side, corresponding to a density of $1.2 \times10^{-5}$$^3$. Their selection criteria are based on
- relative radial velocity between $-250$ and $0$ km s$^{-1}$
- relative tangential velocity smaller than $100$
- separation between 600 and 1000 kpc
- total pair mass in the range $\log(M_{tot}/M_{\odot})$
Applying these very same constraints on our CSs, a total of 75 objects can be identified. Since our search volume is made up of 300 spheres of radius $5$, the density of LG-like pairs is $4.77\times 10^{-4}$$^3$, that is a factor of $40$ larger. Beside the higher production rate, it has to be stressed that each pair is placed within a large scale environment that closely matches the observed one, which is the most relevant feature of the CS method.
![ \[img:candidates\] [[LG-gen]{}]{} isolated halo pairs. Solid lines show the dynamical states compatible with LG observations, determined by $10^7$ Monte Carlo random realisations. Each halo pair that falls within the $95\%$ confidence intervals belongs to the [[LG-dyn]{}]{} sample.](256_512_candidates.png){height="5.5cm"}
To analyze the dynamical properties of this [[LG-gen]{}]{} halo sample, the focus is placed on $e$ and $l$, the reduced energy of the system and its reduced angular momentum. Ideally, in a completely isolated system, these two quantities would be conserved. However, due to the non-ideal nature of the LG this is only partially true. Because of the high degree of isolation implied by our definition of the LG, the expected departure from the ideal case is small. Consequently, it is expected that $e$ and $l$ are almost perfectly conserved, allowing for a more robust classification of the pair. In fact, it could be misleading to look at distances and velocities only, since these quantities are transient by their very nature and might not reveal the fundamental properties of the pair, which might have looked (or will look) much closer to today’s LG values at some other moment in time. By looking at semi conserved quantities, instead, one is able to single out those structures which have most likely formed with the same local initial conditions and compare their histories and evolution. Indeed, the reconstruction in a statistical sense of the history and evolution of the LG is one of the ultimate goals of the CLUES collaboration. As described in the previous paragraph, [[LG-dyn]{}]{} candidates were identified as those halo pairs falling within the Monte Carlo generated $95\%$ confidence intervals in the $e - l$ plane, selected from the previously identified [[LG-gen]{}]{}-type objects located on a filament. These candidates are plotted in [Fig. \[img:candidates\]]{}. From there and from [Table \[tab:lg\]]{} one sees that the number of such object is non-negligible, as one can find 42 pairs, corresponding to a $14\%$ success rate. Again, this is a striking result, since it shows that CSs can produce at a significant pace isolated halo pairs whose global dynamical state is compatible with the observations for the LG. Moreover, these pairs are selected among the larger [[LG-gen]{}]{} sample, which was already constraining their mass and separation, ensuring that none of these [[LG-dyn]{}]{} candidates has properties too far off from realistic values. In fact, it could in principle be possible to produce [[LG-dyn]{}]{}-only pairs with unrealistic values of mass and velocities that could nonetheless conspire and give rise to a compatible $e-l$ combination. It has to be noticed that most of the [[LG-gen]{}]{}-candidates lying outside of the $95\%$ confidence intervals are located towards the upper right corner of the $e - l$ plane, meaning that potential pairs tend to have high angular momenta as well as low binding energies.
As a final test, one can further narrow down the sample to look for [[LG-vel]{}]{}-pairs, with $v_{rad}$ and $v_{tan}$ within $2\sigma$ from the observational values, resulting in a total sample of 6, a factor of ten smaller compared to the total number of [[LG-dyn]{}]{}-type objects embedded in a filament. This lower number is hardly surprising given the fact that these quantities are transients and are thus even more subject to the randomness of the short wave modes. Nonetheless, it is possible to produce a few candidates with the right tangential and radial velocities at $z=0$, which could be the starting point for future high resolution zoom in simulations.
$M_{M31}$ $M_{MW}$ $R$ $V_{rad}$ $V_{tan}$
----------- ---------- ------ ----------- -----------
1.54 0.59 1.45 -101.2 6.6
3.08 1.11 1.01 -116.4 16.5
2.49 1.19 1.32 -99.5 32.3
2.08 0.61 1.48 -119.5 11.3
2.80 0.97 1.03 -117.1 35.1
1.01 0.73 0.59 -102.7 10.3
: Properties of the LG candidates with observationally compatible radial and tangential velocities . $M_{MW}$ and $M_{M31}$ are expressed in $10^{12}$ units and refer to the smaller and larger haloes of the pair. $R$ is the distance between the two halo centers (in ) whereas the tangential and radial velocities are expressed in km s$^{-1}$. []{data-label="tab:lg3"}
Shear tensor analysis
---------------------
Having identified a large number of LG candidates one can extend the analysis of the shear tensor eigenvalues and eigenvectors at its position, along the lines of what was done in [Section \[sec:llse\]]{} at the center of the box, i.e. the *expected* LG position. To do this, each halo belonging to the candidate LG is assigned to the NGP, computing the mean and standard deviation of the three $\lambda$s, to properly account for those pairs whose members belong to different nodes on the cosmic web. Indeed, as shown in the previous section, only 26 out of 146 [[LG-gen]{}]{} candidates were living on different kinds of environment, whereas the remaining 120 pairs both belonged to a filament, though some of them characterized by different sets of eigenvalues and eigenvectors. Moving to the eigenvector basis in each simulation, we then look for correlations to the large scale environment computing the cosines between $\mathbf{e}_1$, $\mathbf{e}_2$, $\mathbf{e}_3$ and $\mathbf{e}_{Virgo}$, the direction to the Virgo cluster. From the results shown in [Table \[tab:lg\_align\]]{} one can see that Virgo is very well aligned with $\mathbf{e}_3$ (which defines the orientation of the filament). This alignment can be explicitly seen by looking at the Aitoff projection on the quadrant of the sky defined by the three eigenvectors at the $MW$ position, in [Fig. \[img:lg\_lss\]]{}. This result is consistent with the findings of [@Libeskind:2015a] who found very similar results using the linear WF/CR reconstruction method on the observational data.
$\mathbf{e}_1$ $\mathbf{e}_2$ $\mathbf{e}_3$
---------------------- ---------------- ---------------- ----------------
$\mathbf{e}_{Virgo}$ $0.28\pm0.24$ $0.30\pm0.21$ $0.92\pm0.10$
: Cosines of the angles between the three eigenvectors at the [[LG-gen]{}]{}positions and $\mathbf{e}_{Virgo}$ (the direction of Virgo from the center of mass of the LG), showing the alignment of the latter to $\mathbf{e}_3$.[]{data-label="tab:lg_align"}
![ \[img:lg\_lss\] Aitoff projection of Virgo (blue dots) in the eigenvector basis at the LG position for each of the 120 [[LG-gen]{}]{} objects located on a filament, showing that $\mathbf{e}_3$ is pointing towards the direction of Virgo.](VirgoVoid_Alignment_LG.png){height="5.0cm"}
To summarize, we have shown that we can use CSs as a *Local Group Factory*: A numerical pipeline for producing isolated halo pairs, living on a filament stretched by a large cluster and pushed by a large nearby void, whose properties can be shown to be broadly compatible with observational data of the real LG.
Conclusions {#sec:end}
===========
We have demonstrated here how the combined Cosmicflows-2 data, the model and the [[WF/RZA/CR]{}]{} methodology effectively give rise to Local Group-like objects, in an environment which closely mimics the actual local neighbourhood, in constrained cosmological simulations. The main improvements, in the methodology and the observational data, introduced since the first generation of the CLUES simulations [@Yepes:2014] have been reviewed. We rely heavily on our recent @Sorce:2016 paper and extend it to focus on the simulation of the LG and its close neighbourhood. We have shown here how the data, cosmological model and the methodology conspire together to yield a robust ’factory’ that produces simulated LG-like objects in abundance.
The simulations reported here are all DM-only and are done within a $100$ box. The model is assumed with the Planck set of cosmological parameters. Three sets of simulations have been performed: [[SimuLN256]{}]{}: designed to study the stability of the large scales with respect to the constrained variance; [[SimuLN512]{}]{}: in which the effect of the mass resolution is studied; and [[SimuLGzoom]{}]{}: where zoom-in techniques is used to produce high resolution LGs and are used to study statistically the simulated LGs.
Our main findings fall into two classes. One concerns the environment of the LG and the other deals with the properties of the two halos that constitute the simulated LGs. It should be emphasized here that essentially all the LGs are embedded in an environment which closely resembles the actual one. A summary of these findings follows:\
1. The local environment, represented here by the Virgo cluster and the local filament are a robust outcome of the CF2 data and the model. Quantitative analysis yields:\
1a. The mean offset in the position of the simulated Virgo’s (from the observed one) is a mere 2.67. The median and scatter of the mass of the simulated Virgo is $2.09 \pm 0.69 \times 10^{14}$. This is roughly smaller by a factor 2 from common estimates of the Virgo mass. The smallness of the present computational box accounts for a factor of $\approx 2$ suppression in the mass of the simulated Virgo [cf. @Sorce:2016].\
1b. The cosmic web is defined here by the V-web, which is based on the analysis of the velocity shear tensor. The simulated local environments recover the directions of the eigenvectors and magnitude of the eigenvalues. Roughly 90% of the simulations recover the local filament at its expected position and 80% of the identified LGs reside in a filament similar to the observed local filament. The $100$ computational box is too small to reproduce the Local Void, but the analysis of the simulated velocity shear tensor recovers its repulsive ’push’ at the position of the LG.\
2. The frequency, or success rate of the production of simulated LGs: Two sets of criteria have been used to identify LG-like objects. The first consists of the masses of the two main halos, their distance and isolation. The other set adds to the first one also kinematic constraints on semi-conserved quantities, namely the energy and orbital angular momentum. The first set of criteria ([[LG-gen]{}]{}) yields isolated halo pairs, neglecting their particular dynamic state, while the second one ([[LG-dyn]{}]{}) singles out candidates whose energy and angular momentum are compatible with observations. We have further refined these criteria to objects with the right phase on their orbit, defined by their energy and angular momentum, so as to capture the observed radial and tangential velocities ([[LG-vel]{}]{}). These sets of criteria yield:\
2a. Out of a total of 300 [[SimuLGzoom]{}]{} runs, a total of 146 [[LG-gen]{}]{}-type objects were identified.\
2b. The fraction of LGs goes down to 17% - 51 out of 300, when adding energy and orbital angular momentum constraints. Out of these 51 [[LG-dyn]{}]{}candidates only 6 are on the same phase on the orbit as the actual LG - these are within $2\sigma$ of the observed radial and tangential velocities.
The LG factory provides a robust tool for producing ensembles of LG-like objects and their environs. The effort of obtaining an ensemble of 100, say, LGs, depends on the definition of a LG. Roughly 200 constrained simulations (of different realizations of the ICs) are need to reproduce 100 LGs defined by their isolation, mass and separation distance. Adding the orbital (energy and angular momentum) constraints the number of required simulations rises to 500. Adding the phase constraint, namely of the radial and tangential relative velocities, 2700 simulations must be performed to obtain an ensemble of 100 LGs with the required properties.
An inherent tension exists between the fields of cosmology and the near field cosmology and between the use of random and constrained (local universe) simulations. It relates to the Copernican question of how typical is the LG, and thereby to what extent the LG and the local neighbourhood constitute a typical realization of the universe at large. The answer to this question will determine whether the terms ’near field’ and ’cosmology’ can be paired together. The LG factory provides a platform for systematic and statistical studies of the problem and thereby sheds light on the relevance of the near field to cosmology at large. One qualitative results can already be drawn from the present study. Our results show that the LG is a likely outcome of the CF2 data and the assumed model. The present paper serves as a proof of concept for our method. It also opens the road to a wide range of applications. It enables the construction of a very large ensemble of constrained simulations which harbour realistic LG-like objects and their environment. Such an ensemble can be used to study the (constrained) nature of the LG. Suitable members of the ensemble can be selected as suitable ICs for simulations. Such simulations can include both DM-only and full hydro high resolution simulations. We intend to pursue these and other applications of our method.
Acknowledgements {#acknowledgements .unnumbered}
================
EC would like to thank the Lady Davis foundation for financial support and Julio Navarro for the fruitful comments and interesting discussions. JS acknowledges support from the Alexander von Humboldt foundation. YH has been partially supported by the Israel Science Foundation (1013/12). SG and YH acknowledge support from DFG under the grant GO563/21-1. GY thanks MINECO (Spain) for financial support under project grants AYA2012-31101 and FPA 2012-34694. SP is supported by the Russian Academy of Sciences program P-41. AK is supported by the [*Ministerio de Economía y Competitividad*]{} (MINECO) in Spain through grant AYA2012-31101 as well as the Consolider-Ingenio 2010 Programme of the [*Spanish Ministerio de Ciencia e Innovación*]{} (MICINN) under grant MultiDark CSD2009-00064. He also acknowledges support from the [*Australian Research Council*]{} (ARC) grants DP130100117 and DP140100198. He further thanks Isaac Hayes for shaft. We thank the anonymous referee for the useful comments and remarks. We also thank the Red Española de Supercomputaci' on for granting us computing time in the Marenostrum Supercomputer at the BSC-CNS where part of the analyses presented in this paper have been performed.
\[lastpage\]
[^1]: E-mail: carlesi@phys.huji.ac.il
[^2]: www.clues-project.org
[^3]: https://github.com/ginnungagapgroup/ginnungagap
|
---
abstract: 'A schematic model for baryon excitations is presented in terms of a symmetric Dirac gyroscope, a relativistic model solvable in closed form, that reduces to a rotor in the non-relativistic limit. The model is then mapped on a nearest neighbour tight binding model. In its simplest one-dimensional form this model yields a finite equidistant spectrum. This is experimentally implemented as a chain of dielectric resonators under conditions where their coupling is evanescent and a good agreement with the prediction is achieved.'
address:
- '$^1$ Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, 72570 Puebla, México'
- '$^2$ Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Av. Universidad s/n, 62210 Morelos, México.'
- '$^3$ Université Nice Sophia Antipolis, CNRS, Laboratoire de Physique de la Matière Condensée, UMR 7336 Parc Valrose, 06100 Nice, France.'
- '$^4$ Centro Internacional de Ciencias, Universidad Nacional Autónoma de México, Av. Universidad s/n, 62210 Morelos, México.'
author:
- 'E. Sadurní$^1$, J. A. Franco-Villafañe$^2$, U. Kuhl$^3$, F. Mortessagne$^3$, and T. H. Seligman$^{2,4}$'
title: 'Schematic baryon models, their tight binding description and their microwave realization'
---
Introduction {#sec:Intro}
============
In the sequel of the surge of interest in graphene [@nov05] and its connection to the Dirac equation, emulations of relativistic equations by analogue systems and their experimental realization have been boosted. There are several realization of artificial graphene [@pol13; @rei13], i.e., a honeycomb lattice structure, like microwave systems [@pel07; @bit10a; @zan10; @bit12; @kuh10a; @bel13a; @bar13a; @bel13b], molecular graphene [@gom12] or ultracold atoms in optical lattices [@tar07]. But not only honeycomb lattices have been realized, also one-dimensional systems, where Klein-Tunneling was observed [@dre12] or the Dirac-Moshinsky oscillator[@sad10a] was realized [@arXfra13]. All those experiments show the interest to investigate relativistic systems in general and realize them in analogue experiments. The relation is based on the symmetry itself which yields the well known Dirac points [@sem80]. Nearest neighbour interaction hamiltonians have been used for a long time and are at the base of many of these models, although in practice both for graphene and most models higher interaction terms complicate the picture to some extent, as was established quite early in [@wal47].
It now seems of interest to find simple covariant models $-$say for particles$-$ that can be realized in classical wave systems, e.g. microwave experiments. Indeed one of the areas of active research in high energy physics is the investigation of the mass spectrum of baryons starting from quantum chromodynamics in a non-perturbative regime[@wil74; @eic78]. The task is not a trivial one, as the efforts to obtain answers in this problem are mainly numerical [@deg06]. On the other hand, exactly solvable models were proposed with relative success from the very beginning: Attempts in this direction include multi-particle systems with relativistic hamiltonians [@isg78; @isg79a; @isg79b; @cha81; @cap86], solvable hamiltonians from a spectrum generating algebra [@bij94; @bij00] and many-particle Dirac-Moshinsky oscillators [@mos90a; @mos91; @mos92; @mos96].
It has not been easy to obtain these models from first principles. This is in particular true for models involving many quarks, either with a fixed or variable number of them. Despite of some conceptual difficulties, such models have had certain phenomenological success and it is desirable to improve our understanding of even the simplest of them. Therefore it is not a useless endeavour to propose similarly simple constructions, but which actually abandon the multiparticle approach and focus more in structural parameters of hadrons such as size, internal spin and moments of inertia. More clues on the necessity of such models are provided by previous attempts to introduce relativistic oscillators or the more precise ‘Cornell’ potentials between two quarks; their spectrum as a function of the orbital angular momentum $l$ and frozen radial motion is roughly $\sqrt{a l + b}$, where $a$ and $b$ are constants. Since the spectrum we are dealing with is not a concave function[^1] of $l$, it appears more sensible to introduce a law of the type $\sqrt{a l^2 + b}$.
The general form of this energy suggests a model hamiltonian which resembles the square root of a non-relativistic rotor. We shall see here that such a system can be represented as a tight-binding model with nearest neighbour interaction only. The model can in turn be realized as a one-dimensional array of resonators, thus fulfilling the program of emulating a covariant Dirac-like equation by a microwave experiment. It has furthermore the attraction of displaying a [*finite*]{} spectrum, and thus it can be realized on a [*finite*]{} array of resonators. We thus need not worry about cut-off effects, and therefore can focus entirely on questions of reducing systematic and statistical deviations as well as on minimizing second neighbour interactions.
We start by presenting a simple comparison of two baryon excitation spectra with those of the solvable symmetric version of a Dirac gyroscope in . In the same section a further simplification is presented which leads to a finite equidistant spectrum. A more complete view and a broader scope is offered in ; this section is not essential to what follows and can be read separately. discusses the tight binding hamiltonian describing such systems as a one-dimensional array of resonators with nearest neighbour interactions only. These resonators are realized as dielectric disks between two metallic plates in a microwave experiment. We fix parameters on an equidistant chain. The distance of the disks for the realization can be calculated from the corresponding relativistic equations and the coupling strengths is obtained from the experiments with equidistant disks. We do get good agreement between experiment and theory. Finally we proceed to a discussion of these results and possible extensions.
A simplified Dirac gyroscope {#sec:SimplGyro}
============================
The gyroscope revisited {#ssec:GyroRevisited}
-----------------------
Based on three postulates for rigidity in a relativistic context [@boh83; @ald83; @ald84; @ald85] given below, one of the authors has studied [@sad09] a relativistic quantum rotor denominated Dirac gyroscope. It generalizes the Dirac equation to a particle with internal structure. The postulates for such a Poincaré invariant formulation are
- Elementary limit (standard Dirac equation), reached when the dimensions of the extended object collapse to zero.
- Consistent classical limit, recovering a classical relativistic equation.
- Consistent non-relativistic limit, reducing the system to a non-relativistic quantum rotor.
In the body-fixed frame of reference [^2], the corresponding hamiltonian is given by [@sad09]
H= c ( |I Ľ ) + Mc\^2, \[original\] where $M$ is the rest mass of the body, ${\mbox{\boldmath$\alpha$\unboldmath}}, \beta$ are Dirac matrices in the laboratory frame, $\bar I = {\rm diag\ }\left( I_{xx}^{-1/2}, I_{yy}^{-1/2}, I_{zz}^{-1/2}\right)$ is the inverse square root of the inertia tensor and $\v L$ is the orbital angular momentum vector in the body frame. In equation (24) of [@sad09] it was also shown that the corresponding Dirac equation in the body frame leads to a tractable eigenvalue problem when the body has axial symmetry. In the fully asymmetric case, on the other hand, one has to rely on matrix diagonalization methods. For a general tensor of inertia, the spectrum can be rich enough to accommodate levels in various forms. Certain values of the moments of inertia may even yield levels which [*decrease *]{} with an increasing orbital angular momentum number $l$.
Even for the exactly solvable symmetric case, we find rich spectra. We illustrate this in , where we compare two particular spectra of such a gyroscope to mass spectra of Nucleons (udd, uud) and Lambda particles (uds)[@nak10]. Indulging the crude identification of quantum numbers, one finds a surprising similarity for low lying levels between a model with axial symmetry and the known data. The simultaneous fit for both spectra can be done by adjusting four parameters. According to the Particle Data Group (PDG) [@nak10], the observables $\sqrt{<R_n^2>}$ (neutron mean square charge radius) and $R_p$ (proton charge radius) have the values $R_p= 0.8 \times 10^{-15}\mbox{m}$, $\sqrt{<R_n^2>} = 0.4 \times 10^{-15}\mbox{m}$. Using the estimate $\hbar c \sqrt{M/I} = \hbar c/ \lambda \sim 1\mbox{GeV}$ for a typical length $\lambda$ of the body, we get $\lambda \sim 10^{-15}\mbox{m}$, which is in the same order of magnitude as the radius of a nucleon.
In the case of Nucleons we choose $M=1\mbox{GeV}$, $I_{xx}/I = \sqrt{13/10}$ and $I M c^2 / \hbar^2 = 1/10$, while in the case of $\Lambda$ baryons we use $I_{xx}/I = \sqrt{145/100}$. The ratios $I_{xx}/I$ for both cases suggest oblate shapes. We have restricted our calculations to an exactly solvable model, but a better fit with the experimental data can be achieved allowing full asymmetry in the body and thus four parameters. Note though, that certain characteristic level inversions between lower and higher angular momenta are present even in this simple model as can be seen in .
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Upper row: Experimental masses of lambda baryons (quark content uds) and nucleons (quark content udd and uud). Lower row: Theoretical levels using only three parameters in a Dirac gyroscope with axial symmetry. For nucleons we choose $M=1\mbox{GeV}$, $I_{xx}/I = \sqrt{13/10}$ and $I M c^2 / \hbar^2 = 1/10$. For lambda baryons we use $I_{xx}/I = \sqrt{145/100}$. The ratios $I_{xx}/I$ for both cases suggest oblate shapes. Note that, in agreement with experiment, for the lambda particles one $l=1$ level lies below the $l=0$ level, while for nucleons two $l=1$ levels and one $l=2$ level have this property.[]{data-label="fig3"}](figure1_1.eps "fig:") ![Upper row: Experimental masses of lambda baryons (quark content uds) and nucleons (quark content udd and uud). Lower row: Theoretical levels using only three parameters in a Dirac gyroscope with axial symmetry. For nucleons we choose $M=1\mbox{GeV}$, $I_{xx}/I = \sqrt{13/10}$ and $I M c^2 / \hbar^2 = 1/10$. For lambda baryons we use $I_{xx}/I = \sqrt{145/100}$. The ratios $I_{xx}/I$ for both cases suggest oblate shapes. Note that, in agreement with experiment, for the lambda particles one $l=1$ level lies below the $l=0$ level, while for nucleons two $l=1$ levels and one $l=2$ level have this property.[]{data-label="fig3"}](figure1_2.eps "fig:")
![Upper row: Experimental masses of lambda baryons (quark content uds) and nucleons (quark content udd and uud). Lower row: Theoretical levels using only three parameters in a Dirac gyroscope with axial symmetry. For nucleons we choose $M=1\mbox{GeV}$, $I_{xx}/I = \sqrt{13/10}$ and $I M c^2 / \hbar^2 = 1/10$. For lambda baryons we use $I_{xx}/I = \sqrt{145/100}$. The ratios $I_{xx}/I$ for both cases suggest oblate shapes. Note that, in agreement with experiment, for the lambda particles one $l=1$ level lies below the $l=0$ level, while for nucleons two $l=1$ levels and one $l=2$ level have this property.[]{data-label="fig3"}](figure1_3.eps "fig:") ![Upper row: Experimental masses of lambda baryons (quark content uds) and nucleons (quark content udd and uud). Lower row: Theoretical levels using only three parameters in a Dirac gyroscope with axial symmetry. For nucleons we choose $M=1\mbox{GeV}$, $I_{xx}/I = \sqrt{13/10}$ and $I M c^2 / \hbar^2 = 1/10$. For lambda baryons we use $I_{xx}/I = \sqrt{145/100}$. The ratios $I_{xx}/I$ for both cases suggest oblate shapes. Note that, in agreement with experiment, for the lambda particles one $l=1$ level lies below the $l=0$ level, while for nucleons two $l=1$ levels and one $l=2$ level have this property.[]{data-label="fig3"}](figure1_4.eps "fig:")
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Simplification of the model to a rotor {#ssec:Rotor}
--------------------------------------
Our purpose is to emulate this relativistic system in one of its simplest forms. We may specialize to limit cases sacrificing neither the relativistic nor the quantum features of the system.
------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(a) (b)
\[-6ex\] ![Energy levels $E(l,m_j)$ of a Dirac gyroscope, shown in a gradual transition from complete symmetry of the inertia tensor to a prolate gyroscope. (a) $I_{xx}/ I = 1$. (b) $I_{xx}/ I = 8/9$. (c) $I_{xx}/ I = 7/9$. (d) $I_{xx}/ I = 2/3$. The levels form bands for each value of $L$. Parameters: $I M c^2 / \hbar^2 = 1/10$.[]{data-label="fig4"}](figure2_1.eps "fig:") ![Energy levels $E(l,m_j)$ of a Dirac gyroscope, shown in a gradual transition from complete symmetry of the inertia tensor to a prolate gyroscope. (a) $I_{xx}/ I = 1$. (b) $I_{xx}/ I = 8/9$. (c) $I_{xx}/ I = 7/9$. (d) $I_{xx}/ I = 2/3$. The levels form bands for each value of $L$. Parameters: $I M c^2 / \hbar^2 = 1/10$.[]{data-label="fig4"}](figure2_2.eps "fig:")
\[2ex\] (c) (d)
\[-6ex\] ![Energy levels $E(l,m_j)$ of a Dirac gyroscope, shown in a gradual transition from complete symmetry of the inertia tensor to a prolate gyroscope. (a) $I_{xx}/ I = 1$. (b) $I_{xx}/ I = 8/9$. (c) $I_{xx}/ I = 7/9$. (d) $I_{xx}/ I = 2/3$. The levels form bands for each value of $L$. Parameters: $I M c^2 / \hbar^2 = 1/10$.[]{data-label="fig4"}](figure2_3.eps "fig:") ![Energy levels $E(l,m_j)$ of a Dirac gyroscope, shown in a gradual transition from complete symmetry of the inertia tensor to a prolate gyroscope. (a) $I_{xx}/ I = 1$. (b) $I_{xx}/ I = 8/9$. (c) $I_{xx}/ I = 7/9$. (d) $I_{xx}/ I = 2/3$. The levels form bands for each value of $L$. Parameters: $I M c^2 / \hbar^2 = 1/10$.[]{data-label="fig4"}](figure2_4.eps "fig:")
------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
For instance, when two elements of the diagonal inertia tensor are large compared to the third - e.g. cigar shaped objects - the terms in the kinetic energy affected by such elements become small and produce a negligible level spacing within certain bands (see ). It is indeed fair to establish an axial symmetry in this case, since the condition $I_{zz} \approx I_{yy}$ is compatible with the usual restrictions on the moments of inertia
I\_[zz]{} + I\_[xx]{} > I\_[yy]{}, I\_[yy]{} + I\_[xx]{} > I\_[zz]{}, I\_[yy]{} + I\_[zz]{} > I\_[xx]{} \[moments\] as well as with the conditions for a prolate rotor I\_[zz]{} I\_[xx]{}, I\_[yy]{} I\_[xx]{}. \[moments2\] With these inequalities, the $L_x$ term in the hamiltonian becomes the dominant part and we denote it by $H_0$. The $L_y$ and $L_z$ terms constitute the perturbative part, and we denote the corresponding summand by $H_{\rm{band}}$. Therefore we have H = H\_0 + H\_[[band]{}]{}, \[zero\] with the dominant part given by H\_0 = c \_x L\_x + Mc\^2 \[one\] and a perturbative part H\_[[band]{}]{} = c ( \_y L\_y + \_z L\_z ). \[two\] The aforementioned bands comprising the nearly degenerate levels can be obtained from the spectrum of the operator $H_0$: By virtue of the Clifford algebra of the Dirac matrices, we have H\_0\^2 = L\_x\^2 + M\^2 c\^4, \[oneprime\] and the position of the bands can be determined by analyzing the spectrum of $L_x$. Interestingly, the details of the spectrum within each band can be described by the perturbative part alone, where again $L_x$ plays an important role. We can show this by computing the square of (\[two\]) and using the definition $I \equiv I_{zz} = I_{yy}$, leading to H\_[[band]{}]{}\^2 = ( L\^2 - L\_[x]{}\^2 - 2 S\_x L\_x ). \[three\] At the end of the day, the Zeeman operator $S_x L_x$ is the essential building block of the problem, giving both the position of the band via $H_0$ and the structure of the band via $H_{\rm{band}}^2$. The resulting eigenvalue problem reduces then to the diagonalization of $S_x L_x$ or $L_x$ alone.
Furthermore, our relativistic hamiltonian allows the possibility of taking the ultra relativistic limit $M \rightarrow 0$ with the length parameter $\sqrt{I_{xx}/ M }$ fixed [^3]. In the extreme case of rods or dumbbells, $H_0$ dominates completely the spectrum and the energy is given directly by $\pm L_{x}$ through the equation c \_x L\_x = E . \[4.1\] Once again, we have arrived at the result that $L_x$ alone determines the energy, in this case for an ultra relativistic prolate body. It is important to stress that we have ended up with a finite equispaced spectrum. Such spectra are of more general interest as recently discussed by [’t Hooft]{} [@hoo10].
Relativistic Rotors and Zeeman interactions: A broader scope {#sec:Zeeman}
============================================================
We have seen so far that a component of the orbital angular momentum is sufficient to produce the energy location of the rotational bands, as well as the internal structure of the levels within each band. Even the hamiltonian of a massless ultra relativistic rotor could be identified with such a component of $\v L$.
In the context of integrable or exactly solvable models with rotational invariance, it is not an exaggeration to state that the operator along the quantization axis $L_z$ controls everything. However, emulating such a diagonal operator with a homogeneous set of resonators represents a challenge. We have reached a solution of the problem by recognizing that the rotated versions of $L_z$ (namely $L_x$ and $L_y$) are tridiagonal operators whose form is readily implemented in a scheme of nearest neighbour couplings.
In all, one may ask whether more complicated or more general models can be realized using similar schemes. Notably, the answer stems from the nature of the Lie algebra $SO(3)$, which is solvable: Given a irreducible matrix representation of the group $SO(3)$, i.e. given $l$, one can use the Cartan basis of the algebra (or ladder operators $L_{\pm}$) in order to construct multidiagonal operators representing first, second and even multiple neighbour couplings.
Such models are limited in number as the size of the operators is always finite, i.e. $L_{\pm}^{2l + 1} = 0$, which is a direct consequence of the group compactness. In this way one may argue straightforwardly that any hamiltonian $H(L^2,L_z)$ can be represented as a polynomial of $L_x$, making plausible its emulation with our constructions.
It is important to recognize that the previous argumentation is valid even in the presence of spin operators or Dirac matrices $-$ the latter are generators of the Minkwoski Clifford algebra and can be obtained from direct products of spin operators. The emulation of Zeeman terms reduce naturally to $S_z L_z$ or its rotated version $S_x L_x$. The group in question is now $SU(2)_{\rm{spin}} \otimes SO(3)_{\rm{orbit}}$. For a given representation of $SU(2)$ (in our case, $s=\ahalf$), it also holds that a hamiltonian $H(L^2,S_x,L_x)$ is also a polynomial of $L_x$ and that the powers of $H$ eventually eliminate the presence of the spin operators by virtue of the Pauli matrices algebra.
We may look at the following example. The dumbbell kinetic energy operator $K = \sigma_{+}L_{-} + \sigma_{-}L_{+} $ satisfies, upon squaring K\^2 &=& \_[+]{}\_[-]{}L\_[-]{}L\_[+]{} + \_[-]{}\_[+]{}L\_[+]{}L\_[-]{}\
&=& ( 1 + \_3 ) ( L\^2 - L\_z\^2-L\_z ) + ( 1 - \_3 ) ( L\^2 - L\_z\^2+L\_z )\
&=& (
[cc]{} L\^2 - L\_z\^2-L\_z & 0\
0 & L\^2 - L\_z\^2+L\_z
). Now the two eigenvalue problems are decoupled and, since $L^2$ is fixed, $L_z$ determines the spectrum of $K$ and it can be emulated by $L_x$.
In conclusion, there are many integrable models in the $SO(3)_{orbital} \times SO(3)_{spin}$ space, but all of them essentially reduce to one tight-binding configuration, which is the emulation of $L_x$ appearing in many fashions either coming from spin or the two signs of the energy in relativistic equations. Most of the arguments we have presented in this section can be applied to any compact Lie algebra, opening the possibility to tight-binding realizations of other physical systems by virtue of a natural map between a Cartan basis and two-point recurrence relations.
Tight-binding realization {#sec:TightBind}
=========================
Arrays of identical resonators with nearest neighbour interactions, e.g., potential wells of equal widths and depths, judiciously located, require specific tridiagonal matrices for their realization. A matrix representation of our angular momentum operators can be readily given by fixing the value of $l$ and thereby the dimension of the corresponding Hilbert space. We have seen that an $L_x$ term appears by itself in the hamiltonian describing the position of the respective bands. The eigenvalue problem (\[4.1\]) can be written conveniently in a spinor basis by recognizing that the presence of $\alpha_x$ only contributes to an overall sign. Using the eigenbasis $\{ D^{l}_{m, m'} \}$ of Wigner rotations for the operators $L^2$ and $L_z$, we obtain
c ( D\^[l]{}\_[m-1,m’]{} + D\^[l]{}\_[m+1,m’]{} ) = E D\^[l]{}\_[m,m’]{}.\
\[4.1.1\] This equation can be compared with a nearest-neighbour tight-binding relation containing the couplings $\Delta_{m}, \Delta_{m+1}$: \_[m]{} \_[m-1]{} + \_[m+1]{} \_[m+1]{} + E\_0 \_m = E \_m, \[4.2\] where $E_0$ is the energy of the resonance in an isolated resonator. The required couplings can be read off as
$$\label{eq:Deltam}
\Delta_{m} = \epsilon \sqrt{(l-m)(l+m+1)}$$
with the convenient definition $\epsilon=c \hbar \sqrt{M/I_{xx}}$, which provides the level spacing. At this point we could consider two possible cases arising from : semi-integer $l$ (the emulation of half integer spin) and integer $l$ (realization of orbital angular momentum). Although it is the second option what fits our scheme in the realization of a rotor, we shall bear both cases in mind for the rest of the paper.
Concrete realizations of tight-binding arrays demand a specific recipe for the engineering of couplings. For the realization we will use, such couplings will typically depend on the spacing between resonators. In the following section we will present the experimental setup, including the distances $d$ between sites using the functional dependence $\Delta(d)$ to induce a specific level spacing $\epsilon$.
Microwave experiments {#sec:MuExp}
=====================
Microwave experiments have been a versatile tool to study questions in non-relativistic quantum mechanics [@stoe99]. They have been used in the context of quantum chaos [@stoe90; @dor90; @sri91; @alt95a; @so95], scattering theory [@alt93b; @kuh05a], spectral statistics [@alt94; @bar99d; @sch01d; @sch03a], disordered systems [@cha00; @lau07; @kuh08a], fidelity [@sch05c; @sch05c; @hoeh08a], absorption [@men03a; @kuh05a] and many others. Recently also the energy spectrum of graphene has been investigated [@pel07; @bit10a; @zan10; @bit12; @kuh10a; @bel13a; @bar13a; @bel13b]. The hamiltonian of graphene around the so called Dirac points resembles a two dimensional relativistic hamiltonian [@sem06]. One microwave realization uses an array of disks with a high index of refraction that are coupled evanescently [@kuh10a; @bel13a; @bar13a; @bel13b]. It is an experimental realization of a tight-binding system, which we will adapt now for the realization of the one-dimensional Dirac gyroscope. In the next subsection we describe the setup and adjust the parameters of the set-up in such a way that we minimize effects of higher order neighbour couplings. For details of the relation between the experiment and a tight binding hamiltonian we refer to Ref. [@bar13a; @bel13b].
Experimental setup and specifications {#ssec:ExpSetup}
-------------------------------------
![\[fig:Setup\] Sketch of the experimental setup showing the metallic top and bottom plate, the kink antenna and a few disks.](figure3.eps){width=".5\columnwidth"}
The main ingredient of the experiment is a disk with a high index of refraction $n_r\approx$6. The disk has a radius of $r_D$=4mm and a height $h_D$=5mm. It is sandwiched between two metallic plates which have a distance $h$ between them (see ). The resonances within the disks are excited using a vector network analyzer connected via a kink antenna to the system. It excites the first TE resonance of the disk [@bar13a]. The disks are coupled by evanescent waves as the resonance frequency of the disk is below the cut-off frequency in air, which is induced by the two metallic plates[@bel13b].
![\[fig:EquiDistDisks\] Three experimental reflection spectra for different heights $h$ and distances $d$. The vertical lines below are the corresponding numerical spectra using only nearest neighbour couplings. As the antenna position is at the central disk the even resonances are strongly suppressed. Even though they are hardly visible in the figure, they still can be extracted from the spectra. The parameters of the shown spectra are marked in .](figure4.eps){width=".8\columnwidth"}
![\[fig:Chi2\] The $\chi^2$ value between the experimental resonances compared to the numerical resonances using only nearest neighbour couplings as a function of the plate distance $h$ and the disk distance $d$. The markings refer to the spectra shown in .](figure5.eps){width=".8\columnwidth"}
To realize the Dirac gyroscope it is necessary that the next-nearest neighbour coupling and all higher order couplings are as small as possible. Thus we first start to adjust the working point. We measured the reflections spectra for 21 equispaced disks for different plate distances $h$ and inter disk distances $d$ (see examples in ). From the measured band width $\delta\nu=\nu_\textrm{max}-\nu_\textrm{min}$ we calculated the next nearest neighbour coupling by $\Delta=\delta\nu/4$ as we have a periodic system. Introducing a tight binding hamiltonian, where the diagonal is set to the eigenfrequency of the disk and the secondary diagonal is set to $\Delta$, we calculated numerically the spectrum. The corresponding eigenfrequencies are indicated by the vertical bars in . Now we calculated the $\chi^2$ deviation between the experimental and numerical resonance positions. The deviations are presented in as a function of the plate distance $h$ and inter disk distance $d$. We observe a minimal plateau around $h$=12-14mm and $d=9-14$mm. Thus for all further measurements we fix the plate distance to $h$=13mm and will stay within a disk distance of 9-14mm. In this range we found that the next-nearest neighbour coupling is less than 7.5% of the nearest neighbour coupling.
![\[fig:Coupling\] Extracted nearest neighbour coupling $\Delta(d)$ of the disks as a function of the disk distance at height $h$=13mm. The blue solid line corresponds to a fit to with $\Delta^\prime_0$=1.53GHz, $\gamma^\prime$=0.148mm$^{-1}$, and $C^\prime$=0.0039GHz and the dashed line to a fit to an approximated exponential with $\Delta_0$=3.94GHz, $\gamma$=0.20mm$^{-1}$ and $C$=0.006 in the range of interest.](figure6.eps){width=".8\columnwidth"}
For $r>r_D$, i.e. outside a single disk, the eigenfunction is described by a modified Bessel function $K_0$. Thus the coupling between two disks can be estimated by [@kuh10a; @bar13a] $$\label{eq:CouplingK}
\Delta(d)= \Delta^\prime_0\left|K_0~\left(\gamma^\prime d\right)\right|^2 + C^\prime,$$ where $d$ is the center to center distance of the disks. The constant $C^\prime$ takes into account that the resonance frequency of the disks are slightly different. Have in mind that at $d=2r_D$, where $r_D$ is the radius of the disks, the disks are touching. Thus the maximal coupling is given $\Delta(2r_D)$. $\gamma^\prime$ depends strongly on the plate distance $h$. shows the nearest neighbour couplings $\Delta(d)$ extracted via the band width as a function of the inter disk distance. The blue line corresponds to a fit of . In the range of interest $d$=9-14mm the coupling can be also approximated by an exponential $$\label{eq:CouplingExp}
\Delta(d) = \Delta_0 \exp \left(-\gamma d\right) + C,$$ which is indicated by the dashed line in . Now we have adjusted the working point and obtained all necessary ingredients to setup the experiment for the Dirac gyroscope.
Spectrum of a Dirac gyroscope {#ssec:ExpSpecGyro}
-----------------------------
As we have shown in we need to adjust the couplings $\Delta_n$ corresponding to . For the sake of simplicity we use here the exponential description of the coupling. Thus we invert first resulting in $$\label{eq:DistanceDelta}
d(\Delta) = -\frac{1}{\gamma} \ln \left(\frac{\Delta}{\Delta_0}\right).$$ Taking into account the relation for the couplings we get the relation for the disk distances $$\label{eq:Distancem}
d_n = d_{l+m+1} = -\frac{1}{\gamma}\ln \left(\frac{\epsilon}{\Delta_0}\sqrt{(l-m)(l+m+1)} \right),$$ where $N=2l+1$ and $m=n-l-1$. In the experiment $\epsilon$ is given in GHz and is defining the level spacing. Have in mind that $n$ is the consecutive disk number ranging from 1 to $N=2l+1$, which is the total number of disks, whereas in we have $l$ and $m$ ranging from $-l$ to $l$. The system is symmetric with respect to the center, where the largest distance is at the center and the minimal distance at the edge is defining the level spacing by $$\label{eq:Epsilon}
\epsilon=\frac{\sqrt{l(l+1)}}{\Delta_0}\exp\left(\gamma d_\textrm{min}\right).$$ The consecutive difference is given by $$\label{eq:deltadm}
\delta d_m=d_{m+1}-d_m=-\frac{1}{\gamma}\log\left(\frac{\sqrt{(l-m+1)(l+m+2)}}{\sqrt(l-m)(l+m+1)}\right)$$ It is important to mention that in experiments involving microwaves we must introduce additional constants $E_0$, which is a global shift that takes into account the resonance frequency of the single disk. In the following we shall fix these quantities according to our setup.
![\[fig:GyroscopeRSpectra\] Experimental reflection spectra $1-|S_{11}|^2$ for three realizations of the gyroscope with different number of disks and the same level spacing $\epsilon = 25 {\rm MHz}$. The bars indicate the eigenvalues of the corresponding prediction from , where the central energy is fixed by means of the lowest experimentally obtained resonance. The spectra are shifted in increasing order and correspond to 7, 9 and 11 disks.](figure7.eps){width=".8\columnwidth"}
We now want to realize a Dirac gyroscope with fixed level spacing for different number of disks $N$. We chose $\epsilon$=25MHz and realized three chains with $N$=7, 9, 11. Using we can calculate the corresponding disk distances $d_n$ necessary to set up the different chains of disks. In we show the three corresponding measured reflection spectra. The bars indicate the expected equidistant Dirac gyroscope spectra, where the central energy is fixed by means of the lowest experimentally obtained resonance. This procedure takes also into account an additional shift induced by higher order couplings. Good agreement between the reflection spectra and theory is found.
![\[fig:GyroscopeNChange\] Eigenfrequencies $\nu_n$ for different numbers of disks with $d=13$mm. The dots are the experimental eigenfrequencies obtained by the maxima of $1-|S_{11}|^2$ and the lines correspond to the theoretical prediction, where the lowest frequencies were adjusted.](figure8.eps){width=".8\columnwidth"}
![\[fig:GyroscopeNChangeRescaled\] Rescaled eigenfrequencies $\nu^\prime_n$ for different numbers of disks with $d=13$mm.](figure9.eps){width=".8\columnwidth"}
Next we fixed the minimal length $d_\textrm{max}$=13mm for $N$=4 disks. We added step by step an additional disk corresponding to . We did this up to $N$=21. In the corresponding experimental resonances are shown as dots, whereas the horizontal bars indicate the levels for the equidistant spectra. The experimental spectra show the expected change of the level spacing, which is due to the fact that, with each additional disk, the minimal distance is reduced, thus also leading to a reduction of the level spacing (see ). Finally we rescale the resonance by $$\label{eq:Rescaling}
\nu'_n= (\nu_n-\nu_0)/\Delta\nu,$$ where $\Delta\nu$ corresponds to the theoretically predicted spacing and $\nu_0$ is the first resonance extracted from the experimental spectra for each number of disks, respectively. Thus $\nu'_n$ should resemble the integer level number if the spectra are evenly spaced. This is shown in fig. \[fig:GyroscopeNChangeRescaled\] and the predicted behaviour is found.
The results show that it is possible to engineer a system with a complete set of quantum numbers using tight-binding arrays. The general spectroscopic structure of the gyroscope has been therefore reproduced: The columns of fixed angular momentum in accommodate an increasing number of states, representing multiplets of such an observable. With our results we establish a connection with the general trend of the different levels of the axi-symmetric case presented in figure \[fig3\]. Furthermore, the baryonic spectra presented also in figure \[fig3\] possess an increasing number of states which can be understood as splittings for low masses, opening the possibility of emulating nearly any bound spectrum of fixed $l$ through deformations of our previous construction.
It is important to note that the phenomenology of the mass states of baryons becomes more involved as the energy increases. The corresponding resonances might be accommodated in new rotational bands, but in such a case any model of rigidity (either from strict or loosened requirements for relativistic systems) should be abandoned, as possible vibrational states should dominate the picture.
In all, our example of an equispaced spectrum can be regarded as a benchmark for more detailed constructions that adjust levels for each observable (e.g. the value of $l$). The experimental results in reproduce such a physical situation. Interestingly, techniques of this type resemble those of chemistry in the form of polyads [@jun02; @her13]: building the skeleton of the spectrum and later enrich it and/or perturb it has proven to be a sensible approach.
Discussion and outlook {#sec:DiscOutlook}
======================
Taking up a suggestion made by one of the authors in a previous paper [@sad09] to use a relativistic rotor or gyroscope as a schematic model for baryons, we have emulated such a system in a microwave setup after mapping it onto a chain of resonators consistent of dimers with successively decreasing coupling. This was implemented by increasing distances between resonators taking advantage of the fact that the selected TE mode was evanescent between the resonators for the selected distance of the covering plate. The fact that we have finite spectra for the relativistic rotor eliminates a priori an important source of errors in the case of infinite spectra namely the inevitable truncation. We have thus achieved an emulation of a relativistic system of relevance and the agreement between experiment and theory is satisfactory.
Wave functions can be analyzed, but the fact that the eigenfunctions are essentially Jacobi polynomials suggests that the rotary structure will be visible when explored in future work. The model relies on the fact that there are nearest neighbour interactions only, while our experimental setup makes it difficult to suppress higher order neighbour interactions, if we wish to use a two-dimensional array. Yet considering that only the coupling strength and the topology determine such a model, we propose to use quantum graphs, e.g. coupling by cables or wave guides of sets of equal resonators that have an isolated resonance in the frequency domain we wish to study. For systems where no cutoff is needed we can expect that the quality of the resonators will determine the quality of the emulation. As to the wavefunctions, it is possible to retrieve them experimentally by measuring the height of the peaks as a function of the disc number. Summarizing we have a promising field to emulate a wide variety of relativistic and non-relativistic systems by more schematical or more realistic schemes.
We are thankful to S. Barkhofen for fruitful discussions.
T. H. S. and J. A. F. V. thank the LPMC for the hospitality during several long term visits and to CONACyT Project Number 44020 and PAPIIT-UNAM project number IG101113 for financial support. E. S. acknowledges support from PROMEP project No. 103.5/12/4367.
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[^1]: One can be convinced of this statement by plotting the data published by the PDG [@nak10] as a function of $l$. See also our .
[^2]: Despite its unusual form, the hamiltonian (\[original\]) emerges from a Poincaré invariant equation containing explicitly the center of mass four-vector and the Pauli-Lubanski vector. A Schrödinger-like equation can be obtained when the center of mass is frozen and the inertia tensor is diagonal.
[^3]: The definition of $I_{xx}$ shows it transparently, as the ratio $I_{xx}/M$ tends to $\langle x^2 \rangle$.
|
---
abstract: 'We study the critical collapse of a massless scalar field with angular momentum in spherical symmetry. In order to mimic the effects of angular momentum we perform a sum of the stress-energy tensors for all the scalar fields with the same eigenvalue $l$ of the angular momentum operator and calculate the equations of motion for the radial part of these scalar fields. We have found that the critical solutions for different values of $l$ are discretely self-similar (as in the original $l=0$ case). The value of the discrete, self-similar period, $\Delta_l$, decreases as $l$ increases in such a way that the critical solution appears to become periodic in the limit. The mass scaling exponent, $\gamma_l$, also decreases with $l$.'
author:
- 'Ignacio (Iñaki) Olabarrieta'
- 'Jason F. Ventrella'
- 'Matthew W. Choptuik'
- 'William G. Unruh'
title: Critical Behavior in the Gravitational Collapse of a Scalar Field with Angular Momentum in Spherical Symmetry
---
Introduction {#sec:introduction}
============
Most studies of black hole critical phenomena (see [@Gundlach:1999cu], [@Gundlach:2002sx] for reviews) to date (or related phenomena in other sets of nonlinear evolution equations) have been performed assuming spherical symmetry as a simplifying assumption (exceptions are [@Abrahams:wa], [@Liebling:2002qp] and more recently [@Choptuik:2003ac]). This simplification has been adopted in most cases because accurate calculation of Type II critical solutions—which exhibit structure at all scales due to their self-similar nature—requires great computational resources. Since spherically symmetric spacetimes do not allow for angular momentum, very little is currently known about the role of angular momentum in critical collapse. For a few cases, most notably the Type II solutions found in spherically symmetric collapse of a massless scalar field [@Garfinkle:1998tt], or certain types of perfect fluid [@Gundlach:1997nb], [@Gundlach:1999cw], perturbative calculations about the spherical critical solutions suggest that non-spherical modes, including those contributing to net angular momentum, are damped as one approaches criticality[^1]. In particular in [@Garfinkle:1998tt], [@Gundlach:1997nb] and [@Gundlach:1999cw] using second order perturbation theory it was predicted that the angular momentum of the black holes produced should have the following dependence as a function of the critical parameter $p$: $$\label{L_scaling_law}
\vec L_{\rm{BH}} = \vec L_0 \left(p-p^\star \right)^\mu,$$ where $\vec L_0$ is family-dependent and $\mu$ is a universal scaling exponent satisfying $\mu > 2\gamma$ ($\gamma$ being the scaling exponent for the black hole mass). Specifically, it was suggested that $\mu\approx0.76$ for the scalar field case, whereas the computations indicated that $\mu$ would depend on the equation of state for perfect fluid collapse. These calculations thus suggest that, at least for small deviations from spherical symmetry, the resulting solutions on the verge of black hole formation should remain spherically symmetric in non-symmetric collapse. We also note that an axisymmetric numerical relativity code has been developed [@hlcp] to study non-perturbatively some effects of angular momentum in the critical collapse of a scalar field. Interestingly, the results found for $\Delta$ and $\gamma$ in the case of a complex scalar field with principal azimuthal “quantum number”, $m=1$ are very close to the results we find in our model for $l=1$, as described in Sec. \[sec:results\].
Here a different approach is taken. Maintaining spherical symmetry, the equations of motion for a massless scalar field are modified by effective terms which mock up some of the effects of angular momentum. As described below, the procedure amounts to performing an angular average over the matter field variables—similar to that done in [@Rein:1998uf], [@Olabarrieta:2001wy] and [@Ventrella:2003fu]—and results in an entire family of models, parameterized by a principal angular “quantum number”, $l$ (we will generally restrict $l$ to take on non-negative integer values, although real-valued $l$’s are also formally possible). We note that since the models remain spherically symmetric, we cannot use them to address the validity of the perturbative calculations mentioned above (e.g. equation (\[L\_scaling\_law\])). Nonetheless, we find interesting results that may shed some light on the effects of angular momentum near the black hole threshold.
Some of the main results that have been found are as follows. First, each value of the angular momentum parameter $l$ apparently defines a distinct critical solution. For $l < 10$, these solutions are found to be discretely self similar, with values of the echoing exponent, $\Delta_l$, that rapidly decrease (approximately exponentially) as $l$ increases. As a result, for large values of $l$, and for the time scales for which we are able to dynamically evolve near criticality, the threshold solutions become approximately [*periodic*]{}. In addition, and as expected for Type II solutions, we find that for $l < 7$ the masses of the black holes formed follow power laws. As with the echoing exponents, for increasing values of $l$ it is found that the mass-scaling exponent, $\gamma_l$, rapidly decreases, again approximately exponentially in $l$.
The remainder of this paper is structured as follows. In the following section we describe the recipe used to calculate the effective equations of motion, along with the regularity and boundary conditions imposed in the solution of these equations. In Sec. \[sec:results\] we briefly describe the numerical code, the way the solutions have been analyzed, and then provide a summary of the results obtained for varying values of $l$. Throughout this paper we use units such that the universal gravitational constant, $G$, and the speed of light in vacuum, $c$, are both unity.
Equations of Motion {#sec:equations}
===================
Equations
---------
In order to derive equations of motion, scalar fields of the following form are considered: \^m\_l(t,r,,) &=& \^[(l)]{}(t,r)Q\_[lm]{}(,),\
&& m = -l, -l+1, , l-1, l, \[sep\] where $Q_{lm}(\theta,\phi)$ are normalized [*real*]{} eigenfunctions of the angular part of the flatspace Laplacian with eigenvalue $l(l+1)$, and the index $m$ labels the $2l+1$ distinct orthonormal eigenfunctions for a given value of $l$.[^2] More explicitly: $$Q_{l m} = \left\{ \begin{array}{l l} Y_{l 0} &{\rm for} \quad m=0,\\
\frac{1}{\sqrt{2}} \left( Y_{l m} + (-1)^m Y_{l -m}\right) & {\rm for} \quad m>0,\\
\frac{1}{i\sqrt{2}} \left( Y_{l |m|} - (-1)^{|m|}Y_{l -|m|}\right) &{\rm for} \quad m<0,
\end{array}
\right.$$ where $Y_{l m}\left(\theta, \phi \right)$ are the regular spherical harmonics. By construction, the scalar fields $\Psi^m_l$ are not, in general, spherically symmetric and we therefore do not study their collapse directly. Instead, our strategy is to find effective equations for the [*single*]{} $(t,r)$-dependent quantity $\psi^{(l)}(t,r)$, which we hereafter denote simply by $\psi$. To do so, for a specific value of $l$, we consider the stress-energy tensors for the $2l+1$ fields $\Psi^m_l$: $$\label{set_Tab}
T^{(l m)}{}_{ab} = \nabla_{a} \Psi^m_l \nabla_{b}\Psi^m_l
- \frac{1}{2}g_{ab}(\nabla^{c}\Psi^m_l\nabla_{c}\Psi^m_l),$$ where $g_{ab}$ is the metric of the spacetime and $\nabla_a$ is the metric-compatible covariant derivative. Again by construction, and as is proven in Appendix \[appendixA\], the sum of these stress tensors $${\mathcal T}^{(l)}{}_{ab}=\sum_m T^{(l m)}{}_{ab},$$ [*is*]{} spherically symmetric, and thus depends only on $\psi(t,r)$, $l$, and the metric $g_{ab}$. We can now compute the effective equation of motion for the field, $\psi(t,r)$, using the fact that the divergence of the total stress energy tensor is zero, as is also proven in Appendix \[appendixA\]: $$\label{conservation}
g^{ac}\nabla_{c}{\mathcal T}^{(l)}{}_{ab}= 0 \, .$$ The equations for the geometric variables are determined from the $3+1$ decomposition of the Einstein field equations. For the current study we adopt Schwarzschild-like (polar-areal) coordinates, in which the metric takes the form: $$\begin{aligned}
\label{polar-areal-metric}
ds^2=-\alpha^2(t,r)dt^2&+&a^2(t,r)dr^2\nonumber\\ &+& r^2d\theta^2 + r^2\sin^2{\theta}d\phi^2 \, .\end{aligned}$$ Here $\alpha(t,r)$ is the lapse function and $a(t,r)$ is the only non-trivial components of the 3-metric (both $\alpha$ and $a$ are positive functions). Using this metric, the non zero components of the stress-energy tensor for a general value of $l$ are $$\begin{aligned}
{\mathcal T}^{(l)}{}^t{}_t &=& -\frac{(2l+1)}{8 \pi}
\left[\frac{1}{a^2}\left(\Pi^2 + \Phi^2 \right)
+l(l+1)\frac{\psi^2}{r^2}\right],\\
{\mathcal T}^{(l)}{}^t{}_r &=& -\frac{(2l+1)}
{8 \pi}\frac{2}{a \alpha}\,\Pi\,\Phi,\\
{\mathcal T}^{(l)}{}^r{}_r &=& \frac{(2l+1)}{8 \pi}
\left[\frac{1}{a^2}\left(\Pi^2 + \Phi^2 \right)
-l(l+1)\frac{\psi^2}{r^2}\right],\\
{\mathcal T}^{(l)}{}^\theta{}_\theta &=&
{\mathcal T}^{(l)}{}^\phi{}_\phi =
\frac{(2l+1)}{8 \pi a^2}\left(\Pi^2 - \Phi^2 \right) \, ,\end{aligned}$$ and the stress-energy trace is $$\begin{aligned}
\label{trace_t}
{\mathcal T}^{(l)}&\equiv&
{\mathcal T}^{(l)}{}^i{}_i \nonumber \\
&=& \frac{(2l+1)}{8\pi}\left[\frac{2}{a^2}\left(\Pi^2-\Phi^2\right)-
2l(l+1)\frac{\psi^2}{r^2}\right]\! .\end{aligned}$$ In the above expressions, we have made use of the auxiliary variables, $\Phi$ and $\Pi$, defined as follows: $$\begin{aligned}
\label{pp_cons}
\Phi\left(t,r\right)&=&\frac{\partial \psi}{\partial r}, \\
\Pi\left(t,r\right)&=&
\frac{a}{\alpha}\frac{\partial \psi}{\partial t}.\end{aligned}$$ The dynamical equations of motion for these fields, which follow from the definition of $\Phi$ as well as the wave equation for $\psi$ (which in turn can be derived from the vanishing of the divergence of the total stress tensor (\[conservation\])) are then: $$\begin{aligned}
\label{pp_evo}
\frac{\partial\Phi}{\partial t}&=&
\frac{\partial}{\partial r}\left(\frac{\alpha}{a}\Pi\right),\\
\label{pi_evo}
\frac{\partial\Pi}{\partial t}&=&
\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\alpha}{a}\Phi\right)-
l(l+1)a\alpha \frac{\psi}{r^2}\, .\end{aligned}$$
Note that the dependence of these equations on $l$ is only through the last term in equation (\[pi\_evo\]) which is proportional to $l(l+1)/r^2$. This term can be thought of as the field-theoretic extension of an analogous term due to the angular momentum potential, $l^2/r^2$, in the 1-dimensional reduced problem of a particle moving in a central potential.
As mentioned above, equations for the geometric variables result from the $3+1$ decomposition of the field equations, as well as from our choice of coordinates. Specifically, we have the following $$\begin{aligned}
\label{hc_chp2}
\frac{1}{a}\frac{\partial a}{\partial r}&=&
\frac{(2l+1)}{2}r\left(\Pi^2+\Phi^2+l(l+1)\frac{a^2}{r^2}\psi^2\right)\nonumber\\
&-&\frac{a^2-1}{2 r},\\
\label{sc}
\frac{1}{\alpha}\frac{\partial \alpha}{\partial r}&=&
\frac{(2l+1)}{2}r\left(\Pi^2+\Phi^2-l(l+1)\frac{a^2}{r^2}\psi^2\right)\nonumber\\
&+&\frac{a^2-1}{2 r},\\
\label{a_evo}
\frac{\partial a}{\partial t}&=&(2l+1) r \alpha \Pi \Phi.\end{aligned}$$ Equation (\[hc\_chp2\]) is the Hamiltonian constraint, which is used to determine the 3-metric component, $a$. Similarly, the slicing condition (\[sc\]) fixes the lapse function $\alpha$ at each instant of time, and is often known as the [*polar slicing condition*]{}. It can be derived from the demand that ${\rm Tr}\left(K_{ab}\right)=K^r{}_r + K^\theta{}_\theta + K^\phi{}_\phi =
K^r{}_r + 2K^\theta{}_\theta = 0$, for all times. The Hamiltonian constraint and slicing condition, with appropriate regularity and boundary conditions, completely fix the geometric variables in this coordinate system. Equation (\[a\_evo\]) is an extra equation derived from the definition of $K^r{}_r$ and the momentum constraint. In our numerical solutions, it is used as a gauge of the accuracy of our calculations, as well as to provide a replacement for the Hamiltonian constraint in certain strong field instances where the numerical constraint solver fails. In addition, we compute the mass aspect function, $M(t,r)$, $$M(t,r) = \frac{r}{2}\left(1-\frac{1}{a^2} \right),$$ which serves as a valuable diagnostic quantity in our calculations. The value of this function as $r\to\infty$ agrees with the ADM mass, and more generally, in a vacuum region of spacetime, measures the amount of (gravitating) mass contained within the 2-sphere of radius $r$ at time $t$. Moreover, $2M(t,r)/r$ is useful since its value approaches $1$ when a trapped surface is developing and hence (modulo cosmic censorship), a black hole would form in the spacetime being constructed. We note that, as is the case with the usual Schwarzschild coordinates for a spherically symmetric black hole, polar-areal coordinates cannot penetrate apparent horizons, and in fact become singular as they come “close to” black-hole regions of spacetime, where $2M(t,r)/r \to 1$. This fact does not present a problem in the study of critical behavior in our models, since the critical solutions [*per se*]{} have $\max_r\left\{2M(t,r)/r\right\}$ bounded away from $1$.
Regularity and Boundary Conditions
----------------------------------
In addition to the above equations of motion, appropriate regularity and boundary conditions are needed. At the origin, $r=0$, regularity is enforced via $$\begin{aligned}
a(t,0) &=&1,\\
\frac{\partial a}{\partial r}(t,0) &=&0,\\
\frac{\partial \alpha}{\partial r}(t,0) &=&0,\\
\label{psi_fall_off}
\psi(t,0) &=&O(r^l),\\
\label{pi_fall_off}
\Pi(t,0) &=&O(r^l),\\
\label{pp_fall_off}
\Phi(t,0) &=& \left\{
\begin{array}{lll}
O(r^{l-1}) & \hbox{\rm for} & l \ge 1,\\
O(r) & \hbox{\rm for} & l=0.
\end{array} \right.
\label{phir0}\end{aligned}$$ In the continuum, our equations of motion are to be solved as a pure Cauchy problem, on the domain $t \ge 0$, $r \ge 0$, with boundary conditions at spatial infinity given by asymptotic flatness (i.e. that the matter fields vanish, and that the metric becomes that of Minkowski spacetime, as $r\to\infty$). Computationally, we solve an approximation to this problem on a finite spatial domain $0 \le r \le r_{\rm max}$, where $r_{\rm max}$ is some arbitrary outer radius chosen sufficiently large that we are confident that the numerical results do not depend significantly on its precise value. At the outer boundary, then, the following condition for $\alpha$ is imposed: $$\label{alpha_bc}
\alpha(t,r_{\rm max})\ a(t,r_{\rm max})=1.$$ This can be viewed as simply providing a convenient normalization for $\alpha$, since given a solution, $\alpha$, of the slicing equation (\[sc\]), $k \alpha$ is also a solution, where $k$ is an arbitrary positive constant. We note that although we have used (\[alpha\_bc\]) in order to perform the calculations, a different normalization convention—i.e. a different, and time dependent, choice of $k$—has been used in order to perform the analysis of the solutions. Specifically, in the analysis we have used central proper time $T$ defined by: $$\label{central_t}
T = \int^{T}_0 \alpha({\tilde t},0) \, d{\tilde t}\,.$$ This definition of time has a natural geometrical interpretation since $r=0$ is invariantly defined by the symmetry of the spacetime. For the scalar field variables, $\Pi$ and $\Phi$, approximate outgoing-radiation boundary conditions (Sommerfeld conditions) are used: $$\begin{aligned}
\frac{\partial \Phi}{\partial t}(t,r_{\rm max})+
\frac{\partial \Phi}{\partial r}(t,r_{\rm max})
+\frac{\Phi(t,r_{\rm max})}{r_{\rm max}}&=&0,\\
\frac{\partial \Pi}{\partial t}(t,r_{\rm max})+
\frac{\partial \Pi}{\partial r}(t,r_{\rm max})
+\frac{\Pi(t,r_{\rm max})}{r_{\rm max}}&=&0.\end{aligned}$$
An important point in the derivation of the equations of motion is the fact that the eigenfunctions in (\[sep\]) are discrete and the allowable values of $l$ are only non-negative integers. Once the equations are obtained we have relaxed that constraint and have allowed $l$ to take non-negative [*real*]{} values. The solutions corresponding to non-integer values of $l$ would have some degree of irregularity at the origin depending on the particular value of $l$ chosen. This implies that only some finite number of derivatives with respect to $r$ will be defined at $r=0$. In our particular numerical implementation, which assumes that second derivatives of the variables are defined, we have been able to study the evolution of these systems as long as $l>3$.
Results {#sec:results}
=======
Numerics
--------
We solve equations (\[pp\_evo\]), (\[pi\_evo\]) for the scalar field gradients, equations (\[hc\_chp2\]), (\[sc\]) for the geometry, and use (\[pp\_cons\]) to reconstruct the field $\psi$. The system is approximated using second order centered finite difference techniques, and coded using RNPL [@rnpl]. Numerical dissipation of the Kreiss-Oliger [@KO] variety was included to damp high frequency modes, and it should be noted that this particular type of dissipation is added at sub-truncation error order, so does not affect the overall accuracy of the scheme as the mesh spacing tends to 0. For the current computations, the damping terms were most useful in regularizing the truncation error estimation procedure that occurs when adaptive mesh refinement (AMR) techniques are used. It was also crucial to impose the correct leading-order regularity conditions close to the origin, $r=0$ (equations (\[pi\_fall\_off\])-(\[pp\_fall\_off\])), in order to keep the solution regular during the evolutions. Most of the calculations were done on a fixed uniform spatial grid $r_j = (j - 1)\Delta r$, $j = 1, 2, \cdots, J$, $J = 1 + r_{\rm max} / \Delta r$ with a typical number of grid points $J=1025$, and the outer boundary of the computational domain typically at $r_{\rm max}=100$. For small values of the angular momentum parameter—specifically for $l\le2$—an AMR algorithm based on that described in [@Choptuik:jv] was used.
Families of Initial Data
------------------------
Our study involved the evolution of $6$ different one parameter families of initial data, each defined by an initial profile $\psi(0,r)$ as listed in Table \[families\_id\_table\], with specific values of the parameters appearing in the profile definitions as given in Table \[initial\_data\_table\]. In addition to $\psi(0,r)$, we need to provide $\Pi(0,r)$ to complete the specification of the initial data. In all cases we chose $\Pi(0,r)$ to produce an approximately in-going pulse at the initial time: $$\Pi(0,r) = \Phi(0,r) = \frac{\partial \psi}{\partial r}(0,r).$$
----------------------------------------------------------
Family Form of initial data, $\psi(0,r)$ $p$
-------- ------------------------------------------- -----
(a) $A\,\exp\left(-(r-r_0)^2/\sigma^2\right)$ $A$
(b) $-2 A\,(r-r_0)/\sigma^2\exp $A$
\left(-(r-r_0)^2/\sigma^2\right)$
(c) $A\,r^2\left( {\mathrm{atan}}(r-r_0)- $A$
{\mathrm{atan}}(r-r_0-\sigma) \right)$
----------------------------------------------------------
: Families of initial data and the parameter $p$ that is tuned to generate a critical solution.[]{data-label="families_id_table"}
As previously mentioned, all of the initial data families listed in Table \[families\_id\_table\] have a single free parameter, $p$, and, as is the usual case in studies of black hole critical phenomena, for any given family we observe two different final states in the evolution, depending on the value of $p$. For values of $p > p^\star$ the maximum value of $2M(t,r)/r$ approaches $1$ implying that an apparent horizon is about to form. On the other hand if $p < p^\star$ the scalar field completely disperses, and leaves (essentially) flat spacetime in its wake. The solution that arises as $p \to p^\star$ then represents the threshold of black hole formation and, by definition, is the critical solution. We note that these critical solutions are [*not*]{} $t\to\infty$ end-states of evolution; rather they persist for only a finite amount of time, and, in fact, are unstable, heuristically representing an infinitely fine-tuned balance between dispersal and gravitational collapse.
Initial Data (F) Family Parameters
------------------ -------- ----------------------------
1 (a) $r_0=70.0$, $\sigma=5.00$
2 (b) $r_0=70.0$, $\sigma=5.00$
3 (c) $r_0=70.0$, $\sigma=5.00$
4 (a) $r_0=40.0$, $\sigma=10.0$
5 (a) $r_0=40.0$, $\sigma=5.00$
6 (a) $r_0=70.0$, $\sigma=10.0$
: Initial data used in our investigations. The family labels are defined in Table I. []{data-label="initial_data_table"}
Analysis
--------
We have calculated $p^\star$ for the different families of initial data described above, and for different values of $l$, via bisection (binary search), tuning $p$ in each case to a typical precision of $\left(p-p^\star\right)/p\approx 10^{-15}$ (which is close to machine precision using 8-byte real floating point arithmetic).
As in the case for $l=0$ (where the equations of motion reduce to those for a single, non-interacting massless scalar field, as studied in [@Choptuik:jv]), the critical solutions for values of $l\le 9.5$ are apparently discretely self similar (DSS). DSS spacetimes are scale-periodic, meaning that any non-dimensional quantity, $Z$, obeys the following equation for some specific values of the parameters $\Delta$ and $T^\star$: $$\label{scaling}
Z\left((T-T^\star),r\right) =
Z\left(e^{n \Delta} (T-T^\star),e^{n \Delta}r\right),$$ where $T$ is central proper time as defined by (\[central\_t\]), and $T^\star$ is the “accumulation time” of the self-similar solution. In (\[scaling\]) the integer $n$ denotes the “echo” number. We also note that due to the discrete $\psi \to -\psi$ invariance that is exhibited both by the equations of motion as well as the critical solutions themselves, if $\Delta$ is the echoing exponent for which formula (\[scaling\]) is satisfied with $Z(T,r) \equiv \psi(T,r)$, then the geometric quantities $a(T,r)$, $\alpha(T,r)$, $2 M(T,r) / r$ obey (\[scaling\]) with an echoing exponent $\Delta/2$.
In order to extract $\Delta$ from our calculations, we use the observation that certain geometric quantities will achieve (locally) extremal values on the spatial domain at discrete central proper times $T_n$ given by $$T_n-T^\star = \left(T_0 - T^\star \right) e^{n \Delta/2 }$$ where $T_0$ is the time at which one starts counting the echoes. Specifically, $\Delta$ and $T^\star$ have been computed by a least squares fit for the times $T_n$ at which $\max_r\left\{2 M(t,r)/r\right\}$ achieves a local maximum in time, i.e. by minimizing: $$\chi^2=\sum_{n=1}^{N} \left\{T_n - T_0 e^{n\Delta /2}+
T^\star \left(e^{n \Delta /2}-1 \right)\right\}^2.$$
$l$ $\Delta_l$ $\gamma_l$
----- --------------------- ---------------------
0 3.43 $\pm$ 0.05 0.376 $\pm$ 0.003
1 0.460 $\pm$ 0.002 0.119 $\pm$ 0.001
2 0.119 $\pm$ 0.003 0.0453 $\pm$ 0.0002
3 0.039 $\pm$ 0.001 0.020 $\pm$ 0.001
3.5 0.0224 $\pm$ 0.0009 0.0127 $\pm$ 0.0008
4 0.0132 $\pm$ 0.0008 0.0082 $\pm$ 0.0008
4.5 0.0077 $\pm$ 0.0007 0.0052 $\pm$ 0.0006
5 0.0044 $\pm$ 0.0007 0.0033 $\pm$ 0.0005
5.5 0.0026 $\pm$ 0.0006 0.0020 $\pm$ 0.0005
6 0.0015 $\pm$ 0.0005 0.0013 $\pm$ 0.0005
6.5 0.0009 $\pm$ 0.0005 0.0008 $\pm$ 0.0005
7 0.0006 $\pm$ 0.0004 -
7.5 0.0004 $\pm$ 0.0004 -
8 0.0003 $\pm$ 0.0004 -
8.5 0.0002 $\pm$ 0.0003 -
9 0.0002 $\pm$ 0.0004 -
9.5 0.0002 $\pm$ 0.0003 -
: Summary of the properties of the critical solutions computed for different values of $l$. Note that both the echoing exponents, $\Delta_l$, and the mass scaling exponents, $\gamma_l$, rapidly decrease as $l$ increases. Quoted errors have been estimated from the variation in values computed across the different families of initial data. Values of $\Delta_l$ have been calculated using central proper time normalization of the lapse function, which is the natural normalization for type-II critical behavior. For $l>6.5$ we have not been able to calculate $\gamma_l$ due to lack of numerical precision. Note that the $l=0$ data agree with the original values calculated in [@Choptuik:jv], and that the $l=1$ data agree with values calculated in [@Liebling:1999ke] and [@Husa:2000kr] using models of completely different origin.[]{data-label="delta_table"}
Results {#results}
-------
Table \[delta\_table\] summarizes the values of $\Delta_l$ we have estimated using this procedure; the data are also graphed in Fig. \[deltas\_fig\]. Again, note that the reported values for $\Delta_l$ have been calculated using central proper time $T$ instead of proper time at infinity (the parameterization used in the numerical evolutions [*per se*]{}). Also the reported uncertainties have been estimated from the deviations in the $\Delta_l$ values computed across the the six different families of initial data. The first entry in Table \[delta\_table\] ($l=0$) corresponds to the original case studied in [@Choptuik:jv]. The second one ($l=1$) is apparently the same solution found for the self-gravitating collapse of an $SO(3)$ non-linear $\sigma$ model, assuming a hedgehog ansatz [@Husa:2000kr], [@Liebling:1999ke]. Interestingly, the values for $\Delta_1$ and $\gamma_1$ also agree quite well with the values obtained from the study of the axisymmetric collapse of a complex-valued scalar field with azimuthal quantum number $m=1$ [@hlcp], where values $\Delta \approx 0.42$ and $\gamma \approx 0.11$ are quoted. However, in the model considered in [@hlcp], the overall solution is clearly different because it is [*not*]{} spherically symmetric. The remainder of the solutions (for the other values of $l$) are, to the best of our knowledge, new.
![ Values of ${\mathrm{log}_{10}}\left( \Delta_l \right)$ versus $l$. In this figure we can see that $\Delta_l$ decreases almost exponentially with $l$. The different lines represent different families of initial data. Assuming universality, the differences between the values calculated for the different families provides one measure of error in our determination of $\Delta_l$. []{data-label="deltas_fig"}](deltas.ps){width="8.0cm"}
Systems exhibiting type II critical behavior, where the critical solution is self-similar, generally also exhibit power-law scaling of dimensionful quantities in near-critical evolutions. For example, we can expect the black hole mass, $M_{\rm BH}$, to scale as $$\label{mass-scaling}
M_{\rm BH} \sim C \left(p-p^\star \right)^{\gamma_l}$$ for super-critical evolutions as $p\to p^\star$ [^3]. Here $C$ is a constant that depends on the family of initial data while $\gamma_l$ is a universal exponent for each value of $l$, i.e. independent of the specific initial data family used to generate the critical solution. We have observed such scaling in at least some of our computations, but, following Garfinkle and Duncan [@Garfinkle:1998va] have found it more convenient to extract $\gamma_l$ by monitoring the maximum value of the trace of the stress tensor, ${\cal T}$, which, from the Einstein equations, is proportional to the maximum value of the Ricci curvature. On dimensional grounds $\cal T$ (defined by (\[trace\_t\])) and $R$ should both scale with an exponent $-2 \gamma$. This technique has the advantage of being more precise than a strategy based directly on (\[mass-scaling\]) since we can calculate the trace of the stress-energy more accurately than the mass of the black hole formed, and can perform the computation using sub-critical evolutions, where the gradients of field variables generally do not become as large as those in the super-critical cases. The values of $\gamma_l$ as a function of $l$ are listed in Table \[delta\_table\] and are plotted in Fig. \[gammas\_fig\].
![Values of ${\mathrm{log}_{10}}\left( \gamma_l \right)$ versus $l$, where $\gamma_l$ is the scaling exponent defined by (\[mass-scaling\]). As for the case of the echoing exponent, $\Delta_l$, $\gamma_l$ also decreases approximately exponentially with $l$. We note that due to lack of numerical precision we can only reliably compute $\gamma_l$ for $l\le6.5$ []{data-label="gammas_fig"}](gammas.ps){width="8.0cm"}
As is characteristic of type-II critical solutions exhibiting discrete self-similarity, $2M(t,r)/r$ oscillates at higher frequencies and on smaller spatial scales during the course of an evolution in the critical regime. As has already been noted, as $l$ increases, the echoing exponent $\Delta_l$ decreases rapidly. This can be observed in Fig. \[raw\_data\] where the evolution of the maximum in $r$ is shown as a function of time for four different values of $L$.
![ Evolution in time of the maximum in $r$ of the function $2M(t,r)/r$ for four different critical solutions with increasing value of $l$ ($l=0$, $l=1$, $l=2$ and $l=4$). The plot shows the evolution during the period of time when each solution shows discrete self similarity. The time coordinate is rescaled by $\Delta_l$ for visualization purposes and is shifted so that the function values coincide at $t=0$. We note how the solutions tend to periodicity with increasing values of $l$. []{data-label="raw_data"}](raw.ps){width="8.0cm"}
In addition, also in Fig. \[raw\_data\], we observe that the maximum and minimum values between which the spatial maximum of $2M(t,r)/r$ oscillates increase with $l$ (this fact is shown for all values of $l$ in Fig. \[tmr\_fig\]) indicating that the critical solutions are becoming increasingly relativistic as the angular momentum barrier becomes more pronounced. The amplitude of the oscillations between these extremal values decreases since $\min_r\left\{2M(t,r)/r\right\}$ increases more rapidly than $\max_r\left\{ 2M(t,r)/r\right\}$ (see Fig. \[tmr\_fig\]).
![$\max_t\left\{\max_r\left\{2M(t,r)/r\right\}\right\} $ in the critical regime as a function of $l$ (solid line) and the same for $\min_t\left\{\max_r\left\{2M(t,r)/r\right\}\right\} $ (dashed line). We see how both the maximum and minimum values of $2M/r$ increase with $l$. On the other hand the amplitude of oscillation, given by their difference, apparently tends to zero with increasing $l$. []{data-label="tmr_fig"}](tmr.ps){width="8.0cm"}
The assumption that the critical solutions are independent of the initial family of initial data implies that the spatial profiles at the same moment during the oscillation for two different families of initial data are the same up to some rescaling of the radial coordinate. In Fig. \[univ\_l=9\_fig\] we show a check of the universality of the spatial profile for the solutions computed with $l=9$. Specifically we compare the spatial profiles at times $T_n$, times at which the local maximum in time is achieved during criticality, for different families $F$ of initial data $F=1,...,6$ given in Table \[initial\_data\_table\]. In order to compare profiles we rescaled the radial coordinate by a constant $K_F$, which depends on the family of initial data. These constants are chosen in such a way that the $\ell_2$-norm[^4] of the difference of the profiles with respect to the one with $F=1$, which is considered to have $K_1=1$, are minimized. We observed that the maximum of the relative difference, i.e. the difference divided by the $\ell_2$-norm of the solution, is of the order of a few percent, providing strong evidence for universality. Similar differences have been observed for other values of the angular momentum parameter.
![ In the top pane we show the spatial profiles (in the region of self-similarity) of the scalar field $\psi$ for different families of initial data, but for fixed angular momentum parameter $l=9$. In particular we show the solutions $\psi_F$ calculated from initial data types $F={1,...,6}$ (see Table \[initial\_data\_table\]) at times when $\psi_F$ reaches maximum amplitude. Each solution is shifted by an amount proportional to its family number for better visualization, with $F=1$ the bottom curve, and $F=6$ the top. The $r$ coordinate is rescaled for each family by a constant factor $K_F$, which is family dependent, in such a way that the difference with respect to the profile obtained for the initial data labeled with $F=1$ (for which we consider $K_1=1$) is minimized. In the bottom pane we show the differences between the rescaled profiles for $F=2,...,6$ and the profile for $F=1$, divided by the $\ell_2$ norm of the solution. The maximum relative difference is of the order of a few percent, providing strong evidence that the critical solution is universal. []{data-label="univ_l=9_fig"}](univ_l=9.ps){width="8.0cm"}
Empirically, we have also found that, as we increase $l$ within a family of initial data, although $\Delta_l \to 0$ and $T^\star_l \to 0$, the product $T^\star_l \Delta_l$ appears to asymptote to a finite value. Note that ostensibly this product is family dependent (see Fig. \[periods\_fig\]), but again that all DSS type-II critical solutions are universal only up to a global scale transformation $(r,t) \to (kr,kt)$, with $k$ an arbitrary positive constant. Choosing $k=k(l)$ for each of the families so that $\max_r\left\{2 M(t,r)/r\right\}$ is attained at some fiducial radius $r_0$, and considering the case $l=10$, we find that the normalized asymptotic oscillation frequency, $f_0$, defined by $$\label{f_0}
f_0 = r_0 / (T^\star \Delta) = 4.35 \pm 0.01$$ agrees for all families to better than 1%. Again, the quoted uncertainty is estimated from the variation of $f_0$ across the different families of initial data. We note that for $l=10$ the near-critical solution stays at a near-constant radial position; our spatial resolution is insufficient to resolve the small changes associated with the extremely small value of $\Delta_l$. The radial location of $\max_r\left\{2 M(t,r)/r\right\}$ in this regime is the value of $r_0$ that we have used in (\[f\_0\]).
![$T^\star_l \Delta_l$ as a function of $l$. The fact that these products remain finite as $T^\star_l \to \infty$ and $\Delta_l \to 0$ is evidence that the critical solutions tend to a [*periodic*]{} solution in the limit $l \to \infty$. \[periods\_fig\] ](periods.ps){width="8.0cm"}
We also note that the observation that $f_0$ is apparently well defined and unique (up to the usual rescalings associated with type-II critical solutions), is consistent with the empirical observation that as $l$ increases, the critical solution becomes ever closer to a [*periodic*]{} solution. In particular, for a periodic solution we have $\Delta \to 0$, and then $$\begin{aligned}
T_n-T^\star &=& \left(T_0 - T^\star \right) e^{n \Delta} \approx
(T_0 - T^\star)\left(1+n\Delta \right) \nonumber\\
&\approx& -\left(T^\star \Delta\right)n-T^\star,\end{aligned}$$ where $T_0$ represents the loosely defined time demarking the onset of the critical regime (and whose precise value is clearly irrelevant in the limit $T^\star\to\infty$) which implies that the maximal value is attained at times $T_n$: $$T_n = -\left(T^\star \Delta \right) n.$$
As shown in Figs. \[l=10\_linear\_fig\] and \[l=10\_dss\_fig\], from our calculations for $l=10$, we cannot ascertain whether the solution is discretely self-similar with $\Delta_l$ very small ($< 0.0002$), or periodic with period $\tau=T^\star \Delta$.
![ Fit of the times $T_n$ at which $\max_r\left\{ 2M(t,r)/r\right\}$ reaches its maximum in time (triangles, left scale) assuming a periodic ansatz. Initial data type $F=1$ was used with angular momentum parameter $l=10$. We also plot the residuals of each data point with respect to the best fit (pentagons, right scale). []{data-label="l=10_linear_fig"}](l=10_linear.ps){width="8.0cm"}
![ Fit of the times $T_n$ at which $\max_r\left\{2M(t,r)/r\right\}$ reaches its maximum in time (triangles, left scale) assuming a self-similar ansatz. As in the previous plot, initial data type $F=1$ was used with angular momentum parameter $l=10$. Again, we also plot the residuals of each data point with respect to the best fit (pentagons, right scale). Notice that the errors in the fit are of the same order as the errors in the fit that assumes periodicity (Fig. \[l=10\_linear\_fig\]), indicating that from our numerical results we are unable to distinguish between the two types of solutions for $l\ge10$. []{data-label="l=10_dss_fig"}](l=10_dss.ps){width="8.0cm"}
Naively at least, we expect that for $l > 10$, distinguishing between discrete self-similarity and periodicity would become even more difficult. However, it is worth noting that for $l=20$ we have [*not*]{} yet seen evidence for (almost)-periodicity, with period $T^\star \Delta$, but have instead seen a more complicated structure near criticality that is not yet understood.
Conclusions {#sec:discussion}
===========
In this paper, we have discussed the results for a model that incorporates some of the effects of angular momentum in the context of critical gravitational collapse. A new family of spherically-symmetric critical solutions, (black hole threshold solutions) labelled by an angular momentum parameter, $l$, has been found. These solutions have similar properties to those for the $l=0$ case originally studied in [@Choptuik:jv]: specifically, the solutions exhibit discrete self-similarity, and have scaling laws for the values of dimensionful quantities in evolutions close to criticality. We have calculated the $l$-dependence of the echoing exponents $\Delta_l$, and the mass-scaling exponents $\gamma_l$, finding that both decrease rapidly with increasing $l$, (at least up to $l\approx 10$). Moreover, we have argued that as $l$ increases, the critical solution approaches a periodic evolution.
Together with the results of [@hlcp], our findings suggest that certain models of collapse may generically admit countable infinities of critical solutions, each member of which can be characterized by distinct near-origin regularity conditions (such as (\[psi\_fall\_off\]-\[pp\_fall\_off\])) that are preserved by dynamical evolution.
As we explained in the introduction, we expect that $\gamma_l = 1/\lambda_l$ where $\lambda_l$ is the Lyapunov exponent associated with the single unstable mode of the critical solution for angular momentum parameter $l$. Therefore since $\gamma_l \rightarrow 0$ with increasing $l$, we apparently have $\lambda_l \rightarrow \infty$. This has the interpretation of increased stability of the critical solution for increasing $l$, i.e. the period of time that a solution can remain close to criticality (for a fixed amount of fine tuning) increases with $l$. We believe that this can be interpreted as an effect of the angular momentum barrier which (partially) stabilizes the collapse to black hole formation.
It is our pleasure to thank the rest of the members of the numerical relativity group at the University of British Columbia and the members of the Hearne Institute in the Department of Physics at LSU for many useful discussions. In addition, special thanks go to A. Nagar and L. Lehner for reading this document. This research was supported by NSERC, the Canadian Institute for Advanced Research and the Government of the Basque Country through a fellowship to I.O. Most of the calculations were performed on the [vn.physics.ubc.ca]{} Beowulf cluster, which was funded by the Canadian Foundation for Innovation, and the British Columbia Knowledge Development Fund.
{#appendixA}
\#1[. \#1 .]{} \#1[. \#1 .]{}
We wish to show that the stress energy tensor ${\mathcal T}^{(l)a}{}_b=\sum_m {T^{(lm)}}^a{}_{b}$ is independent of $\theta $ and $\phi$, where ${T^{(lm)}}^a{}_{b}$ is the stress energy tensor associated with the solution $\psi^{(l)}(t,r)Q_{lm}(\theta,\phi)$, with the same function $\psi^{(l)}(t,r)$, for each value of $m$. For the scalar field, the tensor $T_{a b}$ can be written in terms of the solutions to the wave equation, $\Psi$, as T\^[a]{}\_[b]{} = g\^[a c]{}\_[,c]{}\_[b]{}- [12]{} \^a\_b g\^[dc]{}\_[,c]{}\_[d]{} , and if \_m M\^a\_b&=&g\^[ac]{}[(\^[(l)]{}(t,r)Q\_[lm]{}(,))]{}\_[,c]{}\
&&\_[,b]{} is independent of $\theta,~ \phi$, then so is ${\mathcal T}^{(l)a}{}_b$.
Using the definition of the $Q_{lm}$ this can be written in terms of the $Y_{lm}$ as \_m M\^a\_b&=&g\^[ac]{}[(\^[(l)]{}(t,r)Y\^\*\_[lm]{}(,))]{}\_[,c]{}\
&&\_[,b]{}.
We can write this in terms of the Green’s function P(,,’,’) = \_m Y\^\*\_[lm]{}(,)Y\_[lm]{}(’,’) in the limit as $\theta'\rightarrow\theta$ and $\phi'\rightarrow\phi$.
In bra-ket notation, this is just the operator P=\_m which commutes with all of the angular momentum operators. &=& \_m \[ L\_z, \]\
&=& \_m (m - m)\
&=&0, &=&\_m( .\
&&. - ((L\_x-iL\_y))\^)\
&=& \_m( .\
&& . - )\
&=&0. Thus $\sum_m Y^*_{lm}(\theta,\phi)Y_{lm}(\theta',\phi') $ must be a function of the only rotation invariant function of $\theta,\phi,\theta',\phi'$, which is the angle $\Theta$ defined by ()= ()(’)+()(’)(-’) . $\Theta$ is the angle between the two unit vectors with directions $\theta,\phi$ and $\theta',\phi'$ respectively. Since $P$ depends only on $\Theta$ we can choose $\theta=0$ to evaluate it, which gives P(,,’,’) &=& \_m Y\^\*\_[lm]{}(,)Y\_[lm]{}(’,’)\
&=& Y\^\*\_[l0]{}(0,0)Y\_[l 0]{}(,0)\
&=& [2l+14]{}P\_l(()).
The various components of the tensor $M$ are of three types: ones with no derivatives with respect to $\theta$ or $\phi$ (eg $M_{tt}$), those with one derivative, (for example $M_{t \theta}$) and those with two (eg, $M_{\theta\theta}$). The ones with no derivatives will be functions of $\lim_{\theta',\phi'\rightarrow \theta,\phi}P= \sqrt{(2l+1)/(4\pi)}\,P_l(1)$ which is clearly independent of $\theta,\phi$. The terms with one $\theta,\phi$ derivative will be functions of $$\begin{aligned}
\lim_{\Theta\rightarrow 0}\partial_{\theta,\phi} P_l(\cos(\Theta))&=&
\lim_{\Theta\rightarrow 0}
P_l'\{\sin(\Theta),\sin^2(\theta)\sin(\phi-\phi')\}\nonumber\\&=&0,\end{aligned}$$ and similarly the term M\_ && \_[0]{}\_\_[’]{}P\_l(())\_[’]{}(-’)\
&=&0. Thus the only two terms remaining are M\_&& \_[’]{}\_\_[’]{}P\_l((-’))\
&=&P\_l’((0))(-(0))(-1)\
&=&P\_l’(1),\
M\_[’]{}&& P\_l’(1)()\^2.
Thus the non-zero components of the tensor $M$ are $M_{tt},~M_{tr},~M_{rr},~M_{\phi\phi}=\sin^2(\theta)M_{\theta\theta}$, with only $M_{\phi\phi}$ having $\theta$ dependence. Thus, $M^a{}_b$ will be independent of $\theta,\phi$ and therefore so will ${\mathcal T}^{(l)a}{}_b$, as required.
In addition, from the equations of motion for the individual fields $\Psi_{lm}$, each of the energy momentum tensors for given $l,m$ is conserved in the overall spherically symmetric spacetime. Thus, so is their sum over $m$ for any given $l$, and we have \^[(l)a]{}\_[b;a]{}=0 .
[1]{} C. Gundlach, [*Living Rev. Rel.*]{} [**2**]{}, 4 (1999) \[arXiv:gr-qc/0001046\]. C. Gundlach, [*Phys. Rept.*]{} [**376**]{}, 339 (2003) \[arXiv:gr-qc/0210101\]. A. M. Abrahams and C. R. Evans, [*Phys. Rev. Lett.*]{} [**70**]{}, 2980 (1993). S. L. Liebling, [*Phys. Rev.*]{} [**D66**]{}, 041703, (2002) \[arXiv:gr-qc/0202093\]. S. Hod and T. Piran, [*Phys. Rev.*]{} [**D55**]{}, 440 (1997) \[arXiv:gr-qc/9606087\]. C. Gundlach, [*Phys. Rev.*]{} [**D55**]{}, 695 (1997) \[arXiv:gr-qc/9604019\]. M. W. Choptuik, E. W. Hirschmann, S. L. Liebling and F. Pretorius, [*Phys. Rev.*]{} [**D68**]{}, 044007, (2003) \[arXiv:gr-qc/0305003\]. D. Garfinkle, C. Gundlach and J. M. Martin-Garcia, [*Phys. Rev.*]{} [**D59**]{}, 104012, (1999) \[arXiv:gr-qc/9811004\]. C. Gundlach, [*Phys. Rev.*]{} [**D57**]{}, 7080, (1998) \[arXiv:gr-qc/9711079\]. C. Gundlach, [*Phys. Rev.*]{} [**D65**]{}, 084021, (2002) \[arXiv:gr-qc/9906124\]. M. W. Choptuik, E. W. Hirschmann, S. L. Liebling and F. Pretorius, [*Phys. Rev. Lett.*]{} [**93**]{}, 131101 (2004) \[arXiv:gr-qc/0405101\]. G. Rein, A. D. Rendall and J. Schaeffer, [*Phys. Rev.*]{} [**D58**]{}, 044007, (1998) \[arXiv:gr-qc/9804040\]. I. Olabarrieta and M. W. Choptuik, [*Phys. Rev.*]{} [**D65**]{}, 024007, (2002) \[arXiv:gr-qc/0107076\]. J. F. Ventrella and M. W. Choptuik, [*Phys. Rev.*]{} [**D68**]{}, 044020 (2003) \[arXiv:gr-qc/0304007\]. R. L. Marsa and M. W. Choptuik,\
http://laplace.phas.ubc.ca/users\_guide/users\_guide.html (1995). H. Kreiss and J. Oliger, [*Global Atmospheric Research Programme, Publications Series No. 10.*]{} (1973). M. W. Choptuik, [*Phys. Rev. Lett.*]{} [**70**]{}, 9, (1993). S. L. Liebling, [*Phys. Rev.*]{} [**D60**]{}, 061502, (1999) \[arXiv:gr-qc/9904077\]. S. Husa, C. Lechner, M. Purrer, J. Thornburg and P. C. Aichelburg, [*Phys. Rev.*]{} [**D62**]{}, 104007, (2000) \[arXiv:gr-qc/0002067\]. D. Garfinkle and G. C. Duncan, [*Phys. Rev.*]{} [**D58**]{}, 064024, (1998) \[arXiv:gr-qc/9802061\].
[^1]: There is some numerical evidence for growing [*non-spherical*]{} modes in near-critical collapse, both for perfect fluids [@Gundlach:1999cw] and massless scalar fields [@Choptuik:2003ac]. However, these modes appear to grow so slowly that, in both cases, it is expected that the spherical unstable mode continues to dominate near criticality.
[^2]: Note that, in general, $Q_{l m}(\theta, \phi)$ will [*not*]{} be eigenfunctions of the azimuthal rotation operator $(\partial / \partial \phi)$ since they are real.
[^3]: In accord with the results in [@hod] and [@Gundlach:1997], we expect small amplitude oscillations with period $\Delta$ to be superimposed on the scaling law (\[mass-scaling\]). We have, however, made no attempts to measure this effect in the current work.
[^4]: The $\ell_2$-norm of a vector u defined as $||u|| = \sqrt{\sum_{i=1}^N u_i^2/N}$.
|
---
abstract: 'It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categories over small preadditive categories. In this context, we also show how to deduce parametrizations of hereditary torsion theories of finite type, Abelian recollements by functor categories, and centrally splitting TTFs.'
author:
- 'Carlos Parra[^1]'
- 'Manuel Saorín[^2]'
- 'Simone Virili[^3]'
title: |
Torsion pairs in categories of modules\
over a preadditive category
---
Torsion pair, TTF triple, additive categories, Gabriel topology, Grothendieck topology, idempotent ideal, recollement.\
13D30, 16S90, 18E40, 18E05.
Introduction {#introduction .unnumbered}
============
[**Torsion theories**]{} (also called [**torsion pairs**]{}) were introduced by Dickson [@D] in the general setting of Abelian categories, taking as a model the classical theory of torsion Abelian groups. Given an Abelian category ${\mathcal{C}}$, a torsion pair $({\mathcal{T}},{\mathcal{F}})$ in ${\mathcal{C}}$ is a pair of full subcategories satisfying the following axioms:
1. ${\mathcal{T}}={}^{\perp}{\mathcal{F}}$ (and ${\mathcal{F}}={\mathcal{T}}^{\perp}$); where, for any class $\X$ of objects, we put $$\X^{\perp}:=\{C \in {\mathcal{C}}:{\mathcal{C}}(X,C)=0, \text{ for all }X \in \X\}\quad\text{and}$$ $${}^{\perp}\X:=\{C \in {\mathcal{C}}:{\mathcal{C}}(C,X)=0, \text{ for all }X \in \X\};$$
2. for each object $X$ of ${\mathcal{C}}$, there is an exact sequence $$\label{intro_standard_sequence}
\tag{\dag}\xymatrix{0 \ar[r] & T_X \ar[r] & X \ar[r] & F_X \ar[r] & 0,}$$ with $T_X\in{\mathcal{T}}$ and $F_X\in{\mathcal{F}}$.
Given a torsion pair ${\t}=({\mathcal{T}},{\mathcal{F}})$, the class ${\mathcal{T}}$ (resp., ${\mathcal{F}}$) is said to be a [**torsion**]{} (resp., [**torsionfree**]{}) [**class**]{}. Furthermore, ${\t}$ is said to be [**hereditary**]{} if ${\mathcal{T}}$ is closed under taking subobjects (see, Sec.\[recall\_torsion\_subs\]). (Hereditary) Torsion pairs have become a fundamental tool in the study of Grothendieck categories and their localizations; furthermore, they play an important role in Algebraic Geometry and Representation Theory.
Given a (unitary and associative) ring $R$, it is well-known since Gabriel’s thesis [@Ga] that there is a one-to-one correspondence between [**Gabriel topologies**]{} in $R$ (which are suitable filters of ideals of the ring) and hereditary torsion classes in $\mod R$ (see also [@S Chapter VI]). On the other hand, rings may be regarded as a special case of small [**preadditive categories**]{} (i.e., small categories enriched over Abelian groups, see Sec.\[appendix\_on\_pre\_additive\]). Hence, the category of modules $\mod {\mathcal{A}}$ over a small preadditive category ${\mathcal{A}}$ naturally arises, see Sec.\[subs\_modules\]. There are many sources in the literature which deal with this generalization (see, for example, [@mitchell]). Given a preadditive category ${\mathcal{A}}$, the notion of “linear Grothendieck topology” on ${\mathcal{A}}$ (see Sec.\[grabriel\_subs\]), introduced in [@Lo; @RG], is an additive version of the notion of Grothendieck topology, which is of common use in Algebraic Geometry (see, for example, [@MM]). When applied to a preadditive category with just one object, one obtains the usual notion of Gabriel topology for rings.
Our first general result uses these Grothendieck topologies to extend Gabriel’s classical bijection:
[**Theorem A**]{} (see Thm.\[thm.Gabriel bijection for small categories\])[**.**]{} [ *Let ${\mathcal{A}}$ be a small preadditive category. Then there is an (explicit) one-to-one correspondence between (linear) Grothendieck topologies on ${\mathcal{A}}$ and hereditary torsion pairs in $\mod {\mathcal{A}}$.*]{}
Let us remark that the above theorem could be deduced by the more general statement [@Pr Prop.11.1.11], that applies to locally finitely generated Grothendieck categories; it appears without proof as Proposition 11.1.11 in [@Pr] (the proof follows the same lines of analogous results in [@G; @Po]). Furthermore, Theorem A is in the same spirit of [@AB Prop.2.4 and 3.6], where related characterizations of hereditary torsion classes are given in the more general setting of Grothendieck categories with a projective generator. For an implicit approach to the bijection of the theorem the reader is referred to [@Lo; @L] (see also [@RG Sec.1.1]).
Recall now that, in good enough Abelian categories (e.g., $\mod {\mathcal{A}}$ for a small preadditive category ${\mathcal{A}}$) a class ${\mathcal{T}}$ is a torsion class (resp., a torsionfree class) if and only if it is closed under taking quotients, extensions and coproducts (resp., subobjects, extensions and products). Hence, if we start with a hereditary torsion class ${\mathcal{T}}$ which is also closed under taking products, than ${\mathcal{T}}$ is both a torsion and a torsionfree class, for short, a [**TTF class**]{}. In such case, a triple of the form $(^{\perp}{\mathcal{T}},{\mathcal{T}},{\mathcal{T}}^{\perp})$ is said to be a [**TTF triple**]{} (see, Sec.\[recall\_torsion\_subs\]); these objects have been introduced in categories of modules $\mod R$ over a ring $R$ by Jans [@Jans], who showed that TTF triples are in bijection with idempotent ideals of $R$. Our second general result is to extend Jans’ bijection from rings to small preadditive categories.
[**Theorem B**]{} (see Thm.\[prop.direct proof\])[**.**]{} [ *Let ${\mathcal{A}}$ be a small preadditive category. Then there is an (explicit) one-to-one correspondence between idempotent ideals of ${\mathcal{A}}$ and TTF triples in $\mod {\mathcal{A}}$.*]{}
After extending the Gabriel’s and Jan’s bijections to parametrize hereditary torsion pairs and TTF triples we concentrate on the following problem. [**Recollements**]{} of Abelian categories are particularly nice decompositions of a given Abelian category by two other Abelian categories and, in nice enough situations (e.g., categories like $\mod {\mathcal{A}}$ for a preadditive category ${\mathcal{A}}$), they are known to be in bijection with TTF triples. In particular, given a TTF triple $({\mathcal{C}},{\mathcal{T}},{\mathcal{F}})$ in $\mod {\mathcal{A}}$, one can see $\mod {\mathcal{A}}$ as a recollement by the two Abelian categories ${\mathcal{T}}$ and ${\mathcal{C}}\cap {\mathcal{F}}$. We say that this is a [**recollement by categories of modules**]{} if and only if both ${\mathcal{T}}$ and ${\mathcal{C}}\cap {\mathcal{F}}$ are equivalent to categories of module over some small preadditive categories. The following result extends some of the main results in [@PV], characterizing those idempotent ideals of ${\mathcal{A}}$ that induce recollements by categories of modules:
[**Theorem C**]{} (see Thm.\[main\_thm\_recollement\])[**.**]{}
*Let $\mathcal{A}$ be a small preadditive category. Then there are (explicit) one-to-one correspondences between:*
1. equivalence classes of recollements of $\mod {\mathcal{A}}$ by categories of modules;
2. idempotent ideals of $\mathcal{A}$ that are trace of sets of finitely generated projective modules;
3. the full subcategories of $\proj({\mathcal{A}})$ which are closed under coproducts and summands.
Furthermore, up to replacing ${\mathcal{A}}$ by a Morita equivalent small preadditive category, the idempotent ideals in (2) are generated by a set of idempotent endomorphisms.
Finally, recall that a torsion pair $({\mathcal{T}},{\mathcal{F}})$ is said to be [**split**]{} if, for any object $X$, the canonical sequence splits. Similarly, a TTF triple $({\mathcal{C}},{\mathcal{T}},{\mathcal{F}})$ is said to [**split**]{} if both torsion pairs $({\mathcal{C}},{\mathcal{T}})$ and $({\mathcal{T}},{\mathcal{F}})$ split. Note that it might happen that only one of these torsion pair splits (see [@NS]), withouth the TTF triple being split. By a result of Jans [@Jans], the bijection between TTF triples and idempotent ideals restricts to a second one between central idempotents of a ring $R$ and splitting TTF triples in $\mod R$. As a last result, we extend this correspondence to small preadditive categories:
[**Corollary D**]{} (see Sec.\[coro\_central\_id\])[**.**]{} [ *Let ${\mathcal{A}}$ be a small preadditive category. Then there is an (explicit) one-to-one correspondence between idempotents of the center $Z({\mathcal{A}})$ of $\mathcal{A}$ and split TTF triples in $\mod {\mathcal{A}}$.*]{}
Rings with several objects and their modules
============================================
In this first section we recall some basic results and definitions about small preadditive categories (see Sec.\[appendix\_on\_pre\_additive\]). In particular, starting with a small preadditive category ${\mathcal{A}}$, we construct in a universal way a small additive and idempotent complete category $\widehat{\mathcal{A}}_{\oplus}$, called the Cauchy completion of ${\mathcal{A}}$. In Sec.\[subs\_modules\] we introduce and study the category of modules $\mod {\mathcal{A}}$, showing in particular that two small preadditive categories have equivalent module categories if and only if they have equivalent Cauchy completions. In Sec.\[subs\_traces\_and\_bimodules\] we briefly recall the definition of trace of a class of modules on a given module. Furthermore, after recalling some basic properties of bimodules, we show how to define the trace on bimodules. In Sec.\[subs\_locally\_coh\], we obtain a characterization of when a category of modules over a small preadditive category ${\mathcal{A}}$ is locally coherent. We conclude by recalling in Sec.\[subs\_centre\] the notion of centre of a preadditive category.
Preadditive categories and Cauchy completion {#appendix_on_pre_additive}
--------------------------------------------
We denote by $\Padd$ (resp., ${\mathrm{Add}}$) the ($2$-)category of [**preadditive**]{} (resp., [**additive**]{}) [**categories**]{}. For the rest of this subsection, ${\mathcal{A}}$ will denote a [**small**]{} preadditive category.
We define the [**additive closure**]{} $\widehat {\mathcal{A}}$ of ${\mathcal{A}}$ as follows:
1. $\Ob(\widehat{\mathcal{A}}):=\{(a_1,\dots,a_n): n\in\N,\, a_i\in \Ob({\mathcal{A}})\}$;
2. given $n$, $m\in\N$ and $a=(a_1,\dots,a_n)$, $b=(b_1,\dots,b_m)\in\Ob(\widehat{{\mathcal{A}}})$, $${\widehat{{\mathcal{A}}}}(a,b):=\{(r_{i,j}):r_{i,j}\colon a_i\to b_j, \text{ with } i=1,\dots,n,\, j=1,\dots,m\}.$$
3. composition is given by the usual row-by-column multiplication of matrices.
It is the well-known that $\widehat{{\mathcal{A}}}$ is a small additive category, where the coproduct of two objects $(a_1,\dots,a_n)$ and $(b_1,\dots,b_m)$ is given by $(a_1,\dots,a_n,b_1,\dots,b_m)$. Furthermore, the inclusion $\iota\colon {\mathcal{A}}\to \widehat{{\mathcal{A}}}$ such that $a\mapsto (a)$ is universal in a suitable sense (in particular ${\mathcal{A}}\cong \widehat{{\mathcal{A}}}$ if ${\mathcal{A}}$ was already additive):
\[lemma\_preadditive\_additive closure\] Let ${\mathcal{A}}$ be a small preadditive category and $\B$ an additive category (not necessarily small), then $\iota\colon {\mathcal{A}}\to \widehat{{\mathcal{A}}}$ induces an equivalence of categories (here we are using the $2$-categorical structure of $\Padd$) $$-\circ\iota\colon {\mathrm{Add}}(\widehat{{\mathcal{A}}}, \B)\to \Padd({\mathcal{A}},\B).$$
Let $f\colon {\mathcal{A}}\to \B$ be an additive functor and define $$g\colon \widehat{\mathcal{A}}\to \B\quad \text{as}\quad g(a_1,\dots,a_n):=f(a_1)\sqcup\cdots\sqcup f(a_n).$$ Clearly, $g\circ\iota=f$. Consider now a second functor $g'\colon \widehat{{\mathcal{A}}}\to \B$ and let us sketch an argument to show that there is an isomorphism $$\begin{aligned}
{{\mathrm{Add}}(\widehat{{\mathcal{A}}}, \B)}(g,g')&\to {\Padd({\mathcal{A}},\B)}(f ,g'\circ \iota)\label{map_to_ff_pre_to_add}\\
\notag \alpha&\mapsto \alpha *\iota,\end{aligned}$$ where $*$ denotes the horizontal composition of natural transformations. Indeed, let $\alpha\colon g\to g'$ be such that $\alpha*\iota=0$. For any $a\in \widehat{{\mathcal{A}}}$, $\alpha_a\colon g(a)\to g'(a)$ can be represented by a diagonal matrix and the diagonal entries of this matrix are trivial since $\alpha_a^{i,i}$ is conjugated to $(\alpha*\iota)_{a_i}=0$.
On the other hand, given $\beta \colon g\circ \iota \to g'\circ \iota$, define $\alpha\colon g\to g'$ as follows: for $a=(a_1,\dots,a_n)\in\widehat{{\mathcal{A}}}$, $\alpha_a\colon g(a)\to g'(a)$ is the diagonal $n\times n$ matrix $(\alpha_a^{i,j})$ such that $\alpha_a^{i,i}:=\beta_{a_i}\colon g\iota(a_i)\to g'\iota(a_i)$. It is easy to see that $\alpha$ is a natural transformation and, clearly, $\alpha*\iota=\beta$.
By the above lemma, the $2$-category of small additive categories is reflective in the $2$-category of small preadditive categories, that is, there is a ($2$-)functorial way to make a small preadditive category into an additive category. A second application will be given in the next subsection: we will apply the lemma with $\B=\Ab$ to show that ${\mathcal{A}}$ is Morita equivalent to $\widehat {\mathcal{A}}$.
Suppose now ${\mathcal{A}}$ is an additive category; we define the [**idempotent completion**]{} ${\mathcal{A}}_{\oplus}$ of ${\mathcal{A}}$ as follows:
1. $\Ob({\mathcal{A}}_{\oplus}):=\{(a,r): a\in \Ob({\mathcal{A}}),\, r\colon a\to a \text{ such that }r^2=r\}$;
2. given $(a,r),\, (b,s)\in\Ob ({\mathcal{A}}_\oplus)$, $${{\mathcal{A}}_\oplus}((a,r),(b,s)):=\{t\colon a\to b: t=str\}.$$
3. composition is as expected. Note that the identity of $(a,r)$ is $r\colon a\to a$.
It is well-known, and easy to verify, that ${\mathcal{A}}_\oplus$ is a small additive category where idempotents split due to the following fact: given $a\in \Ob ({\mathcal{A}})$ and an idempotent $r\colon a\to a$, there is the following decomposition in ${\mathcal{A}}_{\oplus}$ $$(a,\id_a)=(a,r)\oplus (a,\id_a-r)$$ where clearly $(\id_a-r)=(\id_a-r)^2$ is an idempotent and the inclusions in the coproduct are given by $r\colon (a,r)\to (a,\id_a)$ and $(\id_a-r)\colon (a,\id_a-r)\to (a,\id_a)$. Furthermore, the inclusion $\iota\colon {\mathcal{A}}\to {\mathcal{A}}_{\oplus}$ such that $a\mapsto (a,\id_a)$ is universal in a suitable sense (in particular, ${\mathcal{A}}\cong {\mathcal{A}}_{\oplus}$ if idempotents split in ${\mathcal{A}}$):
\[lemma\_additive\_idempotent closure\] Let ${\mathcal{A}}$ be a small additive category and $\B$ an idempotent complete additive category, then $\iota\colon {\mathcal{A}}\to {\mathcal{A}}_{\oplus}$ induces an equivalence of categories (here we are using the $2$-categorical structure of ${\mathrm{Add}}$) $$-\circ\iota\colon {\mathrm{Add}}({\mathcal{A}}_{\oplus}, \B)\to {\mathrm{Add}}({\mathcal{A}},\B).$$
Let $f\colon {\mathcal{A}}\to \B$ be an additive functor and define a functor $g\colon {\mathcal{A}}_{\oplus}\to \B$ as follows: given $(a,r)\in \Ob({\mathcal{A}}_{\oplus})$, where $r$ is an idempotent of $a\in {\mathcal{A}}$, then $f(r)$ is an idempotent of $f(a)\in \B$, hence $f(a)=b_1\oplus b_2$ and $f(r)=\pi_1\epsilon_1$, where $\pi_1$ and $\epsilon_1$ are the projection and inclusion relative to $b_1$. We then let $g(a,r):=b_1$. Clearly, $g\circ \iota=f$. Consider a second functor $g'\colon {\mathcal{A}}_\oplus\to \B$ and let us verify that there is an isomorphism $$\begin{aligned}
{{\mathrm{Add}}({\mathcal{A}}_\oplus, \B)}(f,f')&\to {{\mathrm{Add}}({\mathcal{A}},\B)}(f\circ \iota ,f'\circ \iota)\label{map_to_ff_add_to_id}\\
\notag \alpha&\mapsto \alpha *\iota.\end{aligned}$$ Indeed, consider $\alpha\colon f\to f'\colon {\mathcal{A}}_\oplus\to \B$ such that $\alpha*\iota=0$. For any $(a,r)\in \Ob({\mathcal{A}}_{\oplus})$ there is a commutative diagram in ${\mathcal{A}}_{\oplus}$ $$\xymatrix@C=80pt{
(a,r)\ar[r]^{\epsilon_{(a,r)}:=r}\ar[d]_{\id_{(a,r)}:=r}&(a,\id_a)\ar@/_-12pt/[dl]^-{\ \ \ \pi_{(a,r)}:=r}\\(a,r)&}$$ (here $\pi_{(a,r)}$ and $\epsilon_{(a,r)}$ are represented by the same idempotent morphism $r\colon a\to a$ but the former is a morphism $(a,\id_a)\to (a,r)$, while the latter is a morphism $(a,r)\to (a,\id_a)$). Now, $\alpha_{(a,\id_a)}=(\alpha*\iota)_a=0$ and so $f'(\epsilon_{(a,r)})\alpha_{(a,r)}=\alpha_{(a,\id_a)} f(\epsilon_{(a,r)})=0$, so that $\alpha_{(a,r)}=f'(\pi_{(a,r)})f'(\epsilon_{(a,r)})\alpha_{(a,r)}=0$. Finally, let $\beta\colon f\circ \iota \to f'\circ \iota$ and define $\alpha\colon f\to f'$ as follows: given $(a,r)\in \Ob ({\mathcal{A}}_{\oplus})$, let $\alpha_{(a,r)}:=f'(\pi_{(a,r)})\beta_a f(\epsilon_{(a,r)})$. Clearly, $\alpha*\iota=\beta$.
By the above lemma, the $2$-category of idempotent complete small additive categories is reflective in the $2$-category of small additive categories, that is, there is a ($2$-)functorial way to make a small additive category idempotent complete. A second application will be given in the next subsection: we will apply the lemma with $\B=\Ab$ to show that ${\mathcal{A}}$ is Morita equivalent to ${\mathcal{A}}_\oplus$.
Given a preadditive category ${\mathcal{A}}$, the idempotent completion of the additive closure $\widehat {\mathcal{A}}_\oplus$ of ${\mathcal{A}}$ is usually referred to as the [**Cauchy completion**]{} of ${\mathcal{A}}$ (see [@Lawvere]). Furthermore, ${\mathcal{A}}$ is said to be [**Cauchy complete**]{} if it is equivalent to its Cauchy completion.
Modules and Morita equivalence {#subs_modules}
------------------------------
A [**right**]{} (resp., [**left**]{}) [**module**]{} $M$ over a small preadditive category ${\mathcal{A}}$ is an (always additive) functor $M\colon {\mathcal{A}}^{\op}\to \Ab$ (resp., $M\colon {\mathcal{A}}\to \Ab$). A morphism (a natural transformation) $\phi\colon M\to N$ between right ${\mathcal{A}}$-modules consists of a family of morphisms $\phi_a\colon M(a)\to N(a)$ (of Abelian groups), with $a$ ranging in $\Ob ({\mathcal{A}})$, such that the following squares commute for all $(r\colon a\to b)\in {\mathcal{A}}$: $$\xymatrix{
M(a)\ar[r]^{\phi_a}&N(a)\\
M(b)\ar[u]^{M(r)}\ar[r]^{\phi_b}&N(b).\ar[u]_{N(r)}
}$$ We denote by $\mod {\mathcal{A}}$ (resp., $\lmod {\mathcal{A}}$) the category of right (resp., left) ${\mathcal{A}}$-modules. Given two right (resp., left) ${\mathcal{A}}$-modules $M$ and $N$, we denote by $\hom_{\mathcal{A}}(M,N)$ their group of morphisms in $\mod {\mathcal{A}}$ (resp., $\lmod {\mathcal{A}}$). As a natural example of right module over ${\mathcal{A}}$ one can consider the representable modules $$\label{representables_definition}
H_a:={\mathcal{A}}(-,a)\colon {\mathcal{A}}^{\op}\to \Ab;$$ for any $a\in\Ob({\mathcal{A}})$.
\[mod\_is\_groth\] Let ${\mathcal{A}}$ be a small preadditive category. Then $\mod {\mathcal{A}}$ is a Grothendieck category with a family of small projective generators.
The above lemma is well-known but nevertheless let us give a sketchy proof. The zero object $0$ in $\mod {\mathcal{A}}$ is the constant functor $a\mapsto 0$, for all $a\in\Ob({\mathcal{A}})$ and co/kernels are constructed componentwise. As a consequence, a sequence $0\to N\to M\to M/N\to 0$ in $\mod {\mathcal{A}}$ is short exact if and only if $0\to N(a)\to M(a)\to (M/N(a))(=M(a)/N(a))\to 0$ is a short exact sequence in $\Ab$, for all $a\in \Ob({\mathcal{A}})$. Furthermore, given a morphism $\phi$ in $\mod {\mathcal{A}}$, the canonical map $\varphi\colon\coker({\mathrm{Ker}}(\phi))\to {\mathrm{Ker}}(\coker(\phi))$ is an isomorphism since, for any $a\in \Ob({\mathcal{A}})$, the map $\varphi_a$ is an isomorphism in $\Ab$.
Furthermore, arbitrary co/limits are induced componentwise by those in $\Ab$. Hence, $\mod {\mathcal{A}}$ is a bicomplete Abelian category where products and direct limits are exact.
To see that $\mod {\mathcal{A}}$ is Grothendieck, it remains to describe a family of generators. In fact, one can find a family of finitely generated (=finitely presented) projective generators. To give a complete description of such modules, let us introduce the following notation: for a morphism $\alpha \colon x\to y$ in $\mathcal{A}$ we let $$\alpha\mathcal{A}:= {\mathrm{Im}}(\alpha\circ-\colon H_x\to H_y)\leq H_y.$$ In particular, $\id_x{\mathcal{A}}=H_x$, for all $x\in \Ob({\mathcal{A}})$.
\[description\_fpp\_lemma\] Let $\mathcal{A}$ be a small preadditive category, let $P$ be a right $\mathcal{A}$-module, and consider the following assertions:
1. $P$ is finitely generated projective;
2. $P$ is isomorphic to $\epsilon\mathcal{A}$, for some idempotent endomorphism $\epsilon\in\mathcal{A}(x,x)$;
3. $P$ is isomorphic to $H_x$, for some $x\in\Ob({\mathcal{A}})$.
The implications $(3)\Rightarrow(2)\Rightarrow(1)$ always hold true. On the other hand, $(1)\Rightarrow (2)$ holds if ${\mathcal{A}}$ is additive, while $(2)\Rightarrow (3)$ holds if ${\mathcal{A}}$ is idempotent complete. In particular, all the assertions are equivalent if ${\mathcal{A}}$ is Cauchy complete.
(3)$\Rightarrow$(2) is clear since $H_x\cong \id_x{\mathcal{A}}$, and $\id_x$ is an idempotent in ${\mathcal{A}}(x,x)$.
(2)$\Rightarrow$(1). The fact that $H_x$ is finitely generated and projective is a consequence of (the additive version of) the Yoneda Lemma and some standard computations. Hence, it is enough to prove that the inclusion $\iota\colon \epsilon\mathcal{A}\hookrightarrow H_x$ is a section in $\mod {\mathcal{A}}$. Indeed, for each $a\in\Ob({\mathcal{A}})$, we define $\pi_a\colon H_x(a)=\mathcal{A}(a,x)\to (\epsilon\mathcal{A})(a)$ by $\pi_a(\beta)=\epsilon\circ\beta$. It is routine to check that the $\pi_a$’s define a natural transformation $\pi\colon H_x\to \epsilon\mathcal{A}$ that is a retraction for the inclusion.
(1)$\Rightarrow$(2), when ${\mathcal{A}}$ is additive. Being $P$ finitely generated, we have a retraction $$\xymatrix{
\rho\colon \coprod_{i=1}^nH_{x_i}\twoheadrightarrow P.
}$$ By additivity, the coproduct $x=\coprod_{i=1}^nx_i$ exists in $\mathcal{A}$, so we have an isomorphism $\coprod_{i=1}^nH_{x_i}\cong H_x$, and we identify $\rho$ as a retraction $\rho\colon H_x\twoheadrightarrow P$. By choosing a section $\lambda \colon P\to H_x$ for $\rho$, we get an idempotent endomorphism $\epsilon:=\lambda\circ\rho$ of $H_x$ whose image is isomorphic to $P$. By the Yoneda Lemma, we then get $\epsilon=\epsilon^2\in\hom_{\mathcal{A}}(H_x,H_x)\cong (H(x))(x)=\mathcal{A}(x,x)$, so that $P\cong\epsilon\mathcal{A}$.
(2)$\Rightarrow$(3), when ${\mathcal{A}}$ is idempotent complete. As $\epsilon$ is an idempotent, and ${\mathcal{A}}$ is idempotent complete, there are $y\in\Ob({\mathcal{A}})$, $\rho\colon x\to y$ and $\lambda\colon y\to x$ such that $\epsilon=\lambda\circ\rho$ and $\rho\circ\lambda=\id_y$. Hence, $P\cong \epsilon{\mathcal{A}}\cong \rho{\mathcal{A}}=H_y$.
As a consequence of the above lemma one can give a second description of the Cauchy completion $\widehat {\mathcal{A}}_\oplus$ of a preadditive category ${\mathcal{A}}$. For that, let us introduce the following notation: for a class $\S$ of objects we let $${\mathrm{add}}(\S):=\{\text{summands of finite coproducts of objects in $\S$}\}.$$
\[application\_yoneda\] Let ${\mathcal{A}}$ be a small preadditive category and denote by $\proj({\mathcal{A}})$ the class of finitely generated projective ${\mathcal{A}}$-modules. Then,
1. ${\mathcal{A}}$ is equivalent to the full subcategory of $\proj({\mathcal{A}})$ spanned by the representables;
2. $\widehat {\mathcal{A}}$ is equivalent to the full subcategory of $\proj({\mathcal{A}})$ spanned by the finite coproducts of representables;
3. $\widehat{\mathcal{A}}_{\oplus}$ is equivalent to $\proj({\mathcal{A}})={\mathrm{add}}(H_a:a\in\Ob({\mathcal{A}}))$.
\(1) is a consequence of the Yoneda Lemma. In fact, the functor $$y\colon {\mathcal{A}}\to \mod {\mathcal{A}},$$ that maps an object $a\in\Ob({\mathcal{A}})$ to $H_a$, is fully faithful and we have already observed that each of the $H_a$’s is finitely generated and projective.
\(2) Consider a functor $$\widehat y\colon \widehat {\mathcal{A}}\to \mod {\mathcal{A}}$$ that maps an object $(a_1,\dots,a_k)\in \Ob(\widehat {\mathcal{A}})$ to the coproduct $\coprod_{i=1}^kH_{a_k}$, and that sends a morphism in $\Ob(\widehat {\mathcal{A}})$ (which is by definition a suitable matrix) to the morphism between coproducts represented by the corresponding matrix. This functor is clearly fully faithful, so that (2) also follows.
\(3) Let us introduce some notation first: given $(a,r)\in\Ob(\widehat{\mathcal{A}}_{\oplus})$ (that is, $a\in \Ob(\widehat{\mathcal{A}})$ and $r$ an idempotent in $\widehat{\mathcal{A}}(a,a)$) the morphism $\widehat y(r)\colon \widehat y(a)\to \widehat y(a)$ is an idempotent endomorphism in $\proj({\mathcal{A}})$, so we obtain the following epi-mono factorization in $\proj({\mathcal{A}})$: $$\xymatrix@C=50pt@R=3pt{
\widehat y(a)\ar[rr]^{\widehat y(r)}\ar@/_5pt/[dr]_{\pi_r}&& \widehat y(a)\\
&P_r\ar@/_5pt/[ur]_{\iota_r}&
}$$ where $\pi_r\iota_r=\id_{P_r}$, and where $P_r$ is finitely generated projective, as it is a summand of the finitely generated projective object $\widehat y(a)$. We can now define the following functor: $$\widehat y_{\oplus}\colon \widehat {\mathcal{A}}_{\oplus}\to \mod {\mathcal{A}}$$ mapping an object $(a,r)$ in $\widehat {\mathcal{A}}_{\oplus}$ to $P_r$, and a morphism $f\colon (a,r)\to (b,s)$ to $\pi_s\circ\widehat y(f)\circ\iota_r$. Let us verify that $\widehat y_{\oplus}$ is an equivalence. Indeed, any $P\in \proj({\mathcal{A}})$ is a summand of a finite coproduct of representables and so, using the equivalence proved in part (2), there is an object $a$ in $\widehat {\mathcal{A}}$ and an idempotent endomorphism $r\in \widehat{\mathcal{A}}(a,a)$ such that $P= {\mathrm{Im}}(\widehat y(r))$, so that $P=P_r$. To conclude, let $(a,r)$ and $(b,s)$ be objects in $\widehat{\mathcal{A}}_\oplus$ and let us verify that the following homomorphism is bijective: $$\begin{aligned}
\widehat{\mathcal{A}}_\oplus((a,r),(b,s))&\to \hom_{\mathcal{A}}(P_r,P_s)\\
f&\mapsto \pi_s\circ\widehat y(f)\circ\iota_r.\end{aligned}$$ In fact, given $f\in \widehat{\mathcal{A}}_\oplus((a,r),(b,s))$ (so $f=s\circ f\circ r$) such that $ \pi_s\circ\widehat y(f)\circ\iota_r=0$, then $\widehat y(f)= \widehat y(s)\circ \widehat y(f)\circ \widehat y(r)= \iota_s\circ\pi_s\circ \widehat y(f)\circ \iota_r\circ\pi_r=0$, so $f=0$ by (2). On the other hand, given $\phi \in \hom_{\mathcal{A}}(P_r,P_s)$, let $\tilde\phi:=\iota_s\circ \phi\circ \pi_r\colon \widehat y(a)\to \widehat y(b)$ and let $f\colon a\to b$ be such that $\widehat y(f)=\tilde\phi$. Then $f=s\circ f\circ r$, so $f$ can be viewed as a morphism $(a,r)\to(b,s)$ in $\widehat{\mathcal{A}}_\oplus$ and, as such, $\widehat y_\oplus(f)=\pi_s\circ\widehat y(f)\circ\iota_r= \pi_s\circ \iota_s\circ \phi\circ \pi_r\circ\iota_r =\phi$.
Consider now an additive functor $\phi\colon {\mathcal{A}}\to \B$ between two small preadditive categories. Then, $\phi$ induces a [**restriction of scalars functor**]{} $$\phi_*\colon \mod{\B}\to \mod {\mathcal{A}}\qquad\text{such that}\qquad M\mapsto M\circ\phi.$$ It is easy to verify that $\phi_*$ is exact and that it commutes with co/products so, by the Special Adjoint Functor Theorem (see, for example, [@Bo Thm. 3.3.4]), it has a left adjoint, called the [**extension of scalars**]{}, and a right adjoint, called the [**coextension of scalars**]{}, denoted respectively by $\phi^*$ and $\phi^!$. $$\xymatrix{
\mod\B\ar[rr]|{\phi_*}&&\mod {\mathcal{A}}\ar@/_15pt/[ll]_{\phi^!}\ar@/_-15pt/[ll]^{\phi^*}
}$$ As a corollary of Lemm.\[lemma\_preadditive\_additive closure\] and \[lemma\_additive\_idempotent closure\] we can give a precise relation between $\mod{\mathcal{A}}$, $\mod{\widehat{\mathcal{A}}}$, and $\mod {\widehat {\mathcal{A}}_\oplus}$:
\[ADD\_morita\] Given a small preadditive category ${\mathcal{A}}$, consider the inclusions $\iota\colon {\mathcal{A}}\to \widehat{{\mathcal{A}}}$ and $\iota'\colon \widehat{\mathcal{A}}\to \widehat{{\mathcal{A}}}_\oplus$. The restrictions of scalars along $\iota$ and $\iota'$ are both equivalences. As a consequence, two small preadditive categories ${\mathcal{A}}$ and $\B$ are Morita equivalent if and only if there is an equivalence of categories $\widehat{\mathcal{A}}_\oplus\cong\proj({\mathcal{A}})\cong \proj(\B)\cong \widehat\B_\oplus$.
Let us conclude this subsection with the following remark, where an object $a\in\Ob({\mathcal{A}})$ is a [**$\oplus$-generator**]{} if ${\mathcal{A}}={\mathrm{add}}(a)$:
\[rem\_morita\_cauchy\] By the above corollary, we obtain the following bijections: [ $$\xymatrix@R=12pt@C=-60pt{
{\left\{\begin{matrix}\text{(Ab.$3$) Abelian categories with a set}\\ \text{of small projective generators}\\ \text{up to equivalence}\end{matrix}\right\}}\ar@{<->}[rr]^(.55){1:1}\ar@{<->}[rd]^(.65){1:1}&&{\left\{\begin{matrix}\text{Small Cauchy complete}\\ \text{additive categories}\\ \text{up to equivalence}\end{matrix}\right\}}\\
&{\left\{\begin{matrix}\text{Preadditive categories up to Morita equivalence}\end{matrix}\right\}}\ar@{<->}[ru]^(.35){1:1}.
}$$ ]{} Furthermore, these bijections induce, by restriction, the following ones: [ $$\xymatrix@R=15pt@C=-40pt{
{\left\{\begin{matrix}\text{(Ab.$3$) Abelian categories with a}\\ \text{small projective generator}\\ \text{up to equivalence}\end{matrix}\right\}}\ar@{<->}[rr]^(.51){1:1}\ar@{<->}[rd]^(.65){1:1}&&{\left\{\begin{matrix}\text{Small Cauchy complete additive}\\ \text{categories with a $\oplus$-generator}\\ \text{up to equivalence}\end{matrix}\right\}}\\
&{\left\{\begin{matrix}\text{Rings up to Morita equivalence}\end{matrix}\right\}}\ar@{<->}[ru]^(.35){1:1}
}$$ ]{}
Traces and bimodules {#subs_traces_and_bimodules}
--------------------
Given a class $\S$ of ${\mathcal{A}}$-modules and an ${\mathcal{A}}$-module $M$, we can construct a submodule ${\mathrm{tr}}_\S(M)$ of $M$ such that any map $S\to M$, with $S\in \S$, factors through the inclusion ${\mathrm{tr}}_\S(M)\to M$:
Let $\mathcal{S}$ be a class of right $\mathcal{A}$-modules and $M$ a right $\mathcal{A}$-module, then the sum of the submodules of $M$ of the form ${\mathrm{Im}}(f)$, for some morphism $f\colon S\to M$ in $\mod {\mathcal{A}}$, with $S\in\mathcal{S}$, is called the [**trace of $\mathcal{S}$ in $M$**]{} and denoted by ${\mathrm{tr}}_\mathcal{S}(M)$.
In the following lemma we see that the assignment $M\mapsto {\mathrm{tr}}_{\S}(M)$ is in fact functorial:
\[lemma\_traces\_are\_functorial\] Let ${\mathcal{A}}$ be a preadditive category and $\mathcal{S}$ a class of $\mathcal{A}$-modules. Then, the assignment $$M\mapsto{\mathrm{tr}}_\mathcal{S}(M)$$ defines a subfunctor of the identity $\mod {\mathcal{A}}\to\mod {\mathcal{A}}$.
It is enough to show that, given a morphism $\phi \colon M\to N$ in $\mod\mathcal{A}$, then $\phi
({\mathrm{tr}}_\mathcal{S}(M))\leq{\mathrm{tr}}_\mathcal{S}(M)$. But this is clear since, given any morphism $f\colon S\to M$, with $S\in\mathcal{S}$, we have $\phi ({\mathrm{Im}}(f))={\mathrm{Im}}(\phi\circ f)$.
Recall now that an $\mathcal{A}$-[**bimodule**]{} is a bifunctor $$X\colon\mathcal{A}^{\op}\times\mathcal{A}\to \Ab$$ which is additive in each component. The [**regular ${\mathcal{A}}$-bimodule**]{} is the bifunctor ${\mathcal{A}}(-,-)\colon \mathcal{A}^{\op}\times\mathcal{A}\to \Ab$. A [**sub-${\mathcal{A}}$-bimodule**]{} $Y$ of an ${\mathcal{A}}$-bimodule $X$ is just a sub-bifunctor, that is, $Y$ is a bifunctor $\mathcal{A}^{\op}\times\mathcal{A}\to \Ab$ such that
- $Y(a,b)\leq X(a,b)$ is a subgroup, for all $a,\, b\in\Ob({\mathcal{A}})$;
- $Y(\alpha,\beta)$ is the restriction of $X(\alpha,\beta)$, for each morphism $(\alpha,\beta)$ in $\mathcal{A}^{\op}\times\mathcal{A}$.
Given an ${\mathcal{A}}$-bimodule $X$ and $b\in\Ob(\mathcal{A})$, we can define two additive functors $$X_b\colon\mathcal{A}^{\op}\to\Ab\quad\text{and}\quad X^b\colon {\mathcal{A}}\to \Ab, \text{ where:}$$
- $X_b\colon a\mapsto X(a,b)$ and $X^b\colon a\mapsto X(b,a)$, for all $a\in \Ob({\mathcal{A}})$;
- $X_b(f):=(-\circ f)\colon X(a',b)\to X(a,b)$ and $X^b(f):=(f\circ -)\colon X(b,a)\to X(b,a')$, for all $f\colon a\to a'$ in ${\mathcal{A}}$.
Hence, $X_b\in\mod {\mathcal{A}}$ and $X^a\in\lmod {\mathcal{A}}:=\mod {{\mathcal{A}}^{\op}}$. One can check that the assignment $b\mapsto X_b$ (resp., $a\mapsto X^a$) defines an additive functor $\mathcal{A}\to\mod {\mathcal{A}}$ (resp., $\mathcal{A}^{\op}\to\lmod {{\mathcal{A}}}$). Notice that, applying this construction to the regular ${\mathcal{A}}$-bimodule we get ${\mathcal{A}}(-,-)_a=H_a$, for all $a\in\Ob({\mathcal{A}})$.
On the other hand, an additive functor $M\colon\mathcal{A}\to\mod {\mathcal{A}}$ (resp., $L\colon\mathcal{A}^{\op}\to\lmod {{\mathcal{A}}}$) defines an $\mathcal{A}$-bimodule $$X_M\colon\mathcal{A}^{\op}\times\mathcal{A}\to\Ab\qquad\text{(resp., $_LX \colon\mathcal{A}^{\op}\times\mathcal{A}\to\Ab$)},\text{ where}$$
- given $a,\, b\in \Ob({\mathcal{A}})$, $X_M(a,b):=(M(b))(a)$ and ${}_LX(a,b):=(L(a))(b)$;
- given $(f,g)\colon (a,b)\to (a',b')$ in $\mathcal{A}^{\op}\times\mathcal{A}$, we let $X_M(f,g):=(M(b'))(f)\circ M(g)_a$, while ${}_LX(f,g):=L(f)_b\circ (L(a'))(g)$.
This allows us to see $\mathcal{A}$-bimodules either as functors $\mathcal{A}\to\mod {\mathcal{A}}$ or $\mathcal{A}^{\op}\to\lmod {{\mathcal{A}}}$.
\[lemma\_traces\_are\_functorial\_bimodules\] Let ${\mathcal{A}}$ be a preadditive category, $X$ an $\mathcal{A}$-bimodule and $\mathcal{S}$ a class of right $\mathcal{A}$-modules. The assignment $b\mapsto {\mathrm{tr}}_\mathcal{S}(X_b)$ defines a subfunctor of the functor $b\mapsto X_b$ defined in the above discussion. The associated $\mathcal{A}$-bimodule, denoted by ${\mathrm{tr}}_\mathcal{S}(X)$, is then a sub-$\mathcal{A}$-bimodule of $X$.
If $\beta\colon b\rightarrow b'$ is a morphism in $\mathcal{A}$, then $X_\beta:=X(-,\beta)\colon X_b\to X_{b'}$ is a morphism in $\mod {\mathcal{A}}$. Since ${\mathrm{tr}}_\mathcal{S}\colon\mod {\mathcal{A}}\to \mod {\mathcal{A}}$ is a subfunctor of the identity, it follows that $X_\beta ({\mathrm{tr}}_\mathcal{S}(X_b))\leq{\mathrm{tr}}_\mathcal{S}(X_{b'})$, so we get an induced morphism ${\mathrm{tr}}_\mathcal{S}(X_\beta )\colon{\mathrm{tr}}_\mathcal{S}(X_b)\to {\mathrm{tr}}_\mathcal{S}(X_{b'})$ in $\mod {\mathcal{A}}$. We define ${\mathrm{tr}}_\mathcal{S}(X_\beta)$ to be the image of $\beta$ by the desired functor $\mathcal{A}\to \mod {\mathcal{A}}$. The rest of the proof is routine.
Given an ${\mathcal{A}}$-bimodule $X$ and a class of modules $\S$, the ${\mathcal{A}}$-bimodule ${\mathrm{tr}}_\S(X)$ is called the [**trace of $\mathcal{S}$ on the $\mathcal{A}$-bimodule $X$**]{}.
Locally coherent categories of modules {#subs_locally_coh}
--------------------------------------
Given a small preadditive category ${\mathcal{A}}$, a right ${\mathcal{A}}$-module $M$ is said to be [**finitely presented**]{} if the functor $\hom_{\mathcal{A}}(M,-)\colon\mod {\mathcal{A}}\to \Ab$ commutes with direct limits. As a consequence of Coro.\[application\_yoneda\], one can deduce (exactly as one does for categories of modules over a unitary ring) that the category $\mod {\mathcal{A}}$ is [**locally finitely presented**]{}, that is, any right ${\mathcal{A}}$-module can be written as a direct limit of finitely presented modules. In what follows we go one step further and characterize those categories ${\mathcal{A}}$ for which $\mod {\mathcal{A}}$ is also locally coherent, that is, we give a necessary and sufficient condition for $\fpmod {\mathcal{A}}$ (the category of finitely presented modules) to be closed under taking kernels in $\mod {\mathcal{A}}$.
Recall that, given a preadditive category ${\mathcal{C}}$ and a morphism $\phi\colon X\to Y$ in ${\mathcal{C}}$, a morphism $\psi\colon K\to X$ in ${\mathcal{C}}$ is said to be a [**pseudo-kernel**]{} of $\phi$ if, for any $Z\in \Ob({\mathcal{C}})$, the following sequence of Abelian groups is exact: $$\xymatrix@C=15pt{
{{\mathcal{C}}}(Z,K)\ar[rr]^{(Z,\psi)}&&{{\mathcal{C}}}(Z,X)\ar[rr]^{(Z,\phi)}&&{{\mathcal{C}}}(Z,Y).
}$$ [**Pseudo-cokernels**]{} are defined dually. Let us remark that any Abelian or triangulated category has pseudo-kernels and pseudo-cokernels. Pseudo-kernels have been introduced, under the name of “weak kernels”, by Freyd [@Freyd].
\[lem.locally coherent module categories\] Let ${\mathcal{A}}$ be a small preadditive category. The following are equivalent:
1. $\mod{{\mathcal{A}}}$ is a locally coherent Grothendieck category;
2. the subcategory $\proj({\mathcal{A}})$($\cong \widehat{\mathcal{A}}_{\oplus}$) of $\mod{{\mathcal{A}}}$ has pseudo-kernels;
3. the additive closure $\widehat{{\mathcal{A}}}$ has pseudo-kernels.
(1)$\Rightarrow$(2). Let $\phi\colon P\to Q$ be morphism in $\proj({\mathcal{A}})$. Being $\mod{{\mathcal{A}}}$ locally coherent, ${\mathrm{Ker}}(\phi)\in\fpmod{\mathcal{A}}$. Consider an epimorphism $\pi\colon \coprod_{i=1}^nH_{a_i}\to {\mathrm{Ker}}(\phi)$, with $n\in \N$ and $a_i\in \Ob({\mathcal{A}})$; it is not difficult to prove that the composition $\psi\colon \coprod_{i=1}^nH_{a_i}\to {\mathrm{Ker}}(\phi)\to P$ is a pseudo-kernel of $\phi$ in $\proj({\mathcal{A}})$.
(2)$\Rightarrow$(3). By Coro.\[application\_yoneda\] we can identify $\widehat{{\mathcal{A}}}$ with the full subcategory of $\proj({\mathcal{A}})$ of the objects of the form $\coprod_{i=1}^nH_{a_i}$, with $n\in \N$ and $a_i\in \Ob({\mathcal{A}})$. Consider then a morphism $\phi\colon \coprod_{i=1}^n H_{a_i}\to \coprod_{j=1}^m H_{b_j}$ and a pseudo-kernel $\psi\colon K\to \coprod_{i=1}^n H_{a_i}$ of $\phi$ in $\proj({\mathcal{A}})$. Take an epimorphism $\pi\colon \coprod_{l=1}^{n'}H_{c_l}\to K$, with $n'\in \N$ and $c_l\in \Ob({\mathcal{A}})$; it is not difficult to prove that the composition $\psi\colon \coprod_{l=1}^{n'}H_{c_l}\to K\to\coprod_{i=1}^nH_{a_i}$ is a pseudo-kernel in $\widehat{\mathcal{A}}$.
(3)$\Rightarrow$(1) can be proved as in [@Freyd Lem.1.4.5].
The center of a preadditive category {#subs_centre}
------------------------------------
Recall the following definition from [@Ga]: the [**center**]{} $Z({\mathcal{A}})$ of a preadditive category ${\mathcal{A}}$ is the ring of self-natural transformations of the identity functor $\id_{{\mathcal{A}}}$, that is, $$Z({\mathcal{A}}):=(\Padd({\mathcal{A}},{\mathcal{A}}))(\id_{\mathcal{A}},\id_{\mathcal{A}}),$$ where the above formula just means that, in the $2$-category $\Padd$, we consider the category of endomorphisms $\Padd({\mathcal{A}},{\mathcal{A}})$ and, in that category, we take the endomorphism ring of the object $\id_{\mathcal{A}}$.
It is an exercise on the definitions to verify that, given a unitary ring $R$, the commutative ring $Z(R)$ is isomorphic to the subring $\{r\in R:rs=sr,\,\forall s\in R\}$, which is usually called the center of $R$. On the other hand, given a small preadditive category ${\mathcal{A}}$, we can consider both $Z({\mathcal{A}})$ and $Z(\mod {\mathcal{A}})$. In the following proposition we show that both choices give the same ring:
\[prop.Z(A)-vs-Z(Mod-A)\] Given a small preadditive category ${\mathcal{A}}$, $Z({\mathcal{A}})\cong Z(\mod {\mathcal{A}})$.
Consider the following maps: $$\Phi\colon Z({\mathcal{A}})\to Z(\mod {\mathcal{A}})\quad\text{and}\quad\Psi\colon Z(\mod {\mathcal{A}})\to Z({\mathcal{A}}),$$ such that, given $\alpha\colon\id_{\mathcal{A}}\to \id_{\mathcal{A}}$ and $M\in \mod {\mathcal{A}}$, we define $\Phi(\alpha)_M\colon M\to M$ as follows: $\Phi(\alpha)_{M,a}:=M(\alpha_a)\colon M(a)\to M(a)$, for all $a\in \Ob({\mathcal{A}})$. On the other hand, given $\beta\colon \id_{\mod {\mathcal{A}}}\to \id_{\mod {\mathcal{A}}}$ and $a\in {\mathcal{A}}$, we let $\Psi(\beta)_a\colon a\to a$ be the unique morphism such that ${\mathcal{A}}(-,\Psi(\beta)_a)=\beta_{H_a}\colon H_a\to H_a$. It is now an easy exercise to verify that $\Phi$ and $\Psi$ are each other inverse.
As a consequence of the above proposition, one obtains that $Z({\mathcal{A}})\cong Z(\widehat{{\mathcal{A}}})\cong Z(\widehat{{\mathcal{A}}}_\oplus)$. To see this, one can use that ${\mathcal{A}}$, $\widehat {\mathcal{A}}$ and $\widehat{{\mathcal{A}}}_\oplus$ are Morita equivalent categories. More generally, this result shows that the center of a small preadditive category is invariant under Morita equivalence.
Ideals of preadditive categories {#subs_tors_and_id}
================================
The section is organized as follows: we start recalling the definition and some basic facts about (two-sided) ideals of a preadditive categories in Sec.\[Ideals and quotient categories\]; we then specialize the discussion to idempotent ideals in Sec.\[Idempotent ideals and traces of projectives\] and, between them, we give some equivalent characterizations of idempotent ideals that are traces of projective modules; finally, we show in Sec.\[subs\_directsummands\] that the direct sum decompositions of the regular bimodule are all induced by central idempotents of ${\mathcal{A}}$.
Ideals and quotient categories {#Ideals and quotient categories}
------------------------------
Let ${\mathcal{A}}$ be a small preadditive category, a ([**two-sided**]{}) [**ideal**]{} of ${\mathcal{A}}$ is a subfunctor $$\I(-,-)\leq {\mathcal{A}}(-,-)\colon {\mathcal{A}}^{\op}\times {\mathcal{A}}\to \Ab.$$ That is, given $a,\,a',\,b,\,b'\in \Ob({\mathcal{A}})$, and $f\in \I(a,b)$, $r\in {\mathcal{A}}(a',a)$ and $l\in {\mathcal{A}}(b,b')$, the composition $l\circ f\circ r$ belongs in the subgroup $\I(a',b')\leq {\mathcal{A}}(a',b')$. Notice that one can equivalently describe ideals of ${\mathcal{A}}$ as sub-${\mathcal{A}}$-bimodules of the regular bimodule ${\mathcal{A}}(-,-)$.
Given an ideal $\I$ of ${\mathcal{A}}$, we can form a new [**quotient category**]{} ${\mathcal{A}}/\I$ with objects $\Ob({\mathcal{A}}/\I)=\Ob({\mathcal{A}})$ and morphisms defined by $$({\mathcal{A}}/\I)(a,b):={\mathcal{A}}(a,b)/\I(a,b),$$ for all $a,\,b\in \Ob({\mathcal{A}})$, with identities and composition law induced by the ones of ${\mathcal{A}}$. Of course there is a natural functor $\pi_\I\colon {\mathcal{A}}\to {\mathcal{A}}/\I$, that induces a restriction of scalars: $$(\pi_\I)_*\colon \mod{{\mathcal{A}}/\I}\to \mod {\mathcal{A}}\qquad\text{such that}\qquad M\mapsto M\circ\pi_\I.$$ As for the case when ${\mathcal{A}}$ is a ring, one can give a very explicit description of the extension of scalars $(\pi_\I)^*$ (the left adjoint to $(\pi_\I)_*$). Indeed, given $M\in\mod {{\mathcal{A}}}$, we define a subfunctor $M\I\colon {\mathcal{A}}^{\op}\to \Ab$ of $M$ such that $$M\I(a):=\sum_{b\in\Ob({\mathcal{A}}),\,\alpha\in\I(a,b)}{\mathrm{Im}}(M(\alpha)).$$ Then, $(\pi_\I)^*(M)\cong M/M\I$. Let us remark that, by construction, $H_a\I=\I(-,a)$ for each $a\in\Ob({\mathcal{A}})$, so $$\label{extending_representables_eq}
(\pi_\I)^*(H_a)\cong H_a/H_a\I\cong H_a/\I(-,a).$$
Let ${\mathcal{A}}$ be a small preadditive category and let $\I$ be an ideal. The class of right ${\mathcal{A}}$-modules $M$ such that $M\I=0$ coincides with $$\Gen\{H_a/\I(-,a):a\in\Ob(\A)\},$$ that is, with those objects that can be written as a quotient of a coproduct of modules, each isomorphic to some of the $H_a/\I(-,a)$’s. Furthermore, the full subcategory $\Gen\{H_a/\I(-,a):a\in\Ob(\A)\}$ of $\mod {\mathcal{A}}$ is equivalent to $\mod {\mathcal{A}}/\I$.
Let us start by noticing that $\left(H_a/\I(-,a)\right)\I=0$ for all $a\in \Ob({\mathcal{A}})$, so that, if $M\in \Gen\{H_a/\I(-,a):a\in\Ob(\A)\}$, then $M\I=0$. On the other hand, suppose $M\I=0$ and consider an epimorphism $$\phi=(\phi_i)_{I}\colon\coprod_{i\in I}H_{a_i}\to M\to 0.$$ Since $M\I=0$, each $\phi_i$ factors as $\phi_i=\psi_i\pi_i$, where $\pi_i\colon H_{a_i}\to H_{a_i}/\I(-,a_i)$ is the obvious projection. Then the epimorphism $$\psi:=(\psi_i)_{I}\colon\coprod_{i\in I}H_{a_i}/\I(-,a_i)\to M$$ shows that $M\in \Gen\{H_a/\I(-,a):a\in\Ob(\A)\}$.
Given a unitary ring, one can always construct the two-sided ideal generated by a given family of elements. The analogous construction in the more general setting of preadditive categories is given in the following definition:
\[def.principal ideal\] Let $\alpha \colon x\to y$ be a morphism in ${\mathcal{A}}$ and let $\mathcal{M}$ be a set of morphisms. Then define:
- the ([**two-sided**]{}) [**ideal of $\mathcal{A}$ generated by $\alpha$**]{}, denoted by $\mathcal{A}\alpha\mathcal{A}\colon {\mathcal{A}}^{\op}\times {\mathcal{A}}\to \Ab$, is the subfunctor of ${\mathcal{A}}(-,-)$ such that $(\mathcal{A}\alpha\mathcal{A})(a,b)$ is the subgroup of $\mathcal{A}(a,b)$ generated by compositions $\delta\circ\alpha\circ\gamma$, where $\gamma\in\mathcal{A}(a,x)$ and $\delta\in\mathcal{A}(y,b)$;
- the ([**two-sided**]{}) [**ideal of $\mathcal{A}$ generated by $\mathcal{M}$**]{}, denoted by $\mathcal{A}\mathcal{M}\mathcal{A}\colon {\mathcal{A}}^{\op}\times {\mathcal{A}}\to \Ab$, is the sum $$\mathcal{A}\mathcal{M}\mathcal{A}:=\sum_{\mu\in\mathcal{M}}\mathcal{A}\mu\mathcal{A}.$$ That is, $(\mathcal{A}\mathcal{M}\mathcal{A})(a,b)=\sum_{\mu\in\mathcal{M}}(\mathcal{A}\mu\mathcal{A})(a,b)$, where the sum in the second member of the last equality is the sum of subgroups of the Abelian group $\mathcal{A}(a,b)$.
For a set of morphisms $\M$ in ${\mathcal{A}}$, a general element in $(\mathcal{A}\mathcal{M}\mathcal{A})(a,b)$ is a finite sum: $$\label{general_form_of_ideal_generated}
\xymatrix@R=-5pt@C=12pt{
&&&&x_1\ar[rrrr]^{m_1}&&&&y_1\ar[ddrrrr]^{\delta_1}\\
&&&&&&+\\
a\ar[uurrrr]^{\gamma_1}\ar[ddrrrr]_{\gamma_k}&&&&&&\vdots&&&&&&b\\
&&&&&&+\\
&&&&x_k\ar[rrrr]_{m_k}&&&&y_k\ar[uurrrr]_{\delta_k}
}$$ where $m_1,\dots,m_k\in \M$.
In the following lemma we study the invariance of the set of two-sided ideals in ${\mathcal{A}}$ under Morita equivalence:
Let ${\mathcal{A}}$ and $\B$ be two Morita equivalent small preadditive categories. Then,
1. there is a bijection between the ideals of ${\mathcal{A}}$ and those of its Cauchy completion $\widehat {\mathcal{A}}_{\oplus}$;
2. there is a bijection between the sets of ideals of ${\mathcal{A}}$ and $\B$.
\(1) Consider the canonical inclusion ${\mathcal{A}}\to \widehat {\mathcal{A}}_{\oplus}$ and identify ${\mathcal{A}}$ as a full subcategory of $\widehat{\mathcal{A}}_\oplus$. Then there is a map $$F\colon\{\text{Ideals of ${\mathcal{A}}$}\}\to \{\text{Ideals of $\widehat{\mathcal{A}}_\oplus$}\},\qquad I\mapsto \widehat{\mathcal{A}}_\oplus I\widehat{\mathcal{A}}_\oplus,$$ where $\widehat{\mathcal{A}}_\oplus I\widehat{\mathcal{A}}_\oplus$ denotes the ideal of $\widehat{\mathcal{A}}_{\oplus}$ generated by the union $\bigcup_{a,b\in\Ob({\mathcal{A}})}I(a,b)$. On the other hand, one can construct a map in the opposite direction as follows: $$G\colon\{\text{Ideals of $\widehat{\mathcal{A}}_\oplus$}\}\to \{\text{Ideals of ${\mathcal{A}}$}\},\qquad J\mapsto J\restriction_{{\mathcal{A}}^{\op}\times{\mathcal{A}}}.$$ It is now routine to check that these maps are each other inverse.
\(2) This is an application of part (1), just using the fact that ${\mathcal{A}}$ and $\B$ are Morita equivalent if and only if $\widehat {\mathcal{A}}_{\oplus}$ is equivalent to $\widehat \B_{\oplus}$.
Idempotent ideals and traces of projectives {#Idempotent ideals and traces of projectives}
-------------------------------------------
Given two ideals $\I$ and $\mathcal J$ of ${\mathcal{A}}$, we define a new ideal $\I\cdot \mathcal J$ as follows: $$(\I\cdot \mathcal J)(a,b):=\left\{\sum_{i=1}^n\phi_i\circ \psi_i: \phi_i\in \I(c_i,b),\, \psi_i\in \mathcal J(a,c_i)\right\}.$$ An ideal $\I$ is said to be [**idempotent**]{} if $\I\cdot\I=\I$.
Let ${\mathcal{A}}$ be a small preadditive category and let $\I$ be an idempotent ideal. The class of right ${\mathcal{A}}$-modules $M$ such that $M\I=M$ coincides with $$\Gen\{\I(-,a):a\in\Ob(\A)\}.$$ That is, with those objects that can be written as a quotient of a coproduct of modules, each isomorphic to some of the $\I(-,a)$’s.
If $M\in \Gen\{\I(-,a):a\in\Ob(\A)\}$, then $M\I=M$. On the other hand, suppose $M\I=M$, consider an epimorphism $\coprod_{i\in I}H_{a_i}\to M$ and let $K:={\mathrm{Ker}}(\phi)$. Then $$M=M\I\cong \frac{\coprod_{i\in I}H_{a_i}}{K}\I\cong \frac{\coprod_{i\in I}H_{a_i}\I}{K\cap \coprod_{i\in I}H_{a_i}\I}\cong \frac{\coprod_{i\in I}\I(-,{a_i})}{K\cap \coprod_{i\in I}\I(-,{a_i})},$$ and this last module clearly belongs to $\Gen\{\I(-,a):a\in\Ob(\A)\}$.
Notice that Lem.\[lemma\_traces\_are\_functorial\_bimodules\] applies in particular to the regular bimodule $\mathcal{A}(-,-)$. Hence, given a class of right ${\mathcal{A}}$-modules $\S$, the trace of $\mathcal{S}$ in the regular bimodule, called the [**trace of $\mathcal{S}$ in $\mathcal{A}$**]{}, and denoted by ${\mathrm{tr}}_\mathcal{S}(\mathcal{A})$, is a two-sided ideal of $\mathcal{A}$. As for modules over a unital ring, the situation when $\mathcal{S}$ consists of projective $\mathcal{A}$-modules deserves a special attention:
\[lem.trace-of-projectives is idempotent\] Let ${\mathcal{A}}$ be a small preadditive category and let $\S$ be a class of right $\mathcal{A}$-modules. Then,
1. there is a (small) subset $\S'$ of $\S$ such that ${\mathrm{tr}}_\mathcal{S}(\mathcal{A})={\mathrm{tr}}_{T}(\mathcal{A})$, where $T:=\coprod_{\S'}S$;
2. for each $a,\, b\in\Ob({\mathcal{A}})$, the group ${\mathrm{tr}}_T(\mathcal{A})(a,b)$ consists of the morphisms $\gamma\in{\mathcal{A}}(a,b)$ such that the map $H_\gamma\colon H_a\to H_b$ factors through a finite coproduct of copies of $T$;
3. if each $S$ in $\S$ is projective, then the ideal ${\mathrm{tr}}_\mathcal{S}(\mathcal{A})$ is idempotent.
Summarizing, if $\S$ is a class of projectives, then ${\mathrm{tr}}_\mathcal{S}(\mathcal{A})$ is an idempotent ideal such that, for each $a,\,b\in\Ob({\mathcal{A}})$, ${\mathrm{tr}}_\mathcal{S}(\mathcal{A})(a,b)$ consists of those $\gamma\in{\mathcal{A}}(a,b)$ such that the map $H_\gamma$ factors through a finite coproduct of objects in $\S$.
\(1) Given $b\in\Ob(\mathcal{A})$, the submodules $L$ of $H_b:=\mathcal{A}(-,b)$ which are of the form $L={\mathrm{Im}}(f)$, for some morphism $f\colon S\to H_b$, with $S\in\mathcal{S}$, form a set. Therefore we can select a set $\mathcal{S}_b\subseteq\mathcal{S}$ such that ${\mathrm{tr}}_\mathcal{S}(H_b)={\mathrm{tr}}_{\mathcal{S}_b}(H_b)$. Letting $\mathcal{S}':=\bigcup_{b\in\Ob(\mathcal{A})}\mathcal{S}_b$, it is clear that ${\mathrm{tr}}_{\mathcal{S}'}(H_b)={\mathrm{tr}}_\mathcal{S}(H_b)$, for all $b\in\Ob(\mathcal{A})$. Hence, if we put $T:=\coprod_{S\in\mathcal{S}'}S$, then ${\mathrm{tr}}_\mathcal{S}(H_b)={\mathrm{tr}}_\mathcal{S'}(H_b)={\mathrm{tr}}_T(H_b)$, for each $b\in\Ob(\mathcal{A})$.
\(2) Let us denote by $\I_T(-,-)\leq {\mathcal{A}}(-,-)$ the ideal of morphism $\gamma$ such that $H_\gamma$ factors through a finite coproduct of copies of $T$. We then have to verify that, given $a,\,b\in\Ob({\mathcal{A}})$, $\I_T(a,b)= {\mathrm{tr}}_T({\mathcal{A}})(a,b)={\mathrm{tr}}_T(H_b)(a)$. Indeed, consider a morphism $\gamma\colon a\to b$ in ${\mathcal{A}}$, and suppose that $H_\gamma=\sum_{i=1}^kg_if_i$, where $f_i$ and $g_i$ are as in the following picture: $$\label{picture}
\xymatrix@R=-4pt@C=18pt{
&&&&T\ar[ddrrrr]^{g_1}\\
&&&&+\\
\sum_{i=1}^kg_if_i\colon H_a\ar[uurrrr]^(.6){f_1}\ar[ddrrrr]_(.6){f_k}&&&&\vdots&&&&H_b\\
&&&&+\\
&&&&T\ar[uurrrr]_{g_k}
}$$ In particular, by the Yoneda Lemma, we get $\gamma=(H_\gamma)_a(\id_a)=\sum_{i=1}^k(g_i)_a(f_i)_a(\id_a)\leq \sum_{i=1}^k{\mathrm{Im}}(g_i)_a\leq {\mathrm{tr}}_{T}({\mathcal{A}})(a,b)$.\
On the other hand, given an element $\delta\in {\mathrm{tr}}_{T}({\mathcal{A}})(a,b)$, by definition of trace, there exist $g_1,\dots,g_k\colon T\to H_b$, such that $\delta\in \sum_{i=1}^k{\mathrm{Im}}(g_i)_a$. This means that there exist $x_1,\dots,x_k\in T_a$, such that $\delta= \sum_{i=1}^k(g_i)_a(x_i)$. Again, by the Yoneda Lemma, there exist unique morphisms $f_1,\dots,f_k\colon H_a\to T$ such that $(f_i)_a(\id_a)=x_i$. Hence, we are again in the situation of , so that $H_\delta$ factors through $T^k$, since $H_\delta=\sum_{i=1}^kg_if_i$.
\(3) By part (1), it is clear that we can choose a projective module $T$ such that ${\mathrm{tr}}_\S(-,-)={\mathrm{tr}}_T(-,-)$. Consider an epimorphism $p\colon \coprod_{I}H_{x_i}\to T$, with $x_i\in \Ob({\mathcal{A}})$ for all $i\in I$ and, using the projectivity of $T$, choose a section $u\colon T\to \coprod_{I}H_{x_i}$, that is $p\circ u=\id_T$. Now, given $\alpha\in\mathcal{A}(a,b)$ such that $H_\alpha$ factors through $T$, that is, we have $f\colon H_a\to T$ and $g\colon T\to H_b$ such that $H_\alpha=g\circ f$, there exists a finite subset $F\subseteq I$ such that the map $u\circ f\colon H_a\to \coprod_{I}H_{x_i}$ factors through the inclusion $\iota_F\colon \coprod_{F}H_{x_i}\to \coprod_{I}H_{x_i}$; let also $\pi_F\colon \coprod_{I}H_{x_i}\to \coprod_{F}H_{x_i}$ be the corresponding projection. We obtain a commutative diagram as follows: $$\xymatrix@R=15pt@C=30pt{
H_a\ar@/_-20pt/[rrrrrr]|{H_\alpha}\ar@/_+5pt/[dr]_f\ar[rr]&&\coprod_{I}H_{x_i}\ar[r]^{\pi_F}& \coprod_{F}H_{x_i}\ar[r]^{\iota_F}&\coprod_{I}H_{x_i}\ar@/_+5pt/[dr]_p\ar[rr]&&H_b\\
&T\ar@/_+5pt/[ur]_u&&&&T\ar@/_+5pt/[ur]_g
}$$ Now, let also $\pi_k\colon \coprod_{I}H_{x_i}\to H_{x_k}$ and $\iota_k\colon H_{x_k}\to \coprod_{I}H_{x_i}$ be the obvious projection and inclusion, for $k\in F$. Using the Yoneda Lemma, the above discussion shows that $$\begin{aligned}
\alpha&=(H_\alpha)_a(\id_a)\\
&=\sum_{F}((g\circ p\circ \iota_k)\circ(\pi_k\circ u\circ f))_a(\id_a)\\
&=\sum_{F}(g\circ p\circ \iota_k)_{x_k}(\id_{x_k})\circ(\pi_k\circ u\circ f)_a(\id_a).\end{aligned}$$ Now, both $(\pi_k\circ u\circ f)_a(\id_a) \in {\mathrm{tr}}_T({\mathcal{A}})(a,x_k)$ and $(g\circ p\circ \iota_k)_{x_k}(\id_{x_k})\in {\mathrm{tr}}_T({\mathcal{A}})(x_k,b)$, showing that $\alpha\in {\mathrm{tr}}_{T}({\mathcal{A}})^2(a,b)$, as desired.
Given a set $\M$ of morphisms in ${\mathcal{A}}$ we have described in the general form of an element in the ideal ${\mathcal{A}}\M{\mathcal{A}}$. Of particular interest for us is the case when $\mathcal{M}=\mathcal{E}$ is a set of idempotent endomorphisms of objects of $\mathcal{A}$. As in the case of modules over unital rings, the reason is the following proposition:
\[prop.ideals generated by idempotents\] Let $\mathcal{A}$ be a preadditive category and let $\mathcal{E}$ be a set of idempotent endomorphisms of objects of $\mathcal{A}$. Then, letting $\mathcal{P}_\mathcal{E}:=\{\epsilon\mathcal{A}:\epsilon\in\mathcal{E}\}$ (where each $\epsilon\mathcal{A}$ is a finitely generated projective module, see Lem.\[description\_fpp\_lemma\]), $${\mathrm{tr}}_{\P_\E}({\mathcal{A}})={\mathcal{A}}\E{\mathcal{A}}.$$
Let us start by noting that ${\mathrm{tr}}_{\P_\E}({\mathcal{A}})=\sum_{\epsilon\in \E}{\mathrm{tr}}_{\epsilon{\mathcal{A}}}({\mathcal{A}})$ and ${\mathcal{A}}\E{\mathcal{A}}=\sum_{\epsilon\in \E}{\mathcal{A}}\epsilon{\mathcal{A}}$. Hence, our statement will follow if we prove that, for a given idempotent endomorphism $\epsilon\colon x\to x$ in ${\mathcal{A}}$, ${\mathrm{tr}}_{\epsilon {\mathcal{A}}}({\mathcal{A}})={\mathcal{A}}\epsilon{\mathcal{A}}$. For the inclusion “$\geq$”, fix $a,\, b\in\Ob(\mathcal{A})$ and take any composition $$\xymatrix@C=12pt{
a\ar[rr]^{\alpha}&&x\ar[rr]^{\epsilon}&&x\ar[rr]^{\beta}&&b
}$$ of morphisms in $\mathcal{A}$. If we denote by $H'_\beta$ the restriction of $H_\beta \colon H_x\to H_b$ to the submodule $\epsilon\mathcal{A}\leq H_x$, then $(H'_\beta )_a\colon(\epsilon\mathcal{A})(a)\to H_b(a)$ maps $\epsilon\circ\alpha$ onto $\beta\circ\epsilon\circ\alpha$, so that $\beta\circ\epsilon\circ\alpha\in{\mathrm{Im}}(H'_\beta)(a)$. Since each element of $(\mathcal{A}\epsilon\mathcal{A})(a,b)$ is a sum of compositions $\beta\circ\epsilon\circ\alpha$ as indicated, we conclude that $(\mathcal{A}\epsilon\mathcal{A})(a,b)\subseteq{\mathrm{tr}}_{\epsilon\mathcal{A}}(\mathcal{A})(a,b)$, for all $a,b\in\Ob(\mathcal{A})$.
On the other hand, for the inclusion “$\leq$”, recall that ${\mathrm{tr}}_{\epsilon\mathcal{A}}(\mathcal{A})(-,b)={\mathrm{tr}}_{\epsilon\mathcal{A}}(H_b)$ so it is enough to prove that given a morphism $f\colon \epsilon\mathcal{A}\to H_b$, then ${\mathrm{Im}}(f)\leq (\mathcal{A}\epsilon\mathcal{A})(-,b)$. Now $\epsilon\mathcal{A}$ is the image of the idempotent endomorphism $H_\epsilon\colon H_x\to H_x$, so that ${\mathrm{Im}}(f)={\mathrm{Im}}(g\circ H_\epsilon)$, for some morphism $g:H_x\to H_b$. By the Yoneda Lemma, we know that $g=H_\gamma$, for some $\gamma\in\mathcal{A}(x,b)$, and ${\mathrm{Im}}(f)={\mathrm{Im}}(H_\gamma\circ H_\epsilon)={\mathrm{Im}}(H_{\gamma\circ\epsilon})$. Then, for any $a\in\Ob(\mathcal{A})$, $${\mathrm{Im}}(f)(a)=\{\gamma\circ\epsilon\circ\beta:\beta\in\mathcal{A}(a,x)\} \leq\mathcal{A}(a,b)$$ Hence, ${\mathrm{Im}}(f)(a)\leq (\mathcal{A}\epsilon\mathcal{A})(a,b)$, and so ${\mathrm{Im}}(f)\leq (\mathcal{A}\epsilon\mathcal{A})(-,b)$, as desired.
A particular case of the above proposition is when $\E$ is a set of identities of objects in ${\mathcal{A}}$, that is, when there exists a set $\X\subseteq \Ob({\mathcal{A}})$ and $\E=\{\id_x:x\in\X\}$; in this case we let ${\mathcal{A}}\X{\mathcal{A}}:={\mathcal{A}}\E{\mathcal{A}}$. A general element in ${\mathcal{A}}\X{\mathcal{A}}$ is of the form: $$\xymatrix@R=-2.5pt@C=18pt{
&&&&x_1\ar[ddrrrr]^{\delta_1}\\
&&&&+\\
a\ar[uurrrr]^{\gamma_1}\ar[ddrrrr]_{\gamma_k}&&&&\vdots&&&&b\\
&&&&+\\
&&&&x_k\ar[uurrrr]_{\delta_k}
}$$ where $x_1,\dots,x_k\in \X$.
\[prop\_char\_ideals\_trace\_fgp\] Let ${\mathcal{A}}$ be a small preadditive category and consider the following conditions for an ideal $\I$ of ${\mathcal{A}}$:
1. $\I={\mathrm{tr}}_{\P}({\mathcal{A}})$ is the trace of a class of finitely generated projective modules $\P$;
2. $\I={\mathcal{A}}\E{\mathcal{A}}$ is generated by a set $\E$ of idempotent endomorphisms of objects of ${\mathcal{A}}$;
3. $\I={\mathcal{A}}\X {\mathcal{A}}$, where $\X$ is a subset of $\Ob({\mathcal{A}})$.
Then one always has the implications (3)$\Rightarrow$(2)$\Rightarrow$(1), while the implication (1)$\Rightarrow$(2) holds when ${\mathcal{A}}$ is additive, and the implication (2)$\Rightarrow$(3) holds when ${\mathcal{A}}$ is idempotent complete. In particular, the three conditions are all equivalent whenever ${\mathcal{A}}$ is a Cauchy complete additive category.
The implication “(3)$\Rightarrow$(2)” is trivial, while “(2)$\Rightarrow$(1)" follows by Prop.\[prop.ideals generated by idempotents\].\
Suppose now that ${\mathcal{A}}$ is additive and that $\I={\mathrm{tr}}_{\P}({\mathcal{A}})$ is the trace of a class of finitely presented projective modules $\P$. By Lem.\[description\_fpp\_lemma\], for each $P\in \P$ there is $x_P\in \Ob({\mathcal{A}})$ and an idempotent endomorphism $\epsilon_P\colon x_P\to x_P$ such that $P\cong \epsilon_P{\mathcal{A}}$. Letting $\E:=\{\epsilon_P:P\in \P\}$, we obtain by Prop.\[prop.ideals generated by idempotents\] that $\I={\mathrm{tr}}_{\P}({\mathcal{A}})={\mathcal{A}}\E{\mathcal{A}}$, so the implication “(1)$\Rightarrow$(2)" follows when ${\mathcal{A}}$ is additive.\
Finally, suppose that ${\mathcal{A}}$ is idempotent complete and that $\I={\mathcal{A}}\E{\mathcal{A}}$ for a set $\E$ of idempotent endomorphisms of objects of ${\mathcal{A}}$. Then, given $\epsilon\colon x_\epsilon\to x_\epsilon$ in $\E$, there are $y_\epsilon\in\Ob({\mathcal{A}})$ and maps $\iota_\epsilon\colon y_\epsilon\to x_\epsilon$ and $\pi_\epsilon\colon x_\epsilon\to y_\epsilon$ such that $\epsilon=\iota_\epsilon\circ\pi_\epsilon$ and $\id_{y_\epsilon}=\pi_\epsilon\circ\iota_\epsilon$. It is not difficult to verify that ${\mathcal{A}}\E{\mathcal{A}}={\mathcal{A}}\{{y_\epsilon}:\epsilon\in\E\}{\mathcal{A}}$, so that also the implication “(2)$\Rightarrow$(3)” holds when ${\mathcal{A}}$ is idempotent complete.
The above proposition, together with Rem.\[rem\_morita\_cauchy\], allows us to find a characterization for the idempotent ideals which are traces of finitely generated projective modules:
\[coro\_ideals\_vs\_cauchy\_with\_generator\] Let ${\mathcal{A}}$ be a small preadditive category, then there is a bijection: [ $$\xymatrix@R=0pt@C=20pt{
{\left\{\begin{matrix}\text{Full subcategories of $\proj({\mathcal{A}})(\cong\widehat A_{\oplus})$}\\ \text{closed under coproducts and summands}\end{matrix}\right\}}\ar@{<->}[rr]^(.55){1:1}&&{\left\{\begin{matrix}\text{Idempotent ideals of ${\mathcal{A}}$ which}\\ \text{are traces of f.g.\ projectives}\end{matrix}\right\}}\\
\X\ar@{|->}[rr]&&{\mathrm{tr}}_\X({\mathcal{A}})
}$$ ]{} Furthermore, the above bijection restricts to the following one: [ $$\xymatrix@R=0pt@C=20pt{
{\left\{\begin{matrix}\text{Full subcategories of $\widehat A_{\oplus}$ closed}\\ \text{under coproducts and summands}\\
\text{and with a $\oplus$-generator}\end{matrix}\right\}}\ar@{<->}[rr]^(.52){1:1}&&{\left\{\begin{matrix}\text{Idempotent ideals of ${\mathcal{A}}$ which are}\\ \text{traces of a single f.g.\ projective}\end{matrix}\right\}}
}$$ ]{}
Given $\X,\, \Y\subseteq \proj({\mathcal{A}})$, recall that ${\mathrm{add}}(\X)$ is the full subcategory spanned by the summands of finite coproducts of objects in $\X$. Now, if ${\mathrm{add}}(\X)={\mathrm{add}}(\Y)$, it follows by the description of the trace given in Lem.\[lem.trace-of-projectives is idempotent\] that ${\mathrm{tr}}_\X({\mathcal{A}})={\mathrm{tr}}_\Y({\mathcal{A}})$. On the other hand, let us show that ${\mathrm{tr}}_\X({\mathcal{A}})={\mathrm{tr}}_\Y({\mathcal{A}})$ implies ${\mathrm{add}}(\X)={\mathrm{add}}(\Y)$. Indeed, suppose ${\mathrm{tr}}_\X({\mathcal{A}})={\mathrm{tr}}_\Y({\mathcal{A}})$, and let $P\in\mathcal{X}$. Since $P$ is finitely generated projective, there is a finite family $\{b_1,\dots,b_k\}$ in $\Ob(\mathcal{A})$ together with morphisms $u\colon P \rightarrow \coprod_{i=1}^{k} H_{b_i}$ and $p\colon \coprod_{i=1}^{k} H_{b_i} \rightarrow P$ such that $p \circ u=\id_{P}$. This implies that $${\mathrm{tr}}_{\mathcal{X}}(p)\colon {\mathrm{tr}}_{\mathcal{X}}\left(\coprod_{i=1}^{k} H_{b_i}\right) \rightarrow {\mathrm{tr}}_{\mathcal{X}}(P)=P$$ is a split epimorphism. But, ${\mathrm{tr}}_{\mathcal{X}}\left(\coprod_{i=1}^{k} H_{b_i}\right)\cong\coprod_{i=1}^{k} {\mathrm{tr}}_{\mathcal{X}}(H_{b_i})$ and by assumption ${\mathrm{tr}}_{\mathcal{X}}( H_{b_i})={\mathrm{tr}}_{\mathcal{Y}}( H_{b_i})$ for all $i=1,\dots,k$. Hence, $P \in \Gen(\mathcal{Y})$. Using once again the fact that $P$ is finitely generated projective, we deduce that $P \in {\mathrm{add}}(\mathcal{Y})$. This show that $\mathcal{X}\subseteq {\mathrm{add}}(\mathcal{Y})$, and so also ${\mathrm{add}}(\mathcal{X}) \subseteq {\mathrm{add}}(\mathcal{Y})$. By simmetry, we can then conclude that ${\mathrm{add}}(\X)={\mathrm{add}}(\Y)$. Consider now the correspondence in the statement: the assignment $\X\mapsto {\mathrm{tr}}_\X({\mathcal{A}})$ is surjective because, for any set of finitely generated projectives $\X'$, we have verified that ${\mathrm{tr}}_{\X'}({\mathcal{A}})={\mathrm{tr}}_{{\mathrm{add}}(\X')}({\mathcal{A}})$, so it is enough to consider traces of families which are closed under finite coproducts and summands. Furthermore, the assignment is also injective by the first part of the proof.
Direct decompositions and central idempotents {#subs_directsummands}
---------------------------------------------
An ideal $\I$ of a small preadditive category ${\mathcal{A}}$ is said to be a [**direct summand**]{} of ${\mathcal{A}}$ when there is another ideal $\I'$ of ${\mathcal{A}}$ such that, as bi-functors ${\mathcal{A}}^{\op}\times{\mathcal{A}}\to\Ab$, we have a decomposition ${\mathcal{A}}(-,-)\cong\I(-,-)\oplus\I'(-,-)$. When there is no risk of confusion, we just write ${\mathcal{A}}=\I\oplus \I'$ and we call this a [**decomposition of ${\mathcal{A}}$ as a direct sum of ideals**]{}.
\[lemma\_direct\_summands\] Let ${\mathcal{A}}$ be a preadditive category that admits two direct sum decompositions, ${\mathcal{A}}=\I\oplus\I'$ and ${\mathcal{A}}=\I\oplus\I''$, as a direct sum of ideals. The following assertions hold:
1. $\I'=\I''$;
2. $ \I\cdot\I'=0=\I'\cdot\I$;
3. $\I$ is an idempotent ideal;
4. we have a decomposition $M=M\I\oplus M\I''$ in $\mod{\mathcal{A}}$, for all ${\mathcal{A}}$-modules $M$.
\(1) Let $a,b\in\Ob({\mathcal{A}})$ and $\alpha\in\I'(a,b)$. Using the decomposition ${\mathcal{A}}(b,b)=\I(b,b)\oplus\I''(b,b)$, we have that $1_b=e_b+e''_b$, where $e_b\in\I(b,b)$ and $e''_b\in\I''(b,b)$. We then get that $\alpha =1_b\circ\alpha =e_b\circ\alpha +e''_b\circ\alpha$. Furthermore, $e_b\circ\alpha\in\I(a,b)\cap\I'(a,b)=0$ since $\I$ and $\I'$ are both ideals. It then follows that $\alpha =e''_b\circ\alpha\in\I''(a,b)$ since $\I''$ is an ideal of ${\mathcal{A}}$. Therefore, $\I'\subseteq\I''$ and, by symmetry, the converse inclusion also holds.
\(2) This is trivial since $ \I\cdot\I'\subseteq\I\cap\I'\supseteq\I'\cdot\I$, and $\I\cap\I'=0$.
\(3) Taking $\alpha\in\I(a,b)$ and arguing as in the proof of assertion (1), we get a decomposition $\alpha =e_b\circ\alpha +e''_b\circ\alpha$. But the second summand is in $\I\circ\I''=0$. Then $\alpha\in\I^2(a,b)$ since $e_b\in\I(b,b)$.
\(4) If we put $N:=M\I\cap M\I'$, then $N\I=0=N\I'$ using assertion (2). On the other hand, viewing ${\mathcal{A}}$ as an ideal of itself in the obvious way, we have that $M=M{\mathcal{A}}=M(\I+\I'')\subseteq M\I+M\I''$. It then follows that $M=M\I\oplus M\I''$.
As in the case of unital rings, we have the following result.
\[prop.bijection-central-idempotents\] Let ${\mathcal{A}}$ be a small preadditive category and let $$\S_1:=\{\text{idempotent elements in $Z({\mathcal{A}})$}\}\quad\text{and}\quad
\S_2:=\{\text{direct summands of ${\mathcal{A}}$}\}.$$ The assignment $\epsilon\mapsto\I_\epsilon$, where $\I_\epsilon$ is the ideal of ${\mathcal{A}}$ defined as $$\I_\epsilon (a,b):=\{u\in{\mathcal{A}}(a,b):\epsilon_b\circ u=u\text{ (equivalently, }u\circ\epsilon_a=u\text{)}\},$$ for all $a,b\in\Ob({\mathcal{A}})$, defines a bijection $\Phi\colon \S_1\overset{\cong }{\to }\S_2$.
Let us remark that, given an idempotent $\epsilon\in Z(\mathcal{A})$, then $\mathcal{I}_\epsilon =\mathcal{A}\mathcal{E}_\epsilon\mathcal{A}$, where$\mathcal{E}_\epsilon:=\{\epsilon_a : a\in\Ob(\mathcal{A})\}$.
Let $u\in\I_\epsilon (a,b)$ and let us consider morphisms $r\colon a'\to a$ and $l\colon b\to b'$ in ${\mathcal{A}}$. Then, using that $\epsilon\colon \id_{\mathcal{A}}\to \id_{\mathcal{A}}$ is a natural transformation and that $\epsilon_b\circ u=u$, we get the following equalities $$\epsilon_{b'}\circ l\circ u\circ r=l\circ\epsilon_b\circ u\circ r= l\circ u\circ r,$$ proving that $l\circ u\circ r\in\I_\epsilon (a',b')$, so that $\I_\epsilon$ is an ideal of ${\mathcal{A}}$. Note that we have not used yet the idempotency of $\epsilon$. That is, for each $\epsilon\in Z({\mathcal{A}})$, we have a well-defined ideal $\I_\epsilon$ of ${\mathcal{A}}$.
We now consider, for all $a,\, b\in\Ob({\mathcal{A}})$, the subgroup $\I'_\epsilon (a,b)$ of ${\mathcal{A}}(a,b)$ consisting of the morphisms $v\colon a\rightarrow b$ such that $\epsilon_b\circ v=0$ (or, equivalently, $v\circ\epsilon_a=0$). Using again that $\epsilon$ is a natural transformation, one readily sees that the $\I'_\epsilon (a,b)$’s define an ideal $\I'_\epsilon$ of ${\mathcal{A}}$ such that $\I_\epsilon(a,b)\cap\I'_\epsilon (a,b)=0$, for all $a,\, b\in\Ob({\mathcal{A}})$. Moreover, for each $w\in{\mathcal{A}}(a,b)$, we have a decomposition $w=\epsilon_b\circ w +(w-\epsilon_b\circ w)$, where $\epsilon_b\circ w\in\I_\epsilon (a,b)$ and $w-\epsilon_b\circ w\in\I'_\epsilon (a,b)$ due to the idempotency of $\epsilon$. Therefore we have a decomposition as a direct sum of ideals ${\mathcal{A}}=\I_\epsilon\oplus\I'_\epsilon$, so that the assignment $\epsilon\mapsto\I_\epsilon$ gives a well-defined map $\Phi \colon\S_1\to\S_2$.
To check the injectivity of $\Phi$, consider two idempotents $\epsilon ,\, \epsilon'\in Z({\mathcal{A}})$ and suppose that $\I_\epsilon =\I_{\epsilon'}$, i.e. $\Phi(\epsilon )=\Phi (\epsilon')$. We then have that $\epsilon_a,\,\epsilon'_a\in\I_\epsilon (a,a)=\I_{\epsilon'}(a,a)$, for all $a\in\Ob({\mathcal{A}})$. In particular, $\epsilon'_a\circ\epsilon_a=\epsilon_a$ and, using the naturality of $\epsilon$, we also have that $\epsilon'_a\circ\epsilon_a=\epsilon_a\circ\epsilon'_a=\epsilon'_a$ since $\epsilon'_a\in\I_\epsilon (a,a)$. Therefore $\epsilon_a=\epsilon'_a$, for all $a\in\Ob({\mathcal{A}})$, which implies that $\epsilon =\epsilon'$.
For the surjectivity of $\Phi$, let us take a direct summand $\I$ of ${\mathcal{A}}$ and fix a decomposition ${\mathcal{A}}=\I\oplus\I'$, which is unique according to Lemma \[lemma\_direct\_summands\]. We then have a decomposition ${\mathcal{A}}(a,a)=\I(a,a)\oplus\I'(a,a)$ in $\Ab$, for all $a\in\Ob({\mathcal{A}})$. This gives a decomposition $1_a=\epsilon_a+\epsilon'_a$, with $\epsilon_a\in\I(a,a)$ and $\epsilon'_a\in\I'(a,a)$, for all $a\in\Ob({\mathcal{A}})$. Note that, for each $u\in\I(a,b)$, we have $u=1_b\circ u=\epsilon_b\circ u+\epsilon'_b\circ u$, where the second summand belongs in $(\I'\cdot\I)(a,b)=0$. Then $u=\epsilon_b\circ u$.
It remains to check that the collection $\epsilon :=(\epsilon_a:a\rightarrow a)$ defines a natural transformation $\epsilon\colon\id_{\mathcal{A}}\to \id_{\mathcal{A}}$, which will be clearly idempotent and will satisfy that $\Phi (\epsilon )=\I$. Indeed let $u\colon a\to b$ be a morphism in ${\mathcal{A}}$. Bearing in mind that $\I$ and $\I'$ are ideals, the decompositions $u=u\circ\epsilon_a+u\circ\epsilon'_a$ and $u=\epsilon_b\circ u+\epsilon'_b\circ u$ are both the decomposition of $u$ with respect to the decomposition ${\mathcal{A}}(a,b)=\I(a,b)\oplus\I'(a,b)$. By uniqueness, we then get $\epsilon_b\circ u=u\circ\epsilon_a$, and hence $\epsilon$ is a natural transformation.
\[cor.description-TTF\] Let ${\mathcal{A}}$ be a small preadditive category, let $\epsilon=\epsilon^2\in Z({\mathcal{A}})$ be any idempotent element, let $\widehat{\epsilon}$ the element of $Z(\mod{\mathcal{A}})$ corresponding to $\epsilon$ by the bijection of Proposition \[prop.Z(A)-vs-Z(Mod-A)\] and let $\I_\epsilon$ the direct summand of ${\mathcal{A}}$ corresponding to $\epsilon$ by the bijection of Proposition \[prop.bijection-central-idempotents\]. For a right ${\mathcal{A}}$-module $M$, the following assertions hold:
1. $M=M\I_\epsilon$ if, and only if, $\widehat{\epsilon}_M$ is an isomorphism if, and only if, $\widehat{\epsilon}_M=\id_M$;
2. $M\I_\epsilon =0$ if, and only if, $\widehat{\epsilon}_M=0$.
Since we have that $\widehat{\epsilon}_M\circ\widehat{\epsilon}_M=\widehat{\epsilon}_M$, we immediately get that $\widehat{\epsilon}_M$ is an isomorphism if, and only if, $\widehat{\epsilon}_M=\id_M$.
By definition we have that $(M\I_\epsilon)(a)=\sum_{\alpha\in\I_\epsilon (a,b)}{\mathrm{Im}}(M(\alpha))$. By the proof of the last proposition, we know that $\alpha\in\I_\epsilon (a,b)$ if and only if $\epsilon_b\circ\alpha =\alpha$, which is equivalent to say that $\alpha =\alpha\circ\epsilon_a$ due to the naturality of $\epsilon$. It then follows that $M(\alpha )=M(\epsilon_a)\circ M(\alpha)$, for all morphisms $\alpha$ in $\I$ with domain $a$. It easily follows that $(M\I_\epsilon)(a)={\mathrm{Im}}(M(\epsilon_a))$. But, by the proof of Proposition \[prop.Z(A)-vs-Z(Mod-A)\], we know that the evaluation of $\widehat{\epsilon}_M$ at $a$ is $\widehat{\epsilon}_{M,a}=M(\epsilon_a)$. It then follows that $(M\I_\epsilon)(a)={\mathrm{Im}}(\widehat{\epsilon}_{M,a})$. Hence, $M\I_\epsilon =0$ if, and only if, $\widehat{\epsilon}_{M,a}=0$, for all $a\in\Ob({\mathcal{A}})$, verifying assertion (2).
On the other hand, we also have that $M=M\I_\epsilon$ if, and only if, the induced map $\widehat{\epsilon}_{M,a}=M(a)\to M(a)$ is an epimorphism, for all $a\in\Ob({\mathcal{A}})$. But this latter map is an idempotent endomorphism of $M(a)$. Then it is an epimorphism if and only if it is an isomorphism. Hence assertion (1) also follows easily.
\[rem.clarification\] Although, in order to avoid redundancy, we have not said it explicitly in the statement of the above corollary, it is clear from its proof that $M\I_\epsilon=M$ if, and only if, $M(\epsilon_a)$ is an isomorphism (equivalently $M(\epsilon_a)=\id_{M(a)}$), for all $a\in\Ob({\mathcal{A}})$. It is also clear that $M\I_\epsilon =0$ if, and only if, $M(\epsilon_a)=0$ for all $a\in\Ob({\mathcal{A}})$.
The section is organized as follows: we start recalling some basic facts and definitions about torsion pairs in Sec.\[recall\_torsion\_subs\]; we prove our generalization of Gabriel’s bijection in Sec.\[grabriel\_subs\], specializing this result to hereditary torsion pairs of finite type in Sec.\[subs\_torsion\_fin\_type\].
Torsion pairs {#recall_torsion_subs}
-------------
Recall the concept of a torsion pair from the Introduction, that we only consider in the category $\mod{\mathcal{A}}$ in the rest of the paper. If $X$ is an ${\mathcal{A}}$-module and we consider the torsion sequence $$0\rightarrow T_X\to X\to F_X\to 0,$$ then $T_X$ and $F_X$ depend functorially on $X$, and the corresponding functors $t\colon \mod {\mathcal{A}}\to {\mathcal{T}}$ and $(1:t)\colon \mod {\mathcal{A}}\to {\mathcal{F}}$ are called, respectively, the [**torsion radical**]{} and the [**torsion coradical**]{} functors. In fact, $t$ is the right adjoint of the inclusion ${\mathcal{T}}\to \mod {\mathcal{A}}$, while $(1:t)$ is the left adjoint to the inclusion ${\mathcal{F}}\to \mod{\mathcal{A}}$. We can visualize this situation as in the following diagram: $$\xymatrix{
{\mathcal{T}}\ar[rr]|{\text{inclusion}}&&\mod {\mathcal{A}}\ar@/_-12pt/[ll]|t\ar[rr]|{(1:t)}&&{\mathcal{F}}\ar@/_-12pt/[ll]|{\text{inclusion}}
}$$ A torsion pair ${\t}=({\mathcal{T}},{\mathcal{F}})$ is said to be [**hereditary**]{} provided ${\mathcal{T}}$ is closed under taking subobjects.
The following lemma is well-known (see, for example, [@S]):
\[construction\_of\_tor\_rad\] A class ${\mathcal{T}}\subseteq \mod {\mathcal{A}}$ is a torsion class (resp., a torsionfree class) if and only if it is closed under taking quotients, extensions and coproducts (resp., subobjects, extensions and products).
Any torsion pair induces an ideal of ${\mathcal{A}}$, as shown in the following lemma:
\[torsion\_ideal\_lem\] Let $\mathbf{t}=({\mathcal{T}},{\mathcal{F}})$ be a torsion pair in $\mod{{\mathcal{A}}}$. There is an ideal $t({\mathcal{A}})$ of ${\mathcal{A}}$ defined as $t({\mathcal{A}})(a,b):=t(H_b)(a)$, for all $a,b\in\Ob({\mathcal{A}})$.
Let $f\in t({\mathcal{A}})(a,b)$, $r\in {\mathcal{A}}(a',a)$, $l\in {\mathcal{A}}(b,b')$, and let us verify that $f\circ r\in t({\mathcal{A}})(a',b)$ and that $l\circ f\in t({\mathcal{A}})(a,b')$. Indeed, since the torsion radical $t$ is an additive subfunctor of $\id_{\mod{{\mathcal{A}}}}$, then ${\mathcal{A}}(-,l)(t(H_b))\leq t(H_{b'})$. In particular, $f\circ l={\mathcal{A}}(a,l)(f)\in t(H_{b'})(a)=t({\mathcal{A}})(a,b')$. On the other hand, the fact that $f\in t({\mathcal{A}})(a,b)$, means exactly that the image of the map ${\mathcal{A}}(-,f)\colon H_a\to H_b$ is torsion, thus we can corestrict to obtain a map ${\mathcal{A}}(-,f)\colon H_a\to t(H_b)$. It is now clear that also the composition ${\mathcal{A}}(-,f)\circ{\mathcal{A}}(-,r)\colon H_{a'}\to H_b$ takes values in $t(H_b)$, so that $f\circ r\in t({\mathcal{A}})(a',b)$.
Gabriel’s bijection {#grabriel_subs}
-------------------
The following definition appears in [@RG] and earlier, under the name of “linear topology”, in [@Lo], and it can be thought of as an additive version of the notion of Grothendieck topology (see, for example, [@MM]).
\[def.Grothendieck-topology\] A [**(linear) Grothendieck topology**]{} on ${\mathcal{A}}$ is a family $\mathbf{G}=\{\mathbf{G}_a:a\in\Ob({\mathcal{A}})\}$, where $\mathbf{G}_a$ is a set of submodules of the representable ${\mathcal{A}}$-module ${\mathcal{A}}(-,a)$, for each $a\in\Ob({\mathcal{A}})$, satisfying:
- the [**Identity axiom**]{}, $H_a\in\mathbf{G}_a$, for each $a\in\Ob({\mathcal{A}})$;
- the [**Pullback axiom**]{}, for $R\in\mathbf{G}_a$ and $r\colon a'\to a$ in ${\mathcal{A}}$, consider the following pullback square: $$\xymatrix{
r^{-1}R\, \ar@{^(.>}[rr]\ar@{.>}[d]\ar@{}[drr]|{\text{\scriptsize P.B.}}&&H_{a'}\ar[d]^{{\mathcal{A}}(-,r)}\\
R\, \ar@{^(->}[rr]&&H_a.
}$$ Then, $r^{-1}R$ is in $\mathbf{G}_{a'}$;
- the [**Glueing axiom**]{}, given $R\leq H_a$, suppose that there exists $S\in\mathbf{G}_a$ such that, for any $a'\in\Ob({\mathcal{A}})$ and any $r\in S(a')\leq{\mathcal{A}}(a',a)$, one has that $r^{-1}R$ is in $\mathbf{G}_{a'}$. Then, $R\in\mathbf{G}_a$.
Let us remark that, if a ring $R$ is viewed as a preadditive category with one object, then “Grothendieck topology” and “Gabriel topology” on $R$ are synonymous. The following result is part of [@Lo] and [@RG Prop.1.8].
\[lem.axioms-Up-Int\] Let $\mathbf{G}=\{\mathbf{G}_a:a\in\Ob({\mathcal{A}})\}$ be a Grothendieck topology on ${\mathcal{A}}$. The following assertions hold true, for each $a\in\Ob({\mathcal{A}})$:
1. if $R\leq S\leq H_a$ and $R\in\mathbf{G}_a$, then also $S\in\mathbf{G}_a$;
2. if $R_1,\dots ,R_m\in\mathbf{G}_a$, then $R_1\cap\dots \cap R_m\in\mathbf{G}_a$.
We are now going to prove that Grothendieck topologies on ${\mathcal{A}}$ are in bijection with hereditary torsion pairs in $\mod {\mathcal{A}}$ (see Thm.\[thm.Gabriel bijection for small categories\]). In the following lemma we show how a Grothendieck topology induces a hereditary torsion class, while the opposite direction is illustrated in Lem.\[defining\_psi\].
\[defining\_phi\] Let $\mathbf{G}=\{\mathbf G_a:a\in\Ob({\mathcal{A}})\}$ be a Grothendieck topology on ${\mathcal{A}}$ and define $$\mathcal{T}^\mathbf{G}:=\Gen\{H_a/R : a\in\Ob(\A),\, R\in\mathbf{G}_a\}\subseteq \mod{\A}.$$ Then the following statements hold true:
1. an ${\mathcal{A}}$-module $T$ is in $\mathcal{T}^\mathbf{G}$ if, and only if, ${\mathrm{Ker}}(\varphi)\in \mathbf{G}_a$ for every map $\varphi \colon H_a\to T$;
2. ${\mathcal{T}}^\mathbf{G}$ is a hereditary torsion class in $\mod{{\mathcal{A}}}$.
\(1) Given $T\in\mathcal{T}^\mathbf{G}$, there is an epimorphism $p\colon\coprod_{i\in I}(H_{a_i}/R_i)\twoheadrightarrow T$, where $R_i\in\mathbf{G}_{a_i}$, for each $i\in I$. Furthermore, given $\varphi\colon H_a\to T$, one can use that $H_a$ is finitely generated projective to show that there is a morphism $\bar\varphi\colon H_a\to\coprod_{i\in F}(H_{a_i}/R_i)$ for a finite subset $F\subseteq I$, such that $\varphi=p\circ\iota_F\circ \bar\varphi$, where $\iota_F\colon \coprod_{i\in F}(H_{a_i}/R_i)\to \coprod_{i\in I}(H_{a_i}/R_i)$ is the inclusion. Clearly, $\bar\varphi$ is described by a vector $\bar\varphi=(\bar\varphi_i)_{i\in F}$ with $\bar\varphi_j\colon H_a \to (H_{a_i}/R_j)$, for all $j\in F$ and, using again the projectivity of $H_a$ and the Yoneda Lemma, each of the morphisms $\bar\varphi_j$ factors in the form $\bar\varphi_j=p_j\circ {\mathcal{A}}(-,r_j)$, where $p_j\colon H_{a_j}\to H_{a_j}/R_j$ is the projection and $r_j \colon a\to a_j$ is a suitable morphism in ${\mathcal{A}}$. To conclude, notice that ${\mathrm{Ker}}(\bar\varphi)=\bigcap_{i\in F}{\mathrm{Ker}}(\bar\varphi_i)$ so, by Lem.\[lem.axioms-Up-Int\], it is enough to verify that ${\mathrm{Ker}}(\bar\varphi_j)\in\mathbf{G}_{a_j}$, for all $j\in I$. But, with the notation of Def.\[def.Grothendieck-topology\], ${\mathrm{Ker}}(\bar\varphi_j)=r_j^{-1}R_j$, and so ${\mathrm{Ker}}(\bar\varphi_j)\in\mathbf{G}_a$ by the axiom ([**Pb**]{}).
On the other hand, given $T\in\mod {\mathcal{A}}$ such that ${\mathrm{Ker}}(\varphi)\in \mathbf{G}_a$ for every morphism , consider an epimorphism $q\colon\coprod_{i\in I}H_{a_i}\twoheadrightarrow T$ and take the compositions $q\circ\iota_j\colon H_{a_j}{\to}T$, where $\iota_j\colon H_{a_j}\to \coprod_{i\in I}H_{a_i}$ is the inclusion, so that $R_j:={\mathrm{Ker}}(q\circ\iota_j)\in\mathbf{G}_{a_j}$, for all $j\in I$. We then get an induced epimorphism $\coprod_{i\in I}(H_{a_i}/R_i)\twoheadrightarrow T$, showing that $T\in{\mathcal{T}}^\mathbf{G}$.
\(2) follows by part (1) and [@Lo Prop.2.7].
\[defining\_psi\] Let ${\mathcal{T}}$ be a hereditary torsion class in $\mod{{\mathcal{A}}}$ and define $$\mathbf{G}^\mathcal{T}:=\{\mathbf{G}^{\mathcal{T}}_a:a\in\Ob({\mathcal{A}})\},\quad\text{with}\quad \mathbf{G}^{\mathcal{T}}_a:=\{R\leq{\mathcal{A}}(-,a):{\mathcal{A}}(-,a)/R\in{\mathcal{T}}\}.$$ Then, $\mathbf{G}^\mathcal{T}$ is a Grothendieck topology on ${\mathcal{A}}$.
It is immediate to check that $\mathbf{G}$ satisfies axioms ([**Id**]{}) and ([**Pb**]{}) of Def.\[def.Grothendieck-topology\], so we only need to check axiom ([**Glue**]{}). Consider a submodule $R\leq H_a$ and suppose that there exists $S\in\mathbf{G}^{\mathcal{T}}_a$ such that, for any $a'\in\Ob({\mathcal{A}})$ and $r\in S(a')\leq{\mathcal{A}}(a',a)$, $r^{-1}R$ is in $\mathbf{G}^{\mathcal{T}}_{a'}$. Consider the following short exact sequence: $$0\rightarrow S/(S\cap R)\to H_a/R\to H_a/(S+R)\rightarrow 0.$$ Then $H_a/(S+R)\in{\mathcal{T}}$ since it is a quotient of $H_a/S$. Hence, it is enough to verify that ${S}/({S\cap R})\in{\mathcal{T}}$. Indeed, fix an epimorphism $q\colon \coprod_{i\in I}H_{a_i}\twoheadrightarrow S$ and, for each $j\in I$, consider the following composition of $q$ with the obvious inclusions: $$\xymatrix{
\varphi_j\colon H_{a_j}\ar@{->}[r]^{}&\coprod_{i\in I}H_{a_i}\ar@{->>}[r]^{q}& S\leq H_{a}.
}$$ By hypothesis, $\varphi_j^{-1}R\in \mathbf{G}^{\mathcal{T}}_{a_j}$ for any $j\in I$, so that $H_{a_j}/\varphi_j^{-1}R\in {\mathcal{T}}$. Furthermore, there is clearly an epimorphism $\coprod_{i\in I}(H_{a_i}/\varphi_i^{-1}R)\twoheadrightarrow S/(S\cap R)$, so that $S/(S\cap R)\in {\mathcal{T}}$ as desired.
\[thm.Gabriel bijection for small categories\] Let ${\mathcal{A}}$ be a small preadditive category. Then there is a one-to-one correspondence $$\xymatrix@R=0pt{
\Phi:\S_1:=
{\left\{
\begin{matrix}
\text{Grothendieck}\\
\text{topologies on ${\mathcal{A}}$}
\end{matrix}
\right\}}
\ar@{<->}[rr]^{1:1}&&
{\left\{
\begin{matrix}
\text{Hereditary torsion}\\
\text{classes in $\mod{{\mathcal{A}}}$}
\end{matrix}
\right\}}
=:\S_2:\Psi\\
\mathbf{G}\ar@{|->}[rr]&&\Phi(\mathbf{G}):=\mathcal{T}^\mathbf{G}\\
\Psi({\mathcal{T}}):=\mathbf{G}^\mathcal{T}\ar@{<-|}[rr]&&{\mathcal{T}}}$$ where ${\mathcal{T}}^{\mathbf G}$ and $\mathbf G^{{\mathcal{T}}}$ are defined as in Lem.\[defining\_phi\] and \[defining\_psi\], respectively.
The maps $\Phi$ and $\Psi$ are well-defined by Lem.\[defining\_phi\] and \[defining\_psi\]; let us verify that they are inverse bijections. Consider first ${\mathcal{T}}\in \S_2$ and let us verify that ${\mathcal{T}}$ is equal to $\Phi\circ\Psi (\T)=\Gen(H_a/R:R\in \mathbf G_a^\T,\, a\in \Ob(\A))$. In fact, by the very definition of $\mathbf{G}_a^{\mathcal{T}}$, given $R\in\mathbf{G}^{{\mathcal{T}}}_a$, $H_a/R\in{\mathcal{T}}$ so $\Phi\circ\Psi({\mathcal{T}})\subseteq {\mathcal{T}}$. On the other hand, given $T\in {\mathcal{T}}$, consider an epimorphism $\coprod_{i\in I}H_{a_i}\twoheadrightarrow T$, that induces an epimorphism $\coprod_{i\in I}(H_{a_i}/R_i)\twoheadrightarrow T$, where each $H_{a_j}/R_j$ is a subobject of $T$, so it belongs to ${\mathcal{T}}$, that is, $R_j\in G_{a_j}^{\mathcal{T}}$. This shows that $T\in \Phi\circ\Psi({\mathcal{T}})$.
On the other hand, given $\mathbf{G}\in\mathcal{S}_1$, let us show that $\mathbf G=\Psi\circ \Phi (\mathbf G)$, that is $\mathbf G_a=\mathbf G^{{\mathcal{T}}^{\mathbf G}}_a$, for each $a\in \Ob({\mathcal{A}})$. Indeed, given $R\in \mathbf G_a$, it is clear that $H_a/R\in {\mathcal{T}}^{\mathbf G}$, so that $R\in \mathbf G^{{\mathcal{T}}^{\mathbf G}}_a$ and $\mathbf G_a\subseteq\mathbf G^{{\mathcal{T}}^{\mathbf G}}_a$. On the other hand, given $S\in \mathbf G^{{\mathcal{T}}^{\mathbf G}}_a$, that is, given an $S\leq H_a$ such that $H_a/S\in {\mathcal{T}}^{\mathbf G}$, we known by Lem.\[defining\_phi\] that the kernel of any morphism of the form $H_b\to H_a/S$, for some $b\in\Ob({\mathcal{A}})$, is in $\mathbf G_b$. In particular, the kernel of the obvious projection $H_a\to H_a/S$, which is exactly $S$, does belong in $\mathbf G_a$, so that $\mathbf G^{{\mathcal{T}}^{\mathbf G}}_a\subseteq \mathbf G_a$, as desired.
Hereditary torsion classes of finite type {#subs_torsion_fin_type}
-----------------------------------------
Recall that a torsion pair $({\mathcal{T}},{\mathcal{F}})$ in $\mod {\mathcal{A}}$ is said to be of [**finite type**]{} provided ${\mathcal{F}}=\varinjlim {\mathcal{F}}$, that is, if ${\mathcal{F}}$ is closed under taking direct limits. As a well-known consequence of the definition, one can verify that both the torsion radical and the torsion coradical associated with a torsion pair of finite type do preserve direct limits.
\[prop.hereditary of finite type\] Let ${\t}=({\mathcal{T}},{\mathcal{F}})$ be a hereditary torsion pair in $\mod {\mathcal{A}}$. Then ${\t}$ is of finite type if and only if there is a set $\mathcal{S}\subseteq \fpmod {\mathcal{A}}$ (where $\fpmod {\mathcal{A}}$ is the category of finitely presented ${\mathcal{A}}$-modules) such that ${\mathcal{F}}=\S^{\perp}$.
We only need to check the “only if” part, which is an easy adaptation of the proof for modules over a ring (see the proof of implication (2)$\Rightarrow$(1) in [@Go Prop.42.9]), which we just outline. We need to check that each $T\in{\mathcal{T}}\cap{\mathrm{fg}}(\mod {\mathcal{A}})$ (i.e., any finitely generated torsion ${\mathcal{A}}$-module) is a quotient of an object in ${\mathcal{T}}\cap\fpmod {\mathcal{A}}$. Consider an exact sequence $$\label{torsion_fg_torsion_fp}
\xymatrix@C=15pt{
0\ar[r]& R\ar[rr]^{\lambda}&&X\ar[rr]^p&&T\ar[r]&0,
}$$ with $T\in{\mathcal{T}}\cap{\mathrm{fg}}(\mod {\mathcal{A}})$ and $X\in \fpmod {\mathcal{A}}$. Express $R$ as a direct union of its finitely generated subobjects $R=\bigcup_{j\in J}R_j$ and, for any $j\in J$, consider the following diagrams: $$\xymatrix@C=9pt@R=2pt{
&&&&&&&&0\ar[r]& R_j\ar@{=}[dd]\ar[rr]^{\lambda_j'}&&X_j\ar[dd]\ar[rr]^{p_j'}&&t(X/R_j)\ar[dd]\ar[r]&0\\
0\ar[r]& R_j\ar[rr]^{\lambda_j}&&X\ar[rr]^{p_j}&&X/R_j\ar[r]&0\\
&&&&&&&&0\ar[r]& R_j\ar[rr]^{\lambda_j}&&X\ar[rr]^{p_j}&&X/R_j\ar[r]&0
}$$ where the one on the right hand side is obtained with a pullback from the other. Letting $j$ vary in $J$, we obtain two direct systems of short exact sequences $(0\rightarrow R_j\stackrel{}{\rightarrow}X\stackrel{}{\to}X/R_j\rightarrow 0)_{j\in J}$ and $(0\rightarrow R_j\stackrel{}{\rightarrow}X_j\stackrel{}{\to}t(X/R_j)\rightarrow 0)_{j\in J}$ whose direct limit is the sequence (this is clear for the first sequence, while for the second one it is enough to use that the torsion radical preserves direct limits). Now, since $X$ is finitely presented, there is some $k\in J$ such that the canonical map $u_k\colon X_k\to X$ is a retraction, so that we obtain the following commutative diagram with exact rows: $$\xymatrix@C=30pt{
0 \ar[r] & R_k \ar[r]^{} \ar@{^(->}[d]& X_k \ar[r]^{} \ar@{>>}[d]^{u_k} & t(X/R_k) \ar[r] \ar[d]^{\alpha_k} & 0\\
0 \ar[r] & R \ar[r]_{} & X \ar[r]_{} & T \ar[r] & 0.
}$$ Since ${\mathcal{T}}$ is hereditary, ${\mathrm{Ker}}(\alpha _k)\in {\mathcal{T}}$ and, applying the Snake Lemma to the above diagram, we obtain that $R/R_k$ is a quotient of ${\mathrm{Ker}}(\alpha _k)$, so that $R/R_k\in {\mathcal{T}}$. As a consequence, $X/R_k\in{\mathcal{T}}$, since it is an extension of $R/R_k$ and $(X/R_k)/(R/R_k)\cong X/R\in {\mathcal{T}}$. Furthermore, $X/R_k$ is also finitely presented, since $X\in \fpmod {\mathcal{A}}$ and $R_k\in {\mathrm{fg}}(\mod {\mathcal{A}})$, hence $T\cong X/R$ is a quotient of $X/R_k\in \fpmod{\mathcal{A}}\cap {\mathcal{T}}$, as desired.
Recall that a [**basis**]{} for a Grothendieck topology $\mathbf{G}=\{\mathbf{G}_a:a\in\Ob({\mathcal{A}})\}$ on ${\mathcal{A}}$ (see [@Lo]) is a family $\mathbf{B}=\{\mathbf{B}_a:a\in\Ob({\mathcal{A}})\}$ such that
- $\mathbf{B}_a\subseteq\mathbf{G}_a$, for all $a\in\Ob({\mathcal{A}})$;
- for each $R\in\mathbf{G}_a$ there exists $S\in\mathbf{B}_a$ such that $S\leq R$.
We shall say that $\mathbf{B}$ is [**a basis of finitely generated right ideals**]{} of $\mathbf{G}$, when all the right ${\mathcal{A}}$-modules $R\in \mathbf{B}_a$ are finitely generated, for all $a\in \Ob({\mathcal{A}})$. As for modules over associative unital rings and, more generally, in locally finitely presented Grothendieck categories (see [@Pr Prop.11.1.4]), we have:
\[prop.basis of finitely generated ideals\] A hereditary torsion pair ${\t}=({\mathcal{T}},{\mathcal{F}})$ in $\mod{{\mathcal{A}}}$ is of finite type if, and only if, the Grothendieck topology $\mathbf G^{{\mathcal{T}}}$ (see Thm.\[thm.Gabriel bijection for small categories\]) has a basis of finitely generated ideals.
If $\mathbf G^{{\mathcal{T}}}$ has a basis of finitely generated ideals, then let $\S:=\{H_a/R:a\in \Ob({\mathcal{A}}),\, R\in \mathbf{G}_a\cap {\mathrm{fg}}(\mod {\mathcal{A}})\}$ and notice that ${\mathcal{F}}={\S}^{\perp}$, so that ${\t}$ is of finite type. Conversely, let ${\t}$ be of finite type and, for each $a\in\Ob({\mathcal{A}})$, define $\mathbf{B}_a:=\{R\leq H_a:H_a/R\in{\mathcal{T}}\cap \text{mod} {\mathcal{A}})\}=\mathbf{G}_a^{\mathcal{T}}\cap\text{fg}(\mod {\mathcal{A}})$. Let $R\in\mathbf{G}_a^\mathbf{{\mathcal{T}}}$ be arbitrary. The proof of Lem.\[prop.hereditary of finite type\], with $X=H_a$ and $T=H_a/R$, gives a finitely generated subobject $R_k\subseteq R$ such that $H_a/R_k\in{\mathcal{T}}$. This just says that $R_k\in\mathbf{B}_a$, so that $\mathbf{B}:=\{\mathbf{B}_a:a\in\Ob ({\mathcal{A}})\}$ is a basis of $\mathbf{G}^{\mathcal{T}}$ of finitely generated right ideals.
TTF triples, idempotent ideals and recollements
===============================================
The section is organized as follows: we start recalling some basic facts and definitions about TTF triples in Sec.\[sub\_TTF triples and Abelian recollements\], including their relation with Abelian recollements; we then extend Jan’s correspondence in Sec.\[jans\_subs\], showing a bijection between the family of TTF classes in $\mod {\mathcal{A}}$ and idempotent ideals of ${\mathcal{A}}$. In Sec.\[subs\_Abelian recollements of module categories\] we specialize Jan’s Theorem showing that Abelian recollements of $\mod {\mathcal{A}}$ by categories of modules correspond to those idempotent ideals that are traces of finitely generated projectives. Finally, in Sec.\[subs\_centrally\], we show that Jan’s Theorem induces a correspondence between (idempotent ideals generated by) central idempotents and splitting TTF triples.
TTF triples and Abelian recollements {#sub_TTF triples and Abelian recollements}
------------------------------------
A hereditary torsion class ${\mathcal{T}}$ is said to be a [**TTF class**]{} (torsion and torsionfree class), provided it is closed under taking products. By Lem.\[construction\_of\_tor\_rad\], both $({\mathcal{T}},{\mathcal{T}}^\perp=:{\mathcal{F}})$ and $({\mathcal{C}}:={}^{\perp}{\mathcal{T}},{\mathcal{T}})$ are torsion pairs, we denote by $t\colon \mod {\mathcal{A}}\to {\mathcal{T}}$ and by $c\colon \mod {\mathcal{A}}\to {\mathcal{C}}$ the corresponding torsion radicals; the triples of the form $({\mathcal{C}},{\mathcal{T}},{\mathcal{F}})$ are called [**TTF triples**]{}. In this situation, we obtain a diagram as follows: $$\xymatrix{
&&&&{\mathcal{C}}\ar@/_12pt/[lld]|{\text{inclusion}}\\
{\mathcal{T}}\ar[rr]|{\text{inclusion}}&&\mod {\mathcal{A}}\ar@/_-12pt/[ll]|t\ar@/_12pt/[ll]|{(1:c)}\ar@/_10pt/[urr]|{c}\ar@/_-10pt/[rrd]|{(1:t)}&&\\
&&&&{\mathcal{F}}\ar@/_-12pt/[llu]|{\text{inclusion}}\\
}$$ In fact, there is an alternative way to think about TTF triples, that is, these triples are in bijection with (equivalence classes of) the “[Abelian recollements]{}” of $\mod {\mathcal{A}}$.
A [**recollement $\mathcal{R}$ of $\mod{\mathcal{A}}$ by Abelian categories**]{} $\X$ and $\Y$ (also called an [**Abelian recollement**]{}) is a diagram of additive functors $$\xymatrix{
\mathcal{R}:&\Y\ar[rr]|{i_{*}}&& \mod{\mathcal{A}}\ar@/_-12pt/[ll]|{i^{!}}\ar@/_12pt/[ll]|{i^{*}}\ar[rr]|{j^{*}}&&\X\ar@/_-12pt/[ll]|{j_*}\ar@/_12pt/[ll]|{j_!}
}$$ satisfying the following assertions:
1. $(j_{!},j^{*},j_{*})$ and $(i^{*},i_{*},i^{!})$ are adjoint triples;
2. the functors $i_{*},\, j_{!}$ and $j_{*}$ are fully faithful;
3. ${\mathrm{Im}}(i_{*})={\mathrm{Ker}}(j^{*}).$
Two Abelian recollements $\mathcal{R}=(\Y,\mod {\mathcal{A}},\X)$ and $\mathcal{R}^{'}=(\Y',\mod {\mathcal{A}},\X')$ of $\mod{\mathcal{A}}$ are said to be [**equivalent**]{} if, denoting by $j^*\colon \mod{\mathcal{A}}\to \X$ and $(j^*)'\colon \mod{\mathcal{A}}\to \X'$ the functors appearing in the two recollements, respectively, there are equivalences $\Phi\colon \mod {\mathcal{A}}\to \mod{\mathcal{A}}$ and $\Psi\colon \X \to \X^{'}$ such that the following diagram commutes, up to a natural isomorphism: $$\xymatrix@R=15pt@C=50pt{
\mod{\mathcal{A}}\ar[r]^{j^{*}} \ar[d]_{\Phi} & \X \ar[d]^{\Psi}\\
\mod{\mathcal{A}}\ar[r]^{({j^{*}})^{'}} & \X^{'}.
}$$
Now, let us start with a TTF triple $({\mathcal{C}},{\mathcal{T}},{\mathcal{F}})$ in $\mod {\mathcal{A}}$ and let us hint how to construct the associated Abelian recollement. Indeed, one starts considering the Gabriel quotient $q\colon \mod {\mathcal{A}}\to (\mod {\mathcal{A}})/{\mathcal{T}}$ that, by [@BR Prop.I.1.3], is equivalent to the full subcategory ${\mathcal{C}}\cap {\mathcal{F}}$ of $\mod {\mathcal{A}}$. Furthermore, the class ${\mathcal{T}}$ is both localizing and colocalizing, meaning that the quotient functor $\mod {\mathcal{A}}\to (\mod {\mathcal{A}})/{\mathcal{T}}$ has both adjoints, thus the recollement induced by our TTF can be visualized by the following diagram $$\xymatrix{
{\mathcal{T}}\ar[rr]|{\text{inclusion}}&&\mod {\mathcal{A}}\ar@/_-12pt/[ll]|t\ar@/_12pt/[ll]|{(1:c)}\ar[rr]|q&&(\mod {\mathcal{A}})/{\mathcal{T}}\cong {\mathcal{C}}\cap {\mathcal{F}}\ar@/_-12pt/[ll]|{q_*}\ar@/_12pt/[ll]|{q_!}
}$$ The following result is a direct consequence of [@PV Thm.4.3 and Coro.4.4]:
\[prop.bijection-recollements-TTF\] Let $\mathcal{A}$ be a small preadditive category. For each Abelian recollement $$\mathcal{R}:=\xymatrix{
\Y\ar[rr]|{i_{*}}&& \mod {\mathcal{A}}\ar@/_-12pt/[ll]|{i^{!}}\ar@/_12pt/[ll]|{i^{*}}\ar[rr]|{j^{*}}&&\X\ar@/_-12pt/[ll]|{j_*}\ar@/_12pt/[ll]|{j_!}
}$$ there is an associated TTF triple $\tau_\mathcal{R}=({\mathrm{Ker}}(i^*),{\mathrm{Im}}(i_*),{\mathrm{Ker}}(i^!))$ in $\mod \mathcal{A}$. The assignement $\mathcal{R}\mapsto \tau_\mathcal{R}$ defines a bijection from the set of equivalence classes of Abelian recollements of $\mod \mathcal{A}$ to the set of TTF triples in $\mod\mathcal{A}$.
Jan’s Theorem {#jans_subs}
-------------
Analogously to what happens in categories of modules over associative unital rings, TTF triples in $\mod {\mathcal{A}}$ are in bijection with idempotent ideals of ${\mathcal{A}}$ (see Thm.\[prop.direct proof\] below). In the following lemma we show how any TTF triple determines an idempotent ideal of ${\mathcal{A}}$, while the opposite direction is illustrated in Lem.\[idempont\_implies\_TTF\].
\[prop.torsion ideal is two-sided\] Let $({\mathcal{C}},{\mathcal{T}},{\mathcal{F}})$ be a TTF triple in $\mod{{\mathcal{A}}}$, denote by $c\colon \mod{{\mathcal{A}}}\to\mod{{\mathcal{A}}}$ the radical associated with the torsion pair $({\mathcal{C}},{\mathcal{T}})$, and let $c({\mathcal{A}})$ be the ideal defined in Lem.\[torsion\_ideal\_lem\]. Then, the following statements hold true:
1. $c({\mathcal{A}})(a,b)=\mathcal{I}_{\mathcal{T}}(a,b):=\{(r\colon a\to b)\in {\mathcal{A}}: T(r)=0,\, \forall T\in{\mathcal{T}}\}$;
2. the ideal $\mathcal{I}_{\mathcal{T}}$ defined in part (1) is idempotent.
It is clear that $\I_{\mathcal{T}}$ is a two-sided ideal. For a module $M\in\mod {\mathcal{A}}$, let $$\Rej_{\mathcal{T}}(M):=\bigcap\{{\mathrm{Ker}}(\mu):{T\in{\mathcal{T}},\,\mu\in\hom_{\mathcal{A}}(M,T)}\};$$ we claim that $\mathcal{I}_{\mathcal{T}}(-,a)=\Rej_{\mathcal{T}}(H_a)$, for all $a\in\Ob({\mathcal{A}})$. Indeed, by the Yoneda Lemma, ${\mathrm{Hom}}_{\mathcal{A}}(H_a,T)\cong T(a)$ and, identifying each $\mu\colon H_a\to T$ with the corresponding element of $T(a)$, we readily see that $\Rej_{\mathcal{T}}(H_a)(b)=\{\alpha\in{\mathcal{A}}(b,a):T(\alpha)(\mu)=0,\, \forall T\in{\mathcal{T}},\, \mu\in T(a)\}$. That is, $$\Rej_{\mathcal{T}}(H_a)(b)=\{\alpha\in{\mathcal{A}}(b,a):T(\alpha)=0,\, \forall T\in{\mathcal{T}}\}=\mathcal{I}_{\mathcal{T}}(b,a).$$ Being ${\mathcal{T}}$ closed under products and submodules, $M/\Rej_{\mathcal{T}}(M)\in {\mathcal{T}}$, for all $M\in\mod{{\mathcal{A}}}$. In particular, $H_a/\I_{\mathcal{T}}(-,a)\in{\mathcal{T}}$, for all $a\in\Ob({\mathcal{A}})$. Let now $M\in\mod {\mathcal{A}}$, consider an epimorphism $\coprod_{i\in I}H_{a_i}\to M$, and take the composition with the natural projection $M\to M/M \I_{\mathcal{T}}$: $$\varphi\colon \coprod_{i\in I}H_{a_i}\to M\to M/M \I_{\mathcal{T}}.$$ Denote by $\pi\colon {\mathcal{A}}\to {\mathcal{A}}/\I_{\mathcal{T}}$ the projection, then $M/M\I_{\mathcal{T}}\cong \pi_*\pi^*(M)$ and $$\begin{aligned}
\hom_{\mathcal{A}}(H_{a_i},\pi_*\pi^*(M))&\cong \hom_{{\mathcal{A}}/\I_{\mathcal{T}}}(\pi^*H_{a_i},\pi^*(M))\\
&\cong \hom_{{\mathcal{A}}/\I_{\mathcal{T}}}(H_{a_i}/\I_{\mathcal{T}}(-,a_i),M/M\I_{\mathcal{T}}),\end{aligned}$$ see and the discussion there. Thus, each component $\varphi_i:H_{a_i}\to M/M\I_{\mathcal{T}}$ factors through $H_{a_i}/\I_{\mathcal{T}}(-,a_i)$, so $M/M \I_{\mathcal{T}}$ is a quotient of a module of the form $\coprod_{i\in I}(H_{a_i}/\I_{\mathcal{T}}(-,a_i))\in {\mathcal{T}}$, hence, $M/M \I_{\mathcal{T}}\in {\mathcal{T}}$. In particular, $\mathcal{I}_{\mathcal{T}}(-,a)/\mathcal{I}^2(-,a)\in{\mathcal{T}}$ for all $a\in \Ob({\mathcal{A}})$. Considering now the following short exact sequence in $\mod{{\mathcal{A}}}$: $$\xymatrix@C=15pt{
0\ar[r]&{\mathcal{I}_{\mathcal{T}}(-,a)}/{\mathcal{I}_{\mathcal{T}}^2(-,a)}\ar[rr]&&H_a/{\mathcal{I}_{\mathcal{T}}^2(-,a)}\ar[rr]&&H_a/{\mathcal{I}_{\mathcal{T}}(-,a)}\ar[r]& 0
}$$ whose outer terms are in ${\mathcal{T}}$. We then get that $H_a/\mathcal{I}_{\mathcal{T}}^2(-,a)\in {\mathcal{T}}$. Hence, $\Rej_{\mathcal{T}}(H_a)=\mathcal{I}_{\mathcal{T}}(-,a)\subseteq {\mathrm{Ker}}(H_a\to H_a/\mathcal{I}_{\mathcal{T}}^2(-,a))=\mathcal{I}_{\mathcal{T}}^2(-,a)$, proving that $\mathcal{I}_{\mathcal{T}}$ is idempotent.
\[idempont\_implies\_TTF\] Given an idempotent ideal $\mathcal{I}$ of ${\mathcal{A}}$, the following is a TTF class in $\mod{{\mathcal{A}}}$: $${\mathcal{T}}_\I:=\{T\in \mod {\mathcal{A}}:T(\alpha)=0,\,\forall \alpha\in \I(a,b),\, a,b\in{\mathcal{A}}\}.$$
Since products and coproducts in $\mod{{\mathcal{A}}}$ are computed “pointwise”, it is clear that ${\mathcal{T}}_\I$ is closed under taking products, coproducts, submodules and quotients. It only remains to check that it is closed under extensions. Indeed, consider an exact sequence $0\to T'{\to}T{\to}T''\to 0$ in $\mod{{\mathcal{A}}}$, where $T',T''\in{\mathcal{T}}_\I$, and take a morphism $\alpha\in\mathcal{I}(a,b)$. Being $\mathcal{I}$ idempotent, we can write $\alpha =\sum_{j=1}^n\gamma_j\circ\beta_j$, with $\beta_j\in\mathcal{I}(a,c_j)$ and $\gamma_j\in\mathcal{I}(c_j,b)$, with $n\in\N$, $j\in\{1,\dots,n\}$ and $c_j\in \Ob({\mathcal{A}})$. Our goal is to prove that $T(\alpha)=0$, for which it is enough to prove that $T(\gamma_j)\circ T(\beta_j)=0$ for all $j=1,\dots,n$. Therefore, it is not restrictive to assume that $\alpha=\gamma\circ\beta$, for some morphisms $\beta\in\mathcal{I}(a,c)$ and $\gamma\in\mathcal{I}(c,b)$. Due to the definition of ${\mathcal{T}}_\I$, we have the following commutative diagram with exact rows in $\Ab$: $$\xymatrix{
0 \ar[r] & T^{'}(a) \ar[r]^{u_a} \ar[d]^{0} & T(a) \ar[r]^{p_a} \ar[d]^{T(\beta)} & T^{''}(a) \ar[r] \ar[d]^{0} & 0\\
0 \ar[r] & T^{'}(c) \ar[r]_{u_c} & T(c) \ar[r]_{p_c} & T^{''}(c) \ar[r] & 0
}$$ By the universal property of co/kernels, we get a morphism $\varphi \colon T''(a)\to T'(c)$ in $\Ab$ such that $T(\beta)=u_c\circ\varphi\circ p_a$. By a similar argument, we get a morphism $\psi \colon T''(c)\to T'(b)$ such that $T(\gamma)=u_b\circ\psi\circ p_c$. Hence, $T(\alpha )=T(\gamma)\circ T(\alpha)=u_b\circ\psi\circ p_c\circ u_c\circ\varphi\circ p_a$, which is the zero morphism since $p_c\circ u_c=0$.
\[prop.direct proof\] Let ${\mathcal{A}}$ be a small preadditive category. Then there is a one-to-one correspondence $$\xymatrix@R=0pt{
\Phi:\S_1:=\{\text{Idempotent ideals of ${\mathcal{A}}$}\}\ar@{<->}[rr]^(.55){1:1}&&\{\text{TTFs in $\mod{{\mathcal{A}}}$}\}=:\S_2:\Psi\\
\I\ar@{|->}[rr]&&\Phi(\I):={\mathcal{T}}_\I\\
\Psi({\mathcal{T}}):=\mathcal{I}_{\mathcal{T}}\ar@{<-|}[rr]&&{\mathcal{T}}}$$ where $\I_{\mathcal{T}}$ and ${\mathcal{T}}_\I$ are defined as in Lem.\[prop.torsion ideal is two-sided\] and \[idempont\_implies\_TTF\].
Given a TTF class ${\mathcal{T}}$ in $\mod {\mathcal{A}}$ it is not hard to check that ${\mathcal{T}}\subseteq (\Phi\circ\Psi)({\mathcal{T}})$, so let us verify the converse inclusion. Indeed, if $M\in (\Phi\circ\Psi)({\mathcal{T}})$, then we have $M(\alpha)=0$, for any morphism $\alpha\in\mathcal{I}_{\mathcal{T}}(a,b)$, $a,\,b\in\Ob({\mathcal{A}})$; equivalently, any morphism ${\mathcal{A}}(-,b)\to M$ vanishes on $\mathcal{I}_{\mathcal{T}}(-,b)$. This means that $M$ can be written as a quotient of a module of the form $\coprod_{i\in I}(H_{a_i}/\I_{\mathcal{T}}(-,a_i))$ but such a module belongs to ${\mathcal{T}}$ (see the proof of Lem.\[prop.torsion ideal is two-sided\]), so that $M\in {\mathcal{T}}$.
On the other hand, given an idempotent ideal $\I$ of ${\mathcal{A}}$, one checks easily that $\mathcal{I}\subseteq (\Psi\circ\Phi)(\mathcal{I})$. Now, let $\alpha\colon a\to b$ be a morphism in $\alpha\in (\Psi\circ\Phi )(\mathcal{I})(a,b)$, then $T(\alpha)=0$ for all $T\in{\mathcal{T}}_\I=\Psi (\mathcal{I})$. In particular, for $T=H_b/\mathcal{I}(-,b)$, the equality $T(\alpha)=0$ means that the induced map $${{\mathcal{A}}(b,b)}/{\mathcal{I}(b,b)}\to{{\mathcal{A}}(a,b)}/{\mathcal{I}(a,b)}\qquad\text{such that}\qquad\bar{\beta}\mapsto\overline{\beta\circ\alpha},$$ is trivial. Therefore, $\bar{\alpha}=\overline{\id_b\circ\alpha}=\bar{0}$ or, equivalently, $\alpha\in\mathcal{I}(a,b)$. This proves that $(\Psi\circ\Phi)(\mathcal{I})(a,b)\subseteq\mathcal{I}(a,b)$, so that $(\Psi\circ\Phi) (\mathcal{I})=\mathcal{I}$.
As a byproduct of the above proofs we obtain the following: given a TTF triple $({\mathcal{C}},{\mathcal{T}},{\mathcal{F}})$, there is a uniquely associated ideal $\I_{\mathcal{T}}$ such that $$\T=\Gen\{H_a/\I_\T(-,a):a\in \Ob(\A)\}\cong \mod {\A/\I_\T}\quad\text{ and }$$ $$\C=\Gen\{\I_\T(-,a):a\in \Ob(\A)\}.$$
Abelian recollements of module categories {#subs_Abelian recollements of module categories}
-----------------------------------------
Let us start with the following definition:
\[def.TTF triple generated\] Let $\tau=(\mathcal{C},\mathcal{T},\mathcal{F})$ be a TTF triple in $\mod {\mathcal{A}}$ and let $\mathcal{S}$ be a class of right $\mathcal{A}$-modules. We say that $\tau$ is [**generated by $\mathcal{S}$**]{} when the torsion pair $(\mathcal{C},\mathcal{T})$ is generated by $\mathcal{S}$, i.e., when $\mathcal{T}=\mathcal{S}^\perp$. Furthermore, we say that $\tau$ is [**generated by finitely generated projective $\mathcal{A}$-modules**]{} when it is generated by a set of finitely generated projective objects of $\mod {\mathcal{A}}$.
The key result of this section is the following proposition:
\[prop.TTF triples generated by fg projectives\] Let $\mathcal{A}$ be a small preadditive category, let $\tau =(\mathcal{C},\mathcal{T},\mathcal{F})$ be a TTF triple in $\mod {\mathcal{A}}$ and let $\mathcal{I}$ be the associated idempotent ideal of $\mathcal{A}$. The following assertions are equivalent:
1. $\mathcal{C}\cap\mathcal{F}$ is equivalent to the module category over a small preadditive category;
2. $\tau$ is generated by finitely generated projective $\mathcal{A}$-modules;
3. $\mathcal{I}$ is the trace in $\mathcal{A}$ of a set of finitely generated projective right $\mathcal{A}$-modules (see Lem.\[prop\_char\_ideals\_trace\_fgp\] for other equivalent characterizations of these ideals).
Let us fix the following notation for the recollement induced by $\tau$: $$\mathcal{R}:\qquad
\xymatrix{
\mod{({{\mathcal{A}}}/{\I})}\cong\Y\ar[rr]|{i_{*}}&&\mod {\mathcal{A}}\ar@/_-12pt/[ll]|{i^{!}}\ar@/_12pt/[ll]|{i^{*}}\ar[rr]|{j^{*}}&&\X\cong {\mathcal{C}}\cap {\mathcal{F}}\ar@/_-12pt/[ll]|{j_*}\ar@/_12pt/[ll]|{j_!}
}$$
(1)$\Rightarrow$(2). By Rem.\[rem\_morita\_cauchy\], we can fix a set $\mathcal{P}$ of finitely generated projective generators of $\mathcal{C}\cap\mathcal{F}$. Being $j_!\colon\mathcal{C}\cap\mathcal{F}\to\mod\mathcal{A}$ the left adjoint of a functor which preserves all limits and colimits, it preserves finitely generated projective objects, so that $j_!(\P)\subseteq \proj({\mathcal{A}})$; let us show that ${\mathcal{T}}=j_!(\P)^{\perp}$. Indeed, $T\in j_!(\mathcal{P})^\perp$ if and only if $0=\hom_{\mathcal{A}}(j_!P,T)\cong
(\mathcal{C}\cap\mathcal{F})(P,j^*T)$, for all $P\in\P$. But this is equivalent to say that $j^*T=0$ since $\mathcal{P}$ generates $\mathcal{C}\cap\mathcal{F}$. Therefore, $T\in j_!(\mathcal{P})^\perp$ if and only if $T\in{\mathrm{Ker}}(j^*)=\mathcal{T}$. (2)$\Rightarrow$(1). Let $Y\in \mathcal{C}\cap\mathcal{F}$, and fix a set $\mathcal{Q}$ of finitely generated projective $\mathcal{A}$-modules that generates $\tau$. We then have that $\mathcal{C}=\Gen(\mathcal{Q})$, so there is an epimorphism $p\colon\coprod_{\Lambda}Q_\lambda\to j_!(Y)$, for some family $(Q_\lambda)_{\Lambda}\subseteq\mathcal{Q}$. We obtain the following epimorphism in $\mathcal{C}\cap\mathcal{F}$: $$\xymatrix{
\coprod_{\Lambda}j^*(Q_\lambda)\cong j^*(\coprod_{\Lambda}Q_\lambda)\stackrel{j^*(p)}{\longrightarrow} (j^*\circ j_!)(Y)\cong Y.
}$$ We have then reduced our task to prove that $j^*(\mathcal{Q})$ consists of small projective objects of $\mathcal{C}\cap\mathcal{F}$. For this, note that, although kernels and cokernels in $\mathcal{C}\cap\mathcal{F}$ are not computed as in $\mod\mathcal{A}$, epimorphisms and monomorphisms in $\mathcal{C}\cap\mathcal{F}$ are precisely the morphisms which are epimorphisms and monomorphisms, respectively, in $\mod\mathcal{A}$. Hence, $j^*(Q)\cong (1:t)(Q)$ is projective in $\mathcal{C}\cap\mathcal{F}$, for each $Q\in\Q$. Moreover, since coproducts in $\mathcal{C}\cap\mathcal{F}$ are computed as in $\mod\mathcal{A}$, it is clear that $(1:t)(Q)$ is small in $\mathcal{C}\cap\mathcal{F}$. Since $\mathcal{C}\cap\mathcal{F}$ is an (Ab.$5$) (see [@PaV Prop.3.5]), so in particular (Ab.$3$), Abelian category, we conclude that it is equivalent to a category of modules.
(2)$\Rightarrow$(3). Let $\mathcal{P}$ be a set in $\mathcal{C}\cap\proj(\mathcal{A})$ which generates $\tau$, and let $\mathcal{I}':={\mathrm{tr}}_{\P}({\mathcal{A}})$. It is clear that $c(M)={\mathrm{tr}}_\mathcal{P}(M)$, for all $M\in\mod\mathcal{A}$. Then, by definition of $\mathcal{I}'$, we have that $\mathcal{I}'(-,b)={\mathrm{tr}}_\mathcal{P}(H_b)=c(H_b)$. By Lem.\[prop.torsion ideal is two-sided\], $c(H_b)=\mathcal{I}(-,b)$. It follows that $\mathcal{I}'=\mathcal{I}$.
(3)$\Rightarrow$(2). Let $\mathcal{P}$ be a set of finitely generated projective $\mathcal{A}$-modules such that $\mathcal{I}={\mathrm{tr}}_\P({\mathcal{A}})$. We know that $\mathcal{C}=\Gen\{\mathcal{I}(-,a):a\in\text{Ob}(\mathcal{A})\}$ (see the comment after Thm.\[prop.direct proof\]) and each $\mathcal{I}(-,a)={\mathrm{tr}}_\mathcal{P}(H_a)$ is epimorphic image of a coproduct of objects of $\mathcal{P}$. We then get that $\mathcal{C}\subseteq\Gen(\mathcal{P})$, so that $\Gen(\mathcal{P})^{\perp}=\mathcal{P}^\perp \subseteq \mathcal{C}^\perp=\T$.\
For the converse inclusion, let $b\in \Ob({\mathcal{A}})$ and $P\in \P$. By the projectivity of $P$, any morphism $\phi\colon P\to H_{b}/{\mathrm{tr}}_{\P}(H_b)$ lifts to a morphism $P\to H_b$, but any such morphism factors through ${\mathrm{tr}}_\P(H_b)$, so $\phi=0$. We have just verified that $H_{b}/{\mathrm{tr}}_{\P}(H_b)=H_{b}/\mathcal{I}(-,b)\in \P^{\perp}$ for all $b\in\Ob({\mathcal{A}})$. Now, given $T$ in ${\mathcal{T}}$, we know (see once again the comment after Thm.\[prop.direct proof\]) that there is an epimorphism $p\colon \coprod_{\Lambda}H_{b_\lambda}/\mathcal{I}(-,b_\lambda)\to T$. Using the projectivity of the objects in $\mathcal{P}$, we get that $T\in \mathcal{P}^{\perp}$, and so ${\mathcal{T}}\subseteq \mathcal{P}^{\perp}$. Therefore, $\mathcal{T}=\mathcal{P}^\perp$, and hence $\tau$ is generated by $\mathcal{P}$.
We immediately derive the main result of the section.
\[main\_thm\_recollement\] Let $\mathcal{A}$ be a small preadditive category. There are one-to-one correspondences between:
1. the equivalence classes of recollements of $\mod {\mathcal{A}}$ by module categories over small preadditive categories;
2. the TTF triples in $\mod {\mathcal{A}}$ generated by finitely generated projective $\mathcal{A}$-modules;
3. the idempotent ideals of $\mathcal{A}$ which are the trace of a set of finitely generated projective $\mathcal{A}$-modules;
4. the idempotent ideals of the additive closure $\widehat{\mathcal{A}}$ of $\mathcal{A}$ generated by a set of idempotent endomorphisms;
5. the full subcategories of the Cauchy completion $\widehat{\mathcal{A}}_\oplus$ which are closed under coproducts and summands.
The bijection between the families described in parts (1), (2) and (3) is given by Prop.\[prop.TTF triples generated by fg projectives\]. The bijection between the families in (3) and (4) is given by Prop.\[prop\_char\_ideals\_trace\_fgp\]. Furthermore, by Coro.\[ADD\_morita\] we know that $\mod {\mathcal{A}}$, $\mod{\widehat{\mathcal{A}}}$ and $\mod {\widehat {\mathcal{A}}_{\oplus}}$ are equivalent categories, which implies that the sets of TTF triples in these categories coincide, and they are in bijection with the set of idempotent ideals of $\widehat{\mathcal{A}}$ (resp., $\widehat{\mathcal{A}}_{\oplus}$) that are the trace of a set of finitely generated projective modules. By Prop.\[prop\_char\_ideals\_trace\_fgp\], this family is in bijection with the full subcategories of the Cauchy completion $\widehat{\mathcal{A}}_\oplus$ which are closed under coproducts and summands.
In the above theorem we have completely characterized those ideals $\I$ that induce a recollement of $\mod {\mathcal{A}}$ by categories of modules. In the rest of this subsection we are going to describe a standard form for such recollements. Let us start with the following definition:
Let ${\mathcal{A}}$ be a preadditive category and $\E$ a set of idempotent endomorphisms in $\mathcal{A}$. The [**corner category of $\mathcal{E}$ in $\mathcal{A}$**]{}, denoted by $\mathcal{C}_\mathcal{E}$ in the sequel, is defined as follows:
- $\Ob(\mathcal{C}_\mathcal{E}):=\mathcal{E}$;
- given two morphisms $\epsilon\colon x\to x$ and $\epsilon'\colon x'\to x'$ in $\mathcal{E}$, where $x,\, x'\in\Ob(\mathcal{A})$, then $\mathcal{C}_\mathcal{E}(\epsilon,\epsilon')$ is the subgroup of $\mathcal{A}(x,x')$ of those morphisms $\alpha \colon x\to x'$ in $\mathcal{A}$ that admit a decomposition $\alpha =\epsilon'\circ\beta\circ\epsilon$, for some morphism $\beta\in\mathcal{A}(x,x')$;
- composition of morphisms is $\mathcal{C}_\mathcal{E}$ is defined as in $\mathcal{A}$.
We leave to the reader the verification that $\mathcal{C}_\mathcal{E}$ is a well-defined preadditive category. If the ambient category is Cauchy complete, corner categories have a particularly simple description:
Let ${\mathcal{A}}$ be a Cauchy complete preadditive category and let $\E$ be a set of idempotent endomorphisms in $\mathcal{A}$. For each $\epsilon\colon x\to x$ in $\E$, consider the splitting $x\to x_\epsilon\to x$ of $x$ induced by $\epsilon$. Then, ${\mathcal{C}}_\E$ is equivalent to the full subcategory of ${\mathcal{A}}$ having as objects the $x_\epsilon$, with $\epsilon\in\mathcal{E}$.
Let us define a functor $F\colon \X_\E\to {\mathcal{C}}_\E$, defined on objects by the rule $x_\epsilon\mapsto \epsilon$ and that maps a morphism $\alpha\colon x_{\epsilon}\to x_{\epsilon'}$ in $\X_\E$ to $F(\alpha):=\epsilon'\circ\alpha\circ \epsilon$. This functor is clearly essentially surjective and it is an exercise to verify that it is also fully faithful.
We have now the following version of [@PV Thm.5.3] for preadditive categories:
\[cor.PV generalization\] Let $\mathcal{A}$ be a small preadditive category. Let $\tau=({\mathcal{C}},{\mathcal{T}},{\mathcal{F}})$ be a TTF associated to a recollement of $\mod {\mathcal{A}}$ by categories of modules over small preadditive categories. Then, this recollement is equivalent to one of the form $$\mathcal{R}:\qquad
\xymatrix{
\mod{({\B}/{\B\E\B})}\ar[rr]|{i_{*}}&&\mod \B\ar@/_-12pt/[ll]|{i^{!}}\ar@/_12pt/[ll]|{i^{*}}\ar[rr]|{j^{*}}&&\mod{{\mathcal{C}}_\E}\ar@/_-12pt/[ll]|{j_*}\ar@/_12pt/[ll]|{j_!}
}$$ where $\mathcal{B}$ is a preadditive category Morita equivalent to $\mathcal{A}$ and $\mathcal{E}$ is a set of idempotent endomorphisms in $\mathcal{B}$.
Let $\B:=\widehat{\mathcal{A}}_{\oplus}$ be the Cauchy completion of ${\mathcal{A}}$. Then, by Prop.\[prop.TTF triples generated by fg projectives\], we know that there is a full subcategory $\X$ of $\B$, closed under summands and coproducts, such that our recollement is induced by the idempotent ideal $\I:=\B\X\B$. Notice that ${\mathcal{C}}_\X\cong \X$ is a (Cauchy complete) preadditive category; to conclude the proof it is enough to show that ${\mathcal{C}}\cap {\mathcal{F}}\cong \mod \X$. For this, it is enough to show that the full subcategory of finitely generated projective objects in ${\mathcal{C}}\cap {\mathcal{F}}$ is generating and equivalent to $\X$. We know that $\X$ is equivalent to the full subcategory of $\mod {\mathcal{A}}$ spanned by $\{H_x:x\in \X\}$. Now, one can prove exactly as in the implication “(2)$\Rightarrow$(1)” of Prop.\[prop.TTF triples generated by fg projectives\], that the full subcategory $\{c(H_x):x\in \X\}$ of ${\mathcal{C}}\cap {\mathcal{F}}$ is again equivalent to $\X$, it generates ${\mathcal{C}}\cap {\mathcal{F}}$, and it consists of finitely generated projectives.
It is natural to ask what happens when the side categories in the recollement of the above corollary are actually module categories over unital rings:
\[cor.sides are unital modcats\] Let $\mathcal{A}$ be a small preadditive category whose module category admits a recollement $$\mathcal{R}:\qquad
\xymatrix{
\mod{B}\ar[rr]|{i_{*}}&&\mod {\mathcal{A}}\ar@/_-12pt/[ll]|{i^{!}}\ar@/_12pt/[ll]|{i^{*}}\ar[rr]|{j^{*}}&&\mod{C}\ar@/_-12pt/[ll]|{j_*}\ar@/_12pt/[ll]|{j_!}
}$$ where $B$ and $C$ are associative and unital rings. Then there is an associative unital ring $A$ such that $\mod {\mathcal{A}}$ is equivalent to $\mod A$. Hence, any such recollement is equivalent to a recollement as the one described in [@PV Thm.5.3].
Up to equivalence, we can suppose that ${\mathcal{A}}$ is Cauchy complete and we can let $\B\cong \proj(B)$ and ${\mathcal{C}}\cong \proj(C)$ be the Cauchy completions of $B$ and $C$, respectively. Now, by Rem.\[rem\_morita\_cauchy\], we have $\oplus$-generators $b$ and $c$ in $\B$ and ${\mathcal{C}}$, respectively. In fact, we can identify ${\mathcal{C}}$ with a full subcategory (closed under coproducts and summands) of ${\mathcal{A}}$ and, up to this identification, $\B\cong {\mathcal{A}}/{\mathcal{A}}{\mathcal{C}}{\mathcal{A}}$ (where ${\mathcal{A}}{\mathcal{C}}{\mathcal{A}}$ is the ideal of ${\mathcal{A}}$ generated by the identities of objects in ${\mathcal{C}}$). Identifying $\B$ with ${\mathcal{A}}/{\mathcal{A}}{\mathcal{C}}{\mathcal{A}}$ we can consider both $b$ and $c$ as objects in ${\mathcal{A}}$. To conclude, it is enough to verify that $a:=b\oplus c$ is an $\oplus$-generator in ${\mathcal{A}}$, so that $\mod {\mathcal{A}}\cong \mod {{\mathcal{A}}(a,a)}$. For this, let $x\in \Ob({\mathcal{A}})=\Ob({\mathcal{A}}/{\mathcal{A}}{\mathcal{C}}{\mathcal{A}})$, then $x$, when viewed as an object in ${\mathcal{A}}/{\mathcal{A}}{\mathcal{C}}{\mathcal{A}}$, is a summand of a finite coproduct of copies of $b$, that is, there is $n\in\N$ such that $$\xymatrix@C=10pt{
x\ar[rr]^(.4){\id_x}&&x& =& x\ar[rr]^{}&&b^n\ar[rr]^{}&&x,&&\text{in ${\mathcal{A}}/{\mathcal{A}}{\mathcal{C}}{\mathcal{A}}$.}
}$$ This means exactly that there exists $c'\in\Ob({\mathcal{C}})$ such that $$\xymatrix@R=2pt@C=10pt{
&&&&&&b^n\ar@{}[dd]|+\ar[drr]^{}\\
x\ar[rr]^(.4){\id_x}&&x& =& x\ar[drr]\ar[rru]^{}&&&&x,&&\text{in ${\mathcal{A}}$.}\\
&&&&&&c'\ar[urr]
}$$ Now, since $c$ is a $\oplus$-generator in ${\mathcal{C}}$, $c'$ is a summand of $c^m$ for some $m\in \N$. Hence, the identity of $x$ factors through $(b\oplus c)^{\max\{m,n\}}$, so that $b\oplus c$ is a $\oplus$-generator for ${\mathcal{A}}$.
Centrally splitting TTF’s {#subs_centrally}
-------------------------
\[coro\_central\_id\]
In the following proposition we show that the TTF triples that arise from a central idempotent are exactly the split ones. This proves Corollary D in the Introduction. For a more general version of the following result, in the setting of idempotent complete additive categories, we refer to [@N Prop.1.7.4]. We include here a complete proof, in our particular setting, since it easily follows as a consequence of the results of the previous subsections.
Consider a TTF triple $({\mathcal{C}},{\mathcal{T}},{\mathcal{F}})$ in $\mod {\mathcal{A}}$, denote by $\I$ the associated idempotent ideal, and denote by $c\colon \mod {\mathcal{A}}\to {\mathcal{C}}$ and $t\colon \mod {\mathcal{A}}\to {\mathcal{T}}$ the associated torsion radicals. The following are equivalent
1. for any module $M$, there is a decomposition $M=c(M)\oplus t(M)$;
2. there is an idempotent element $\epsilon\in Z({\mathcal{A}})$ such that $\I=\I_\epsilon$;
3. ${\mathcal{C}}={\mathcal{F}}$.
In this case we have $${\mathcal{C}}=\{M: M(\epsilon_a)\text{ is an iso, for all }a\in\Ob({\mathcal{A}})\}\quad\text{and}$$ $${\mathcal{T}}=\{M: M(\epsilon_a)=0\text{, for all }a\in\Ob({\mathcal{A}})\}.$$
(3)$\Rightarrow$(1). Notice first that $M=c(M)+t(M)$, in fact, the inclusion ${\mathcal{F}}\subseteq {\mathcal{C}}$ is equivalent to say that $c(M/t(M))=M/t(M)$. Hence, $$\frac{M}{t(M)}=c\left(\frac{M}{t(M)}\right)=\frac{M}{t(M)}\cdot\I=\frac{M\I+t(M)}{t(M)}=\frac{c(M)+t(M)}{t(M)}.$$ Furthermore, ${\mathcal{T}}$ is always hereditary, while ${\mathcal{C}}={\mathcal{F}}$ is hereditary because ${\mathcal{F}}$ is closed under taking submodules, hence $c(M)\cap t(M)\in {\mathcal{T}}\cap {\mathcal{C}}={\mathcal{T}}\cap {\mathcal{F}}=0$.
(1)$\Rightarrow$(3). Let $M\in\mod {\mathcal{A}}$; by the decomposition $M\cong t(M)\oplus c(M)$ we can see that $M\in {\mathcal{F}}$ if and only if $t(M)=0$ if and only if $M\cong c(M)$, if and only if $M\in{\mathcal{C}}$.
(1)$\Rightarrow$(2). We have a decomposition ${\mathcal{A}}(-,a)=c({\mathcal{A}}(-,a))\oplus t({\mathcal{A}}(-,a))$, natural in $a$, for all $a\in\Ob({\mathcal{A}})$. It then follows a decomposition ${\mathcal{A}}=c({\mathcal{A}})\oplus t({\mathcal{A}})$ as ideals of ${\mathcal{A}}$. By Prop.\[prop.bijection-central-idempotents\], there exists a unique $\epsilon =\epsilon^2\in Z({\mathcal{A}})$ such that $\I=c({\mathcal{A}})=\I_\epsilon$.
(2)$\Rightarrow$(1). It is a consequence of Thm.\[prop.direct proof\], Prop.\[prop.bijection-central-idempotents\] and Lem.\[lemma\_direct\_summands\].
For the last statement apply Coro.\[cor.description-TTF\] and Rem.\[rem.clarification\].
[AAA]{}
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Manuel Saorín – `msaorin@um.es`\
[Departamento de Matemáticas, Universidad de Murcia, Aptdo.4021, 30100 Espinardo, Murcia, SPAIN]{}
Carlos E. Parra – `carlos.parra@uach.cl`\
[Instituto de Ciencias Físicas y Matemáticas, Edificio Emilio Pugin, Campus Isla Teja, Universidad Austral de Chile, 5090000 Valdivia, CHILE]{}
Simone Virili – `s.virili@um.es` or `virili.simone@gmail.com`\
[Departamento de Matemáticas, Universidad de Murcia, Aptdo.4021, 30100 Espinardo, Murcia, SPAIN]{}
[^1]: The first named author was supported by CONICYT/FONDECYT/Iniciación/11160078
[^2]: The second named author was supported by the research projects from the Ministerio de Economía y Competitividad of Spain (MTM2016-77445-P) and the Fundación ‘Séneca’ of Murcia (19880/GERM/15), both with a part of FEDER funds.
[^3]: The third named author was supported by the Ministerio de Economía y Competitividad of Spain via a grant ‘Juan de la Cierva-formación’. He was also supported by the Fundación ‘Séneca’ of Murcia (19880/GERM/15) with a part of FEDER funds.
|
---
abstract: 'We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighbourhoods of curves in the plane. If the reference curve is complete and finite (a “curved annulus”), then the strip itself is a Cheeger set and the Cheeger constant equals the inverse of the half-width of the strip. The latter holds true for unbounded strips as well, but there is no Cheeger set. Finally, for strips about non-complete finite curves, we derive lower and upper bounds to the Cheeger set, which become sharp for infinite curves. The paper is concluded by numerical results for circular sectors.'
author:
- 'David Krejčiř[í]{}k$\,^{a, b}$[^1] and Aldo Pratelli$\,^c$'
date: |
*$a)$ Basque Center for Applied Mathematics, Bizkaia Technology Park,\
Building 500, 48160 Derio, Spain; krejcirik@ujf.cas.cz*\
*$b)$ IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain*\
*$c)$ Department of Mathematics “F. Casorati”, University of Pavia,\
Via Ferrata 1, 27100 Pavia, Italy; pratelli@unipv.it*\
15 November 2010
title: '**The Cheeger constant of curved strips** '
---
Introduction
============
Let $\Omega$ be an open connected set in the plane $\R^2$. The *Cheeger constant* of $\Omega$ is defined as $$\label{Ch.constant}
h(\Omega) := \inf_{S\subseteq\Omega} \frac{P(S)}{|S|} \,,$$ where the infimum is taken over all sets $S\subseteq \Omega$ of finite perimeter. Here and in the following, $P(S)$ and $|S|$ denote the perimeter and the area of $S$, respectively. Any minimizer of , if it exists, is called *Cheeger set* of $\Omega$ and denoted by $\C_\Omega$.
The problems of existence, uniqueness and regularity of Cheeger sets have been widely studied in last years, for instance one may look at [@Kawohl-Fridman_2003; @Hebey-Saintier_2006; @Saintier; @CCN]. We briefly list and discuss here some of the general known properties.
\[genprop\]
- While for a general $\Omega$ neither existence nor uniqueness are guaranteed, there is always some Cheeger set if $\Omega$ is a bounded open set.
- If $\Omega_1\subseteq\Omega_2$, then $h(\Omega_1)\geq h(\Omega_2)$, but the strict inclusion does not imply the strict inequality.
- The boundary of any Cheeger set $\C_\Omega$ intersects the boundary of the set $\Omega$.
- The part of $\partial\C_\Omega$ which is inside $\Omega$ is made by arcs of circle, all of radius $1/h(\Omega)$, and each of which starts and ends touching the boundary of $\Omega$.
- A Cheeger set cannot have corners (i.e., discontinuities in the tangent vector to the boundary giving rise to an angle smaller than $\pi$). In particular, the arcs of circle of $\partial\C\cap \Omega$ must intersect the boundary of $\Omega$ tangentially or in “open corners” (i.e., angles bigger than $\pi$).
- If there is some Cheeger set, then in particular there must be some connected one.
Concerning property (i), examples of non-existence or non-uniqueness can be found for instance in [@Kawohl-Lachand-Robert_2006], while the existence is immediate by the compactness results for BV functions (see for instance [@Evans-Gariepy; @AFP]). Property (ii) is immediate by the definition (\[Ch.constant\]), and examples for the non strict inequality can be found again in [@Kawohl-Lachand-Robert_2006]. Property (iii) comes from a standard variational argument (see for instance [@Kawohl-Fridman_2003 Remark 9]). Property (iv) comes immediately by a rescaling of $\C$ with a factor bigger than $1$, since this lowers the ratio in (\[Ch.constant\]). Property (v) comes directly by noticing that “cutting a corner” of a small length $\eps$ decreases $|\C_\Omega|$ of at most $C\eps^2$ and the perimeter of at least $c\eps$. Here, and in the sequel, by “corner” we mean a point of the boundary where the tangent vector is discontinuous and makes an angle smaller than $\pi$ (with respect to the internal part of $\Omega$, of course). In the case of angles bigger than $\pi$, we talk about “open corners”, and they cannot be excluded from $\partial\C$, since for instance, as pointed out in [@Kawohl-Lachand-Robert_2006], there are open corners (or “reentrant corners” in their terminology) in an $\mathrm{L}$-shaped set. Finally, property (vi) is immediate because if a Cheeger set has different connected components, each of them must be also a Cheeger set thanks to the characterization (\[Ch.constant\]).
Apart from the above-mentioned general properties, it is usually a difficult task to find the Cheeger constant or the Cheeger set of a given domain $\Omega$. The situation is simplified when $\Omega$ is a bounded convex set, which is a well-studied particular situation. In fact, in this case it is known that there is a unique open Cheeger set, which is again convex (see [@Alter-Caselles-Chambolle_2005a; @Kawohl-Lachand-Robert_2006; @CCN]). Moreover, it is also possible to give the following characterization.
\[Thm.convex\] Let $\Omega$ be a bounded convex subset of $\R^2$. For $r \geq 0$, define $$\Omega^r := \{x \in \Omega \ | \ \mathrm{dist}(x,\partial\Omega) > r \}\,.$$ There exists a unique value $r=r^*>0$ such that $$\label{distances}
|\Omega^{r}|=\pi r^2 \,.$$ Then $h(\Omega)=1/r^*$ and the Cheeger set of $\Omega$ is the Minkowski sum $\C_\Omega = \Omega^{r^*} + B_{r^*}$, with $B_{r^*}$ denoting the disc of radius $r^*$.
This theorem can be used to find explicitly $h(\Omega)$ and $\C_\Omega$ in some cases, for example for discs, rectangles and triangles – in particular, the Cheeger sets of rectangles and triangles are obtained by suitably “cutting the corners”. Furthermore, it provides a constructive algorithm for the determination of the Cheeger constant and Cheeger set for general convex domains, in particular for convex polygons.
Unfortunately, there is no such a constructive method for non-convex domains. Only one particular case seems to be explicitly known in the literature, namely the annulus, for which it is known that $\C_\Omega=\Omega$. In general, while a trivial strategy to find upper estimates for $h(\Omega)$ is to choose a suitable “test domain” $S$ in (\[Ch.constant\]), it is less clear how to obtain lower estimates. One possibility is given by the following result concerning “test vector fields”.
\[Thm.Grieser\] Let $V:\Omega\to\R^2$ be a smooth vector field on $\Omega$, $h\in\R$, and assume that the pointwise inequalities $|V| \leq 1$ and $\div V \geq h$ hold in $\Omega$. Then $h(\Omega) \geq h$.
An example of applicability of this criterion is the above-mentioned result for the annulus, which can be obtained by employing the vector field of [@Bellettini-Caselles-Novaga_2002 Sec. 11, Ex. 4] (see also Remark \[Rem.test\] below, where the corresponding vector field can be found explicitly). However, for a general set $\Omega$ it is not easy at all to find a vector field producing non-trivial lower bounds by this criterion.
The purpose of the present paper is to introduce a class of non-convex planar domains for which the Cheeger constant and the Cheeger set can be determined explicitly, namely, the curved strips. This class of sets has been intensively studied in the last two decades as an effective configuration space for curved quantum waveguides (see the survey papers [@DE; @KK] and the references therein).
More precisely, we call “curved strip” a tubular neighbourhood of a curve without boundary in the plane. There are then few possibilities: a “curved annulus”, a “finite curved strip”, an “infinite curved strip” or a “semi-infinite curved strip” – see Figure \[figstrip\] (we leave the formal definitions to Section \[GP\]).
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![The four possible types of strips.[]{data-label="figstrip"}](f-strip1.eps "fig:"){width="30.00000%"} ![The four possible types of strips.[]{data-label="figstrip"}](f-strip2.eps "fig:"){width="30.00000%"}
infinite curved strip semi-infinite curved strip
![The four possible types of strips.[]{data-label="figstrip"}](f-strip3.eps "fig:"){width="30.00000%"} ![The four possible types of strips.[]{data-label="figstrip"}](f-strip4.eps "fig:"){width="30.00000%"}
finite curved strip curved annulus
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Our main results, Theorems \[Thm.Cheeger\] and \[Thm.bounded\], describe the situation in all these cases. In particular, for a curved annulus the situation is analogous to the standard annulus, that is, the strip itself is the unique Cheeger set and the Cheeger constant only depends on the width of the strip, irrespectively of the curvature of the curve – more precisely, the Cheeger constant is the inverse of the half-width (Theorem \[Thm.Cheeger\], part (i)). For an infinite or a semi-infinite curved strip, again the Cheeger constant equals the inverse of the half-width of the strip, but there is no Cheeger set (Theorem \[Thm.Cheeger\], part (ii)). Finally, for a finite curved strip, the situation is analogous to the standard rectangle, that is, there exists a Cheeger set, which is not the whole strip because of the corners, and the Cheeger constant is strictly bigger than the inverse of the half-width. Moreover, in this last case we can also give a (sharp) upper and a lower bound, which only depend on the width and on the length of the strip (Theorem \[Thm.bounded\]).
We conclude this introductory section with a couple of comments. First of all, it is to be mentioned that, in the study of the Cheeger problem, an important role is played by those sets $\Omega$ which are Cheeger sets of themselves. This is what happens in many situations, such as the discs and the annuli and, as we show in the present paper, the “curved annuli”. Those sets are called *calibrable* and are intensively studied in the image processing literature, see for instance [@Bellettini-Caselles-Novaga_2002].
A second remark has to be done on the connection between the Cheeger constant and the eigenvalue problems. In fact, the *Cheeger inequality* tells that $$\label{Ch.bound}
\lambda_p(\Omega) \geq \left(\frac{h(\Omega)}{p}\right)^p$$ for any $p\in(1,\infty)$, where $\lambda_p(\Omega)$ is the first eigenvalue of the $p$-Laplacian. Moreover, as shown in [@Kawohl-Fridman_2003], $h(\Omega)=\lim_{p\searrow 1} \lambda_p(\Omega)$. At this regards, it is interesting to notice one property of the curved strips. In fact, it is well known that the first eigenvalue of the Dirichlet Laplacian (or, more in general, the infimum of the Rayleigh quotient, in the case of unbounded strips for which there might be no eigenvalues) for a curved strip strongly depends on its curvature (see for instance [@DE; @EFK; @KK]). On the other hand, the Cheeger constant is much less sensitive, since we will show, for instance, that for infinite and semi-infinite curve strips, as well as for curved annuli, the Cheeger constant does not depend at all on the curvature of the strip, but only on its width.
The geometrical setting\[GP\]
-----------------------------
In this section we set the notations for the geometrical situation that we will consider throughout the paper. Let $\Gamma$ be a ${\rm C}^2$, connected curve in $\R^2$ (i.e., the homeomorphic image of $(0,1)$ or of $\mathbb S^1$ under a ${\rm C^2}$ function), and let us denote by $|\Gamma| = \int_\Gamma dq$ its length, $dq$ being the arclength element of $\Gamma$. Let also $N:\Gamma\to\R^2$ be a ${\rm C}^1$ vector field giving the normal vector in the points of $\Gamma$, and let $\kappa:\Gamma\to\R$ be the associated curvature (notice that the sign of $\kappa$ depends on the choice of the orientation of $N$). We recall that to define $\kappa$ it is enough to take a unit-speed parametrization $\gamma$ of $\Gamma$, and hence it is $$\label{frenet}
\kappa(q) = \ddot\gamma\big(\gamma^{-1}(q)\big)\cdot N(q)\,,$$ where the dot denotes the standard scalar product in $\R^2$. Now, we introduce a mapping $\L$ from $\Gamma\times \R$ to $\R^2$ by $$\L(q,t) := q + t \, N(q)\,,$$ and for any positive $a$ we introduce the set $$\Omega_{\Gamma,a} := \L\big(\Gamma\times(-a,a)\big) \,.$$ We are interested in the sets $\Omega_{\Gamma,a}$ which are non-self-intersecting tubular neighbourhoods of $\Gamma$. More precisely, we will always make the assumption that $$\label{Ass.basic}
\hbox{$\L$ is injective in $\Gamma\times [-a,a]$}\,,$$ hence the set is as in Figure \[figgeomsetting\]. Using the expression for the bilinear form $$\label{metric}
d\L^2 = \big(1-\kappa(q)\,t\big)^2 \, dq^2 + dt^2$$ that follows from (\[frenet\]), by the Inverse Function Theorem we can easily notice that the assumption (\[Ass.basic\]) forces $a$ to be small compared to the curvature. More precisely, (\[Ass.basic\]) implies that $\big|\kappa(q)\big|\,a \leq 1$ for any $q\in\Gamma$, that the boundary of $\Omega_{\Gamma,a}$ is ${\rm C}^{1,1}$, and that $\L$ is in fact a ${\rm C}^1$ diffeomorphism between $\Gamma\times (-a,a)$ and $\Omega_{\Gamma,a}$.
![The geometry of a curved strip $\Omega_{\Gamma,a}$ and the corresponding curve $\Gamma$; the parallel lines correspond to the curves $s \mapsto \L(s,t)$ with fixed $t\in(-a,a)$.[]{data-label="figgeomsetting"}](f-geometry.eps){width="50.00000%"}
Summing up, under the hypothesis $\Omega_{\Gamma,a}$ has the geometrical meaning of an open non-self-intersecting strip, contained between the parallel curves $q \mapsto q \pm a \, N(q)$, with $q\in\Gamma$, and it can be identified with the Riemannian manifold $\Gamma\times(-a,a)$ equipped with the metric .
In this paper, we will call *curved strip* any set $\Omega_{\Gamma,a}$ satisfying the assumption (\[Ass.basic\]). Notice that when $\Gamma$ is contained in a line then $\Omega$ reduces to a rectangle, but the most interesting situation is when $\Gamma$ has a more complicated geometry, since then the associated set is not convex, hence not covered by the preceding known results for the Cheeger problem. It is easy to characterize the four possible situations occurring for a curved strip, to each of which we will associate a name to fix the ideas. The four kinds of strips are shown in Figure \[figstrip\]. First of all, if the curve $\Gamma$ is not finite, it may be either infinite or semi-infinite (that is, not finite but complete, or not finite and not complete, respectively). We will call *infinite curved strip* and *semi-infinite curved strip* the corresponding sets $\Omega_{\Gamma,a}$. On the other hand, if the curve is finite, then it can be either compact or not compact (then homeomorphic to a circle or to an open segment, respectively). In the first case, we will speak about a *curved annulus*, the annulus corresponding to the case when $\Gamma$ is exactly a circle, and in the other case about a *finite curved strip*.
The main geometrical results
============================
In this Section we will give some general technical properties, which will be used later to show our main results. First of all, we can easily obtain an upper bound for the curved strips. In the next result, for a curve $\Gamma$ which is not finite we consider a unit-speed parametrization $\gamma:(0,+\infty)\to\R^2$ (respectively, $\gamma:(-\infty,+\infty)\to\R^2$) if the strip is semi-infinite (respectively, infinite). Moreover, we will denote by $\Gamma_L$ the subset of $\Gamma$ given by $\gamma(0,L)$ or $\gamma(-L,L)$ for the semi-infinite or infinite case, respectively.
\[Lem.upper\] Let $\Gamma$ be infinite or compact (*i.e.*, $\Omega_{\Gamma,a}$ is a semi-infinite or infinite curved strip, or a curved annulus). Then $$h(\Omega_{\Gamma,a}) \leq \frac{1}{a}\,.$$ In particular, if $\Omega_{\Gamma,a}$ is a curved annulus, then $$\frac{P(\Omega_{\Gamma,a})}{|\Omega_{\Gamma,a}|} = \frac{1}{a}\,,$$ while if $\Omega_{\Gamma,a}$ is a semi-infinite or infinite curved strip, then $$\frac{P\big(\Omega_{\Gamma_L,a}\big)}{\big|\Omega_{\Gamma_L,a}\big|} \xrightarrow[L\to\infty]{} \frac{1}{a}\,.$$
If $\Omega_{\Gamma,a}$ is a curved annulus, then we take the whole $S=\Omega_{\Gamma,a}$ as test domain in . Recalling (\[metric\]), we have then $$\frac{P(S)}{|S|}
= \frac{\int_\Gamma (1+\kappa(q)\,a) \, dq + \int_\Gamma (1-\kappa(q)\,a) \, dq} {\int_\Gamma \int_{-a}^a (1-\kappa(q)\,t) \, dt \, dq}
= \frac{2|\Gamma|}{2 a |\Gamma|}
= \frac{1}{a}\,.$$ Notice that, by the symmetry of the set $S$, the curvature term cancels both in the numerator and in the denominator.
On the other hand, if $\Gamma$ is not finite, then the whole strip is not admissible because it has both infinite area and perimeter. However, for any $L>0$, we can consider the finite curved strip $S=\Omega_{\Gamma_L,a}$, which is of course contained in $\Omega_{\Gamma,a}$. Therefore, one can easily evaluate $$\label{approximation}
\frac{P(S)}{|S|}
= \frac{4a + \int_{\Gamma_L} (1+\kappa(q)\,a) \, dq+ \int_{\Gamma_L} (1-\kappa(q)\,a) \, dq}{\int_{\Gamma_L} \int_{-a}^a (1-\kappa(q)\,t) \, dt \, dq}
= \frac{4a + 2|\Gamma_L|}{2 a |\Gamma_L|}
\xrightarrow[L\to\infty]{} \frac{1}{a}\,.$$ In the formula for the perimeter, notice the term $4a$ corresponding to the two “vertical” parts of $\partial S$ at the start and at the end. Thanks to the definition (\[Ch.constant\]), the two above estimates give the thesis.
The lower bound is much more complicated to obtain. To find it, we will introduce an operation which, in a sense, fills in the “holes” and the “bays” in the test domains $S$. More precisely, let us take an open set $S\subseteq \Omega_{\Gamma,a}$, and define first $$\Gamma_S := \Big\{ q \in \Gamma:\, \L\Big(\{q\}\times (-a,a)\Big)\cap S \neq \emptyset \Big\}\,.$$ Notice that if $S$ is connected, then of course so is $\Gamma_S$. Now, we define $f_\pm : \Gamma_S\to [-a,a]$ as $$\begin{aligned}
f_-(q) := \inf \Big\{ t\in (-a,a):\, (q,t)\in S\Big\}\,, &&
f_+(q) := \sup \Big\{ t\in (-a,a):\, (q,t)\in S\Big\}\,.\end{aligned}$$ Therefore, $S$ is contained between the two graphs of $f_+$ and $f_-$. Finally, we can give the following definition.
\[defS\*\] Let $S$ be an open subset of $\Omega_{\Gamma,a}$ with finite perimeter, and let $\Gamma_S$ and $f_\pm$ be defined as above. We define then $$S^* := \Big\{\L(q,t)\in \Omega_{\Gamma,a}:\, q\in\Gamma_S,\, f_-(q)<t<f_+(q) \Big\}\,.$$
We can now show the main property of the set $S^*$, which will be fundamental for our purposes.
\[Lem.technical\] Let $S$ be an open, bounded and connected subset of $\Omega$ of finite perimeter. Then $$\begin{aligned}
|S^*| \geq |S|\,, && P(S^*) \leq P(S) \,.\end{aligned}$$ Moreover, $f_\pm \in BV\big(\Gamma_S\big)$, and the following formula $$\label{stimaper}\begin{split}
P(S^*)&=\int_{\Gamma_S}\sqrt{\big(1-\kappa(q)\,f_+(q)\big)^2+f_+'(q)^2}\,dq+\int_{\Gamma_S}\sqrt{\big(1-\kappa(q)\,f_-(q)\big)^2+f_-'(q)^2}\, dq\\
&\hspace{60pt}+ \big|D_s f_+\big|(\Gamma_S)+\big|D_s f_-\big|(\Gamma_S)+ \Big( f_+(q_0)-f_-(q_0)\Big)+\Big( f_+(q_1)-f_-(q_1)\Big)
\end{split}$$ holds, being $f_\pm'\,dq$ the absolute continuous part of $Df_\pm$ and $D_s f_\pm$ its singular part.
First of all, the fact that $|S^*|\geq |S|$ is obvious, since by definition $S^* \supseteq S$. Concerning the inequality for the perimeter, we start noticing that, by standard arguments, it is admissible to assume that $S$ is smooth. In fact, by the Compactness Theorem for BV functions (see for instance [@AFP]), we can take a sequence $S_j$ of smooth sets converging in the $L^1$ sense to $S$ in such a way that $P(S_j)\to P(S)$. By definition, the corresponding sets $S^*_j$ converge to $S^*$, and by the lower semicontinuity of the perimeter this yields $P(S^*)\leq \liminf P(S^*_j)$. As an immediate consequence, once we establish the validity of this lemma for smooth sets, it will directly follow also in full generality.
The inequality $P(S^*)\leq P(S)$ for smooth sets is very easy to guess, but a bit boring to prove. For simplicity, we will divide the proof in some steps. In this first step, we underline the following very easy topological fact. Here, by $\pi_1:\R^2\to \R$ we denote the first projection. $$\label{stepIold}\begin{array}{c}
\hbox{\emph{Let $q_0\in \R$, let $\gamma_{1,2}\subseteq \R^2$ be two non-intersecting continuous curves in the plane such}}\\
\hbox{\emph{that $\min \pi_1\gamma_1=\min \pi_1\gamma_2 = q_0$. If $t_1:=\max\{t: (q_0,t)\in \gamma_1\}>\max\{t: (q_0,t)\in \gamma_2\}=:t_2$,}}\\
\hbox{\emph{then for all $q\in \pi_1\gamma_1\cap\pi_1\gamma_2$ one has $\max\{t: (q,t)\in \gamma_1\}>\max\{t: (q,t)\in \gamma_2\}$.}}
\end{array}$$ The meaning of this claim is very simple: if one has two continuous and non-intersecting curves in the plane, and the least abscissa of points in the two curves coincide (otherwise, it is obvious that the claim is false), then the curve which starts above always remains above.
To show the validity of the claim, suppose it is not true, and let $\bar q\in \pi_1\gamma_1\cap\pi_1\gamma_2$ be a point for which $$\bar t_1:= \max\{t: (\bar q,t)\in \gamma_1\}<\max\{t: (\bar q,t)\in \gamma_2\} =: \bar t_2$$ (notice that the equality can not hold true, since the curves do not intersect). Figure \[Fig.mixed\](a) shows the situation.
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![Figures clarifying some steps of the proof of Lemma \[Lem.technical\].[]{data-label="Fig.mixed"}](fig1.eps "fig:"){width="47.00000%"} ![Figures clarifying some steps of the proof of Lemma \[Lem.technical\].[]{data-label="Fig.mixed"}](fig2.eps "fig:"){width="40.00000%"}
\(a) The situation of Step I. \(b) A possible $\partial S^+$ in Step III.
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The curve $\gamma_1$, then, is contained by definition in $$A := \Big\{ (q,t) \in \R^2\setminus \gamma_2:\, q\geq q_0,\, \Big\} \setminus \Big\{ (\bar q, t) \in \R^2 :\, t > \bar t_1 \Big\} \,.$$ This is a contradiction with the continuity of the curve $\gamma_1$, since the points $\big(q_0,t_1\big)$ and $\big(\bar q, \bar t_1 \big)$ are in $\gamma_1$ but belong to two distinct connected components of $A$. Therefore, the claim (\[stepIold\]) is proved. We can immediately observe some easy consequences of Step I. First of all, since $S$ is connected and bounded, then so is $\Gamma_S$, and we can define the left and right extrema $l,\, r\in \R$, in the sense that $\Gamma_S=\mathcal{L}\big((l,r)\times\{0\}\big)$. Moreover, $\partial S$ is the union of four disjoint connected curves, namely the part of $\partial S$ connecting $\L(l,f_+(l))$ and $\L(r, f_+(r))$, the part connecting $\L(r,f_+(r))$ and $\L(r,f_-(r))$, the part connecting $\L(r,f_-(r))$ and $\L(l,(f_-(l))$, and the last one connecting $\L(l,f_-(l))$ and $\L(l,f_+(l))$. We will denote these four parts as $\partial S^+$, $\partial S^r$, $\partial S^-$ and $\partial S^l$, respectively. We also underline that, since $\partial S$ is closed, then for any $q\in \Gamma_S$ one has $$\begin{aligned}
\L\big( q, f_+(q) \big) \in \partial S^+\,, && \L\big( q, f_-(q) \big) \in \partial S^-\,.\end{aligned}$$ More precisely, again the closedness of $\partial S$ tells us that, for any $\bar q\in (l,r)$ and any $t\in [-a,a]$, one has that $\L(\bar q,t)\in \partial S^+$ if and only if $$\liminf_{q\to \bar q} f_+(q)\leq t \leq \limsup_{q\to \bar q} f_+(q)\,.$$ In this step we show that the curve $\partial S^+$ reaches all the points $\L(q,f_+(q))$ in the “correct order”. This means that, if we parametrize $\partial S^+$ as $\gamma([0,1])$ with $\gamma(0)=\L(l,f_+(l))$ and $\gamma(1)=\L(r,f_+(r))$, then $$\label{stepIIIold}\begin{array}{c}
\hbox{\emph{If $\gamma(\sigma_1)=\L\big(q_1,f_+(q_1)\big)$ and $\gamma(\sigma_2)=\L\big(q_2,f_+(q_2)\big)$, one has $\sigma_1< \sigma_2 \Longleftrightarrow q_1<q_2$.}}
\end{array}$$ Notice that this fact is not trivial, since the curve $\partial S^+$ does not have to be a graph on $\Gamma_S$, hence it can, sometimes, move towards left, as in Figure \[Fig.mixed\](b). However, the figure itself suggests that the points $(q,f_+(q))$ are in any case reached “from left to right”. Let us now show (\[stepIIIold\]). To do so, suppose by contradiction that it is not true. Hence, there exist $\sigma_1, \, \sigma_2,\, q_1$ and $q_2$ in such a way that $\gamma(\sigma_i)=\L(q_i,f_+(q_i))$ for $i=1,2$ but one has $\sigma_1>\sigma_2$ and $q_1<q_2$. We can then give the following definitions, being $\pi$ the projection from $\Omega$ to $\Gamma$. $$\begin{split}
\sigma_3 &= \min \Big\{ \sigma \in (\sigma_1,1): \pi(\gamma(\sigma)) = q_2\Big\} \,, \\
q^* &= \min \Big\{ \pi(\gamma(\sigma)): \sigma\in (\sigma_1,\sigma_3)\Big\} \,, \\
\sigma_0 &= \max \Big\{ \sigma\in (0,\sigma_2): \pi(\gamma(\sigma))=q^*\Big\}\,.
\end{split}$$ Notice that by construction one has $0<\sigma_0<\sigma_2<\sigma_1<\sigma_3<1$, as well as $q^*\leq q_1<q_2$. Now, consider the two curves $\gamma_1=\L^{-1}\big(\gamma_{|[\sigma_0,\sigma_2]}\big)$ and $\gamma_2=\L^{-1}\big(\gamma_{|[\sigma_1,\sigma_3]}\big)$, which are continuous and non-intersecting. Moreover, $\min\pi_1\gamma_1= \min \pi_1 \gamma_2 = q^*$, hence we can apply Step I to derive that $\gamma_1$ is either “always above” or “always below” $\gamma_2$, in the sense of (\[stepIold\]). By checking $q=q_1$, one observes that $\gamma_1$ is below $\gamma_2$, since $\max\{\sigma: (q_1,\sigma)\in \gamma_2\}=f_+(q_1)$ is surely greater than $\max\{\sigma: (q_1,\sigma)\in \gamma_1\}$, by definition of $f_+$. On the other hand, by checking $q=q_2$, the very same reason shows that $\gamma_1$ is above $\gamma_2$, being $\max\{\sigma: (q_2,\sigma)\in \gamma_1\}=f_+(q_2)$. The contradiction shows the validity of (\[stepIIIold\]), hence this step is concluded. Let us fix an arbitrary $N\in\N$, and an arbitrary sequence $l=q_0<q_1< \cdots < q_N <q_{N+1} = r$ in $\Gamma_S$. We claim that $$\label{stepIVold}
\sum_{i=0}^N \big| f_+(q_i) - f_+(q_{i+1}) \big| \leq \H^1\big( \partial S^+ \big)\,,$$ being $\H^1$ the Hausdorff measure of dimension $1$, that is, the length. Notice that this inequality would show that $f_+\in BV(\Gamma_S)$, since $S$ is of finite perimeter.
To show the estimate, let us call $\gamma_i$ the part of the curve $\partial S^+$ which connects $\L(q_i,f_+(q_i))$ with $\L(q_{i+1},f_+(q_{i+1}))$. By the preceding steps, we know that $\partial S^+$ consists of the disjoint union of the curves $\gamma_i$, so that $$\H^1 \big( \partial S^+ \big) = \sum_{i=0}^N \H^1 \big(\gamma_i\big)\,.$$ Hence, (\[stepIVold\]) will follow at once as soon as we observe that for any $i=0, \,\dots\,, \, N$ one has $$\label{showthenfollows}
\H^1\big(\gamma_i\big) \geq \Big| \L\big( q_i,f_+(q_i)\big)-\L\big( q_{i+1},f_+(q_{i+1})\big) \Big| > \big| f_+(q_i) - f_+(q_{i+1}) \big|\,.$$ The first inequality is trivial, since it just says that the length of the curve $\gamma_i$ is greater than the distance of its extreme points. Concerning the strict inequality, instead, let us denote for brevity $$\begin{aligned}
P:= \L \big(q_i, f_+(q_i) &\big)\,, \qquad
Q:= \L \big(q_{i+1}, f_+(q_{i+1}) \big)\,, \qquad
Q':= \L \big(q_i, f_+(q_{i+1}) \big)\,, \\
&S':= \L\big(q_i,0 \big)\,, \qquad\qquad\qquad
S:= \L\big(q_{i+1},0\big)\,.\end{aligned}$$ Hence, assuming that $f_+(q_i)\geq f_+(q_{i+1})\geq 0$ (it is then trivial to modify the argument to cover the other cases), one has $$\overline{PQ'}+\overline{Q'S'} = \overline{PS'} < \overline{PS} < \overline{PQ} + \overline{QS} = \overline{PQ} + \overline{Q'S'}\,,$$ where the first inequality is due to the fact that, by definition, $S'$ is the closest point to $P$ inside $\Gamma$. The inequality above says that $\overline{PQ'} < \overline{PQ}$, which is precisely the missing inequality in (\[showthenfollows\]). As explained above, this implies the validity of (\[stepIVold\]), hence the fact that $f_+\in BV(\Gamma_S)$.
Of course, the very same argument shows that also $f_-\in BV(\Gamma_S)$. Let us define $\{q_i,\, i\in \N\}\subseteq \Gamma_S$ the jump points of $f_+$, which are countably many since $f_+\in BV(\Gamma_S)$. For any $i$, moreover, let us call $$\begin{aligned}
f_+^l(q_i) = \lim_{q\uparrow q_i} f_+(q)\,, &&
f_+^r(q_i) = \lim_{q\downarrow q_i} f_+(q)\,:\end{aligned}$$ being $f_+\in BV(\Gamma_S)$, these two limits exist and correspond to the $\liminf$ and the $\limsup$ of $f_+$ for $q\to q_i$. In particular, one has that $$\partial \big( {S^*}^+\big) = \Big\{ \L\big(q,f_+(q)\big):\, q\in \Gamma_S \Big\} \cup \bigcup_{i\in\N} J_i\,,$$ where $J_i$ is the segment joining $\L\big(q_i,f_+^l(q_i)\big)$ and $\L\big(q_i,f_+^r(q_i)\big)$. Let us fix now $\eps>0$, so that there exists $N\in \N$ such that $$\sum_{i> N} \big| J_i \big| < \eps\,.$$ For simplicity, we can assume that the points $q_i$ are ordered so that $l < q_1 < \cdots < q_n < r$. We can now pick, for any $1\leq i\leq N$, two points $q_i^l<q_i < q_i^r$ in $\Gamma_S$ in such a way that
- the different intervals $(q_i^l,q_i^r)$ are disjoint;
- for any $i$ one has $$\big|f_+(q_i^l) - f_+^l(q_i) \big| + \big|f_+(q_i^r) - f_+^r(q_i) \big| \leq \frac \eps N\, ;$$
- one has $$\H^1\Big( \big( \partial {S^*}^+\big) \cap \L \big( (q_i^l,q_i^r) \times (-a,a)\big)\Big)
\leq \big| J_i \big| + \frac \eps N
= \big| f_+^l(q_i) - f_+^r(q_i) \big| + \frac \eps N\,.$$
Now, we can consider the “bad” intervals $B_i = (q_i^l, q_i^r)$, where there are high jumps, and the “good” intervals $G_i= (q_i^r, q^l_{i+1})$, where there are not. Define also $G_0=(l,q_1^l)$, while $G_N=(q_N^r,r)$. Therefore, we have decomposed $\Gamma_S=\cup_{i\leq N} B_i \cup G_i$. For any good interval $G_i$, one has $$\partial {S^*}^+ \cap \L \big( G_i \times (-a,a) \big) = \Big\{ \L\big(q,f_+(q)\big):\, q\in G_i \Big\} \cup \bigcup_{j\in\N} \widetilde J_{i,j}\,,$$ where $\widetilde J_{i,j}$ are the jumps of $f_+$ contained in the interval $G_i$. Of course all the jumps $\widetilde J_{i,j}$, varying $0\leq i\leq N$ and $j\in\N$, correspond to different jumps $J_i$ for $i>N$. For any bad interval $B_i$, moreover, call $\gamma_i$ the part of the curve $\partial S^+$ from $\L\big(q_i^l,f_+(q_i^l)\big)$ to $\L\big(q_i^r,f_+(q_i^r)\big)$. Thanks to Step III, all the curves $\gamma_i$ are disjoint, and in particular $\L(q,f_+(q))$ belongs to $\gamma_i$ if and only if $q\in B_i$. Since we know that $\L(q,f_+(q))\in \partial S^+$ for all $q\in\Gamma_S$, this implies that $$\H^1\big( \partial S^+\big) \geq \H^1\Big(\big\{\L(q,f_+(q)):\, q\in \bigcup\nolimits_{i=0}^N G_i \big\} \Big) +\sum_{i=1}^N \H^1\big(\gamma_i\big)\,.$$ Notice also that, as shown with (\[showthenfollows\]) in Step IV, one has for each $1\leq i\leq N$ that $$\H^1\big( \gamma_i \big) > \big| f_+(q^l_i) - f_+(q^r_i)\big|\,.$$ So, we can finally conclude, using all the properties listed above, that $$\begin{split}
\H^1\big( \partial {S^*}^+\big) & =
\sum_{i=0}^N \H^1 \Big( \partial {S^*}^+\cap\L \big( G_i \times (-a,a)\big) \Big)+\sum_{i=1}^N \H^1 \Big( \partial {S^*}^+\cap\L \big( B_i \times (-a,a)\big) \Big)\\
&\leq\sum_{i=0}^N \bigg(\H^1 \Big(\big\{\L\big(q,f_+(q)\big):\, q\in G_i\big\}\Big)+\sum_{j\in\N} \big| \widetilde J_{i,j}\big|\bigg)
+\sum_{i=1}^N \bigg( \big| f_+^l(q_i) - f_+^r(q_i) \big| + \frac \eps N\bigg)\\
&\leq \H^1 \Big(\big\{\L\big(q,f_+(q)\big):\, q\in \bigcup\nolimits_{i=0}^N G_i\big\}\Big) + \sum_{i>N} \big| J_i\big|
+\sum_{i=1}^N \bigg( \big| f_+(q^l_i) - f_+(q^r_i) \big| + 2\,\frac \eps N\bigg)\\
&\leq \H^1 \Big(\big\{\L\big(q,f_+(q)\big):\, q\in \bigcup\nolimits_{i=0}^N G_i\big\}\Big) + \eps
+\sum_{i=1}^N \bigg( \H^1\big( \gamma_i\big) + 2\,\frac \eps N\bigg)\\
&\leq \H^1\big( \partial S^+\big) + 3 \eps\,.
\end{split}$$ Since $\eps>0$ was arbitrary, this step is concluded. By Step II, we know that $$\partial S = \partial S^+ \cup \partial S^- \cup \partial S^l \cup \partial S^r\,,$$ and the union is disjoint. Similarly, we have $$\partial S^* = \partial {S^*}^+ \cup \partial {S^*}^- \cup \partial {S^*}^l \cup \partial {S^*}^r\,.$$ By Step V we know that $\H^1\big(\partial S^+\big)\geq \H^1\big(\partial {S^*}^+\big)$, and in the very same way of course $\H^1\big(\partial S^-\big)\geq \H^1\big(\partial {S^*}^-\big)$. Let us then focus for a moment on $\partial S^l$ and on $\partial {S^*}^l$. While the first one is a curve between $\L(l,f_-(l))$ and $\L(l,f^+(l))$, the second one is the segment joining the same points. Hence, of course $\H^1\big(\partial S^l\big)\geq \H^1\big(\partial {S^*}^l\big)$, and similarly $\H^1\big(\partial S^r\big)\geq \H^1\big(\partial {S^*}^r\big)$. Adding up the four inequalities, we finally get that $\H^1\big(\partial S\big) \geq \H^1\big(\partial S^*\big)$.
Concerning formula (\[stimaper\]), it is immediate to obtain it for smooth functions $f_-$ and $f_+$, while the generalization for $BV$ functions is standard.
With the above result at hand, it will be quite easy to obtain the lower bound.
\[Lem.lower\] For a curved strip $\Omega_{\Gamma,a}$ of any kind, one has $$h(\Omega_{\Gamma,a}) \geq \frac{1}{a}\,.$$ Moreover, if the inequality above is an equality and there is a Cheeger set, then this Cheeger set must be $\Omega_{\Gamma,a}$ itself.
Let $S$ be any open connected set of finite perimeter in $\Omega_{\Gamma,a}$, and let $S^*$ be as in Definition \[defS\*\]. Denoting by $$\begin{aligned}
t_-:= \inf \big\{ f_-(q) :\, q\in\Gamma_S \big\}\,, &&
t_+:= \sup \big\{ f_+(q):\, q\in\Gamma_S \big\}\,,\end{aligned}$$ we can easily estimate $$\label{exp}\begin{split}
|S^*| & = \int_\Gamma \int_{f_-(q)}^{f_+(q)} \big(1-\kappa(q)\,t\big) \, dt \, dq= \int_\Gamma \big(f_+(q)-f_-(q)\big)
\left(1-\kappa(q)\,\frac{f_+(q)+f_-(q)}{2}\right) dq\\
& \leq (t_+-t_-) \int_\Gamma \bigg(1-\kappa(q)\,\frac{f_+(q)+f_-(q)}{2}\bigg)\, dq \,.
\end{split}$$ On the other hand, by (\[stimaper\]) it is easy to estimate the perimeter of $S^*$ as $$\begin{split}
P(S^*)&=\int_{\Gamma_S}\sqrt{\big(1-\kappa(q)\,f_+(q)\big)^2+f_+'(q)^2}\, dq+\int_{\Gamma_S}\sqrt{\big(1-\kappa(q)\,f_-(q)\big)^2+f_-'(q)^2}\,dq\\
&\hspace{60pt}+ \big|D_s f_+\big|(\Gamma_S)+\big|D_s f_-\big|(\Gamma_S)+ \Big( f_+(q_0)-f_-(q_0)\Big)+\Big( f_+(q_1)-f_-(q_1)\Big)\,,\\
&\geq 2 \int_{\Gamma_S} \bigg(1-\kappa(q)\,\frac{f_+(q)+f_-(q)}{2}\bigg)\,dq\,,
\end{split}$$ simply by neglecting both the absolutely continuous and the singular part of $Df$. Hence, thanks to Lemma \[Lem.technical\] we can readily deduce that $$\frac{P(S)}{|S|} \geq \frac{P(S^*)}{|S^*|} \geq \frac{2}{\ t_+-t_-} \geq \frac{1}{a} \,,$$ where the last inequality is due to the trivial bounds $-a \leq t_- < t_+ \leq a$. Finally, if $h(\Omega_{\Gamma,a})=1/a$ and there is some Cheeger set $\C=\C_{\Omega_{\Gamma,a}}$, then all the preceding inequalities must be equalities for $S=\C$, from which it immediately follows that $f_+$ and $f_-$ are constant, and that $t_\pm = \pm a$, thus $\C=\Omega_{\Gamma,a}$.
[As a consequence of for $p=2$, from the above result we get the lower bound $$\lambda_2(\Omega_{\Gamma,a}) \geq \frac{1}{4a^2} \,,$$ which is in fact weaker that the bound $$\lambda_2(\Omega_{\Gamma,a}) \geq \frac{j_{0,1}^2}{4a^2}$$ known from [@EFK]. Here $j_{0,1} \approx 2.4$ denotes the first positive zero of the Bessel function $J_0$. In fact, even a better bound, reflecting the local geometry of $\Gamma$ and valid in arbitrary dimensions, is established in [@EFK].]{}
\[Rem.test\][It is possible to establish the lower bound of Lemma \[Lem.lower\] directly from Theorem \[Thm.Grieser\], without using the “stripization” procedure $S^*$ of Definition \[defS\*\] and its properties stated in Lemma \[Lem.technical\]. Indeed, inspired by the formula of [@Bellettini-Caselles-Novaga_2002 Sec. 11, Ex. 4] for the annulus, let us introduce the function $V_t:\Gamma\times(-a,a)\to\R$ by $$V_t(q,t) :=
\left\{
\begin{aligned}
&\frac{(1-\kappa(q)\,a)\,(1+\kappa(q)\,a)-(1-\kappa(q)\,t)^2}
{2 \, a \, \kappa(q) \, (1-\kappa(q)\,t)}
&& \mbox{if} \quad \kappa(q)\not=0 \,,
\\
&\frac{t}{a}
&& \mbox{if} \quad \kappa(q)=0 \,.
\end{aligned}
\right.$$ Note that the value for vanishing curvature corresponds to taking the limit $\kappa(q) \to 0$ in the formula for positive curvatures. One easily checks that the vector field $V(q,t):=(0,V_t(q,t))$, where the components are considered with respect to the coordinates $(q,t)$, satisfies $\|V\|_{L^\infty(\Gamma\times(-a,a))} = 1$ and $$(\div V)(q,t)
= \frac{1}{1-\kappa(q)\,t} \, \partial_t \big[(1-\kappa(q)\,t) \, V_t(q,t)\big]
= \frac{1}{a}$$ for every $(q,t) \in \Gamma\times(-a,a)$. Hence, the searched lower bound is a consequence of Theorem \[Thm.Grieser\]. However, Lemma \[Lem.technical\] is needed to establish some finer properties of the Cheeger constant and Cheeger set.]{}
The main results
================
This section is devoted to show our two main results, namely Theorem \[Thm.Cheeger\], which deal with the case of curved annuli or not finite curved strips, and Theorem \[Thm.bounded\], which deals with finite curved strips.
The case of a curved annulus and that of a not finite curved strip
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\[Thm.Cheeger\] Let $\Gamma$ be compact, infinite or semi-infinite. Then $$\label{strip.complete}
h(\Omega_{\Gamma,a}) = \frac{1}{a} \,.$$ In particular:
1. If $\Gamma$ is compact (*i.e.* $\Omega_{\Gamma,a}$ is a curved annulus), then the infimum of is attained and the unique Cheeger set is $\C_{\Omega_{\Gamma,a}}=\Omega_{\Gamma,a}$.
2. If $\Gamma$ is infinite or semi-infinite (*i.e.* $\Omega_{\Gamma,a}$ is an infinite or semi-infinite curved strip), then the infimum of is not attained, but the sequence $\Omega_{\Gamma_L,a}$ of Lemma \[Lem.upper\] is an optimizing sequence for $L\to \infty$.
The equality \[strip.complete\] follows directly from the upper estimate of Lemma \[Lem.upper\] and the lower estimate of Lemma \[Lem.lower\].
From the characterization of Lemma \[Lem.lower\], moreover, we know that the unique possible Cheeger set is the whole $\Omega_{\Gamma,a}$. Since this set has an infinite area and perimeter in the case of an infinite or semi-infinite curved strip, we get the non-existence result of a minimizer for the case (ii), while the fact that $\Omega_{\Gamma_L,a}$ is a minimizing sequence for $L\to \infty$ follows by Lemma \[Lem.upper\]. On the other hand, in case (i) we know by compactness that some Cheeger set must exist, hence the existence and uniqueness of the whole $\Omega_{\Gamma,a}$ as a Cheeger set again come by Lemma \[Lem.lower\].
The case of a finite curved strip
---------------------------------
\[Thm.bounded\] Let $\Gamma$ be non-complete and finite (hence, $\Omega_{\Gamma,a}$ is a finite curved strip). Then there exists a positive dimensionless constant $c$ such that $$\label{strip.bounded}
\frac{1}{a} + \frac{c}{|\Gamma|}
\leq h(\Omega_{\Gamma,a}) \leq
\frac{1}{a} + \frac{2}{|\Gamma|} \,.$$ For instance, one may take $c=1/400$. Moreover, the infimum in is attained for some connected set $\C_{\Omega_{\Gamma,a}}\subsetneq \Omega_{\Gamma,a}$.
Concerning the existence of a Cheeger set $\C=\C_{\Omega_{\Gamma,a}}$, and in particular of a connected one, this follows by Theorem \[genprop\]. From the same Theorem, we know also that $\partial \C\cap \Omega_{\Gamma,a}$ is made by arcs of circle of radius $h(\Omega_{\Gamma,a})^{-1}$, and it cannot coincide with the whole set $\Omega_{\Gamma,a}$ again by Theorem \[genprop\], since $\C$ may not have corners. As a consequence, by the characterization of Lemma \[Lem.lower\] we deduce that $h(\Omega_{\Gamma,a})>1/a$. To conclude, we have then only to give a proof of the bounds (\[strip.bounded\]), which will be done in some steps. Obtaining the upper bound is very easy: it is enough to remind that $$\begin{aligned}
P\big(\Omega_{\Gamma,a}\big)= 2 |\Gamma| + 4a\,, && \big|\Omega_{\Gamma,a}\big|= 2 a\big|\Gamma\big|\,,\end{aligned}$$ as already checked for instance in (\[approximation\]), and then $$h(\Omega_{\Gamma,a}) \leq \frac{P\big(\Omega_{\Gamma,a}\big)}{\big|\Omega_{\Gamma,a}\big|}
=\frac{2 |\Gamma| + 4a}{2a\big|\Gamma\big|}
=\frac{1}{a} + \frac{2}{|\Gamma|}
\,.$$ Thanks to Theorem \[genprop\], we know that $\partial\C$ can not have corners. Hence, $\partial \C\cap \Omega_{\Gamma,a}$ is not empty, and it is done by some arcs of circle, all of radius $1/h(\Omega_{\Gamma,a})$, hence strictly smaller than $a$ as noticed above, such that all the four corners of $\Omega_{\Gamma,a}$ are ruled out from $\C$. Denoting by $q_0$ and $q_1$ the extreme points of $\Gamma$, let us call for simplicity “up”, “down”, “left” and “right” the four parts of $\partial \Omega_{\Gamma,a}$ given by the points of the form $\L(q,a)$, $\L(q,-a)$, $\L(q_0,t)$ and $\L(q_1,t)$ for $q\in\Gamma$ and $t\in (-a,a)$ respectively. We aim to show the following claim: $$\label{stepII}\begin{array}{c}
\hbox{\emph{All the arcs of circle of $\partial \C\cap\Omega_{\Gamma,a}$ connect two points of $\Omega_{\Gamma,a}$,}}\\
\hbox{\emph{at least one of which is either in the left or in the right part.}}
\end{array}$$ To show this claim, we have to exclude the case of an arc of circle starting and ending in the upper part, and the case of an arc connecting the up and the down (the case of an arc starting and ending in the bottom part is exactly the same as the first one).
Suppose first that there is an arc of circle connecting the points $P$ and $Q$, both in the upper part, thus $P=\L(q',a)$ and $Q=\L(q'',a)$. By Theorem \[genprop\], we know that the circle is tangent to $\partial \Omega_{\Gamma,a}$ at $P$ and $Q$, hence its centre $O$ is the intersection between the two lines which are normal to $\partial\Omega_{\Gamma,a}$ at $P$ and $Q$, which are $t\mapsto \L(q',t)$ and $t\mapsto \L(q'',t)$. Since the radius $r$ of these circles is at most $a$, the two lines must intersect in the point $\L(q',a-r)\equiv \L(q'',a-r)$, while this is impossible for any $r\leq 2a$ because $\L$ is one-to-one.
A very similar argument works assuming that an arc of circle connects the point $P=\L(q',a)$ in the upper part with the point $Q=\L(q'',-a)$ in the lower part. Indeed, again the circle would be tangent to $\partial\Omega_{\Gamma,a}$ at both $P$ and $Q$, so that its centre would be in the intersection of the segments $t\mapsto \L(q',t)$ and $t\mapsto \L(q'',t)$. This is impossible if the circle has radius smaller than $2a$ for $q'\neq q''$, but it is impossible also for a radius strictly smaller than $a$ in the case $q'=q''$. This completely shows (\[stepII\]).
Notice now that by definition the left and the right part of $\partial\Omega_{\Gamma,a}$ are segments, so also the case of a circle starting and ending in the left is impossible, as well as an arc starting and ending in the right. As a conclusion, we now know that there can be either $2$, or $3$, or $4$ arcs of circle in $\partial\C$. The simplest case is when there are four arcs, each of which making a “rounded corner”. This happens for instance for a rectangle (*i.e.*, if $\Gamma$ is a segment), and more in general if $a$ is sufficiently small with respect to $\big|\Gamma\big|$. However, it is also possible that there are only three arcs, one of which connecting the left and the right part of the boundary. This happens for instance whenever the upper or the lower part of the boundary are very short due to a big (but still admissible) curvature of $\Gamma$. An example of this situation is a sector of an annulus with very small inner radius, which then is very similar to a triangle: in this case the boundary of $\C$ does not touch the inner circle (some examples of this kind are shown in the next section). Concerning the last possibility, namely only two arcs of circle both connecting left and right, we have no example in mind and it is maybe impossible, but we do not need to exclude this case within this proof. Indeed, in Steps III and IV we will show the Theorem in the case of the four rounded corners, while in the last Step V we will show how it is always possible to reduce to this case.
To show the lower bound, we start from the case when $\C$ has four rounded corners. Let us recall that, as shown by (\[stimaper\]) in Lemma \[Lem.technical\], one has $$\label{stimaper2}\begin{split}
P(\C) &= \int_\Gamma \sqrt{\big(1-\kappa(q)\,f_+(q)\big)^2+f_+'(q)^2} \, dq + \int_\Gamma \sqrt{\big(1-\kappa(q)\,f_-(q)\big)^2+f_-'(q)^2} \, dq\\
&\hspace{60pt}+ \big|D_s f_+\big|(\Gamma)+\big|D_s f_-\big|(\Gamma)+ \Big( f_+(q_0)-f_-(q_0)\Big)+\Big( f_+(q_1)-f_-(q_1)\Big)\,,
\end{split}$$ where $f_\pm'\,dq$ is the absolute continuous part of $Df_\pm$ and $D_s f_\pm$ its singular part (notice that, in the language of Definition \[defS\*\], we have $\Gamma_{\C}=\Gamma$ thanks to Step II). Hence, in particular $$P(\C) \geq \int_\Gamma \bigg( 2-\kappa(q)\Big(f_+(q)+f_-(q)\Big) \bigg)\, dq\,.$$ We claim that, at least in the case when $\C$ has the four corners, $$\label{claim}
P(\C) \geq \int_\Gamma \bigg(2-\kappa(q)\Big(f_+(q)+f_-(q)\Big)\bigg) \, dq + \frac{1}{50} \,a\,.$$ We will prove this estimate in next step, now we show how this implies the thesis. In fact, we can easily estimate, as in (\[exp\]), the area of $\C$ as $$\begin{split}
\big|\C\big| &= \int_\Gamma \int_{f_-(q)}^{f_+(q)} \big(1-\kappa(q)\,t\big) \, dt \, dq
= \int_\Gamma \Big(f_+(q)-f_-(q)\Big) \bigg( 1 -\kappa(q)\,\frac{f_+(q)+f_-(q)}{2}\bigg)\,dq\\
&\leq a \int_\Gamma \bigg(2-\kappa(q)\,\Big(f_+(q)+f_-(q)\Big) \bigg)dq\,.
\end{split}$$ Hence, using (\[claim\]) we get (\[strip.bounded\]) because, recalling that $a \|\kappa\|_{L^\infty(\Gamma)} \leq 1$ (as pointed out in Section \[GP\]) $$\label{cp}\begin{split}
h(\Omega_{\Gamma,a}) &= \frac{P\big(\C\big)}{\big|\C\big|}
\geq \frac{\bal \int_\Gamma \bigg(2-\kappa(q)\Big(f_+(q)+f_-(q)\Big)\bigg) \, dq + \frac{1}{50} \,a\eal}{\bal a \int_\Gamma \bigg(2-\kappa(q)\,\Big(f_+(q)+f_-(q)\Big)\bigg)\, dq\eal}\\
&= \frac{1}{a}+ \frac{1}{50\bal \int_\Gamma \bigg(2-\kappa(q)\Big(f_+(q)+f_-(q)\Big) \bigg)\, dq\eal}
\geq \frac{1}{a}+ \frac{1}{50\big|\Gamma\big|\big(2+2 a \|\kappa\|_{L^\infty(\Gamma)} \big)}
\geq \frac{1}{a} + \frac{1}{200\,|\Gamma|}\,.
\end{split}$$
Here we show that, assuming that $\partial\C\cap \Omega_{\Gamma,a}$ consists of four arcs of circle, the claim (\[claim\]) holds. This will be done by considering a single arc. To choose it, we start noticing that (\[stimaper2\]) already trivially implies (\[claim\]) if $f_+(q_1) - f_-(q_1) \geq a/50$. As a consequence, we can assume that $$\label{upperright}
f_+(q_1) \leq \frac{1}{100}\, a$$ and we concentrate on the arc of circle corresponding to the “upper right corner”. Of course, if (\[upperright\]) were not true, then one could assume $f_-(q_1)\geq - a/100$ and then make the completely symmetric considerations on the “lower right corner”. As shown in Figure \[figTHREE\], we call $\gamma$ the arc of circle that we are considering, and we can also look $\gamma$ in the reference rectangle, where of course it is no more part of a circle. We will call $\eta$ as in the Figure.
![The situation (both in $\Omega$ and in the reference configuration) of Step IV.[]{data-label="figTHREE"}](fig3.eps){width="99.00000%"}
Calling $\Gamma_\gamma$ the part of $\Gamma$ related to the curve $\gamma$, hence the subset of $\Gamma$ such that $$\gamma = \Big\{ \big(q,f_+(q)\big):\, q\in \Gamma_\gamma \Big\}\,,$$ we can subdivide $\Gamma_\gamma$ in two parts, namely $$\begin{aligned}
\Gamma_1:= \bigg\{ q\in\Gamma_\gamma:\, \big|f_+'(q)\big|<\frac 15 \bigg\} \,, && \Gamma_2:= \bigg\{ q\in\Gamma_\gamma:\, \big|f_+'(q)\big|\geq \frac15 \bigg\}\,.\end{aligned}$$ Notice that the above subdivision makes sense because $f_+$ has no singular part inside $\Gamma_\gamma$ (since the image of its graph under $\L$ is an arc of circle). By definition, $$\label{1pic}
\int_{\Gamma_1} \big|f_+'(q)\big|\, dq \leq \frac{\big| \Gamma_1 \big|}{5} \leq \frac{\eta}{5} \leq \frac 25 \,a\,.$$ In the last inequality we used that $\eta\leq 2a$, which in turn is true because $\gamma$ is an arc of circle or radius smaller than $a$ (keep in mind that by Lemma \[Lem.lower\] we already know that $h(\Omega_{\Gamma,a})\geq 1/a$) and the lengths in the reference rectangle are at most the double of the true lengths (recall also that the arcs of circle touch $\partial\Omega_{\Gamma,a}$ tangentially, as we know by Theorem \[genprop\]). But then, thanks to (\[upperright\]), one has $$\int_{\Gamma_\gamma} \big|f_+'(q)\big| \, dq \geq \frac{99}{100}\,a\,,$$ so that by (\[1pic\]) we get $$\label{use1}
\int_{\Gamma_2} \big|f_+'(q)\big|\, dq \geq \bigg(\frac{99}{100}-\frac 25\bigg)\,a=\frac{59}{100}\,a\,.$$ Recalling again that $0<1-\kappa(q)f_+(q) < 2$, a trivial calculation ensures that for any $q\in\Gamma_2$ it is $$\label{use2}
\sqrt{\big(1-\kappa(q)\,f_+(q)\big)^2+f_+'(q)^2} \geq \big(1-\kappa(q)\,f_+(q)\big) +\frac {1}{25}\, \big|f_+'(q)\big|\,.$$ Hence, thanks to (\[use1\]) and (\[use2\]) we can estimate the length of $\gamma$ as $$\begin{split}
\big| \gamma \big| &= \int_{\Gamma_\gamma} \sqrt{\big(1-\kappa(q)\,f_+(q)\big)^2+f_+'(q)^2} \, dq
\geq \int_{\Gamma_\gamma} \big(1-\kappa(q)\,f_+(q)\big)\, dq +\frac 1{25} \int_{\Gamma_2} \big| f_+'(q) \big|\,dq\\
&\geq \int_{\Gamma_\gamma} \big(1-\kappa(q)\,f_+(q)\big)\, dq +\frac{59}{25\cdot 100}\,a\,.
\end{split}$$ Recalling now formula (\[stimaper2\]) for the perimeter of $\C$, we finally conclude $$P(\C) \geq \int_\Gamma \bigg(2-\kappa(q)\Big(f_+(q)+f_-(q)\Big) \bigg)\, dq + \frac{59}{25\cdot 100}\,a\,,$$ thus finally getting (\[claim\]).
In this last step we conclude the proof of the Theorem. Thanks to the above steps, we already know that the result holds in the case of four rounded corners, hence we can now assume that $\partial\C$ has only two or three arcs. In this case, there exist two maximal numbers $a^\pm\leq a$ such that $\C \subseteq \L\big(\Gamma\times (-a^-,a^+)\big)$. Let us now introduce a new strip $\Omega_{\widetilde\Gamma,\tilde a}$ by $$\begin{aligned}
\bar t:=\frac{a^+-a^-}{2}\,,&& \widetilde \Gamma:=\L\Big(\Gamma\times \{\bar t\}\Big)\,, && \tilde a := \frac{a^++a^-}{2}\,.\end{aligned}$$ Notice that there is a bijective map $\varphi:\Gamma\to\widetilde\Gamma$ given by $\varphi(q) = \L\big(q,\bar t\big)$, and that since $\widetilde\Gamma$ is by construction parallel to $\Gamma$, then the normal vector $N(q)$ to $\Gamma$ at $q$ coincides with the normal vector $\widetilde N\big(\varphi(q)\big)$ to $\widetilde \Gamma$ at $\varphi(q)$. Thus, being $\Omega_{\widetilde\Gamma,\tilde a}$ a subset of $\Omega_{\Gamma,a}$, the injectivity condition (\[Ass.basic\]) trivially holds also for $\widetilde\Gamma$ and $\tilde a$, and we can conclude that the strip $\Omega_{\widetilde\Gamma,\tilde a}$ is admissible for our purposes.
By construction, we have $\C\subseteq \Omega_{\widetilde \Gamma,\tilde a}$, hence $\C$ is also the Cheeger set of $\Omega_{\widetilde\Gamma,\tilde a}$. Moreover, by maximality of $a^\pm$ we know that $\C$ touches all the four parts of the boundary of $\Omega_{\widetilde\Gamma,\tilde a}$, so the preceding steps, and in particular (\[cp\]), allow to deduce that $$h\big(\Omega_{\Gamma,a}\big) = \frac{P(\C)}{|\C|} = h\big(\Omega_{\widetilde\Gamma,\tilde a}\big) \geq \frac{1}{\tilde a} + \frac{1}{200\big|\widetilde \Gamma\big|}\,.$$ Finally, by definition $\tilde a\leq a$, while $$\big| \widetilde\Gamma\big| = \int_\Gamma \Big(1 -\bar t\kappa(q)\Big)\,dq \leq 2 \big|\Gamma\big|\,.$$ Thus, we get (\[strip.bounded\]) with the constant $c=1/400$.
Solvable models
===============
In this section we discuss our results on the basis of several examples of curved strips about circles and circular arcs. They are referred to as *solvable models* since the determination of the Cheeger constant and the Cheeger set is reduced to solving an explicit algebraic equation. Where the exact solution is not available, we have solved the problem with help of standard numerical tools.
Annuli
------
Probably the simplest example is given by annuli, *i.e.* strips built about (full) circles, see Figure \[Fig.annulus\]. Then the Cheeger set is the strip itself and the Cheeger constant equals the half of the distance between the boundary curves. It follows from out Theorem \[Thm.Cheeger\] that exactly the same situation holds for general curved annuli. Let us remark that also discs can be thought as examples of curved strips. Indeed, a disc with its central point removed has the same Cheeger set (up to the point) and Cheeger constant as the disc, and the former set can be considered as the limit case of the annulus built about the circle of radius $a+\eps$ when $\eps \to 0+$.
![The annulus and the disc considered as its limit case[]{data-label="Fig.annulus"}](f-annulus.eps "fig:"){width="20.00000%"} ![The annulus and the disc considered as its limit case[]{data-label="Fig.annulus"}](f-disc.eps "fig:"){width="20.00000%"}
Rectangles
----------
The rectangle $\mathcal{R}_{a,b}:=(-b,b)\times(-a,a)$, with $a,b > 0$, can be considered as a strip built about the segment $\Gamma:=(-b,b)\times\{0\}$. Using Theorem \[Thm.convex\], it is easy to find its Cheeger constant explicitly: $$h(\mathcal{R}_{a,b}) =
\frac{a + b + \sqrt{(a - b)^2 + \pi a b}}{2 a b}
\,.$$ Notice the scaling $h(\mathcal{R}_{a,b})=h(\mathcal{R}_{1,b/a})$. The procedure also determines the Cheeger set of $\mathcal{R}_{a,b}$ as the rectangle with its corners rounded off by circular arcs of radius $h(\mathcal{R})^{-1}$, see Figure \[Fig.rectangle\].
![The rectangle and its Cheeger set (light gray) for $b/a=3$[]{data-label="Fig.rectangle"}](f-rectangle.eps){width="40.00000%"}
The Cheeger constant can be written as $$h(\mathcal{R}_{a,b}) = \frac{1}{a} + \frac{k(a,b)}{|\Gamma|}
\,, \qquad \mbox{where} \qquad
k(a,b) :=
\frac{a - b + \sqrt{(a - b)^2 + \pi a b}}{a}$$ and $|\Gamma|=2b$. Notice the scaling $k(a,b)=k(1,b/a)$. It is straightforward to check that $b/a \mapsto k(a,b)$ is a decreasing function with the limits $k(a,b) \to 2$ as $b/a \to 0$ and $k(a,b) \to \pi/2$ as $b/a \to \infty$. Hence the upper bound of Theorem \[Thm.bounded\] becomes sharp in the limit of very narrow rectangles. The dependence of the Cheeger constant $h$ and of the quantity $k$ on rectangle parameters is shown in Figure \[Fig.rectangle.graphs\].
![The Cheeger constant $h$ and the quantity $k$ for rectangles with $a=1$ []{data-label="Fig.rectangle.graphs"}](f-rect1.eps "fig:"){width="40.00000%"} ![The Cheeger constant $h$ and the quantity $k$ for rectangles with $a=1$ []{data-label="Fig.rectangle.graphs"}](f-rect2.eps "fig:"){width="40.00000%"}
Sectors
-------
Let $\Gamma_a$ be the circle of curvature $\kappa=a^{-1}$ and consider its part $\Gamma_a^{\alpha}$ of length $|\Gamma_a^\alpha| = \alpha a$, with any $\alpha\in(0,2\pi)$, see Figure \[Fig.sector\]. The corresponding strip $\Omega_a^\alpha := \Omega_{\Gamma_a^\alpha,a}$ does not satisfy the assumption . However, since $\L$ is in fact injective in $\Gamma_a\times (-a,a)$, it can be considered as a limit case of admissible strips along corresponding parts of the circle of radius $a+\eps$ when $\eps \to 0+$.
![The sector of a a disc considered as a strip built about the $(\frac{\alpha}{2\pi})^\mathrm{th}$-part of a circle []{data-label="Fig.sector"}](f-sector.eps){width="40.00000%"}
The Cheeger constant and the Cheeger set of $\Omega_a^\alpha$ can be found as follows. Firstly, we construct a family of domains $S_r$, with $r\in(0,a)$, defined by rounding off the corners in $\Omega_a^\alpha$ of angle smaller than $\pi$ by circular arcs of radius $r$. This can be done by a straightforward usage of elementary geometric rules. Secondly, we minimize the quotient $P(S_r)/|S_r|$ with respect to $r$, which is done with help of a numerical optimization. The minimum of the quotient corresponds to the Cheeger constant and the minimizer is the Cheeger set. The procedure is equivalent to using Theorem \[Thm.convex\], which seems to remain valid also for $\alpha > \pi$, corresponding to non-convex sectors.
In view of the obvious scaling $h(\Omega_a^\alpha) = h(\Omega_1^\alpha) / a$, one can restrict to $a=1$, without loss of generality. The dependence of the Cheeger constant on $\alpha$ is shown in Figure \[Fig.sector.graphs\]. Table \[Tab.sector\] contains numerical values for some specific angles.
Writing the Cheeger constant as $$h(\Omega_a^\alpha) = \frac{1}{a} + \frac{k(\alpha)}{|\Gamma_a^\alpha|}
\,,$$ we also study the dependence of the constant $k(\alpha)$ on $\alpha$, see Figure \[Fig.sector.graphs\] and Table \[Tab.sector\]. The third value of $\alpha$ in Table \[Tab.sector\] corresponds to the maximal point of the curve $\alpha \mapsto k(\alpha)$ from Figure \[Fig.sector.graphs\]. In any case, we see that the upper bound of Theorem \[Thm.bounded\] is quite good for all the sectors. Finally, Figure \[Fig.sector.sets\] shows a numerical approximation of the Cheeger sets for some annuli.
![The Cheeger constant $h$ and the constant $k$ for sectors with $a=1$[]{data-label="Fig.sector.graphs"}](f-sector1.eps "fig:"){width="40.00000%"} ![The Cheeger constant $h$ and the constant $k$ for sectors with $a=1$[]{data-label="Fig.sector.graphs"}](f-sector2.eps "fig:"){width="40.00000%"}
$\alpha$ $\pi/10$ $\pi/2$ $0.656749 \, \pi$ $3\pi/4$ $\pi$ $3\pi/2$ $2\pi$
---------------------- ---------- --------- ------------------- ---------- --------- ---------- ---------
$h(\Omega_1^\alpha)$ 5.92687 2.16358 1.89111 1.77915 1.57714 1.37582 1.27722
$k(\Omega_1^\alpha)$ 1.54782 1.82774 1.83856 1.83583 1.81315 1.77101 1.74184
: The Cheeger constant $h$ and the constant $k$ for sectors with $a=1$ []{data-label="Tab.sector"}
------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------
![The sectors and their Cheeger sets (light gray) []{data-label="Fig.sector.sets"}](f-s1.eps "fig:"){width="30.00000%"} ![The sectors and their Cheeger sets (light gray) []{data-label="Fig.sector.sets"}](f-s2.eps "fig:"){width="20.00000%"} ![The sectors and their Cheeger sets (light gray) []{data-label="Fig.sector.sets"}](f-s3.eps "fig:"){width="30.00000%"}
$\alpha=\pi/10$ $\alpha=\pi/2$ $\alpha=3\pi/4$
![The sectors and their Cheeger sets (light gray) []{data-label="Fig.sector.sets"}](f-s4.eps "fig:"){width="30.00000%"} ![The sectors and their Cheeger sets (light gray) []{data-label="Fig.sector.sets"}](f-s5.eps "fig:"){width="30.00000%"} ![The sectors and their Cheeger sets (light gray) []{data-label="Fig.sector.sets"}](f-s6.eps "fig:"){width="30.00000%"}
$\alpha=\pi$ $\alpha=3\pi/2$ $\alpha=2\pi$
------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------
Acknowledgement {#acknowledgement .unnumbered}
---------------
The authors acknowledge the hospitality of the Nuclear Physics Institute ASCR in Řež and the University of Pavia. This work was partially supported by the Czech Ministry of Education, Youth and Sports within the project LC06002, by the Spanish Ministry of Education and Science within the project MTM2008-03541, and by the ERC within the Advanced Grant n. 226234 and the Starting Grant n. 258685.
[100]{} F. Alter, V. Caselles, and A. Chambolle, *Evolution of characteristic functions of convex sets in the plane by the minimizing total variation flow*, Interfaces Free Bound. **7** (2005), 29–53. L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford University Press (2000). G. Bellettini, V. Caselles, and M. Novaga, *The total variation flow in [$\mathbb{R}^N$]{}*, J. Differential Equations **184** (2002), 475–525. V. Caselles, A. Chambolle, and M. Novaga, Uniqueness of the Cheeger set of a convex body, Pacific J. Math. **232** (2007), 77–90. P. Duclos and P. Exner, *[C]{}urvature-induced bound states in quantum waveguides in two and three dimensions*, Rev. Math. Phys. **7** (1995), 73–102. L. C. Evans and R. F. Gariepy, *Measure theory and fine properties of functions*, CRC Press, Boca Raton, 1992. P. Exner, P. Freitas, and D. Krejčiřík, *A lower bound to the spectral threshold in curved tubes*, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. **460** (2004), no. 2052, 3457–3467. D. Grieser, *The first eigenvalue of the [L]{}aplacian, isoperimetric constants, and the [M]{}ax [F]{}low [M]{}in [C]{}ut [T]{}heorem*, Arch. Math. **87** (2006), 75–85. E. Hebey and N. Saintier, *Stability and perturbation of the domain for the first eigenvalue of the [$1$]{}-[L]{}aplacian*, Archiv der Math. **86** (2006), 340–351. B. Kawohl and V. Fridman, *Isoperimetric estimates for the first eigenvalue of the [$p$]{}-[L]{}aplace operator and the [C]{}heeger constant*, Comment. Math. Univ. Carolinae **44** (2003), 659–667. B. Kawohl and T. Lachand-Robert, *Characterization of [C]{}heeger sets for convex subsets of the plane*, Pacific J. Math. **225** (2006), 103–118. D. Krejčiřík and J. Kříž, *On the spectrum of curved quantum waveguides*, Publ. RIMS, Kyoto University **41** (2005), no. 3, 757–791. N. Saintier, *Shape derivative of the first eigenvalue of the [$1$]{}-[L]{}aplacian*, arXiv:0706.0873v1 \[math.AP\] (2007).
[^1]: On leave from *Department of Theoretical Physics, Nuclear Physics Institute ASCR, 25068 Řež, Czech Republic*
|
[ ]{}\
IFT-UAM/CSIC-11-37\
September 20, 2011
[**On supersymmetric Lorentzian Einstein-Weyl spaces**]{}\
[**Patrick Meessen**]{}$^{\aleph}$, [**Tomás Ortín**]{}$^{\sharp}$ and [**Alberto Palomo-Lozano**]{}$^{\sharp}$\
$^{\aleph}$ [*HEP Theory Group, Departamento de Física, Universidad de Oviedo\
Avda. Calvo Sotelo s/n, E-33007 Oviedo, Spain.*]{}\
$^{\sharp}$ [*Instituto de Física Teórica UAM/CSIC, C/ Nicolás Cabrera 13-15\
Ciudad Universitaria de Cantoblanco, E-28049 Madrid, Spain.*]{}\
[In loving memory of Cte. Lozano Cid]{}\
> [<span style="font-variant:small-caps;">Abstract.</span> [ We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor, implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to supergravity theories to characterise those Lorentzian EW geometries that allow for a weighted parallel spinor, calling the resulting geometries supersymmetric. The overall result is that they are either conformally related to ordinary geometries admitting parallel spinors (w.r.t. the Levi-Cività connection) or are conformally related to certain Kundt spacetime. A full characterisation is obtained for the 4 and 6 dimensional cases. ]{} ]{}
Over the last decades, spinorial fields parallelised by some [(generalised)]{} covariant derivative[ (we shall call such spinorial fields Killing spinors),]{}[^1] have become a prominent tool in physics as well as mathematics. In physics, such spinorial fields are usually related to supersymmetry and can be used to prove the positivity of the energy in physical systems, the stability of objects that preserve some residual supersymmetry or the non-renormalisability of the mass-charge relation for the so-called BPS objects which is of the utmost importance in, for example, String Theory’s microscopic explanation of the entropy of supersymmetric black holes. In mathematics, one application which also appears frequently in the physics literature, is the link established by Hitchin between manifolds admitting parallel spinors and them having a special holonomy group [@Hitchin:1974], but can also be applied to more general settings, such as Weyl geometry [@art:moroianu1996a; @Buchholz:1999a].[^2]
Seeing the importance of such spinors it should not be surprising that in the last decade techniques were developed to extract the geometric information contained in the so-called [*Killing spinor equations*]{} (KSEs)[ [*i.e.*]{} the equations imposing the parallelity of the spinorial field under the generalised connection ]{}. The first systematic approach was made by Tod in ref. [@Tod:1983pm], taking leads from earlier work by Gibbons and Hull [@Gibbons:1982fy], who used the Newman-Penrose spinorial techniques [@Penrose:1985jw] to obtain the supersymmetric solutions to a 4-dimensional supergravity theory usually referred to as minimal (or *pure*) $N=2$ supergravity.[^3] In ref. [@Gauntlett:2002nw], Gauntlett [*et al.*]{} overcame the inherent 4-dimensional restriction of the Newman-Penrose formalism by introducing the [*spinor bilinear method*]{} and classifying the supersymmetric solutions of 5-dimensional minimal $N=1$ supergravity. This seminal article was the starting shot for a period of feverish activity in the supergravity literature, during which the supersymmetric solutions of the majority of supergravity theories were characterised, and even more powerful techniques, such as Gillard [*et al.*]{}’s spinorial geometry method [@Gillard:2004xq], were developed.
The process of the spinor bilinear characterisation is basically split into two parts: first, given a rule for the parallel propagation of the spinor in terms of the relevant supergravity fields, one deduces the most general form of those fields compatible with the existence of a non-vanishing Killing spinor; the form of the fields thus obtained is called a supersymmetric field configuration. Seeing that the KSEs are linear in derivatives and the equations of motion (EOMs) are of second order, one cannot hope to obtain a recipe for solutions of the EOMs straight-away, and instead one uses the supersymmetric configurations as Ansätze to find (supersymmetric) solutions. In this sense, an observation made by Gauntlett [*et al.*]{} in ref. [@Gauntlett:2002nw] (which was formalised in ref. [@Bellorin:2005hy]) reduces the amount of work necessary to find the conditions that a supersymmetric field configuration needs to fulfill, in order to give rise to a supersymmetric solution. The basic observation is that the fact that a solution preserves some supersymmetry means that there are relations between components of the equations of motion, meaning that there is a minimal set of independent components of the EOMs that, once satisfied, implies that all EOMs are satisfied. This observation is in fact completely general and depends only on a subset of the integrability conditions for the KSEs under consideration and on the spinorial structure used [@Bellorin:2005hy].
The first ones to realise that these techniques could be applied outside the realm of supersymmetry were the authors of ref. [@Grover:2008jr]. They considered a KSE similar to the one used in 5-dimensional minimal gauged supergravity, but with a De Sitter-like cosmological constant.[^4] As explained above, the integrability condition of their KSE places a contraint on the Ricci tensor corresponding to the Einstein’s equations of motion, which then follow automatically from a solution to the KSE. The article goes on to classify the *timelike* solutions of the constructed theory, which turn out to show a four-dimensional hyper-Kähler torsion (HKT) base space dependence.
The work we present here follows similar lines, since we also consider a ‘novel’ KSE (in the sense that such KSE is not related *a priori* to any supersymmetric setting previously treated), and whose relevance becomes apparent once one analyses its integrability condition. Our motivation, however, is different from that of characterisations of solutions to (f)SUGRA theories. We are interested in classifying Lorentzian Einstein-Weyl spaces of arbitrary dimension, and the KSE is chosen in such a way that the integrability condition resembles the geometric constraint for a manifold to be Einstein-Weyl. The tools we will use for this work are the same ones as used in the programme of classification of solutions to supergravity theories, and we will split the problem at hand according to whether they employ a timelike or null vector field. The characterisation we give is of those EW spaces that arise from the existence of a *Killing spinor* [*i.e.*]{} a spinor that fulfills the KSE we propose, and it is in that sense that we refer to them as supersymmetric.
Section (\[sec:maths\]) introduces the spinorial rule, its integrability condition (which resembles the geometric constraint for Einstein-Weyl spaces) and a short manipulation on a vector bilinear valid for all dimensions and cases. Section (\[sec:timelike\]) analyses all possible timelike cases, showing their triviality. Section (\[sec:N1D4\]) describes the [*null*]{} solutions for the $N=1$, $n=4$ case, while section (\[sec:D6chiral\]) treats the $n=6$ null case and section (\[sec:general\]) the remaining cases. Section (\[sec:conclusions\]) recapitulates the work done. Three appendices are presented at the end for reference and completeness. Appendix (\[appsec:Weyl\]) gives some basic knowledge (by no means exhaustive) of Weyl geometry and Einstein-Weyl spaces. Appendix (\[appsec:spinors\]) presents the spinorial notation we use in the article. Appendix (\[appsec:Kundt\]) gives the geometrical description for Kundt waves, as they turn out to be relevant.
Covariant rule and the Einstein-Weyl condition {#sec:maths}
==============================================
Consider the following rule for the covariant derivative of some spinor, which we shall take to be Dirac, $$\label{eq:1}
\nabla_{a}\epsilon \; =\; \textstyle{\frac{4-n}{4}}\ A_{a}\epsilon \;
+\; \textstyle{1\over 2} \gamma_{ab}A^{b}\epsilon \; ,$$ where $n$ is the number of spacetime dimensions and $A$ is just some real 1-form, which at this point is completely unconstrained. We will call the solutions $\epsilon$ of this equation Killing spinors and the corresponding metric and 1-form, a supersymmetric field configuration. Observe that with our choice of Dirac conjugate, the above rule implies $$\label{eq:38}
\nabla_{a}\overline{\epsilon} \; =\; \textstyle{\frac{4-n}{4}}\
A_{a}\overline{\epsilon} \;
-\; \textstyle{1\over 2}A^{b}\ \overline{\epsilon}\gamma_{ab} \; .$$
A straightforward calculation of the integrability condition leads to $$\label{eq:2}
\textstyle{1\over 2} \gamma_{a}\slashed{F}\ \epsilon \; =\; \textstyle{1\over 2}\ \mathrm{W}_{(ab)}\gamma^{b}\epsilon \; ,$$ where $F\equiv dA$ is called the Faraday tensor and $$\label{eq:14}
\mathrm{W}_{(ab)} \ =\ \mathtt{R}(g)_{ab}
\ -\ (n-2) \nabla_{(a}A_{b)}
\ -\ (n-2)\ A_{a}A_{b}
\ -\ g_{ab}\ \left[
\nabla_{c}A^{c}
\ -\ (n-2)\ A_{c}A^{c}
\right] \; ,$$ which is readily identified with (the symmetric part of) the Ricci tensor in Weyl geometry (see appendix \[appsec:Weyl\] for a small introduction).
Contracting the above integrability condition with $\gamma^{a}$ one finds that $$\label{eq:3}
n\ \slashed{F}\epsilon \; =\; \mathrm{W}\ \epsilon \; ,$$ which when combined with eq. (\[eq:2\]) leads to $$\label{eq:4}
\textstyle{1\over 2}\left[\
\mathrm{W}_{(ab)} \; -\; \textstyle{1\over n}\mathrm{W}g_{ab}
\ \right]\ \gamma^{b}\epsilon \; =\; 0\; .$$ In the Riemannian setting the above is enough to conclude that if we find a spinor $\epsilon$ satisfying eq. (\[eq:1\]), then the underlying geometry is Einstein-Weyl. In the non-Riemannian setting this conclusion is not true: experience from the classification of supersymmetric solutions to supergravity theories shows instead that there are two quite different cases to be considered, namely the timelike or the null case. The sexer of these two cases is the norm of a particular vector-bilinear built out of the Killing spinor, which can be shown to be either zero or positive, hence the naming of the cases. The minimal set of equations of motion that need to be imposed in order to guarantee that all EOMs are satisfied, is different in each case: in the timelike case a supersymmetric field configuration automatically satisfies the EW condition, whereas in the null case the minimal set consists of only one component of the EW condition, namely the one lying in the double direction of the null vector-bilinear.
Seeing the similarity of the integrability condition of the spinorial rule with the geometric constraint for EW spaces, it should not come as a surprise that eq. (\[eq:1\]) is invariant under the following Weyl transformations $$\label{eq:16}
\begin{array}{lclclcl}
g & =& e^{2w}\tilde{g} &\hspace{.4cm},\hspace{.4cm}& e^{a} & =& e^{w}\tilde{e}^{a} \; ,\\
A & =& \tilde{A}+dw & ,& \theta_{a} & =& e^{-w}\tilde{\theta}_{a} \; , \\
\epsilon & =& e^{\alpha w}\tilde{\epsilon} & ,& \alpha & =& \textstyle{\frac{4-n}{4}} \; .
\end{array}$$ This Weyl symmetry can in fact be used to obtain the r.h.s. of eq. (\[eq:1\]), which would otherwise have to be wild-guessed: the Weyl connection, eq. (\[eq:W2\]), in the spinorial representation is given by $\mathtt{D}_a=\nabla_a-\frac{1}{2}\gamma_{ab}A^b$ , allowing us to rewrite eq. (\[eq:1\]) as $\mathtt{D}_a\epsilon=\frac{4-n}{4}\,A_a\,\epsilon$. In other words, we are dealing with a weighted Killing spinor in Weyl geometry.
The next step in the analysis is to define the bilinear $\hat{L} =
L_{\mu}dx^{\mu} = \overline{\epsilon}\gamma_{\mu}\epsilon\ dx^{\mu}$, which (as shown in appendix \[appsec:spinors\]) is a real 1-form and, for a Lorentzian spacetime, is either timelike, $g(L,L)>0$ in our conventions, or null, $g(L,L)=0$. Independently of these details, however, we can always derive from the spinorial equation (\[eq:1\]) the following differential rule for the bilinear $$\label{eq:33}
\nabla_{a}L_{b} \; =\ \frac{4-n}{2}\ A_{a}L_{b} \ -\ L_{a}A_{b} \ +\ \imath_{L}A\ g_{ab}\; ,$$ whose totally antisymmetric part reads $$\label{eq:6}
d\hat{L} \; =\; \frac{6-n}{2}\ A\wedge \hat{L} \; ,$$ singling out the $n=6$ case as special, as $\hat{L}$ is then closed.
We shall start the analysis by considering the timelike case.
Timelike solutions {#sec:timelike}
==================
Suppose that $L$ is timelike and define $f\equiv g(L,L)$. We can straightforwardly use eq. (\[eq:33\]) to find $$\label{eq:34}
df \; =\; (4-n)\ A\ f \; ,$$ so that, as long as $n\neq 4$, the Weyl structure is exact and any supersymmetric EW-space is equivalent to a metrical space allowing for a parallel spinor. Bryant [@art:Bryant] has classified all the pseudo-Riemannian spaces admitting covariantly-constant spinors for a different number of dimensions. Then, this prescribes the *timelike* Einstein-Weyl metrics with Lorentzian signature in dimensions three (flat), five and six ($g=\mathfrak{R}^{1,n-5} \times \tilde{g}$, where $\tilde{g}$ is a 4-dimensional Ricci-flat Kähler manifold). A general study for the remaining dimensions is still an open problem, as far as we know. However, Galaev & Leistner [@art:galaev2008] provide a partial answer by giving a blueprint for the geometry of simply-connected, complete Lorentzian spin manifolds that admit a Killing spinor (see theorem 1.3 therein).
For the $n=4$ case, we use the same building blocks as in ref. [@Meessen:2009ma] to set up the whole calculus of spinor bilinears. We deal with the spinor structure of $N=2$ $n=4$ supersymmetry, which allows us to decompose a Dirac spinor in $n=4$ as a sum of two Majorana spinors, which we can project onto the anti-chiral part, denoted $\epsilon_{I}$ ($I=1,2$), and the chiral part, denoted by $\epsilon^{I}$. Here the position of the $I$-index indicates exclusively the chirality, and the chiralities are interchanged by complex conjugation [*i.e.*]{} $(\epsilon_{I})^{*}=\epsilon^{I}$, so the theory has two independent spinors. Doing this decomposition, the rule eq. (\[eq:1\]) can be written as $$\label{eq:35}
\nabla_{a}\epsilon_{I} \; =\; \textstyle{1\over 2}\gamma_{ab}A^{b}\epsilon_{I}
\hspace{.5cm}\mbox{and}\hspace{.5cm}
\nabla_{a}\epsilon^{I} \; =\; \textstyle{1\over 2}\gamma_{ab}A^{b}\epsilon^{I} \; .$$ Using the spinors one can then construct (see ref. [@Meessen:2009ma]) a complex scalar $X\equiv\frac{1}{2}\varepsilon^{IJ}\bar{\epsilon}_I\epsilon_J$, 3 complex 2-forms $\Phi^{x}$ ($x=1,2,3$) that will not play any rôle in what follows, and 4 real 1-forms $V^{a}=i\bar{\epsilon}^I \gamma^a
\epsilon_I$. These 4 1-forms form a linearly independent base and can be used to write the metric, $g$, as $$\label{eq:36}
4|X|^{2}\ g \; =\; \eta_{ab}\ V^{a}\otimes V^{b} \; ,$$ whence $V^{0}\sim L$. Given the definitions of the bilinears we can calculate $$\begin{aligned}
\label{eq:37}
dX & =& 0\; ,\\
\label{eq:37a}
dV^{a} & =& A\wedge V^{a} \; ,\end{aligned}$$ meaning that $X$ is just a complex constant. The integrability condition of eq. (\[eq:37a\]) is $F\wedge V^{a} = 0$ which, due to the linear-independency of the $V^{a}$ implies that $F=0$. Locally, then, we can transform $A$ to zero and introduce coordinates $x^{a}$ such that $V^{a}= 4|X|^{2}\ dx^{a}$, resulting in a Minkowski metric. Whence, in $n=4$ a *timelike* supersymmetric Lorentzian EW space is locally conformal to Minkowski space.\
The conclusion then w.r.t. the *timelike* solutions to the rule (\[eq:1\]) is that they are trivial in the sense that they are always related by a Weyl transformation to a Lorentzian space admitting Killing spinors, [*i.e.*]{} spinors satisfying the rule $\nabla_{a}\epsilon =0$.
The analysis of the null cases is more involved, mainly due to a lack of systematics in the bilinears, the exception being the vector bilinear $L$ as one can see from eq. (\[eq:33\]), but also because the bilinear approach to classification of supersymmetric solutions becomes unwieldy for $n>6$. In stead of attempting to do a complete analysis in all the cases where the bilinear approach can be applied, we shall analyse the cases $n=4$ and $n=6$ explicitly, and then give some generic comments in section (\[sec:general\]).
Null $N=1$ $n=4$ solutions {#sec:N1D4}
==========================
The natural starting point, seeing the explicit case treated in the foregoing section, would be the null case in $n=4$ $N=2$. Prior experience with this case in supergravity, however, shows that this case is related to the simpler case of $n=4$ $N=1$ supergravity [@art:ortin2008], a theory for which the vector bilinear $L$ is automatically a null vector. In $n=4$ $N=1$ sugra the spinor is a Weyl spinor, and one can see that the KSE (\[eq:1\]) is compatible with the truncation of $\epsilon$ to a chiral spinor, and in this section we shall henceforth take $\epsilon$ to be a Weyl spinor.
The first rule we can derive for the bilinear is $$\label{eq:5}
\nabla_{a}L_{b} \; =\; - L_{a}A_{b} \ +\ \imath_{L}A\ g_{ab} \; ,$$ which is already enough to see that $L^{\flat}$ is a geodesic null vector. The antisymmetric and symmetric parts of the above equation read $$\begin{aligned}
\label{eq:7}
d\hat{L} & =& A\wedge \hat{L} \; , \\
\label{eq:8}
\nabla_{(a} L_{b)} & =& - A_{(a}L_{b)} \; +\; \textstyle{1\over 3}\ \nabla\cdot L\ g_{ab}\; .\end{aligned}$$
There is another bilinear that can be constructed [@art:ortin2008], which is a 2-form defined as $\Phi_{ab}=\overline{\epsilon}\gamma_{ab}\epsilon$ and using the propagation rule we can deduce $$\label{eq:17}
\nabla_{a}\Phi_{bc} \; =\; 2\Phi_{a[b}A_{c]} \ -\ 2 g_{a[b}\Phi_{c]d}A^{d} \; ,$$ which through antisymmetrisation gives rise to $$\label{eq:18}
d\Phi \; =\; 2A\wedge\Phi \; .$$
Eq. (\[eq:7\]) implies that $\hat{L}\wedge d\hat{L} =0$, whence $\hat{L}$ is hypersurface orthogonal, and we can use the Frobenius theorem to introduce two real functions $u$ and $P$ such that $\hat{L}=e^{P}du$. Since by eq. (\[eq:7\]) above $\hat{L}$ has gauge charge 1 under $A$, we can perform a Weyl-gauge transformation to take $P=0$, as to obtain $\hat{L}=du$. This further implies that $A=\Upsilon\ \hat{L}$, where $\Upsilon$ is a real function whose coordinate dependence needs to be deduced, and also $\imath_{L}A=0$. Furthermore, we see that $d^{\dagger} \hat{L}=0$ and $\nabla_{L}L=0$, [*i.e.*]{} $L$ is the tangent vector to an affinely parametrised null geodesic.
Observe that we can apply the same reasoning for eq. (\[eq:6\]) in dimensions different from six: as long as $n\neq 6$ we can always use a Weyl transformation as to fix $\hat{L}=du$ and write $A=\Upsilon\ \hat{L}$. The fact that in the case $n=6$ the 1-form $\hat{L}$ is automatically closed has profound implications, as will be shown in section (\[sec:D6chiral\]).
Having fixed the Weyl symmetry, we can introduce a normalised null tetrad [@Penrose:1985jw] and a corresponding coordinate representation by $$\label{eq:19}
\begin{array}{lclclcl}
\hat{L}
& = &
du &\hspace{.4cm},\hspace{.4cm}& L & =& \partial_{v} \; , \\
\hat{N}
& = &
dv + Hdu +\varpi dz + \bar{\varpi}d\bar{z} & ,& N & =& \partial_{u}\ -\ H\partial_{v} \; , \\
\hat{M}
& = &
Udz & ,& M & =& -\bar{U}^{-1}\left( \partial_{\bar{z}} \ -\ \bar{\varpi}\partial_{v}\right) \; ,\\
\hat{\overline{M}}
& = &
\bar{U}d\bar{z} & ,& \overline{M} & =& -U^{-1}\left( \partial_{z} \ -\ \varpi\partial_{v}\right) \; ,
\end{array}$$ for which the metric reads $$\begin{aligned}
\label{eq:20}
g & = &
\hat{L}\otimes \hat{N} \ +\ \hat{N}\otimes \hat{L} \ -\
\hat{M}\otimes\hat{\overline{M}} \ -\ \hat{\overline{M}}\otimes \hat{M}
\hspace{1cm}\longrightarrow
\nonumber \\
ds^{2} & = &
2du\left( dv + Hdu + \varpi dz + \bar{\varpi}d\bar{z}\right) \ -\ 2|U|^{2} dzd\bar{z} \; .\end{aligned}$$ A straightforward calculation shows that the constraint (\[eq:8\]) implies that $$\label{eq:21}
\Upsilon = -\partial_{v}H \;\;\; ,\;\;\;
\partial_{v}\varpi \ =\ 0 \;\;\; ,\;\;\;
\partial_{v}\bar{\varpi} \ =\ 0 \;\;\; ,\;\;\;
\partial_{v}|U|^{2} \ =\ 0 \; ,$$ so that the only $v$-dependence resides in the function $H$, and we determined the gauge field $A$ in terms of $H$.
In $N=1$ $n=4$ one can see that $\Phi = \hat{L}\wedge\hat{\overline{M}}$ (see [*e.g.*]{} [@Meessen:2009ma eq. (70)]). Combining this with eq. (\[eq:18\]) we see that $$\label{eq:22}
0 \ =\ \hat{L}\wedge d\hat{\overline{M}} \ =\
d\bar{U}\wedge d\bar{z}\wedge du \hspace{1cm}\mbox{whence}\;\;
\bar{U} \ =\ \bar{U}(u,\bar{z}) \; .$$ This result means that we can take $U=1$ by a suitable coordinate transformation $Z=Z(u,z)$ such that $\partial_{z}Z = U$, which leaves the chosen form of the metric invariant.
In order to finish the analysis, let us investigate eq. (\[eq:17\]). As $A\sim \hat{L}$ we have that $\imath_{A}\Phi \sim \imath_{L}\Phi = 0$ and we find that $\nabla_{a}\Phi_{bc} = 2\Upsilon\ \Phi_{a[b}L_{c]}$. Combining this with $\Phi_{ab} = 2L_{[a}\overline{M}_{b]}$ we find that $$\label{eq:23}
0 \; =\; L_{[b|}\nabla_{a}\overline{M}_{|c]} \; ,$$ which can be evaluated on the chosen coordinate basis to give $$\label{eq:24}
0\; =\; \partial_{\bar{z}}\varpi \ -\ \partial_{z}\bar{\varpi}
\hspace{1cm}\mbox{which implies:}\;\;
\varpi \ =\ \partial_{z}B \; ,\; \bar{\varpi}\ =\ \partial_{\bar{z}}B \; ,$$ where $B$ is a real function. As is well-known, one can then get rid of $\varpi$ altogether by a suitable shift of the coordinate $v \rightarrow v-B$.
The end result of this analysis is that, given the fact that the spinor $\epsilon$ is taken to be a Weyl spinor, any solution[^5] to the equation (\[eq:1\]) is related by a Weyl transformation to $$\begin{aligned}
\label{eq:10}
ds^{2}_{(4)} & = &
2du\left( dv \ +\ Hdu\right) \; -\; 2dzd\bar{z} \; , \\
A & = &
-\partial_{v}H\ du \; ,\end{aligned}$$ Actually, this metric is a special case of a more-general metric, referred to as a Kundt metric in the physics literature (see appendix (\[appsec:Kundt\]) for more information), a type of metric that appears naturally in the null case of not only supergravity [@Brannlund:2008zf] solutions, but also fake supergravity solutions[, see [*e.g.*]{} refs. [@Meessen:2009ma; @Gutowski:2009vb] and [@Grover:2009ms]]{}.
At this point we would like to recall what was mentioned in section (\[sec:maths\]) above about pseudo-Riemannian signatures and certain EOMs (the EW conditions in this case) not having to be explicitly checked. Since we are trying to give a prescription for EW spaces, we obviously need to satisfy eq. (\[eq:W5\]). An explicit calculation shows that the integrability conditions (\[eq:4\]) are automatically satisfied, with the only non-trivial component being $\mathrm{W}(N,N)\
L_{c}\gamma^{c}\epsilon$. Adapting the Fierz identities to the null case scenario, one obtains the constraint $L_{c}\gamma^{c}\epsilon =0$ (see [ *e.g.*]{} eq. (5.12) of ref. [@Bellorin:2005]), satisfying this way the integrability condition, and hence we see that we have a solution to the KSE.
However, we still need to ensure that the local geometry (\[eq:10\]) indeed solves all EW conditions (\[eq:W5\]), and we must therefore impose by hand that $\mathrm{W}(N,N)=0$. A small calculation shows that this implies that $H$ must satisfy the following differential equation $$\label{eq:11}
\partial_{u}\partial_{v}H \ -\ H\partial_{v}^{2}H \; =\; \partial\bar{\partial}H \; .$$ We can find a four-dimensional generalisation of the Weyl-scalar-flat EW geometry obtained by Calderbank & Dunajski in [@art:Calderbank01SFlat] by using a function $H$ of the form $$\label{eq:12b}
H\ =\ v\partial F +v\bar{\partial} \bar{F}+\bar{z}\partial_u F +z \partial_u \bar{F} \qquad \text{where } F=F(u,z)\ .$$ It gives rise to a non-trivial EW space as long as $\partial^2 F\neq 0$.
Null $N=(1,0)$ $n=6$ solutions {#sec:D6chiral}
==============================
As in the foregoing section we will consider the spinor $\epsilon$ to be chiral which not only implies that the vector bilinear is null, but also that we can use the results of Gutowski [*et al.*]{} [@Gutowski:2003rg], who classified the supersymmetric solutions of ungauged chiral supergravity in 6 dimensions, [*i.e.*]{} minimal $n=6$ $N=(1,0)$ supergravity. This theory is in itself quite curious, and so are the spinor bilinears: there is only a null vector $L$ and a triplet of selfdual 3-forms $\Phi^{r}_{(3)}$ ($r=1,2,3$). These bilinears are defined by $$\label{eq:40}
\begin{array}{lclclcl}
L_{a} & \equiv&
-\varepsilon^{IJ}\ \epsilon_{I}^{c}\gamma_{a}\epsilon_{J} &\;\;\; , \;\;\;&
\epsilon_{I}^{c}\gamma_{a}\epsilon_{J} & =&
-\textstyle{1\over 2}\ \varepsilon_{IJ}\ L_{a} \; ,\\
&& && && \\
\Phi^{r}_{abc} &\equiv& i\left[\sigma^{r}\right]^{IJ}\
\epsilon_{I}^{c}\gamma_{abc}\epsilon_{J} & ,&
\epsilon_{I}^{c}\gamma_{abc}\epsilon_{J} & =&
\textstyle{i\over 2}\ \left[\sigma^{r}\right]_{IJ}\ \Phi^{r}_{abc} \; ,
\end{array}$$ where $\epsilon^{c}=\epsilon^{T}\mathcal{C}$ means the Majorana conjugate.
These bilinears satisfy the following Fierz-relations $$\begin{aligned}
\label{eq:39}
L_{a}L^{a} & =& 0 \; , \\
\label{eq:39a}
\imath_{L}\Phi^{r}_{(3)} & =& 0 \hspace{.5cm}\longrightarrow\hspace{.5cm} \hat{L}\wedge\Phi^{r}_{(3)} \ =\ 0 \; , \\
\label{eq:39b}
\Phi^{r\ fab}\Phi^{s}_{fcd} & =& 4\delta^{rs}\ L^{[a}L_{[c}\ \eta^{b]}_{d]}
\ -\ \varepsilon^{rst}L^{[a|}\Phi^{t\ |b]}{}_{cd}
\ +\ \varepsilon^{rst}L_{[c}\Phi^{t\ ab}{}_{d]} \; .\end{aligned}$$ Seeing eqs. (\[eq:39a\]) and (\[eq:39b\]) we find that $\Phi^{r}_{(3)} =
\hat{L}\wedge\mathsf{K}^{r}_{(2)}$ with $\imath_{L}\mathsf{K}_{(2)}^{r}=0$.
Using the definitions of the bilinears we can use the rule eq. (\[eq:1\]) to calculate the effect of parallel-transporting them. The results is that for an arbitrary vector field $X$ we have $$\begin{aligned}
\label{eq:43}
\nabla_{X}\hat{L} & = &
-\imath_{X}A\ \hat{L} \ -\ \imath_{X}\hat{L}\ A \ +\ \imath_{L}A\ \hat{X} \; ,\\
\label{eq:43a}
\nabla_{X}\Phi^{r} & = &
-\imath_{X}A\ \Phi^{r} \ +\ \hat{X}\wedge\imath_{A^{\flat}}\Phi^{r} \ -\ A\wedge\imath_{X}\Phi^{r} \; ,\end{aligned}$$ From eq. (\[eq:43\]) it is clear that $L$ is a null geodesic, [*i.e.*]{} $\nabla_{L}L=0$, and, as we already knew from (\[eq:6\]), $d\hat{L}=0$.
At this point then we can, as before, introduce a Vielbein adapted to the null nature of $L$ in terms of the natural coordinates $v$, $u$ and $y^{m}$ ($m=1,\ldots ,4$) as $$\label{eq:KundtCurv1a}
\begin{array}{lclclcl}
E^{+}
& = &
du &\hspace{.4cm},\hspace{.4cm}& \theta_{+} & =& \partial_{u} \ -\
H \partial_{v} \; ,
\\
E^{-}
& = &
dv + Hdu + S_{m}dy^{m} & ,& \theta_{-} & =& \partial_{v} \; ,
\\
E^{i}
& = & {e_{m}}^{i}\ dy^{m} & ,& \theta_{i} & =& {e_{i}}^{m}\left[ \partial_{m} \ -\ S_{m}\partial_{v}\right] \; ,
\end{array}$$ where $\hat{L}\equiv E^{+}$ and $L\equiv \theta_{-}$. As usual we can then define the metric on the base space by $\mathsf{h}_{mn}\equiv
{e_{m}}^{i}{e_{n}}^{i}$ and we can write the full 6-dimensional Kundt metric as $$\label{eq:44}
ds_{(6)}^{2} \; =\; 2du\left( dv\ +\ Hdu\ +\ \hat{S}\right) \ -\ \mathsf{h}_{mn}\,dy^{m}dy^{n} \; .$$
We can expand the 2-forms as $2\ \mathsf{K}^{r}\equiv \mathsf{K}^{r}_{ij}
E^{i}\wedge E^{j}$ w.r.t. the above Vielbein, and by choosing the light-cone directions such that $\varepsilon^{+-1234}=1=\varepsilon^{1234}$, we see that $\star_{(4)}\mathsf{K}^{r} = -\mathsf{K}^{r}$. Defining the $(1,1)$-tensors $\mathsf{J}^{r}$ by means of $\mathsf{h}(\mathsf{J}^{r}X,Y)\equiv
\mathsf{K}^{r}(X,Y)$, we can see that eq. (\[eq:39b\]) implies $$\label{eq:41}
\mathsf{J}^{r}\mathsf{J}^{s} \; =\; -\delta^{rs} \, +\, \varepsilon^{rst}\ \mathsf{J}^{t} \; ,$$ so that the 4-dimensional base space is always going to be an [*almost quaternionic manifold*]{}.
At this point we will fix part of the Weyl gauge symmetry by imposing the gauge-fixing condition $\imath_{L}A=0$ and consequently we can expand the gauge field as $$\label{eq:45}
A \; =\; \Upsilon\ \hat{L} \; +\; \mathsf{A}_{m}dy^{m} \; .$$ Using this expansion and the explicit form of the Vielbein in terms of the coordinates, we can analyse eq. (\[eq:43\]), resulting in $$\begin{aligned}
\label{eq:46}
\Upsilon & =& -\textstyle{1\over 2}\ \partial_{v}H \; , \\
\label{eq:46a}
\partial_{v}\hat{S} & =& -2\ \mathsf{A}\; ,\\
\label{eq:46b}
0 & =& \partial_{v}\mathsf{h}_{mn} \; .\end{aligned}$$ Contrary to what is usually the case in (fake) supergravities, we do not know the full $v$-dependence of $H$ and therefore we cannot completely fix the $v$-dependence of the unknowns. The above results comprise all the information contained in eq. (\[eq:43\]).\
In order to analyse the content of eq. (\[eq:43a\]) we first take $X=L$ to find that $\nabla_{L}\Phi^{r}=0$, which when evaluated in the chosen coordinate system implies $\partial_{v}\mathsf{K}^{r}_{mn}=0$. This innocuous result fixes, however, the $v$-dependence of $\mathsf{A}$: from the totally antisymmetric part of eq. (\[eq:43a\]) one obtains $$\label{eq:47}
d\Phi^{r}\; =\; 2A\wedge\Phi^{r} \;\longrightarrow\;
0\ =\ \hat{L}\wedge\left[ \mathsf{d}\mathsf{K}^{r}\ -\
2\mathsf{A}\wedge\mathsf{K}^{r}\ \right] \; ,$$ where we introduced the exterior derivative on the base space $\mathsf{d}\equiv dy^{m}\partial_{m}$. As the $\mathrm{K}$s are $v$-independent and $\hat{L}=du$, we see that the consistency of the above equation requires $\mathsf{A}$ to be $v$-independent. Then, we also obtain from eq. (\[eq:46a\]) that $$\label{eq:48}
\hat{S} \; =\; -2v\ \mathsf{A} \, +\, \varpi \hspace{2cm}(\partial_{v}\varpi_{m}=0 )\; .$$ It should be clear from eq. (\[eq:47\]) that the $y$-dependence of the $\mathrm{K}$s is given by the equation $$\label{eq:49}
\mathsf{d}\mathsf{K}^{r}\ =\ 2\mathsf{A}\wedge\mathsf{K}^{r}
\hspace{1cm}\mbox{whose integrability condition reads}\hspace{1cm}
\mathsf{F}\wedge\mathsf{K}^{r} \ =\ 0\; ,$$ where we defined $\mathsf{F}=\mathsf{d}\mathsf{A}$. Actually, the last equation implies, as one can easily verify, that $\mathsf{F}$ is selfdual, [*i.e.*]{} $\star_{(4)}\mathsf{F}=\mathsf{F}$, whence $\mathsf{A}$ is a selfdual connection or in physics-speak an $\mathbb{R}$-instanton.
The analysis of eq. (\[eq:43a\]) in the direction $X=\theta_{+}$ is straightforward and leads to the following constraints on the spin connection $$\begin{aligned}
\label{eq:51}
\omega_{+-k}\ \mathsf{K}^{r}_{kj}
& = &
-\mathsf{A}_{k}\ \mathsf{K}^{r}_{kj} \; ,
\\
\label{eq:51a}
0 & = &
\omega_{+i}{}^{k}\ \mathrm{K}^{r}_{kj} \; +\; \omega_{+j}{}^{k}\ \mathrm{K}^{r}_{ik} \; .\end{aligned}$$ By using the results in appendix \[appsec:Kundt\], we see that eq. (\[eq:51\]) is automatically satisfied. A small investigation in eq. (\[eq:51a\]) shows that it implies the base space 2-form $\omega_{+ij}\
E^{i}\wedge E^{j}$ to be selfdual! Coupling this observation with eq. (\[eq:KundtCurv2d\]) and taking into account $\mathsf{F}$’s selfduality, we see that the base space 2-form $2\Omega = \Omega_{ij}\ E^{i}\wedge E^{j}$, whose components are defined by $$\label{eq:52}
\Omega_{ij}\; \equiv\; 2\mathsf{D}_{[i}\varpi_{j]} \ +\ 2e_{[i}{}^{m}\partial_{u}e_{j]m}
\hspace{2cm}(\mbox{where:}\;\;
\mathsf{D}\varpi \equiv \mathsf{d}\varpi\ -\ 2\mathsf{A}\wedge\varpi )\; ,$$ has to be selfdual, [*i.e.*]{} $\star_{(4)}\Omega = \Omega$.
In order to completely drain eq. (\[eq:43a\]) of information we need to consider $X$ lying on the base space. Let $\mathsf{X}$ be such a vector. Then, we find that $$\label{eq:50}
\nabla^{(\lambda )}_{\mathsf{X}}\ \mathrm{K}^{r} \; =\;
\mathsf{X}^{\sharp}\wedge\ \star_{(4)}\left[\ \mathsf{A}\wedge\mathsf{K}^{r}\ \right]
\ -\ \mathsf{A}\wedge \imath_{\mathsf{X}}\mathsf{K}^{r} \; ,$$ where $\nabla^{(\lambda )}$ is the ordinary spin connection on the base space using the $\lambda$s in eq. (\[eq:KundtCurv2d\]). Following ref. [@Grover:2008jr] we can then introduce a torsionful connection $\overline{\nabla}_{\mathsf{X}}\mathsf{Y}\equiv \nabla^{(\lambda
)}_{\mathsf{X}}\mathsf{Y}\ -\ \mathsf{S} _{\mathsf{X}}\mathsf{Y}$ with the torsion being totally antisymmetric and proportional to the Hodge dual of the $\mathbb{R}$-gauge field, [*i.e.*]{} $$\label{eq:53}
\mathsf{h}\left( \mathsf{S}_{\mathsf{X}}\mathsf{Y},\mathsf{Z}\right) \;
\equiv\;
-\left[\star_{(4)}\mathsf{A}\right]\ \left( \mathsf{X},\mathsf{Y},\mathsf{Z}\right) \; ,$$ such that eq. (\[eq:50\]) can be written compactly as $\overline{\nabla}\mathsf{K}^{r}=0$. Almost quaternionic manifolds admitting a torsionful connection parallelising the almost quaternionic structure are called [*Hyper-Kähler Torsion manifolds*]{}, HKT manifolds for short, a name that first appeared in [@Howe:1996] to describe the geometry of supersymmetric sigma-model manifolds with torsion [@Gates:1984].
As pointed out in ref. [@Grover:2008jr], we can make use of the residual Weyl symmetry in eq. (\[eq:16\]) with $w=w(y)$, [*i.e.*]{} a Weyl transformation depending only on the coordinates of the base space, to gauge-fix the condition $\mathsf{d}^{\dagger}\mathsf{A}=0$. This immediately implies that the torsion $\mathsf{S}$ is closed, and the resulting mathematical 4-dimensional structure is called a [*closed HKT manifold*]{}. Let us mention, even though it will not be needed, that the coordinate transformation $v\rightarrow v+\Lambda (y)$, induces the ’gauge’ transformation $\varpi\rightarrow \varpi + \mathsf{D}\Lambda$.\
Thus far, the analysis has shown that the pair $(g,A)$ admits a solution to eq. (\[eq:1\]) iff $g$ is the metric of a Kundt wave whose base space is a $u$-dependent family of HKT-spaces. Given such a family of HKT spaces we can find the 1-form $\varpi$ by imposing selfduality of the 2-form $\Omega$ in eq. (\[eq:52\]) and then the only indeterminate element of the metric is the wave profile $H$. This analysis has given us the necessary conditions for the existence of a non-null spinor satisfying eq. (\[eq:1\]). It remains to be checked that they are also sufficient by direct substitution into eq. (\[eq:1\]).
A quick calculation of the $(-)$ component, leads to $\theta_{-}\epsilon =0$, whence the spinor is $v$-independent. The $(+)$-component leads, after using the constraint $\gamma^{+}\epsilon =0$, to $$\label{eq:25}
\partial_{u}\epsilon \; =\; -\textstyle{1\over 4}\ T_{ij}\ \gamma^{ij}\epsilon \; =\; 0 \; ,$$ where the last step follows from the selfduality of $T$ (see eq. (\[eq:9\])) and the chirality of the spinor $\epsilon$. We conclude that the spinor is also $u$-independent. Giving the $i$ components of eq. (\[eq:1\]) a similar treatment we end up with $$\label{eq:55}
\nabla_{i}^{(\lambda )}\epsilon \; =\; \textstyle{1\over 2}\ \tilde{\gamma}_{ij}\ \mathsf{A}_{j}\epsilon \; ,$$ where we have defined $\tilde{\gamma}^{i}\equiv i\gamma^{i}$, so $\{\tilde{\gamma}^{i},\tilde{\gamma}^{j}\} = 2\delta^{ij}$, in order to obtain a purely Riemannian spinorial equation.
As one can readily see from eq. (\[eq:1\]), the above equation is nothing more than its Riemannian version for four-dimensional spaces: this kind of spinorial equations was studied by Moroianu in ref. [@art:moroianu1996a] who investigated Riemannian Weyl geometries admitting spinor fields parallel w.r.t. the Weyl connection. For $n\neq 4$ he found that any such Weyl structure was closed, whereas in $n=4$ he found the HKT structure outlined above. Furthermore, he showed that, if the 4-dimensional space is compact, then the HKT structure is conformally related to either a flat torus, a K3 manifold or the Hopf surface $S^{1}\times S^{3}$ with the standard, locally flat metric (see [*e.g.*]{} [@Gibbons:1997iy]).
The integrability condition of eq. (\[eq:55\]) implies that the Ricci tensor of the metric $\mathsf{h}$ has to satisfy $$\label{eq:58}
\mathsf{R}(\mathsf{h})_{ij} \; =\; 2\nabla^{(\lambda )}_{(i}\mathsf{A}_{j)} \ +\ 2\mathsf{A}_{i}\mathsf{A}_{j}
\ +\ \mathsf{h}_{ij}\left(\ \nabla^{(\lambda )}_{i}\mathsf{A}_{i}\ -\ 2\mathsf{A}^{2}\ \right) \; ,$$ which, by comparison with eq. (\[eq:14\]), is equivalent to saying that the pair $(\mathsf{h},\mathsf{A})$ forms a Ricci-flat Weyl geometry [*i.e.*]{} $\mathrm{W}_{(ij)}=0$.\
As we did in section (\[sec:N1D4\]), we impose the Einstein-Weyl equations in those directions in which it is not trivially satisfied, [*i.e.*]{} in the ($++$)-direction. This, in turn, fixes the function $H$, which was otherwise unknown. At this point, however, we would like to impose the simplifying restriction that the HKT structure on the base space does not depend on $u$. The motivation for this simplifying adjustment has to do with the difficulty of finding analytic solutions to the differential equation resulting from a $u$-dependent base space. A calculation of the $(++)$-components of the E-W equations then shows that $$\label{eq:59}
2\theta_{+}\theta_{-}H \ +\ \left(\theta_{-}H\right)^{2} \; =\;
\left(\ \nabla^{(\lambda )}_{i} - S_{i}\theta_{-} - 4\mathsf{A}_{i}\ \right)
\left(\ \partial_{i} - S_{i}\theta_{-} -2\mathsf{A}_{i}\ \right)\ H \; ,$$ where we have allow for a $u$-dependence of $H$.
To summarise, any solution to the $N=(1,0)$ $n=6$ null scenario is once again prescribed by a Kundt wave of the form eq. (\[eq:44\]) constrained by eqs. (\[eq:48\]), (\[eq:52\]) and (\[eq:59\]), whose 4-dimensional base space is given by a $v$-independent, closed HKT manifold subject to eqs. (\[eq:58\]), and the gauge connection being that of an $\mathbb{R}$-instanton.
Remaining null cases {#sec:general}
====================
Having treated the null cases in $n=4$ and $n=6$, we are ready to make some general comments on the null case in other dimensions. First of all, as was pointed out in section (\[sec:N1D4\]), as long as $n\neq 6$ we can use a Weyl transformation to introduce a coordinate $u$ such that $\hat{L}=du$ and then also $A=\Upsilon \hat{L}$. Choosing the coordinate $v$ to be aligned with the flow of $L\ (=\partial_{v})$, we can introduce the base space coordinates $y^{m}$ ($m=1,\ldots , n-2$) and a Vielbein similar to the one in eq. (\[eq:KundtCurv1\]), so that the metric is always of the form $$\label{eq:15}
ds^{2}_{(n)} \; =\; 2du\left( dv \ +\ Hdu\ +\ S_{m}dy^{m}\right) \; -\;
\mathsf{h}_{mn}\,dy^{m}dy^{n}\ ,$$ where $\mathsf{h}_{mn}\equiv e^i_m e^i_n$. This is again a Kundt metric, and evaluating the symmetric part of eq. (\[eq:33\]) in this coordinate system, we get the following restrictions $$\label{eq:26}
\Upsilon \ =\ -\frac{2}{n-2}\ \partial_{v}H
\;\; ,\;\; \partial_{v}S_{m} \ =\ 0
\;\; ,\;\; \partial_{v}\mathsf{h}_{mn} \ =\ 0 \; ,$$ so that the whole $v$-dependence resides in $H$ and $\Upsilon$ only. Following the convention in section (\[sec:D6chiral\]), we shall call the $v$-independent part of $\hat{S}$ by $\varpi$, so that in the $n\neq 6$ case we have $\hat{S}=\varpi$.
With this information, and the constraint of $u$-independence imposed, we can proceed to analyse the spinorial rule. The KSE in the $v$-direction is automatically satisfied ([*i.e.*]{} $\partial_v \epsilon=0$) and the remaining directions are $$\begin{aligned}
\label{eq:56}
0 & =& \nabla_{i}^{(\lambda )}\ \epsilon \; , \\
\label{eq:56a}
\partial_{u}\epsilon & =& \textstyle{1\over 8}\ \left[ \mathsf{d}\varpi\right]_{ij}\gamma^{ij}\epsilon \; .\end{aligned}$$ Eq. (\[eq:56\]) clearly states that the base space must be a Riemannian manifold of special holonomy. The integrability condition of the above two equations then is that $$\label{eq:54}
0\; =\; \left[ \nabla^{(\lambda )}_{i} (\mathsf{d}\varpi )_{kl}\right]\ \gamma^{kl}\epsilon
\hspace{.4cm}\mbox{which implies}\hspace{.4cm}
\left[ \mathsf{d}^{\dagger}\mathsf{d}\varpi\right]_{i}\gamma^{i}\epsilon \ =\ 0 \; ,$$ so that $\mathsf{d}^{\dagger}\mathsf{d}\varpi =0$.[^6] Using the coordinate transformation $v\rightarrow v +\Lambda
(y)$ we can always take $\mathsf{d}^{\dagger}\varpi =0$, whence $\varpi\in
\mathrm{Harm}^{1}(\mathcal{B})$, [*i.e.*]{} $\varpi$ is a harmonic 1-form on the base space.[^7]
Given this input, the condition for such a pair $(g,A)$ to be an Einstein-Weyl manifold is $$\label{eq:57}
2\partial_{u}\partial_{v}H \ -\
2H\partial_{v}^{2}H \ +\
2\textstyle{n-4\over n-2}\ \left(\partial_{v}H\right)^{2} \; =\;
-\left( \nabla^{(\lambda )} - \varpi\right)^{i}\ \theta_{i}H \; .$$ The factor on the r.h.s. of the above equations becomes, in the $\varpi =0$ limit, the d’Alembertian on the base space, and we make contact with eq. (\[eq:11\]). This shows that the $n=4$ case is a subcase of the general one studied in this section, where one was allowed to use the 2-form $\Phi$ to get rid of $\hat{S}$. $n=6$, however, is an independent case where the characteristic behaviour of the theory in that dimension (see [*e.g.*]{} eq. (\[eq:6\])) nurtures the HKT structure.
Summary and conclusions {#sec:conclusions}
=======================
In this work we have presented a characterisation of supersymmetric Einstein-Weyl spaces with Lorentzian signature in $n$ arbitrary dimensions. We have done this by making use of the techniques developed for the classification of supergravity solutions. In particular, we assumed the existence of a spinor $\epsilon$ satisfying eq. (\[eq:1\]). It is in this sense that our solutions have a supersymmetric character. We then proceeded to build and analyse the bilinears that can be constructed from $\epsilon$, which shape the resulting geometry.
We have found that (for most dimensions) those spaces arising from a vector bilinear which is timelike are trivial, in the sense that they are conformally related to a space admitting a Killing spinor. The odd duck in the pond is the 4-dimensional case, for which the only *timelike* solution actually turns out to be Minkowski space, which coincides with which was already know for parallel spinors [@art:Bryant]. The null case solutions are given by a Kundt metric and a prescribed Weyl gauge field. It is worth mentioning that the special structure of the $n=6$ case determines that the base space is given by a closed [*Hyper-Kähler Torsion*]{} manifold.
As a closing paragraph let us consider the case $n=3$: in that case one can see that eq. (\[eq:57\]), once one takes into account the fact that one perform coordinate transformations such that $\mathsf{h}=1$ and $\varpi =0$, corresponds to the dispersionless Kadomtsev-Petviashvili equation. As shown in ref. [@boek:Dunajski sec. 10.3.1.3], the thus obtained class of 3-dimensional EW spaces is the unique class of 3-dimensional EW spaces of Lorentzian signature admitting a weighted covariantly constant null vector. Furthermore, the supersymmetric class can be obtained by the Jones-Tod construction on a conformal space of neutral signature admitting an anti-selfdual Null-Kähler structure [@boek:Dunajski], a geometric structure which admits a parallel spinor. Evidently, there are n-dimensional EW spaces, as there are 3-dimensional examples, that are not supersymmetric, and it would be interesting to get a better handle on them.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported in part by the Spanish Ministry of Science and Education grant FPA2009-07692, a C.S.I.C. scholarship JAEPre-07-00176, a Ramón y Cajal fellowship RYC-2009-05014, the Principáu d’Asturies grant IB09- 069, the Comunidad de Madrid grant HEPHACOS S2009ESP-1473, and the Spanish Consolider-Ingenio 2010 program CPAN CSD2007-00042. PM wishes to thank J. de Medinaceli and L. Fernández Seivane for useful discussions and TO wishes to thank M.M. Fernández for her unfaltering support. AP would like to dedicate this work to the memory of his grandfather, Ramón Lozano Cid, who passed away while this article was being completed, and whose great sense of responsability and care for the family -even on his ultimate moments- is something AP would like to live up to.
A short introduction to Einstein-Weyl geometry {#appsec:Weyl}
==============================================
A Weyl manifold is a manifold $\mathcal{M}$ of dimension $n$, a conformal class $[g]$ of metrics on $\mathcal{M}$, and a torsionless connection $\mathtt{D}$, which preserves the conformal class, [*i.e.*]{} $$\label{eq:W1}
\mathtt{D}\ g \; =\; 2 A \otimes g \; ,$$ for a chosen reference $g\in [g]$ and $A \in \Omega(\mathcal{M})$. Using the above definition, we can express the connection $\mathtt{D}_{X}Y$ as $$\label{eq:W2}
\mathtt{D}_{\mu}Y_{\nu} \; =\; \nabla^{g}_{\mu}Y_{\nu} \ +\
\gamma_{\mu\nu}{}^{\rho}\ Y_{\rho} \hspace{.3cm}\mbox{with}\hspace{.3cm}
\gamma_{\mu\nu}{}^{\rho} \ =\
g_{\mu}{}^{\rho}A_{\nu} \ +\ g_{\nu}{}^{\rho}A_{\mu} \ -\ g_{\mu\nu}A^{\rho}
\; ,$$ where $\nabla^{g}$ is the Levi-Cività connection for the chosen $g\in
[g]$. We define the curvature of this connection as usual, [*i.e.*]{} $\left[ \mathtt{D}_{\mu},\mathtt{D}_{\nu}\right] Y_{\rho} =
-\mathtt{W}_{\mu\nu\rho}{}^{\sigma}Y_{\sigma}$, and which we can use to define the associated Ricci curvature as $\mathtt{W}_{\mu\nu} \equiv
\mathtt{W}_{\mu\sigma\nu}{}^{\sigma}$. A calculation shows that the Ricci tensor is not symmetric, which was to be suspected as we have a connection with non-vanishing contorsion, and we have $$\begin{aligned}
\label{eq:W3a}
\mathtt{W}_{[\mu\nu]} & =& -\textstyle{n\over 2}\
F_{\mu\nu}\hspace{2.5cm}\mbox{other notations:}\, F=dA
=\rho^{\mathtt{D}} \; , \\
& & \nonumber \\
\label{eq:W3b}
\mathtt{W}_{(\mu\nu)} & =& \mathtt{R}(g)_{\mu\nu}
\ -\ (n-2) \nabla_{(\mu}A_{\nu)}
\ -\ (n-2)\ A_{\mu}A_{\nu}
\ -\ g_{\mu\nu}\left[
\nabla_{\sigma}A^{\sigma}
\ -\ (n-2)\ A_{\sigma}A^{\sigma}
\right] \; .\end{aligned}$$ The Ricci-scalar is then of course defined as $\mathtt{W}\equiv
\mathtt{W}_{\sigma}{}^{\sigma}$, which explicitly reads $$\label{eq:W4}
\mathtt{W} \; =\; \mathtt{R}(g) \ -\ 2(n-1)\ \nabla_{\sigma}A^{\sigma}
\, +\, (n-1)(n-2)\
A_{\sigma}A^{\sigma} \; .$$ The 1-form $A$ acts as gauge field gauging an $\mathbb{R}$-symmetry, and this is also the reason why we have been talking about a conformal class of metrics on $\mathcal{M}$. In fact, under a transformation $g_{\mu\nu}\rightarrow
e^{2w}\ g_{\mu\nu}$, we have that $A\rightarrow A +dw$ and $\mathtt{W}\rightarrow e^{-2w}\mathtt{W}$, whereas $\mathtt{W}_{\mu\nu\rho}{}^{\sigma}$ and $\mathtt{W}_{\mu\nu}$ are conformally invariant. In this sense, we say that an EW structure is trivial if the field strength $F=dA=0$, [*i.e.*]{} locally the Weyl connection is conformally vanishing.\
A Weyl manifold is said to be Einstein-Weyl if the curvatures satisfy $$\label{eq:W5}
\mathtt{W}_{(\mu\nu)} \; =\; \frac{1}{n}\ g_{\mu\nu}\ \mathtt{W} \; .$$
A metric $g$ in the conformal class is said to be [*standard*]{} or [ *Gauduchon*]{} if it is such that $$\label{eq:W4a}
d\star A \ =\ 0 \hspace{.4cm}\mbox{or equivalently}\hspace{.4cm}
\nabla_{\sigma} A^{\sigma}\ =\ 0\; ,$$ where the $\star$ is taken w.r.t. the chosen metric. Gauduchon [@art:gauduchon1984a] showed that on a compact EW manifold there always exists a standard metric, and Tod [@art:Tod1992] then went on to show that on compact EW manifolds this implies that $A^{\flat}$ is a Killing vector of the metric, [*i.e.*]{} it generates an isometry of $g$.
Spinors in $\mathrm{SO}(1,d-1)$ {#appsec:spinors}
===============================
On $\mathbb{R}^{1,n-1}$ we shall put the mostly negative metric $\eta
=\mathrm{diag}(+,[-]^{n-1})$ and take the $\gamma$-matrices to satisfy $$\label{eq:27}
\left\{\ \gamma_{a},\gamma_{b}\ \right\} \; =\; 2\eta_{ab} \; .$$ We use a unitary representation of the $\gamma$-matrices, which implies that $\gamma_{0}^{\dagger} =\gamma_{0}$ and $\gamma_{i}^{\dagger} =-\gamma_{i}$. Choosing the Dirac conjugation matrix $\mathcal{D}=\gamma_{0}$, we define the Dirac conjugate of a spinor $\psi$ by $\overline{\psi}\equiv \psi^{\dagger}\mathcal{D}$ and find that $$\label{eq:28}
\mathcal{D}\gamma_{a}\mathcal{D}^{-1} \; =\; \gamma_{a}^{\dagger}
\hspace{.5cm}\mbox{and}\hspace{.5cm}
\mathcal{D}\gamma_{ab}\mathcal{D}^{-1} \; =\; -\gamma_{ab}^{\dagger}$$
Defining the 1-form $L=L_{a}\ e^{a}$ by means of $L_{a}\equiv
\overline{\psi}\gamma_{a}\psi$ which is then automatically real: $$\label{eq:29}
L_{a}^{*} \ =\ \overline{\ \overline{\psi}\gamma_{a}\psi\ }
\ =\ \psi^{T} \left(\mathcal{D}\gamma_{a}\right)^{*}\ \psi^{*}
\ =\ \psi^{\dagger} \left(\mathcal{D}\gamma_{a}\right)^{\dagger}\ \psi
\ =\ \overline{\psi}\mathcal{D}^{-1}\gamma_{a}^{\dagger}\mathcal{D}^{\dagger}\psi
\ =\ \overline{\psi}\gamma_{a}\psi \ =\ L_{a} \; ,$$ where a perhaps expected $-1$ sign in the third step is absent as we are dealing with classical (commuting) spinors.
In terms of the components we have that $L_{a}=\epsilon^{\dagger}\mathcal{D}\gamma_{a}\epsilon$ and it is clear that $L_{0}=\epsilon^{\dagger}\epsilon$. Furthermore, we can always rotate the spatial components of $L$ in such a way that only the first component is non-vanishing. This then implies that $$\label{eq:31}
g(L,L) \ =\ L_{0}^{2}\ -\ L_{1}^{2} \; .$$ $L_{1}=\epsilon^{\dagger}\gamma_{01}\epsilon$ and if we combine this with $\gamma_{01}^{\dagger}=\gamma_{01}$, $\gamma_{01}^{2}=1$ and $\mathrm{Tr}(\gamma_{01} )=0$ we can use a $\mathrm{SO}(\lfloor n/2\rfloor )$ rotation to write $\gamma_{01}= \mathrm{diag}([+]^{ \lfloor n/2\rfloor } ,
[-]^{\lfloor n/2\rfloor })$. Decomposing the spinor w.r.t. the structure of $\gamma_{01}$ as $\epsilon^{t}=(v,w)$, where $v$ and $w$ are vectors in $\mathbb{C}^{\lfloor n/2\rfloor }$, we see that $$\label{eq:32}
L_{0} \ =\ |v|^{2}+|w|^{2} \; ,\;
L_{1} \ =\ |v|^{2}-|w|^{2} \;\;\longrightarrow\;\;
g(L,L) \ =\ 4|v|^{2}|w|^{2} \; ,$$ which implies the positive-definiteness of $|L|^2$.\
In the derivation of the spinorial rule eq. (\[eq:1\]) we have not made any particular assumption about the nature of the spinor $\epsilon$ which has been taken to be a (general) plain Dirac spinor. In the construction of the bilinears, however, it is wise to impose a bit more structure on $\epsilon$; this naturally leads one to investigate the compatibility of eq. (\[eq:1\]) with the conditions for the existence of a Weyl, Majorana, Majorana-Weyl [*[&]{}c.*]{} spinor, a question that is answered affirmatively.
Kundt metrics {#appsec:Kundt}
=============
[ A Kundt metric is a type of wave-like metric that allows for an expansion, shear and twist-free geodesic null-vector [@Coley:2005sq] and were first studied in the arbitrary-$n$ case in refs. [@Podolsky:2008ec] and [@art:juist]. The line-element can always be taken to be ]{} $$\label{eq:KundtMetric}
ds^2=\hat{E}^+ \otimes \hat{E}^-+\hat{E}^- \otimes \hat{E}^+ - \hat{E}^x \otimes \hat{E}^x$$ where generically we introduce the light-cone-frame by[^8] $$\label{eq:KundtCurv1}
\left\{
\begin{array}{lclclcl}
E^{+} & =& du &\hspace{.4cm},\hspace{.4cm}& \theta_{+} & =& \partial_{u} \ -\ H \partial_{v} \; , \\
E^{-} & =& dv + Hdu + S_{m}dy^{m} & ,& \theta_{-} & =& \partial_{v} \; ,\\
E^{x} & =& {e_{m}}^{x}\ dy^{m} & ,& \theta_{x} & =& {e_{x}}^{m}\left[ \partial_{m} \ -\ S_{m}\partial_{v}\right] \; ,
\end{array}\right.$$ where the Vielbein on the base space $e_{i}^{x}$ is independent of $v$; the only $v$-dependence resides in $H$ and $\hat{S}\equiv S_{m}dy^{m}$.\
This is the kind of metric that appeared in the characterisations of the null cases above, eqs. (\[eq:44\]) and (\[eq:15\]), where we defined the correspondence between the $(n-2)$-bein and the base space metric as $\mathsf{h}_{mn}\equiv e^i_m e^i_n$.\
Defining the spin-connection $\omega^{a}{}_{b}\equiv E^{c}\
\omega_{c}{}^{a}{}_{b}$ by means of $dE^{a}=\omega^{a}{}_{b}\wedge E^{b}$ and imposing it to be metric compatible $\omega_{(ab)}=0$, leads to $$\begin{aligned}
\label{eq:KundtCurv2a}
\omega_{+-} & =& -\theta_{-}H\ E^{+}
\, -\ \textstyle{1\over 2}\ \theta_{-}S_{x}\ E^{x} \; ,\\
\label{eq:KundtCurv2b}
\omega_{+x} & =& -\left( \theta_{x}H \ -\ e_{x}^{m}\theta_{+}S_{m}\right)\ E^{+}
\, +\ \textstyle{1\over 2}\theta_{-}S_{x}\ E^{-}
\ - \left[\ T_{yx}
+ e_{(y}^{m}\theta_{+}e_{x)m}
\right]\, E^{y}\; ,\\
\label{eq:KundtCurv2c}
\omega_{-x} & =& \textstyle{1\over 2}\theta_{-}S_{x}\ E^{+} \; ,\\
\label{eq:KundtCurv2d}
\omega_{xy} & =& -\lambda_{zxy}\ E^{z}
\ -\ \left[\ T_{xy}
\ -\ e_{[x}^{m}\theta_{+}e_{y]m}
\right]\ E^{+} \; , \end{aligned}$$ where we defined $\mathsf{d} E^{x} = \lambda^{x}{}_{y}\wedge E^{y}$ and also defined $\lambda_{zy}=\delta_{zx}\lambda^{x}{}_{y}$, whereas $\omega_{xy}=\eta_{xz}\omega^{z}{}_{y}$ so that the sign difference is paramount.[^9] Furthermore, we defined $$\label{eq:9}
T_{xy} \; \equiv\; e_{[x}{}^{m}\theta_{y]}S_{m} \hspace{.7cm}\mbox{which for $n=6$ reads:}\hspace{.4cm}
T_{ij} \ =\ v\ \mathsf{F}_{ij} \; -\; \textstyle{1\over 2}\ \left[ \mathsf{D}\varpi\right]_{ij} \; .$$
If we impose that the only $u$-dependency resides in $H$, the non-vanishing components of the Ricci tensor become $$\begin{aligned}
\label{eq:KundtCurv3a}
R_{++} & =& -\nabla^{(\lambda )}_{x}\partial_{x}H
\ +\ \theta_{-}H\ \nabla^{(\lambda )}_{x}S_{x}
\ -\ H\ \nabla^{(\lambda )}_{x}\theta_{-}S_{x} \nonumber \\
& & +\ 2S_{x}\ \partial_{x}\theta_{-}H
\ -\ \theta_{-}S_{x}\ \partial_{x}H
\ -\ S_{x}S_{x}\ \theta_{-}^{2}H \; ,\\
\label{eq:KundtCurv3b}
R_{+-} & =& -\theta_{-}^{2}H\ -\ \textstyle{1\over 2}\theta_{-}S_{x}\theta_{-}S_{x} \ +\ \textstyle{1\over 2} \nabla^{(\lambda )}_{x}\ \theta_{-}S_{x} \; ,\\
\label{eq:KundtCurv3c}
R_{+x} & =& -\theta_{x}\theta_{-}H
\ -\ \nabla^{(\lambda )}_{y}T_{xy}
\ +\ S_{y}\theta_{-}T_{xy}
\ +\ T_{xy}\theta_{-}S_{y} \; , \\
\label{eq:KundtCurv3d}
R_{xy} & =& \mathsf{R}(\lambda )_{xy} \ -\ \nabla^{(\lambda )}_{(x|}\ \theta_{-}S_{|y)} \ +\ \textstyle{1\over 2}\ \theta_{-}S_{x}\theta_{-}S_{y} \; ,\end{aligned}$$ The Ricci scalar is then given by $$\label{eq:KundtCurv4}
R \; =\; -2\theta_{-}^{2}H
\ -\ \textstyle{3\over 2}\ \theta_{-}S_{x}\theta_{-}S_{x}
\ +\ 2\nabla^{(\lambda )}_{x}\theta_{-}S_{x}
\ -\ \mathsf{R}(\lambda ) \; .$$
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[^1]: Observe that the concept of Killing spinor [(]{}equation[)]{} in the physics literature has a far broader meaning than in the mathematical literature.
[^2]: (Global) Spinors also have their applications and can [*e.g.*]{} be used to find generalised instanton equations [@Detournay:2009mp].
[^3]: [ In the supergravity literature it is customary to refer to specific theories by indicating the dimension of spacetime, $d$, and the number of minimal spinors, $N$, used to generate the supersymmetry transformations; the theory we just mentioned is therefore known as $d=5$ $N=2$ supergravity. However, in order not to give too many different meanings to $d$ we will use the number $n$ to mean dimensionality of spacetime, and will use the non-standard nomenclature “$N=\sharp$ $n=\sharp$ supergravity”. ]{}
[^4]: This is in general incompatible with supersymmetry. Sometimes these theories are referred to as *fake* supergravities or (f)SUGRA.
[^5]: By solution we refer to a geometry that arises from the existence of a spinor that fulfills eq. (\[eq:1\]).
[^6]: The same constraint can be obtained through explicit evaluation of the Einstein-Weyl equations.
[^7]: Bochner’s theorem states that any harmonic 1-form on a compact, oriented Ricci-flat manifold is parallel, which implies that in that case the Killing spinor is $u$-independent. In the non-compact case, however, there is no such theorem as can be envisaged by taking the base space to be $\mathbb{R}^{n-2}$ and to take $2\varpi \equiv
f_{mn}x^{m}dx^{n}$, where the $f_{nm}$’s are constants.
[^8]: In order not to confuse the reader we define the directional derivatives $\theta_{a}$ to be the duals of the frame 1-forms $E^{a}$, [*i.e.*]{} we have $E^{a}(\theta_{b})= \delta^{a}{}_{b}$. We shall reserve the notation $\partial_{x}$ for the directional derivative on the base space, namely $\partial_{x}\equiv e_{x}{}^{m}\partial_{m}$.
[^9]: Observe that a similar condition holds for defining $e_{mx}=e_{m}{}^{x}$.
|
---
abstract: |
The observation of a large cross-section for the $\alpha + d$ channel compared to breakup into the $\alpha + t$ channel from an exclusive measurement for the $^{7}$Li+$^{65}$Cu system at 25 MeV is presented. A detailed analysis of the angular distribution using coupled channels Born approximation calculations has provided clear evidence that the observed $\alpha + d$ events arise from a two step process, i.e. direct transfer to the 2.186 MeV ($3^+$) resonance in the $\alpha + d$ continuum of $^6$Li followed by breakup, and are not due to final state interaction effects.
PACS: 25.70.Mn, 25.70.Hi, 25.70.Bc, 25.70.Ef, 24.10.Eq
[*Keywords:* ]{} Exclusive breakup, Transfer-breakup, Elastic scattering, Weakly bound nuclei, Coupled channels calculations.
address:
- 'Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India'
- 'DSM/DAPNIA/SPhN, CEA Saclay, F-91191 Gif sur Yvette, France'
- 'DNAP, Tata Institute of Fundamental research, Mumbai 400005, India'
author:
- 'A. Shrivastava [^1], A. Navin[^2], N. Keeley, K. Mahata, K. Ramachandran, V. Nanal, V.V. Parkar, A. Chatterjeeand S. Kailas'
title: 'Evidence for transfer followed by breakup in $^{7}$Li + $^{65}$Cu'
---
Introduction
============
Nuclear reactions involving unstable/weakly bound nuclei that have low breakup thresholds and exotic structures display features remarkably different from those of well-bound stable nuclei. The advent of recent ISOL facilities, apart from opening new avenues in this field, has also caused a revival in the study of stable but weakly bound nuclei like $^{6,7}$Li and $^9$Be. Measurements with these nuclei, with better understood cluster structures, are relatively easier, due to the larger available beam intensities. At energies around the barrier the effect of breakup on the fusion process and also the measurement of associated direct processes are topics of current interest.
Recent interpretations of measurements of products arising from the excited compound nucleus (fission or evaporation residues) show the need for distinguishing between various mechanisms leading to the same final product prior to deriving any conclusions about the effect of weak binding on other reaction processes [@raa04; @nav04]. Measurements involving $^6$He and $^{6,7}$Li [@nav04; @dip04; @pak03; @kol02; @kel00], having an alpha + [*x*]{} configuration, show significantly large cross-sections for alpha particle production. It is non-trivial to disentangle the contributions to the alpha yield arising from different reaction mechanisms that include fusion-evaporation, transfer, transfer followed by evaporation/ breakup of the ejectile and direct (non capture) breakup of the projectile, especially from inclusive measurements. It is a well known observation that the alpha yield is much larger than the corresponding triton/deuteron yield in reactions involving $^{6,7}$Li beams and some possible scenarios have been suggested [@nav05; @sig03; @cas78; @fle78; @uts83].
Exclusive measurements of alpha particles provide a tool to understand the deconvolution of these processes. Such measurements, along with differential cross-sections for various direct processes like elastic scattering and few nucleon transfer reactions, provide important constraints for coupled channels calculations necessary for a consistent understanding of reactions with weakly bound nuclei near the Coulomb barrier. Apart from the above stated motivations, one of the key aspects of the present work is to investigate the two step reaction mechanism, namely, one nucleon transfer to a resonant state followed by breakup, for the case of a $^7$Li projectile. This complex process needs the simultaneous understanding of both the breakup and transfer reactions. Further, such measurements also provide information on the $^7$Li wave function in terms of the $^6$Li(ground/resonance)$+n$ configuration. In an earlier measurement for the $^7$Li+$^{197}$Au system [@que74], the authors speculated on the possibility of such a process and placed limits on the cross-section. Recently, analysing powers for this type of reaction have been reported [@nd04]. In more recent exclusive measurements of $\alpha - n $ for the $^6$He +$^{209}$Bi system, the contributions of single and two neutron transfer to the continuum followed by evaporation have been reported [@kol04; @kol05]. The present work is the first quantitative measurement of differential cross-section for the transfer-breakup reaction.
We report in this letter exclusive measurements of alpha particles along with $d/t$ to identify different reaction mechanisms of alpha emission, and a detailed study of the transfer-breakup mechanism of $^7$Li on a medium mass target at energies near the Coulomb barrier. These have been compared with similar measurements using a $^6$Li beam. Elastic scattering angular distributions and nucleon transfer angular distributions have also been measured. Extensive coupled channels Born approximation (CCBA) calculations along with continuum discretized coupled channels calculations (CDCC) to simultaneously explain the large number of observables are presented.
Experimental details and results
================================
The measurements were performed at the 14UD BARC-TIFR Pelletron, Mumbai using 25 MeV $^{6,7}$Li beams on an enriched $^{65}$Cu target. The target thickness used for measuring the elastic scattering and nucleon-transfer differential cross-sections was 1.0 mg/cm$^2$, while for the exclusive breakup it was 2.5 mg/cm$^2$. Typical beam currents were around 2 pnA. The coincidence detection system consisted of three $\Delta$E(30, 40 and 47$\mu$m )–E(2mm) Si surface barrier telescopes and a $10\times 10\times 10$mm$^3$ CsI(Tl) charged particle detector, kept 20$^{\circ}$ apart. With the present setup it was possible to measure fragment kinetic energies corresponding to unbound states of $^{6,7}$Li up to an excitation energy of 5.5 MeV. From the measured kinetic energy of each fragment and the breakup $Q$ value, the relative energies between $\alpha-d(t)$ for $^6$Li($^7$Li) breakup were deduced using three body final state correlations [@sch77]. The detectors were calibrated using discrete energy alpha particles in the range of 12 to 26 MeV [@ser73], produced in the reaction with $^7$Li beam of energies 20 and 25 MeV on a 50 $\mu$g/cm$^2$ thick $^{12}$C target. The elastic scattering, transfer and alpha emission angular distribution measurements were performed in a separate experiment employing three Si-surface barrier telescopes ($\Delta$E– 10, 20 and 25 $\mu$m and E– 1mm) covering an angular range of 10$^{\circ}$ to 140$^{\circ}$. A 300 $\mu$m thick Si- detector at forward angles, for both the exclusive and inclusive measurements, was used for monitoring the number of incident beam particles.
The data were collected in an event by event mode, with the trigger generated from fast coincidences between adjacent detectors. In Fig. 1 alpha and deuteron coincidence spectra for both the systems at $\theta_{\alpha d}$ = 20$^{\circ}$ are shown. The two localized contributions in the spectra are identified using three body kinematics [@sch77], and correspond to the sequential breakup of the first resonant state in $^6$Li. The full curves in Figs. 1a and 1b show the calculated kinematic correlations for the breakup of the 3$^+$ state in $^6$Li. In Fig. 1b, the relative energy ( E$_{\alpha
d}$) of 0.71 corresponding to the 3$^+$ resonance state is indicated. Fig. 1c shows the projection of Fig. 1b along the alpha energy axis. The alpha particles moving forward (backward) in the $\alpha-d$ center of mass system corresponds to the high (low) energy peak in the spectrum.
Shown in Fig. 2a are the angular distributions for the breakup of $^6$Li proceeding through the 3$^+$ and 2$^+$ resonant states at 2.186 and 4.312 MeV, respectively. Differential cross-sections were computed from $\alpha$ + $d$ coincidence yields, assuming isotropic emission of breakup fragments in the $^6$Li$^*$ frame and are shown in Fig. 2. The Jacobian of transformation and the center of mass angle of the scattered $^6$Li$^*$ were obtained as described in references [@mei85; @fuc82]. The ground state and first resonant state (3$^+$, 2.186 MeV) of $^6$Li are well described with a $\alpha + d$ configuration [@til02]. The decay of the second resonant state (0$^+$, 3.56 MeV) of $^6$Li to $\alpha + d$ is forbidden due to parity considerations; however, a decay through the $t$ + $^3$He channel is possible. The cross-section for this state was inferred to be negligible as $^3$He + $t$ coincidences were not observed.
The importance of the two step process can be seen from Fig. 1d which shows the $\alpha $+ $d$ coincidences for the $^7$Li+$^{65}$Cu system at 25 MeV. Two clear peaks are seen in the deuteron vs alpha energy correlation plot, indicating breakup of $^6$Li formed after one neutron stripping ($Q$ = -0.185 MeV) of $^7$Li and clearly not breakup of $^7$Li into $\alpha+d+n$ ($Q$ = -8.8 MeV). From the relative energy plot shown in Fig. 1e, the two peaks can be identified as arising from the breakup of $^6$Li via its 2.18 MeV state, formed after a one neutron transfer reaction. The transfered neutron can populate various states in $^{66}$Cu depending on their spin and spectroscopic factors. The kinematic curves shown in the figure are for transfer of the neutron to the ground, 1.15 MeV and 2.14 MeV states of $^{66}$Cu with $^6$Li in its 3$^+$ resonance state. As can be seen from the figure, there is very good agreement of the data with these kinematic plots. The corresponding angular distribution, integrated over excited states of $^{66}$Cu , is displayed in Fig. 2b. Also shown in the figure are the angular distributions for the breakup of $^7$Li $\rightarrow$ $\alpha$ + $t$ through its first resonance state at 4.63 MeV. From the known cluster structure of $^7$Li one would have naively expected a much larger cross-section for the latter compared to the $\alpha$ + $d$ coincidence yield.
The measured elastic angular distributions for the $^6$Li + $^{65}$Cu and $^7$Li + $^{65}$Cu systems are shown in Figs. 2c and 2d, respectively. The errors on the data points shown in the figure are statistical only. The angular distribution for the one neutron stripping of $^7$Li + $^{65}$Cu (Q = -0.185 keV) populating $^6$Li in its ground state (as $^6$Li has no bound excited states) and $^{66}$Cu (E$^*$ up to 5 MeV) was obtained independently from the inclusive data. The integrated cross-section obtained assuming a Gaussian shape for the angular distribution is listed in Table 1. As can be seen from Table 1, this cross-section is larger than that for all the other direct processes listed. The errors on the measured cross-sections are from uncertainties in the fitting procedure (6 to 8%) and statistics (3 to 6%). The 1$n$-pickup and $t$-stripping cross-section for $^6$Li + $^{65}$Cu are also listed in Table 1. For both isotopes the exclusive breakup cross-section is observed to be much smaller than the inclusive cross-section for alpha emission. The contribution of alpha particles evaporated from the compound nucleus is estimated to be less than 30% (Table 1) of the total inclusive alpha yield. These were obtained by fitting the measured backward angle alpha angular distribution using the statistical model code PACE [@gav80]. This suggests that the majority of the alpha particle yields arise from processes where the deuteron (triton) is transfered or captured by the target [@nav05; @sig03; @cas78; @fle78; @uts83] after breakup of the $^6$Li($^7$Li) in field of the target. The alpha emission cross-section from the inclusive and exclusive data for the $^6$Li projectile is larger than that for $^7$Li, as expected from the difference in the breakup thresholds. Similar results for the inclusive alpha emission cross-sections for $^6$Li and $^7$Li were obtained in references [@kel00; @pfe73].
Calculations {#sec:calcs}
============
Two distinct sets of calculations were carried out to describe the data. Those for the elastic scattering and breakup processes were performed within the CDCC formalism using a cluster folding model [@buc77] for the structure of $^{6,7}$Li. Calculations for the transfer breakup employed the CCBA framework, i.e. CDCC in the entrance and exit channel and DWBA for the transfer step, utilizing the potentials that explained the elastic scattering data. Both the CCBA and CDCC calculations described here were performed using the code FRESCO [@ijt].
The CDCC calculations for $^6$Li ($^7$Li) were similar to those described in [@kee03; @kee02] and assumed an $\alpha$+$d(t)$ cluster structure. The binding potentials between $\alpha$+$d$ and $\alpha$+$t$ were taken from [@kh] and [@bm], respectively. The $\alpha$+$d(t)$ continuum was discretized into a series of momentum bins of width $\Delta k = 0.2~ $fm$^{-1}$ (up to $k$ = 0.8 fm$^{-1}$) for $L$ = 0, 1, 2 for $^6$Li and $L$ = 0, 1, 2, 3 for $^7$Li , where $\hbar$$k$ denotes the momentum of the $\alpha + d(t)$ relative motion. All couplings, including continuum–continuum couplings, up to multipolarity $\lambda = 2$ were incorporated. Optical potentials for $\alpha$+$^{65}$Cu and $d(t)$+$^{65}$Cu are required as input for the cluster-folded $^6$Li($^7$Li)+$^{65}$Cu potential. The potential for $\alpha$+$^{65}$Cu was obtained by adjusting the real and imaginary depths of the global $\alpha$ optical potential of Avrigeanu [*et al.*]{} [@av], in order to match the measured $\alpha$+$^{65}$Cu elastic scattering data of reference [@cos]. In the absence of suitable elastic scattering data, the global parameters of [@bg; @pp] were employed for the $t$+$^{65}$Cu and $d$+$^{65}$Cu optical potentials. The real and imaginary strengths of the cluster-folded potentials were adjusted to describe the elastic scattering data.
Results of the calculations for the elastic scattering are shown in Figs. 2c and 2d. The two different curves are results of the calculation performed with (solid lines) and without (dashed lines) couplings. The calculated angular distributions for the resonant states, 3$^+$ and 2$^+$ of $^6$Li (Fig. 2a) and 7/2$^-$ of $^7$Li (Fig. 2b) show good agreement with the data. The results of the full CDCC calculation are listed in Table 1. The angle integrated cross-sections of the resonant states obtained by fitting to a Gaussian distribution show good agreement with the calculation. The total calculated breakup cross-sections for $^6$Li and $^7$Li were obtained by integrating contributions from the states in the continuum up to 11 MeV. As can be seen in Table 1, the total $^7$Li($^6$Li) $\rightarrow$ $\alpha$ + $t(d)$ breakup provides a negligible contribution to the total reaction cross-section. The fusion cross-sections listed in Table 1, for $^{6,7}$Li + $^{65}$Cu were obtained using the barrier penetration model, as described in reference [@nick02].
As mentioned earlier, for the $^7$Li + $^{65}$Cu system a significant number of $\alpha$+$d$ coincidence events consistent with decay of the 2.18 MeV $^6$Li($3^+$) resonance were observed. The simplest reaction mechanism for producing these events is a transfer–breakup process, with direct neutron stripping to the unbound $3^+$ resonance and/or neutron stripping to the $^6$Li $1^+$ ground state followed by excitation to the $3^+$ resonance through final state interaction. Due to the high density of states in the residual $^{66}$Cu nucleus and the experimental resolution, the angular distribution of $^6$Li($3^+$) resonance events was integrated over the residual $^{66}$Cu excitation energy up to 2.5 MeV. The $^{66}$Cu could thus be left in any one of up to 40 states [@dp69].
It was not possible to incorporate this process directly into the full CDCC calculation. Thus, to establish the dominant reaction mechanism for the observed $\alpha$+$d$ coincidences in $^7$Li + $^{65}$Cu (direct transfer to the continuum or transfer to ground state followed by breakup), a series of CCBA calculations employing a much reduced coupling scheme in the entrance and exit partitions, shown in Fig. 3 was performed. The potentials used were taken from the CDCC calculation as explained in the previous paragraph. For the $^6$Li+ $^{66}$Cu exit partition, only coupling to the 2.18 MeV ($3^+$) of $^6$Li was retained. The optical potential for $\alpha$+$^{66}$Cu was again calculated from the global parameters of Avrigeanu [*et al.*]{} [@av], renormalized by the same factors needed to fit the 14 MeV $\alpha$+$^{66}$Cu data of Costa [*et al.*]{} [@cos]. The optical potential for $d$+$^{66}$Cu was the central part of the potential of Bieszk and Knutson [@bie] for 9 MeV $d$+$^{63}$Cu. Spectroscopic factors for the $^7$Li $\rightarrow$ $^6$Li+$n$ overlaps were taken from Cohen and Kurath [@ck]. The neutron was bound in a Woods–Saxon well of radius 1.25 $\times$ A$^{1/3}$ fm and diffuseness 0.65 fm, the depth being adjusted to yield the correct binding energy. The spectroscopic factors for $^{66}$Cu $\rightarrow$ $^{65}$Cu+$n$ were taken from Daehnick and Park [@dp69]. The neutron was again bound in a Woods–Saxon well of radius 1.25 $\times$ A$^{1/3}$ fm and diffuseness 0.65 fm, as used in ref. [@dp69]. The transfer part of the calculations was performed using the post–form DWBA and included the full complex remnant term.
The CCBA calculations were carried out for transfers leaving the residual $^{66}$Cu in levels up to 1.43 MeV in excitation, partly due to the uncertain nature of many of the spin assignments above 1.5 MeV and the presence of unresolved doublets. As the reaction Q-value for $^{65}$Cu($^7$Li,$^6$Li)$^{66}$Cu is slightly negative (-0.185 MeV), the population of states near the ground state of $^{66}$Cu will be favoured to some extent due to Q-matching considerations. Hence the sum of the present CCBA calculations covers most of the observed $\alpha$+$d$ coincidence cross-section. In the cases where the spin assignments of Daehnick and Park differ from those of the compilation [@bhat], the latter has been followed. The shape of the calculated sum of the CCBA angular distributions is in good agreement with the measurement (Fig. 2b), although the magnitude is lower due to the omission of $^{66}$Cu states above 1.43 MeV. The results of the calculation confirm the transfer/breakup mechanism for the observed $\alpha$+$d$ coincidences. Normalising the summed CCBA calculations to the data yields a total cross-section for $^6$Li($3^+$) production of about 9 mb, nearly twice that for the measured breakup of $^7$Li via 7/2$^-$ state. The seperately measured $^{65}$Cu($^7$Li,$^6$Li)$^{66}$Cu reaction leaving $^{6}$Li in its ground state is also well described. The dominant peak in the spectrum for this transfer is centred on the 1.15 MeV $6^-$ state in $^{66}$Cu, and the experimental value for the total cross section integrated over a bin of width 400 keV centred at 1.15 MeV is $4.2 \pm 0.5$ mb, while the summed total cross section from the CCBA calculation for $^{66}$Cu states in the same energy range is 3.9 mb.
Having established that the $^6$Li($3^+$) resonance is populated by the transfer–breakup mechanism, a further distinction between direct transfer to the unbound $3^+$ resonance in $^6$Li (transfer to the continuum) and transfer to $^6$Li in its $1^+$ ground state followed by excitation to the $3^+$ (final state interaction) was investigated. Calculations omitting the direct transfer step showed that the final state interaction process provides a negligible contribution (10%) except at extreme forward angles. It can thus be concluded that the main reaction mechanism for the observed large $\alpha$ + $d$ exclusive cross-sections is direct transfer followed by breakup of the unbound $3^+$ resonant state in the $^6$Li continuum.
Discussion
==========
The exclusive breakup cross-sections for the resonant states of $^{6,7}$Li could be explained well by CDCC calculations performed using potentials that fit the elastic scattering angular distributions. The total non capture breakup cross-section for $^6$Li was found to be larger than that for $^7$Li mainly due to the lower alpha binding energy in $^6$Li compared to $^7$Li. The exclusive breakup cross-sections are a very small fraction of the reaction cross-sections for both $^6$Li and $^7$Li (Table 1). The exclusive breakup for $^{6,7}$Li + $^{65}$Cu contributes less than 10% and compound nucleus evaporation less than 30% towards the observed large alpha-singles cross-section. The origin of the large alpha yield in $^6$Li($^7$Li) induced reactions seems to be mainly due to deuteron(triton) capture/deuteron(triton) transfer as discussed in [@nav05; @sig03; @cas78; @fle78; @uts83].
In a recent study with $^{6,7}$Li on a medium mass target, $^{64}$Zn, very large cross-sections for the break up (where both the fragments survive) have been indirectly inferred by subtracting the complete and incomplete fusion cross-sections from the reaction cross-section [@gom04]. This could arise from a neglect of other direct reaction processes, for instance; nucleon transfer, inelastic excitation of the target/projectile etc. Before arriving at any conclusion on the role of breakup on other reaction channels unambiguous information on the breakup cross-section is necessary. The present work clearly shows that exclusive measurements for the breakup cross-sections are essential and indirect methods can be unreliable.
Conclusion
===========
The present work reports a detailed study of the multi-step reaction mechanism, namely transfer-breakup. The origin of the large yields for $\alpha + d$ events from the coincidence data for $^7$Li breakup has been identified as transfer followed by breakup of the excited $^6$Li via its 3$^+$ resonant state in the continuum. To get a deeper insight into the mechanism behind this reaction – direct transfer of the neutron to the $^6$Li–continuum or transfer to the ground state of $^6$Li followed by excitation to the continuum – CCBA calculations were performed. The results of the calculations have established that the main reaction mechanism is direct transfer to the continuum.
Reactions with low energy unstable radioactive ion beams from newly available facilities are expected to be of similar complexity. Identification of the reaction processes and development of theoretical understanding for such multi-step reactions is a challenging task. The present study with weakly bound stable nuclei on breakup and transfer–breakup mechanism along with extensive theoretical analysis is an attempt in this direction.
[99]{} R. Raabe, [*et al.*]{}, Nature 431 (2004) 823, and references therein. A. Navin, [*et al.*]{}, Phy. Rev. C 70 (2004) 044601. A. di Pietro, [*et al.*]{}, Europhys. Lett. 64 (2004) 309. A. Pakou, [*et al.*]{}, Phys. Rev. Lett. 90 (2003) 202701. J.J. Kolata, Eur. Phys. J. A 13 (2002) 117. G.R. Kelly, [*et al.*]{}, Phy. Rev. C 63 (2000) 024601. V. Tripathi, [*et al.*]{}, Phy. Rev. C 72 (2005) 0170601. C. Signorini, [*et al.*]{}, Phy. Rev. C 67 (2003) 044607. C. M. Castaneda, [*et al.*]{}, Phys. Lett. B 77 (1978) 371. J.G. Fleissner, [*et al.*]{}, Phys. Rev. C 17 (1978) 1001. H. Utsunomiya, [*et al.*]{}, Phys. Rev. C 28 (1983) 1975. J.L. Quebert, [*et al.*]{}, Phys. Rev. Lett. 32 (1974) 1136. N. Davis, [*et al.*]{}, Phys. Rev. C 69 (2004) 064605. J.P. Bychowski, [*et al.*]{}, Phys. Lett. B 596 (2004) 26. P.A. DeYoung, [*et al.*]{}, Phy. Rev. C 71 (2005) 051601. D. Scholz, [*et al.*]{}, Nucl. Phys. A 288 (1977) 351. I. Tserruya, B. Rosner and K. Bethge, Nucl. Phys. A 213 (1973) 22. R.J.de. Meijer and R. Kamermans, Rev. Mod. Phys. 57 (1985) 147. H. Fuchs, Nucl. Instr. and Meth. 190 (1981) 75. D.R. Tilley, [*et al.*]{}, Nucl. Phys. A 708 (2002) 3. A. Gavron, Phy. Rev. C 21 (1980) 230. K.O. Pfeiffer, [*et al.*]{}, Nucl. Phys. A 206 (1973) 545. B. Buck and A.A. Pilt, Nucl. Phys. A 280 (1977) 133. I.J. Thompson, Comp. Phys. Rep. 7 (1988) 167. N. Keeley, [*et al.*]{}, Phys. Rev. C 68 (2003) 054601. N. Keeley, K.W. Kemper, and K. Rusek, Phys. Rev. C 66 (2002) 044605. K.-I. Kubo and M. Hirata, Nucl. Phys. A 187 (1972) 186. B. Buck and A.C. Merchant, J. Phys. G 14 (1988) L211. V. Avrigeanu, P.E. Hodgson, and M. Avrigeanu, Phys. Rev.C 49 (1994) 2136. S. Costa, [*et al.*]{}, Nuovo Cim. 72 A (1982) 40. F.D. Becchetti, Jr. and G.W. Greenlees, [*Polarization Phenomena in Nuclear Reactions*]{} (H.H. Barschall and W. Haeberli, eds.) p. 682, The University of Wisconsin Press, Madison, Wis. (1971); C.M. Perey and F.G. Perey, Atomic Data and Nucl. Data Tables 17 (1976) 1. C.M. Perey and F.G. Perey, Phys. Rev. 132 (1963) 755; C.M. Perey and F.G. Perey, Atomic Data and Nucl. Data Tables 17 (1976) 1. N. Keeley, K.W. Kemper and K. Rusek, Phys. Rev. C 65 (2002) 014601. W.W. Daehnick and Y.S. Park, Phys. Rev. 180 (1969) 1062. J.A. Bieszk and L.D. Knutson, Nucl. Phys. A 349 (1980) 445. S. Cohen and D. Kurath, Nucl. Phys. A 101 (1967) 1. M.R. Bhat, Nucl. Data Sheets 83 (1998) 789. P.R.S. Gomes, [*et al.*]{}, Phys. Lett B 601 (2004) 20; P.R.S. Gomes [*et al.*]{}, Phy. Rev. C 71 (2005) 034608.
[@lllllll]{} $^6$Li + $^{65}$Cu& &\
Channel & $\sigma_{exp}$ (mb) & $\sigma_{cal}$ (mb)\
$^6$Li$^*$(2.186 MeV)$\rightarrow$ $\alpha+d$ & 22 $\pm$2 & 19.5\
$^6$Li$^*$(4.31 MeV)$\rightarrow$ $\alpha+d$ & 4.3 $\pm$ 0.5 & 3.9\
$^6$Li$^*$(5.65 MeV)$\rightarrow$ $\alpha+d$ & - & 0.8\
$^6$Li$^*$(upto 11 MeV) $\rightarrow$ $\alpha+d$ & - &48\
$^7$Li (1-neutron pickup) & 14.7 $\pm$ 2.0 & -\
$^3$He (triton stripping) & 3.3 $\pm$ 0.5 & -\
$\alpha$ (CN evaporation) & 177 $\pm$ 20 & -\
$\alpha$ (inclusive) & 612 $\pm$ 40 & -\
Fusion & - & 1199\
Total reaction & - & 1492\
\
$^7$Li + $^{65}$Cu& &\
Channel & $\sigma_{exp}$ (mb) & $\sigma_{cal}$ (mb)\
$^7$Li$^*$(4.652 MeV)$\rightarrow$ $\alpha+t$ & 4.5 $\pm$ 0.6 & 5.1\
$^7$Li$^*$(7.454 MeV)$\rightarrow$ $\alpha+t$ & - & 0.4\
$^7$Li$^*$(upto 11 MeV)$\rightarrow$ $\alpha+t$ & - & 20.9\
$^6$Li$^*$(2.186 MeV)$\rightarrow$ $\alpha+d$ & 9 $\pm$ 1 &5.6\
$^6$Li (1-neutron stripping) & 44 $\pm$ 4 & 9.3\
$^6$He (1-proton stripping) & 7.8 $\pm$ 1.0 & -\
$\alpha $ (CN evaporation) & 110 $\pm$ 18 & -\
$\alpha$ (inclusive) & 422 $\pm$ 33 & -\
Fusion & - & 1061\
Total reaction & - & 1401\
 Alpha - deuteron correlations for $^{6,7}$Li + $^{65}$Cu systems. For the $^6$Li projectile the data are for $\alpha$ particles detected at 65$^{\circ}$ and deuteron at 45$^{\circ}$ plotted as (a)E$_d$ vs E$_{\alpha}$ (b) the relative energy E$_{\alpha d}$ vs E$_{\alpha}$ and (c) projection of the $\alpha$ particle energy for data shown in (b). The solid curves in (a) and (b) are results of three body kinematical calculations. Similar plots for $^7$Li projectile for the breakup of $^6$Li$^*$ after a 1n stripping reaction are shown in (d), (e) and (f) for $\alpha$ particles detected at 26$^{\circ}$ and deuteron at 46$^{\circ}$. The solid, dashed and dot-dashed curves in (d) and (e) are the same as above corresponding to $^{66}$Cu in the ground and excited states at 1.15 and 2.14 MeV respectively.](fig1.eps "fig:"){width="30pc"} **
![\[fig2\] Differential cross-sections for resonant, transfer breakup and elastic scattering. (a) Breakup via resonant states 2.18 MeV (3$^+$) and 4.31 MeV (2$^+$) in $^6$Li (CDCC calculations are shown as solid and dash-dot lines). (b) Breakup via resonant state 4.63 (7/2$^-$) MeV in $^7$Li along with CDCC calculations (dash-dot lines) and data for transfer–breakup reaction, $^7$Li + $^{65}$Cu$\rightarrow$ $^6$Li$^*$(3$^+$) + $^{66}$Cu$^*$ (0 to 2.5 MeV) along with CCBA calculations. (c) and (d) The ratio of the elastic scattering to the Rutherford cross-section as a function of angle for $^6$Li + $^{65}$Cu and $^7$Li + $^{65}$Cu. CDCC calculations are shown as solid (coupled) and dashed (uncoupled) lines.](fig2.eps){width="30pc"}
**
![\[fig3\] Reduced coupling scheme for the projectile used in the CCBA calculations (see text).](fig3.eps){width="30pc"}
**
[^1]: email:aradhana@apsara.barc.ernet.in
[^2]: Permanent address: GANIL, Bd. Henri Becquerel, BP 55027, Cedex 5, 14076, Caen, France
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abstract: 'Real world prediction problems often involve the simultaneous prediction of multiple target variables using the same set of predictive variables. When the target variables are binary, the prediction task is called multi-label classification while when the target variables are real-valued the task is called multi-target regression. Although multi-target regression attracted the attention of the research community prior to multi-label classification, the recent advances in this field motivate a study of whether newer state-of-the-art algorithms developed for multi-label classification are applicable and equally successful in the domain of multi-target regression. In this paper we introduce two new multi-target regression algorithms: multi-target stacking (MTS) and ensemble of regressor chains (ERC), inspired by two popular multi-label classification approaches that are based on a single-target decomposition of the multi-target problem and the idea of treating the other prediction targets as additional input variables that augment the input space. Furthermore, we detect an important shortcoming on both methods related to the methodology used to create the additional input variables and develop modified versions of the algorithms (MTSC and ERCC) to tackle it. All methods are empirically evaluated on 12 real-world multi-target regression datasets, 8 of which are first introduced in this paper and are made publicly available for future benchmarks. The experimental results show that ERCC performs significantly better than both a strong baseline that learns a single model for each target using bagging of regression trees and the state-of-the-art multi-objective random forest approach. Also, the proposed modification results in significant performance gains for both MTS and ERC.'
author:
- 'Eleftherios Spyromitros-Xioufis'
- Grigorios Tsoumakas
- William Groves
- Ioannis Vlahavas
bibliography:
- 'citations.bib'
title: 'Multi-Label Classification Methods for Multi-Target Regression'
---
Introduction {#sec:intro}
============
Learning from multi-label data has recently received increased attention by researchers working on machine learning and data mining for two main reasons. The first one is the ubiquitous presence of multi-label data in application domains ranging from multimedia information retrieval to tag recommendation, query categorization, gene function prediction, medical diagnosis, drug discovery and marketing. The other reason is a number of challenging research problems involved in multi-label learning, such as dealing with label rarity, scaling to large number of labels and exploiting label relationships (e.g. hierarchies), with the most prominent one being the explicit modeling of label dependencies [@dembczynski2012].
Multi-label learning is closely related to multi-target regression, also known as multivariate or multi-output regression, which aims at predicting multiple real-valued target variables instead of binary ones. Despite that multi-target regression is a less popular task, it still arises in several interesting domains, such as predicting the wind noise of vehicle components [@kuznar2009], stock price prediction and ecological modeling [@kocev2009]. Multi-label learning is often treated as a special case of multi-target regression in statistics [@izenman2008]. However, we could more precisely state that both are instances of the more general learning task of predicting multiple targets, which could be real-valued, binary, ordinal, categorical or even of mixed type. The baseline approach of learning a separate model for each target applies to both learning tasks. Furthermore, there exist techniques, such as [@blockeel1998; @sieger2005; @lebanon2012], that can naturally handle both tasks. Most importantly, they share the same core challenge of modeling dependencies among the different targets.
Given this tight connection between these two learning tasks, it would be interesting to investigate whether recent advances in the more popular multi-label learning task can be successfully transferred to multi-target regression. In particular, this paper adapts two multi-label learning methods that successfully model label dependencies [@godbole2004; @read2011] to multi-target regression. The results of an empirical evaluation on several real-world datasets, some firstly introduced here, show that the benefits of these two multi-label algorithms apply also to the multi-target regression setting.
Furthermore, by studying the relationship between these two learning tasks, this work aims to increase our understanding in their shared challenge of modeling target dependencies and widen the applicability of existing specialized techniques for either task. This kind of abstraction of key ideas from solutions tailored to related problems offers additional advantages, such as improving the modularity and conceptual simplicity of learning techniques and avoiding reinvention of the same solutions[^1].
The rest of the paper is organized as follows: Section \[sec:related\] discusses related work in the field of multi-label classification and gives some insight into which multi-label classification methods would be more appropriate for multi-target regression. Section \[sec:methods\] gives detailed descriptions of the proposed methods and Section \[sec:setup\] presents the evaluation methodology and introduces the datasets used in the empirical evaluation. The experimental results are shown in Section \[sec:results\] and finally, Section \[sec:conclusions\] concludes our study and points to directions for future work.
From Multi-Label Classification to Multi-Target Regression {#sec:related}
==========================================================
Multi-label learning methods are often categorized into those that [*adapt*]{} a specific learning approach (e.g. $k$ nearest neighbors, decision tree, support vector machine) for handling multi-label data and those that [*transform*]{} the multi-label task into one or more single-label tasks that can be solved with off-the-shelf learning algorithms [@tsoumakas2010b]. The latter can be further categorized to those that model single labels, pairs of labels and sets of labels [@zhang2011].
Approaches that model single labels include the typical one-versus-all (also known as binary relevance) baseline, methods based on stacked generalization [@godbole2004; @tsoumakas2009a; @cheng2009] and the classifier chains algorithm [@read2011; @dembczynski2010a]. Such approaches are almost straightforward to adapt to multi-target regression by employing a regression instead of a classification algorithm. It is this kind of approaches that we extend in this work.
Approaches that model pairs of labels [@furnkranz2008] follow the paradigm of the one-versus-one decomposition for using binary classifiers on multi-class learning tasks. This concept however is not transferable to multi-target regression. The same holds for approaches that consider sets of labels as different values of a single-label multi-class task [@read2008b; @tsoumakas2011a].
The extension of multi-label algorithm adaptation methods for handling multi-target data mainly depends on how easy it is for the underlying learning algorithm to handle classification and regression data interchangeably. For example, decision trees can handle both classification and regression data through different functions for calculating the impurity of internal nodes and the output of leaves. It thus comes as no surprise that there exist decision tree algorithms for both multi-label classification [@clare2001] and multi-target regression [@appice2007], with the most representative and well-developed ones being based on the predictive clustering trees framework [@blockeel1998; @kocev2013]. It is interesting to note that the predictive clustering framework is an example of a technique that originally focused on multi-target regression, and only recently showcased its effectiveness on multi-label classification [@vens2008; @madjarov2012].
Several multi-label algorithm adaptation methods are based on the definition and optimization of alternative loss functions, compared to the algorithms they extend. For example, the core idea in [@zhang2006] was the optimization of a ranking loss function that takes into account label pairs, instead of the typical logistic loss of neural networks that looks at individual labels. Similarly, [@crammer2003] defines a family of online algorithms based on different loss functions considering label relationships. To the best of our knowledge, this type of reasoning for algorithm design has not been transferred to multi-target regression. We believe that it could be an interesting avenue to investigate. For example, consider an application of food sales prediction [@zliobaite2009], in particular pastry sales prediction for a patisserie in order to minimize the amount of pastries with short expiration date that are thrown away. In this application, we may like to minimize the sum of prediction errors, but we might also want to minimize the maximum individual prediction error, to avoid for example an early run-out of any of the pastries.
Finally, we would like to note that there exist approaches, such as [@sieger2005; @lebanon2012], that have recognized this dual applicability (classification and regression) of their multi-target approach and have given a general formulation of their key ideas.
Methods {#sec:methods}
=======
We first formally describe the multi-target regression task and provide the notation that will be used subsequently for the description of the methods. Let $X$ and $Y$ be two random vectors where $X$ consists of $d$ input variables $X_1,..,X_d$ and $Y$ consists of $m$ target variables $Y_1,..,Y_m$. We assume that samples of the form $(x,y)$ are generated iid by some source according to a joint probability distribution $P(X,Y)$ on $\mathcal{X} \times \mathcal{Y}$ where $\mathcal{X}=R^d$ and $\mathcal{Y}=R^m$ are the domains of $X$ and $Y$ and are often referred to as the input and the output space. In a sample $(x,y)$, $x=[x_1,..,x_d]$ is the input vector and $y=[y_1,..,y_m]$ is the output vector which are realizations of $X$ and $Y$ respectively. Given a set $D=\{(x^1,y^1),..,(x^n,y^n)\}$ of $n$ training examples, the goal in multi-target regression is to learn a model $h:\mathcal{X}\rightarrow\mathcal{Y}$ which given an input vector $x^q$, is able to predict an output vector $\hat{y}^q = h(x^q)$ which best approximates the true output vector ${y}^q$.
In the baseline Single-Target (ST) method, a multi-target model $h$ is comprised of $m$ single-target models $h_j:\mathcal{X} \rightarrow R$ where each model $h_j$ is trained on a transformed training set $D_j = \{(x^{1},y_{j}^{1}),..,(x^{n},y_{j}^{n})\}$ to predict the value of a single target variable $Y_j$. This way, target variables are predicted independently and potential relations between them cannot be exploited.
Multi-Target Stacking (MTS) {#subsec:mtrs}
---------------------------
The first method that we consider is inspired by [@godbole2004] where the idea of stacked generalization was applied in a multi-label classification context. That method works by expanding the original input space of each training example with $m$ additional binary variables, corresponding to the predictions of $m$ binary classifiers, one for each label, for that example. This expanded training set is fed to a second layer of binary classifiers that produce the final decisions. The second-layer classifiers can exploit label dependencies. Variations of this core idea appear in [@cheng2009; @tsoumakas2009a; @cherman2012].
Here, we adapt this method for multi-target regression and denote it as Multi-Target Stacking (MTS). The training of MTS consists of two stages. In the first stage, $m$ independent single-target models $h_j:\mathcal{X} \rightarrow R$ are learned as in ST. However, instead of directly using these models for prediction, MTS involves an additional training stage where a second set of $m$ meta models $h^*_j: \mathcal{X} \times R^{m-1} \rightarrow R$ are learned, one for each target $Y_j$. Each meta model $h^*_j$ is learned on a transformed training set $D^*_j=\{(x^{*1},y^1_j),..,(x^{*n},y^n_j)\}$, where $x^{*i}_j=[x^i_1,..,x^i_n,\hat{y}^i_1,..,\hat{y}^{i}_{j-1},\hat{y}^{i}_{j+1},..,\hat{y}^i_m]$ are expanded input vectors consisting of the original input vectors of the training examples augmented by $m-1$ predictions (estimates) of the rest of the target variables obtained by the first stage models. We intentionally differentiate slightly our method from the multi-label method to the point of not including the predictions of the first stage model for target variable $Y_j$ in the input space of the second stage model for this variable as in [@cherman2012], since this would add redundant information to the second stage models. To obtain predictions for an unknown instance $x^q$, the first stage models are first applied and an output vector $\hat{y}^q=[\hat{y}^q_1,..,\hat{y}^q_m]=[h_1(x^q),..,h_m(x^q)]$ is obtained. Then the second stage models are applied on a transformed input vector $x^{*q}_j =[x^q_1,..,x^q_n,\hat{y}^q_1,..,\hat{y}^q_{j-1},\hat{y}^q_{j+1},..,\hat{y}^q_m]$ to produce the final output vector $\hat{\hat{y}}^q_j=[h^*_1(x^{*q}),...,h^*_m(x^{*q})]$.
Ensemble of Regressor Chains (ERC) {#subsec:erc}
----------------------------------
The second method that we consider is inspired by the recently proposed Classifier Chains (CC) method [@read2011] and we henceforth denote it as Regressor Chains (RC). RC is based on the idea of chaining single-target models. The training of RC consists of selecting a random chain (permutation) of the set of target variables and then building a separate regression model for each target. Assuming that the default chain $C = \{Y_1 , Y_2 , .. , Y_m\}$ (where $C$ is represented as an ordered set) is selected, the first model concerns the prediction of $Y_1$, has the form $h_1: \mathcal{X} \rightarrow R$ and is the same as the model built by the ST method for this target. The difference in RC is that subsequent models $h_{j,j>1}$ are trained on transformed datasets $D^*_j = \{(x^{*1}_j,y^{1}_{j}),..,(x^{*n}_j,y^{n}_{j})\}$, where $x^{*i}_j = [x^i_1,..,x^i_1,y^{i}_{1},..,y^{i}_{j-1}]$ are transformed input vectors consisting of the original input vectors of the training examples augmented by the actual values of all previous targets in the chain. Thus, the models built for targets $Y_{j,j>1}$ have the form $h_j: \mathcal{X} \times R^{j-1} \rightarrow R$. Given such a chain of models, the output vector $\hat{y}^q$ of an unknown instance $x^q$ is obtained by sequentially applying the models $h_j$, thus $\hat{y}^q = [h_1(x^q),h_2(x^{*q}_2),..,h_m(x^{*q}_m)]$ where $x^{*q}_{j,j>1}= [x^q_1,..,x^q_d,\hat{y}^q_1,..,\hat{y}^q_{j-1}]$. Note that since the true values $y^q_1,..,y^q_{j-1}$ of the target variables are not available at prediction time, the method relies on estimates of these values obtained by applying the models $h_1,..,h_{j-1}$.
One notable property of RC is that it is sensitive in the selected chain ordering. The main problem arising from the use of a single random chain, is that targets which appear earlier in a chain cannot model potential statistical relationships with targets appearing later in that chain. Additionally, prediction error is likely to by propagated and amplified along a chain when making predictions for a new test instance. To mitigate these effects, [@read2011] proposed an ensemble scheme called Ensemble of Classifier Chains where a set of $k$ (typically $k$=10) CC models with differently ordered chains are built on bootstrap samples of the training set and the final predictions come from majority voting. This scheme has proven to consistently improve the accuracy of a single CC in the classification domain. We apply the same idea (without sampling) on RC and compute the final predictions by taking the mean of the $k$ estimates for each target. The resulting method is called Ensemble of Regressor Chains (ERC).
MTS & ERC Corrected {#sec:methods:meta}
-------------------
Both MTS and ERC are based on the same core idea of treating the other prediction targets as additional input variables that augment the original input space. These meta-variables differ from ordinary input variables in the sense that while their actual values are available at training time, they are missing during prediction. Thus, during prediction, both methods have to rely on estimates of these values which come either from ST (in MTS) or from RC (in ERC) models built on the training set. An important question, which is answered differently by each method, is what type of values should be used at training time for these meta-variables. MTS uses estimates of these variables obtained by applying the first stage models on the training examples, while ERC uses the actual values of these variables to train the RC models. In both cases, *a core assumption of supervised learning is violated*: that the training and testing data should be identically and independently distributed. In the MTS case, the in-sample estimates that are used to form the second stage training examples, will typically be more accurate than the out-of-sample estimates that are used at prediction time. The situation is even more problematic in the case of RC where the actual target values are used at training time. In both cases, the underlying learning algorithm is trained using meta-variables that become noisy (or noisier in the MTS case) at prediction time. As a result, the actual importance and relationship of these meta-variables to the prediction target is falsely estimated. The impact that this discrepancy in the distributions has on CC’s accuracy was recently studied in [@senge2013], where factors such as the length and order of the chain, the accuracy of the binary classifiers and the degree of dependence between the labels were identified. The latter two factors also apply to MTS.
An alternative to the above approaches is using only a part of the training set for learning the first stage ST models (in MTS) or the RC models (in ERC) and then applying these models to the hold-out part in order to obtain out-of-sample estimates of the meta-variables. While the distribution of the estimates obtained using this approach is expected to be more representative of the distribution of the estimates obtained during prediction, it would lead to reduced second stage training sets for MTS as only the examples of the hold-out set can be used for training the second stage models. The same holds for ERC where the chained RC models would be trained on training sets of decreasing size.
Here, we propose modifications of the original MTS and ERC methods that alleviate the aforementioned problem by employing a procedure that resembles that of $f$-fold cross-validation in order to obtain unbiased out-of-sample estimates of the meta-variables. The cross-validation approach avoids the problem of the hold-out approach as all training examples are used in the second stage ST models of MTS and the RC models of ERC. We expect that compared to using the actual values or in-sample estimates, the cross-validation estimates will approximate the distribution of the estimates obtained at prediction time more accurately and thus will be more useful for the model. The training procedures the proposed corrected versions denoted as *MTS Corrected* (MTSC) and *RC Corrected* (RCC) are outlined in Algorithms \[algorithm:mts:training\] and \[algorithm:rc:training\]. ERCC consists of simply repeating the RCC procedure $k$ times with $k$ randomly ordered chains. The prediction procedures are the same for all variants of the methods and are presented in Algorithms \[algorithm:mts:prediction\] and \[algorithm:rc:prediction\]. In Section \[sec:results\] we empirically evaluate the original MTS and ERC methods with the proposed corrected versions (MTSC) and (ERCC).
Experimental Setup {#sec:setup}
==================
This section describes our experimental setup. We first present the participating methods and their parameters and provide details about their implementation in order to facilitate reproducibility of the experiments. Next, we describe the evaluation measure and justify the process that was followed for the statistical comparison of the methods. Finally, we present the datasets that we used, 8 of which are firstly introduced in this paper.
Methods, Parameters and Implementation {#sec:params}
--------------------------------------
The empirical evaluation compares the performance of the basic MTS and ERC methods against the performance of their corrected versions MTSC and ERCC. All the proposed methods are also compared against the ST baseline as well as the state-of-the-art multi-objective random forest algorithm [@kocev2013] (MORF).
Since all the proposed methods (including ST) transform the multi-target prediction problem into several single-target problems, they have the advantage of being combinable with any single-target regression algorithm. To facilitate a fair comparison and to simplify the analysis we use bagging [@breiman1996] of 100 regression trees as the base regression algorithm in all methods. MTSC and ERCC are run with $f=10$ internal cross-validation folds and the ensemble size of ERC and ERCC is set to 10 models, each one trained using a different random chain (in cases where the number of distinct chains is smaller than 10 we create exactly as many models as the number of distinct label chains). In order to ensure a fair comparison of ERC and ERCC with the non-ensemble methods, we do not perform bootstrap sampling (i.e. each ensemble model is build using all training examples). Finally, for MORF we use an ensemble size of 100 trees and the values suggested in [@kocev2013] for the rest of its parameters.
All the proposed methods and the evaluation framework were implemented within the open-source multi-label learning Java library Mulan[^2] [@tsoumakas2011b] by expanding the library to handle multi-target regression tasks. Mulan is built on top of Weka[^3] [@witten2011], which includes implementations of bagging and regression tree (via the REPTree class). A wrapper of the CLUS[^4] software, which includes support for MORF, was also implemented and included in Mulan, enabling the evaluation of all methods under a single software framework. In support of open science and to ease replication of the experimental results of this paper, we have included a class called ExperimentMTR in Mulan’s experiments package.
Evaluation {#sec:eval}
----------
We use Relative Root Mean Squared Error (RRMSE) to measure the accuracy of a multi-target model on each target variable. The RRMSE of a multi-target model $h$ that has been induced from a train set $D_{train}$ is estimated based on a test set $D_{test}$ according to the following equation: $$RRMSE(h,D_{test})
=
\sqrt
{
\frac
{
\sum_{(x,y_{j}) \in D_{test}} (\hat{y}_{j}-y_{j})^{2}
}
{
\sum_{(x,y_{j}) \in D_{test}} (\bar{Y}_{j}-y_{j})^{2}
}
}$$ where $\bar{Y}_{j}$ is the mean value of target variable $Y_{j}$ over $D_{train}$ and $\hat{y}_{j}$ is the estimation of $h(x)$ for $Y_{j}$. More intuitively, RRMSE for a target is equal to the Root Mean Squared Error (RMSE) for that target divided by the RMSE of predicting the average value of that target in the training set. The RRMSE measure is estimated using the hold-out approach for large datasets, while 10-fold cross-validation is employed for small datasets.
To test the statistical significance of the observed differences between the methods, we follow the methodology suggested in [@demsar2006]. When comparing two methods on multiple datasets we use the Wilcoxon signed-ranks test. When the comparison involves multiple methods we first apply the non-parametric Friedman test that operates on the average ranks of the methods and checks the validity of the hypothesis (null-hypothesis) that all methods are equivalent. If the null-hypothesis is rejected, we proceed to the Nemenyi post-hoc test that computes the critical distance (between average ranks) required in order for two methods to be considered significantly different. Finally, we graphically present the results with appropriate diagrams which plot the average ranks of the methods and show groups of methods whose average rank differences are less than the critical distance for a $p$-value of 0.05.
As the above methodology requires a single performance measurement for each method on each dataset, it is not directly applicable to multi-target evaluation where we have multiple performance measurements (one for each target) for each method on each dataset. One option is to take the average RRMSE (aRRMSE) across all target variables within a dataset as a single performance measurement. However, this may not always be a meaningful choice since: a) different targets may represent different things and b) we may not be always interested into the best average performance (see patisserie example in Section \[sec:related\]). Another option is to treat the RRMSE performance of each method on each different target as a different measurement. In this case, however, Friedman’s test assumption of independence between performance measurements might be violated. In the absence of a better solution, we perform the two dimensional analysis of [@aho2009], where statistical tests are conducted using both aRRMSE but also considering the RRMSE value per target as an independent measurement.
Datasets {#sec:data}
--------
Despite the numerous interesting applications of multi-target regression, there are few publicly available datasets of this kind, perhaps because most applications are industrial. In fact, among the datasets used in other multi-target regression studies, e.g. [@aho2009], only Solar Flare [@asuncion+newman:2007], Water Quality [@dzeroski2000] and EDM [@karalic1997] are publicly available while the rest are proprietary and could not be acquired. This lack of available data, motivated the collection of real-world multi-target regression data and the composition of 8 new benchmark datasets which have been made publicly available[^5]. A description of each dataset follows, while Table \[tbl:data sets\] summarizes their statistics.
--------- ---------- ----- ----- --------- ----------- ----- -----
Dataset Examples $d$ $m$ Dataset Examples $d$ $m$
EDM 154 16 2 OES97 334 263 16
SF1 323 10 3 OES10 403 298 16
SF2 1066 10 3 ATP1d 337 411 6
WQ 1060 16 14 ATP7d 296 411 6
RF1 4108/5017 64 8
RF2 4108/5017 576 8
SCM1d 8145/1658 280 16
SCM20d 7463/1503 61 16
--------- ---------- ----- ----- --------- ----------- ----- -----
: Statistics of the datasets used in the evaluation. Those on the right are firstly introduced in this paper. A two-value entry for the Examples column indicates the sample counts for the train and test set respectively, $d$ denotes the number of input variables and $m$ denotes the number of target variables.[]{data-label="tbl:data sets"}
### EDM
The Electrical Discharge Machining dataset [@karalic1997] represents a two-target regression problem. The task is to shorten the machining time by reproducing the behavior of a human operator which controls the values of two variables. Each of the target variables takes 3 distinct numeric values ($-1$,$0$,$1$) and there are 16 continuous input variables.
### SF1 & SF2
The Solar Flare dataset [@asuncion+newman:2007] has 3 target variables that correspond to the number of times 3 types of solar flare (common, moderate, severe) are observed within 24 hours. There are two versions of this dataset. SF1 contains data from year 1969 and SF2 from year 1978.
### WQ
The Water Quality dataset [@dzeroski2000] has 14 target attributes that refer to the relative representation of plant and animal species in Slovenian rivers and 16 input attributes that refer to physical and chemical water quality parameters.
### OES97 & OES10
The Occupational Employment Survey datasets were obtained from years 1997 (OES97) and 2010 (OES10) of the annual Occupational Employment Survey compiled by the US Bureau of Labor Statistics[^6]. Each row provides the estimated number of full-time equivalent employees across many employment types for a specific metropolitan area. There are 334 and 403 cities in the 1997 and May 2010 datasets, respectively. The input variables in these datasets are a randomly sequenced subset of employment types (i.e. doctor, dentist, car repair technician) observed in at least 50% of the cities (some categories had no values for particular cities). The targets for both years are randomly selected from the entire set of categories above the 50% threshold. Missing values in both the input and the target variables were replace by sample means for these results. To our knowledge, this is the first use of the OES dataset for benchmarking of multi-target prediction algorithms.
### ATP1d & ATP7d
The Airline Ticket Price dataset concerns the prediction of airline ticket prices. The rows are a sequence of time-ordered observations over several days. Each sample in this dataset represents a set of observations from a specific observation date and departure date pair. The input variables for each sample are values that may be useful for prediction of the airline ticket prices for a specific departure date. The target variables in these datasets are the next day (ATP1d) price or minimum price observed over the next 7 days (ATP7d) for 6 target flight preferences (any airline with any number of stops, any airline non-stop only, Delta Airlines, Continental Airlines, Airtrain Airlines, and United Airlines). The input variables include the following types: the number of days between the observation date and the departure date (1 feature), the boolean variables for day-of-the-week of the observation date (7 features), the complete enumeration of the following 4 values: 1) the minimum price, mean price, and number of quotes from 2) all airlines and from each airline quoting more than 50% of the observation days 3) for non-stop, one-stop, and two-stop flights, 4) for the current day, previous day, and two days previous. The result is a feature set of 411 variables. For specific details on how these datasets are constructed please consult [@groves2011]. The nature of these datasets is heterogeneous with a mixture of several types of variables including boolean variables, prices, and counts.
### SCM1d & SCM20d
The Supply Chain Management datasets are derived from the Trading Agent Competition in Supply Chain Management (TAC SCM) tournament from 2010. The precise methods for data preprocessing and normalization are described in detail in [@groves2011b]. Some benchmark values for prediction accuracy in this domain are available from the TAC SCM Prediction Challenge [@pardoe2008], these datasets correspond only to the “Product Future” prediction type. Each row corresponds to an observation day in the tournament (there are 220 days in each game and 18 tournament games in a tournament). The input variables in this domain are observed prices for a specific tournament day. In addition, 4 time-delayed observations are included for each observed product and component (1,2,4 and 8 days delayed) to facilitate some anticipation of trends going forward. The datasets contain 16 regression targets, each target corresponds to the next day mean price (SCM1d) or mean price for 20-days in the future (SCM20d) for each product in the simulation. Days with no target values are excluded from the datasets (i.e. days with labels that are beyond the end of the game are excluded).
### RF1 & RF2
The river flow datasets concern the prediction of river network flows for 48 hours in the future at specific locations. The dataset contains data from hourly flow observations for 8 sites in the Mississippi River network in the United States and were obtained from the US National Weather Service. Each row includes the most recent observation for each of the 8 sites as well as time-lagged observations from 6, 12, 18, 24, 36, 48 and 60 hours in the past. In RF1, each site contributes 8 attribute variables to facilitate prediction. There are a total of 64 variables plus 8 target variables.The RF2 dataset extends the RF1 data by adding precipitation forecast information for each of the 8 sites (expected rainfall reported as discrete values: 0.0, 0.01, 0.25, 1.0 inches). For each observation and gauge site, the precipitation forecast for 6 hour windows up to 48 hours in the future is added (6, 12, 18, 24, 30, 36, 42, and 48 hours). The two datasets both contain over 1 year of hourly observations ($>$9,000 hours) collected from September 2011 to September 2012. The training period is the first 40% of observations, and the test period is the remainder. The domain is a natural candidate for multi-target regression because there are clear physical relationships between readings in the contiguous river network.
Results {#sec:results}
=======
Table \[tbl:results\_arrmse\] shows the aRRMSE obtained by each multi-target method on each dataset while per target RRMSE results are shown in Table \[tbl:results\_rrmse\] (Appendix). The two last rows of Table \[tbl:results\_arrmse\] show the average rank of each method calculated over dataset RRMSE averages (aRRMSE) and over per target RRMSE results.
[l \*[6]{}[r]{}]{} Dataset & MORF & ST & MTS & MTSC & ERC & ERCC\
EDM & **73.38** & 74.21 & 74.30 & 73.96 & 74.35 & 74.07\
SF1 & 128.25 & 113.54 & 112.70 & 106.80 & **105.01** & 108.87\
SF2 & 142.48 & 114.94 & **94.48** & 105.53 & 105.32 & 108.79\
WQ & **89.94** & 90.83 & 91.10 & 90.95 & 90.97 & 90.59\
OES97 & 54.90 & 52.48 & 52.59 & 52.43 & 52.54 & **52.39**\
OES10 & 45.18 & 42.00 & 42.01 & 42.05 & 42.02 & **41.99**\
ATP1d & 42.22 & 37.35 & 37.16 & 37.17 & **37.10** & 37.24\
ATP7d & 55.08 & 52.48 & 51.43 & **50.74** & 53.43 & 51.24\
SCM1d & 56.63 & 47.75 & 47.41 & 47.01 & 47.09 & **46.63**\
SCM20d & 77.75 & 77.68 & 78.62 & 78.54 & 77.55 & **75.97**\
RF1 & 85.13 & **69.63** & 82.37 & 69.82 & 79.47 & 69.89\
RF2 & 91.89 & **69.64** & 81.75 & 69.86 & 79.61 & 69.82\
Avg. rank d & 5.00 & 3.42 & 4.08 & 2.83 & 3.42 & **2.25**\
Avg. rank t & 4.37 & 3.55 & 3.75 & 3.28 & 3.35 & **2.70**\
We observe that in both types of analyses ERCC obtains the lowest average rank, followed by MTSC. Surprisingly, the ST baseline obtains a lower average rank than MORF which has the worst average rank in both cases. We also see that both MTSC and ERCC obtain lower average ranks than the non-modified versions. When the Friedman test is applied to compare the six algorithms, it finds significant differences between them at $p=0.01$ in both the per dataset and the per target analysis. Thus, in both cases we proceed to the Nemenyi post-hoc test, whose results at $p=0.05$ are presented in the average rank diagrams of Figures \[fig:nemenyi\_average\] and \[fig:nemenyi\_pertarget\]. In the per dataset analysis case, the performance of ERCC is significantly better than that of MORF while the experimental data is not sufficient to reach statistically significant conclusions regarding the performance of the other methods. In the per target analysis case, the following significant performance differences are found: $ERCC > \{MORF, MTS, ST\}$ and $MORF < \{ST, ERC, MTSC\}$ where $>$ ($<$) denotes a statistically significant performance improvement (degradation).
To further study the impact of the proposed modification on MTS and ERC we apply the Wilcoxon signed-ranks test between MTS and MTSC and between ERC and ERCC. In the MTS/MTSC comparison, the p-value is 0.0425 and 0.0445 in the per dataset and the per target analysis respectively suggesting that the differences are statistically significant for a = 0.05. In the ERC/ERCC comparison, the p-value is 0.1763 and 0.0098 in the per dataset and the per target analysis respectively suggesting that the differences are not found statistically significant when the analysis is performed per dataset while in the per target analysis ERCC is found significantly better for a = 0.01.
Summarizing the comparative results, we see that the novel multi-label-inspired approaches obtain a better performance than the state-of-the-art MORF method in 10 out of 12 datasets and 82 out of 114 targets, indicating that the knowledge transfer between the two domains was successful. Nevertheless, the fact that there is a large variation in relative performance between the proposed methods across different prediction targets, suggests that their performance requires a further analysis especially compared to the performance of the ST baseline that without taking any target dependencies into account obtains the best performance in 2 out of 12 datasets and 12 out of 114 targets.
Conclusions and Future Work {#sec:conclusions}
===========================
This paper attempts to highlight the connection between multi-label classification and multi-target regression and investigates the applicability of methods proposed to solve the former task into the later task. The analysis of Section \[sec:related\] reveals that transformation multi-label methods that model each label independently are directly applicable to multi-target regression, simply by employing regression models.
In Section \[sec:methods\] we introduce two new techniques for multi-target regression, MTS and ERC, through straightforward adaptation of two corresponding multi-label learning methods that model label dependencies by using targets as inputs to meta-models. Furthermore, we detect an important shortcoming of these techniques, namely the discrepancy between the distribution of the target variables during training and during prediction, and propose modified versions of MTS (MTSC) and ERC (ERCC), that use cross-validation estimates of the targets during training.
As revealed by the empirical evaluation, this modification has a significant positive impact on the performance of the two methods. Equipped with this modification, MTSC and ERCC exhibit better performance than both the ST baseline and the state-of-the-art MORF approach. In particular, ERCC has the best overall performance and is found statistically significantly more accurate than both ST and MORF (when the analysis is performed per target).
Another important contribution of this paper is the introduction of 8 new publicly available multi-target datasets. This is important as most multi-target datasets mentioned in the literature are kept private due to their industrial nature, a fact that hinders benchmarking techniques and advancing state-of-the-art.
As future work, we would like to perform an analysis into why different multi-target methods perform better on different targets/datasets and to study the connection of these performance differences with dataset characteristics such as the degree and type of dependence between target variables. It would also be interesting to explore whether we could complete the circle back to multi-label classification, by investigating to what extend the discrepancy problem in MTS and ERC affects the corresponding multi-label classification algorithms and whether our modification can increase the generalization performance in this case as well.
Appendix {#sec:appendix .unnumbered}
========
[m[3cm]{} \*[6]{}[m[1.5cm]{}]{}]{} Dataset
------------------------------------------------------------------------
& & & & & &\
Target
------------------------------------------------------------------------
&&&&&&\
EDM
------------------------------------------------------------------------
&&&&&&\
dflow & **77.54** & 81.53 & 81.68 & 81.23 & 81.79 & 81.37\
dgap & 69.22 & 66.89 & 66.92 & **66.70** & 66.90 & 66.77\
SF1
------------------------------------------------------------------------
&&&&&&\
c-class & 103.46 & 101.68 & 103.72 & 100.08 & **100.03** & 100.67\
m-class & 121.16 & 109.63 & 113.68 & 100.50 & **99.18** & 103.35\
x-class & 160.12 & 129.29 & 120.70 & 119.81 & **115.84** & 122.61\
SF2
------------------------------------------------------------------------
&&&&&&\
c-class & 99.59 & 98.01 & 97.45 & **96.36** & 97.33 & 96.43\
m-class & 115.95 & 107.54 & 99.35 & **99.20** & 102.18 & 102.98\
x-class & 211.89 & 139.28 & **86.64** & 121.04 & 116.46 & 126.97\
WQ
------------------------------------------------------------------------
&&&&&&\
17300 & **89.53** & 90.21 & 92.10 & 91.39 & 90.63 & 90.79\
19400 & **82.79** & 83.42 & 82.93 & 82.94 & 83.55 & 82.85\
25400 & **92.38** & 92.45 & 93.05 & 92.81 & 92.95 & 92.72\
29600 & **97.63** & 98.65 & 98.62 & 98.38 & 98.54 & 98.26\
30400 & **94.20** & 94.52 & 94.49 & 95.11 & 94.37 & 94.61\
33400 & **89.32** & 91.20 & 90.37 & 90.23 & 91.21 & 90.18\
34500 & 95.95 & 96.95 & 96.14 & 96.34 & 96.50 & **95.71**\
37880 & 85.08 & 85.57 & 84.82 & **84.69** & 85.68 & 84.87\
38100 & **90.70** & 91.20 & 90.77 & 90.80 & 91.24 & 91.05\
49700 & **79.31** & 79.48 & 81.49 & 80.82 & 79.90 & 79.57\
50390 & **89.15** & 89.16 & 90.11 & 89.91 & 89.21 & 89.19\
55800 & **90.32** & 92.36 & 93.11 & 92.88 & 92.59 & 92.40\
57500 & **89.63** & 91.76 & 91.73 & 91.86 & 91.75 & 91.42\
59300 & **93.10** & 94.67 & 95.66 & 95.09 & 95.42 & 94.69\
OES97
------------------------------------------------------------------------
&&&&&&\
13008 & **27.17** & 33.94 & 33.97 & 33.74 & 33.92 & 33.79\
15014 & 46.50 & 35.83 & 36.54 & **35.16** & 35.88 & 35.24\
15017 & 40.25 & 36.49 & 36.96 & 36.79 & **36.47** & 36.61\
21114 & **29.07** & 30.69 & 30.72 & 30.82 & 30.70 & 30.73\
27108 & 63.64 & 58.02 & **57.63** & 57.66 & 58.07 & 57.70\
27311 & **56.59** & 60.08 & 59.47 & 59.51 & 59.90 & 59.80\
32314 & 70.43 & 61.41 & 61.54 & 61.72 & **61.29** & 61.52\
32511 & 75.16 & 71.67 & 71.50 & **71.37** & 71.58 & 71.42\
53905 & 60.21 & **56.89** & 57.41 & 57.17 & 57.00 & 57.00\
58028 & **26.46** & 33.62 & 33.48 & 33.69 & 33.67 & 33.64\
65032 & **53.46** & 55.20 & 55.43 & 55.01 & 55.18 & 55.14\
85110 & 63.60 & 56.69 & 57.45 & 56.83 & 56.68 & **56.63**\
92965 & 69.84 & 64.76 & 64.84 & 64.73 & 64.92 & **64.72**\
92998 & 70.27 & 64.36 & **64.16** & 64.42 & 64.69 & 64.25\
98502 & 71.49 & **66.49** & 66.61 & 66.54 & 67.05 & 66.50\
98902 & 54.28 & 53.59 & 53.72 & 53.63 & 53.63 & **53.52**\
\
[m[3cm]{} \*[6]{}[m[1.5cm]{}]{}]{} Dataset
------------------------------------------------------------------------
& & & & & &\
Target
------------------------------------------------------------------------
&&&&&&\
OES10
------------------------------------------------------------------------
&&&&&&\
119032 & 40.17 & 36.97 & **36.89** & 37.09 & 36.97 & 37.01\
151131 & 45.70 & 43.05 & 43.14 & 43.07 & 43.10 & **43.03**\
151141 & 49.61 & **43.64** & 43.88 & 44.05 & 43.80 & 43.80\
172141 & 59.30 & **49.44** & 49.63 & 49.56 & 49.49 & 49.45\
291051 & 26.65 & 26.46 & 26.72 & 26.58 & 26.51 & **26.40**\
291069 & 65.01 & 61.64 & 61.70 & **61.63** & 61.72 & 61.64\
291127 & **44.16** & 46.15 & 45.28 & 45.60 & 46.08 & 45.78\
292037 & 38.24 & **33.49** & 33.70 & 33.69 & 33.53 & 33.55\
292071 & 39.32 & 38.50 & 38.27 & **38.16** & 38.70 & 38.28\
392021 & **39.72** & 41.22 & 41.25 & 41.36 & 41.26 & 41.24\
412021 & 46.15 & 37.66 & 37.85 & 37.88 & **37.63** & 37.79\
419022 & 65.44 & 64.41 & 64.44 & 64.44 & 64.43 & **64.40**\
431011 & **22.72** & 26.86 & 26.72 & 27.09 & 26.84 & 26.90\
432011 & 39.69 & **39.15** & 39.22 & 39.22 & 39.19 & 39.17\
513021 & 45.34 & 43.81 & 43.79 & 43.72 & **43.64** & 43.79\
519061 & 55.66 & **39.48** & 39.67 & 39.66 & 39.52 & 39.55\
ATP1d
------------------------------------------------------------------------
&&&&&&\
acominpa & 35.63 & 24.19 & **23.32** & 24.16 & 24.17 & 23.94\
adlminpa & 42.44 & 41.57 & 40.46 & 40.96 & **39.76** & 41.20\
aflminpa & 48.69 & 47.12 & 47.97 & 47.30 & **46.97** & 47.31\
allminp0 & 43.65 & 42.97 & **42.00** & 42.20 & 42.64 & 42.79\
allminpa & **47.36** & 48.23 & 48.51 & 48.19 & 48.54 & 48.18\
auaminpa & 35.57 & **20.03** & 20.71 & 20.19 & 20.52 & 20.05\
ATP7d
------------------------------------------------------------------------
&&&&&&\
acominpa & 43.71 & 31.62 & **26.90** & 27.81 & 29.36 & 29.05\
adlminpa & **52.39** & 54.64 & 53.32 & 53.70 & 55.29 & 54.21\
aflminpa & **67.44** & 68.92 & 72.84 & 70.04 & 70.74 & 69.38\
allminp0 & **60.16** & 66.99 & 67.22 & 65.30 & 67.50 & 65.99\
allminpa & 63.56 & 64.07 & **61.89** & 62.93 & 69.57 & 62.60\
auaminpa & 43.20 & 28.61 & 26.43 & **24.65** & 28.10 & 26.23\
SCM1d
------------------------------------------------------------------------
&&&&&&\
lbl & 51.18 & 37.19 & 38.01 & **36.58** & 37.28 & 36.87\
mtlp10 & 51.57 & 45.13 & **40.33** & 44.24 & 41.83 & 44.56\
mtlp11 & 52.21 & 47.31 & **43.74** & 46.34 & 44.40 & 46.56\
mtlp12 & 56.19 & 43.06 & 43.80 & 43.21 & **42.93** & 42.96\
mtlp13 & 49.34 & 39.18 & 39.30 & 39.09 & **38.67** & 38.86\
mtlp14 & 45.45 & 37.56 & 38.27 & 37.37 & 37.23 & **37.17**\
mtlp15 & 45.39 & 39.62 & 39.05 & 39.95 & **38.99** & 39.54\
mtlp16 & 44.17 & 40.97 & **39.28** & 40.08 & 39.72 & 40.30\
mtlp2 & 50.47 & 38.18 & 37.06 & 37.32 & **36.61** & 37.19\
mtlp3 & 53.12 & 46.85 & 44.02 & **43.59** & 44.01 & 44.10\
mtlp4 & 56.22 & 46.98 & 50.71 & 48.48 & 47.34 & **46.89**\
mtlp5 & 72.79 & 59.56 & 63.64 & 58.87 & 64.17 & **58.20**\
mtlp6 & 76.07 & 66.05 & 65.82 & 64.95 & 65.63 & **64.77**\
mtlp7 & 73.23 & 64.51 & 64.42 & 63.59 & 63.82 & **61.05**\
mtlp8 & 77.48 & 70.56 & 69.13 & 67.74 & 70.40 & **66.17**\
mtlp9 & 51.11 & 41.31 & 42.07 & 40.82 & **40.35** & 40.83\
\
[m[3cm]{} \*[6]{}[m[1.5cm]{}]{}]{} Dataset
------------------------------------------------------------------------
& & & & & &\
Target
------------------------------------------------------------------------
&&&&&&\
SCM20d
------------------------------------------------------------------------
&&&&&&\
lbl & 65.64 & 62.86 & 64.90 & 62.28 & 64.47 & **61.76**\
mtlp10a & 76.43 & 74.16 & 74.04 & 77.84 & **73.94** & 74.76\
mtlp11a & 77.15 & 73.82 & 72.48 & 76.30 & 74.24 & **72.43**\
mtlp12a & 81.80 & 74.52 & 77.69 & 78.00 & 77.25 & **73.03**\
mtlp13a & 77.53 & 72.92 & **71.98** & 72.00 & 72.76 & 72.22\
mtlp14a & 75.35 & 72.49 & **71.41** & 73.78 & 71.48 & 71.93\
mtlp15a & 74.51 & 70.22 & 72.80 & 72.19 & 71.28 & **69.19**\
mtlp16a & 75.02 & 73.92 & 73.98 & 73.81 & 73.35 & **71.91**\
mtlp2a & 65.68 & 64.91 & 63.84 & 64.58 & 64.96 & **62.20**\
mtlp3a & 64.21 & 71.19 & 65.77 & 67.85 & **62.58** & 65.17\
mtlp4a & 69.04 & 71.74 & 74.78 & 72.93 & **67.33** & 69.11\
mtlp5a & **91.85** & 92.67 & 103.83 & 93.72 & 98.98 & 92.68\
mtlp6a & **94.61** & 97.17 & 99.64 & 99.38 & 99.02 & 95.80\
mtlp7a & **89.68** & 100.13 & 97.33 & 96.79 & 96.72 & 94.43\
mtlp8a & **94.01** & 99.00 & 103.54 & 101.70 & 100.99 & 99.27\
mtlp9a & 71.51 & 71.12 & 69.90 & 73.44 & 71.43 & **69.62**\
RF1
------------------------------------------------------------------------
&&&&&&\
chsi2 & 54.98 & 49.32 & 53.25 & **47.64** & 50.98 & 48.65\
clkm7 & 87.23 & 73.81 & 74.72 & **72.89** & 73.71 & 73.53\
dldi4 & 94.55 & **92.55** & 96.02 & 100.65 & 93.67 & 96.60\
eadm7 & 55.52 & 52.36 & **49.79** & 50.50 & 50.04 & 51.85\
napm7 & 119.36 & 58.01 & 62.89 & **57.48** & 59.52 & 57.83\
nasi2 & 101.19 & 103.84 & **100.82** & 102.33 & 102.83 & 103.15\
sclm7 & 75.09 & 66.82 & **62.56** & 64.95 & 62.93 & 65.86\
vali2 & 93.17 & **60.34** & 158.93 & 62.15 & 142.05 & 61.62\
RF2
------------------------------------------------------------------------
&&&&&&\
chsi2 & 59.04 & 49.30 & 53.06 & **47.56** & 51.06 & 48.63\
clkm7 & 83.62 & 74.09 & 75.01 & **73.57** & 74.00 & 74.02\
dldi4 & **92.33** & 92.51 & 96.00 & 100.47 & 93.72 & 96.43\
eadm7 & 56.91 & 52.42 & 50.32 & 50.35 & **50.08** & 51.88\
napm7 & 148.90 & 58.01 & 62.41 & **57.09** & 59.44 & 57.81\
nasi2 & **100.55** & 103.90 & 100.76 & 102.49 & 103.01 & 103.25\
sclm7 & 95.89 & 67.41 & **62.50** & 66.12 & 62.81 & 66.38\
vali2 & 97.87 & **59.50** & 153.91 & 61.21 & 142.76 & 60.15\
[^1]: See the motivation of the NIPS 2011 workshop on relations among machine learning problems at <http://rml.anu.edu.au/>
[^2]: <http://mulan.sourceforge.net>
[^3]: <http://www.cs.waikato.ac.nz/ml/weka>
[^4]: <http://dtai.cs.kuleuven.be/clus/>
[^5]:
[^6]: <http://www.bls.gov/>.
|
---
abstract: |
According to the recent rulings of the Federal Communications Commission (FCC), TV white spaces (TVWS) can now be accessed by secondary users (SUs) after a list of vacant TV channels is obtained via a geo-location database. Proper business models are therefore essential for database operators to manage geo-location databases. Database access can be simultaneously priced under two different schemes: the registration scheme and the service plan scheme. In the registration scheme, the database reserves part of the TV bandwidth for registered White Space Devices (WSDs). In the service plan scheme, the WSDs are charged according to their queries.
In this paper, we investigate the business model for the TVWS database under a hybrid pricing scheme. We consider the scenario where a database operator employs both the registration scheme and the service plan scheme to serve the SUs. The SUs’ choices of different pricing schemes are modeled as a non-cooperative game and we derive distributed algorithms to achieve Nash Equilibrium (NE). Considering the NE of the SUs, the database operator optimally determines pricing parameters for both pricing schemes in terms of bandwidth reservation, registration fee and query plans.
author:
- 'Xiaojun Feng, Qian Zhang, Jin Zhang, [^1] [^2]'
title: |
A Hybrid Pricing Framework for TV\
White Space Database
---
TV White Space, Geo-location Database, Pricing, Game Theory, Contract Theory.
Introduction
============
Recently, the FCC has released the TVWS for secondary access [@FCCrule10][@FCCrule12] under a database-assisted architecture, where there are several geo-location databases providing spectrum availability of the TV channels. These databases will be managed by database operators (DOs) approved by the FCC. To manage the operation costs, proper business models are essential for DOs.
The FCC allows DOs to determine their own pricing schemes [@FCCrule10] and there are two different ways in which SUs can assess the TVWS with the help of a database. SUs can register their WSDs in the database in a soft-licence style [@crwoncom12]. Part of the available TV spectrum is then reserved for the registered SUs [@ReserveFCC][@ReserveOfcom]. Unregistered SUs can also access TVWS in a purely secondary manner. For instance, an SU can first connect to the database and upload WSD information such as location and transmission power and then obtain a list of available channels from the database.
As a result, two different pricing schemes can be employed, one for registered and the other for unregistered SUs, respectively. Registered SUs pay a registration fee to DOs and access the reserved bandwidth exclusively. This pricing scheme can be referred to as the *registration scheme*. Unregistered SUs query the database only when they are in need of TV spectrum. DOs charge them according to the number of database queries they make. This pricing scheme is referred to as the *service plan scheme*. The co-existence of multiple pricing channels allows DOs to better manage their costs and provides different service qualities to different types of SUs. For example, the registration scheme can be adopted by SUs providing rural broadband or smart metering services, since the reserved bandwidth may suffer from less severe interference. On the other hand, the service plan scheme suits the temporary utilization of TVWS such as home networking.
To harvest the advantages of both pricing schemes and maximize the profit of DOs, two challenges need to be addressed. First, how should DOs determine pricing parameters for each scheme? With limited available TV bandwidth, DOs need to decide how much to allocate to each pricing channel. Also, the registration fee in the registration scheme and the price for a certain amount of queries should be determined. Second, how should SUs choose between the two schemes? Both schemes have their pros and cons for different types of SUs. The two challenges are coupled together. The decisions of SUs on which pricing scheme to choose can affect the profit obtained by DOs while the pricing parameters designed by DOs ensure SUs have different preferences for either scheme.
In this paper, we focus on DO’s hybrid pricing scheme design considering both the registration and the service plan scheme. To the best of our knowledge, there are no existing works on geo-location database considering a hybrid pricing model [@database_dyspan08]-[@database_icc12]. We consider one DO and multiple types of SUs. The SUs can strategically choose between the two pricing schemes. Unlike many existing works consider no SU strategies when there exist multiple pricing schemes, in this paper, we assume users can have their own choices other than being directly classified into either pricing scheme. We argue that especially for a new service like database-based networking, users will consider seriously of the benefits from each scheme and other players’ responses.
In this paper, we consider a two-stage pricing framework. At Stage I, the DO announces the amount of bandwidth to be reserved and the registration fee for the registered SUs and then the SUs choose whether to register or not. At Stage II, the DO announces a set of service plans for the unregistered SUs to choose from.
For the SUs, they decide which pricing scheme to choose given the announced pricing schemes. If the service plan scheme is adopted, SUs should further decide one particular plan to buy. In this paper, we consider two different SU scenarios. In the non-strategic case, SUs have fixed their pricing scheme choices. In the strategic case, SUs can compete with each other in choosing either pricing schemes. In the later case, the competition among the SUs is modeled with the non-cooperative game theory since the choices of the other SUs can affect an SU’s utility owing to the sharing of TVWS.
For the DO, the problem is to optimally allocate bandwidth and design pricing parameters based on estimated actions of the SUs. In this paper, we consider the DO has either complete or incomplete information of the SUs. By complete information, we mean that the DO knows the exact *type* of each SU. The type of SU relates with its channel quality, valuation for the spectrum et. al. In the case of incomplete information, the DO knows only a distribution of the types. We model the optimal service plan design problem with contract theory. Different service plans are considered to be different contract items and an optimal contract is determined based on the knowledge of the SU types. To optimally choose pricing parameters, one challenge to tackle is how the DO estimate the possible actions of the SUs, especially for the strategic case. There may be multiple possible equilibriums existed in the game. We solve this challenge by exploring the nature of our problem and propose computationally feasible algorithm to estimate the outcome of the game.
The major contributions of this paper are summarized as follows:
- We propose a hybrid pricing scheme for the DO considering the heterogeneity of SUs’ types. As far as we know, it is the first work considering hybrid pricing schemes for TVWS database.
- We model the competition among the strategic SUs as a non-cooperative game. We prove the existence of the Nash Equilibrium (NE) under both the complete and incomplete information cases. By exploring the nature of the problem, we design computational feasible distributed algorithms for the SUs to achieve an NE with bounded iterations.
- We propose algorithms for the DO to optimally decide pricing parameters. We formulate the service plan design with contract theory and derive optimal contract items under the complete information and sub-optimal items for and incomplete information scenario.
The rest of this paper is organized as follows. In Section \[sec:model\], we describe the pricing framework and detailed system model. Problem formulation is presented in Section \[sec:formulation\]. In Section \[sec:non-strategic\_complete\] we study the optimal pricing solution for non-strategic SUs under the complete information scenario as a baseline case. Then in Section \[sec:strategic\_complete\] and Section \[sec:strategic\_incomplete\], we consider the SUs to be strategic players, under complete and incomplete information scenario, respectively. Numerical results are given in Section \[sec:simulation\]. Related works are further reviewed in Section \[sec:related\_works\]. Finally, Section \[sec:conclusion\] concludes the paper.
System Model and Pricing Framework {#sec:model}
==================================
In this section, after an overview of the system parameters, we introduce the big picture of the proposed pricing framework. Then we further detail the model for the DO and the SUs, respectively. Key notations are summarized in Table \[tab:Notations\].
System Parameters {#subsec:parameter}
-----------------
We consider the scenario with one DO and $N$ SUs denoted as $\mathcal{N}=\{u_1, u_2, \cdots, u_N\}$. We assume $N$ is public information available to the DO and the SUs. For the ease of analysis, we assume all the SUs are in the same contention domain [@contention_domain]. FCC requires the SUs to periodically access the database [@FCCrule10]. In this paper, we consider a time duration of $M$ periods. In each period, before accessing the TV channels, the SUs should connect to the database to obtain a list of available channels. We assume the total available TV channels in each period have an expected bandwidth of $B$ and the bandwidth of a channel is $b_0$. The number of available channels is then $B/b_0$.
$N$ The number of SUs
------------- ----------------------------------------------------------------------------------------------
$M$ The number of time periods considered
$u_i$ The secondary user with index $i$
$a_i$ The pricing scheme selected by $u_i$, 0: the registration scheme; 1: the service plan scheme
$\theta_i$ The $type$ of $u_i$
$B$ The expected bandwidth of available TVWS
$B_R$ The expected bandwidth reserved for registered SUs
$r$ The registration fee
$q_i$ The number of database queries in a query plan $(q_i, p_i)$
$p_i$ The wholesale price in the query plan $(q_i, p_i)$
$\mu_0$ The number of the registered SUs
$\mu_1$ The number of the unregistered SUs
$\phi_0(b)$ The cost to reserve $b$ bandwidth
$\phi_1(q)$ The maintenance cost for $q$ database queries
$T$ The number of different SUs’ types
$\theta^i$ The $i$’th smallest user type
$\beta^i$ The percentage of SUs with type $\theta^i$
$\gamma^i$ The percentage of SUs with type $\theta^i$ with registration scheme
: Key notations in this paper.[]{data-label="tab:Notations"}
Pricing Framework {#subsec:framework}
-----------------
The DO offers two different pricing schemes, the registration scheme and the service plan scheme, simultaneously.
In the registration scheme, the SUs are registered with a uniform registration fee $r$. The DO reserves a fraction of $B_R/B$ ($0\leq B_R \leq B$) of the total TV bandwidth for them in each period. The expected number of reserved channels is $B_R/b_0$. The DO further divides the reserved channels equally to serve each registered SU. To make the problem solvable, in this paper, we consider only uniform registration fee. In a more flexible setting, the registered SUs can pay different amount of registration fees and enjoy different shares of the reserved bandwidth. We also assume an SU can refuse to pay the registration fee to abort the deal.
In the service plan scheme, the DO offers the SUs several query plans, which are a set of $K+1$ query-price combinations denoted as $\mathcal{P}=\{(q_i, p_i), i=0,1,\cdots, K\}$. During the $M$ periods, the SUs with chosen plan $(q_i, p_i)$ can access the database $q_i$ times with a wholesale price $p_i$. Note that $q_i$s are integers and $0\leq q_i\leq M$. We assume there is a plan $(q_0=0, p_0=0)$ for SUs choosing neither scheme. The unregistered SUs shared the unreserved band in a time division manner. For example, these SUs can leverage CSMA in the MAC layer for channel access as today’s Wi-Fi.
The pricing procedure is conducted at the beginning of the $M$ periods and can be viewed as two stages. At stage I, the DO announces parameters $B_R$ and $r$ for the registration scheme. Then the SUs decide whether to register. At stage II, the DO first announces $\mathcal{P}$ for all unregistered SUs and then these SUs specify which particular query plan to buy. An SU can always choose $(q_0, p_0)$ to abort the deal.
The procedure of pricing and interactions between the DO and the SUs are summarized in Fig. \[fig:procedure\].
![The procedure of pricing and access of TVWS.[]{data-label="fig:procedure"}](procedure.pdf "fig:"){width="3.2in"}\
DO Model {#subsec:DO}
--------
We assume the DO has two different sources of database maintenance cost. On one hand, to reserve bandwidth for the registered SUs, DO needs to pay a bandwidth reservation fee to the regulators. We assume the cost to reserve $b$ MHz bandwidth is in the form $\phi_0(b)=\epsilon_0\cdot b^{\alpha}$, $\alpha\geq 1$. The reservation cost is a convex function on $b$. The key reason is that spectrum bandwidth is a limited resource. Reservation must be approved by the regulator and there can be multiple databases trying to reserve bandwidth. When the total demand is higher, the equilibrium price can be also higher [@reservation_price]. Furthermore, to avoid unnecessary reservations, the regulator may consider charge higher for more reserved bandwidth [@ReserveOfcom]. On the other hand, there is also the cost for the DO to calculate the available bandwidth and respond to the queries of unregistered SUs. Here we adopt a linear cost function on the query number. We assume for $q$ queries, the DO has to pay a marginal maintenance cost $\phi_1(q)=\epsilon_1\cdot q$.
We define the *utility* of the DO, denoted by $U_{DO}$ as the difference between the payment received from all the SUs and the total cost. For convenience, we assume the number of SUs choosing query plan $i$ is $N_i, (i=0, 1, \cdots, K)$, the number of registered SUs is $\mu_0$ and the number of unregistered users is $\mu_1$ then $$\begin{aligned}
\label{eqn:u_do}
U_{DO}&=&\mu_0\cdot r+\sum_{i=0}^K N_i\cdot p_i-\phi_0(B_R)-\phi_1(\sum_{i=0}^K N_i\cdot q_i).\end{aligned}$$
SU Model {#subsec:SU}
--------
According to the pricing framework, the SUs choosing the same pricing scheme share the spectrum evenly. For $u_i$, if there are $X$ SUs sharing the same bandwidth of $W$, its capacity is given by the Shannon-Hartley theorem as: $
C_i(X, W)=\frac{1}{X}\cdot W\log_2\left(1+\frac{S_i}{n_0}\right)
$, where $S_i$ is the average received signal power over bandwidth $W$ at $u_i$’s receiver, $n_0$ is the average noise and interference power over the bandwidth. $S_i$ relates to the transmission power of $u_i$ and the path loss between $u_i$’s transmitter and receiver. Without loss of generality, we assume $n_0$ is identical for all SUs.
We assume SUs have linear valuations for any expected capacity they achieve. We define the valuation function of $u_i$ as: $
V_i(X, W)=w_iC_i(X, W)
$, where $w_i$ is the valuation parameter of $u_i$. $w_i>0$ and its physical meaning is that the valuation of the unit channel capacity contributes to the overall benefit. $w_i$ relates to the personal information of $u_i$ such as the possible transmission duration in each period.
We define the *utility* of $u_i$, denoted as $U_i$, as the difference between its valuation of the guaranteed capacity and the price charged by the DO. If $u_i$ chooses the registration scheme, it can occupy part of the TV bandwidth by itself. Its utility is $$U_i=V_i(1, \frac{B_R}{\mu_0})-r=w_i\cdot \frac{B_R}{\mu_0}\log_2\left(1+\frac{S_i}{n_0}\right)-r.$$ If $u_i$ chooses the service plan scheme, it should share the bandwidth with other unregistered SUs. There are at most $\mu_1$ unregistered SUs sharing, any one of them is guaranteed with a valuation of $V_i(\mu_1, B-B_R)$ in one access. Moreover, $u_i$ can only access at most $q_j$ periods after selecting query plan $(q_j, p_j)$. Therefore, its utility is $$\begin{aligned}
U_i&=&V_i(\mu_1, B-B_R)\cdot\frac{q_j}{M}-p_j \\
&=&w_i\cdot \frac{B-B_R}{\mu_1}\log_2\left(1+\frac{S_i}{n_0}\right)\cdot \frac{q_j}{M}-p_j.\end{aligned}$$
From the model, we can see that $S_i$, $w_i$ are personal parameters for $u_i$. We further define $\theta_i=w_i\log_2\left(1+\frac{S_i}{n_0}\right)$ as the *type* of $u_i$ and $v(q)=\frac{B-B_R}{\mu_1}\frac{1}{M}q$ as the *valuation* of $q$ queries. The physical meaning of $\theta_i$ is the capacity of $u_i$ that can turn unit bandwidth into revenue. We use $a_i$ to denote the pricing scheme selected by $u_i$. $a_i=0$ represents the registration scheme and $a_i=1$ represents the service plan scheme. The utility function of $u_i$ can be summarized as $$\label{equ:u_su}
U_i=\left\{
\begin{array}{ll}
\frac{B_R}{\mu_0}\theta_i-r, & \textrm{if $a_i=0$} \\
\theta_iv(q_j)-p_j, & \textrm{if $a_i=1$ and the plan is $(q_j, p_j)$}
\end{array}
\right.$$
We assume there are a total of $T$ different types and we denote the types by the set $\Theta=\{\theta^1, \theta^2, \cdots, \theta^T\}$. Without loss of generality, we assume $0<\theta^1< \theta^2< \cdots <\theta^T$.
Problem Formulation {#sec:formulation}
===================
In this section, we formulate the pricing problem according to the two-stage pricing framework.
SU Strategy
-----------
In this paper, we consider two different cases of SUs based on their abilities to act strategically.
- *Non-strategic SUs*: in this case, for $u_i$, $a_i$ is fixed before the pricing procedure. SUs are categorized with pricing scheme choices. An SU will not change the choice of the pricing scheme strategically according to the choices of other users.
- *Strategic SUs*: in this case, for $u_i$, $a_i$ is not fixed before the pricing procedure. $u_i$ can determine its pricing scheme dynamically according to the choices of other users.
Actually, we argue that both assumptions are reasonable in real world. In matured services [@icdcs12segmentation], different pricing schemes have clear pros and cons. Users tend to have their preferences. In emerging services, the SUs may have no experiences or preferences.
In the case of non-strategic SUs, $a_i$ is fixed. We assume the fraction of SUs with type $\theta^i$ choosing $a_i=0$ is $\gamma^i$ ($i=1,\cdots, T$). And we assume the ratio of SUs of type $\theta^i$ among all SUs is $\beta^i$. Therefore, the total number of SUs with $a_i=0$ is $N\sum_{i=1}^T\beta^i\gamma^i$ and the total number of SUs with $a_i=1$ is $N\sum_{i=1}^T\beta^i(1-\gamma^i)$.
In the case of strategic SUs, $a_i$ is not fixed. SUs strategically choose either pricing scheme considering other SUs’ strategies. Therefore, at stage I, given $B_R$ and $r$, the SUs compete for pricing schemes. The interactions among the SUs form a *Non-cooperative Database Registration Game (NDRG)* $G=(\mathcal{N}, \mathcal{S}=\{1,0\}, \{U_i\}_{u_i\in \mathcal{N}})$, where $\mathcal{N}$ is the set of players, $\mathcal{S}$ is the set of strategies, and $U_i$ is the payoff function of play $u_i$. Let $\mathbf{a}=(a_1, a_2, \cdots, a_N)$ be the strategy-tuple of all the SUs. Let $a_{-i}=(a_1, a_2, \cdots, a_{i-1}, a_{i+1}, \cdots, a_N)$ be the strategy-tuple of all the SUs expect $u_i$.
A strategy vector $\mathbf{a^*}=(a_1^*, a_2^*, \cdots, a_N^*)$ is an NE of the game NDRG if $
a_i^*={\operatornamewithlimits{argmax}}_{a_i\in S}U_i(a_i, a_{-i}^*), \forall u_i \in\mathcal{N}.
$
Information Scenario
--------------------
In this paper, we also study the following two different information scenarios.
- *Complete information*: in this case, the DO and the SUs are perfectly informed about the type of each SU.
- *Incomplete information*: in this case, the DO and the SUs know only the distribution of the SUs’ types.
In the complete information scenario, the DO can treat each SU separately and offer a type-specific contract item. However, in the incomplete information scenario, the DO cannot observe the type of each SU, it has to offer the same set of query plans to all SUs.
To tackle the two different information scenarios, we model the design of query plans with contract theory [@contract_book]. The set of query plans $\mathcal{P}$ can be viewed as the set of contact items. The DO is the seller and the SUs are the buyers. The goods offered by the seller are the database queries and the wholesale price for $q_i$ queries is $p_i$. The buyers’ types are their private information. According to the revelation principle [@contract_book][@revelation], it is sufficient to design at most $T$ contract items, one for each type of the SUs, to enable them to truthfully reveal their types. Therefore, we assume $K=T$ and an unregistered SU of type $\theta^i$ will eventually choose plan $(q_i, p_i)$.
Backward Induction for the Two-stage Pricing Framework
------------------------------------------------------
Based on the two-stage pricing procedure, the design of the optimal pricing schemes can be analyzed with backward induction. More specifically, we first show how should the DO design the query plans for the SUs at stage II, given $B_R$, $r$ and SUs’ strategy-tuple $\mathbf{a}$. Then we study how the SUs choose the pricing scheme given $B_R$, $r$ announced at stage I. Finally, based on the knowledge of the possible pricing scheme selection of the SUs, the DO optimally selects $B_R$ and $r$ at stage I.
In this paper, we will first consider a baseline scenario where SUs are non-strategic players and the DO processes complete information of all the SUs. Then, we extend our analysis to strategic SUs under both complete and incomplete information scenarios. By comparing the first and the second scenario, we show the impact of SU strategy on the pricing design. By comparing the second and the third scenario, we show the impact of information completeness. Due to the page limit and similarity in analysis, we omit the case of non-strategic SUs under incomplete information.
Pricing Solution for Non-strategic SUs in Complete Information Scenario {#sec:non-strategic_complete}
=======================================================================
In this section, we consider a baseline case where the SUs are non-strategic players and the DO has complete information of all the SUs.
SU Behavior Given Pricing Parameters
------------------------------------
Since the SUs fix their choices of pricing schemes, we can analyze the registered and unregistered SUs separately.
At stage I, some SUs with $a_i=0$ may refuse to pay the fee if it is too high for them. Given $B_R$ and $r$, $u_i$ pays the registration fee if and only if the following condition holds: $
\frac{B_R}{\mu_0^*}\theta_i-r\geq 0,
$ where $\mu_0^*=N\sum_{i=1}^T\beta^i\gamma^i$ is the prior knowledge of the number of registered SUs. Non-strategic SUs decide whether to pay the fee only based on the prior knowledge. The actual number of registered SUs can then be calculated as $$\label{eqn:mu0}
\mu_0=\sum_{i\in\left\{i|\frac{B_R}{\mu_0^*}\theta^i-r\geq 0\right\}}N\beta^i\gamma^i.$$
At stage II, there are $\mu_1=N\sum_{i=1}^T\beta^i(1-\gamma^i)$ unregistered SUs. Since any $u_i$ can be guaranteed with non-negative utility by choosing $(q_0, p_0)$, to enable $u_i$ to select query plan $(q_i, p_i)$, $q_i, p_i$ should satisfy $\theta_iv(q_i)-p_i\geq 0, \ \forall 1\leq i\leq N$. The constraints are usually referred to as the *Individual Rationality* (IR). The following lemma says that all the IR constraints are bind under complete information.
\[lemma:IR\_complete\] At stage II, if the DO has complete information of $\theta_i$ for each $u_i$, the optimal query plan $(q_i, p_i)$ offered to $u_i$ satisfies: $\theta_iv(q_i)-p_i=0, \ 1\leq i\leq N$.
We prove by contradiction. Suppose in the optimal pricing design, the DO offers $(q_i^*, p_i^*)$ to $u_i$ at stage II to optimize $U_{DO}$ and $\theta_iv(q_i^*)-p_i^*>0$. However, when the DO increases the price $p_i^*$ by $\Delta p_i^*=\theta_iv(q_i)^*-p_i^*$ and keeps all the other pricing parameters unchanged. $U_{DO}$ can be increased by $\Delta p_i^*$, which contradicts with the optimality of $(q_i^*, p_i^*)$. Therefore, we must have $\theta_iv(q_i^*)-p_i^*=0$.
According to Lemma \[lemma:IR\_complete\], we have: $$\label{eqn:IR}
\theta_iv(q_i)-p_i = 0, \quad \forall 1\leq i\leq N.$$
Optimal Parameter Selection of DO
---------------------------------
By substituting (\[eqn:mu0\]) and (\[eqn:IR\]) into (\[eqn:u\_do\]) we can rewrite $U_{DO}$ as $$\begin{aligned}
\label{eqn:UDO0}
U_{DO}(B_R, r, \mathcal{P})&=&\mu_0\cdot r + \sum_{i=0}^KN_ip_i-\phi_0(B_R)-\phi_1(\sum_{i=0}^KN_iq_i) \nonumber \\
&=&\sum_{i\in\left\{i|\frac{B_R}{\mu_0^*}\theta^i-r\geq 0\right\}}N\beta^i\gamma^i\cdot r - \epsilon_0B_R^{\alpha} \nonumber\\
&+&\sum_{i=1}^TN\beta^i\left(1-\gamma^i\right)\left(\frac{B-B_R}{\mu_1M}\theta^i-\epsilon_1\right)\cdot q_i.\end{aligned}$$
The optimization problem for DO is $$\begin{aligned}
\label{eqn:max_udo0}
&&\max_{B_R, r, \{q_i\}} U_{DO}(B_R, r, \mathcal{P}) \nonumber\\
&\textrm{subject to:} & 0\leq q_i\leq M (\forall 1\leq i\leq T), \quad\textrm{and}\quad 0\leq B_R\leq B, \nonumber\\
&\textrm{variables:} & \{q_i(\forall 1\leq i\leq T), B_R, r\}.\end{aligned}$$
Problem (\[eqn:max\_udo0\]) is a non-convex optimization problem and there is no closed-form solution. The first term $\sum_{i\in\left\{i|\frac{B_R}{\mu_0^*}\theta^i-r\geq 0\right\}}N\beta^i\gamma^i$ in Eqn. (\[eqn:UDO0\]) is a piecewise constant function of $B_R$. The $M$ points $\frac{r}{\theta^{i}}\mu_0^*$ divide the domain of $B_R$ into $M+1$ different intervals. It is a quadratic programming problem to optimize (\[eqn:UDO0\]) in each interval. The problem (\[eqn:max\_udo0\]) thus can be decomposed into $M+1$ quadratic programming problems, which may be NP-hard in the general form [@nonlinear].
Considering the nature of our problem, the physical meaning of $B_R$ is the bandwidth reserved for registered SUs and $B_R$ is measured in channels. The possible values of $B_R/b_0$ are integers in the range $[0, B/b_0]$. Also, we observe that $U_{DO}$ has a linear relationship for all $\{q_i\}$, then $q_i$ will either be 0 or $M$. $$\label{eqn:q0}
q_i=\left\{
\begin{array}{ll}
0 & \textrm{if $\frac{B-B_R}{\mu_1}\frac{1}{M}\theta^i-\epsilon_1\leq0$} \\
M & \textrm{otherwise}
\end{array}
\right.$$ Furthermore, the range and scale of the registration price $r$ are also limited in reality. If we also restrict the range of $r$ to be $[\emph{\b{r}}, \emph{\={r}}]$ and its minimum scale to be $r_0$, the DO needs only to explore a total of $\frac{B}{B_R}\cdot\frac{\emph{\={r}}-\emph{\b{r}}}{r_0}$ combinations to find the solution. In this paper, we will leverage this simpler but efficient two-dimensional search method to find the pricing parameters.
Therefore DO tries to solve the following optimization problem $$\begin{aligned}
\label{eqn:max_udo1}
&&\max_{B_R, r, \{q_i\}} U_{DO}(B_R, r, \mathcal{P}) \nonumber\\
&\textrm{subject to:}& \quad (\ref{eqn:q0}) \quad\textrm{and}\quad 0\leq B_R\leq B.\end{aligned}$$ We provide the numerical results in Section \[sec:simulation\].
Pricing Solution for Strategic SUs in Complete Information Scenario {#sec:strategic_complete}
===================================================================
In this section, we consider the SUs to be strategic players.
Stage II: Optimal Contract Design under Complete Information
------------------------------------------------------------
Same as the case discussed in Section \[sec:non-strategic\_complete\], when the DO processes complete information of the SUs, any SU choosing the service plan scheme will have zero utility. The contract items ${(q_i, p_i)}$ satisfy the condition in Eqn. (\[eqn:IR\])
Stage I: Database Registration Game of SUs
------------------------------------------
### Existence of an NE
At stage I, SUs compete against each other in the NDRG considering the outcome at stage II given the parameters $B_R$ and $r$. The utility function of $u_i$ can be rewritten as $$\begin{aligned}
U_i(a_i, a_{-i})=\left\{
\begin{array}{ll}
\frac{B_R}{\mu_0}\theta_i-r & \textrm{if $a_i=0$} \\
0 & \textrm{if $a_i=1$}
\end{array}
\right. \label{eqn:u_complete_info}\end{aligned}$$
We show that the NDRG under complete information is a *Unweighted Congestion Game* [@congestion_game].
A non-cooperative game satisfying the following condition is referred to as the unweighted congestion game:
- The players share a common set of strategies.
- The payoff the $i^{th}$ player receives for playing the $j^{th}$ strategy is a monotonically non-increasing function of the total number $\mu_j$ of players playing the $j$th strategy.
\[lemma:NDRG\_is\_CG\] NDRG under complete information is an unweighted congestion game.
It is easy to see that all the players in NSDG share a common set of strategies $\mathcal{S}$. In the scenario of complete information, the payoff function of player $u_i$ is defined as $U_i$ in (\[eqn:u\_complete\_info\]). For strategy 1, $U_i$ is constant. For strategy 0, $U_i$ is a monotonically non-increasing function with $\mu_0$.
The nice properties of the unweighted congestion game are summarized as lemma [@congestion_game].
\[lemma:property\_of\_CG\] Every unweighted congestion game $\Gamma$ possesses an NE in pure strategies. Given an arbitrary strategy-tuple $\mathbf{a}$ in $\Gamma$, there exists a best-reply improvement path [@congestion_game] with $L$ steps and $L\leq s{{n+1}\choose 2}$, where $s$ is the number of all strategies and $n$ is the number of players in $\Gamma$.
### Distributed Database Registration Algorithm
Lemma \[lemma:property\_of\_CG\] confirms the existence of an NE in NDRG under complete information. Starting from any strategy-tuple, an NE can be achieved with $L\leq s{{n+1}\choose 2}$ steps. In our scenario, $s=2$ and $n=N$. $L$ is therefore bounded by $N(N+1)$. Observe that when selecting the registration scheme, SUs of higher types have higher utility. When more SUs choosing the registration scheme, the utility of SUs of lower types becomes non-positive sooner. Therefore, the best strategy for an SU is to make sure that no more SUs with higher types will choose the registration scheme. Otherwise, if an SU chooses registration and more SUs with higher types also choose, the SU will be under the risk of non-positive utility. We can then design a distributed algorithm for SUs to achieve an NE without any improvement steps. The key idea is to allow the SUs to choose the registration scheme in a descending order of their types until no more SUs have incentives to join.
We assume that each SU is assigned a unique ID by the DO when connecting with the database and it knows the number of existing registered SUs at stage I, by checking the database. Without loss of generality, we assume the ID of $u_i$ is $i$. In the complete information scenario, the SUs know the exact value of the type and the ID of each other. Therefore, all the SUs can independently calculate a threshold of user type denoted as $\theta_{i_M}$. When SUs of type higher than $\theta_{i_M}$ are registered, their utilities are positive. But when SUs with type equal to $\theta{i_M}$ also register, these SUs can only have non-positive utility. Therefore, only part of the SUs with type equal to $\theta{i_M}$ can choose registration. After the calculation, the SUs with type higher than $\theta_{i_M}$ then choose the registration scheme. And the SUs of $\theta_{i_M}$ choose to register in the order of the assigned ID. Finally the remaining SUs with even lower types choose the service plan scheme. The decisions of the SUs are made at the same time. The procedure is summarized in Algorithm \[alg:registration\_complete\_info\].
$a_i=1, \forall 1\leq i\leq N$; $\quad i_M=T$; $\quad R=0$; $R=R+N\cdot\beta^{i_M}$; $\quad i_M=i_M-1$; **if** [$i_M=0$]{} **then** BREAK out of WHILE; $a_i=0, \forall \theta_i> \theta^{i_M}$; $R=R+1$; $\quad a_i=0$;
\[thm:algorithm1\_property\] Under complete information, Algorithm \[alg:registration\_complete\_info\] is guaranteed to achieve an NE of NDRG. Its time complexity is $O(N)$.
We first prove that all the SUs with the registration scheme have positive payoffs. In Algorithm \[alg:registration\_complete\_info\], suppose the last SU selecting the registration scheme is $u_i$, $u_i$ has positive utility since $B_R\theta_{i}>\mu_0\cdot r$ ($\mu_0$ is the final number of registered SUs). Any other registered SU $u_j$, must has $\theta_{j}\geq \theta_{i}$. Therefore, we have $B_R\theta_{j}\geq B_R\theta_{i}>\mu_0\cdot r$.
Then we show that there is no SU with the service plan scheme has the incentive to change its strategy. Any unregistered SU $u_k$ with $\theta_k=\theta_i$ will not change strategy since the condition $B_R\theta_{k}>(\mu_0+1)\cdot r$ is not satisfied. Therefore, its payoff will not improve by changing to the registration scheme. Also, for any other unregistered SU $u_l$ with $\theta_l<\theta_k$, $B_R\theta_{l}\leq B_R\theta_{k}\leq(\mu_0+1)\cdot r$.
In summary, no player has the incentive to deviate from the strategy provided by Algorithm \[alg:registration\_complete\_info\]. It is also easy to see the number of operations in the algorithm is $O(N)$.
Each SU can independently implement Algorithm \[alg:registration\_complete\_info\] by acquiring the number of currently registered SUs. Note that the converged NE is not unique if there are multiple SUs with the same type $\theta^{i_M}$. Due to the NE output proved in Theorem \[thm:algorithm1\_property\], no SU has the incentive to deviate unilaterally from the algorithm.
### Stage I: Optimal Parameter Selection of DO
At stage I, DO selects pricing parameters considering the NE of NDRG. From Algorithm \[alg:registration\_complete\_info\], DO can get an estimation of $\mu_0$ and $\mu_1$ as $
\mu_0=\mathcal{R}(B_R, r)=R_0+R_1, \quad\mu_1=N-\mu_0,
$ where $R_0=\sum_{i=i_M+1}^T N\cdot\beta^i$, $R_1=\arg\max_k\left\{\frac{B_R\theta^{i_M}}{r}>R_0+k\right\}$ and $i_M=\arg\min_k\left\{\frac{B_R\theta^k}{r}>\sum_{i=k}^T N\cdot\beta^i\right\}-1$. $R_0$ is the total number of SUs with type larger than $\theta^{i_M}$. $R_1$ is the number of registered SUs of type $\theta^{i_M}$.
Together with Eqn. (\[eqn:IR\]), we can rewrite $U_{DO}$ in (\[eqn:u\_do\]) as: $$\begin{aligned}
U_{DO}(B_R, r, \mathcal{P})&=&\left(\mu_0r-\epsilon_0B_R^{\alpha}\right) + \sum_{i=1}^{i_M-1} N\beta^i\left(\frac{B-B_R}{\mu_1M}\theta^i-\epsilon_1\right) q_i \nonumber\\
&+&\left(N\beta^{i_M}-R_1\right)\left(\frac{B-B_R}{\mu_1M}\theta^{i_M}-\epsilon_1\right) q_{i_M}.\end{aligned}$$
Here $U_{DO}$ also has a linear relationship for all $\{q_i\},\forall i<i_M$, we have: $$\label{eqn:q}
q_i=\left\{
\begin{array}{ll}
0 & \textrm{if $\frac{B-B_R}{\mu_1}\frac{1}{M}\theta^i-\epsilon_1\leq0$} \\
M & \textrm{otherwise}
\end{array}
\right.$$ The optimization problem for the DO becomes $$\begin{aligned}
\label{eqn:max_udo2}
&&\max_{B_R, r, \{q_i\}} U_{DO}(B_R, r, \mathcal{P}) \nonumber\\
&\textrm{subject to:}& \quad (\ref{eqn:q}) \quad\textrm{and}\quad 0\leq B_R\leq B.\end{aligned}$$ Similarly, we restrict the range of $r$ to be \[**, *r*\] and use a two-dimensional exhaustive search to find the solution. We provide the numerical results in Section \[sec:simulation\].
Pricing Solution for Strategic SUs in Incomplete Information Scenario {#sec:strategic_incomplete}
=====================================================================
In this section, we study the pricing problem under incomplete information for strategic SUs. Due to hidden information, at stage II, the DO cannot design query plans for each SU to extract all their revenue. Instead, the DO should offer a set of contract items for each type of SUs to choose from. Since SUs can obtain non-negative utility choosing the service plan scheme, the analysis of the NDRG will be much more complicated.
Stage II: Contract Design under Incomplete Information
------------------------------------------------------
In the incomplete information scenario, to ensure SUs of type $\theta^i$ have the incentive to select query plan $(q_i, p_i)$, another set of constraints called *Incentive Compatibility* (IC) should be satisfied: $$\theta^iv(q_i)-p_i\geq \theta^iv(q_j)-p_j, \forall 1\leq i,j\leq T.$$ The physical meaning of IC is that an SU with type $\theta^i$ achieves the maximum utility when choosing the corresponding contract item $(q_i, p_i)$. Note that there are a total of $N(N-1)$ IC constraints.
To analyze the design of optimal contract items, we first relax the condition that $q_i (1\leq i\leq T)$ are integer numbers and allow $q_i$s to be any real number in the range $[0, M]$. With the relaxed $q_i$s, the optimal contract design problem satisfies the following condition.
Spence-Mirrlees single-crossing condition (SMC) is satisfied if the user’s utility function $U(q, p, \theta)$ satisfies:$\frac{\partial}{\partial\theta}\left[-\frac{\partial U/\partial q}{\partial U/\partial p}\right]>0$, where $q$ is quantity of items, $p$ is the price for $q$ items and $\theta$ is the type of user.
When SMC is satisfied, the $N(N-1)$ IC constraints can be reduced to a set of tractable equivalent constraints [@contract_book].
\[lemma:IC\_incomplete\] If SMC is satisfied, the necessary and sufficient conditions for the satisfaction of all the IC constraints are: $$\begin{aligned}
\label{eqn:monotone_q}
\left\{
\begin{array}{l}
\theta^1v(q_1)-p_1=0 \\
\theta^iv(q_i)-p_i=\theta^iv(q_{i-1})-p_{i-1}, \quad 1 < i \leq T\\
q_i\geq q_j \quad \textrm{where} \quad \theta^i\geq\theta^j
\end{array}
\right.\end{aligned}$$
It is easy to verify that the SMC condition holds when $q_i$s are real numbers. Based on Lemma \[lemma:IC\_incomplete\], introducing a $\theta^0=0$, we can express the query plans as $$\begin{aligned}
\label{eqn:pk}
\left\{
\begin{array}{l}
p_k=\sum_{i=1}^k \left[\theta^iv(q_i)-\theta^iv(q_{i-1})\right],\quad \forall 1\leq k\leq T\\
q_i\geq q_j \quad \forall i\geq j
\end{array}
\right.\end{aligned}$$ Substituting (\[eqn:pk\]) and $N_i=\mu_1\cdot\beta^i$ into (\[eqn:u\_do\]) and putting the terms related to the same query variable together, we have: $$\begin{aligned}
U_{DO}(\mathcal{P}) &=&(\mu_0r-\epsilon_0B_R^{\alpha}) \\
&+& \sum_{i=1}^T\left[
\mu_1\beta^i\theta^iv(q_i)-\Delta^iv(q_i)\sum_{j=i+1}^T\mu_1\beta^j - \mu_1\beta^i\epsilon_1q_i\right] \\
&=&(\mu_0r-\epsilon_0B_R^{\alpha}) + \sum_{i=1}^T g_i(\mu_1)q_i,\end{aligned}$$ where $\Delta^i=\theta^{i+1}-\theta^i,\forall i<T$, $\Delta^T=0$ and $g_i(\mu_1)=\beta^i\theta^i\frac{B-B_R}{M}-\Delta^i\frac{B-B_R}{M}\sum_{j=i+1}^T\beta^j - \mu_1\beta^i\epsilon_1$. The optimal contract design problem for DO can then be expressed as $$\begin{aligned}
&U_{DO}(\mathcal{P})&=(\mu_0r-\epsilon_0B_R^{\alpha}) + \sum_{i=1}^Tg_i(\mu_1)q_i \nonumber\\
&\max_{\{q_i\}}& U_{DO}(\mathcal{P}) \nonumber\\
&\textrm{subject to:}& 0\leq q_i\leq M, \quad q_i\geq q_j,\,\forall i\geq j.\end{aligned}$$
$i_S(\mu_1)=T$; $i_S(\mu_1)=i_S(\mu_1)-1$; $\hat{q_i}(\mu_1)=0, \forall i\leq i_S(\mu_1)$; $\quad \hat{q_i}(\mu_1)=M, \forall i> i_S(\mu_1)$;
Now, we consider $q_i$s to be integers. Since factor $g_iq_i$ only relates to $q_i$ given $\mu_1$, we can then find the optimal value for each $q_i$ independently. We use $\tilde{q_i}(\mu_1)$ to denote ${\operatornamewithlimits{argmax}}_{0\leq q_i\leq M} g_iq_i$. The set of $\{\tilde{q_i}(\mu_1)\}$ can be achieved at the boundary points (0 or $M$). It is worth noting that $\{\tilde{q_i}(\mu_1)\}$ is a function of $\mu_1$ since $g_i$ is a function of $\mu_1$. However, $\tilde{q_i}$ may not be monotonically nondecreasing on $i$ (requirement of Eqn. (\[eqn:monotone\_q\])). For example, if for some $i^*$, $g_{i^*}>0$ and $g_{i^*+1}<0$ then $q_{i^*}=M$ and $q_{i^*+1}=0$. We design Algorithm \[alg:valid\_q\] to obtain valid $\{q_i\}$. We first find the largest $i_S(\mu_1)$, such that $g_{i_S(\mu_1)}<0$ and set all $q_{j}, j\leq i_S(\mu_1)$ to be 0. Note that $q_i$s obtained in Algorithm \[alg:valid\_q\] can be 0 or $M$, which is the same as that when $q_i$s are real numbers. Therefore, it is valid for us to replace the IC constraints with Eqn. \[eqn:monotone\_q\].
Suppose the valid queries obtained are $\hat{q_i}$ for type $\theta^i$ SUs, the optimal query plans at stage II are $$\begin{aligned}
\left\{
\begin{array}{lll}
\mathcal{P}&=&\{(q_i, p_i)\}\bigcup\{(q_0=0, p_0=0)\},\quad 1\leq i \leq T \\
q_i&=&\hat{q_i}(\mu_1)\\
p_i&=&\hat{p_i}(\mu_1)=\sum_{j=1}^i \left[\theta^jv(\hat{q}_j(\mu_1))-\theta^jv(\hat{q}_{j-1}(\mu_1))\right]
\end{array}
\right.\end{aligned}$$
Note that Algorithm \[alg:valid\_q\] finds a suboptimal sequence of valid $\{q_i\}$. Because Algorithm \[alg:valid\_q\] has restricted the range of $\{q_i\}$ to be only boundary points. One way to obtain optimal $\{q_i\}$ is the brunching and ironing algorithm [@contract_book]. Instead of restricting the range of $\{q_i\}$, this algorithm first finds the optimal sequence of $\{q_i\}$ ignoring the condition of Eqn. (\[eqn:monotone\_q\]). Then it continues to find pairs of $q_i, q_{i+1}$ with $q_i<q_{i+1}$. After that, it finds a new $q_i^*\in[q_i, q_{i+1}]$ such that the payoff is maximized when setting $q_i=q_{i+1}=q_i^*$. It continues until the condition of Eqn. (\[eqn:monotone\_q\]) is satisfied. The algorithm is named since it “iron” out decreasing sub-sequence in $\{q_i\}$. However, the optimal algorithm is of high complexity and cannot guarantee the existence of an NE in the NDRG at stage I. It is difficult to prove theoretical the gap between the optimal and suboptimal solution in this paper, we will evaluate the price of the suboptimal $\{q_i\}$ in Section \[sec:simulation\].
Stage I: Database Registration Game of SUs
------------------------------------------
We now consider the NDRG between SUs under incomplete information scenario.
### Existence of an NE
At stage II, Algorithm \[alg:valid\_q\] is published by the DO to the SUs. Assuming the type of $u_i$ is $\theta^{\sigma(i)}$, the utility of $u_i$ when choosing the service plan scheme can be computed as: $
U_i=\theta^{\sigma(i)}v(\hat{q}_{\sigma(i)}(\mu_1))-\hat{p}_{\sigma(i)}(\mu_1)
$. Note that $\mu_1$ is the number of SUs choosing the service plan scheme. We can therefore rewrite the utility of $u_i$ considering the two possible strategies as $$\begin{aligned}
\label{eqn:u_imcomplete_info}
U_i(a_i, a_{-i})=\left\{
\begin{array}{ll}
\frac{B_R}{\mu_0}\theta_i-r & \textrm{if $a_i=0$} \\
\theta^{\sigma(i)}\frac{B-B_R}{\mu_1M}(\hat{q}_{\sigma(i)}(\mu_1))-\hat{p}_{\sigma(i)}(\mu_1) & \textrm{if $a_i=1$}
\end{array}
\right. \label{eqn:u_incomplete_info}\end{aligned}$$ From the utility function of $u_i$, we can verify that NDRG under incomplete information is also a congestion game when using Algorithm \[alg:valid\_q\] to obtain $\{q_i\}$.
\[lemma:NDRG\_is\_CG2\] NDRG under incomplete information is an unweighted congestion game.
All the players in NSDG share a common set of strategies $\mathcal{S}$. The payoff function of player $u_i$ is defined as $U_i$ in (\[eqn:u\_incomplete\_info\]). For strategy $0$, $U_i$ is a monotonically non-increasing function with $\mu_0$. For strategy $1$, $g_i(\mu_1)$ is a non-increasing function of $\mu_1$. Therefore, $i_S(\mu_1)$ is also a non-increasing function of $\mu_1$. As a result, $\hat{q_i}(\mu_1)$ is a non-increasing function of $\mu_1$. When $\hat{q_i}(\mu_1)=0$, $U_i=0$ and when $\hat{q_i}(\mu_1)=M$, $U_i\geq0$. Therefore $U_i$ is also a monotonically non-increasing function with $\mu_1$.
As a result, we can also find an NE with an improvement path with a finite number of improvement steps. According to Lemma \[lemma:property\_of\_CG\] and Lemma \[lemma:NDRG\_is\_CG2\], we have:
\[thm:NDRG\_property2\] Under incomplete information, NDRG has a pure strategy NE which can be achieved with an improvement path of at most $N(N+1)$ steps starting from any strategy-tuple.
SUs randomly select one pricing scheme and record in the database; DO assigns each SU a unique ID in the range $[0, N - 1]$; Time duration counter $t=0$; Active SU ID in $t$ is $i = t\mod N$; Update strategy from $a_i$ to $1-a_i$ within $t$; $t=t+1$;
### Distributed Database Registration Algorithm
In the case of incomplete information, some SUs of higher types are guaranteed with non-negative payoff with the service plan scheme. These SUs may prefer the service plan scheme.
Unlike the complete information case, the SUs now have to make the pricing scheme choice repeatedly. The SUs can act according to Algorithm \[alg:registration\_incomplete\_info\] to achieve an NE in a distributed manner via repeated improvements. Algorithm \[alg:registration\_incomplete\_info\] is based on the improvement path. Since all the SUs are connected with the database, we assume at stage I, the SUs are synchronized with the database. Also, we assume there are predefined *time slots*. SUs can determine whether to change their strategies within a time slot one by one according to the order of their assigned IDs. In each time slot, only one SU may update his strategy to optimize its own utility given the new situation of other SUs’ choices. It is easy to see that in Algorithm \[alg:registration\_incomplete\_info\], every update is a step in the improvement path. Therefore, Algorithm \[alg:registration\_incomplete\_info\] is guaranteed to find an NE within $N(N+1)$ updates. Note that the found NE by the algorithm is also not unique if there are many SUs with the same type.
Stage I: Optimal Parameter Selection of DO
------------------------------------------
### Estimate the SUs’ choices in an NE
Estimating $\mu_0$ and $\mu_1$ is essential for the DO to optimize its pricing parameters. However, due to the randomness of the SUs’ strategies the NE found by Algorithm \[alg:registration\_incomplete\_info\] is not unique. Therefore, it is impossible to get an accurate estimation. We assume the DO uses the expected value as an estimation of $\mu_0$ and $\mu_1$. However, to calculate the expectation, the DO needs to generate all possible strategy-tuples and check all possible NEs. Therefore the DO has to generate $2^N$ different strategy-tuples, which is computationally infeasible. However, by exploring the nature of SUs in our scenario, we are able to dramatically reduce the computation complexity of the DO. The following lemma provides a way for the DO to estimate $\mu_0$ and $\mu_1$ with linear complexity with $N$.
\[lemma:NDRG\_property\_incomplete\] For every NE strategy-tuple $\mathbf{a}$, there exists another NE $\mathbf{\bar{a}}$ with the same number of registered SUs. In $\mathbf{\bar{a}}$, there is only one $\theta^{i_S}, 1\leq i_S\leq T$, such that $\forall i\neq i_S, \forall u_j, u_k$, if $\,\theta^j=\theta^k=i, a_j=a_k$.
Suppose in an NE, two of the user types are $\theta^i$ and $\theta^j$ and the strategy tuple is $\mathbf{a}$. Suppose $n_i^0$ of the $\theta^i$ users are choosing $0$ and $n_i^1$ of the $\theta^i$ SUs are choosing $1$. $n_j^0$ of the $\theta^j$ SUs are choosing $0$ and $n_j^1$ of the $\theta^j$ SUs are choosing $1$. Assume one $\theta^i$ SU $u_i$ changes its strategy from $0$ to $1$ and one $\theta^j$ SU $u_j$ changes its strategy from $1$ to $0$ at the same time. After the above strategy inter-changing, the strategy tuple, denoted as $\mathbf{a'}$ is also an NE. That is because $\mu_0$ and $\mu_1$ are the same in both $\mathbf{a}$ and $\mathbf{a'}$. And none of the other SUs except $u_i$ and $u_j$ will have the incentive to change strategy in the previous NE tuple $\mathbf{a}$. We can also see that $u_i$ and $u_j$ have no incentive to change strategies since no other $\theta^i$ and $\theta^j$ SUs want to change from $0$ to $1$ or from $1$ to $0$.
Similarly, another equivalent NE can be achieved if some of the type $\theta^i$ SUs change from $1$ to $0$ and some of the type $\theta^j$ SUs change from $0$ to $1$ at the same time. By interchanging strategy choices of SUs in an NE scenario, we can obtain an NE tuple $\mathbf{\bar{a}}$ in which there is at most one type of SUs, say, $i_S$, satisfying the conditions in Lemma \[lemma:NDRG\_property\_incomplete\].
Lemma \[lemma:NDRG\_property\_incomplete\] says all SUs with the same type other than the type $\theta^{i_S}$ have the same strategies in $\mathbf{\bar{a}}$. As a result, the DO can use Algorithm \[alg:estimate\_SU\_incomplete\] to estimate the number of $\mu_0$ and $\mu_1$. It first chooses one type to be the $\theta^{i_S}$ and generates all possible strategy combinations of the other types. Then the DO checks whether it is possible to be an NE under the chosen $\theta^{i_S}$ by varying the number of $\theta^{i_S}$ type SUs’ choices on strategy $0$ and $1$. In the algorithm, the DO only needs to check no more than $2^T\cdot N$ different strategy-tuples which is linear with $N$.
Multi-set $R_{ALL}=\{\}$; // all possible values of $\mu_0$ $a_{-i}$= the bits in $Counter$’s binary representation; Assign strategy of SUs with type $\theta\neq\theta^i$ with $a_{-i}$; Let $j$ of the type $\theta^i$ SUs choose $0$; Let $N\cdot\beta^i-j$ of the type $\theta^i$ SUs choose $1$; **if** [It is an NE]{} **then** $R_{ALL} = R_{ALL}\bigcup \{\textrm{current}\ \mu_0\}$; $\mu_0=$ the most common value in the multi-set $R_{ALL}$; $\mu_1=N-\mu_0$;
### Optimization Parameter Selection
Utilizing the estimation of $\mu_0$ and $\mu_1$ obtained from Algorithm \[alg:estimate\_SU\_incomplete\], the DO now needs to solve the following optimization problem: $$\begin{aligned}
\label{eqn:max_udo3}
&U_{DO}(B_R, r)&=(\mu_0(B_R, r)r-\epsilon_0B_R^{\alpha}) + \sum_{i=1}^Tg_i(B_R, r)\hat{q}_i(B_R, r) \nonumber\\
&\max_{B_R, r}& U_{DO}(B_R, r) \nonumber\\
&\textrm{subject to:}& 0\leq B_R\leq B.\end{aligned}$$ where $
g_i(B_R, r)=N\beta^i\theta^i\frac{B-B_R}{\mu_1(B_R, r)M} - \Delta^i\frac{B-B_R}{\mu_1(B_R, r)M}\sum_{j=i+1}^TN\beta^j -N\beta^i\epsilon_1
$. Note that $\hat{q}_i$ is a function of $B_R, r$. Again, the problem can be solved using two-dimensional exhaustive search to find the solution when we restrict the range of $r$ to be \[**, *r*\]. We provide numerical results in Section \[sec:simulation\].
Numerical Results {#sec:simulation}
=================
In this section, we use numerical results to evaluate the proposed hybrid pricing scheme. We first introduce the simulation setup and then discuss the results.
Simulation Setup
----------------
We assume there are a total of $N=100-1000$ SUs in the network. The default value of $N$ is 100. There are $M=100$ periods. The expected available TVWS is $B=60$ MHz [@TVWS_availability1][@TVWS_availability2] We assume there are $T=5-30$ types, with 10 as the default value. For $T=10$, we generate 5 different possible type distributions summarized in table \[tab:distributions\]. For all $T=5-30$, we also have a random type distribution where the number of SUs in each type is randomly generated.
$\theta^i$ 1 2 3 4 5 6 7 8 9 10
------------ ---- ---- ---- ---- ---- ---- ---- ---- ---- ----
Distr. 1 10 10 10 10 10 10 10 10 10 10
Distr. 2 1 3 5 7 9 11 13 15 17 19
Distr. 3 19 17 15 13 11 9 7 5 3 1
Distr. 4 2 6 10 14 18 18 14 10 6 2
Distr. 5 18 14 10 6 2 2 6 10 14 18
: $N=100$ SUs in $T=10$ types.[]{data-label="tab:distributions"}
When applying a two-dimensional search, we need to fix the range of $r$. Recall that $r$ is the money paid by the registered SUs to the DO. When $B_R=60, \mu_0=1$, the highest revenue a SU can obtain is $B_R/\mu_0\times\theta^{10}=600$. So $r$ should be within $[0, 600]$. We assume the smallest change in $r$ is $r_0=1$ unit. We assume the total TV bandwidth can only be allocated in the unit of $b_0=6$ MHz (the same as the spectrum span of a TV channel in the US). So the possible values of $B_R$ are $\{0, 6, 12, \cdots, 60\}$.
For the database maintenance cost we set $\alpha=1.2$ in $\phi_0(b)=\epsilon_0\cdot b^{\alpha}$ and we vary the parameter $\epsilon_0$ in the range $[0, 7]$. If $\epsilon_0$ is too high, DO will not have the incentive to consider the registration scheme. Since the marginal cost of one database access is very small compared with the bandwidth reservation cost, we set the default value of $\epsilon_1$ to be 0.
In the case of non-strategic SUs, we assume $\gamma^i$s are the same among all types. We use a single value $\gamma$ to denote the fraction of SUs preferring the registration scheme. We will set $\gamma$ to be either $0.2$ or $0.5$ in our simulations.
In this section, for convenience we refer to the Complete Information Scenario as *CIS* and the Incomplete Information Scenario as *IIS*.
Simulation Results
------------------
### Impact of SU strategy
In this section, we compare the two cases: non-strategic SUs in CIS and strategic SUs in CIS to show the impact of SU strategy on the pricing solution.
First, in Fig. \[fig:epsilon0-B\_R\_non-strategic\], we fixed the SU type distribution to be Distr. 1. We set $\epsilon_1$ to be either 0.05 or 0.1. We then vary the reservation cost $\epsilon_0$ and plot the optimal $B_R$ obtained from three cases: non-strategic SUs with $\gamma=0.2$, non-strategic SUs with $\gamma=0.5$ and strategic SUs, all under CIS. From Fig. \[fig:epsilon0-B\_R\_non-strategic\], we can see that with non-strategic SUs, the DO tends to reserve less bandwidth for the reservation scheme under the same $\epsilon_0$ and $\epsilon_1$. That is because when the SUs are non-strategic players, they decide whether to pay the registration fee only based on the prior knowledge of the number of SUs in the registration scheme which is predetermined. Therefore, many SUs will overestimate the registration fee, so that the DO cannot increase the registration fee to obtain higher profit. As a result, the DO prefers the service plan scheme and reserves less bandwidth.
Second, in Fig. \[fig:strategic\_vs\_non-strategic\], we fixed the SU type distribution to be Distr. 1 and set $\epsilon_1$ to be 0.05. We vary the reservation cost $\epsilon_0$ and plot the utility of DO and the average utility of SUs given the optimal pricing parameters. From Fig. \[fig:strategic\_vs\_non-strategic\_UDO\], we can see that in both scenarios, the DO utility first decreases and then remains the same after $B_R=0$. When $\epsilon_0$ is higher, the DO utility in the non-strategic scenario is higher compared with that in the strategic SU scenario. That’s because when $\epsilon_0$ is higher, in the strategic scenario, the profit DO makes from the registered SUs become smaller however, most of DO profit in the non-strategic SU scenario comes from the non-registered SUs. We can also see from Fig. \[fig:strategic\_vs\_non-strategic\_USU\] that when SUs are strategically player, their average utility are still non-zero when $\epsilon_0>0.5$. That is because, in the non-strategic SU scenario, when $\epsilon_0>0.5$, $B_R=0$, all SUs can only get zero utility. However, in the strategic SU scenario, some SUs can choose the registered scheme to obtain non-zero utility. From the comparison, we can conclude that with more intelligent SU strategy, SUs achieve higher average utility.
### Benefit of Hybrid Pricing Scheme
We compared the DO’s revenue when applying hybrid and uniform pricing schemes. There are two uniform pricing schemes: registration only or service plan only. We set $epsilon_0=3.0$. In the case of registration scheme only, $B_R=60$ while in the case of service plan scheme only, $B_R=0$. We show the maximum possible revenues the DO can obtain in type distributions 1-3 and random distribution in Fig. \[fig:hybrid\]. The results from random distribution is an average of 100 runs. We can see that the hybrid pricing scheme provides higher revenue for the DO. By offering hybrid pricing schemes, the DO has a new degree of freedom to tune the bandwidth segmentation to increase its profit. We will evaluate the impact of $epsilon_0$ in the next evaluation.
### Impact of the bandwidth reservation cost
To show the impact of the bandwidth reservation cost on the optimal bandwidth reservation $B_R$, we vary $\epsilon_0$ from 0 to 5.2 and plot the optimal $B_R$ under each $\epsilon_0$ in both CIS and IIS in Fig. \[fig:epsilon0-B\_R\_scenario0\] and Fig. \[fig:epsilon0-B\_R\_scenario1\], respectively. In both information scenarios, $B_R$ decreases with the increase of $\epsilon_0$. That is because with greater $\epsilon_0$, the reservation cost for the same bandwidth is higher, so the DO prefers to allocate more bandwidth for unregistered users. We can see that when $\epsilon_0\geq 5.2$, $B_R=0$ for all cases, which means no bandwidth is reserved for the registration scheme.
We can also observe from Fig. \[fig:epsilon0-B\_R\] that in Distr. 2 $B_R$ decreases the fastest among the three distributions. That is because in Distr. 2, there are more SUs of higher types, who are more likely to have $g_i>0$ in Eqn. (\[eqn:max\_udo3\]), thus DO can benefit more from the service plan scheme.
It is also worth noting that in the CIS, $B_R$ decreases to zero sooner compared with that in the IIS. That is because in CIS, DO can design the query plans for individual unregistered SUs to make more profit.
{width="1.9in"}
### Impact of the DO’s knowledge of SUs’ personal information
In Fig. \[fig:information\], we show the impact of the DO’s knowledge of the SUs’ personal information on the utility of the DO and the SUs. The SUs’ types are set to distribution 1.
We can see from Fig. \[fig:information\_UDO\] that the DO has higher utility in the CIS than that in the IIS, when $\epsilon_0\geq 1.7$. That is because when $\epsilon_0\geq 1.7$, $B_R<60$ in CIS. As a result, the bandwidth for the service plan scheme is non-zero and the DO can make more revenue from unregistered SUs in the CIS.
We can see from Fig. \[fig:information\_USU\] that the SUs have higher average utility in the IIS than that in CIS, when $\epsilon_0>2.7$. That’ because when $\epsilon_0>2.7$, $B_R<60$ in IIS. As a result, the bandwidth for the service plan scheme is non-zero and the SUs can get non-negative utility when choosing the service plan scheme.
We can also notice that when $\epsilon_0<1.7$ in Fig. \[fig:information\_UDO\] and when $\epsilon_0<2.7$ in Fig. \[fig:information\_USU\], $B_R=60$, which means all bandwidth is reserved for registered SUs. As the DO design registration scheme to admit only registered SUs with the highest type, the utilities obtain in both information scenario are the same.
In summary, with more knowledge of the SUs’ information, the DO enjoys higher utility. On the other hand, the hidden information provides the SUs higher utility.
### Impact of the SUs’ type distribution
To show the impact of the SUs’ type distribution on the contract design, we fixed $B_R=30$, $r=200$, $\epsilon_0=3.0$ and compute the $q_i$ and $p_i$ in the service plans for 50 unregistered SUs in the 5 different SU type distributions. Note that the parameters may not be the optimal pricing parameters for each distribution, the use of the same parameters for all the distributions allows us to see the impact of SU type distribution separately. Fig. \[fig:contract\] shows the results for $q_i$ and $p_i$. In Fig. \[fig:contract\_query\], we can see that when the number of SUs of lower types is higher, (compared distr. 3 and distr. 1), DO tends to assign non-zero queries for lower type SUs. That is because a query $q_i$ is determined by a factor of $\sum_{j=i+1}^TN\beta^j$ in Eqn. (\[eqn:max\_udo3\]). In Fig. \[fig:contract\_price\] we can see that if non-zero $q_i$ is assigned to a lower type, a lower price $p_i$ should be charged. This is because, the price should follow the individual rationality constraint (Eqn. (\[eqn:IR\])) for such SUs.
### Impact of the suboptimal query assignment
We implemented the brunching and ironing algorithm [@contract_book] for the contract design. We assume at stage I, the DO and the SUs still use Algorithm \[alg:valid\_q\] to estimate the contracts to ensure the existence of an NE. We generate random SU type distributions with $N=100$, $T=5-30$ and set $B_R=0$, which indicates that all the $N$ SUs choose the service plan scheme. We repeated the test for 100 times. We show the average utility of the DO obtained from query assignment algorithms in Fig. \[fig:suboptimal\]. For a different $T$, Algorithm \[alg:valid\_q\] achieves the DO utility within 90% of that obtained from the optimal algorithm.
### Convergence time to an NE under incomplete information scenario
To check the convergence time to NE via Algorithm \[alg:registration\_incomplete\_info\], we fix $B_R=30$, $r=200$ and generate $N=100-1000$ SUs with uniformly distributed types. Under each $N$, we repeat Algorithm \[alg:registration\_incomplete\_info\] 200 times. Fig. \[fig:convergence\] shows the average steps of improvement needed to achieve an NE. It is clear that the number of steps is bounded by $N\cdot(N+1)$, which verifies Theorem \[thm:NDRG\_property2\]. We may prove a tighter bound for the convergence time in our future work.
![Convergence time to an NE under IIS.[]{data-label="fig:convergence"}](suboptimal.pdf){width="1.6in"}
![Convergence time to an NE under IIS.[]{data-label="fig:convergence"}](convergence.pdf){width="1.6in"}
Discussion and Future Works
===========================
In this paper, to make the problem solvable, we have made several simplifications in the system model. There are several interesting ways to further extend this work.
First, the registration scheme considered in this paper admits a uniform registration fee. In a more flexible setting, the registered SUs can pay different amounts of registration fees and enjoy different shares of the reserved bandwidth. Similarly with the query plan scheme, the DO can offer several levels of registration fees.
Second, the pricing framework considered in this paper is not dynamic. The pricing parameters are determined at the beginning once and will not change. In a more dynamic setting, the DO can redesign new pricing parameters for the following time durations and the SUs can also re-select the pricing scheme given the observation in previous time durations and the newly announced pricing parameter. In this case, the interactions between the DO and the SUs become a repeated game. Also, the SUs may dynamically join and leave the query plan. Therefore, the SUs should consider the expectation of future SU behaviors when choosing their pricing schemes.
Related Works {#sec:related_works}
=============
Most existing works on geo-location databases can be classified into two categories. Some works focus on the design of geo-location database to protect primary users. In [@database_dyspan08], Gurney et al. discussed the methods to calculate the protection area for TV stations. In [@database_dyspan11], Murty et al. designed a database-driven white space network based on measurement studies and terrain data. Some other works focused on the networking issue with the assumption that the database is already set up. In [@database], Feng et al. presented a white space system utilizing a database. In [@database_icdcs12], Chen et al. considered the channel selection and access point association problem. One recent work [@database_icc12] also address the business model related to the geo-location database. In [@database_icc12], the authors proposed that the geo-location database acts as a spectrum broker reserving the spectrum from spectrum licensees. They considered only one pricing scheme which is similar to the registration scheme discussed in our paper. Compared to our previous work [@databasePricing], in this paper, we further extend the scenario to non-strategic SUs and compared the pricing schemes with non-strategic and strategic SUs under complete information scenario. We also extend our theoretical analysis and numerical evaluations.
Many works also focus on the economic issue of dynamic spectrum sharing. In [@add1], the pricing-based spectrum access control is investigated under secondary users competitions. In [@add2], spectrum pricing with spatial reuse is considered. Contract theory is utilized in the scenarios where the spectrum buyers have hidden information. In [@JSAC10], Gao et al. leveraged contract theory to analyze the spectrum trading between primary operator and SUs. In [@dyspan11Duan], contract theory is applied to the cooperative communication scenario. In this paper, we also model the service plan design with contract theory. However, due to the co-existence of hybrid pricing schemes, there is uncertainty about the number of SUs choosing the contract items, which is different from existing works.
There are some works focus on the hybrid pricing of other limited resources. In [@icdcs12segmentation]. Wang et.al study the problem of capacity segmentation for two different pricing schemes for cloud service providers. One key difference between our work and [@icdcs12segmentation] is that the strategic SUs considered in our paper can dynamically choose between pricing schemes. While in [@icdcs12segmentation], the users are pre-categorized into different pricing scheme before designing the pricing schemes.
Conclusions {#sec:conclusion}
===========
In this paper, we consider a hybrid pricing model for TVWS database. The SUs can choose between the registration and the service plan scheme. We investigate scenarios where the SUs can be either non-strategic or strategic players and the DO has either complete and incomplete information of the SUs. In the non-strategic SU scenario, we model the competitions among the SUs as non-cooperative game and prove the existence of an NE in two different scenarios by showing that the game is an unweighted congestion game. We model the pricing for unregistered SUs with contract theory and derive suboptimal query plans for different types of SUs. Based on the SUs’ pricing scheme choices, the DO optimally determines the bandwidth segmentation and pricing parameters to maximize its profit. We have conducted extensive simulations to obtain numerical results and verify our theoretical claims.
Acknowledgement {#acknowledgement .unnumbered}
===============
The research was support in part by grants from 973 project 2013CB329006, China NSFC under Grant 61173156, RGC under the contracts CERG 622410, 622613, HKUST6/CRF/12R, and M-HKUST609/13, as well as the grant from Huawei-HKUST joint lab.
[1]{}
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[^1]: X. Feng and Q. Zhang are with the Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Hong Kong. e-mail: {xfeng, qianzh}@cse.ust.hk.
[^2]: J. Zhang is with the HKUST Fok Ying Tung Research Institute, Hong Kong. e-mail: jinzh@ust.hk
|
---
abstract: 'We present a new proof that the statement ’every sugbroup of a free group is free’ implies the Axiom of Choice for finite sets.'
author:
- 'Philipp Kleppmann[^1]'
bibliography:
- 'bibliography.bib'
title: 'Nielsen-Schreier implies the finite Axiom of Choice'
---
Introduction
============
In 1921, Nielsen [@Nielsen1921NS] proved that every subgroup of a finitely generated free group is free. This result was generalised to arbitrary free groups by Schreier [@Schreier1927Untergruppen] in 1927, giving us the following result.
[NS]{}[Nielsen-Schreier]{} If $F$ is a free group and $K\leq F$ is a subgroup, then $K$ is a free group.
Since every proof of $\mathsf{NS}$ uses the Axiom of Choice, it is natural to ask whether it is equivalent to the Axiom of Choice. The first step was made by Läuchli [@Lauchli1962Auswahlaxiom], who showed that $\mathsf{NS}$ cannot be proved in $\mathsf{ZF}$ set theory with atoms. Jech and Sochor’s embedding theorem [@Jech2008Choice] allows this result to be transferred to standard $\mathsf{ZF}$ set theory. It was improved in 1985 by Howard [@Howard1985Subgroups], who showed that $\mathsf{NS}$ implies $\mathsf{AC}_{fin}$, the Axiom of Choice for finite sets:
[Axiom of Choice for finite sets]{} Every set of non-empty finite sets has a choice function.
Another Choice principle used in this article is the Axiom of Choice for pairs:
[Axiom of Choice for pairs]{} Every set of 2-element sets has a choice function.
The purpose of this paper is to provide a new and shorter proof of Howard’s result.
Nielsen-Schreier implies $\mathsf{AC}_{fin}$ {#section: Nielsen-Schreier implies ACfin}
============================================
Before beginning the proof, we must fix some notation and terminology. If $X$ is a set, let $X^-=\{x^{-1}:x\in X\}$ be a set of formal inverses of $X$. It does not matter what the elements of $X^-$ are, as long as $X^-$ is disjoint from $X$. Members of $X^\pm=X\cup X^-$ are called $X$-*letters*. Finite sequences $x_1\cdots x_n$ with $x_1,...,x_n\in X^\pm$ are $X$-*words*. An $X$-word $x_1\cdots x_n$ is $X$-*reduced* if $x_i\not=x_{i+1}^{-1}$ for $i=1,...,n-1$. If $\alpha$ is an $X$-word, the $X$-*reduction* of $\alpha$ is the $X$-reduced $X$-word obtained by performing all possible cancellations within $\alpha$. For notational simplicity, we don’t distinguish between $X$-words and their $X$-reductions. Reference to $X$ is omitted if $X$ is clear from the context.
If $G$ is a group and $S\subseteq G$, then $\langle S\rangle$ is the subgroup of $G$ *generated by* $S$.
Let $X$ be a set. The *free group on* $X$, written $F(X)$, consists of all reduced $X$-words. The group operation is concatenation followed by reduction, and the identity is the empty word ${\bf 1}$.
A group $G$ is *free* if it is isomorphic to $F(X)$ for some $X\subseteq G$. If this is the case, $X$ is a *basis* for $G$.
The following proofs will start with a family $Y$ of non-empty sets and construct a choice function $c:Y\rightarrow\bigcup Y$. Without loss of generality, we assume that the members of $Y$ are pairwise disjoint. We then define $X=\bigcup Y$ to be the basis of the free group $F=F(X)$. With every $y\in Y$ we associate a function $\sigma_y:F\rightarrow\mathbb{Z}$ which counts the number of occurrences of $y$-letters in words $\alpha\in F$ as follows.
> Write $\alpha=x_1^{\epsilon_1}\cdots x_n^{\epsilon_n}$ as an $X$-reduced word with $x_1,...,x_n\in X$ and $\epsilon_1,...,\epsilon_n\in\{\pm1\}$. Then define $$\sigma_y(\alpha)=|\{i:x_i\in y\land\epsilon_i=1\}|-|\{i:x_i\in y\land\epsilon_i=-1\}|.$$ It is easily checked that, for each $y\in Y$, $\sigma_y$ is a group homomorphism from the free group $F$ to the additive group of integers.
Before proving theorem \[theorem: NS implies ACfin\] we handle a special case in lemma \[lemma: NS implies AC2\]. Its proof serves as an introduction to ideas used in the proof of the main theorem.
\[lemma: NS implies AC2\] $\mathsf{ZF}\vdash\mathsf{NS}\Rightarrow\mathsf{AC}_2$
Let $Y$ be a family of 2-element sets. Without loss of generality, assume that the members of $Y$ are pairwise disjoint.
Let $X=\bigcup Y$, let $F=F(X)$ be the free group on $X$, and define the subgroup $K\leq F$ by $$K=\langle\{wx^{-1}:(\exists y\in Y)w,x\in y\}\rangle.$$ By the Nielsen-Schreier theorem, $K$ has a basis $B$. Note that $$\label{equation: K is in the kernel}
\sigma_y(\alpha)=0\text{ for all }y\in Y\text{ and all }\alpha\in K.$$
We will construct a choice function for $Y$, i.e. a function $c:Y\rightarrow X$ satisfying $c(y)\in y$ for each $y\in Y$.
Let $y\in Y$. Define the function $s_y:y\rightarrow y$ to swap the two elements of $y$. For any choice of $x\in y$, $y=\{x,s_y(x)\}$. To simplify notation, we set $x_i=s_y^i(x)$ for all $i\in\mathbb{Z}$; hence $y=\{x_0,x_1\}$. Express $x_0x_1^{-1}$ and $x_1x_0^{-1}$ as reduced $B$-words: $$\begin{aligned}
x_0x_1^{-1}&=&b_{0,1}\cdots b_{0,l_0}\\
x_1x_0^{-1}&=&b_{1,1}\cdots b_{1,l_1},\end{aligned}$$ where $b_{i,j}\in B^\pm$ for all $i,j$. As $x_0x_1^{-1}=(x_1x_0^{-1})^{-1}$, it follows that $l_0=l_1=l$, say, and that $$\label{equation: cancellation in NS=>AC2}
b_{1,1}=b_{0,l}^{-1},...,b_{1,l}=b_{0,1}^{-1}.$$
There are two cases:
(i) $l$ is odd:
Let $m=(l-1)/2$. The middle $B$-letter of $x_0x_1^{-1}$ is $b_{0,m+1}$, whereas the middle $B$-letter of $x_1x_0^{-1}$ is $b_{1,m+1}=b_{0,m+1}^{-1}$ by (\[equation: cancellation in NS=>AC2\]). One of these two is in $B$, while the other is in $B^-$. Define $c(y)$ to be the unique element $x\in y$ such that the middle $B$-letter of $xs_y(x)^{-1}$ is a member of $B$.
(ii) $l$ is even:
Let $m=l/2$. The following two functions are the key to the proof. $$\begin{aligned}
f_y:&y\rightarrow K:&x_i\mapsto b_{i,1}\cdots b_{i,m}\\
g_y:&y\rightarrow F:&x\mapsto f_y(x)^{-1}\cdot x\end{aligned}$$
The idea of $f_y$ is to map $x_i$ to the ’first half’ of $x_ix_{i+1}^{-1}$ in terms of the new basis $B$. $f_y(x)$ is intended to represent $x$ in $K$.
Using (\[equation: cancellation in NS=>AC2\]), we obtain $$\begin{split}
\label{equation: f is well-behaved}
f_y(x_i)f_y(x_{i+1})^{-1}&=b_{i,1}\cdots b_{i,m}b_{i+1,m}^{-1}\cdots b_{i+1,1}^{-1}\\
&=b_{i,1}\cdots b_{i,m}b_{i,m+1}\cdots b_{i,2m}\\
&=x_ix_{i+1}^{-1}.
\end{split}$$
It follows that $g_y(x_0)=g_y(x_1)$. Hence the image of $y$ under $g_y$ has a single member, $\alpha_y$, say. Note that $$\label{equation: sigma of alpha is 1}
\begin{split}
\sigma_y(\alpha_y)& =\sigma_y(g_y(x_0))\\
& =\sigma_y(f_y(x_0)^{-1}x_0)\\
& =\sigma_y(f_y(x_0)^{-1})+\sigma_y(x_0)\\
& =0+1\text{ using (\ref{equation: K is in the kernel}), }f_y(x_0)\in K\text{, and }x_0\in y
\end{split}$$ is non-zero. This means that $\alpha_y$ mentions at least one $y$-letter. So we define $c(y)$ to be the $y$-letter which appears first in the $X$-reduction of $\alpha_y$.
We are now ready to prove the general case:
\[theorem: NS implies ACfin\] $\mathsf{ZF}\vdash\mathsf{NS}\Rightarrow\mathsf{AC}_{fin}$.
Let $Z$ be a family of non-empty finite sets. Without loss of generality, assume that the members of $Z$ are pairwise disjoint. We form a new family $$Y=\{y:y\not=\emptyset\land(\exists z\in Z)y\subseteq z\},$$ i.e. the closure of $Z$ under taking non-empty subsets. Since $Z\subseteq Y$, any choice function for $Y$ immediately gives a choice function for $Z$.
Let $X=\bigcup Y$, let $F=F(X)$ be the free group on $X$, and let $K\leq F$ be the subgroup defined by $$K=\langle\{wx^{-1}:(\exists y\in Y)w,x\in y\}\rangle.$$ By the Nielsen-Schreier theorem, $K$ has a basis $B$.
For each $n<\omega$, let $Y^{(n)}=\{y\in Y:|y|=n\}$ and $Y^{(\leq n)}=\{y\in Y:|y|\leq n\}$. By induction on $n$, we will find a choice function $c_n$ on $Y^{(\leq n)}$ for each $2\leq n<\omega$. By construction, the $c_n$ will be nested, so that $\bigcup_{2\leq n<\omega}c_n$ is a choice function for $Y$.
A choice function $c_2$ on $Y^{(\leq2)}$ is guaranteed by lemma \[lemma: NS implies AC2\].
Assume that $n\geq3$ and that there is a choice function $c_{n-1}$ for $Y^{(\leq n-1)}$. For every $y\in Y^{(n)}$ we define a function $s_y$ by $$s_y:y\rightarrow y:x\mapsto c_{n-1}(y\setminus\{x\}).$$ Note that, as $Y$ is closed under taking non-empty subsets, $y\setminus\{x\}\in Y^{(n-1)}$, so $c_{n-1}(y\setminus\{x\})$ is defined. There are four cases:
(i) $s_y$ is not a bijection:
In this case, $|\{s_y(x):x\in y\}|\leq n-1$, so defining $$c_n(y)=c_{n-1}(\{s_y(x):x\in y\})$$ gives a choice for $y$.
(ii) $s_y$ is a bijection with at least two orbits[^2]:
Since there are at least two orbits, each orbit has size $\leq n-1$. Moreover, as $s_y(x)\not=x$ for all $x\in y$, the number of orbits is also $\leq n-1$. So choosing one point from each orbit, and then choosing one point from among the chosen points gives a single element of $y$. More specifically, if we write $orb(x)$ for the orbit of $x\in y$ under $s_y$, we define $$c_n(y)=c_{n-1}(\{c_{n-1}(orb(x)):x\in y\}).$$
(iii) $s_y$ is a bijection with one orbit, and $n$ is even:
If $n$ is even, $s_y^2$ is a bijection with two orbits. Remembering that we are assuming $n\geq 3$, this gives us $\leq n-1$ orbits of size $\leq n-1$ each. A choice is made as in the previous case.
(iv) $s_y$ is a bijection with one orbit, and $n$ is odd:
Notice that, for any $x\in y$, $y=\{x,s_y(x),s_y^2(x),...,s_y^{n-1}(x)\}$. $s_y(x)$ may be viewed as the successor of $x$. For simplicity, we set $x_i=s_y^i(x)$ for $i\in\mathbb{Z}$, so that $y=\{x_0,x_1,...,x_{n-1}\}$.
In order to further simplify our notation, we shall assume that the elements of $Y^{(n)}$ are pairwise disjoint. Of course, this is not possible when $Y$ is constructed as above. But replacing every $y\in Y^{(n)}$ with $y\times\{y\}$ makes no difference to the argument, so the proof carries over without any changes.
Recall the basis $B$ of the subgroup $K$ defined earlier in the proof. We may write $$\begin{aligned}
x_0x_1^{-1}&=&b_{0,1}\cdots b_{0,l_0}\\
x_1x_2^{-1}&=&b_{1,1}\cdots b_{1,l_1}\\
&...&\\
x_{n-1}x_0^{-1}&=&b_{n-1,1}\cdots b_{n-1,l_{n-1}}\end{aligned}$$ as reduced $B$-words, with $b_{i,j}\in B^\pm$ for all $i,j$. First, we make two simplifications:
(a) If it is *not* the case that $l_0=...=l_{n-1}$, let $l=min\{l_i:i=0,...,n-1\}$. Then $\{x_i:l_i=l\}$ is a proper non-empty subset of $y$, and we define $$c_n(y)=c_{n-1}(\{x_i:l_i=l\}).$$ From now on it is assumed that $l_0=...=l_{n-1}=l$, say.
(b) Note that $$(x_0x_1^{-1})(x_1x_2^{-1})\cdots(x_{n-1}x_0^{-1})={\bf 1},$$ i.e. $$\label{equation: cyclic cancelling}
(b_{0,1}\cdots b_{0,l})(b_{1,1}\cdots b_{1,l})\cdots(b_{n-1,1}\cdots b_{n-1,l})={\bf 1}.$$
For $i=0,...,n-1$, let $k_i$ be the number of $B$-cancellations in $$\label{equation: number of cancellations}
(b_{i,1}\cdots b_{i,l})(b_{i+1,1}\cdots b_{i+1,l}).$$ If it is *not* the case that $k_0=...=k_{n-1}$, let $k=min\{k_i:i=0,...,n-1\}$. Then $\{x_i:k_i=k\}$ is a proper non-empty subset of $y$, and we define $$c_n(y)=c_{n-1}(\{x_i:k_i=k\}).$$ From now on it is assumed that $k_0=...=k_{n-1}=k$, say.
As letters always cancel in pairs, (\[equation: cyclic cancelling\]) implies that $nl$ is even.[^3] Since we are assuming that $n$ is odd, it follows that $l$ is even. Define $m=l/2$, and note that $k\geq m$: if not, then complete cancellation in (\[equation: cyclic cancelling\]) would not be possible. This allows us to define functions $f_y$ and $g_y$, as in the proof of lemma \[lemma: NS implies AC2\]: $$\begin{aligned}
f_y:&y\rightarrow K:&x_i\mapsto b_{i,1}\cdots b_{i,m}\\
g_y:&y\rightarrow F:&x\mapsto f_y(x)^{-1}x.\end{aligned}$$ Since there are $k\geq m$ cancellations in (\[equation: number of cancellations\]), we have $b_{i+1,1}=b_{i,l}^{-1},...,b_{i+1,m}=b_{i,l-m+1}^{-1}=b_{i,m+1}^{-1}$ for all $i$. By the same calculation as in (\[equation: f is well-behaved\]), it follows that $$f_y(x_i)f_y(x_{i+1})^{-1}=x_ix_{i+1}^{-1}$$ for all $i$, and hence that $g_y(x_i)=g_y(x_{i+1})$ for all $i$. So $g_y:y\rightarrow F$ is a constant function, taking a single value $\alpha_y$, say. The same calculation as (\[equation: sigma of alpha is 1\]) yields $$\sigma_y(\alpha_y)=1.$$ So we set $c_n(y)$ to be the first $y$-letter occurring in the $X$-reduction of $\alpha_y$.
Whether or not the Nielsen-Schreier theorem is equivalent to the Axiom of Choice still remains an open question. A positive answer might be obtainable by adapting the proof of theorem \[theorem: NS implies ACfin\]. Finiteness of the sets was used to define the choice function recursively, splitting up in cases (i) – (iv). Cases (i) – (iii) were easily dealt with. Case (iv) gave us a cyclic ordering on the finite set – enough structure to use the basis of the subgroup $K$ to choose a single element.
[^1]: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK. Email: P.Kleppmann@dpmms.cam.ac.uk.
[^2]: Thanks to Thomas Forster for suggesting a simplification of this part of the proof
[^3]: I would like to thank John Truss and Benedikt Löwe for finding an error in this proof and suggesting a solution.
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---
abstract: 'We provide an algorithmic approach to the construction of point transformations for scalar ordinary differential equations that admit three-dimensional symmetry algebras which lead to their respective canonical forms.'
---
H. Azad$^*$, Ahmad Y. Al-Dweik$^*$, F. M. Mahomed$^{**}$ and M. T. Mustafa$^{***}$\
[$^*$Department of Mathematics & Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia]{}\
[$^{**}$DST-NRF Centre of Excellence in Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa\
[$^{***}$Department of Mathematics, Statistics and Physics, Qatar University, Doha, 2713, State of Qatar]{}\
]{}
hassanaz@kfupm.edu.sa, aydweik@kfupm.edu.sa, Fazal.Mahomed@wits.ac.za and tahir.mustafa@qu.edu.qa.
Keywords: ODEs, three-dimensional symmetry algebras, point transformations, canonical forms.
Introduction
============
This is a contribution to the algorithmic theory of differential equations in the sense of Schwarz \[6\]. In the first five sections we provide results needed to construct algorithms, which are illustrated in detail in section 6 of the paper.
Not all the three-dimensional algebras have invariant differential equations of second-order in the sense that they are the full symmetry algebra of the equation. For this reason, we have given higher-order invariant equations for such types of three-dimensional algebras.
Although all the results of this paper were independently arrived at, the main ideas are already in Lie [@Lie1891]. The justification for placing this in the public domain is its brevity, clarity and a uniform treatment of compact and non-compact algebras.
The realizations of two- and three-dimensional Lie algebras as vector fields in $\mathbb{C}^2$ is given in e.g., Ibragimov [@Ibragimov1999 page 163]. The details, as well as the invariant second-order ordinary differential equations (ODEs) are given in Lie [@Lie1891 pages 479-542]. The two-dimensional algebras are essentially distinguished by their ranks. It is thus desirable to give a similar description of three-dimensional Lie algebras over the reals.
The main aim of this work is to give such a description and an algorithmic procedure that systematically utilizes the structural information more explicitly and which is programmable. This program reduces any given ODE that admits a three-dimensional algebra to its simplest form.
This is achieved by proving a version of the Lie-Bianchi classification in an algorithmic way and giving the realizations of the algebras as vector fields in $\mathbb{R}^2$.
In every case, there is an invariantly defined two-dimensional abelian algebra - which is not always a subalgebra of the given algebra - and its rank determines the coordinates that reduces the equation to its simplest form.
As far as the form of the invariant differential equations is concerned, one can use the result in [@Azad2015 pages 69-76] that reduces the computation of local joint invariants of any finite number of vector fields algorithmically to that of an abelian algebra of vector fields.
A few words regarding the importance of low-dimensional algebras for differential equations are in order.
Lie \[3,4\] obtained the complete explicit classification of scalar second-order ODEs that possess non-similar (not transformable into each other via point transformation) complex Lie algebras of dimension $r$, where $r=0,\ldots,8$. He showed that the complex Lie algebra of vector fields acting in the plane admitted by a given second-order ODE can only be of dimensions $0,1,2,3$ or $8$. He also proved that if a second-order equation admits an eight-dimensional algebra, it is linearizable by means of a point transformation and it is then equivalent to the simplest equation, viz. the free particle equation.
It is well-known that one- and two-dimensional algebras have identical structures over the reals as well as over the complex numbers. As a consequence, the Lie symmetry algebra classification of scalar second-order ODEs over the reals is precisely the same as that over the complex numbers for one- and two-dimensional Lie-algebras.
If a second-order equation admits a single generator of symmetry, then in general its order can be reduced by one \[4\]. Moreover, Lie \[4\] showed that scalar second-order equations possessing two generators of symmetry have four canonical forms. These are well-known now as the Lie canonical forms for the vector fields and their representative second-order equations. Lie \[4\] also proved that the rank one algebras result in linearization of the associated second-order ODE.
The situation is different for three- and higher-dimensional Lie algebras as there are fewer complex than real algebras of dimension three or more. This arose in the Bianchi \[2\] classification of Lie algebras. Two of the complex Lie algebras of dimension three in the real plane each split up into two real non-isomorphic Lie algebras. Therefore, there are two more non-isomorphic real three-dimensional Lie algebras than complex algebras.
Due to the above considerations on Lie algebras in the real plane for higher dimensions, there are additional three-dimensional algebras of vector fields acting in the real plane than in the complex plane. These were deduced by Mahomed and Leach \[5\]. These yield additional non-similar scalar second-order equations that admit real Lie algebras \[5\].
In summary, this is a contribution to the algorithmic Lie theory of scalar ODEs in the sense of Schwarz \[6\]. Schwarz \[6\] utilized janet bases in the representation of the determining equations of the symmetry generators. The main difference in our approach is to use canonical forms of the symmetry algebra in order to construct the requisite point transformations that bring a given ODE with known symmetry algebra to its canonical form.
The reader is referred to Ibragimov \[7\] for the background on Lie’s theory of symmetries of differential equations.
The Lie-Bianchi Classification of three-dimensional solvable algebras
=====================================================================
We begin with a formulation of the Lie-Bianchi \[1,2\] classification of three-dimensional Lie algebras.
\[th111\] Let $G$ be a three-dimensional Lie algebra. If $G$ is solvable, then $G'$ is abelian. Moreover,
\(a) if $G'$ is two-dimensional, then the structure of $G$ is completely determined by the eigenvalues and multiplicities of [ad]{} $(X)$ as a linear transformation of $G'$, where $X$ is any representative of $G/G'$ in $G$,
\(b) if $G'$ is one-dimensional, then the structure of $G$ is completely determined by the dimension of the centralizer of $G'$ in $G$.
In general, by Lie’s theorem on complex solvable linear Lie algebras [@Neeb2012 page 106], if $G$ is a solvable Lie algebra then the algebra ad $(G^{\mathbb{C}})$ is nilpotent, where $G^{\mathbb{C}}=G+\sqrt{-1}~G$. Therefore, the commutator $G'$ of $G$ is nilpotent.
Now assume that $G$ is solvable and of dimension three.
\(a) If $H$ is a two-dimensional algebra with basis $\{X,Y\}$, then its commutator is generated by $[X,Y]$. If $H$ is nonabelian, extend $U=[X,Y]$ to a basis $\{U,V\}$ of $H$. Then, scaling $V$, we have the canonical representation of $H$ by the relations $[V,U]=U$; such an algebra is not nilpotent.
Therefore if $G'$ is two-dimensional, it must be abelian. Take a basis $\{X,Y\}$ of $G'$ and extend it to a basis $\{X,Y,Z\}$ of $G'$. Then ad $(Z)$ operating on $G'$ does not have 0 as an eigenvalue and the eigenvalues and their multiplicities of ad $(Z)$ operating on $G'$ completely determine the structure of $G$.
\(b) Assume that $G'$ is one-dimensional. Let $\{U\}$ be a basis of $G'$. Extend it to a basis of $\{X,Y,U\}$ of $G$.
So $[X,Y]=aU, [X,U]=bU,[Y,U]=cU$. Now dim $Z_{A}(U)\geq 1.$ Suppose it is one. Then $b,c \neq 0$. By scaling $X,Y$ suitably, we then have $[X,Y]=aU,[X,U]=U,[Y,U]=U$ so $[X-Y,U]=0$. Therefore $X-Y, U$ are in $Z_G(U)$ and dim $Z_G (U)$ is at least two.
\(i) Suppose dim $Z_{G}(U)=2.$ Choose a basis $\{Y,U\}$ of $Z_{G}(U)$ and extend it to a basis $\{X,Y,U\}$ of $G$. So $[X,Y]=aU,[X,U]=bU,[Y,U]=0$, with $b\neq 0$. By scaling, we have that $[X,Y]=aU,[X,U]=U,[Y,U]=0$.
Now, for any $\lambda$ we have $$[X,Y+\lambda U]=aU+\lambda U, [X,U]=U, [Y+\lambda U,U]=0.$$ Choosing $a+\lambda =0$ and renaming $Y-aU$ as $Y$ we have the canonical relations $[X,Y]=0,[X,U]=U,[Y,U]=0$ and $Z_{G}(U)=\langle U,Y\rangle$.
\(ii) Suppose dim $Z_{G}(U)=3$. Take any basis, $\{X,Y,U\}$ of $G$. We then have $$[X,Y]=aU,[X,U]=0,[Y,U]=0, \,\,\mbox{with}\, a\neq 0.$$ By scaling, we can assume that $[X,Y]=U,[X,U]=0,[Y,U]=0$.
Therefore the canonical representations of $G$ when dim $G'$ is one are: $$[X,Y]=0,[X,U]=U,[Y,U]=0\,\,\mbox{(dim}\,Z_{G}(G')\,\mbox{is}\,
2)$$ and $$[X,Y]=U,[X,U]=0,[Y,U]=0\,\,\mbox{(dim}\,
Z_{G}(G')\,\mbox{is}\, 3).$$
If $G'$ is two-dimensional then it is abelian and there is a basis $\{X,Y,Z\}$ of $G$ with $\{X,Y\}$ a basis of $G'$ with [ad]{} $(Z)$ operating on $G'$ as follows - recall that [ad]{} $(Z)$ does not have $0$ as an eigenvalue of $G'$:
\(i) [ad]{} $(Z)$ has real and distinct eigenvalues.
If $X,Y$ are eigenvectors, then as [ad]{} ($Z$) does not have 0 as an eigenvalue of $G'$, by scaling $Z$ we have $[Z,X]=X,[Z,Y]=cY,c\neq 0,1$.
\(ii) [ad]{} $(Z)$ has only one eigenvalue and the corresponding eigenspace is two-dimensional.
In this case, scaling $Z$, we have the canonical representation $$[Z,X]=X,[Z,Y]=Y.$$
\(iii) [ad]{} $(Z)$ has only one real eigenvalue and the corresponding eigenspace is one-dimensional. There is a basis $\{X,Y\}$ of $G'$ with $[Z,X]=X+Y,[Z,Y]=Y$.
\(iv) [ad]{} $(Z)$ has a complex eigenvalue $\lambda$. The canonical relations - after scaling $Z$ - are $$[Z,X]=\cos \theta ~X+ \sin \theta ~Y,[Z,Y]=-\sin \theta ~X+\cos \theta ~Y,$$ where $X,Y$ are the real and imaginary parts of an eigenvector for the eigenvalue $\lambda.$
Only cases (iii) and (iv) need proofs.
Case (iii): ad $(Z)$ has only one real eigenvalue, which is non-zero and the null-space of ad $(Z)-\lambda I$ is one-dimensional. Let $V$ be an eigenvector for the eigenvalue $\lambda$. Extend $V$ to a basis $\{X,V\}$ of $G'$. Then $X$ is a generalized eigenvector of $Z$ and $\{X,Y=$ ad $(Z-\lambda I)\,(X)\}$ is a basis of $G'$.
We have $[Z,X]=\lambda X+Y,[Z,Y]=\lambda Y$. Dividing $Z$ by $\lambda$ and relabeling it $Z$ we have the relations $[Z,X]=X+cY,(c\neq 0),[Z,Y]=Y$. Finally, replacing $Y$ by $cY$, we have the canonical relations $[Z,X]=X+Y,[Z,Y]=Y$.
Case (iv): if one of the eigenvalues of ad ($Z$) is complex but not real, then the other eigenvalue is its conjugate. Denote by $\lambda$ any one of these eigenvalues. Then the eigenvectors live in the Lie algebra $G+\sqrt{-1}~G$ - the complexification of $G$. We will denote the complexification of any Lie algebra $G$ by $G^{\mathbb{C}}$. The eigenvectors of ad $(Z)$ live in $G^{'\mathbb{C}}$. Find an eigenvector $e$ for ad $(Z$).
The real and imaginary parts of $e$ are $\displaystyle
{\rm Re}(e)=\frac{e+\bar{e}}{2}, {\rm Im}(e)= \frac{e-\bar{e}}{2i}$.
By scaling $Z$ by $\frac{1}{|\lambda|}$, we have the canonical relations $$[Z, {\rm Re}(e)]=\cos \theta \,{\rm Re}(e)+\sin \theta \,{\rm Im}(e)$$ $$[Z, {\rm Im}(e)]=-\sin \theta \, {\rm Re}(e)+\cos \theta \, {\rm Im}(e).$$ Take $X={\rm Re}(e)$ and $Y={\rm Im}(e)$.
Local classification of commuting Lie algebras of vector fields in $\mathbb{R}^2$
=================================================================================
The algorithms for bringing a given ODE to its canonical form by point transformations ultimately reduce to constructive classifications of abelian Lie algebras of vectors fields. This section is devoted to an algorithmic procedure for finding canonical forms of such algebras.
It is well-known that if $X$ is a vector field and $X(p)\neq 0$, then near $p$ we can introduce coordinates in which $X=\partial_{x}$. To find such a canonical coordinate, one uses the method of characteristics [@Ibragimov1999 page 142], to find a basic invariant function $y$ of $X$ and choose a function $x$ functionally independent from $y$. Then in these coordinates $X=f(x,y)\,\partial_{x},$ with $f$ non-vanishing near $p$.
We want functions $\tilde{x},\tilde{y}$ with $X(\tilde{x})=1,\,
X(\tilde{y})=0$. Then the requirements become $f(x,y)\,\partial_{x}(\tilde{x})=1,
f(x,y)\,\partial_{x}\,(\tilde{y})=0.$ A solution of this system is $\displaystyle \tilde{x}=\int
\frac{dx}{f(x,y)},\,\tilde{y}=y$.
Now suppose that $X,Y$ are commuting vector fields with rank 2 near a point $p$; so that $X(q),Y(q)$ are linearly independent near $p$; say $X(p)\neq 0$. We may therefore assume that in some neighbourhood of $p, X(q)\neq 0$ and $X,Y$ of rank 2 in this neighbourhood.
Choose local coordinates $x,y$ in possibly a smaller neighbourhood with $X=\partial_{x}$. As $Y$ operates on invariants of $X$, $Y(y)=g(y)$. Now as $Y$ operates non-trivially on invariants of $X$ because of the rank condition, we can find a function $\tilde{y}$ of $y$ with $Y(\tilde{y})=1$. By change of notation, we now have coordinates $x,y$ with $X=\partial_{x},Y=\xi
(y)\,\partial_{x}+\partial_{y}$. We want new coordinates $\tilde{x},\tilde{y}$ with $X(\tilde{x})=1,
X(\tilde{y})=0, \, Y(\tilde{x})=0,\, Y(\tilde{y})=1.$ The system to solve now is $\displaystyle \frac{\partial
\tilde{x}}{\partial x}=1,\frac{\partial \tilde{y}}{\partial x}
=0\,\,\xi (y)+\frac{\partial \tilde{x}}{\partial
y}=0,\frac{\partial
\tilde{y}}{\partial y}=1.$ The solution is given by $\displaystyle \tilde{x}=x+\varphi (y),
\, \frac{\partial \tilde{x}}{\partial y}=\varphi'(y)=-\xi
(y),\,\tilde{y}=y.$ If the fields are commuting of rank 1, and $X=\partial_{x}$ in local coordinates, then $Y=f(y)X,$ with $f'(y)\neq 0$. Then in the variables $x,\tilde{y}=f(y)$ the canonical form of the fields is $X=\partial_{x},
Y=\tilde{y}\,\partial_{x}$.
Realizations of three-dimensional algebras as vector fields in $\mathbb{R}^2$
===============================================================================
We now turn to realizations of the algebras occurring in Theorem \[th111\] and its corollary as algebras of vector fields in $\mathbb{R}^2$. This was done by Lie for complex Lie algebras that arise as symmetries of second-order ODEs: see [@Ibragimov1999 page 164] and [@Lie1891 pages 479-530]. Mahomed and Leach \[5\] extended this study for real Lie algebras. Similar ideas apply to vector fields in $\mathbb{R}^3$.
$\mathbf{Outline~of~the~argument}$: Using Theorem \[th111\], the canonical forms for solvable three-dimensional algebras with nontrivial commutator can be easily obtained algorithmically as one has just to put a two-dimensional algebra, depending on its rank, in canonical form. The reason is that if $G'$ is two-dimensional then it is abelian and its rank determines the canonical form of $G$. If $G'$ is of dimension one and its centralizer has dimension two, then the rank of the centralizer determines the canonical form of $G$. Finally, if $G'$ is one-dimensional and its centralizer has dimension three, then rank of $G$ must be 2 and choosing a field supported outside of $G'$ gives an abelian two-dimensional algebra of rank 2 which determines the canonical form of $G$.
If $G'=G$, then picking any nonzero element $X$ of $G$, the eigenvalues of ad $(X$) determine the canonical form of $G$ and of the vector fields also - as detailed below in section \[4.3\].
$\mathbf{Details~of~the~classification}$: Assume that $G$ is three-dimensional and abelian. Then its rank is at most 2 and it cannot be 2 as the centralizer of $\{
\partial_{x},\partial_{y}\}$ is $\langle
\partial_{x},\partial_{y}\rangle$. Therefore it is of rank 1. Pick any non-zero element $X$ of $G$ and find canonical coordinates $x,y$ so that $X=\partial_{x}$ and extend it to a basis $\{X,Y,Z\}$. Since rank of $G$ is 1, necessarily $Y=f(y)\,\partial_{x},\,Z=g(y)\,\partial_{x}$. Since $f$ is not a constant, we can take $\tilde{y}=f(y)$. So the basis becomes $\{\partial_{x},\tilde{y}\partial_{x},\,g(f^{-1}\,
(\tilde{y}))\,\partial_{x}=h(\tilde{y})\,\partial_{x}\}$, where $h(\tilde{y})$ is linearly independent of $\{1,\tilde{y}\}$.
For the canonical realizations of the algebras occurring in Theorem 1, one needs to solve equations of the type $[Z,X]=aX+bY,
\,[Z,Y]=cX+dY$, where $X=\partial_{x}$ and $Y=\partial_{y}$ or $y
\,\partial_{x}$. In case $Y=\partial_{y}$, this is straightforward. However, when $Y=y\partial_{x}$, we have $Z=\xi \partial_{x}+\eta
\,\partial_{y}$, where $$\xi =-ax-bxy+\varphi (y), \,\eta =c+(d-a)y-by^2.$$ We want a change of variables $\tilde{x},\tilde{y}$ so that $\partial_{x}=\partial_{\tilde{x}},\,
y\,\partial_{x}=\tilde{y}\,\partial_{\tilde{x}}$ and $$Z=(-a\tilde{x}-b\tilde{x}\,\tilde{y})\,\partial_{\tilde{x}}+(c+(d-a)\,\tilde{y}-b\tilde{y}^{2})\, \partial_{\tilde{y}}\label{*}$$ Then necessarily $\tilde{y}=y,\,\tilde{x}=x+\psi (y),
\,\partial_{\tilde{y}}=-\psi^{'}(y)\,\partial_{x}+\partial_{y}$. Substituting these expressions in the equation (\[\*\]) we arrive at equation $$\varphi (y)=\psi(y)(-a-by)-\psi^{'}(y)(c+(d-a)y-by^2)\label{**}$$ Solving this differential equation for $\psi$ removes the term $\varphi (y)$ in the field $Z$ in the new variables.
This gives the following realizations of the algebras occurring in Theorem 1 as vector fields in the plane:
${\rm dim}\,(G\,')=1$
---------------------
I: $\mbox{dim}\, Z_G(G\,')=2,\,[X,Y]=0,\,[X,U]=U,[Y,U]=0,$ rank $Z_{G}\,(G\,')=1$\
$U=\partial_{y}, Y=f(x)\,\partial_{y},$ where $f$ is not a constant. Making a change of variables $\tilde{x}=f(x),$ relabeling $\tilde{x}=x,$ using (\[\*\]), (\[\*\*\])- with $Z$ replaced by $X$ - we obtain the representation $$U=\partial_{y}, Y=x\,\partial_{y}, X=-y\partial_{y}-x\,\partial_{x}$$
II: $\mbox{dim}\, Z_G(G\,')=2,\,[X,Y]=0,\,[X,U]=U,[Y,U]=0,$ rank $Z_{G}\,(G\,')=2,$ $$U=\partial_{y},\, Y=\partial_{x}, X=-y\partial_{y}$$
III: dim $Z_G(G\,')=3,[X,Y]=U,[X,U]=0,[Y,U]=0$\
In this case rank of $G$ must be 2. In the canonical coordinates for $U$, we have $U=\partial_{y},$ and one of $X$ or $Y$ is supported outside $\partial_{y},$ otherwise $[X,Y]$ would be 0. By symmetry between $X$ and $Y$, we may suppose that $X$ is supported outside $\partial_{y}$ and therefore rank $\langle X,U\rangle=2.$ We may then suppose that $X=\partial_{x}.$ This determines $Y=x\,\partial_{y}.$ The canonical realization is thus $X=\partial_{x}, U=\partial_{y},\, Y=x\,\partial_{y}.$
${\rm dim}\,(G\,')=2$
---------------------
\(a) There is a basis $\{X,Y,Z\}$ of $G$ with $\{X,Y\}$ a basis of $G'$ with ad $(Z)$ operating on $G\,'$ having real and distinct eigenvalues. Computing the eigenvectors for these eigenvalues and labeling them $X,Y$ and dividing $Z$ by the eigenvalue for the eigenvector $X$, and relabeling it $Z$ we have the relations $[Z,X]=X, [Z,Y]=cY,\, c\neq 0,1.$
IV: rank $(G\,')=2.$\
Choosing coordinates with $X=\partial_{x},\,
Y=\partial_{y}$ we get $Z=-x\,\partial_{x}-cy\,\partial_{y}$
V: rank $(G\,')=1.$\
Choosing coordinates in which $X=\partial_{y},\, Y=x\,\partial_{y}$ and using (\[\*\]) and (\[\*\*\]) we obtain - by change of notation $Z=(c-1)\, x\,\partial_{x}-y\,\partial_{y}$.
\(b) There is a basis $\{X,Y,Z\}$ of $G$ with $\{X,Y\}$ a basis of $G\,'$ with ad $(Z)$ operating on $G\,'$ having a real eigenvalue with the corresponding eigenspace of dimension two. Computing the eigenvectors for these eigenvalues and labeling them $X,Y$ and dividing $Z$ by the eigenvalue and relabeling it $Z$ we have the relations $$[Z,X]=X,\,[Z,Y]=Y.$$
VI: rank $(G\,')=2.$\
Choosing coordinates with $X=\partial_{x},\,Y=\partial_{y}$ we deduce $Z=-x\,\partial_{x}-y\,\partial y$
VII: rank $(G\,')=1.$\
Choosing coordinates with $X=\partial_{y},\,Y=x\,\partial_{y}$ and using (\[\*\]) and (\[\*\*\]) we get - by change of notation - $Z=-y\,\partial_{y}.$
\(c) There is a basis $\{X,Y,Z\}$ of $G$ with $\{X,Y\}$ a basis of $G\,'$ with ad $(Z)$ operating on $G\,'$ having only one eigenvalue $\lambda$ with the corresponding eigenspace of dimension one. Find the corresponding eigenvector and let $X$ be a vector linearly independent from this eigenvector. Then $X$ is a generalized eigenvector and $\{X,Y=(\mbox{ad}\,(Z)-\lambda I)\,X\}$ is a basis of $G'$. Dividing $Z$ by $\lambda$ and relabeling it $Z$, we have $ [Z,X]=X+\frac{1}{\lambda}\, Y, [Z,Y]=Y.$ Replacing $Y$ by $\frac{1}{\lambda}Y$ in the second equation and finally denoting $ \frac{1}{\lambda}Y$ by $Y$ we have the relations $[Z,X]=X+Y,\,[Z,Y]=Y,\,[X,Y]=0.$
VIII: rank $(G\,')=2.$\
Choosing coordinates with $X=\partial_{x},\, Y=\partial_{y}$ we arrive at $$Z=-x\,\partial_{x}-(x+y)\,\partial_{y}$$
IX: rank $(G\,')=1.$\
Choosing coordinates in which $X=x\,\partial_{y},\, Y=\partial_{y}$ and using (\[\*\]) and (\[\*\*\]) we find - by change of notation - $Z=\partial_{x}-y\,\partial_{y}$.
\(d) There is a basis $\{X,Y,Z\}$ of $G$ with $\{X,Y\}$ a basis of $G\,'$ and ad $(Z)$ operating on $G\,'$ having a non-real complex eigenvalue $\lambda.$ Let $e$ be an eigenvector. Dividing $Z$ by $\frac{1}{|\lambda|},$ denoting it again by $Z$ we have the canonical relations $$[Z,\,{\rm Re}(e)]=\cos \theta \,{\rm Re}(e)-\sin \theta \, {\rm Im}(e)$$ $$[Z,{\rm Im}(e)]=\sin \theta \,{\rm Re}(e)+\cos \theta \, {\rm Im}(e).$$ Take $X={\rm Re}(e)$ and $Y={\rm Im}(e).$
X: rank $(G\,')=2.$\
Choosing coordinates with $X=\partial_{x}, \,
Y=\partial_{y}.$ Then $$Z=(-x \cos \theta - y \sin \theta)\,\partial_{x}+(x\,\sin \theta - y \, \cos \theta)\,\partial_{y}.$$
XI: rank $(G\,')=1.$\
Choosing coordinates in which $X=\partial_{y},\,Y=x\,\partial_{y}$ and using (\[\*\]) and (\[\*\*\]) we obtain - by change of notation - $Z=(1+x^2)\,\sin \theta \,
\partial_{x}+y( x\sin \theta-\cos\theta )\,\partial_{y}$
${\rm dim}\, (G\,')=3$ {#4.3}
----------------------
Let $G$ be a three-dimensional Lie algebra with $G'=G$. If $I$ is any non-zero vector in $G$, then its centralizer consists of multiples of $I$; for if $U$ is linearly independent of $I$ and it centralizes $I$, then extending $I$ to a basis of $G$ we see that $G'$ is at most two-dimensional. Therefore, the non-zero eigenvalues of ad $(I)$ occur in pairs $\lambda,-\lambda$ - as the trace of ad $(I)$ is zero.
Case (i): The non-zero eigenvalues of ad $(I)$ are real given as $\pm \lambda$. Then we can find eigenvectors $U,V$ of $I$ with $[I,U]=\lambda~
U,[I,V]=-\lambda V,[U,V]=cI$ where $c\ne 0$ as $[U,V]$ centralizes $I$. Setting $X=U,~Y=\frac{2}{c~\lambda}~V,~Z=\frac{2}{\lambda}~I$, we have the standard relations of $sl(2,\mathbb{R})$ given by $$[Z,X]=2X,[Z,Y]=-2Y,[X,Y]=Z.$$ In this case the Killing form is non-degenerate and indefinite.
Case (ii): ad $(I)$ has a non-real eigenvalue. In this case, the nonreal eigenvalues must be purely imaginary. Scaling $I$, we may suppose that the non-zero eigenvalues of ad $(I)$ are $\pm \sqrt{-1}$. Take an eigenvector $e$ of ad $(I)$ in the complexification of $G$ with eigenvalue $\sqrt{-1}$. Write $e=U-\sqrt{-1}~V$. Then $[I,U]=V$ and $[I,V]=-U.$ Now $[e,\bar{e}]$ commutes with $I$ and $[e,\bar{e}]=2~\sqrt{-1}~[U,V]$. Therefore $[U,V]$ commutes with $I$. We can scale these generators such that $[I,U]=V,[I,V]=-U$ and $[U,V]=\epsilon I$, where $\epsilon^2=1$. If $\epsilon=-1$, then we have the generators with $[I,U]=V,[I,V]=-U$ and $[U,V]=-I$. So $[U,V+I]=-(V+I)$ and $[U,V-I]=V-I$ and we are back to case (i). In this case the Killing form is non-degenerate and indefinite. If $\epsilon=1$, then we have the generators with $[I,U]=V,[I,V]=-U$ and $[U,V]=I$. The Killing form is negative definite. These are the standard relations of $so(3)$.
Case (iii): All the eigenvalues of ad $(I)$ are zero. In this case ad $(I)$ is nilpotent and in the normalizer $N(I)$ of $I$ there must be an element with real non-zero eigenvalues, specifically, any element $H$ in $N(I)$ complementary to $I$ must have non-zero eigenvalues, so $[H,I]=\lambda I,[H,U]=-\lambda U$ for some element $U$ and we are back to case (i).
If the Killing form is definite, then the Lie algebra must be $so(3)$. So its Killing form must in any case be negative definite. In this case the non-zero eigenvalues of [ad]{} $(I)$ must be purely imaginary. Arguing exactly as above, given a non-zero element $I$ of $L$, we can find generators $U,V$ with $[I,U]=V,[I,V]=-U$ and $[U,V]=cI$, where $c=\pm 1$. Now $c=-1$ would give an indefinite Killing form, so $[I,U]=V,[I,V]=-U$ and $[U,V]=I$ which give the relations for $so(3)$ for the triple $U,V,I$. In this case clearly $[I,\sqrt{-1}~U]=\sqrt{-1}~V,[I,\sqrt{-1}~V]=-\sqrt{-1}~U$ and $[\sqrt{-1}~U,\sqrt{-1}~V]=-I$ which are the relations for $sl(2,\mathbb{C})$. On the other hand, if $[I,U]=
U,[I,V]=-V,[U,V]=I,$ the non-zero eigenvalues of [ad]{} $(U+V)$ are purely imaginary and working with the corresponding eigenvectors we obtain a basis for $so(3,\mathbb{C})$. For this reason, over the complex numbers, there is a correspondence between $sl(2,\mathbb{R})$ and $so(3)$ invariant equations.
### Realizations of $sl(2,\mathbb{R})$ as vector fields in ${\mathbb{R}}^2$
Here we discuss the indefinite case. So let $L$ be a three-dimensional algebra of vector fields with indefinite Killing form. Then $L$ has a basis $\{X, Y, Z\}$ such that $$[Z,X]=2X,[Z,Y]=-2Y,[X,Y]=Z.$$ Find coordinates in which $Z=\partial_x$. Then necessarily $X$ has the following form $$\label{te6}
\begin{array}{cc}
X=e^{2 x}\left( f_1(y)\partial_{x}+f_2(y)\partial_{y} \right).\\
\end{array}$$ As $\{\partial_x, f_1(y)\partial_{x}+f_2(y)\partial_{y}\}$ are commuting vectors, we have the following two cases:
Case A: the rank of $\{\partial_x, f_1(y)\partial_{x}+f_2(y)\partial_{y}\}$ is 2. There is a change of variables in which $\partial_x=\partial_{\tilde{x}}$ and $f_1(y)\partial_{x}+f_2(y)\partial_{y}=\partial_{\tilde{y}}$. Using section 2, such a change of variables can be given explicitly as follows: $$\begin{array}{lll}
\tilde{x}=x-\int{\frac{f_1}{f_2}dy},& \tilde{y}&=\int{\frac{1}{f_2}dy}.\\
\end{array}$$ Therefore in these coordinates $$\label{te7}
\begin{array}{cc}
X=e^{2 (\tilde{x}+f(\tilde{y}))} \partial_{\tilde{y}},\\
\end{array}$$ where $f(\tilde{y})=\int{\frac{f_1}{f_2}dy}.$ In the coordinates $\tilde{\tilde{x}}$, $\tilde{\tilde{y}}$ in which $\tilde{\tilde{x}}=\tilde{x}$ and $e^{2 f(\tilde{y})}
\partial_{\tilde{y}}=\partial_{\tilde{\tilde{y}}}$ given by $$\begin{array}{lll}
\tilde{\tilde{x}}=\tilde{x},& \tilde{\tilde{y}}&=\int{e^{-2 f(\tilde{y})} d\tilde{y}},\\
\end{array}$$ we have $$\label{te8}
\begin{array}{cccc}
Z=\partial_{\tilde{\tilde{x}}},&X=e^{2 \tilde{\tilde{x}}} \partial_{\tilde{\tilde{y}}},&Y=e^{-2\tilde{\tilde{ x}}}\left( (\tilde{\tilde{y}}+c_1)\partial_{\tilde{\tilde{x}}}+\left( {(\tilde{\tilde{y}}+c_1)}^2+\epsilon~\lambda^2\right)\partial_{\tilde{\tilde{y}}} \right),\\
\end{array}$$ where $\epsilon \in \{0,1,-1\}$. So, the new coordinates $\bar{x}=\tilde{\tilde{x}},~\bar{y}=\frac{1}{\lambda}(\tilde{\tilde{y}}+c_1)$ transform $X,Y$ and $Z$ to $$\label{te9}
\begin{array}{cccc}
Z=\partial_{\bar{x}},&X=\frac{1}{\lambda}e^{2 \bar{x}} \partial_{\bar{y}},&Y=\lambda e^{-2\bar{x}}\left( \bar{y}\partial_{\bar{x}}+\left( {\bar{y}}^2+\epsilon\right)\partial_{\bar{y}} \right),\\
\end{array}$$ where $\epsilon \in \{0,1,-1\}$. Finally, the transformation $\hat{x}=e^{-2\bar{x}},\hat{y}=
\bar{y} e^{-2\bar{x}}$ transforms $X,Y$ and $Z$ to the polynomial form $$X=\partial_{\hat{y}}, Y=-2\hat{x}\hat{y}\partial_{\hat{x}}+({-\hat{y}}^2+\epsilon~{\hat{x}}^2)\partial_{\hat{y}}, Z= -2\hat{x}\partial_{\hat{x}}-2\hat{y}\partial_{\hat{y}},$$ where $\epsilon \in \{0,1,-1\}$.
Case B: the rank of $\{\partial_x, f_1(y)\partial_{x}+f_2(y)\partial_{y}\}$ is 1. In this case the vector $X$ should have the form $$\label{te10}
\begin{array}{cc}
X=e^{2 x} f_1(y)\partial_{x}.\\
\end{array}$$ If $f_1(y)$ is constant, then $$\begin{array}{cccc}
Z=\partial_{x},&X=e^{2 x} \partial_{x}.\\
\end{array}$$ Using $[X,Y]=Z$, gives $$\begin{array}{cccc}
Y=-\frac{1}{4}e^{-2x} \partial_{x}.\\
\end{array}$$ If $f_1(y)$ is not a constant, we can introduce a change of variables $$\begin{array}{lll}
\tilde{x}=x,& \tilde{y}&=f_1(y).\\
\end{array}$$ Therefore in these coordinates $$\label{te11}
\begin{array}{cc}
X=e^{2 (\tilde{x}+f(\tilde{y}))} \partial_{\tilde{x}},\\
\end{array}$$ where $f(\tilde{y})=\frac{1}{2}\ln{\tilde{y}}.$ Finally, in the coordinates $\tilde{\tilde{x}}$, $\tilde{\tilde{y}}$ given by $$\begin{array}{lll}
\tilde{\tilde{x}}=\tilde{x}+\frac{1}{2}\ln{\tilde{y}},& \tilde{\tilde{y}}&=\tilde{y},\\
\end{array}$$ we have $$\label{te12}
\begin{array}{cccc}
Z=\partial_{\tilde{\tilde{x}}},&X=e^{2 \tilde{\tilde{x}}} \partial_{\tilde{\tilde{x}}}.\\
\end{array}$$ Using $[X,Y]=Z$, gives $$\label{te13}
\begin{array}{cccc}
Y=-\frac{1}{4}e^{-2\tilde{\tilde{ x}}} \partial_{\tilde{\tilde{x}}}.\\
\end{array}$$ Finally, the transformation $\hat{x}=\tilde{\tilde{y}},\hat{y}=-\frac{1}{2}
e^{-2\tilde{\tilde{x}}}$ transforms $X,Y$ and $Z$ to the polynomial form $$X=\partial_{\hat{y}}, Y=-{\hat{y}}^2\partial_{\hat{y}}, Z=- 2\hat{y}\partial_{\hat{y}}.$$
### Realizations of $so(3)$ as vector fields in ${\mathbb{R}}^2$
Let $L$ be a three-dimensional algebra of vector fields with negative definite Killing form. Then $L$ has a basis $\{X, Y, Z\}$ such that $$[X,Y]=Z,[Y,Z]=X,[Z,X]=Y.$$ Find coordinates in which $X=\partial_x$. Then necessarily $Y-\sqrt{-1}~Z$ has the following form $$\label{te0}
\begin{array}{cc}
Y-\sqrt{-1}~Z=e^{\sqrt{-1} x}\left[ \left( f_1(y)\partial_{x}+f_2(y)\partial_{y}\right) + \sqrt{-1} \left( f_3(y)\partial_{x}+f_4(y)\partial_{y}\right)\right].\\
\end{array}$$ Since the rank of $\{\partial_x,
f_1(y)\partial_{x}+f_2(y)\partial_{y},
f_3(y)\partial_{x}+f_4(y)\partial_{y}\}$ cannot be 1, we can assume without loss of generality that rank of $\{\partial_x,
f_1(y)\partial_{x}+f_2(y)\partial_{y}\}$ is 2. We may therefore assume that $f_2(y)\ne 0$. As $\partial_x, f_1(y)\partial_{x}+f_2(y)\partial_{y}$ are commuting vectors, there is a change of variables in which $\partial_x=\partial_{\tilde{x}}$ and $f_1(y)\partial_{x}+f_2(y)\partial_{y}=\partial_{\tilde{y}}$. Using section 2, such a change of variables can be given explicitly as follows: $$\begin{array}{lll}
\tilde{x}=x-\int{\frac{f_1}{f_2}dy},& \tilde{y}&=\int{\frac{1}{f_2}dy},\\
\end{array}$$ Therefore in these coordinates $$\label{te1}
\begin{array}{cc}
Y-\sqrt{-1}~Z=e^{\sqrt{-1}(\tilde{x}+f(\tilde{y}))}\left[\partial_{\tilde{y}} + \sqrt{-1} \left( A(\tilde{y})\partial_{\tilde{x}}+B(\tilde{y})\partial_{\tilde{y}}\right)\right],\\
\end{array}$$ where $f(\tilde{y})=\int{\frac{f_1}{f_2}dy},~A(\tilde{y})=
\frac{f_2f_3-f_1f_4}{f_2},$ and $B(\tilde{y})=\frac{f_4}{f_2}.$ Using the fact that $[Y,Z]=\partial_{\tilde{x}}$ if and only if $[Y-\sqrt{-1}~Z,Y+\sqrt{-1}~Z]=2~\sqrt{-1}\partial_{\tilde{x}}$, the necessary and sufficient conditions for $f(\tilde{y}), A(\tilde{y})$ and $B(\tilde{y})$ to give a representation of $so(3)$ are $$\label{te2}
\begin{array}{cc}
A^2+ABf'+A'=-1,\\
(AB+B')+(1+B^2)f'=0.\\
\end{array}$$ To reduce the form (\[te1\]) to the simplest form, we look at the classical Bianchi representation of vector fields on $\mathbb{P}^2(\mathbb{R})$ induced by the rotations on $\mathbb{R}^3$. It is given by $$L_{3:9}:~X=\partial_{\tilde{\tilde{x}}}, Y=\tilde{\tilde{y}}\sin
\tilde{\tilde{x}}
\partial_{\tilde{\tilde{x}}}+(1+{\tilde{\tilde{y}}}^2)\cos \tilde{\tilde{x}} \partial_{\tilde{\tilde{y}}}, Z=\tilde{\tilde{y}}\cos \tilde{\tilde{x}}
\partial_{\tilde{\tilde{x}}}-(1+{\tilde{\tilde{y}}}^2)\sin \tilde{\tilde{x }}\partial_{\tilde{\tilde{y}}}.$$ So $$\label{te3}
\begin{array}{cc}
Y-\sqrt{-1}~Z=e^{\sqrt{-1}\tilde{\tilde{x}}}\left[(1+{\tilde{\tilde{y}}}^2)\partial_{\tilde{\tilde{y}}}- \sqrt{-1}\tilde{\tilde{y}}\partial_{\tilde{\tilde{x}}}\right].\\
\end{array}$$ The conditions that the form (\[te1\]) can be written in the form (\[te3\]) with $\partial{\tilde{\tilde{x}}}=\partial{\tilde{x}}$ and $\tilde{
\tilde{y}}=\psi(\tilde{y})$ are exactly the equations (\[te2\]). This gives the transformation $$\begin{array}{lll}
\tilde{\tilde{x}}=\tilde{x}+f+{\tan}^{-1}B,& \tilde{\tilde{y}}&=-\frac{A}{\sqrt{1+B^2}}.\\
\end{array}$$ So the transformation $$\begin{array}{lll}\label{te4}
\tilde{\tilde{x}}=x+{\tan}^{-1}\left(\frac{f_4}{f_2}\right),&\tilde{ \tilde{y}}&=\frac{f_1f_4-f_2f_3}{\sqrt{f_2^2+f_4^2}}.\\
\end{array}$$ maps the form (\[te0\]) to the form (\[te3\]).
In case $f_2(y)=0$, $f_4(y)\ne 0$, the formula becomes $$\begin{array}{lll}\label{te5}
\tilde{\tilde{x}}=x-{\tan}^{-1}\left(\frac{f_2}{f_4}\right)+\frac{\pi}{2},&\tilde{ \tilde{y}}&=\frac{f_1f_4-f_2f_3}{\sqrt{f_2^2+f_4^2}}.\\
\end{array}$$ Hence, up to change of coordinates, there is only one realization as vector fields in ${\mathbb{R}}^2$.
Summary of the results
======================
Based on the discussion in the previous section and using the notations in ref. [@Mahomed1989], we can state the following theorem:
\[th1\] Every three-dimensional Lie algebra has one of the following $17$ realizations in $\mathbb{R}^2$:\
A) [dim]{} $G'=0$:\
Then $G$ is abelian of rank 1 and there are infinitely many realizations:\
$L_{3;1}:~X=\partial_y, Y=x\partial_y,Z=f(x)\partial_{y}$ where $f(x)$ is linearly independent of $\{1,x\}$.\
B) [dim]{} $(G') =2$:\
Then the eigenvalues of $G/G'$ on $G'$ never zero and there are the following eight cases:\
1) [rank]{} $(G')=1$ and the eigenvalues of $G/G'$ on $G'$ are real and distinct.\
$L^{II}_{3:6}:~X=\partial_y, Y=x\partial_y,Z=(c-1)x\partial_{x}-y\partial_{y}, c\neq 0, 1$\
with the nonzero commutators $[Z,X]=X,[Z,Y]=cY, c\neq 0,1.$\
2) [rank]{} $(G')=2$ and the eigenvalues of $G/G'$ on $G'$ are real and distinct.\
$L^{I}_{3:6}:~X=\partial_x, Y=\partial_y,Z=-x\partial_{x}-cy\partial_{y}, c\neq 0, 1$\
with the nonzero commutators $[Z,X]=X,[Z,Y]=cY, c\neq 0,1.$\
3) [rank]{} $(G')=1$ and the eigenvalues of $G/G'$ on $G'$ are real and repeated with eigenspace of dimension 2.\
$L^{II}_{3:5}:~X=\partial_y, Y=x\partial_y,Z=-y\partial_{y}$\
with the nonzero commutators $[Z,X]=X,[Z,Y]=Y$.\
4) [rank]{} $(G')=2$ and the eigenvalues of $G/G'$ on $G'$ are real and repeated with eigenspace of dimension 2.\
$L^{I}_{3:5}:~X=\partial_x, Y=\partial_y,Z=-x\partial_{x}-y\partial_{y}$\
with the nonzero commutators $[Z,X]=X,[Z,Y]=Y$.\
5) [rank]{} $(G')=1$ and the eigenvalues of $G/G'$ on $G'$ are real and repeated with eigenspace of dimension 1.\
$L^{II}_{3:3}:~X=x\partial_y, Y=\partial_y,Z=\partial_x-y\,\partial_y$\
with the nonzero commutators $[Z,X]=X+Y, [Z,Y]=Y$\
6) [rank]{} $(G')=2$ and the eigenvalues of $G/G'$ on $G'$ are real and repeated with eigenspace of dimension 1.\
$L^{I}_{3:3}:~X=\partial_x, Y=\partial_y,Z=-x\partial_x-(x+y)\,\partial_y$\
with the nonzero commutators $[Z,X]=X+Y, [Z,Y]=Y$\
7) [rank]{} $(G')=1$ and the eigenvalues of $G/G'$ on $G'$ are complex.\
$L^{II}_{3:7}:~X=\partial_y, Y=x\partial_y, Z=\sin \theta(1+x^2)\, \partial_{x}+y(x\sin \theta- \,\cos \theta)\,\partial_{y}$\
with the nonzero commutators $[Z,X]=\cos \theta~X-\sin \theta~Y,[Z,Y]=\sin \theta~X+ \cos \theta~Y$.\
8) [rank]{} $(G')=2$ and the eigenvalues of $G/G'$ on $G'$ are complex.\
$L^{I}_{3:7}:~X=\partial_x, Y=\partial_y, Z=(-x \cos \theta -y \sin \theta)\, \partial_{x}+(x\sin \theta-y \,\cos \theta)\,\partial_{y}$\
with the nonzero commutators $[Z,X]=\cos \theta~X-\sin \theta~Y,[Z,Y]=\sin \theta~X+ \cos \theta~Y$.\
C) [dim]{} $(G')=1$\
1) The centralizer $Z_G(G')$ is 2 dimensional, then there are the two cases:\
(i) [rank]{} $(Z_G(G'))$ is 1.\
$L^{II}_{3:4}:~X=-x\partial_x-y\partial_y, Y=x\partial_y,Z=\partial_y$\
with the nonzero commutator $[X,Z]=Z$.\
(ii) [rank]{} $(Z_G(G'))$ is 2.\
$L^{I}_{3:4}:~X=-y\partial_y, Y=\partial_x,Z=\partial_y$\
with the nonzero commutator $[X,Z]=Z$.\
2) The centralizer $Z_G(G')$ is three-dimensional:\
$L_{3:2}:~X=\partial_x, Y=x\partial_y,Z=\partial_y$\
with the nonzero commutator $[X,Y]=Z$.\
D) $G'=G$:\
1) The Killing form is negative definite.\
The Lie algebra is $so(3)$ and there is one realization:\
$L_{3:9}:~X=\partial_x, Y=y\sin x \partial_x+(1+y^2)\cos x \partial_y, Z=y\cos x \partial_x-(1+y^2)\sin x \partial_y$\
with the nonzero commutators $[X,Y]=Z,[Y,Z]=X,[Z,X]=Y$.\
2) The Killing form is indefinite.\
The Lie algebra is $sl(2,R)$ and there are two cases:\
(i) The rank of the generators is 2 and there is one of the following three realizations.\
$L^{I}_{3:8}:~ X=\partial_y, Y=-2xy\partial_x-y^2\partial_y, Z= -2x\partial_x-2y\partial_y,$\
$L^{II}_{3:8}:~ X=\partial_y, Y=-2xy\partial_x+(-y^2+x^2)\partial_y, Z= -2x\partial_x-2y\partial_y,$\
$L^{III}_{3:8}:~X=\partial_y, Y=-2xy\partial_x-(y^2+x^2)\partial_y, Z= -2x\partial_x-2y\partial_y,$\
with the nonzero commutators $[Z,X]=2X,[Z,Y]=-2Y,[X,Y]=Z$.\
(ii) The rank of the generators is 1 and there is one realization.\
$L^{IV}_{3:8}:~X=\partial_y, Y=-y^2\partial_y, Z=- 2y\partial_y,$\
with the nonzero commutators $[Z,X]=2X,[Z,Y]=-2Y,[X,Y]=Z$.\
Illustrative examples on the 17 Bianchi types
=============================================
The equations considered here have been obtained by determining joint invariants of appropriate order for all the Lie-Bianchi types and transforming the equations by simple point transformations. The point of these examples is to recover something close to the inverse of these transformations algorithmically.
By computing the joint invariants of the realizations of Bianchi types, we see that there are no second-order invariant ODEs when rank $G'=1$. However, there are higher-order invariant ODEs for such cases. Even when rank $G'=2$, not all of Bianchi types have second-order invariant ODEs. For this reason, we give a procedure illustrated by examples given below for each of the Bianchi types which works in principle for any ODE of arbitrary order that admits a three-dimensional symmetry algebra to reduce it to its canonical form.
[$L_{3:1}$]{} \[ex1\]\
Consider the ODE $$\label{1e1}
v^{(4)}=\frac{1}{{v'}^{5}}\left({-v{v'}^{10}+10\,{v'}^{4}v''v'''-15\,{v'}^{3}{v''}^{3}-{v'}^{2}{v'''}^{2}+6\,v'{v''}^{2}v'''-9\,{v''}^{4}}\right)\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{1e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial u},&Y_2=v\frac{\partial}{\partial u},&Y_3=v^2\frac{\partial}{\partial u},\\
\end{array}$$ Since $G$ is abelian of rank 1, using Theorem \[th1\], the fourth-order ODE (\[1e1\]) can be transformed to the canonical form of $L_{3:1}$ via a point transformation.
In order to construct such a point transformation, one needs to match the the symmetries with the realizations of the Lie algebra of $L_{3:1}$ given by Theorem \[th1\] in the following way: $$\label{1e5}
\begin{array}{ccc}
X=Y_1,&Y=Y_2,&Z=Y_3.\\
\end{array}$$ Applying this correspondence to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{1e6}
\begin{array}{cccccc}
\phi_y=1,&x\phi_y=\psi,&f(x)\phi_y={\psi}^2,&\psi_y=0,&x\psi_y=0,&f(x)\psi_y=0.\\
\end{array}$$ The solution of the system (\[1e6\]) gives the following point transformation $$\label{1e7}
\begin{array}{cc}
u=y,& v=x,\\
\end{array}$$ for $f(x)=x^2$ which transforms ODE (\[1e1\]) to its canonical form $$\label{1e8}
y^{(4)}=\left(\frac{f^{(4)}}{f''}\right)y''+g\left(x,y'''-\left(\frac{f'''}{f''}\right)y''\right),\\$$ with $g(z,w)=z+w^2$.
[$L_{3:2}$]{} \[ex2\]\
Consider the ODE $$\label{2e1}
v'''=-\left( v'-v'' \right) ^{3}{{\rm e}^{- 3\,u}}+{{\rm e}^{3\,u}}- 2\,v'+3\,v'',\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{2e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial v},&Y_2=e^{u}\frac{\partial}{\partial v},&Y_3=e^{-u}\frac{\partial}{\partial u},\\
\end{array}$$ with the nonzero commutators $$\label{2e3}
\begin{tabular}{llllll}
$[Y_{2}, Y_{3}] =-Y_{1}$.\\
\end{tabular}$$ Since dim $G'=1$, dim $Z_G(G')=3$, using Theorem \[th1\], the third-order ODE (\[2e1\]) can be transformed to the canonical form of $L_{3:2}$ via a point transformation.
In order to construct such a point transformation, one needs to match any vector from $G'=<Y_1>$ with $Z=\frac{\partial
}{\partial y}$ and any vector which is functionally independent of $G'$ with $X=\frac{\partial }{\partial x}$ in the following way: $$\label{2e5}
\begin{array}{ccc}
Z=Y_1,&X=Y_3.\\
\end{array}$$ Applying this correspondence to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{2e6}
\begin{array}{cccc}
\phi_x=e^{-\phi},&\phi_y=0,&\psi_x=0,&\psi_y=1.\\
\end{array}$$ The solution of the system (\[2e6\]) gives the following point transformation $$\label{2e7}
\begin{array}{cc}
u=\ln{x},& v=y,\\
\end{array}$$ which transforms ODE (\[2e1\]) to its canonical form $$\label{2e8}
y'''=f\left(y''\right),$$ with $f(z)=z^3+1$.
[$L^{I}_{3:3}$]{} \[ex3\]\
Consider the ODE $$\label{3e1}
v''=-\frac{1}{3}(v'-2)^3\exp{\left(\frac{v'+1}{v'-2}\right)}\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{3e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial v},&Y_2=\frac{\partial}{\partial u},&Y_3=(5u-v)\frac{\partial}{\partial u}+(4u+v)\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{3e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{3}] = Y_{1}-Y_{2}$,& $[Y_{2}, Y_{3}] = 4Y_{1}+5Y_{2}$.\\
\end{tabular}$$ Here dim $G'=2$, rank $(G')=2$ and the adjoint action of $G/G'=<\overline{Y}_3>$ on $G'=<Y_1,Y_2>$ given by $$\label{3e4}
{\rm ad}\, (\overline{Y}_3)= \left({
\begin{array}{cc}
-1 & -4 \\
1 & -5 \\
\end{array}
}\right)$$ has $\lambda=-3$ as a repeated real eigenvalue. The vector $2Y_1+Y_2$ is an eigenvector and $Y_1$ is a generalized eigenvector because, in two dimensions, any vector linearly independent of the eigenvector is a generalized eigenvector. Using Theorem \[th1\], the second-order ODE (\[3e1\]) can be transformed to the canonical form of $L^{I}_{3:3}$ via a point transformation.
In order to construct such a point transformation, one needs to match the the generalized eigenvector with $X=\frac{\partial
}{\partial x}$ and the scaled eigenvector by $\frac{1}{\lambda}$ with $Y=\frac{\partial }{\partial y}$, in the following way: $$\label{3e5}
\begin{array}{ccc}
X=Y_1,&Y=-\frac{1}{3}(2Y_1+Y_2).\\
\end{array}$$ Applying this correspondence to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{3e6}
\begin{array}{cccc}
\phi_x=0,&\phi_y=-\frac{1}{3},&\psi_x=1,&\psi_y=-\frac{2}{3}.\\
\end{array}$$ The solution of the system (\[3e6\]) gives the following point transformation $$\label{3e7}
\begin{array}{cc}
u=-\frac{1}{3}y,& v=x-\frac{2}{3}y,\\
\end{array}$$ which transforms ODE (\[3e1\]) to its canonical form $$\label{3e8}
y''=C \exp{(-y')}$$ with $C=-e$.
[$L^{II}_{3:3}$]{} \[ex4\]\
Consider the ODE $$\label{4e1}
v'''=\frac{1}{u}\left({{\rm e}^{-u}}\ln \left( \left( uv''+2\,v'\right) { {\rm e}^{u}}-u \right) -{{\rm e}^{-u}}u-3\,v''\right)\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{4e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial v},&Y_2=\frac{1}{u}\frac{\partial}{\partial v},&Y_3=\frac{\partial}{\partial u}+(\frac{e^{-u}-v-uv}{u})\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{4e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{3}] = -Y_{2}-Y_{1}$,& $[Y_{2}, Y_{3}] = -Y_{2}$.\\
\end{tabular}$$ Here dim $G'=2$, rank $(G')=1$ and the adjoint action of $G/G'=<\overline{Y}_3>$ on $G'=<Y_1,Y_2>$ given by $$\label{4e4}
{\rm ad}\,(\overline{Y}_3)= \left({
\begin{array}{cc}
1 & 0 \\
1 & 1 \\
\end{array}
}\right)$$ has $\lambda=1$ as a repeated real eigenvalue. Here $Y_2$ is an eigenvector and $Y_1$ is a generalized eigenvector for the same reason as explained in example 6.3. Using the Theorem \[th1\], the second-order ODE (\[4e1\]) can be transformed to the canonical form of $L^{II}_{3:3}$ via a point transformation.
In order to construct such a point transformation, one needs to match the the generalized eigenvector with $X=x\frac{\partial
}{\partial y}$ and the eigenvector scaled by $\frac{1}{\lambda}$ with $Y=\frac{\partial }{\partial y}$ in the following way: $$\label{4e5}
\begin{array}{ccc}
X=Y_1,&Y=Y_2.\\
\end{array}$$ Applying this correspondence to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{4e6}
\begin{array}{cccc}
\phi_y=0,&x\phi_y=0,&\psi_y=\frac{1}{\phi},&x\psi_y=1.\\
\end{array}$$ The solution of the system (\[4e6\]) gives the following point transformation $$\label{4e7}
\begin{array}{cc}
u=x,& v=\frac{y}{x},\\
\end{array}$$ which transforms the vector $Y_3$ which is linearly independent of $G'$ to $$\label{4e8}
\begin{array}{cc}
Z=\frac{\partial}{\partial x}+(-y+f(x))\frac{\partial}{\partial y},\\
\end{array}$$ with $f(x)=e^{-x}$. Such a function $f(x)$ can be absorbed using the transformation $$\label{4e9}
\begin{array}{cc}
\tilde{x}=x,& \tilde{y}=y-e^{-x}\int e^x f(x) dx=y-xe^{-x}.\\
\end{array}$$ Finally, the composition of the transformations (\[4e7\]) and (\[4e9\]) transforms ODE (\[4e1\]) to its canonical form $$\label{4e10}
\tilde{y}'''=e^{-\tilde{x}}g\left(e^{\tilde{x}}\tilde{y}''\right),$$ with $g(z)=-3+\ln (z-2)$.
[$L^{I}_{3:4}$]{} \[ex5\]\
Consider the ODE $$\label{5e1}
v'''=-{\frac { {v'}^{4}-3\,{v''}^{2} { }}{{v'} }}-{\frac { { {\rm e}^{-\,v}} \left({v'}^{2}-v'' \right) ^{3} { }}{{v'}^{2} \left( uv'+1 \right) ^{3}}}\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{5e2}
\begin{array}{lll}
Y_1=-u\frac{\partial}{\partial u}+\frac{\partial}{\partial v},&Y_2=e^{-v}\frac{\partial}{\partial u},&Y_3=-ue^{-v}\frac{\partial}{\partial u}+e^{-v}\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{5e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{3}] =-Y_{3}$.\\
\end{tabular}$$ Since dim $G'=1$, dim $Z_G(G')=2$ and rank $(Z_G(G'))=2$, using Theorem \[th1\], the third-order ODE (\[5e1\]) can be transformed to the canonical form of $L^{I}_{3:4}$ via a point transformation.
In order to construct such a point transformation, one needs to match any vector from $G'=<Y_3>$ with $Z=\frac{\partial
}{\partial y}$ and any vector from $Z_G(G')=<Y_3,Y_2>$ which is linearly independent of $G'$ with $Y=\frac{\partial }{\partial x}$ in the following way: $$\label{5e5}
\begin{array}{ccc}
Z=Y_3,&Y=Y_2.\\
\end{array}$$ Applying this correspondence to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{5e6}
\begin{array}{cccc}
\phi_x=e^{-\psi},&\phi_y=-\phi e^{-\psi},&\psi_x=0,&\psi_y=e^{-\psi}.\\
\end{array}$$ The solution of the system (\[5e6\]) gives the following point transformation $$\label{5e7}
\begin{array}{cc}
u=\frac{x}{y},& v=\ln{y},\\
\end{array}$$ which transforms ODE (\[5e1\]) to its canonical form $$\label{5e8}
y'''={y'}f\left(\frac{y''}{{y'}}\right),$$ with $f(z)=z^3+3z^2$.
[$L^{II}_{3:4}$]{} \[ex6\]\
Consider the ODE $$\label{6e1}
v'''={\frac { \left( 32\,{v}^{6}-1 \right){v'}^{8}-8\,{v}^{4}{v'}^{5}v''+12\,{v}^{5}{v'}^{3}{v''}^{2}-3\,{v}^{3}{v''}^{3}}{4\,{v}^{5}{v'}^{4}-{v}^{3}v'v''}}\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{6e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial u},&Y_2=v\frac{\partial}{\partial u},&Y_3=(u-v^4)\frac{\partial}{\partial u}+v\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{6e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{3}] =Y_{1}$.\\
\end{tabular}$$ Since dim $G'=1$, dim $Z_G(G')=2$ and rank $(Z_G(G'))=1$, using Theorem \[th1\], the third-order ODE (\[6e1\]) can be transformed to the canonical form of $L^{II}_{3:4}$ via a point transformation.
In order to construct such a point transformation, one needs to match any vector from $G'=<Y_1>$ with $Z=\frac{\partial
}{\partial y}$ and any vector from $Z_G(G')=<Y_1,Y_2>$ which is linearly independent of $G'$ with $Y=x\frac{\partial }{\partial
y}$, in the following way: $$\label{6e5}
\begin{array}{ccc}
Z=Y_1,&Y=Y_2.\\
\end{array}$$ Applying this correspondence to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{6e6}
\begin{array}{cccc}
\phi_y=1,&x\phi_y=\psi,&\psi_y=0,&x\psi_y=0.\\
\end{array}$$ The solution of the system (\[6e6\]) gives the following point transformation $$\label{6e7}
\begin{array}{cc}
u=y,& v=x,\\
\end{array}$$ which transforms the vector $-Y_3$ which is linearly independent of $Z_G(G')$ to $$\label{6e8}
\begin{array}{cc}
X=-x\frac{\partial}{\partial x}+(-y+f(x))\frac{\partial}{\partial y},\\
\end{array}$$ with $f(x)=x^4$. Such a function $f(x)$ can be absorbed using the transformation $$\label{6e9}
\begin{array}{cc}
\tilde{x}=x,& \tilde{y}=y+x\int \frac{f(x)}{x^2}dx=y+\frac{1}{3}x^4.\\
\end{array}$$ Finally, the composition of the transformations (\[6e7\]) and (\[6e9\]) transforms ODE (\[6e1\]) to its canonical form $$\label{6e10}
\tilde{y}'''=\frac{1}{\tilde{x}^2}g\left(\tilde{x}\tilde{y}''\right),$$ with $g(z)=\frac{1}{z}$.
[$L^{I}_{3:5}$]{} \[ex7\]\
Consider the ODE $$\label{7e1}
v'''= \frac{1}{{{v'}^{4}}}\left({- \left( {v''}^{2}+v'' \right) ^{2}{{\rm e}^{-3\,v}}+2\,{v'}^{7}+3\,v''{v'}^{5}+3\,{v'}^{3}{v''}^{2}}\right)\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{7e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial u},&Y_2=u\frac{\partial}{\partial u}+\frac{\partial}{\partial v},&Y_3=e^{-v}\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{7e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{2}] = Y_{1}$,& $[Y_{2}, Y_{3}] = -Y_{3}$.\\
\end{tabular}$$ Here dim $G'=2$, rank $(G')=2$ and the adjoint action of $G/G'=<\overline{Y}_2>$ on $G'=<Y_1,Y_3>$ is given by $$\label{7e4}
{\rm ad}\,(\overline{Y}_2)= \left({
\begin{array}{cc}
-1 & 0 \\
0 & -1 \\
\end{array}
}\right).$$ So $\lambda=-1$ is a repeated real eigenvalue with eigenspace of dimension 2. Using Theorem \[th1\], the third-order ODE (\[7e1\]) can be transformed to the canonical form of $L^{I}_{3:5}$ via a point transformation.
In order to construct such a point transformation, one needs to match any two linearly independent vectors of $G'$ with $X=\frac{\partial }{\partial x}$ and $Y=\frac{\partial }{\partial
y}$. For example, one can try the obvious choices: $$\label{7e5}
\begin{array}{ccc}
X=Y_1,&Y=Y_3,\\
\end{array}$$ or the opposite $$\label{7e6}
\begin{array}{ccc}
X=Y_3,&Y=Y_1.\\
\end{array}$$ Applying the correspondence (\[7e5\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{7e7}
\begin{array}{cccc}
\phi_x=1,&\phi_y=0,&\psi_x=0,&\psi_y=e^{-\psi}.\\
\end{array}$$ The solution of the system (\[7e7\]) gives the following point transformation $$\label{7e8}
\begin{array}{cc}
u=x,& v=\ln{y},\\
\end{array}$$ which transforms ODE (\[7e1\]) to its canonical form $$\label{7e9}
y'''=f(y') {y''}^2$$ with $f(z)=\frac{3z^3-1}{z^4}$.
Similarly, applying the correspondence (\[7e6\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{7e10}
\begin{array}{cccc}
\phi_x=0,&\phi_y=1,&\psi_x=e^{-\psi},&\psi_y=0.\\
\end{array}$$ The solution of the system (\[7e10\]) gives the following point transformation $$\label{7e11}
\begin{array}{cc}
u=y,& v=\ln{x},\\
\end{array}$$ which transforms ODE (\[7e1\]) to its canonical form $$\label{7e12}
y'''=f(y') {y''}^2$$ with $f(z)=z^2$.
[$L^{II}_{3:5}$]{} \[ex8\]\
Consider the ODE $$\label{8e1}
v^{(4)}={\frac {4\,v{v'}^{12}-4\,v{v'}^{9}v''+v{v'}^{6}{v''}^{2}+10\,{v'}^{2}v''{v'''}^{2}-45\,v'{v''}^{3}v'''+45\,{v''}^{5}}{{v'}^{2} \left( v'''v'-3\,{v''}^{2} \right) }}\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{8e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial u},&Y_2=v\frac{\partial}{\partial u},&Y_3=(u+v^2)\frac{\partial}{\partial u},\\
\end{array}$$ with the nonzero commutators $$\label{8e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{3}] =Y_{1}$,& $[Y_{2}, Y_{3}] =Y_{2}$.\\
\end{tabular}$$ Here dim $G'=2$, rank $(G')=1$ and the adjoint action of $G/G'=<\overline{Y}_3>$ on\
$G'=<Y_1,Y_2>$ is given by $$\label{8e4}
{\rm ad}\, (\overline{Y}_3)= \left({
\begin{array}{cc}
-1 & 0 \\
0 & -1 \\
\end{array}
}\right).$$ We have $\lambda=-1$ as a repeated real eigenvalue with eigenspace of dimension 2. Using Theorem \[th1\], the forth-order ODE (\[8e1\]) can be transformed to the canonical form of $L^{II}_{3:5}$ via a point transformation.
In order to construct such a point transformation, one needs to match any two linearly independent vectors of $G'$ with $X=\frac{\partial }{\partial y}$ and $Y=x\frac{\partial }{\partial
y}$. For example, one can try the obvious choice: $$\label{8e5}
\begin{array}{ccc}
X=Y_1,&Y=Y_2,\\
\end{array}$$ Applying the correspondence (\[8e5\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{8e7}
\begin{array}{cccc}
\phi_y=1,&x\phi_y=\psi,&\psi_y=0,&x\psi_y=0.\\
\end{array}$$ The solution of the system (\[8e7\]) gives the following point transformation $$\label{8e8}
\begin{array}{cc}
u=y,& v=x,\\
\end{array}$$ which transforms the vector $-Y_3$ which is linearly independent of $G'$ to $$\label{8e9}
\begin{array}{cc}
Z=(-y+f(x))\frac{\partial}{\partial y},\\
\end{array}$$ with $f(x)=-x^2$. The function $f(x)$ can be absorbed using the transformation $$\label{8e10}
\begin{array}{cc}
\tilde{x}=x,& \tilde{y}=y-f(x)=y+x^2.\\
\end{array}$$ Finally, the composition of the transformations (\[8e8\]) and (\[8e10\]) transforms ODE (\[8e1\]) to its canonical form $$\label{8e11}
\tilde{y}^{(4)}=\tilde{y}''~g\left(x,\frac{\tilde{y}'''}{\tilde{y}''}\right),\\$$ with $g(z,w)=\frac{z}{w}$.
[$L^{I}_{3:6}$]{}\[ex9\]\
Consider the ODE $$\label{9e1}
v''=\frac{1}{9}v^{-4}\left(18\,{v_{{}}}^{3}{v'}^{2}-\sqrt{{v_{{}}}^{2}-2\,v'} \left( {v_{{}}}^{2}+v' \right) ^{5/2}\right)\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{9e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial u},&Y_2=v^2\frac{\partial}{\partial v},&Y_3=(u+\frac{4}{5v})\frac{\partial}{\partial u}-(\frac{2}{5}uv^2+\frac{7}{5}v)\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{9e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{3}] = Y_{1}-\frac{2}{5}Y_{2}$,& $[Y_{2}, Y_{3}] = -\frac{4}{5}Y_{1}+\frac{7}{5}Y_{2}$.\\
\end{tabular}$$ We have dim $G'=2$, rank $(G')=2$ and the adjoint action of $G/G'=<\overline{Y}_3>$ on $G'=<Y_1,Y_2>$ given by $$\label{9e4}
{\rm ad}\, (\overline{Y}_3)= \left({
\begin{array}{cc}
-1 & \frac{4}{5} \\
\frac{2}{5} & -\frac{7}{5} \\
\end{array}
}\right)$$ has $\lambda_1=-\frac{3}{5}$ and $\lambda_2=-\frac{9}{5}$ as distinct real eigenvalues. The corresponding eigenvectors are $2Y_1+Y_2$ and $Y_2-Y_1$ respectively. Using the Theorem \[th1\], the second-order ODE (\[9e1\]) can be transformed to the canonical form of $L^{I}_{3:6}$ via a point transformation.
In order to construct such a point transformation, one needs to match the the two eigenvectors of ad $(G/G')$ on $G'$ with constant multiples $X=\frac{\partial }{\partial x}$ and $Y=\frac{\partial
}{\partial y}$ respectively, in the following way: $$\label{9e5}
\begin{array}{ccc}
X=r(2Y_1+Y_2),&Y=s(Y_2-Y_1),& r, s\in \mathbb{R}\setminus{\{0\}}.\\
\end{array}$$ Applying the correspondence (\[9e5\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{9e6}
\begin{array}{cccc}
\phi_x=2r,&\phi_y=-s,&\psi_x=r{\psi}^2,&\psi_y=s{\psi}^2.\\
\end{array}$$ The solution of the system (\[9e6\]) gives the following point transformation $$\label{9e7}
\begin{array}{cc}
u=2rx-sy,& v=\frac{-1}{rx+sy},\\
\end{array}$$ which transforms ODE (\[9e1\]) to its canonical form $$\label{9e8}
y''=C y'^{\frac{c-2}{c-1}}, c\ne 0,\frac{1}{2},1,2,$$ with $C=r^{\frac{3}{2}}{(-s)}^{-\frac{1}{2}}$ and $c=\frac{\lambda_2}{\lambda_1}=3$.
[$L^{II}_{3:6}$]{} \[ex10\]\
Consider the ODE $$\label{10e1}
v'''= \frac{1}{2} \left(-4\,v'+6\,v''-8 \right)+ \frac{1}{2}\left(v'-v''+2 \right) ^{ 2} {{\rm e}^{2\,u}}\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{10e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial v},&Y_2=e^u\frac{\partial}{\partial v},&Y_3=\frac{\partial}{\partial u}-(4u+2v)\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{10e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{3}] =-2 Y_{1}$,& $[Y_{2}, Y_{3}] = -3Y_{2}$.\\
\end{tabular}$$ Here dim $G'=2$, rank $(G')=1$ and the adjoint action of $G/G'=<\overline{Y}_3>$ on $G'=<Y_1,Y_2>$ has $\lambda_1=2$ and $\lambda_2=3$ as distinct real eigenvalues with eigenvector $Y_1$ and $Y_2$ respectively. Using Theorem \[th1\], the third-order ODE (\[10e1\]) can be transformed to the canonical form of $L^{II}_{3:6}$ via a point transformation.
In order to construct such a point transformation, one needs to match the the two eigenvectors of ad $(G/G')$ on $G'$ with $X=\frac{\partial }{\partial y}$ and $Y=x\frac{\partial }{\partial
y}$ respectively, in the following way: $$\label{10e5}
\begin{array}{ccc}
X=Y_1,&Y=Y_2.\\
\end{array}$$ Applying the correspondence (\[10e5\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{10e6}
\begin{array}{cccc}
\phi_y=0,&x\phi_y=0,&\psi_y=1,&x\psi_y=e^{\phi}.\\
\end{array}$$ The solution of the system (\[10e6\]) gives the following point transformation $$\label{10e7}
\begin{array}{cc}
u=\ln x,& v=y,\\
\end{array}$$ which transforms the vector $\frac{1}{2}Y_3$ which is linearly independent of $G'$ to $$\label{10e8}
\begin{array}{cc}
Z=(c-1)x\frac{\partial}{\partial x}+(-y+f(x))\frac{\partial}{\partial y},\\
\end{array}$$ with $f(x)=-2\ln x$ and $c=\frac{\lambda_2}{\lambda_1}=\frac{3}{2}$. The function $f(x)$ can be absorbed using the transformation $$\label{10e9}
\begin{array}{cc}
\tilde{x}=x,& \tilde{y}=y+\frac{1}{1-c}~x^{\frac{1}{1-c}}\int f(x) x^{\frac{2-c}{c-1}}dx=y+2\ln x-1.\\
\end{array}$$ Finally, the composition of the transformations (\[10e7\]) and (\[10e9\]) transforms ODE (\[10e1\]) to its canonical form $$\label{10e10}
\tilde{y}'''=\tilde{x}^{\frac{2-3c}{c-1}}g\left(\tilde{y}''\tilde{x}^{\frac{2c-1}{c-1}}\right), c\ne 0,1\\$$ with $g(z)=\frac{1}{2}z^2$ and $c=\frac{3}{2}$.
[$L^{I}_{3:7}$]{}\[ex11\]\
Consider the ODE $$\label{11e1}
v''=\frac{1}{9} {(2{v'}^2-2v'+5)}^{\frac{3}{2}}\exp{\left(3\arctan{\left(\frac{ v'-2}{v'+1}\right)}\right)}\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{11e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial u},&Y_2=\frac{\partial}{\partial v},&Y_3=(4u+v)\frac{\partial}{\partial u}+(5v-\frac{5}{2}u)\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{11e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{3}] =4 Y_{1}-\frac{5}{2}Y_{2}$,& $[Y_{2}, Y_{3}] = Y_{1}+5Y_{2}$.\\
\end{tabular}$$ Here dim $G'=2$, rank $(G')=2$ and the adjoint action of $G/G'=<\overline{Y}_3>$ on $G'=<Y_1,Y_2>$ is given by $$\label{11e4}
{\rm ad}\, (\overline{Y}_3)= \left({
\begin{array}{cc}
-4 & -1 \\
\frac{5}{2} &-5\\
\end{array}
}\right).$$ The eigenvalues are $-\frac{9}{2}\pm\frac{3}{2}\sqrt{-1}$ with eigenvectors $(\frac{1}{5}\pm\frac{3}{5}\sqrt{-1})Y_1+Y_2$ respectively. Using the Theorem \[th1\], the second-order ODE (\[11e1\]) can be transformed to the canonical form of $L^{I}_{3:7}$ via a point transformation.
In order to construct such a point transformation, one needs to match the real and imaginary parts of an eigenvector of ad $(G/G')$ on $G'$ with $X=\frac{\partial }{\partial x}$ and $Y=\frac{\partial }{\partial y}$ respectively in the following way: $$\label{11e5}
\begin{array}{cc}
X=\frac{1}{5}Y_1+Y_2,&Y=\frac{3}{5}Y_1.\\
\end{array}$$ Applying the correspondence (\[11e5\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{11e6}
\begin{array}{cccc}
\phi_x=\frac{1}{5},&\phi_y=\frac{3}{5},&\psi_x=1,&\psi_y=0.\\
\end{array}$$ Solving the system (\[11e6\]) gives the following point transformation $$\label{11e7}
\begin{array}{cc}
u=\frac{1}{5}x+\frac{3}{5}y,& v=x,\\
\end{array}$$ which transforms ODE (\[11e1\]) after simplification using the identity $$\tan^{-1} x-\tan^{-1}
\left(\frac{2x-1}{x+2}\right)=c=\tan^{-1}
\left(\frac{1}{2}\right)$$ to its canonical form $$\label{11e8}
y''=C {(1+{y'}^2)}^{\frac{3}{2}}\exp{(b\arctan{ y'})}\\$$ with $C=-\frac{e^{3c}}{\sqrt{5}}$ and $b=\cot{\theta}=-3$ where $\theta=arg(-\frac{9}{2}+\frac{3}{2}\sqrt{-1})$.
[$L^{II}_{3:7}$]{} \[ex12\]\
Consider the ODE $$\label{12e1}
v'''={\frac {4{{\rm e}^{-8\,\arctan \left( \frac{1}{u} \right)}}}{ \left( {u}^{2}+1 \right) ^{\frac{5}{2}}u \left( 1-4 \left( {u}^{2}+1 \right) ^{\frac{3}{2}} \left( uv''+2\,v' \right) \right) }}-3\,{\frac {2{u}^{2}v''+2v'u+\,v''}{ \left( {u}^{2}+1 \right)u}}\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{12e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial v},&Y_2=\frac{1}{u}\frac{\partial}{\partial v},&Y_3=(u^2+1)\frac{\partial}{\partial u}+\frac{1}{u}(4uv-v-\sqrt{u^2+1})\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{12e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{3}] =4 Y_{1}-Y_{2}$,& $[Y_{2}, Y_{3}] =Y_{1}+4Y_{2}$.\\
\end{tabular}$$ Here dim $G'=2$, rank $(G')=1$ and the adjoint action of $G/G'=<\overline{Y}_3>$ on $G'=<Y_1,Y_2>$ is given by $$\label{12e4}
{\rm ad}\,(\overline{Y}_3)= \left({
\begin{array}{cc}
-4 & -1 \\
1 &-4\\
\end{array}
}\right).$$ The eigenvalues are $-4\pm \sqrt{-1}$ with eigenvectors $Y_2\pm
\sqrt{-1} ~Y_1$ respectively. Using the Theorem \[th1\], the third-order ODE (\[12e1\]) can be transformed to the canonical form of $L^{II}_{3:7}$ via a point transformation.
To construct such a point transformation, one needs to match the real part and imaginary part of an eigenvector of ad $(G/G')$ on $G'$ with $X=\frac{\partial }{\partial y}$ and $Y=x\frac{\partial
}{\partial y}$ in the following way: $$\label{12e5}
\begin{array}{cc}
X=Y_2,&Y=Y_1.\\
\end{array}$$ Applying the correspondence (\[12e5\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{12e6}
\begin{array}{cccc}
\phi_y=0,&x\phi_y=0,&\psi_y=\frac{1}{\phi},&x\psi_y=1.\\
\end{array}$$ Solving the system (\[12e6\]) gives the following point transformation $$\label{12e7}
\begin{array}{cc}
u=x,& v=\frac{y}{x},\\
\end{array}$$ which transforms the vector $Y_3$ which is linearly independent of $G'$ to $$\label{12e8}
\begin{array}{cc}
(\frac{1}{\sin \theta})Z=(1+x^2)\, \partial_{x}+(y(x-b)+f(x))\,\partial_{y},\\
\end{array}$$ with $f(x)=-\sqrt{x^2+1}$ and $b=\cot{\theta}=-4$. The function $f(x)$ can be absorbed using the transformation $$\label{12e9}
\begin{array}{lll}
\tilde{x}=x,& \tilde{y}&=y-\sqrt{x^2+1} ~e^{-b\tan^{-1}x}\int \frac{e^{b \tan^{-1}x}}{{(x^2+1)}^{\frac{3}{2}}} f(x) dx=y-\frac{1}{4}\sqrt{x^2+1}.\\
\end{array}$$ Finally, the composition of the transformations (\[12e7\]) and (\[12e9\]) transforms ODE (\[12e1\]) after simplification using the identity $\tan^{-1}
x+\tan^{-1}\frac{1}{x}=\frac{\pi}{2}$ to its canonical form $$\label{12e10}
\tilde{y}'''=\frac{\tilde{y}''}{1+\tilde{x}^2}\left(f\left(\tilde{y}''{(\tilde{x}^2+1)}^{\frac{3}{2}}e^{b\tan^{-1}\tilde{x}}\right)-3\tilde{x}\right)\\$$ with $f(z)=-\frac{e^{4\pi}}{z^2}$ and $b=-4$.
[$L^{I}_{3:8}$]{}\[ex13\]\
Consider the ODE $$\label{13e1}
v''= {(v+v')}^{3}-\frac{1}{2}v-\frac{3}{2}v'\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{13e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial u},&Y_2=\exp(-u)\frac{\partial}{\partial v},&Y_3=v\exp(u)\frac{\partial}{\partial u}-\frac{1}{2}v^2\exp(u)\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{13e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{2}] = -Y_{2}$,& $[Y_{1}, Y_{3}] = Y_{3}$,& $[Y_{2}, Y_{3}] = Y_{1}$.\\
\end{tabular}$$ Since dim $G'=3$, the Killing form is indefinite and rank $G=2$, using the Theorem \[th1\], the second-order ODE (\[13e1\]) can be transformed to one of the three canonical forms $L^{I}_{3:8}$, $L^{II}_{3:8}$ and $L^{III}_{3:8}$ via a point transformation.
Since the eigenvalues of ad $(Y_1)$ are $\pm 1$, then by a scaling, as explained in section \[4.3\], one can get the change of basis $$\label{13e4}
\begin{array}{cccc}
X=Y_3,& Y= -2Y_2,&Z=2Y_1.\\
\end{array}$$ This maps the nonzero commutators (\[13e3\]) to the standard relations given by $$[Z,X]=2X,[Z,Y]=-2Y,[X,Y]=Z.$$ Applying the correspondence (\[13e4\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{13e5}
\begin{array}{llll}
\phi_y=\exp(\phi)\psi,&\psi_y=-\frac{1}{2}\exp(\phi){\psi}^2,\\
-2xy\phi_x+(\epsilon~x^2-y^2)\phi_y=0,&-2xy\psi_x+(\epsilon~x^2-y^2)\psi_y=-2\exp(-\phi),\\
x\phi_x+y\phi_y=-1,&x\psi_x+y\psi_y=0,\\
\end{array}$$ for some $\epsilon \in \{0,1,-1\}$. Since the system (\[13e5\]) is consistent for $\epsilon=0$, its solution $$\label{13e6}
\begin{array}{cc}
u=\ln{\left(\frac{x}{y^2}\right)}+c_1,& v=-2e^{-c_1}\left(\frac{y}{x}\right),\\
\end{array}$$ transforms ODE (\[13e1\]) to the canonical form $L^{I}_{3:8}$ $$\label{13e7}
xy''=C{y'}^3-\frac{1}{2}~y',\\$$ with $C=4e^{-2c_1}$.
[$L^{II}_{3:8}$]{} \[ex14\]\
Consider the ODE $$\label{14e1}
u^4 v v''= {({v'}^2 u^4+1)}^{\frac{3}{2}}-1-2v v' u^3-{v'}^2 u^4\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{14e2}
\begin{array}{lll}
Y_1=u^2\frac{\partial}{\partial u},&Y_2=-u\frac{\partial}{\partial u}+v\frac{\partial}{\partial v},&Y_3=(u^2v^2-1)\frac{\partial}{\partial u}+\frac{2v}{u}\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{14e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{2}] = Y_{1}$,& $[Y_{1}, Y_{3}] = -2Y_{2}$,& $[Y_{2}, Y_{3}] = Y_{3}$.\\
\end{tabular}$$ Since dim $G'=3$, the Killing form is indefinite and rank $G=2$, using the Theorem \[th1\], the second-order ODE (\[14e1\]) can be transformed to one of the three canonical forms $L^{I}_{3:8}$, $L^{II}_{3:8}$ and $L^{III}_{3:8}$ via a point transformation.
Since the eigenvalues of ad $(Y_2)$ are $\pm 1$, by a scaling, as explained in section \[4.3\], one can get the change of basis $$\label{14e4}
\begin{array}{cccc}
X=Y_3,& Y= Y_1,&Z=2Y_2.\\
\end{array}$$ This maps the nonzero commutators (\[14e3\]) to the standard relations given by $$[Z,X]=2X,[Z,Y]=-2Y,[X,Y]=Z.$$ Applying the correspondence (\[14e4\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{14e5}
\begin{array}{llll}
\phi_y={\phi}^2{\psi}^2-1,&\psi_y=\frac{2\psi}{\phi},\\
-2xy\phi_x+(\epsilon~x^2-y^2)\phi_y={\phi}^2,&-2xy\psi_x+(\epsilon~x^2-y^2)\psi_y=0,\\
x\phi_x+y\phi_y=\phi,&x\psi_x+y\psi_y=-\psi,\\
\end{array}$$ for some $\epsilon \in \{0,1,-1\}$. Since the system (\[14e5\]) is consistent for $\epsilon=1$, its solution $$\label{14e6}
\begin{array}{cc}
u=-\frac{x^2+y^2}{y},& v=\frac{x}{x^2+y^2},\\
\end{array}$$ transforms ODE (\[14e1\]) to the canonical form $L^{II}_{3:8}$ $$\label{14e7}
xy''=y'+{y'}^3+C(1+{y'}^2)^{\frac{3}{2}},\\$$ with $C=1$.
[$L^{III}_{3:8}$]{} \[ex15\]\
Consider the ODE $$\label{15e1}
u^4 v^3 v''= 3v^2{v'}^2u^4-v^6-2v^3v'u^3+{({v'}^2u^4-v^4)}^{\frac{3}{2}}\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{15e2}
\begin{array}{lll}
Y_1=u^2\frac{\partial}{\partial u},&Y_2=-u\frac{\partial}{\partial u}-v\frac{\partial}{\partial v},&Y_3=(1+\frac{u^2}{v^2})\frac{\partial}{\partial u}+\frac{2v}{u}\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{15e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{2}] = Y_{1}$,& $[Y_{1}, Y_{3}] = 2Y_{2}$,& $[Y_{2}, Y_{3}] = Y_{3}$.\\
\end{tabular}$$ Here again the Killing form is non-degenerate and indefinite and rank $G=2$. Using the Theorem \[th1\], the second-order ODE (\[15e1\]) can be transformed to one of the three canonical forms $L^{I}_{3:8}$, $L^{II}_{3:8}$ and $L^{III}_{3:8}$ via a point transformation.
Since the eigenvalues of ad $(Y_2)$ are $\pm 1$, then by a scaling, as explained in section \[4.3\], one has the change of basis $$\label{15e4}
\begin{array}{cccc}
X=Y_3,& Y= -Y_1,&Z=2Y_2.\\
\end{array}$$ This maps the nonzero commutators (\[15e3\]) to the standard relations given by $$[Z,X]=2X,[Z,Y]=-2Y,[X,Y]=Z.$$ Applying the correspondence (\[15e4\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{15e5}
\begin{array}{llll}
\phi_y=1+\frac{{\phi}^2}{{\psi}^2},&\psi_y=\frac{2\psi}{\phi},\\
-2xy\phi_x+(\epsilon~x^2-y^2)\phi_y=-{\phi}^2,&-2xy\psi_x+(\epsilon~x^2-y^2)\psi_y=0,\\
x\phi_x+y\phi_y=\phi,&x\psi_x+y\psi_y=\psi,\\
\end{array}$$ for some $\epsilon \in \{0,1,-1\}$. Since the system (\[15e5\]) is consistent for $\epsilon=-1$, its solution $$\label{15e6}
\begin{array}{cc}
u=\frac{y^2-x^2}{y},& v=\frac{y^2-x^2}{x},\\
\end{array}$$ transforms ODE (\[15e1\]) to the canonical form $L^{III}_{3:8}$ $$\label{15e7}
xy''=y'-{y'}^3+C(1-{y'}^2)^{\frac{3}{2}},\\$$ with $C=-1$.
[$L^{IV}_{3:8}$]{} \[ex16\]\
Consider the ODE $$\label{16e1}
v'''=\frac{3}{2}\frac{{v''}^2}{v'}-\frac{{v'}^3}{v^2}\\$$ that admits the three-dimensional point symmetry $\mathbf{subalgebra}$ generated by $$\label{16e2}
\begin{array}{lll}
Y_1=u\frac{\partial}{\partial u},&Y_2=\frac{\partial}{\partial u},&Y_3=\frac{1}{2}u^2\frac{\partial}{\partial u},\\
\end{array}$$ with the nonzero commutators $$\label{16e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{2}] = -Y_{2}$,& $[Y_{1}, Y_{3}] = Y_{3}$,& $[Y_{2}, Y_{3}] = Y_{1}$.\\
\end{tabular}$$ Here dim $G'=3$, the Killing form is indefinite and rank $G=1$. Using the Theorem \[th1\], the third-order ODE (\[16e1\]) can be transformed to the canonical form $L^{IV}_{3:8}$ via a point transformation.
Since the eigenvalues of ad $(Y_1)$ are $\pm 1$, then by a scaling, as explained in section \[4.3\], one can get the change of basis $$\label{16e4}
\begin{array}{cccc}
X=Y_3,& Y= -2Y_2,&Z=2Y_1.\\
\end{array}$$ This maps the nonzero commutators (\[13e3\]) to the standard relations given by $$[Z,X]=2X,[Z,Y]=-2Y,[X,Y]=Z.$$ Applying the correspondence (\[16e4\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{16e5}
\begin{array}{llll}
\phi_y=\frac{1}{2}{\phi}^2,&\psi_y=0,\\
-y^2\phi_y=-2,&-y^2\psi_y=0,\\
y\phi_y=-\phi,&y\psi_y=0.\\
\end{array}$$ The solution of the system (\[16e5\]) gives a point transformation $$\label{16e6}
\begin{array}{cc}
u=-\frac{2}{y},& v=x,\\
\end{array}$$ that transforms ODE (\[16e1\]) to the canonical form $L^{IV}_{3:8}$ $$\label{16e7}
y'''=\frac{3}{2}\frac{{y''}^2}{y'}+f(x)y',\\$$ with $f(x)=\frac{1}{x^2}$.
[$L_{3:9}$]{} \[ex17\]\
Consider the ODE $$\label{17e1}
v''=-{v'}^3\cos u \sin u-2v'\cot u+\csc u{({v'}^2\sin^2 u +1)}^\frac{3}{2}\\$$ that admits the three-dimensional point symmetry algebra generated by $$\label{17e2}
\begin{array}{lll}
Y_1=\frac{\partial}{\partial v},&Y_2=\sin v\frac{\partial}{\partial u}+\cos v\cot u\frac{\partial}{\partial v},&Y_3=\cos v\frac{\partial}{\partial u}-\sin v\cot u\frac{\partial}{\partial v},\\
\end{array}$$ with the nonzero commutators $$\label{17e3}
\begin{tabular}{llllll}
$[Y_{1}, Y_{2}] =Y_{3}$,& $[Y_{1}, Y_{3}] =-Y_{2}$,& $[Y_{2}, Y_{3}] =Y_{1}$.\\
\end{tabular}$$ Since the Killing form is negative definite, using Theorem \[th1\], the second-order ODE (\[17e1\]) can be transformed to the canonical form of $L_{3:9}$ via a point transformation.
In order to construct such a point transformation, pick any vector like $Y_1$ and find its non-zero eigenvalues. Here ad $(Y_1)$ has $\pm \sqrt{-1}$ as eigenvalues with eigenvectors $Y_3\pm
\sqrt{-1}~ Y_2$ respectively. One needs to match the vector $Y_1$ with $X$ and a multiple of eigenvector $Y_3+ \sqrt{-1}~ Y_2$ with the vector $Y-\sqrt{-1}~Z$ such that $[Y,Z]=X$ in the following way: $$\label{17e5}
\begin{array}{ccc}
X=Y_1,&Y=Y_3,&Z=-Y_2.\\
\end{array}$$ Applying the correspondence (\[17e5\]) to the point transformation $u=\phi(x,y), v=\psi(x,y)$ yields the system $$\label{17e6}
\begin{array}{llll}
\phi_x=0,&\psi_x=1,\\
y\sin x ~\phi_x + \left( {y}^{2}+1\right) \cos x ~\phi_y =\cos \psi,&y\sin x ~\psi_x + \left( {y}^{2}+1 \right) \cos x ~\psi_y =-\sin \psi \cot \phi,\\
y\cos x ~\phi_x - \left( {y}^{2}+1\right) \sin x ~\phi_y =-\sin\psi,&y\cos x ~\psi_x - \left( {y}^{2}+1 \right) \sin x ~\psi_y =-\cos \psi \cot \phi.\\
\end{array}$$ Solution of the system (\[17e6\]) gives the required point transformation. A systematic way of solving such a nonlinear system is as follows:
One can match the vector $Y_1$ with $X$ through the canonical coordinates of $Y_1$ as $$\label{17e7}
\begin{array}{cc}
u=y,& v=x.\\
\end{array}$$ This transforms the vector $Y-\sqrt{-1}~Z$ to $$\label{17e8}
\begin{array}{cc}
Y-\sqrt{-1}~Z=Y_3+\sqrt{-1}~Y_2=e^{\sqrt{-1}~x}\left[ \left( f_1(y)\partial_{x}+f_2(y)\partial_{y}\right) + \sqrt{-1} \left( f_3(y)\partial_{x}+f_4(y)\partial_{y}\right)\right],\\
\end{array}$$ with $f_1(y)=0,f_2(y)=1,f_3(y)=\cot{y}$ and $f_4(y)=0$. Now using the formula (\[te4\]), the vector $Y-\sqrt{-1}~Z$ can be transformed using the transformation $$\label{17e9}
\begin{array}{lll}
\tilde{x}=x+{\tan}^{-1}\left(\frac{f_4}{f_2}\right)=x,& \tilde{y}&=\frac{f_1f_4-f_2f_3}{\sqrt{f_2^2+f_4^2}}=-\cot y.\\
\end{array}$$ to the canonical form $$\label{17e10}
\begin{array}{cc}
Y-\sqrt{-1}~Z=e^{\sqrt{-1}\tilde{x}}\left[ (1+{\tilde{y}}^2)\partial_{\tilde{y}} - \sqrt{-1}~\tilde{y}\partial_{\tilde{x}}\right],\\
\end{array}$$ Finally, the composition of the transformations (\[17e7\]) and (\[17e9\]) given by $$\label{17e11}
\begin{array}{cc}
u=-\cot^{-1}{\tilde{y}},& v=\tilde{x},\\
\end{array}$$ transforms ODE (\[17e1\]) to its canonical form $$\label{17e12}
\tilde{y}''=C{\left( \frac{\tilde{y}^{'2}+\tilde{y}^2+1}{1+\tilde{y}^2}\right)}^{\frac{3}{2}}-\tilde{y}\\$$ with $C=1$.
Moreover, a solution of the nonlinear system (\[17e6\]) is $$\label{17e13}
\begin{array}{cc}
u=-\cot^{-1}{y},& v=x.\\
\end{array}$$
Conclusion
==========
The Lie-Bianchi classification of three-dimensional algebras and their realizations as vector fields in ${\mathbb{R}}^2$ are recovered in an algorithmic way. This is done in such a way that one can read off the type of the algebra from its invariants like the dimension of its commutator or the centralizer of its commutator and its rank.
The compact and non-compact Lie algebras are treated uniformly in a manner that makes their realizations as vector fields in the plane transparent.
The algorithms are illustrated by examples for each type of three dimensional algebras. The procedures works in principle for any ODE which admits a three-dimensional subalgebra of symmetries.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The authors would like to thank the King Fahd University of Petroleum and Minerals for its support and excellent research facilities. FM is grateful to the NRF of South Africa for research funding support.
[99]{} Lie, S, Theorie der Transformationsgruppen, Vol 1-3, Leipzig, 1888, 1890, 1893.
Bianchi, L, Lezioni sulla teoria dei gruppi continui finiti di transformazioni, Pisa, Spoerri, 1918.
Lie, S, Klassifikation und Integration von gew¨onlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestaten, Archiv der Mathematik 1883; VIII, IX, 187.
Lie, Sophus, Vorlesungen $\ddot{\textrm{u}}$ber Differentialgleichungen mit bekannten infinitesimalen Transformationen, BG Teubner, 1891.
Mahomed FM and Leach PGL, Lie algebras associated with second-order ordinary differential equations, Journal of Mathematical Physics 1989; 30, 2770.
Schwarz, F, Janet bases of 2nd order ordinary differential equations. Proceedings of the ISSAC’96, Lakshman R. (ed.). ACM: New York, 1996; 179.
Ibragimov, NH, Elementary Lie Group Analysis and Ordinary Differential Equations. Wiley: Chichester, 1999.
H. Azad, I. Biswas, R. Ghanam and M. T. Mustafa, On Computing Joint Invariants of vector fields Journal of Geometry and Physics, 2015; 97.
Hilgert, Joachim and Karl-Hermann Neeb, Structure and Geometry of Lie Groups. Springer: New York, 2012; 79-132.
|
[Gauge independence of magnetic moment and vanishing charge of Dirac neutrinos: an exact one-loop demonstration]{}\
[Wen-Tao Hou$^{a,b}$, Yi Liao$^a$[^1], and Hong-Jun Liu$^a$]{}
[*$^a$ School of Physics, Nankai University, Tianjin 300071, China\
$^b$ Department of Modern Physics, University of Science and Technology of China,\
Hefei 230026, Anhui, China*]{}
[**Abstract**]{}
The magnetic moment and vanishing charge of a Dirac neutrino are physically observable quantities and must not depend on the choice of gauge in a consistent quantum field theory. We verify this statement explicitly at the one loop level in both $R_\xi$ and unitary gauges of the minimally extended standard model. We accomplish this by manipulating directly the integrands of loop integrals and employing simple algebraic identities and integral relations. Our result generally applies for any masses of the relevant particles and unitary neutrino mixing.
PACS: 14.60.Lm, 13.40.Em, 12.15.Lk
Introduction {#sec:intro}
============
The fundamental properties of a particle like its charge and electromagnetic dipole moments are physical quantities that in principle are experimentally measurable, for reviews, see Refs. [@Pospelov:2005pr; @Giunti:2008ve]. These quantities can be unambiguously calculated in a quantum-mechanically consistent theory like the standard model (SM), and confronted with the measurements to decide whether the theory is correct or not. That said, the practical calculation and demonstration of its result being independent of computational methods are not always trivial. We have witnessed a similar circumstance recently, concerning the one-loop contribution of the charged weak gauge bosons $W^\pm$ to the two-photon decay rate of the Higgs boson. A new computation in unitary gauge [@Gastmans:2011ks; @Gastmans:2011wh] claimed an answer that is different from the well-spread result obtained long ago [@Ellis:1975ap; @Ioffe:1976sd; @Shifman:1979eb; @Vainshtein:1980ea] in a special gauge, i.e., the ’t Hooft-Feynman gauge ($\xi=1$) among the class of renormalizable $R_\xi$ gauges. Subsequent studies by various methods, including computing in both $R_\xi$ and unitary gauges, see for instance Refs. [@Marciano:2011gm; @Shao:2011wx], confirmed the old result, and taught us a great deal on computational subtleties in a theory that is nontrivial in the high energy regime.
In this work we examine a similar problem in the neutrino sector, i.e., the charge and magnetic moment of a Dirac neutrino in SM that is minimally extended by the introduction of right-handed neutrinos. We show explicitly at the one-loop level in both $R_\xi$ and unitary gauges that the neutrino charge vanishes and its magnetic moment is a gauge independent quantity. The issue has been partially studied in the literature. Early works [@Bardeen:1972vi; @Lee:1977tib; @Beg:1977xz; @Marciano:1977wx; @Fujikawa:1980yx] assumed a massless neutrino or expanded the quantities to the leading order in the small masses of neutrinos and charged leptons, ignored the lepton mixing, or computed in a special gauge. A further step was taken some years ago [@CabralRosetti:1999ad; @Dvornikov:2003js; @Dvornikov:2004sj]. It was found [@Dvornikov:2003js], for instance, that up to the second order in the expansion of small neutrino masses the charge vanishes and the magnetic moment is $\xi$-independent, and that the charge vanishes exactly in the ’t Hooft-Feynman gauge. Here we cope directly with the integrands of loop integrals, and demonstrate manifestly that both quantities are gauge independent for any masses of the relevant particles and for any unitary lepton mixing.
In the next section we set up our notations and suggest how to calculate in a nice way to isolate the terms that potentially contribute to the charge and magnetic moment. We describe in some detail in sec \[sec:Rgau\] our calculation in $R_\xi$ gauge. Our one-loop exact result for the magnetic form factor at vanishing momentum transfer is shown in Eq.(\[eq\_F20\]). This is followed by a short discussion in sec \[sec:Ugau\] on the calculation in unitary gauge. We summarize briefly in the last section.
Computational strategy {#sec:pre}
======================
The charge and magnetic moment of a Dirac particle can be defined by the amplitude of a process in which it radiates a photon, $$\begin{aligned}
\bar u(p_-)i{{\cal A}}_\mu(q)u(p_+)=(-ie)\bar u(p_-)\Big[\gamma_\mu
F_1(q^2)-\frac{1}{2m}i\sigma_{\mu\nu}q^\nu F_2(q^2)
+\cdots\Big]u(p_+).
\label{eq_A}$$ Here $p_\pm=p\pm q/2$ are the momenta of the incoming and outgoing particle of mass $m$, and $q$ is the photon’s outgoing momentum. The above decomposition in terms of the standard form factors is based on Lorentz covariance and electromagnetic gauge invariance, and assumes that the Dirac particle in both initial and final states is physical: $$\begin{aligned}
{/\!\!\!\!\!p}_\pm u(p_\pm)=mu(p_\pm),~p_\pm^2=m^2.
\label{eq_EoM}$$ The dots in Eq. (\[eq\_A\]) stand for two more form factors that are irrelevant here; one corresponds to the electric dipole moment that cannot occur at one loop in the minimally extended SM (as can also be seen from sec \[sec:Rgau\]), and the other is the so-called anapole whose Lorentz structure is quadratic in $q$. The form factors at an arbitrary $q^2$ are generally not measurable quantities, since the above (unphysical) amplitude appears as part of the complete contribution to a physical process. Nevertheless, $F_1(0)$ and $F_2(0)$ are physical quantities because they correspond to the charge and anomalous magnetic moment of the particle. Our convention is such that the electron has the charge $eF_1(0)=e<0$ and the magnetic moment vector, $\vec\mu=(e/m)[F_1(0)+F_2(0)]\vec S$ with $\vec S$ being its spin vector, that appears, e.g., in the interaction potential of the dipole with an external magnetic field $\vec B$, $V=-\vec\mu\cdot\vec B$.
The charge, $F_1(0)$, is relatively easy to isolate. Setting $q=0$ removes all other Lorentz structures, and allows us to employ the equations of motion (EoMs) in the limit $q\to 0$ for both initial and final particles, ${/\!\!\!\!\!p}u(p)=mu(p)$, to reduce the amplitude completely to the $\gamma_\mu$ form. There are several ways to work out the anomalous magnetic moment $F_2(0)$. One could isolate by brute force terms contributing to the form factor $F_2(q^2)$ and take its value at $q^2=0$. Most studies in the literature follow this approach. In the second approach, one employs a projection operator, and expresses $F_2(0)$ as a combination of Dirac traces [@Roskies:1990ki; @Czarnecki:1996if; @Czarnecki:1996rx]. Here we take a third approach, which might be the best to observe the cancellation of gauge dependence among various Feynman graphs. As we will show in the next section, the cancellation happens at the level of loop integrands. In this approach, we take the derivative of the amplitude with respect to the photon momentum, $i\partial_\nu^q{{\cal A}}_\mu(q)$, antisymmetrize it in the Lorentz indices $\mu$ and $\nu$, and then evaluate it at $q=0$. Since all form factors are smooth at $q^2=0$, only the magnetic moment term survives the procedure and yields $-e/(2m)\sigma_{\mu\nu}F_2(0)$. \[We remind once again that the electric dipole term vanishes at one loop but would appear at higher orders.\] Comparison of the two gives the answer for $F_2(0)$.
An important point in implementing the above procedure should be noted. We mentioned that the decomposition in eq (\[eq\_A\]) is possible only upon using EoMs (\[eq\_EoM\]). When computing $F_2(0)$, we are essentially expanding ${{\cal A}}_\mu(q)$ in small $q$ and isolating its linear terms. A term that is manifestly linear in $q$ cannot avoid our eyes, for which we are free to apply the limiting EoMs, ${/\!\!\!\!\!p}u(p)=mu(p)$, because the difference to the exact ones does not affect $F_2(0)$. With terms of apparently zeroth order in $q$ we should be careful. For these terms, when necessary, we must apply the exact equations (\[eq\_EoM\]) since the difference now is exactly what we are interested in and may enter $F_2(0)$. Ignoring this will result in an incorrect, gauge-dependent answer. Another point is more technical. Although antisymmetrization in Lorentz indices is not mandatory since it will come out automatically upon finishing the calculation, one can simplify the algebra by doing antisymmtrization at an early stage.
(400,180)(0,0)
(10,120) (0,0)(20,0)(20,0)(80,0)(80,0)(100,0)(20,0)(50,40)[3]{}[7]{}(80,0)(50,40)[-3]{}[7]{}(50,40)(50,55)[-3]{}[2.5]{}(0,-8)\[r\][$\nu_i$]{}(50,-8)\[\][$\ell_\alpha$]{} (100,-8)\[l\][$\nu_i$]{}(10,6)\[\][$p_+$]{}(50,6)\[\][$k\!\!+\!\!p$]{}(90,6)\[l\][$p_-$]{}(55,50)\[l\][$q\!\uparrow$]{}(50,60)\[\][$\gamma~_\mu$]{}(22,25)\[\][$W^-$]{}(82,25)\[\][$W^-$]{}(38,18)\[l\][$k_-$]{}(65,18)\[r\][$k_+$]{}(22,15)\[\][$_\rho$]{}(82,15)\[\][$_\sigma$]{} (35,35)\[\][$_\alpha$]{}(65,35)\[\][$_\beta$]{} (50,-25)\[\][$(a)$]{}
(140,120) (0,0)(20,0)(20,0)(80,0)(80,0)(100,0)(20,0)(50,40)[3]{}(80,0)(50,40)[3]{}(50,40)(50,55)[-3]{}[2.5]{}(22,25)\[\][$G^-$]{}(82,25)\[\][$G^-$]{}(50,-25)\[\][$(b)$]{}
(270,120) (0,0)(20,0)(20,0)(80,0)(80,0)(100,0)(20,0)(50,40)[3]{}(80,0)(50,40)[-3]{}[7]{}(50,40)(50,55)[-3]{}[2.5]{}(50,-25)\[\][$(c)$]{}
(10,20) (0,0)(20,0)(20,0)(80,0)(80,0)(100,0)(80,0)(50,40)[3]{}(20,0)(50,40)[3]{}[7]{}(50,40)(50,55)[-3]{}[2.5]{}(50,-25)\[\][$(d)$]{}
(140,20) (0,30)(20,30)(20,30)(50,30)(50,30)(80,30) (80,30)(100,30)(50,30)(30,180,360)[3]{}[10]{} (50,30)(50,55)[-3]{}[3]{}(10,36)\[\][$p_+$]{}(33,36)\[\][$k\!\!+\!\!p_+$]{} (67,36)\[\][$k\!\!+\!\!p_-$]{}(90,36)\[l\][$p_-$]{}(50,-20)\[\][$(e)$]{}
(270,20) (0,30)(20,30)(20,30)(50,30)(50,30)(80,30) (80,30)(100,30)(50,30)(30,180,360)[3]{}(50,30)(50,55)[-3]{}[3]{}(50,-20)\[\][$(f)$]{}
Fig. 1 Feynman diagrams contributing at one loop to the vertex function $i\Gamma_\mu(q)$. Wavy (dashed, dotted, solid) lines stand for the gauge boson (scalar, ghost, fermion) fields.
(350,170)(0,0)
(10,120)(30,5)(0,0)[2]{}[4]{}(30,5)(60,0)[2]{}[4]{}(30,25)(20,0,360)[2]{}[18]{} (0,-8)\[r\][$A_\mu$]{}(60,-8)\[l\][$Z_\nu$]{} (10,10)\[r\][$_\rho$]{}(50,10)\[l\][$_\sigma$]{}(55,25)\[l\][$W^-$]{}(30,-25)\[\][$(a)$]{}
(90,120) (30,5)(0,0)[2]{}[4]{}(30,5)(60,0)[2]{}[4]{}(30,25)(20,0,360)[3]{} (55,25)\[l\][$G^-$]{}(30,-25)\[\][$(b)$]{}
(170,120)(15,20)(0,20)[2]{}[2.5]{}(55,20)(70,20)[-2]{}[2.5]{} (35,20)(20,0,360)[2]{}[18]{}(35,30)\[\][$W^-$]{}(15,10)\[r\][$_\rho$]{}(15,30)\[r\][$_\alpha$]{}(55,10)\[l\][$_\sigma$]{}(55,30)\[l\][$_\beta$]{}(35,-25)\[\][$(c)$]{}
(250,120)(15,20)(0,20)[2]{}[2.5]{}(55,20)(70,20)[-2]{}[2.5]{} (35,20)(20,0,360)[3]{}(35,30)\[\][$G^-$]{}(35,-25)\[\][$(d)$]{}
(10,30)(15,20)(0,20)[2]{}[2.5]{}(55,20)(70,20)[-2]{}[2.5]{} (35,20)(20,0,180)[2]{}[9]{}(35,20)(20,180,360)[3]{}(35,30)\[\][$W^\mp$]{}(35,10)\[\][$G^\mp$]{}(15,30)\[r\][$_\alpha$]{}(55,30)\[l\][$_\beta$]{} (35,-25)\[\][$(e)$]{}
(90,30)(15,20)(0,20)[2]{}[2.5]{}(55,20)(70,20)[-2]{}[2.5]{} (35,20)(20,0,180)[1.5]{}(35,20)(20,180,360)[1.5]{}(35,30)\[\][$c^\pm$]{}(35,-25)\[\][$(f)$]{}
(170,30)(15,20)(0,20)[2]{}[2.5]{}(55,20)(70,20)[-2]{}[2.5]{} (35,20)(20,0,180)(35,20)(20,180,360)(35,30)\[\][$f$]{}(35,-25)\[\][$(g)$]{}
(250,30)(0,20)(15,20)[2]{}[3]{}(25,20)[10]{}(35,20)(50,20)[2]{}[3]{} (60,0)(50,20)(50,20)(60,40)
Fig. 2 Feynman diagrams contributing at one loop to the $\gamma Z$ mixing energy $i\Pi_{\mu\nu}(q)$.
There are two classes of Feynman graphs in SM that contribute at one loop to the amplitude $i{{\cal A}}_\mu(q)$, through the proper vertex $i\Gamma_\mu(q)$ in Fig. 1 and the photon-$Z$ boson mixing energy in Fig. 2, $i\Pi_{\mu\nu}(q)$, attached to the tree level neutrino-$Z$ vertex (see the last graph in Fig. 2). While the former contributes to both $F_1(0)$ and $F_2(0)$, the latter contributes only to $F_1(0)$ through $$\begin{aligned}
i\Pi_{\mu\nu}(0)\frac{i}{m_Z^2}\frac{ig_2}{2c_W}\gamma^\nu P_L.
\label{eq_mixing}$$ Here we use the standard notations of SM: $m_{W,Z}$ are the masses of the $W^\pm$ and $Z$ bosons, $g_2$ is the gauge coupling of $SU(2)_L$, $c_W=\cos\theta_W$ and $s_W=\sin\theta_W$ with $\theta_W$ being the weak mixing angle, and $P_{L,R}=(1\mp\gamma_5)/2$. We display here the contributions from individual graphs. Working in $d$-dimensions, we write $$\begin{aligned}
i\Gamma_\mu(q)&=&\frac{1}{2}eg_2^2|V_{\alpha i}|^2\sum_{x=a}^f\int_k(1x),\label{eq_Gamma}\\
i\Pi_{\mu\nu}(0)&=&\frac{eg_2}{c_W}\sum_{x=a}^f
\int_k(2x),~\int_k\equiv\int\frac{d^dk}{(2\pi)^d},
\label{eq_mixingenergy}$$ where, denoting $k_\pm=k\pm q/2$, from Fig. 1, $$\begin{aligned}
(1a)&=&+\gamma_\sigma P_L({/\!\!\!\!\!k}+{/\!\!\!\!\!p}+m_\alpha)\gamma_\rho P_L
\Gamma_{\alpha\beta\mu}(-k_-,k_+,-q)P^{\alpha\rho}(k_-)P^{\beta\sigma}(k_+)P^{-1},
\\
(1b)&=&+m_W^{-2}\frac{(m_iP_L-m_\alpha
P_R)({/\!\!\!\!\!k}+{/\!\!\!\!\!p}+m_\alpha)(m_iP_R-m_\alpha P_L)(k_-+k_+)_\mu}
{(k_+^2-\xi_W m_W^2)(k_-^2-\xi_W m_W^2)P},
\\
(1c)&=&+\frac{\gamma_\sigma P_L
({/\!\!\!\!\!k}+{/\!\!\!\!\!p}+m_\alpha)(m_iP_R-m_\alpha P_L)
P^{\mu\sigma}(k_+)}{[k_-^2-\xi_W m_W^2]P},
\\
(1d)&=&+\frac{(m_i P_L-m_\alpha P_R)({/\!\!\!\!\!k}+{/\!\!\!\!\!p}+m_\alpha)
\gamma_\rho P_LP^{\mu\rho}(k_-)}{[k_+^2-\xi_W m_W^2]P},
\\
(1e)&=&-\frac{\gamma_\sigma
P_L({/\!\!\!\!\!k}+{/\!\!\!\!\!p}_-+m_\alpha)\gamma_\mu
({/\!\!\!\!\!k}+{/\!\!\!\!\!p}_++m_\alpha)\gamma_\rho P_LP^{\rho\sigma}(k)}
{[(k+p_+)^2-m_\alpha^2][(k+p_-)^2-m_\alpha^2]},
\\
(1f)&=&+m_W^{-2}\frac{(m_iP_L-m_\alpha P_R)
({/\!\!\!\!\!k}+{/\!\!\!\!\!p}_-+m_\alpha)\gamma_\mu({/\!\!\!\!\!k}+{/\!\!\!\!\!p}_++m_\alpha)(m_iP_R-m_\alpha
P_L)} {[(k+p_+)^2-m_\alpha^2][(k+p_-)^2-m_\alpha^2]Q_2},\end{aligned}$$ and from Fig. 2, $$\begin{aligned}
(2a)&=&-c_W^2[2g_{\rho\sigma}g_{\mu\nu}-g_{\rho\mu}g_{\sigma\nu}
-g_{\rho\nu}g_{\sigma\mu}]P^{\rho\sigma},
\\
(2b)&=&-(c_W^2-s_W^2)g_{\mu\nu}Q_2^{-1},
\\
(2c)&=&+c_W^2\Gamma_{\rho\alpha\mu}(-k,k,0)
\Gamma_{\beta\sigma\nu}(-k,k,0)P^{\alpha\beta}P^{\rho\sigma},
\\
(2d)&=&+2(c_W^2-s_W^2)k_\mu k_\nu Q_2^{-2},
\\
(2e)&=&+2s_W^2m_W^2P_{\mu\nu}Q_2^{-1},
\\
(2f)&=&-2c_W^2k_\mu k_\nu Q_2^{-2}.\end{aligned}$$ Note that the fermion loop in Fig. 2(g) is transverse and drops out at $q=0$. We have defined the shortcuts for the propagators and triple-gauge vertex: $$\begin{aligned}
&&\Gamma_{\alpha\beta\mu}(p_1,p_2,p_3)=(p_2-p_3)_\alpha g_{\beta\mu}
+(p_3-p_1)_\beta g_{\mu\alpha}+(p_1-p_2)_\mu g_{\alpha\beta},
\label{eq_triple}
\\
&&P_{\mu\nu}(p)=g_{\mu\nu}[p^2-m_W^2]^{-1}-\delta_Wp_\mu
p_\nu[p^2-\xi_Wm_W^2]^{-1}[p^2-m_W^2]^{-1},
\label{eq_PropW}\\
&&P=(k+p)^2-m_\alpha^2,~Q_1=k^2-m_W^2,~Q_2=k^2-\xi_Wm_W^2,
\label{eq_prop}$$ with $P_{\alpha\beta}=P_{\alpha\beta}(k)$ and $\delta_W=1-\xi_W$. $m_\alpha$ ($m_i$) is the mass of the charged lepton $\ell_\alpha$ (neutrino $\nu_i$), and $V_{\alpha i}$ is the lepton mixing matrix appearing in the charged current interaction. A summation over all $\ell_\alpha$ is always implied. The identical initial and final neutrino satisfies EoMs (\[eq\_EoM\]) where now $m=m_i$. The above loop integrands will be manipulated in the next two sections.
Evaluation in $R_\xi$ gauge {#sec:Rgau}
===========================
Charge
------
Let us start with the charge. Setting $q=0$ simplifies significantly the expressions of $(1x)$. Using $({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma_\mu({/\!\!\!\!\!k}+{/\!\!\!\!\!p})=-(k+p)^2\gamma_\mu+2(k+p)_\mu({/\!\!\!\!\!k}+{/\!\!\!\!\!p})$, $\partial_\mu Q_2^{-1}=-2k_\mu Q_2^{-2}$, and $\partial_\mu
P^{-1}=-2(k+p)_\mu P^{-2}$, $(1b)$ and $(1f)$ sum to a total derivative: $$\begin{aligned}
[(1b)+(1f)]_0&=&-m_W^{-2}\partial_\mu\big\{\big[({/\!\!\!\!\!k}+{/\!\!\!\!\!p})(m_i^2P_R+m_\alpha^2 P_L)-m_im_\alpha^2\big](PQ_2)^{-1}\big\},\end{aligned}$$ where the subscript $0$ denotes evaluation at $q=0$. Considering the relation $$\begin{aligned}
\Gamma_{\alpha\beta\mu}(-k,k,0)P^{\alpha\rho}P^{\beta\sigma}
=(k^\rho P^\sigma_\mu+k^\sigma P^\rho_\mu)Q_2^{-1}+
\partial_\mu P^{\rho\sigma},\end{aligned}$$ we combine the pure $W^\pm$-loop graphs, $$\begin{aligned}
[(1a)+(1e)]_0&=&+\partial^\mu\big\{\gamma_\sigma({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma_\rho
P_LP^{-1}P^{\rho\sigma}\big\}\nonumber
\\
&&+P_R\gamma_\sigma({/\!\!\!\!\!k}+{/\!\!\!\!\!p}){/\!\!\!\!\!k}P^{\sigma\mu}(PQ_2)^{-1}+{/\!\!\!\!\!k}({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma_\rho P_LP^{\rho\mu}(PQ_2)^{-1}.
\label{eq_1aplus1e0}$$ The last two terms in the above are summed with the remaining two graphs to yield $$\begin{aligned}
[(1a)+(1e)+(1c)+(1d)]_0&=&+\partial^\mu\big\{\gamma_\sigma({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma_\rho
P_LP^{-1}P^{\rho\sigma}\big\}\nonumber
\\
&&+P_R\gamma_\sigma
\big[({/\!\!\!\!\!k}+{/\!\!\!\!\!p})({/\!\!\!\!\!k}+m_i)-m_\alpha^2)\big]P^{\mu\sigma}(PQ_2)^{-1}
\nonumber
\\
&&+\big[({/\!\!\!\!\!k}+m_i)({/\!\!\!\!\!k}+{/\!\!\!\!\!p})-m_\alpha^2\big]\gamma_\rho
P_LP^{\mu\rho}(PQ_2)^{-1}.\end{aligned}$$ Since the above expression is sandwiched between the spinors of the initial and final states, it is tempting to replace $m_i$ in the last two terms by ${/\!\!\!\!\!p}$. But this is not legitimate as emphasized in the last section. Instead, $m_i$ should be replaced by $({/\!\!\!\!\!p}\pm{/\!\!\!\!\!q}/2)$ on the rightmost (leftmost), in terms of the exact EoMs (\[eq\_EoM\]): $$\begin{aligned}
&&\bar u(p_-)[(1a)+(1e)+(1c)+(1d)]_0u(p_+)
\nonumber
\\
&=&\bar u(p_-)\big(
\partial^\mu\big\{\gamma_\sigma({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma_\rho
P_LP^{-1}P^{\rho\sigma}\big\}+2\gamma_\rho P_LP^{\mu\rho}Q_2^{-1}\big)u(p_+)\nonumber
\\
&&+\bar u(p_-)(1cd)_qu(p_+),\end{aligned}$$ where the last term linear in $q$ does not contribute to the charge but may contribute to the magnetic moment, $$\begin{aligned}
(1cd)_q&=&\frac{1}{2}P_R\gamma_\sigma({/\!\!\!\!\!k}+{/\!\!\!\!\!p}){/\!\!\!\!\!q}P^{\mu\sigma}(PQ_2)^{-1}-\frac{1}{2}{/\!\!\!\!\!q}({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma_\rho
P_LP^{\mu\rho}(PQ_2)^{-1}.
\label{eq_1cd}$$ In summary, leaving aside the $(1cd)_q$ term, we have $$\begin{aligned}
\sum_{x=a}^f(1x)_0&=&+\partial_\mu\big\{-m_W^{-2}\big[({/\!\!\!\!\!k}+{/\!\!\!\!\!p})(m_i^2P_R+m_\alpha^2P_L)-m_im_\alpha^2\big](PQ_2)^{-1}\nonumber
\\
&&+\gamma_\sigma({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma_\rho P_LP^{\rho\sigma}P^{-1}\big\}+2\gamma_\rho P_LP_\mu^\rho Q_2^{-1}.\end{aligned}$$ The total derivative can be dropped in regularized loop integrals, so that Fig. 1 contributes to the $F_1(0)$ term in Eq. (\[eq\_A\]) the following: $$\begin{aligned}
+eg_2^2\sum_\alpha|V_{\alpha i}|^2
\int_k\gamma_\rho P_LP_\mu^\rho Q_2^{-1}=+eg_2^2\int_k\gamma_\rho P_LP_\mu^\rho Q_2^{-1},
\label{eq_charge1}$$ where unitarity of $V$ is used to finish the sum as the integrand is independent of $m_\alpha$.
Now we manipulate $i\Pi_{\mu\nu}(0)$. First of all, $(2b)$ and $(2d)$ form a total derivative: $$\begin{aligned}
(2b)+(2d)&=&-(c_W^2-s_W^2)\partial_\mu\big(k_\nu Q_2^{-1}\big).\end{aligned}$$ Using the shortcuts in Eqs. (\[eq\_triple\],\[eq\_PropW\]), we have $$\begin{aligned}
(2a)&=&2c_W^2\big[\delta_\xi(k^2g_{\mu\nu}-k_\mu k_\nu)(Q_1Q_2)^{-1}
-g_{\mu\nu}(d-1)Q_1^{-1}\big],
\label{eq_2a}\\
(2c)&=&2c_W^2\big[\xi_W(k^2g_{\mu\nu}-k_\mu k_\nu)(Q_1Q_2)^{-1}
+2(d-1)k_\mu k_\nu(Q_1)^{-2}\big].
\label{eq_2c}$$ The last terms in $(2a)$ and $(2c)$ already form a total derivative. In the first terms, we decompose $k^2(Q_1Q_2)^{-1}=Q_2^{-1}+m_W^2(Q_1Q_2)^{-1}$, and then sum judiciously with $(2f)$ to arrive at the result $$\begin{aligned}
(2a)+(2c)+(2f)&=&2c_W^2\Big\{-(d-1)\partial_\mu(k_\nu Q_1^{-1})
+g_{\mu\nu}\big[Q_2^{-1}+m_W^2(Q_1Q_2)^{-1}\big]
\nonumber\\
&&+k_\mu k_\nu\big[Q_2^{-2}-(Q_1Q_2)^{-1}\big]-2k_\mu k_\nu
Q_2^{-2}\Big\}
\nonumber\\
&=&2c_W^2\Big\{\partial_\mu\big[k_\nu Q_2^{-1}-(d-1)(k_\nu
Q_1^{-1})\big] +m_W^2Q_2^{-1}P_{\mu\nu}\Big\}.\end{aligned}$$ Thus, using $c_W^2+s_W^2=1$, the sum of all graphs is $$\begin{aligned}
\sum_{x=a}^f(2x)&=&\partial_\mu\big\{k_\nu
Q_2^{-1}-2c_W^2(d-1)(k_\nu Q_1^{-1})\big\}
+2m_W^2Q_2^{-1}P_{\mu\nu}.
\label{eq_sum2x}$$ Dropping the regularized total derivative and using Eqs. (\[eq\_mixing\], \[eq\_mixingenergy\]), its contribution to the $F_1(0)$ term in Eq. (\[eq\_A\]) is as follows, $$\begin{aligned}
-eg_2^2\int_kQ_2^{-1}P_{\mu\nu}\gamma^\nu P_L, \label{eq_charge2}\end{aligned}$$ which cancels Eq. (\[eq\_charge1\]). The vanishing charge is thus established at one loop in $R_\xi$ gauge.
Magnetic moment {#subsec:mag}
---------------
Moving to the magnetic moment, we follow the computational procedure proposed in sec \[sec:pre\]. Now only the graphs in Fig. 1 contribute. Since $(1b)$ is quadratic in $q$ when expanding in $q$, it drops out. The next simplest is $(1f)$. Taking a derivative with respect to $q^\nu$, setting $q=0$ and making it manifestly antisymmetric in $\mu$ and $\nu$ (denoted by the pair of square brackets below), we have $$\begin{aligned}
\big[\partial_\nu^q(1f)_0\big]&=&+\frac{1}{4m_W^2}\frac{1}{P^2Q_2}\Big(
(K_{\mu\nu}^0+K_{\mu\nu}^1)(m_i^2P_R+m_\alpha^2P_L)-2m_im_\alpha^2[\gamma_\mu,\gamma_\nu]\Big),\end{aligned}$$ where $$\begin{aligned}
K_{\mu\nu}^0&=&{/\!\!\!\!\!p}[\gamma_\mu,\gamma_\nu]+[\gamma_\mu,\gamma_\nu]{/\!\!\!\!\!p},
\\
K_{\mu\nu}^1&=&{/\!\!\!\!\!k}[\gamma_\mu,\gamma_\nu]+[\gamma_\mu,\gamma_\nu]{/\!\!\!\!\!k}.\end{aligned}$$ Anticipating that $\big[\partial_\nu^q(1f)_0\big]$ is to be sandwiched between the initial and final spinors and noting that the ${/\!\!\!\!\!k}$ in $K_{\mu\nu}^1$ will yield a ${/\!\!\!\!\!p}$ upon loop integration, we can apply the limiting EoMs ${/\!\!\!\!\!p}u=m_iu$ after the moment has been isolated. The above is thus reduced to $$\begin{aligned}
\big[\partial_\nu^q(1f)_0\big]&\leftrightharpoons&+\frac{1}{m_W^2}\frac{1}{8P^2Q_2}
\Big(K^1_{\mu\nu}(m_i^2+m_\alpha^2)+2m_i[\gamma_\mu,\gamma_\nu](m_i^2-m_\alpha^2)\Big),
\label{eq_dipole1f}$$ where, from now on, $\leftrightharpoons$ means equality when sandwiched between the spinors or under the loop integration or both. All factors of $P_{L,R}$ are removed in a similar fashion, confirming that the electric dipole moment does not arise at the one loop.
Figs. $(1c)$ and $(1d)$ should be treated together for symmetry reasons. There are two sources of terms, one from those explicitly linear in $q$ and the other from the remaining terms (\[eq\_1cd\]) when computing the charge. Putting them together and taking the derivative, we have $$\begin{aligned}
\partial_\nu^q(1c+1d)_0&=&\big[m_i({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma^\sigma P_L
-P_R\gamma^\sigma m_i({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\big]
\big[P_{\nu;\mu\sigma}Q_2^{-1}-k_\nu
Q_2^{-2}P_{\mu\sigma}\big]P^{-1}
\nonumber\\
&&+\frac{1}{2}\big[\gamma^\sigma({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma^\rho
-\gamma^\rho({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma^\sigma\big]P_L
g_{\rho\nu}P_{\mu\sigma}(PQ_2)^{-1},\end{aligned}$$ where $$\begin{aligned}
P_{\nu;\alpha\beta}&=&-\frac{1}{2}\partial_\nu P_{\alpha\beta}.\end{aligned}$$ Antisymmetrization and applying EoMs yield, after some algebra, $$\begin{aligned}
\big[\partial_\nu^q(1c+1d)_0\big]&\leftrightharpoons&m_iK_{\mu\nu}^2
\bigg(\frac{\delta_\xi}{4PQ_1Q_2^2}
-\frac{1}{4PQ_1^2Q_2}+\frac{1}{4PQ_1Q_2^2}\bigg)
\nonumber\\
&&
-\big(K_{\mu\nu}^1+2m_i[\gamma_\mu,\gamma_\nu]\big)\frac{1}{8PQ_1Q_2},
\label{eq_dipole1cd}$$ where $$\begin{aligned}
K^2_{\mu\nu}&=&k_\mu[{/\!\!\!\!\!k},\gamma_\nu]+k_\nu[\gamma_\mu,{/\!\!\!\!\!k}].\end{aligned}$$ In deriving the above result, we used identities such as $$\begin{aligned}
\gamma_\mu{/\!\!\!\!\!k}\gamma_\nu-\gamma_\nu{/\!\!\!\!\!k}\gamma_\mu
&=&-\frac{1}{2}({/\!\!\!\!\!k}[\gamma_\mu,\gamma_\nu]+[\gamma_\mu,\gamma_\nu]{/\!\!\!\!\!k}),
\label{eq_identity1}\\
{/\!\!\!\!\!p}{/\!\!\!\!\!k}\gamma_\nu-\gamma_\nu{/\!\!\!\!\!k}{/\!\!\!\!\!p}&=&+\frac{1}{2}({/\!\!\!\!\!p}[{/\!\!\!\!\!k},\gamma_\nu]+[{/\!\!\!\!\!k},\gamma_\nu]{/\!\!\!\!\!p}).
\label{eq_identity2}$$
Now we manipulate $(1e)$. Taking the derivative, plugging in the propagator $P^{\rho\sigma}$ and doing antisymmetrization, one obtains $$\begin{aligned}
\big[\partial_\nu^q(1e)_0\big]&=&-\big(K_{\mu\nu}^0+K_{\mu\nu}^1\big)P_L\frac{1}{2P^2Q_1}+E_{\mu\nu}P_L\frac{\delta_\xi}{4P^2Q_1Q_2},\end{aligned}$$ where, using $p^2=m_i^2$ and the identity $$\begin{aligned}
{/\!\!\!\!\!k}[\gamma_\mu,\gamma_\nu]{/\!\!\!\!\!k}&=&k^2[\gamma_\mu,\gamma_\nu]-2([\gamma_\mu,{/\!\!\!\!\!k}]k_\nu+[{/\!\!\!\!\!k},\gamma_\nu]k_\mu),
\label{eq_identity3}$$ the second term is recast as follows: $$\begin{aligned}
E_{\mu\nu}&=&{/\!\!\!\!\!k}\big(({/\!\!\!\!\!k}+{/\!\!\!\!\!p})[\gamma_\mu,\gamma_\nu]
+[\gamma_\mu,\gamma_\nu]({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\big){/\!\!\!\!\!k}\nonumber\\
&=&[(k+p)^2-m_i^2]K_{\mu\nu}^1-k^2K_{\mu\nu}^0
\nonumber\\
&&+2\big\{k_\mu({/\!\!\!\!\!p}[{/\!\!\!\!\!k},\gamma_\nu]+[{/\!\!\!\!\!k},\gamma_\nu]{/\!\!\!\!\!p})
+k_\nu({/\!\!\!\!\!p}[\gamma_\mu,{/\!\!\!\!\!k}]+[\gamma_\mu,{/\!\!\!\!\!k}]{/\!\!\!\!\!p})\big\}.\end{aligned}$$ Application of the limiting EoMs gives finally $$\begin{aligned}
\big[\partial_\nu^q(1e)_0\big]&\leftrightharpoons&-\big(K^1_{\mu\nu}
+2m_i[\gamma_\mu,\gamma_\nu]\big)\frac{1}{4P^2Q_1}\nonumber\\
&&+\big(PK^1_{\mu\nu}+(m_\alpha^2-m_i^2)K^1_{\mu\nu}
-2m_ik^2[\gamma_\mu,\gamma_\nu]+4m_iK^2_{\mu\nu}\big)\frac{\delta_\xi}{8P^2Q_1Q_2}.
\label{eq_dipole1e}$$
The graph Fig. 1(a) involves the triple gauge coupling and double gauge boson propagators, making it the most complicated to evaluate. We outline how this is accomplished. Taking the derivative and doing antisymmetrization we have $$\begin{aligned}
[\partial_\nu^q(1a)_0]&=&+P^{-1}\gamma^\sigma({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma^\rho
P_LA_{\rho\sigma;\mu\nu},\end{aligned}$$ where $$\begin{aligned}
A_{\rho\sigma;\mu\nu}&=&-\frac{1}{2}\delta_\xi
k^2\bigg(\frac{G_{\mu\nu,\rho\sigma}^0}{Q_1^2Q_2}
+\frac{G_{\mu\nu,\rho\sigma}^2}{Q_1^2Q_2^2}\bigg)
-\delta_\xi\frac{G_{\mu\nu,\rho\sigma}^2}{Q_1^2Q_2}
-\frac{3}{2}\frac{G_{\mu\nu,\rho\sigma}^0}{Q_1^2},\end{aligned}$$ and $$\begin{aligned}
G_{\mu\nu,\rho\sigma}^0&=&g_{\nu\sigma}g_{\mu\rho}-g_{\nu\rho}g_{\mu\sigma},
\\
G_{\mu\nu,\rho\sigma}^2&=&k_\mu(g_{\nu\rho}k_\sigma-g_{\nu\sigma}k_\rho)
-k_\nu(g_{\mu\rho}k_\sigma-g_{\mu\sigma}k_\rho).\end{aligned}$$ The contraction with $G^0$ is standardized using Eq. (\[eq\_identity1\]) into $(K^0_{\mu\nu}+K_{\mu\nu}^1)P_L$, while the contraction with $G^2$ yields, by making use of Eq. (\[eq\_identity2\]), $$\begin{aligned}
\big(k_\mu({/\!\!\!\!\!p}[{/\!\!\!\!\!k},\gamma_\nu]+[{/\!\!\!\!\!k},\gamma_\nu]{/\!\!\!\!\!p})
+k_\nu({/\!\!\!\!\!p}[\gamma_\mu,{/\!\!\!\!\!k}]+[\gamma_\mu,{/\!\!\!\!\!k}]{/\!\!\!\!\!p})\big)P_L,\end{aligned}$$ which reduces to $m_iK_{\mu\nu}^2$ using EoMs. The final form is $$\begin{aligned}
[\partial_\nu^q(1a)_0]&\leftrightharpoons&-\big(K^1_{\mu\nu}
+2m_i[\gamma_\mu,\gamma_\nu]\big) \bigg(\frac{\delta_\xi
k^2}{8PQ_1^2Q_2}+\frac{3}{8PQ_1^2}\bigg)
\nonumber\\
&& +2m_iK_{\mu\nu}^2\bigg(\frac{\delta_\xi k^2}{8PQ_1^2Q_2^2}
+\frac{\delta_\xi}{4PQ_1^2Q_2}\bigg).
\label{eq_dipole1a}$$
To summarize our calculation thus far, the terms relevant to the neutrino magnetic moment are given in Eqs. (\[eq\_dipole1a\],\[eq\_dipole1cd\],\[eq\_dipole1e\],\[eq\_dipole1f\]). The next task is to demonstrate the $\xi_W$ cancellation among those terms. We first decompose $\delta_\xi k^2=Q_2-\xi_WQ_1$ to remove $k^2$ from numerators in Eqs. (\[eq\_dipole1a\],\[eq\_dipole1e\]). The $K_{\mu\nu}^2$ terms sum to $$\begin{aligned}
\frac{1}{2}m_i\delta_\xi K_{\mu\nu}^2\bigg(\frac{1}{PQ_1^2Q_2}
+\frac{1}{PQ_1Q_2^2}+\frac{1}{P^2Q_1Q_2}\bigg).\end{aligned}$$ For any of the three terms in the above, the $k_\mu$ and $k_\nu$ factors in $K_{\mu\nu}^2$ may be simultaneously replaced by $(k+p)_\mu$ and $(k+p)_\nu$, because the resulted additional terms, upon the loop integration, will be proportional to $$\begin{aligned}
p_\mu[{/\!\!\!\!\!p},\gamma_\nu]+p_\nu[\gamma_\mu,{/\!\!\!\!\!p}],\end{aligned}$$ which vanishes when sandwiched between the initial and final spinors. We make this replacement for the last term in the sum. Using again $\partial_\mu Q_j^{-1}=-2k_\mu Q_j^{-2}$ and $\partial_\mu P^{-1}=-2(k+p)_\mu P^{-2}$, the sum becomes $$\begin{aligned}
&&-\frac{1}{4}m_i\delta_\xi\big([{/\!\!\!\!\!k},\gamma_\nu]\partial_\mu
+[\gamma_\mu,{/\!\!\!\!\!k}]\partial_\nu\big)(PQ_1Q_2)^{-1}
\nonumber\\
&=&-\frac{1}{4}m_i\delta_\xi\Big[\partial_\mu\big([{/\!\!\!\!\!k},\gamma_\nu](PQ_1Q_2)^{-1}\big)
-(\mu\leftrightarrow\nu)\Big]
+\frac{1}{2}m_i\delta_\xi[\gamma_\mu,\gamma_\nu](PQ_1Q_2)^{-1}.
\label{eq_K2}\end{aligned}$$ The apparently $\xi_W$-dependent terms in the sum $\displaystyle\sum_{x=a}^f[\partial_\nu^q(1x)_0]$, including the one in Eq. (\[eq\_K2\]) but dropping total derivatives, are collected below: $$\begin{aligned}
\sum_{x=a}^f[\partial_\nu^q(1x)_0]_\xi&\leftrightharpoons&
m_i[\gamma_\mu,\gamma_\nu]\frac{\delta_\xi}{4PQ_1Q_2}
\nonumber\\
&&+\frac{m_i}{m_W^2}\frac{1}{4P^2Q_2}
\big(K^1_{\mu\nu}m_i+[\gamma_\mu,\gamma_\nu](m_i^2-m_\alpha^2+\xi_Wm_W^2)\big).
\label{eq_dipolexi}$$ The integral of the above second term is simplified using Eq. (\[eq\_integral1\]) and EoMs, while the first one is split by $\delta_\xi(Q_1Q_2)^{-1}=m_W^{-2}(Q_1^{-1}-Q_2^{-1})$, so that the $\xi_W$ dependence disappears completely from the sum: $$\begin{aligned}
\sum_{x=a}^f[\partial_\nu^q(1x)_0]_\xi&\leftrightharpoons&
m_i[\gamma_\mu,\gamma_\nu]\frac{1}{4m_W^2}\bigg(\frac{1}{PQ_1}-\frac{1}{P^2}\bigg).
\label{eq_dipolexi0}\end{aligned}$$ Adding the above with the terms that are explicitly $\xi_W$-independent, we obtain the final sum of terms contributing to the neutrino magnetic moment: $$\begin{aligned}
\sum_{x=a}^f[\partial_\nu^q(1x)_0]&\leftrightharpoons&
m_i[\gamma_\mu,\gamma_\nu]\frac{1}{4m_W^2}\bigg(\frac{1}{PQ_1}-\frac{1}{P^2}\bigg)
-\big(K^1_{\mu\nu}+2m_i[\gamma_\mu,\gamma_\nu]\big)\frac{1}{2PQ_1^2}
\nonumber\\
&&+\big([m_W^{-2}(m_\alpha^2-m_i^2)-2]K^1_{\mu\nu}
-6m_i[\gamma_\mu,\gamma_\nu]\big)\frac{1}{8P^2Q_1}.
\label{eq_sum}$$
From Eqs. (\[eq\_A\],\[eq\_Gamma\],\[eq\_sum\]) and the loop integrals defined in the appendix, we obtain for the neutrino $\nu_i$ the magnetic form factor at the vanishing momentum transfer, $$\begin{aligned}
F_2(0)&=&-\frac{g_2^2}{(4\pi)^2}\frac{2m_i^2}{m_W^2}
\sum_\alpha|V_{\alpha i}|^2\bigg[\frac{1}{4}I_1+J_2 -\frac{1}{2}K_2
+\frac{3}{4}J_1-\frac{1}{8}(2-x_\alpha+y_i)K_1\bigg],
\label{eq_F20}$$ where $I_1,~J_{1,2},~K_{1,2}$ are functions of the mass ratios $x_\alpha=m_\alpha^2/m_W^2$ and $y_i=m_i^2/m_W^2$. This result is indeed manifestly gauge independent in the class of $R_\xi$ gauges.
Evaluation in unitary gauge {#sec:Ugau}
===========================
Working in unitary gauge means that the limit $\xi_W\to\infty$ is taken before the loop integrals are evaluated. Since $\xi_W$ appears exclusively in the propagators of the $W^\pm$ gauge bosons, would-be Goldstone bosons $G^\pm$ and the ghosts $c^\pm$, only the gauge boson propagator survives the limit, $$\begin{aligned}
P_{\mu\nu}(k)\to\bar P_{\mu\nu}(k)=(g_{\mu\nu}-m_W^{-2}k_\mu k_\nu)
Q_1^{-1},\end{aligned}$$ and thus only the pure-$W^\pm$ graphs $(a,~e)$ in Fig. 1 and $(a,~c)$ in Fig. 2 remain. We have presented our calculation in $R_\xi$ gauge in a way that can be easily adapted for unitary gauge.
For the charge contribution from Fig. 1 we take the limit $\xi_W\to\infty$ in the integrand (\[eq\_1aplus1e0\]) where only the total derivative term survives: $$\begin{aligned}
[(1a)+(1e)]_0&\to&\partial_\mu\big\{\gamma_\sigma({/\!\!\!\!\!k}+{/\!\!\!\!\!p})\gamma_\rho
P_LP^{-1}\bar P^{\rho\sigma}\big\},\end{aligned}$$ whose integral vanishes in dimensional regularization. The relevant terms from the photon-$Z$ mixing energy are obtained from Eqs. (\[eq\_2a\],\[eq\_2c\]), or more readily from Eq.(\[eq\_sum2x\]), $$\begin{aligned}
[(2a)+(2c)]&\to&\partial_\mu\big\{-2c_W^2(d-1)(k_\nu
Q_1^{-1})\big\},\end{aligned}$$ whose integral again vanishes. Thus the vanishing of charge at one loop occurs in unitary gauge in a stronger manner: each of the contributions from the proper vertices and the mixing energy vanishes separately.
The magnetic form factor $F_2(0)$ can also be obtained from intermediate steps in subsec \[subsec:mag\]. We can sum Eqs. (\[eq\_dipole1a\],\[eq\_dipole1e\]) and take the limit $\xi_W\to\infty$, or cope directly with the total of all graphs since we know only Figs. 1$(a,e)$ survive the limit. The latter point can also be seen from explicit results in Eqs.(\[eq\_dipole1cd\],\[eq\_dipole1f\]). Dropping the total derivatives and sending $\xi_W\to\infty$, the potentially $\xi_W$-dependent part of the total in $R_\xi$ gauge, Eq. (\[eq\_dipolexi\]), goes exactly to Eq. (\[eq\_dipolexi0\]) without additional manipulations. The result in Eq. (\[eq\_F20\]) is thus recovered in unitary gauge.
Summary {#sec:summary}
=======
The electromagnetic properties of neutrinos are an interesting topic that is potentially relevant to various astrophysical phenomena and laboratory measurements. Although we know from principles that the charge and dipole moments of a Dirac neutrino are physical quantities and cannot depend on computational methods or the choice of gauge in a consistent theory, this has never been explicitly examined before in a satisfactory manner even at one loop. We have studied this issue in the minimally extended standard model that incorporates neutrinos masses and mixing. We demonstrated at one loop in both $R_\xi$ and unitary gauges that the magnetic moment and vanishing charge are indeed gauge-independent quantities. This statement is exact in the sense that it is true for any values of various masses and the lepton mixing matrix as long as the latter is unitary. We have accomplished this by manipulating directly the integrands of loop integrals and employing simple algebraic identities like (\[eq\_identity1\],\[eq\_identity2\],\[eq\_identity3\]) and integral relations like (\[eq\_integral1\]). We believe this approach is advantageous over the one that handles the results of loop integration, and may be useful in other contexts. Finally, we mention that various approximations to our exact one-loop result for the magnetic moment in Eq. (\[eq\_F20\]) are possible. For instance, when all neutrinos and charged leptons are much lighter than the weak gauge bosons as is the case in SM, we have from the explicit results in the appendix that $F_2(0)\approx
-(3G_Fm_i^2)/(4\pi^2\sqrt{2})$, where the mixing matrix drops out from the leading term, so that the interaction potential of the neutrino $\nu_i$ of mass $m_i$ and spin $\vec S$ with an external magnetic moment is, $V\approx(3eG_Fm_i)/(4\pi^2\sqrt{2})\vec
S\cdot\vec B$, recovering the well-known result in the literature.
[**Acknowledgement**]{}
WTH would like to thank the members of theory group at Nankai University for hospitality during a long-term visit when this work was conducted. This work was supported in part by the grant NSFC-11025525 and by the Fundamental Research Funds for the Central Universities No.65030021.
[**Appendix: some useful integrals**]{} We list some loop integrals relevant to our evaluation of the magnetic moment. The following relation is used in sec \[sec:Rgau\] for reduction of terms: $$\begin{aligned}
2p^2\int_k\frac{k_\alpha}{D_1^{n_1}D_2^{n_2}}
&=&p_\alpha\int_k\bigg(\frac{1}{D_1^{n_1-1}D_2^{n_2}}
-\frac{1}{D_1^{n_1}D_2^{n_2-1}}+\frac{m_1^2-m_2^2-p^2}{D_1^{n_1}D_2^{n_2}}\bigg),
\label{eq_integral1}$$ where $D_1=(k+p)^2-m_1^2,~D_2=k^2-m_2^2$. Using notations in Eq. (\[eq\_prop\]) with $p^2=m_i^2$, the basic integral is $$\begin{aligned}
&&\int_k\bigg(\frac{1}{Q_1^2}-\frac{1}{PQ_1}\bigg)=\frac{i}{(4\pi)^2}I(x_\alpha,y_i),\end{aligned}$$ where $x_\alpha=m_\alpha^2/m_W^2$ and $y_i=m_i^2/m_W^2$. For simplicity, we also define the integrals $$\begin{aligned}
&&\int_k\bigg(\frac{1}{PQ_1}-\frac{1}{P^2}\bigg)=\frac{i}{(4\pi)^2}I_1(x_\alpha,y_i),
\\
&&\int_k\frac{1}{P^2Q_1}=-\frac{i}{(4\pi)^2}\frac{1}{m_W^2}J_1(x_\alpha,y_i),
\\
&&\int_k\frac{1}{PQ_1^2}=-\frac{i}{(4\pi)^2}\frac{1}{m_W^2}J_2(x_\alpha,y_i),
\\
&&\int_k\frac{2k_\mu}{P^2Q_1}=\frac{i}{(4\pi)^2}\frac{p_\mu}{m_W^2}K_1(x_\alpha,y_i),
\\
&&\int_k\frac{2k_\mu}{PQ_1^2}=\frac{i}{(4\pi)^2}\frac{p_\mu}{m_W^2}K_2(x_\alpha,y_i).\end{aligned}$$ The parametric integral for $I(s,t)$ is $$\begin{aligned}
I(x,y)=\int_0^1dt~\ln\big[xt+(1-t)-yt(1-t)-i0^+\big].\end{aligned}$$ The other functions are related to it by $$\begin{aligned}
I_1(x,y)&=&\ln x-I(x,y),
\\
J_1(x,y)&=&\frac{\partial}{\partial x}I(x,y),
\\
J_2(x,y)&=&J_1(1/x,y/x),
\\
K_1(x,y)&=&y^{-1}[I_1(x,y)+(1+y-x)J_1(x,y)],
\\
K_2(x,y)&=&y^{-1}[I(x,y)+(1+y-x)J_2(x,y)].\end{aligned}$$ Note that the singularity at $y=0$ is spurious since the original integrals are smooth there.
The analytic result for $I$ is known for all parameter regions, but we only record it for the case relevant to SM, i.e., for $0\le
y<x\ll 1$, $$\begin{aligned}
I(x,y)&=&-2-\frac{1}{2y}(1-x-y)\ln x +\frac{\lambda}{2y}\ln R,\end{aligned}$$ where $$\begin{aligned}
\lambda=(1+x^2+y^2-2x-2y-2xy)^{1/2},~R=\frac{1+x-y-\lambda}{1+x-y+\lambda}.\end{aligned}$$ The other two functions are $$\begin{aligned}
J_1(x,y)&=&-\frac{1-x+y}{2y\lambda}\ln R +\frac{1}{2y}\ln x,
\\
J_2(x,y)&=&-J_1(x,y)-\frac{1}{\lambda}\ln R.\end{aligned}$$
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[^1]: liaoy@nankai.edu.cn
|
---
abstract: 'We identify the precise hallmarks of the local magnetic moment formation and its Kondo screening in the frequency structure of the generalized charge susceptibility. The sharpness of our identification even pinpoints a novel criterion to determine the Kondo temperature of strongly correlated systems on the two-particle level, which only requires calculations at the [*lowest*]{} Matsubara frequency. We showcase its strength by applying it to the single impurity and the periodic Anderson model as well as to the Hubbard model. Our results represent a significant progress for the general understanding of quantum field theory at the two-particle level and allow for tracing the limits of the physics captured by perturbative approaches in correlated regimes.'
author:
- 'P. Chalupa$^a$'
- 'T. Schäfer$^{b,c}$'
- 'M. Reitner$^{a}$'
- 'S. Andergassen$^d$'
- 'A. Toschi$^a$'
bibliography:
- 'main.bib'
date:
title: |
Fingerprints of the local moment formation and its Kondo screening\
in the generalized susceptibilities of many-electron problems
---
[*Introduction.*]{} The goal of any successful theory is to extract essential features of the phenomena of interest from the complexity of the physical world, neglecting all superfluous pieces of information. This objective is particularly crucial for the cutting-edge quantum field theory (QFT) approaches designed to describe complex many-electron systems in the presence of strong correlations.
Presently, one can rely on a solid textbook interpretation[@Abrikosov1975; @Mahan2000] of the QFT formalism describing the single-particle (1P) processes, measurable e.g. by the (angular resolved) direct and inverse photoemission[@PhotoRMP2003] or the scanning tunneling microscopy[@STMRMP1987; @STMRMP2007]. Crucial information about the metallic or insulating nature of a given many-electron problem, as well as quantitative information about the electronic mass renormalization $Z$ and quasiparticle lifetime $\tau$ is unambiguously encoded in the momentum/energy dependence of the electronic self-energy $\Sigma$. If the temperature $T$ is low enough, even a quick glance at the low-energy behavior of $\Sigma$, either in real or in Matsubara frequencies, can yield a qualitatively reliable estimate of the most important physical properties.
The situation is clearly very different on the two-particle (2P) level. Due to the higher complexity of the physical mechanisms at play, the related textbook knowledge is mostly limited to general definitions[@Abrikosov1975; @Mahan2000]. For this reason corresponding analytical/numerical calculations are often performed with significant approximations or with a black-box treatment of the 2P processes. However, the last decade has seen a rapid development of methods at the forefront of the many-electron theory[@Maier2005; @Metzner2012; @Rohringer2018], for which generalized 2P correlations functions are the key ingredient. This is reflected in a increasing effort to develop the corresponding formal aspects and algorithmic procedures[@Kunes2011; @Rohringer2012; @Metzner2012; @Hafermann2014; @Gunnarsson2015; @Wentzell2016; @Kaufmann2017; @Tagliavini2018; @vanLoon2018; @Rohringer2018; @Maier2005; @Kugler2018; @Krien2019SBE; @vanLoon2020; @Reitner2020]. At the same time, our rather poor physical understanding of the 2P processes remains largely behind the requirements of the most advanced QFT methods. Interesting progress has been recently reported[@Krien2019; @Kotliar2020] on the relation of the 1P Fermi-liquid parameters to the scattering functions. Ideally, however, one would like to be able to interpret the physics encoded at the 2P level with a similar degree of confidence as for the 1P processes. In our paper, we make a significant step forward in this direction: We identify the fingerprints of two major hallmarks of strong correlations in the generalized charge susceptibility. In particular, we pinpoint the frequency structures encoding the formation of local magnetic moments as well as of their Kondo screening. In this perspective, we also show how the Kondo temperature $T_K$ corresponds to a specific property of the generalized charge susceptibility, allowing for an alternative, simple path of extracting its value directly from the lowest Matsubara frequency data.
We recall that the Kondo problem[@Hewson1993] provides a paradigm for a variety of physical effects involving strong electronic correlations, ranging from mesoscopic electron transport to heavy-fermion materials and high-temperature superconductors. In this work we focus on the formation of a local magnetic moment and its screening due to the interplay with a metallic surrounding, which can be described by effective Kondo problems, as it happens, e.g., in dynamical mean-field theory (DMFT)[@Georges1996], with the auxiliary Anderson impurity model (AIM). Learning how to extract physical information from the generalized susceptibility represents a substantial improvement for the understanding of quantum many electron physics at the 2P level. Further, having this information at hand also enables us to draw conclusions on two important theoretical questions: (i) The relation of the recently reported multifaceted manifestations[@Gunnarsson2017] of the breakdown of perturbation theory, such as the divergences of the irreducible vertex functions[@Schaefer2013; @Janis2014; @Schaefer2016c; @Gunnarsson2016; @Ribic2016; @Chalupa2018; @Vucicevic2018; @Thunstroem2018; @Springer2019] and the crossing of multiple solutions[@Kozik2015; @Stan2015; @Schaefer2016c; @Gunnarsson2017; @Tarantino2018; @Thunstroem2018; @Vucicevic2018] of the Luttinger-Ward functional, with the local moment physics and its Kondo screening; (ii) the built-in limit of advanced perturbative approaches to describe this kind of physics.
[*How to read two-particle quantities.*]{} We start from the definition of the generalized local susceptibility[@Rohringer2012] $$\chi_{\sigma\sigma'}^{\nu \nu'}(\Omega) \! = \! G^{(2)}_{\sigma\sigma'}(\nu, \nu',\Omega) \! - \! T^{-1}G(\nu)G(\nu')\delta_{\Omega0}\delta_{\sigma\sigma'}
\label{eq:chigen}$$ in terms of the 2P ($G^{(2)}$) and 1P ($G$) Green’s functions, where $\nu,\nu'$ and $\Omega$ are fermionic and bosonic Matsubara frequencies, and $\sigma,\sigma'\!=\!\{\uparrow,\downarrow\}$ spin indices. As we show in the following, for repulsive interactions, the generalized [*charge*]{} susceptibility $\chi^{\nu \nu'}(\Omega)\!=\!\chi_{\uparrow \uparrow}^{\nu \nu'}(\Omega) + \chi_{\uparrow\downarrow}^{\nu \nu'}(\Omega)$ allows for the best readability of the underlying physics at the 2P level. We recall that from the generalized susceptibility the physical response function can be readily obtained by summing over the fermionic Matsubara frequencies: the static charge response $\chi(\Omega\! = \! 0)$ is obtained in the following way $$\chi = T^2 \sum\limits_{\nu\nu'}\chi^{\nu\nu'} = T^2 \sum\limits_{\nu\nu'}{(\chi_{\uparrow\uparrow}^{\nu\nu'}+\chi_{\uparrow\downarrow}^{\nu\nu'})}.
\label{eq:chiphys}$$
We first analyze the arguably simple case of an isolated atom with a repulsive interaction $U$ (Hubbard atom, HA), where analytic expressions are available[@Rohringer2012; @Thunstroem2018]. In Fig. \[fig:1\] (upper panels), we show an intensity plot of $\chi^{\nu\nu'}$ (normalized by $T^2$) for $U\!=\!5.75$, half filling (where $\chi^{\nu \nu'}$ is real) and different temperatures. At high temperature ($T_{\text{high}}\!=\!2$, left panel), the overall frequency structure is dominated by a positive-valued diagonal (yellow/red). This corresponds to a typical [*perturbative*]{} behavior, dominated by the diagonal bubble term $\chi_0^{\nu \nu'} \!=\! - \delta_{\nu \nu'} G(\nu)^2/T $: Correlation effects are washed out for $T \! \sim \! U$, consistently with the feasibility of high-$T$ expansions. The situation changes radically when reducing $T$: in the intermediate ($T_{\rm int}\!=\!0.1$) and low ($T_{\rm low}\!=\!1/60\!\approx\!0.017$) temperature regime (central and right panels), one observes a strong [*damping*]{} of all diagonal elements of $\chi^{\nu\nu'}$. The effect is more pronounced at low frequencies, as $\chi^{\nu \nu}$ becomes even [*negative*]{} (bluish colors) for $|\nu|\!\lesssim\!U$[@Thunstroem2018] (black square). This major feature is accompanied by the appearance of [*positive*]{} off-diagonal elements (yellow), smaller in size w.r.t. the diagonal ones. The net effect is a suppression of the physical susceptibility, see Eq. (\[eq:chiphys\]), which occurs when the thermal energy is no longer large enough ($T\! \sim\! \nu < U$) to counter the formation of a local moment driven by $U$, eventually yielding and exponentially small $\chi \sim \! e^{-U/2T}$ for $T\rightarrow0$. In fact, the low-$T$ data of the HA (right panel) provides a perfect pedagogical illustration of how the onset of a “pure" local moment is encoded in the charge sector. Note that the progressive emergence of a sign structure in $\chi^{\nu \nu'}$, inverted w.r.t. the perturbative one, is responsible for all problematic manifestations[@Gunnarsson2017] of the breakdown of perturbative expansions: The divergences[@Schaefer2013; @Janis2014; @Schaefer2016c; @Ribic2016; @Gunnarsson2016; @Chalupa2018; @Vucicevic2018; @Thunstroem2018; @Springer2019] of irreducible vertex functions $\Gamma^{\nu \nu'}\!=\! [\chi^{\nu\nu'}]^{-1}\!-\![\chi_0^{\nu\nu'}]^{-1}$, that directly reflect the sign changes of the eigenvalues of $\chi^{\nu\nu'}$, and the corresponding multivaluedness[@Kozik2015; @Stan2015; @Schaefer2016c; @Tarantino2018; @Gunnarsson2017; @Thunstroem2018; @Vucicevic2018] of the Luttinger-Ward functional.
Let us now examine how this picture changes when the HA system is connected to an electronic bath (here: with a flat DOS of half bandwidth $D \!=\!10$ and hybridization $V\!=\!2\!<\!U\!=\!5.75$[^1]), corresponding to the well-known Anderson impurity model (AIM). By comparing the results of $T^2\chi^{\nu\nu'}$ (central-row panels of Fig. \[fig:1\] computed with w2dynamics[^2]) to those of the HA, we observe almost no difference at $T_{\rm high}$. This is not surprising as thermal fluctuations prevail over both correlation ($U$) and hybridization ($V$) effects in this case. Lowering $T$ to $T_{\rm int}$, we enter the [*local moment*]{} regime of the AIM. This is reflected in a qualitatively similar evolution as seen in the HA: a progressive suppression of the diagonal entries of $\chi^{\nu\nu'}$, becoming negative in the low-energy sector (black square), partly compensated by positive off-diagonal contributions. This is how the formation of a local moment affects the charge sector, suppressing local density fluctuations. Due to the screening effects of the bath, its action gets weakened, explaining the quantitative differences to the HA (e.g., the reduced size of the black square, and the less negative values on the diagonal, see [@Suppl] for a comparison).
The most interesting situation is observed by reducing $T$ further down to $T_{\rm low}\gtrsim T_K$ (right panel), where Kondo screening effects induce qualitative differences w.r.t. the HA. We clearly see that at low frequencies (white square) the sign of $\chi^{\nu \nu'}$ along the diagonal is flipped back to positive, similarly as in the perturbative regime. Physically, this nicely illustrates how the Kondo screening of the local moment acts [*energy-selectively*]{} in the charge sector, mitigating the suppression of density fluctuations at the lowest frequencies. Hence, the fingerprint of the Kondo regime is the [*onion*]{}-like frequency structure of $\chi^{\nu \nu'}$, which is clearly recognizable in the rightmost central panel of Fig. \[fig:1\]: (i) a high-frequency perturbative asymptotic, (ii) a local moment driven structure (with suppressed diagonal) at intermediate frequencies, (iii) an inner core (with a similar sign structure as (i)) induced by the Kondo screening. A quick glance at the overall sign structure of $\chi^{\nu\nu'}$ therefore allows for an immediate understanding of the underlying physics.
[*How to extract $T_K$.*]{} The behavior described above is also reflected in the temperature evolution of the lowest frequency entries of $\chi^{\nu\nu'}$: the diagonal $\chi^{D}\!=\!T^2\chi^{\pi T, \, \pi T}$ and the off-diagonal $\chi^{O}\!=\!T^2\chi^{\pi T, -\pi T}$, shown in the lowest panel of Fig. \[fig:1\]. Evidently, we can readily trace the sign changes marking the three regimes discussed above, associating the (negative) minimum of $\chi^{D}$ with the temperature at which the strongest local moment effects are observed. The screening induced enhancement of $\chi^{D}$ at lower $T$ has [*remarkable consequences*]{}: We find that crossing the Kondo temperature ($T_K\!=\!1/65\!\approx\!0.015$ at $U\!=\!5.75$ for the AIM[^3]) matches, with high accuracy, the equality of $\chi^{D}$ and $\chi^{O}$ observed at low $T$ (s. inset of Fig. \[fig:1\], marked by black triangle). We emphasize that this criterion holds more generally. As shown in the phase diagram of the AIM in Fig. \[fig:2\] (left panel), the condition $\chi^{D}\!=\!\chi^{O}$ (black triangles) traces perfectly $T_K$ (blue line)[^4] in the [*whole*]{} local moment regime $T, V < U$, i.e., where the definition of a Kondo scale is actually meaningful. Note that this is not the case for other criteria one could naturally think of, such as $\chi^{D}\!=\!-\chi^{O}$ or $\chi^{D}\!=\!0$, see [@Suppl].
Moreover, our simple 2P definition of $T_K$ holds also beyond the single impurity problem. In Fig. \[fig:2\], we show DMFT calculations for the periodic Anderson model on a square lattice with nearest-neighboring hopping $t$ (PAM, central) and for a Hubbard model on a Bethe lattice with unitary half-bandwidth (HM, right). We observe that for the PAM, the [*same matching*]{} of the condition $\chi^{D}\!=\!\chi^{O}$ (black triangles) and $T_K$ [@Schaefer2019; @ladderDGA] (blue line) is found in the local moment regime (i.e., when $V < t$).
In the HM, the Kondo temperature characterizing the auxiliary AIM associated with the self-consistent DMFT solution, depends on the temperature itself: $T_K^{HM}(T)$. Hence, $\chi^{D}\!=\!\chi^{O}$ (black triangles) indicates that the temperature equals the effective Kondo temperature, i.e. $T_K^{HM}(T) \! = \! T$. Physically, it is natural to associate this condition to the onset of low-energy electronic coherence: For all temperatures below the $\chi^{\rm D}\!=\!\chi^{\rm O}$ condition line, a conventional Fermi-liquid behavior of the physical response is expected (e.g.: $\rho(T) \propto T^2, c_V(T) \propto T$, etc.[@Abrikosov1975]). This would be also consistent with the $\chi^{D}\!=\!\chi^{O}$ condition approaching the Mott Hubbard MIT at $U_{MIT}(T\!=\! 0)\!=\!U_{c2}$ in the low-$T$ limit (see also recent DMFT studies of the physics in the proximity of the MIT[@Terletska2011; @Vucicevic2013]). The equality of the elements of the innermost $2\times 2$ submatrix of $\chi^{\nu \nu'}$ represents therefore a very simple, clear-cut criterion for determining $T_K$ at the 2P level.
[*A non-perturbative Fermi liquid.*]{} Beyond its physical relevance, our improved 2P understanding sheds light onto the nontrivial relation with the breakdown of perturbation theory[@Gunnarsson2017]. At high $T$, where $\nu_0 \!=\! \pi T \gtrsim V, U, t$, the $2\times 2$ submatrix encodes all relevant energy scales, the rest being nonsingular high-frequency asymptotics. In this case $\chi^{D}\!=\!\chi^{O}$ corresponds to a singular eigenvalue of the [*whole*]{} $\chi^{\nu \nu'}$ and hence to a divergence of the irreducible vertex function $\Gamma^{\nu \nu'}\!=\! [\chi^{\nu\nu'}]^{-1}\!-\![\chi_0^{\nu\nu'}]^{-1}$, specifically to the first (I) one encountered by increasing the interaction (red line in all plots)[@Schaefer2013; @Schaefer2016c; @Chalupa2018; @Springer2019].
At much lower temperatures $T\sim T_K$, one finds again $\chi^D > \chi^O$ as in the perturbative regime (s. Fig. \[fig:1\], lowest panel). Here, however, because of the onion-like structure of $\chi^{\nu \nu'}$, the positive-definiteness (and thus the invertibility) is guaranteed only for an inner submatrix describing the Fermi liquid regime, but not of the full $\chi^{\nu \nu'}$. This explains why divergences of irreducible vertex functions can occur also at low temperatures[@Chalupa2018] even in the presence of a Fermi liquid ground state. Indeed, such vertex divergences mark the distinction between a Fermi liquid in the weak- and in the strong-coupling regime.
[*Limitations of perturbative approaches.*]{} The identification of the 2P fingerprints of local moments and their screening allows also to identify the limits to the physical description possible via perturbative approaches. As discussed above, the strong suppression of charge fluctuations in the local moment regime is encoded in negative diagonal and enhanced off-diagonal entries of $\chi^{\nu \nu'}$, which is associated[@Gunnarsson2017] to several negative eigenvalues. Hence, the local moment physics, as well as its Kondo screening, cannot be accessed without crossing several divergences of $\Gamma$[@Schaefer2016c; @Chalupa2018]. This requirement is beyond perturbative approaches, where -per construction- $\Gamma$ is finite[^5]. We substantiate this statement by considering two of the most advanced perturbative schemes, the functional renormalization group (fRG)[^6] and the parquet approximation (PA)[@Bickersbook2004; @Yang2009; @Tam2013; @Valli2015; @Wentzell2016; @Li2016; @Janis2019; @Kauch2020]. The results obtained for the AIM with $U\!=\!5.75$[^7] and $T \! =\! T_{\rm int}$ are shown in Fig. \[fig:3\]. $\chi^{\nu \nu'}$ computed by the fRG and PA appears qualitatively different from the (numerically) exact one of Fig. \[fig:1\] (AIM, central): the diagonal elements are all [*positive*]{}, and substantially larger than the vanishing off-diagonal ones, see also [@Suppl]. This ensures the positive-definiteness of the whole $\chi^{\nu \nu'}$, ruling out, at the same time, the suppression effects of the charge response characterizing the local moment regime. This drawback qualitatively affects the physical description. In particular, the temperature dependence of the static susceptibility $\chi$ (Fig. \[fig:3\] lower panel) exhibits a [*clear minimum*]{} for intermediate $T_K < T < T_{\rm high}$. This emerges from the competition between the suppression effects of the local moment (see the extreme HA case) and the low-energy screening. Both features are not captured by the fRG and PA, which display a monotonous behavior. [*Conclusions.*]{} We have shown how fundamental physical properties of correlated systems, i.e. the local moment formation and its Kondo screening, can be directly read in the Matsubara frequency structure of the generalized susceptibility $\chi^{\nu \nu'}$. In particular, the competition between localization effects and metallic screening are encoded in a clearly recognizable “onion-like” fingerprint in $\chi^{\nu \nu'}$, emerging in the Kondo regime. The thorough inspection of the latter even discloses an alternative, simpler, route to extract $T_K$ directly from Matsubara frequency space. Eventually, our improved understanding of the 2P processes sets clear-cut limits to the physics accessible by means of perturbative approaches.\
[*Acknowledgements*]{} - We thank M. Capone, S. Ciuchi, J. von Delft, K. Held, C. Hille, F. Krien, F.B. Kugler, and E. van Loon, G. Sangiovanni and D. Springer for insightful discussions. The authors also want to thank the Simons Foundation for hospitality. The present work was supported by the Austrian Science Fund (FWF) through the Project I 2794-N35 and Erwin-Schrödinger Fellowship J 4266 - “[*Superconductivity in the vicinity of Mott insulators*]{}” (SuMo, T.S.), the Deutsche Forschungsgemeinschaft (DFG) through Project No. AN 815/6-1, as well as the European Research Council for the European Union Seventh Framework Program (FP7/2007-2013) with ERC Grant No. 319286 (QMAC, T.S.).
[^1]: For the AIM the interaction U is often put into reference with $\Delta\!=\!V^2\pi\rho_0$, where in our case $\Delta\!=\!\pi/5$. Hence $U/\Delta\!\approx\!9.15$.
[^2]: Calculations of the 1P and 2P impurity Green’s functions have been performed with a CT-QMC algorithm[@Gull2011a] in the hybridization expansion, i.e. the w2dynamics code[@w2dynamics]
[^3]: \[TKAIMExtract\] To extract the Kondo temperature we follow an approach presented in Ref. [@Krishnamurthy1980], which is also applied for the same model and parameters in the appendix B of Ref. [@Chalupa2018].
[^4]: For different interaction values $T_K$ is extracted\[TKAIMExtract\] (blue crosses), and then a fit based on the analytical expression for $T_K$ in the wide-band limit[@Hewson1993] ($A\sqrt{U\Delta}\exp{(-B\frac{U}{\Delta} + C\frac{\Delta}{U})}$) is used.
[^5]: With the only exception of second-order phase transitions to long-range ordered phases, which is not relevant here.
[^6]: We use the one-loop flow equations for the self-energy and two-particle vertex, see Ref. for a review. For the details of the implementation we refer to the Supplemental Material [@Suppl].
[^7]: The value of $U\!=\!5.75$ ($U/\Delta\!\approx\!9.15$) has been chosen in such a way that the local magnetic moment is already well defined, and at the same time is still accessible within PA and fRG in order to allow for a direct comparison to Fig. \[fig:1\]. Similar results are obtained for other values of $U$, see [@Suppl].
|
---
author:
- 'Mohammad Khosravi[^1] and Roy S. Smith [^2] [^3]'
bibliography:
- 'mybib.bib'
title: '**Convex Nonparametric Formulation for Identification of Gradient Flows**'
---
[^1]: Corresponding author
[^2]: This research project is part of the Swiss Competence Center for Energy Research SCCER FEEB&D of the Swiss Innovation Agency Innosuisse.
[^3]: The authors are with Automatic Control Lab, ETH Zurich, Switzerland\
[{khosravm,rsmith}@control.ee.ethz.ch]{}
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abstract: 'We detected extended, curved stellar tidal tails emanating from the sparse, disrupting halo globular cluster Pal 5, which cover $10^{\circ}$ on the sky. These streams allow us to infer the orbit of Pal 5 and to ultimately constrain the Galactic potential at its location.'
author:
- 'Michael Odenkirchen, Eva K. Grebel, Hans-Walter Rix'
- Walter Dehnen
- Heidi Jo Newberg
- 'Constance M. Rockosi'
- Brian Yanny
title: |
The Extended Tidal Tails of Palomar 5:\
Tracers of the Galactic Potential
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
\#1 1.25in .125in .25in
Palomar 5: A Globular Cluster Torn Apart by the Milky Way
=========================================================
Palomar 5 is an extraordinarily sparse globular cluster in the outer halo of the Milky Way, at a distance of 23 kpc from the Sun (Fig.1). Its peculiar properties (e.g., very low mass, large core, relatively flat luminosity function) fostered the idea that this cluster might be a likely victim of disruptive Galactic tides.
Using deep multi-color photometry from the Sloan Digital Sky Survey (SDSS; York et al. 2000, Gunn et al. 1998) we found unambiguous, direct evidence for the suspected tidal disruption of Pal5 (Odenkirchen et al. 2001; Rockosi et al. 2002): For the first time, two massive tails of stellar debris with well-defined S-shape geometry were detected, emanating in opposite directions from the cluster.
As the SDSS is scanning more and more of the sky we have now extended our search over an area of $\sim$87 deg$^2$. Contaminating objects were removed by eliminating extended sources and by applying an optimized smooth color-magnitude-dependent weighting function. This optimized weighting enhances the density contrast between cluster and field stars by almost a factor of 20 and provides a least-squares solution for the spatial distribution of the cluster population. The resulting surface density map of Pal5 stars is shown in Fig. 2.
A Narrow, Curved, 10 Degree Stream of Debris
============================================
We find that the tidal tails extend over an arc of at least 10$^\circ$ on the sky and form a narrow stream with a FWHM of only $18'$. This corresponds to a projected length of $\simeq$ 4 kpc in space, and a projected FWHM of 120 pc. The northern tail is visible out to 6.5$^\circ$ from the cluster. The southern tail is traced over 3.5$^\circ$ but probably continues beyond the border of the currently available field (Fig. 2). The stellar mass in the tails adds up to 1.2 times the mass of stars in the cluster, i.e., the tails contain more mass than what is left in the cluster. Pal5 thus presents a text-book example of a tidally disrupting globular cluster. It is so far the only known stellar system besides the Sagittarius dwarf galaxy that demonstrates the formation of a halo stream within the Milky Way.
The tails have a clumpy structure (Fig. 2). This implies that the mass loss has been episodic, and suggests that it was triggered by disk and/or bulge shocks. Indeed Pal5 passes through the Galactic disks at intervals of a few 100 Myr (Fig. 1). In Fig. 3 we present the radial profile of the stellar surface density (i.e., the azimuthally averaged surface density as a function of distance from the cluster center) from the core of the cluster out to the current end points of the tails. The profile shows a characteristic break near the cluster’s tidal radius at about $16'$. Inside this radius, the profile decreases approximately like $r^{-3}$. Beyond this limit the profile is flatter and approximately follows an $r^{-1.5}$ power law. The overall decrease thus differs from a simple $r^{-1}$ power law that would result from a constant linear density along the tails.
Clues on the Cluster’s Orbit and Mass Loss Rate
===============================================
Location and curvature of the tails are direct tracers of the cluster’s Galactic motion and hence provide unique information about the orbit of Pal5. The best-fit local orbit is shown as dashed line in Fig. 2. The direction of the cluster’s motion is determined by its orientation with respect to the Galactic center: The southern, leading tail and the northern, trailing tail indicate that Pal5 is on a prograde orbit. Using a standard three-component model for the Galactic potential we infer that Pal5 is observed close to the tails’ maximum distance from the Galactic disk and has recently passed through apogalacticon (implying that the tidal stream is currently relatively dense). In about 100 Myr the cluster will cross the disk at a distance of only 7 kpc from the Galactic center (see Fig. 1). This will produce a strong tidal shock that might lead to complete disruption.
The amount by which the tails are offset from the orbit of the cluster is directly related to the velocity at which the tidal debris drifts away from the cluster. The observed mean offset (about 75 pc in projection), the parameters of our model orbit, and the total amount of stellar mass seen in the tails lead to an estimate of the mean mass loss rate of about $5\,M_\odot$ Myr$^{-1}$. Assuming this rate to be more or less constant (as suggested by numerical simulations) we conclude that 10 Gyr ago Pal5 may have had a mass of about $5 \cdot 10^4 M_\odot$. This is about ten times as much as it has today, but still considerably less than the mass of an average present-day Galactic globular cluster.
The current data allow us to predict the tangential velocity of Pal5 as a function of the parameters of the Galactic potential. GAIA and SIM will allow us to accurately measure the velocities and proper motions of stars in the cluster and in the extended tails, fully characterizing the kinematics of the stream independent of any galactic model. In return, these kinematics impose strong constraints on the Galactic potential at the location of the cluster.
Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS website is http://www.sdss.org/.
Gunn, J.E., et al. 1998, AJ, 116, 3040 Odenkirchen, M., et al. 2001, ApJ, 548, L165 Rockosi, C.M., et al. 2002, AJ, 124, 349 York, D.G., et al. 2000, AJ, 120, 1579
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abstract: 'We investigate the photo-induced spin dynamics of single nitrogen-vacancy (NV) centres in diamond near the electronic ground state level anti-crossing (GSLAC), which occurs at an axial magnetic field around 1024 G. Using optically detected magnetic resonance spectroscopy, we first find that the electron spin transition frequency can be tuned down to 100 kHz for the [$^{14}$NV]{} centre, while for the [$^{15}$NV]{} centre the transition strength vanishes for frequencies below about 2 MHz owing to the GSLAC level structure. Using optical pulses to prepare and readout the spin state, we observe coherent spin oscillations at 1024 G for the [$^{14}$NV]{}, which originate from spin mixing induced by residual transverse magnetic fields. This effect is responsible for limiting the smallest observable transition frequency, which can span two orders of magnitude from 100 kHz to tens of MHz depending on the local magnetic noise. A similar feature is observed for the [$^{15}$NV]{} centre at 1024 G. As an application of these findings, we demonstrate all-optical detection and spectroscopy of externally-generated fluctuating magnetic fields at frequencies from 8 MHz down to 500 kHz, using a [$^{14}$NV]{} centre. Since the Larmor frequency of most nuclear spin species lies within this frequency range near the GSLAC, these results pave the way towards all-optical, nanoscale nuclear magnetic resonance spectroscopy, using longitudinal spin cross-relaxation.'
author:
- 'David A. Broadway'
- 'James D. A. Wood'
- 'Liam T. Hall'
- Alastair Stacey
- Matthew Markham
- 'David A. Simpson'
- 'Jean-Philippe Tetienne'
- 'Lloyd C. L. Hollenberg'
bibliography:
- 'bib.bib'
title: 'Spin dynamics of diamond nitrogen-vacancy centres at the ground state level anti-crossing and all-optical low frequency magnetic field sensing'
---
Introduction
============
Detection and identification of spin species using established techniques such as magnetic resonance spectroscopy proves to have a host of applications in materials science, chemistry and biology. However, these techniques are limited in sensitivity, and thus require macroscopic ensembles of spins in order to produce a measurable signal [@Blank2003]. A variety of techniques have been developed over the last decade to extend magnetic resonance spectroscopy to the nanometre scale [@Poggio2010; @Artzi2015; @Bienfait2016; @Rondin2014; @Schirhagl2014]. Notably, methods based on the nitrogen-vacancy (NV) centre in diamond [@Doherty2013] have attracted enormous interest owing to their ability to operate under conditions compatible with biological samples [@Schirhagl2014; @McGuinness2011]. While significant progress has been made with NV-based sensing in the last few years [@Rondin2014], spectroscopy at the single nuclear spin level remains a major challenge.
The detection of external spins with the NV centre is generally achieved through measuring the longitudinal spin relaxation rate ($T_1$ processes) [@Steinert2013; @Tetienne2013; @Kaufmann2013; @Ermakova2013; @Sushkov2014] or transverse spin relaxation rate (dephasing, or $T_2$ processes) [@Ermakova2013; @Maze2008; @DeLange2011; @McGuinness2013] of the NV’s electron spin, as they are sensitive to the magnetic field fluctuations produced by the target spins [@Cole2009; @Hall2009; @Laraoui2010]. To obtain spectral information on the target spins, most studies so far have focussed on using $T_2$-based techniques, which have been applied to the spectroscopy of small ensembles of either electronic [@Grotz2011; @Laraoui2012; @Mamin2012; @Knowles2013] or nuclear spins [@Mamin2013; @Staudacher2013; @Loretz2014; @Muller2014; @DeVience2015]. However, spectroscopy can also be achieved by relying on $T_1$ processes, via cross-relaxation between a probe spin (the NV centre’s electron spin) and the target spins [@Jarmola2012; @Wang2014; @VanderSar2015; @Hall2016; @Wood2016]. The $T_1$-based approach to spectroscopy introduced in Ref. [@Hall2016] allows for nanoscale, all-optical, wide-band magnetic resonance spectroscopy [@Wood2016]. As such, it represents a promising alternative to $T_2$-based approaches, as the latter require radiofrequency (RF) driving of the probe and/or the target, which poses various technical challenges in addition to limiting the accessible frequency range.
In cross-relaxation spectroscopy, the $T_1$ of the NV spin is monitored while varying the strength of an applied axial magnetic field, $B_z$ (Fig. \[Fig:intro\]a). When the transitional energy between two of the NV eigenstates (generally [$\left| 0_e \right>$]{} and [$\left| -1_e \right>$]{}, where the number refers to the electron spin projection $m_e$) is equal to that of a target spin, cross-relaxation occurs, which results in an increase in the relaxation rate, $1/T_1$, of the NV [@Hall2016; @Wood2016]. This increase can be measured by purely optical means, even for a single NV centre [@Tetienne2013; @Kaufmann2013; @Ermakova2013; @Sushkov2014]. By scanning across a range of magnetic field strengths, a resonance spectrum of the target spins can be obtained, which can be deconvolved to produce the target spin spectrum [@Hall2016]. This technique has been recently used to measure electronic spin resonance (ESR) spectra of P1 centres within the diamond at magnetic fields of 460-560 G, corresponding to transition frequencies of 1300-1600 MHz [@Hall2016; @Wood2016]. In order to detect nuclear magnetic resonances (NMR), the NV transition frequency must be matched to the Larmor frequency of the target nuclear spins, which is generally of order a few MHz. This occurs when the states [$\left| 0_e \right>$]{} and [$\left| -1_e \right>$]{} approach degeneracy, at a magnetic field $B_z\approx1024$ G (Fig. \[Fig:intro\]b). However, in this region the NV experiences a complex Ground State Level Anti-Crossing (GSLAC, Figs. \[Fig:intro\]c,d) [@He1993] due to hyperfine interaction of the NV electron spin with its own nuclear spin ($^{14}$N or $^{15}$N), which also has a coupling strength of several MHz. The hyperfine interaction causes spin mixing, which may prevent the NV electron spin from being initialised and read out [@Epstein2005]. Therefore, a detailed understanding of the spin dynamics at the GSLAC is required in order to assess the potential for performing $T_1$-based spectroscopy of nuclear spins as initially explored in Ref. [@Wood2016]. While the GSLAC of the NV centre has been previously studied and exploited in several works [@He1993; @Wei1999; @Wilson2003; @Epstein2005; @Fuchs2011; @Wang2013; @Wang2015a; @Wickenbrock2016], there is little knowledge about how $T_1$ varies and how the spin dynamics (including optical initialization) behaves at transition frequencies relevant to NMR, close to the GSLAC.
In this paper, we investigate the spin dynamics at the GSLAC for both [$^{14}$NV]{} and [$^{15}$NV]{} spin systems in various diamond samples. We begin by looking at the computed energy spectra of both spin systems and compare them to optically detected magnetic resonance (ODMR) measurements at their respective GSLACs. The short time spin dynamics of the [$^{14}$NV]{} spin at the GSLAC are probed using optical pulses, revealing a feature at $B_z\approx1024$ G which manifests itself in either coherent spin oscillations, or in a simple polarisation drop depending on the sample. This feature is explained by spin mixing induced by residual transverse magnetic fields. The [$^{15}$NV]{} centre shows a similar feature at $B_z\approx 1024$ G. In addition to these narrow features, we find that the spin polarisation and $T_1$ time remains constant across the GSLAC, implying that the NV centre can be used to detect magnetic signals at low frequencies via $T_1$ measurements. Finally, we demonstrate one such application by performing all-optical spectroscopy of fluctuating magnetic fields generated at known frequencies, mimicking those of nuclear spins. This suggests that it is possible to perform NMR spectroscopy via longitudinal cross-relaxation near the GSLAC.
Energy levels of the NV centre at the GSLAC
===========================================
![ (a) Schematic view of the system under study: the NV centre in diamond comprises a nuclear spin and an electron spin, which can be initialised and read out optically using a confocal microscope equipped with an avalanche photodiode (APD); it is surrounded by target spins located within the diamond or external to it. (b) Energy structure of the NV electronic ground state, showing the Zeeman splitting under a magnetic field applied along the NV symmetry axis. The three electronic spin levels are labelled by their spin projection $m_e$. (c,d) Calculated hyperfine structure for the [$^{14}$NV]{} (c) and [$^{15}$NV]{} centre (d), plotted near the GSLAC where the two branches $m_e=0$ and $m_e=-1$ cross. The different levels are labelled by the nuclear spin projection $m_n$ of the unperturbed states (away from the GSLAC). Due to hyperfine interaction, some of the levels exhibit an avoided crossing at the GSLAC.[]{data-label="Fig:intro"}](introduction.pdf){width="45.00000%"}
The NV spin system consists of a nitrogen atom adjacent to a vacancy in the carbon lattice of diamond. It comprises a pair of electrons (forming a spin-1) and a nuclear spin, which is a spin-1 for [$^{14}$NV]{} and spin-1/2 for [$^{15}$NV]{} (Fig. \[Fig:intro\]a). Due to spin-spin interaction, the electronic spin states [$\left| \pm 1_e \right>$]{} are split from [$\left| 0_e \right>$]{} by $D/2\pi\approx2.87$ GHz. The degeneracy of the [$\left| \pm 1_e \right>$]{} can be lifted by the application of an external magnetic field along the NV centre’s symmetry axis, defined as the $z$ axis, as shown in Fig. \[Fig:intro\]b. The [$\left| -1_e \right>$]{} and [$\left| 0_e \right>$]{} states cross at a field around $B_z=D/\gamma_e\approx 1024$ G (where $\gamma_e$ is the electron gyromagnetic ratio), shown as a black dot in Fig. \[Fig:intro\]b. However, hyperfine interaction with the nitrogen nuclear spin causes an avoided crossing, the GSLAC, which is the main focus of this paper.
The energy spectrum of the NV electronic ground state near the GSLAC can be found by solving for the eigenvalues of the spin Hamiltonian. The relevant Hamiltonians for the [$^{14}$NV]{} and [$^{15}$NV]{} cases, expressed in units of angular frequencies, are $$\begin{aligned}
{\cal H}(\text{{\ensuremath{^{14}}NV}}) &= D S_z^2 + \gamma_e B_z S_z - \gamma_n B_z I_z + Q I_z^2 \\
&\qquad \qquad + A_{\parallel} S_z I_z + A_{\perp}\left( S_x I_x + S_y I_y \right), \notag\\
{\cal H}(\text{{\ensuremath{^{15}}NV}}) &= D S_z^2 + \gamma_e B_z S_z - \gamma'_n B_z I'_z \\
&\qquad\qquad + A_{\parallel}^\prime S_z I'_z + A_{\perp}^\prime \left( S_x I'_x + S_y I'_y \right), \notag \end{aligned}$$ where ${\bf S}=(S_x,S_y,S_z)$ is the electron spin operator, [**I**]{} is the nuclear spin operator, and $\gamma_n$ is the nuclear gyromagnetic ratio. The magnetic field is aligned along the NV axis, with strength $B_z$. The primed symbols refer to the [$^{15}$NV]{} case. The longitudinal and transverse hyperfine parameters are denoted as $A_\parallel$ and $A_\perp$, whose values are $A_\parallel/2\pi = -2.14$ MHz and $A_\perp/2\pi = -2.7$ MHz for [$^{14}$NV]{}, $A_\parallel^\prime/2\pi = 3.03$ MHz and $A_\perp^\prime/2\pi = 3.65$ MHz for [$^{15}$NV]{} [@Felton2009]. In addition, the [$^{14}$NV]{} has a quadrupole coupling with strength $Q/2\pi = -5.01$ MHz [@Felton2009].
The energy spectrum is obtained by evaluating the eigenvalues of the Hamiltonian at different axial field strengths $B_z$ and is shown in Fig. \[Fig:intro\]c for [$^{14}$NV]{} and Fig. \[Fig:intro\]d for [$^{15}$NV]{}. The manifold associated with the spin projection [$\left| +1_e \right>$]{} is not shown as it lies about 6 GHz above the manifold spanned by [$\left| 0_e \right>$]{} and [$\left| -1_e \right>$]{} and does not contribute to the effects discussed in this paper. We thus consider only the 6 lower-energy states for [$^{14}$NV]{}, and the 4 lower states for [$^{15}$NV]{}. Away from the GSLAC, the eigenstates have well-defined spin projections along the $z$ axis. In the electron-nuclear spin space, we denote states as [$\left| m_e,m_n \right>$]{} where $m_e$ ($m_n$) is the electronic (nuclear) spin projection along the $z$ axis. The hyperfine coupling between the NV nuclear and electron spins results in a splitting of the nuclear spin states for each electronic spin state, due to the longitudinal component $A_\parallel$. Near the GSLAC, the perpendicular component of the hyperfine ($A_\perp$) induces a mixing of some of the $z$-basis states. This effect is noticeable when the quantization energy between said states becomes of order $A_\perp$. In the case of the [$^{14}$NV]{}, the states [$\left| 0, +1 \right>$]{} and [$\left| -1, -1 \right>$]{} do not mix as they have no hyperfine coupling to any other state, and as such, they remain eigenstates. The rest of the NV states are mixed, that is, the eigenstates are superpositions of $z$-basis states, creating an avoided crossing. The [$^{15}$NV]{} spin also exhibits mixing at the GSLAC. In particular the [$\left| 0,-1/2 \right>$]{} and [$\left| -1,+1/2 \right>$]{} states become mixed while the [$\left| 0,+1/2 \right>$]{} and [$\left| -1,-1/2 \right>$]{} states remain eigenstates.
Optically detected magnetic resonance at the GSLAC {#Sec:ODMR}
==================================================
![ODMR spectra of a [$^{14}$NV]{} (left panels) and [$^{15}$NV]{} (right panels) centre, measured as a function of the axial magnetic field strength near the GSLAC. The top panels show the electron spin transitions $|0_e\rangle\rightarrow|+1_e\rangle$, while the bottom panels show $|0_e\rangle\rightarrow|-1_e\rangle$. Overlaid on the graph are the theoretical frequencies of all allowed or partly allowed (via spin mixing) transitions. However, dynamic nuclear spin polarisation makes some of the transitions dominant (shown as solid lines), the other transitions having comparatively small or vanishing contrast (dashed lines).[]{data-label="Fig:ODMR"}](ODMR.pdf){width="47.00000%"}
Experimentally, one can probe the NV energy spectrum using optically detected magnetic resonance (ODMR) spectroscopy [@Gruber1997]. This is achieved by measuring the photoluminescence (PL) intensity of the NV centre while varying the frequency of an applied RF field, using a purpose-built confocal microscope with green laser excitation (Fig. \[Fig:intro\]a). The laser serves both to initialize the NV in the electronic spin state [$\left| 0_e \right>$]{}, and read out the spin state following an RF pulse, exploiting the fact that [$\left| \pm1_e \right>$]{} emit less PL on average than [$\left| 0_e \right>$]{} [@Manson2006]. Thus, ODMR allows us to probe the electron spin transitions $|0_e\rangle\rightarrow|\pm 1_e\rangle$. We recorded ODMR spectra for magnetic fields varied from 1018 G to 1032 G, for single [$^{14}$NV]{} and [$^{15}$NV]{} centres in a high-purity diamond grown by chemical vapour deposition (CVD). The results are shown in Fig. \[Fig:ODMR\], where the top (bottom) panels show the [$\left| 0_e \right>$]{}$\rightarrow$ [$\left| +1_e \right>$]{} ([$\left| 0_e \right>$]{}$\rightarrow$ [$\left| -1_e \right>$]{}) transitions. The left (right) panels correspond to a [$^{14}$NV]{} ([$^{15}$NV]{}) centre. Comparing with the theoretical expectations for the allowed transitions (shown as black lines), it can be seen that only a limited number of expected transitions are observed experimentally. This is because the nuclear spin is efficiently polarised under optical pumping near the GSLAC, owing to hyperfine-induced spin mixing. This effect has been well documented at the excited state level anti-crossing (ESLAC, [@Jacques2009; @Ivady2015]), but has not been previously quantified experimentally across the GSLAC [@Fuchs2011]. Using the relative strengths of the [$\left| 0_e \right>$]{}$\rightarrow$[$\left| +1_e \right>$]{} transitions (top panels in Fig. \[Fig:ODMR\]), we find that the nuclear spin is polarised to $>90\%$ into [$\left| +1_n \right>$]{} for [$^{14}$NV]{} across the whole range of fields scanned here. For the [$^{15}$NV]{} investigated in Fig. \[Fig:ODMR\], the nuclear spin is polarised to $>90\%$ in [$\left| +1/2_n \right>$]{} up to $B_z\approx 1026$ G, but becomes completely unpolarised above $B_z\approx 1028$ G.
We now discuss the ODMR spectrum of the [$\left| 0_e \right>$]{}$\rightarrow$[$\left| -1_e \right>$]{} transition (bottom panels in Fig. \[Fig:ODMR\]). For [$^{14}$NV]{}, the NV is efficiently polarised in the state [$\left| 0,+1 \right>$]{}, which remains an eigenstate at all fields. Away from the GSLAC, the only transition allowed is to the state with the same nuclear spin projection, [$\left| -1,+1 \right>$]{}. However, near the GSLAC the state [$\left| -1,+1 \right>$]{} becomes mixed with [$\left| 0,0 \right>$]{}, which creates two eigenstates of the form ${\ensuremath{\left| \alpha_\pm \right>}} = \alpha_{1\pm}{\ensuremath{\left| -1,+1 \right>}} + \alpha_{2\pm}{\ensuremath{\left| 0,0 \right>}}$. This results in an avoided crossing feature centred at $B_z\approx 1022$ G, around the transition frequency $\omega\approx 5$ MHz which corresponds to the quadrupole coupling $Q$. This leads to little mixing at the allowed crossing at $B_z\approx1024$ G, which means the dominant transition here is [$\left| 0,+1 \right>$]{}$\rightarrow$[$\left| -1,+1 \right>$]{}. We observed clear ODMR signatures with resonance frequencies down to 100 kHz in this sample. This implies that the NV spin could be resonantly coupled to most nuclear spin species, which have Larmor frequencies ranging typically from 500 kHz to 5 MHz at this magnetic field. However, the minimum observable transition frequency was found to be highly sample dependent, as will be discussed in section \[Sec:Dynamics\].
On the other hand, the [$^{15}$NV]{} exhibits a very different spectrum near the GSLAC (Fig. \[Fig:ODMR\], right-hand panels). Under optical pumping, it is efficiently polarised in the state [$\left| 0,+1/2 \right>$]{}, which remains an eigenstate at all fields. The states it can transit to under RF driving are superpositions ${\ensuremath{\left| \beta_\pm \right>}} = \beta_{1\pm}{\ensuremath{\left| 0,-1/2 \right>}} + \beta_{2\pm}{\ensuremath{\left| -1,+1/2 \right>}}$, where the avoided crossing is centred approximately around the initial state [$\left| 0,+1/2 \right>$]{} (see Fig. \[Fig:intro\]d). As a result, the ODMR plot shows two transitions that bend upon approaching vanishing frequencies. They cross at a field $B_z\approx 1024$ G and a transition frequency $\omega'_{\times}/2\pi\approx2.65(2)$ MHz given by $$\begin{aligned}
\omega'_{\times} = \sqrt{\frac{A_\perp^{\prime 2}}{2}+\left(\frac{\gamma'_n}{\gamma_e-\gamma'_n}\right)^2\left(D-\frac{A'_\parallel}{2}\right)^2}.\end{aligned}$$ Incidentally, this allows the perpendicular component of the hyperfine interaction ($A'_\perp$) to be measured directly on a single [$^{15}$NV]{} centre, which gives here $A'_{\perp}/2\pi = 3.69(3)$ MHz, in excellent agreement with the ensemble-averaged value of 3.65(3) MHz reported in Ref. [@Felton2009]. The peculiar GSLAC structure of the [$^{15}$NV]{} centre has important consequences for sensing. In particular, the contrast of the transitions decreases rapidly for frequencies below $\omega'_{\times}$ as they become forbidden. As a result, the [$^{15}$NV]{} is unsuited to detecting resonances below about 2 MHz under typical conditions. Most nuclear spin species have transitions within this range, with an exception being hydrogen ($^1$H), which has a Larmor frequency of about 4.4 MHz at this field and could be in principle detected via cross-relaxation with an [$^{15}$NV]{}.
Photo-induced spin dynamics at the GSLAC {#Sec:Dynamics}
========================================
![(a) Depiction of the laser pulse sequence used to measure the spontaneous spin dynamics, containing an initialisation pulse and a readout pulse after a wait time $t$. (b) Time trace measured for a single [$^{14}$NV]{} centre at a field $B_z=1000$ G. Here a decay with a characteristic time $T_1\approx5$ ms is observed, associated with phonon relaxation. (c) Set of PL scans as a function $B_z$ for various wait times $t$, under a small residual transverse field $B_\perp$, showing a narrow feature at $B_z\approx1024$ G. The wait times $t=2$ $\mu$s (grey dots), 10 $\mu$s (light blue) and 1 ms (purple) are indicated as dashed lines in (b) with exaggerated positions for ease of viewing. (d) Time traces recorded at $B_z\approx1024$ G. The top curve was measured with a detuning $\delta B_z=0.2$ G, showing no variation in the range $t=0-15~\mu$s. The other curves are recorded with no detuning, but with a transverse magnetic field $B_\perp$ increasing from $\approx0$ G to $\approx0.4$ G (top down). The curves are vertically offset from each other for clarity. Inset: Averaged time traces $\langle P_0(t)\rangle$ computed from Eq. (\[Eq:osc\]) where $\delta B_z$ and $B_\perp$ are normally distributed, with means $\langle\delta B_z\rangle=0$ and $\langle B_\perp\rangle=0$ G (orange curve) or 0.2 G (green curve) and variances $\sigma_{B_z}=1.3~\mu$T and $\sigma_{B_\perp}=\sqrt{3/4}\sigma_{B_z}$. The dashed blue line is the approximate envelope $e^{-(t/T_\times)^2}$ with $T_\times=(\gamma_e\sigma_{B_\perp})^{-1}=5~\mu$s.[]{data-label="Fig:C12"}](C12_2.pdf){width="47.00000%"}
We now investigate the spin population dynamics near the GSLAC. Our aim is to assess the possibility of measuring the longitudinal spin relaxation time ($T_1$), as required in order to perform cross-relaxation spectroscopy and detect nearby nuclear spins [@Wood2016]. The $T_1$ time is typically measured by using laser pulses to initialise the NV into [$\left| 0_e \right>$]{}, and read out the remaining population of [$\left| 0_e \right>$]{} after a variable delay $t$ (Fig. \[Fig:C12\]a). In practice, the PL signal is integrated at the start of the readout pulse ($I_s$) and normalised by the PL from the back of the pulse ($I_r$). The normalised signal $I_s/I_r$ can be expressed as [@Manson2006] $$\begin{aligned}
\frac{I_s}{I_r}(t)=a+bP_0(t)\end{aligned}$$ where $P_0(t)=|\langle 0_e|\psi(t)\rangle|^2$ is the population in [$\left| 0_e \right>$]{} of the current spin state $|\psi(t)\rangle$, and $a\approx 1$ and $b\approx 0.3$ are constants. The resulting time trace $\frac{I_s}{I_r}(t)$ therefore allows us to estimate the initial population $P_0(0)$, which approaches unity under normal conditions [@Manson2006], as well as the evolution of the spin state in the dark. In general (away from the GSLAC or ESLAC), the spin population exhibits a simple exponential decay towards a thermal mixture, i.e. $P_0(t)=\frac{1}{3}+\frac{2}{3}e^{-t/T_1}$ assuming perfect initialisation (see an example in Fig. \[Fig:C12\]b). In the following, we measure $\frac{I_s}{I_r}(t)$ for different wait times $t$, as a function of the axial magnetic field $B_z$.
[$^{14}$NV]{} centres
---------------------
We first performed measurements of [$^{14}$NV]{} centres in a ultra-high purity CVD-grown diamond with isotopically purified carbon content (\[[$^{12}$C]{}\]$>99.99\%$). In this sample, the main source of magnetic noise comes from the bath of remaining [$^{13}$C]{} impurities [@Balasubramanian2009]. A scan across the GSLAC for a representative [$^{14}$NV]{} centre is shown in Fig. \[Fig:C12\]c, where we probed three time points $t=2$ $\mu$s, 10 $\mu$s, and 10 ms. When the NV is far from the GSLAC crossing, an exponential decay is observed as shown in the full time trace in Fig. \[Fig:C12\]b. This decay corresponds to phonon-induced relaxation, with a characteristic time $T_1\approx5$ ms [@Jarmola2012]. At the crossing at $B_z\approx1024$ G, however, a sharp variation in signal is observed (Fig. \[Fig:C12\]c). Here, the NV spin undergoes population oscillations, as indicated by the 2 $\mu$s time point dropping below the 10 ms point.
To understand this oscillation, we look at the reduced Hamiltonian, ${\cal H}_R$, in the basis that contains the states that cross, $\{|0,+1\rangle,|-1,+1\rangle\}$. In a magnetic field ${\bf B}=(B_x,B_y,B_z)$, this Hamiltonian is given by $$\begin{aligned}
{\cal H}_R =
\begin{pmatrix}
0 & \gamma_e B_\perp\frac{e^{-i\theta}}{\sqrt{2}} \\
\gamma_e B_\perp\frac{e^{+i\theta}}{\sqrt{2}} & \gamma_e\delta B_z
\end{pmatrix} ,\end{aligned}$$ where we introduced the longitudinal detuning from the crossing, $\gamma_e \delta B_z=D-\gamma_e B_z-A_\parallel$, the transverse magnetic field, $B_\perp=\sqrt{B_x^2+B_y^2}$, and the angle $\theta$ defined by $\tan\theta=B_y/B_x$. The transverse field causes a mixing between $|0,+1\rangle$ and $|-1,+1\rangle$ and opens an energy gap associated with a level avoided crossing. Assuming that optical pumping always initialises the NV in the [$\left| 0,+1 \right>$]{} state, and reads out the population in that same state, we then expect oscillations between [$\left| 0,+1 \right>$]{} and [$\left| -1,+1 \right>$]{} that are mirrored in the PL when in the presence of a transverse magnetic field. Under these conditions, the probability of occupying the state [$\left| 0,+1 \right>$]{} after a wait time $t$ following initialisation is given by $$\begin{aligned}
\label{Eq:osc}
P_0(t) = \frac{\delta B_z^2 + B_\perp^2 \left[1 + \cos \left(\gamma_e t \sqrt{\delta B_z^2 + 2 B_\perp^2}\right)\right]}{\delta B_z^2 + 2 B_\perp^2}.\end{aligned}$$ The amplitude of the oscillation vanishes when the detuning is much larger than the transverse field ($\delta B_z\gg B_\perp$), far from the avoided crossing region. This is illustrated in Fig. \[Fig:C12\]d (top curve), which was recorded with a detuning $\delta B_z\approx0.2$ G $\gg B_\perp$. On the other hand, near the avoided crossing where the amplitude is maximal, the frequency of the oscillation is expected to increase as $B_\perp$ is increased. This effect was tested through a series of measurements with varying transverse fields. Experimentally this involved using a permanent magnet to align the field at the 1024 G crossing so that no oscillations are detected. The permanent magnet is then moved in the transverse direction ($x$ or $y$) to add a transverse field. The results are shown in Fig. \[Fig:C12\]d where $B_\perp$ is increased from top down, resulting in faster oscillations.
Damping of the oscillations is attributed to noise in the local magnetic field. In this sample, the noise comes predominantly from the bath of [$^{13}$C]{} impurities. Examination of this interaction via the rotating wave approximation shows that only the $x$-$z$, $y$-$z$ and $z$-$z$ components of the dipole-dipole coupling to the NV spin need be considered [@Hall2014]. As such, the effective magnetic noise from the environment may be regarded as static over the short wait times, $t$, considered here. We assume that the field components $\delta B_z$ and $B_\perp$ are normally distributed with means $\langle\delta B_z\rangle$ and $\langle B_\perp\rangle$ and variances $\sigma^2_{B_z}$ and $\sigma^2_{B_\perp}$, respectively. Averaging Eq. (\[Eq:osc\]) over these distributions, we find numerically that the decay envelope of $\langle P_0(t)\rangle$ is well approximated by a gaussian function $e^{-(t/T_\times)^2}$, where the characteristic time $T_\times$ is given by $T_\times^{-1}=\gamma_e\sigma_{B_\perp}$, regardless of the means $\langle\delta B_z\rangle$ and $\langle B_\perp\rangle$ (see inset in Fig. \[Fig:C12\]d). In other words, the damping of the 1024 G oscillations is mainly caused by the fluctuations in the transverse magnetic field. It is interesting to link this damping time $T_\times$ to the dephasing time $T_2^*$ measured in a free induction decay (FID) experiment [@Hall2014; @Maze2012]. Under the same assumptions, the FID envelope takes the form $e^{-(t/T_2^*)^2}$ where $(T_2^*)^{-1}=\gamma_e\sigma_{B_z}/\sqrt{2}$. Moreover, a bath of randomly placed spins around the NV centre leads to $\sigma^2_{B_\perp}\approx\frac{3}{4}\sigma^2_{B_z}$ on average [@Hall2014], which gives the relation $$\begin{aligned}
T_\times \approx T_2^*\sqrt{\frac{2}{3}}.\end{aligned}$$ For the NV centre studied in Fig. \[Fig:C12\], the damping time of the 1024 G oscillations is $T_\times\approx5-10~\mu$s, estimated from the curves shown in Fig. \[Fig:C12\]d, hence $\sqrt{\frac{3}{2}}T_\times\approx6-12~\mu$s. This is significantly shorter than the dephasing time $T_2^*>50~\mu$s. We attribute the discrepancy mainly to drifts in the magnetic field applied during the measurements, which leads to overestimating the damping rate $1/T_\times$.
We now consider the case where the magnetic field is aligned along the NV axis, i.e. $\langle B_\perp\rangle=0$. At the crossing when $\delta B_z = 0$, the averaged population $\langle P_0(t)\rangle$ does not oscillate but still decays with a characteristic time $T_\times$ (see inset in Fig. \[Fig:C12\]d). However, the amplitude of the decay decreases as $\delta B_z$ is increased. We define the width of the 1024 G feature, denoted $\Delta B_z^\times$, as twice the detuning $\delta B_z$ to apply to obtain a maximum population drop of 20%. We find numerically that $\Delta B_z^\times\approx 4\sigma_{B_\perp}$, which can also be expressed as a function of $T_2^*$ according to $$\begin{aligned}
\label{Eq:width}
\gamma_e\Delta B_z^\times\approx \frac{4}{T_\times}\approx \frac{2\sqrt{6}}{T_2^*}.\end{aligned}$$ For the NV studied here, we predict a width $\Delta B_z^\times<1~\mu$T for a perfectly aligned background field. We note however that in the measurements of Fig. \[Fig:C12\]c, the width is instead given by the residual transverse field ($B_\perp\approx0.3$ G in Fig. \[Fig:C12\]c), which could not be maintained to significantly smaller values for extended periods of time due to drifts in the applied magnetic field.
The observation of coherent oscillations at the GSLAC suggests a direct application to DC magnetometry. Indeed, the frequency of the oscillation is directly proportional to the strength of the transverse field according to Eq. (\[Eq:osc\]), assuming $\delta B_z\ll B_\perp$. For photon shot noise limited measurements, the magnetic sensitivity is similar to that obtained by FID measurements [@Balasubramanian2009], with the advantage of being an all-optical technique (no microwave or RF field is required).
![(a) PL scan across the GSLAC for a shallow [$^{14}$NV]{} centre in a CVD diamond, with wait times $t=2~\mu$s, $10~\mu$s and 1 ms. Here no oscillation is detected but an overall decrease in spin population is observed at $B_z\approx1024$ G. (b) Time traces recorded at the crossing feature ($B_z\approx1024$ G, blue dots) and away from it ($B_z\approx1020$ G, grey dots). (c) PL scans across the GSLAC recorded with [$^{14}$NV]{} centres in various diamond samples: (i) deep NV in isotopically purified CVD diamond as in Fig. \[Fig:C12\], (ii) deep NV in natural isotopic content CVD diamond, (iii) shallow NV in CVD diamond as in (a,b), (iv) deep NV in type-Ib diamond. The curves are vertically offset from each other for clarity.[]{data-label="Fig:GSLACcomp"}](GSLAC_comp.pdf){width="47.00000%"}
We now compare the spin dynamics of [$^{14}$NV]{} centres at the GSLAC in different diamond samples. Of particular relevance to sensing applications are NV centres implanted close to the diamond surface. We performed measurements of shallow NV centres in a CVD-grown diamond with natural isotopic concentration (\[[$^{13}$C]{}\]$=1.1\%$). The NV centres were created by implantation of N$^+$ ions with an energy of 3.5 keV followed by annealing, resulting in NV centres at a mean depth of 10 nm [@Lehtinen2016]. Fig. \[Fig:GSLACcomp\]a shows a scan across the GSLAC for a particular [$^{14}$NV]{} centre. A reduction in the PL is observed at a field $B_z\approx1024$ G, corresponding to the crossing discussed before. However, full time traces (shown in Fig. \[Fig:GSLACcomp\]b) now reveal a simple offset of the PL at the crossing, with no obvious oscillatory behaviour. This can be understood by the large magnetic noise originating from the surface, which results in a decay time $T_\times$ shorter than the minimum probe time of $t=1~\mu$s (limited by the lifetime of the singlet state [@Manson2006]). The width of the feature in Fig. \[Fig:GSLACcomp\]a is $\approx1$ G (or $\approx3$ MHz). By measuring various shallow NV centres in the same sample, we found a range of widths of the 1024 G feature from 1 to 3 G (or 3 to 9 MHz). This variability is attributed to different local environments, especially because each NV centre sits at a different distance from the surface. For applications to $T_1$-based NMR spectroscopy as proposed in Ref. [@Wood2016], this implies that nuclear spin species with large gyromagnetic ratios such as $^1$H (Larmor frequency $\approx4.4$ MHz at 1024 G) can be resonantly coupled to a shallow [$^{14}$NV]{} such as that measured in Fig. \[Fig:GSLACcomp\]a. However, species with smaller gyromagnetic ratios such as [$^{13}$C]{} (Larmor frequency $\approx1.1$ MHz) are generally within the width of the crossing feature in the present sample, and could therefore hardly be detected via cross-relaxation. This motivates further progress in optimising the coherence properties of shallow NV spins, or devising ways to mitigate the effect of dephasing in $T_1$ measurements.
Finally, we measured the properties of [$^{14}$NV]{} centres at the GSLAC in two other settings: (1) deep [$^{14}$NV]{} centres in a CVD diamond with \[[$^{13}$C]{}\]$=1.1\%$, where decoherence is dominated by the [$^{13}$C]{} bath rather than surface effects; (ii) deep [$^{14}$NV]{} centres in type-Ib diamond grown by the high-pressure high-temperature method, where the main source of decoherence is the bath of electronic spins associated with nitrogen impurities [@Hanson2008]. Example scans across the GSLAC are shown in Fig. \[Fig:GSLACcomp\]c. Deep NVs in CVD diamond showed line widths of the 1024 G feature smaller than $\approx0.3$ G for most NVs ($\approx1$ MHz). By contrast, line widths in the type-Ib diamond are of the order of 10-20 MHz, which makes such diamonds unsuited to $T_1$-based NMR spectroscopy.
[$^{15}$NV]{} centres
---------------------
![(a) PL scan across the GSLAC for a shallow [$^{15}$NV]{} centre in a CVD diamond, with wait times $t=2~\mu$s, $10~\mu$s and 3 ms. A feature at $B_z\approx1024$ G is observed, attributed to spin mixing induced by transverse magnetic fields. (b,c) Time traces recorded at the crossing feature ($B_z\approx1024$ G, orange dots) and away from it ($B_z\approx1020$ G, blue dots). The long and short time scales are shown in (b) and (c), respectively. (d) Energy level structure of the [$^{15}$NV]{} centre in the presence of a transverse magnetic field $B_\perp=0.3$ G, showing an induced avoided crossing (indicated by the arrow).[]{data-label="Fig:N15dynamics"}](N15.pdf){width="47.00000%"}
As previously discussed, the energy structure of the [$^{15}$NV]{} centre at the GSLAC precludes it from accessing transition frequencies below about 2 MHz. Although this limits the range of nuclear spin species that could be resonantly coupled to the [$^{15}$NV]{}, the highly relevant $^1$H remains accessible. It is therefore important to test the ability to measure the $T_1$ of [$^{15}$NV]{} centres near the GSLAC.
As in the [$^{14}$NV]{} case, we recorded PL scans across the GSLAC with different wait times, $t$. For this study the measurements were performed on shallow [$^{15}$NV]{} centres in a CVD diamond only, as this is the most relevant sample for sensing applications. The implantation energy was 3.5 keV and the concentration of [$^{13}$C]{} is 1.1%, similar to the diamond used in Fig. \[Fig:GSLACcomp\]a. Fig. \[Fig:N15dynamics\]a shows a scan obtained for a particular [$^{15}$NV]{} centre. The spin population remains essentially constant across the GSLAC, except at a magnetic field $B_z\approx1024$ G where a sharp change is observed. Time traces at and away from the feature are shown in Figs. \[Fig:N15dynamics\]b and \[Fig:N15dynamics\]c. While the long time scale reveals an exponential decay with a characteristic time $T_1\approx2$ ms independent from the magnetic field (Fig. \[Fig:N15dynamics\]b), the contrast of the decay is significantly smaller at $B_z\approx1024$ G. This is due to the initial population being lower, as can be seen from the drop of signal at short time scales (Fig. \[Fig:N15dynamics\]c). This 1024 G feature was consistently seen in most NVs investigated, exhibiting a variety of amplitudes and widths. At this field, the dominant NV transition has a frequency of $\approx4.3$ MHz. This is 1 G beyond the 1023 G crossing observed in the ODMR (see Fig. \[Fig:ODMR\]), when the NV transition frequency is $\omega'_\times\approx2.65$ MHz.
To understand this 1024 G feature, we consider the energy level structure shown in Fig. \[Fig:intro\]d. As mentioned before, under optical pumping near the GSLAC the [$^{15}$NV]{} centre is efficiently polarised in the state $|0,+1/2\rangle$. This state crosses the state $|-1,-1/2\rangle$ precisely at 1024 G. These two states cannot be coupled directly by a transverse magnetic field because they have distinct nuclear spin projections. However, they are indirectly coupled to each other via transverse-field-enabled coupling to the other two hyperfine-mixed states, which are superpositions of $|0,-1/2\rangle$ and $|-1,+1/2\rangle$. This is illustrated in Fig. \[Fig:N15dynamics\]d, which shows the computed energy levels as a function of $B_z$ in the presence of a finite transverse field, here $B_\perp=0.3$ G. The transverse field opens a gap between $|0,+1/2\rangle$ and $|-1,-1/2\rangle$ at 1024 G. As a consequence, they become mixed states which can give rise to coherent spin oscillations since optical pumping initialises the NV in the $|0,+1/2\rangle$ state. This situation is reminiscent of the [$^{14}$NV]{} case, where the 1024 G feature was due to an avoided crossing between two states coupled via a transverse magnetic field. The main difference here is that the coupling is indirect, mediated by two intermediate states. In the presence of magnetic noise, the coherent oscillations between the two coupled states are expected to be averaged out and appear as a decrease of the initial spin population, as we observed experimentally in this sample (Fig. \[Fig:N15dynamics\]c). We note that a transverse-field-induced coupling also occurs at $B_z\approx1027$ G, between $|0,+1/2\rangle$ and ${\ensuremath{\left| \beta_+ \right>}}\approx|0,-1/2\rangle$ (see Fig. \[Fig:intro\]d). This coupling explains why the dynamic nuclear spin polarisation becomes ineffective around this field, as discussed in section \[Sec:ODMR\] (see Fig. \[Fig:ODMR\]).
An unfortunate consequence of the 1024 G feature of the [$^{15}$NV]{} centre is that resonant coupling with a $^1$H spin would normally occur very close to 1024 G, since the Larmor frequency of $^1$H is $\approx4.36$ MHz at this field. Therefore, any signature of [$^{15}$NV]{}-$^1$H coupling would be overwhelmed by this strong intrinsic feature. It should be noted however that cross-relaxation resonances with nuclear spins should occur on both sides of the GSLAC [@Wood2016], so that $^1$H can still be detected before the GSLAC, at a magnetic field $B_z\approx1022$ G. Moreover, improving the coherence properties (i.e., reducing the noise) of shallow NV centres should significantly reduce the width and amplitude of the 1024 G feature.
All-optical magnetic noise spectroscopy
=======================================
![PL scans across the GSLAC recorded with a wait time $t=10~\mu$s on the same [$^{14}$NV]{} centre as in Fig. \[Fig:C12\]. For each scan, a magnetic noise was generated at a central frequency of 8 MHz, 4 MHz, 1 MHz and 500 kHz, respectively (from top down). The root-mean-square amplitude of the applied field is 1 $\mu$T. The curves are offset from each other for clarity.[]{data-label="Fig: noise"}](Noise_musurements.pdf){width="47.00000%"}
By scanning the magnetic field across the GSLAC, we have shown that the transition frequency of the [$^{14}$N]{} centre can be tuned down to frequencies as low as 100 kHz in diamond samples with low intrinsic magnetic noise. This is approximately an order of magnitude below the transition frequencies exhibited by nuclear species (e.g., [$^{13}$C]{}) at 1024 G, thus opening the possibility to perform all-optical NMR spectroscopy by detecting cross-relaxation events between a probe NV spin and target nuclear spins [@Wood2016]. When the NV transition frequency is matched to the nuclear Larmor frequency, the fluctuating nuclear field would cause the NV spin to relax faster, translating into a decreased longitudinal relaxation time, $T_1$.
In order to test the possibility of detecting fluctuating magnetic fields near the GSLAC, we generated a local magnetic field by running an oscillating current through a wire placed in proximity to the diamond. To mimic nuclear spin detection, we applied signals at various frequencies: 8 MHz, 4 MHz ($\sim$ $^1$H or $^{19}$F), 1 MHz ($\sim$ [$^{13}$C]{}), and 500 kHz ($\sim$ $^2$H or $^{17}$O). The alternating current was modulated in amplitude and phase to ensure that the NV is not coherently driven but experiences noise from a randomly fluctuating current around a given frequency, similar to the signal from a possible target nuclear spin. The amplitude of the current was adjusted to obtain a root-mean-square field strength of 1 $\mu$T, which corresponds approximately to the field generated by a dense organic sample of nuclear spins located at a 5 nm stand-off distance [@Staudacher2013]. The probe time was set to $t=10~\mu$s to maximise the PL contrast. The resulting spectra (PL as a function of $B_z$), measured on a deep [$^{14}$N]{} centre in an isotopically-purified CVD diamond, are shown in Fig. \[Fig: noise\]b. While the 8 MHz, 1 MHz and 500 kHz detections are clear on both sides of the 1024 G crossing, the 4 MHz detection is weaker before the GSLAC. This is due to the NV transition being very weak in this region because of spin mixing (associated with an avoided crossing), as discussed previously (see Fig. \[Fig:ODMR\]). Past the GSLAC, however, there is no issue measuring any of the signals and thus, NMR spectroscopy would be possible in this region for most commonly found nuclear spin species. We note that the width of the resonances is governed here by the amplitude of the applied field (1 $\mu$T) through power broadening. Weaker signals will produce narrower lines, down to the limit of spectral resolution imposed by spin dephasing, characterised by $T_2^*$ [@Hall2016]. Reaching this limit was not possible in our experiment due to limited precision and stability of the applied magnetic field. This could be improved by using, e.g., an electromagnet [@Wickenbrock2016].
Conclusions
===========
In this work we have investigated the photo-induced spin dynamics of NV centres near the GSLAC. For the [$^{14}$NV]{} centre, the spin transition frequency can be tuned down to values as low as 100 kHz in high purity diamond. At the crossing (1024 G), we observe coherent spin oscillations caused by spin mixing induced by residual transverse magnetic fields. This, in turn, limits the minimum accessible transition frequency exhibited by the environment. Measurements with shallow [$^{14}$NV]{} centres showed that frequencies compatible with nuclear spin signals (1-5 MHz) are within reach. For the [$^{15}$NV]{} centre, the minimum transition frequency practically accessible is of order 2 MHz, governed by the avoided crossing intrinsic to the [$^{15}$NV]{} hyperfine structure. The [$^{15}$NV]{} also exhibits a crossing feature at 1024 G, which is induced by transverse magnetic fields via an indirect hyperfine-mediated process. With this detailed understanding of the low frequency spin dynamics around the GSLAC, we have demonstrated all-optical spectroscopy of externally-generated magnetic noise with frequencies ranging from 8 MHz down to 500 kHz, mimicking signals produced by precessing nuclear spins. This work thus paves the way towards all-optical, nanoscale NMR spectroscopy via cross-relaxation.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank L. McGuinness for experimental assistance with the diamond samples. This work was supported in part by the Australian Research Council (ARC) under the Centre of Excellence scheme (project No. CE110001027). L.C.L.H. acknowledges the support of an ARC Laureate Fellowship (project No. FL130100119).
|
IJS-TP-95/13\
NUHEP-TH-95-12\
September 1995\
[**$c\to u \gamma$ in Cabibbo suppressed D meson radiative weak decays\
**]{}
[**B. Bajc $^{a}$, S. Fajfer $^{a}$ and Robert J. Oakes $^{b}$\
**]{}
[*a) J. Stefan Institute, University of Ljubljana, 61111 Ljubljana, Slovenia\
*]{}
[*b) Department of Physics and Astronomy, Northwestern University, Evanston, Il 60208 U.S.A.*]{}
**ABSTRACT**
We investigate Cabibbo suppressed $D^{0}$, $D^{+}$ and $D^{+}_{s}$ radiative weak decays in order to find the best mode to test $c\to u \gamma$ decay. Combining heavy quark effective theory and the chiral Lagrangian approach we determine the decay widths. We calculate $\Gamma(D^{0}\to \rho^{0}/\omega\gamma)/
\Gamma(D^{0}\to {\bar K}^{*0} \gamma)$ previously proposed to search for possible New Physics. However, we notice that there are large, unknown, corrections within the Standard Model. We propose a better alternative, the ratio $\Gamma(D_{s}^{+}\to K^{*+} \gamma)/
\Gamma(D_{s}^{+}\to \rho^{+} \gamma)$, and show that it is less sensitive to the Standard Model.
[**1 Introduction**]{}
According to the Standard Model, the physics of charm mesons is not as exciting as the physics of bottom mesons [@BIGI1; @BIGI2; @BIGI3]: the relevant CKM matrix elements $V_{cs}$ and $V_{cd}$ are well known, the $D^{0} -{\bar D}^{0}$ oscillations and CP asymmetries are small, weak decays of D mesons are difficult to investigate due to the strong final state interactions, and very small branching ratios are expected for rare decays. However, authors [@BIGI1; @BIGI2; @BIGI3] have noticed that the $D^{0} -{\bar D}^{0}$ oscillation and $c \to u \gamma$ decays obtain contributions coming from non-minimal supersymmetry which are not present within the Standard Model. Therefore, these two phenomena might be guides for a signal of New Physics. In ref. [@BIGI2] it was observed that New Physics can generate $c\to u \gamma$ transitions leading to a deviation from
$$\label{r1}
R_{\rho/\omega}\equiv{\Gamma(D^0 \to \rho^0 /\omega \gamma)
\over\Gamma(D^0 \to {\bar K}^{*0} \gamma)}=
{tan^{2} \theta_{c}\over 2}$$
(the factor $\frac{1}{2}$ was overlooked in refs. [@BIGI1; @BIGI2]). Motivated by the importance of this signal we investigate Cabibbo suppressed radiative weak decays in which $c \to u \gamma$ transition occurs. As a theoretical framework we use a hybrid theory: a combination of heavy quark effective theory (HQET) and chiral Lagrangians (CHL) [@MW; @BD; @YCC; @G2; @BFO]. This approach, accompanied by the factorization hypothesis, enables us to use the Standard Model results for electroweak processes. It is possible to apply other approaches like for example [@burdman], but the result which indicates the deviation from $tan^{2} \theta_{c}$ cannot be very different from ours obtained with HQET + CHL. In fact, our results agree with \[9\] within the uncertainties.
We calculate the ratios between various Cabibbo suppressed and Cabibbo allowed charm meson radiative weak decays. Analysing them we notice that the relation (\[r1\]) can be badly violated already in the Standard Model framework, while a similar relation for $D_{s}^{+}$ radiative decays, i.e.
$$\label{r2}
R_{K}\equiv{\Gamma(D_s^+ \to K^{*+} \gamma)
\over\Gamma(D_s^+ \to \rho^+ \gamma)}=
tan^{2} \theta_{c}$$
offers a much better test for $c \to u \gamma$.
The paper is organised as follows: in Sect. 2 we sketch the relevant theoretical framework for radiative decays; in Sect. 3 we give results for the branching ratios of the widths for Cabibbo suppressed and Cabibbo allowed radiative decays; finally we make a short discussion of our results in Sect. 4.
[**2 Theoretical framework**]{}
Experimentally radiative decays of $D$ mesons have not yet been measured, while the known branching ratios of $D^{*}$ radiative decays [@pdg; @expr] can be described using the combination HQET + CHL [@BFO; @CG; @ABJ].
The initial HQET ideas [@IW; @HG] were implemented with the chiral Lagrangian formalism for light pseudoscalar mesons first in [@MW; @BD; @YCC], and the electromagnetic interaction included in [@CG; @ABJ; @LLY]. Consequently, the light vector mesons were introduced [@G2], following the hidden symmetry approach [@BKY]. We will follow the model described in [@BFO], where in addition to [@G2] the electromagnetic (EM) interaction was introduced.
Let us briefly describe the relevant terms (for the charm meson radiative weak decays) of the Lagrangian [@BFO]. The main contribution comes from the odd-parity Lagrangian
$$\begin{aligned}
\label{odd}
{\cal L}_{odd} = &-&4 e {\sqrt 2}
\frac{C_{V\pi\gamma}}{f} \epsilon
^{\mu \nu \alpha \beta}
Tr (\{ \partial_{\mu}{\rho}_{\nu},
\Pi \} Q \partial_{\alpha} B_{\beta})\nonumber\\
&-&4 \frac{C_{VV\Pi}}{f} \epsilon
^{\mu \nu \alpha \beta}Tr (\partial_{\mu}
{\rho}_{\nu} \partial_{\alpha}{\rho}_{\beta} \Pi)\nonumber\\
&-& \lambda^{\prime} e Tr [H_{a}\sigma_{\mu \nu}
F^{\mu \nu} (B) {\bar H_{a}}]\nonumber\\
&+& i {\lambda} Tr [H_{a}\sigma_{\mu \nu}
F^{\mu \nu} (\hat \rho)_{ab} {\bar H_{b}}]\;,\end{aligned}$$
where $C_{VV\Pi} = 0.423$, $C_{V\Pi\gamma} = -3.26\times
10^{-2}$[@BOS; @FSO], $f=132$ MeV is the pseudoscalar decay constant, while the phenomenological parameters $\lambda$ and $\lambda'$ are constrained by the analysis [@BFO]:
$$\begin{aligned}
\label{lambda}
|\lambda'+{2\over 3}\lambda | & = &(0.50 \pm 0.15)
\;\hbox{GeV}^{-1}\;,\\
|\lambda'-{\lambda\over 3}| & < & 0.19
\;\hbox{GeV}^{-1}\;.\end{aligned}$$
In (\[odd\]) $\Pi$ and $\rho_{\mu}$ are the usual pseudoscalar and vector Hermitian matrices
$$\begin{aligned}
\label{pseudoscalar}
\Pi = \pmatrix{
{\pi^0\over\sqrt{2}}+{\eta_8\over\sqrt{6}}+
{\eta_0\over\sqrt{3}}&\pi^+&K^+\cr
\pi^-&-{\pi^0\over\sqrt{2}}+{\eta_8\over\sqrt{6}}+
{\eta_0\over\sqrt{3}}&K^0\cr
K^-&{\bar K^0}&-2{\eta_8\over\sqrt{6}}+
{\eta_0\over\sqrt{3}}\cr}\;,\end{aligned}$$
$$\begin{aligned}
\label{vector}
\rho_\mu = \pmatrix{
{\rho^0_\mu + \omega_\mu \over \sqrt{2}} & \rho^+_\mu & K^{*+}_\mu \cr
\rho^-_\mu & {-\rho^0_\mu + \omega_\mu \over \sqrt{2}} & K^{*0}_\mu \cr
K^{*-}_\mu & {\bar K^{*0}}_\mu & \Phi_\mu \cr}\end{aligned}$$
with $\eta=\eta_8\cos{\theta}-\eta_0\sin{\theta}$, $\eta'=\eta_8\sin{\theta}+\eta_0\cos{\theta}$ and $\theta=-23^o$ [@pdg] is the $\eta-\eta'$ mixing angle. $Q=diag(2/3,-1/3,-1/3)$ is the light quark sector charge matrix,
$$\label{defha}
H_a={1 \over 2} (1 + \!\!\not{\! v}) (\sqrt{m_{D^{a*}}}
D_\mu^{a*}\gamma^{\mu} - \sqrt{m_{D^a}} D^a \gamma_{5})\;,$$
where $D_\mu^{a*}$ and $D^a$ annihilate, respectively, a spin-one and spin-zero meson $c{\bar q}^{a}$ ($q^a=u$, $d$ or $s$) of velocity $v_{\mu}$ and ${\bar H}_a\equiv\gamma^0H_a^\dagger\gamma^0$. Finally, $F_{\mu\nu}({\hat \rho})=
\partial_\mu{\hat\rho}_\nu-
\partial_\nu{\hat\rho}_\mu+
[{\hat\rho}_\mu,{\hat\rho}_\nu]$, ${\hat\rho}_\mu=ig_V\rho_\mu/\sqrt{2}$ with $g_V=5.8$ [@G2], and $F_{\mu\nu}(B)=\partial_\mu B_\nu-
\partial_\nu B_\mu$ with $B_\mu$ being the photon field with the EM coupling constant $e$.
The first (third) term in (\[odd\]) describes the anomalous direct emission of the photon by the light (heavy) meson, while the second (fourth) term, together with the vector meson dominance (VMD) coupling
$$\begin{aligned}
\label{vmd}
{\cal L}_{V-\gamma} & = & - m_V^2 {e \over g_V} B_{\mu}
(\rho^{0\mu} + {1 \over 3} \omega^{\mu} -
{\sqrt{2} \over 3} \Phi^{\mu})\end{aligned}$$
describes a two step photon emission, with an intermediate neutral vector meson with mass $m_V$ which transforms to the final photon.
A charged charm meson can emit a real photon also through the usual electromagnetic coupling
$$\label{emheavy}
{\cal L}_{EM}=-e v^\mu B_\mu Tr[H_a(Q-2/3)_{ab}{\bar H}_b]\;,$$
while a charged light vector meson can produce through
$$\label{vvv}
{\cal L}_{VVV}={1\over 2 g_V^2}tr[
F_{\mu\nu}({\hat\rho})
F^{\mu\nu}({\hat\rho})]$$
first a neutral vector meson, which subsequently transforms via VMD (\[vmd\]) to a photon.
The weak Lagrangian is approximated by the current-current type interaction
$$\label{fermi}
{\cal L}_{W}^{eff}(\Delta c = 1) =
-{G_F\over\sqrt{2}}
[a_{1}({\bar u}d')^\mu({\bar s}'c)_\mu+
a_{2}({\bar s}'d')^\mu({\bar u}c )_\mu]\;,$$
where $({\bar q}_1 q_2)^\mu\equiv
{\bar q}_1\gamma^\mu(1-\gamma^5)q_2$, $G_F$ is the Fermi constant and $a_{1,2}$ are the QCD Wilson coefficients, which depend on the energy scale $\mu$. One expects the scale to be the heavy quark mass and we take $\mu \simeq 1.5$ GeV which gives $a_{1} = 1.2$ and $a_{2} = -0.5$, with an approximate $20\%$ error. Since we are interested only in the physics of the first two generations, we can express the weak eigenstates $d'$, $s'$ with the mass eigenstates $d$, $s$ using the Cabibbo angle instead of the CKM matrix:
$$\begin{aligned}
\label{dstransf}
\pmatrix{ d' \cr
s' \cr } =
\pmatrix{ \cos{\theta_c} & -\sin{\theta_c} \cr
\sin{\theta_c} & \cos{\theta_c} \cr }
\pmatrix{ d \cr
s \cr }\end{aligned}$$
with $\sin{\theta_c}=0.222$. Possible contributions caused by the penguin type diagrams are found to be very small [@burdman].
Many heavy meson weak nonleptonic amplitudes [@KXC; @WSB1; @G4] have been calculated using the factorization approximation. In this approach the quark currents are approximated by the “bosonised" currents [@MW; @G2; @BFO], of which only
$$\begin{aligned}
\label{ethirty}
({\bar q}^a c)_\mu&=&i(m_{D^{*a}}f_{D^{*a}}D^{*a}_\mu-
m_{D^a}f_{D^a}v_\mu D^a)\;,\\
({\bar q}_b q_a)^{\mu}&=&-f\partial^\mu\Pi_{ab}+
m_V f_V\rho^\mu_{ab}\end{aligned}$$
will contribute to our amplitudes. The numerical values for the masses will be taken from [@pdg] and for the decay constants from [@G4].
It is now straightforward to calculate the decay widths. The result, of course, depends on the numerical values we take for $(\lambda' +
2 \lambda /3)$ and $(\lambda'-\lambda/3)$.
[**3 Cabibbo suppressed radiative weak decays in HQET + CHL**]{}
Apart from the Cabibbo allowed decays $D^0\to{\bar K}^{*0}\gamma$ and $D_s^+\to\rho^+\gamma$, five once Cabibbo suppressed ($D^{0}\to\rho^{0}\gamma$, $D^{0}\to\omega\gamma$, $D^{0}\to\phi\gamma$, $D^{+}\to\rho^{+} \gamma$, $D_s^{+}\to K^{*+} \gamma$) and two doubly Cabibbo suppressed ($D^0\to K^{*0}\gamma$ and $D^+\to K^{*+}\gamma$) decays are possible.
According to [@WSB1] the weak amplitudes can be categorized into two groups: quark decays and weak annihilations. As these authors have noticed, the factorization works much better for the quark decays. The decays of $D_s^+$ and $D^+$ are of the quark decay type (their amplitudes are proportional to $a_1$, see below), and therefore the results for these decays are more trustworthy than for the $D^0$ $D$ decays, which proceed through weak annihilation diagrams (proportional to $a_2$).
We write the amplitude for the $D^{q} \to V^{q} \gamma$, where $q$ stands for the charge of D meson ($q= 1$ stands for $+$ charge, while $q = 0$ is for the neutral D mesons)
$$\begin{aligned}
\label{ampl}
A(D^{q} &\to &V^{q} \gamma) =
e{G_F\over\sqrt 2} K_c a(q)
[ C^{(1)}_{D V \gamma} \epsilon _{\mu \nu \alpha \beta}
k^{\mu} \epsilon_{\gamma}^{\nu *} v^{\alpha} \epsilon_{V}^{\beta *}
\nonumber\\
& + &iC^{(2)}_{D V \gamma} m_{V} ( \epsilon_{\gamma}^{*}
\cdot \epsilon_{V}^{*} - \frac{\epsilon_{\gamma}^{*} \cdot p_{V}
\epsilon_{V}^{*}\cdot k}
{p_{V} \cdot k} )]\end{aligned}$$
with $a(+1)=a_{1}$ and $a(0)=a_{2}$. $(k,\epsilon_\gamma)$ and $(p_V,\epsilon_V)$ are the $4$-momenta and polarization vectors of the photon and vector meson respectively, while $v$ is the $4$-velocity of the heavy meson.
The overall factor $K_c$ contains the Cabibbo angle and is equal to $\cos{\theta_c}^2$ for allowed decays, to $+\sin{\theta_c}\cos{\theta_c}$ (when there is no $s$ quark or antiquark in the final $V$) or $-\sin{\theta_c}\cos{\theta_c}$ (when there is at least one $s$ quark or antiquark in the final $V$) for once suppressed decays and to $-\sin{\theta_c}^2$ for double suppressed decays. The coefficients $C^{(i)}$ in (\[ampl\]) can be written as
$$\begin{aligned}
\label{c1}
C^{(1)}_{D V \gamma} & = &
( C_{VV\Pi} \frac{1}{g_{V}} + C_{V \Pi \gamma})
4 {\sqrt 2} f_{D} m_{D}^{3} b^V\nonumber\\
&+ &4 [\lambda^{\prime} + (\frac{2}{3} -q) \lambda] f_{D^{*}} f_{V}
\frac{m_{D^{*}}m_{V}}{m_{D^{*}}^{2} - m_{V}^{2}}
\sqrt{m_{D}m_{D^{*}}}b_0^V\;,\\
C^{(2)}_{D V \gamma} & = & q f_D f_V\;.\end{aligned}$$
The coefficient $b^V$ is equal to $(2/3-q)/(m_D^2-m_P^2)$ for $V=({\bar K}^{*0}$, $K^{*0}$, $\rho^+$, $K^{*+})$, for which $P=({\bar K}^0$, $K^0$, $\pi^+$, $K^+)$. For the remaining final state vector mesons this coefficient is expressed as
$$\label{defbv}
b^V=\sum_{i=1}^{3} \frac{b^{VP_i}}{m_{D}^{2} - m_{P_{i}}^{2}}\;,$$
where the pole coefficients $b^{VP_i}$ are given in Table 1. Furthermore we have $b_0^V=-1/\sqrt{2}$ for $V=\rho^0$, $b_0^V=1/\sqrt{2}$ for $V=\omega$ and $b_0^V=1$ otherwise.
In ref. [@BIGI1; @BIGI2] it was noticed that a nice bonus can be obtained by measuring the charm meson decay width $D \to \rho / \omega \gamma$ which is generated by $c\to u \gamma$ transitions. Namely, the authors claim that observing the violation of equation (\[r1\]) would then eventually signal New Physics [@BIGI1]. Using our model, which pretends to describe the low energy meson physics within the Standard Model, we find that this relation does not exactly hold due to U(3) breaking. We assume that the leading effect of this breaking is to change the values of the masses and decay constants for different members of the same multiplets and between octet and singlet. However, one would naively expect deviations from this limit in the Standard Model of the order of $20-30\%$. We will see that this is not true for the $D^0\to\rho^0/\omega\gamma$ decay, but it is correct for $D_s^+$ Cabibbo suppressed radiative weak decay.
Within our framework $\lambda$ and $\lambda^{\prime}$ are the most important parameters for charm meson radiative decays [@BFO], and therefore we present the ratios of the decay widths as functions of combinations of $\lambda$ and $\lambda^{\prime}$. Our result for $R_\rho$ (\[r1\]) is showed on Fig. 1: if the combination of $(\lambda^{\prime} + \frac{2}{3} \lambda)$ turns out to be negative, the ratio $R_\rho$ can approach $0$. As it is known from $D^{0} \to {\bar K}^{*0} \gamma$ [@BFO], the negative values $(\lambda^{\prime} + \frac{2}{3} \lambda)$ cause a destructive interference between the photon emission by the heavy meson and the photon emission by the light meson. A similar effect is possible also in the decay $D^{0} \to \rho^0 \gamma$, only that the $0$ is achieved at a different value of $(\lambda^{\prime} + \frac{2}{3} \lambda)$, because the model parameters are here slightly different due to $U(3)$ breaking. It is obvious that such a large sensitivity to the model parameters does not allow us to conclude anything about some New Physics. If $(\lambda^{\prime} + \frac{2}{3} \lambda)$ turns out to be positive, the decays are much easier to detect experimentally, and also the theoretical situation is clearer, since the curve is approaching the ideal theoretical value. A large disagreement with the theoretical prediction (\[r1\]) would give in this case some sign of New Physics. But even here one should be careful, since in this case the amplitudes are approximately proportional to the decay constants of the final vector meson. This can be seen, if we calculate the decay $D^{0} \to \omega \gamma$ with the values of the light vector decay constants taken from [@G4] : $f_{K*} = f_{\rho} = 221$ MeV and $f_{\omega} = 156$ MeV. In this case we get for $R_\omega$ a similar curve as in Fig. 1, but for large positive values $(\lambda^{\prime} + \frac{2}{3} \lambda)$ the ratio is approaching a value of approximately $0.5$ instead of 1. The fact can be explained by the difference in the decay constants, i.e. $(f_{\omega}/f_{K*})^{2} \simeq 0.5$.
The ratio $\Gamma(D^0\to\Phi\gamma)/\Gamma(D^0\to
{\bar K}^{*0}\gamma)$ would indicate the deviation from $\tan{\theta_c}^2$ instead of $\tan{\theta_c}^2/2$ like for the $\rho,\omega$ case. When calculated, it exhibit a similar behaviour like $D^0\to\omega\gamma$, and therefore we find it is not useful to understand $c\to u\gamma$ physics.
The decays $D^+\to\rho^+\gamma$ is also not of great importance in our purpose to find New Physics, since the $D^+$ does not have Cabibbo allowed decays.
Contrary to the above cases we find that the decay $D_{s}^{+} \to
K^{*+} \gamma$ offers a much better chance to test New Physics. Using the general formulas for the amplitudes (\[ampl\]) it is easy to derive a deviation from equation (\[r2\]), which is exactly correct only in the U(3) limit. The result for $R_K$ as a function of $(\lambda^{\prime} -\frac{1}{3} \lambda)$ is presented on Fig. 2 (note the changed scale with respect to Fig. 1). We notice that the result is rather stable within the allowed region for $(\lambda^{\prime} -\frac{1}{3} \lambda)$. The discrepancy to relation (\[r2\]) is due to U(3) breaking and is of order $30\%$, as usually expected. If the the experimental results are found to be far away from the curve Fig. 2, one can interpret it as a sign of New Physics.
We point out that it is difficult to observe all these decays. In fact the Cabibbo allowed decays are already rare: the branching ratio for $D^0\to{\bar K}^{*0}\gamma$ is smaller than $0.3\times 10^{-4}$ for $(\lambda'+2\lambda /3)<0$ and around $(2-4)\times10^{-4}$ for $(\lambda'+2\lambda /3)>0$, while for $D_s^+\to\rho^+\gamma$ the branching ratio is around $(2-7)\times 10^{-4}$ [@BFO].
[**4 Conclusions**]{}
We determine amplitudes of Cabibbo suppressed radiative decays using the combination of heavy quark symmetry and chiral symmetry, which builds an effective strong, weak and electromagnetic Lagrangian. This theoretical framework just illustrates the characterictics of these amplitudes in the Standard Model. In our framework two parameters, $\lambda$ and $\lambda^{\prime}$, are not well known. We show the dependence of the ratio between the Cabibbo suppressed and Cabibbo allowed decay widths on the combination of $\lambda$ and $\lambda^{\prime}$. We find that is better to search for a signal of New Physics coming from $c \to u \gamma$ decays from the ratio $\Gamma (D^{+}_{s} \to K^{*+} \gamma)/
\Gamma (D^{+}_{s} \to \rho^{+} \gamma)$ instead of the proposed ratio $\Gamma(D^{0} \to \rho^{0}/\omega \gamma) /
\Gamma(D^{0} \to {\bar K}^{*0} \gamma) $ [@BIGI1; @BIGI2; @BIGI3].
0.5cm [*Acknowledgement.*]{} This work was supported in part by the Ministry of Science and Technology of the Republic of Slovenia (B.B. and S.F.) and by the U.S. Department of Energy, Division of High Energy Physics, under grant No. DE-FG02-91-ER4086 (R.J.O.).
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FIGURES
Fig. 1: The ratio $2 R_{\rho/\omega}/\tan{\theta_c}^2$ as function of the combination $\lambda'+2\lambda/3$. The full (dashed/dot-dashed) lines denote the experimentally allowed (forbidden) values for this combination. In the U(3) symmetry limit of the Standard Model this ratio is equal $1$.
0.5cm Fig. 2: The ratio $R_K/\tan{\theta_c}^2$ as function of the combination $\lambda'-\lambda/3$. The full (dashed) lines denote the experimentally allowed (forbidden) values for this combination. In the U(3) symmetry limit of the Standard Model this ratio is equal $1$.
$\enspace$ $\pi^{0}$ $ \eta$ $\eta^{\prime}$
------------ ----------------------- --------------------------------------- ---------------------------------------
$\rho^0$ $\frac{1}{3\sqrt{2}}$ $ -\frac{1}{\sqrt{2}}c(c-\sqrt{2} s)$ $ -\frac{1}{\sqrt{2}}s(\sqrt{2}c+s)$
$\omega$ $\frac{1}{{\sqrt 2}}$ $ -\frac{1}{3\sqrt{2}}c(c-\sqrt{2}s)$ $ -\frac{1}{3\sqrt{2}}s(\sqrt{2}c+s)$
$\phi$ $0$ $ \frac{\sqrt{2}}{3}c(\sqrt{2}c+s)$ $ -\frac{\sqrt{2}}{3}s(c-\sqrt{2}s)$
: The $b^{VP_i}$ coefficents defined in relation (19), where $s=\sin{\theta}$, $c=\cos{\theta}$ and $\theta$ is the $\eta-\eta'$ mixing angle.
|
---
abstract: |
We outline how the coupled cluster method of microscopic quantum many-body theory can be utilized in practice to give highly accurate results for the ground-state properties of a wide variety of highly frustrated and strongly correlated spin-lattice models of interest in quantum magnetism, including their quantum phase transitions. The method itself is described, and it is shown how it may be implemented in practice to high orders in a systematically improvable hierarchy of (so-called LSUB$m$) approximations, by the use of computer-algebraic techniques. The method works from the outset in the thermodynamic limit of an infinite lattice at all levels of approximation, and it is shown both how the “raw” LSUB$m$ results are themselves generally excellent in the sense that they converge rapidly, and how they may accurately be extrapolated to the exact limit, $m \rightarrow \infty$, of the truncation index $m$, which denotes the [*only*]{} approximation made. All of this is illustrated via a specific application to a two-dimensional, frustrated, spin-half $J^{XXZ}_{1}$–$J^{XXZ}_{2}$ model on a honeycomb lattice with nearest-neighbor and next-nearest-neighbor interactions with exchange couplings $J_{1}>0$ and $J_{2} \equiv
\kappa J_{1} > 0$, respectively, where both interactions are of the same anisotropic $XXZ$ type. We show how the method can be used to determine the entire zero-temperature ground-state phase diagram of the model in the range $0 \leq \kappa \leq 1$ of the frustration parameter and $0 \leq \Delta \leq 1$ of the spin-space anisotropy parameter. In particular, we identify a candidate quantum spin-liquid region in the phase space.
author:
- 'R. F. Bishop'
- 'P. H. Y. Li'
- 'C. E. Campbell'
bibliography:
- 'bib\_general.bib'
title: 'Highly frustrated spin-lattice models of magnetism and their quantum phase transitions: A microscopic treatment via the coupled cluster method'
---
[ address=[School of Physics and Astronomy, Schuster Building, The University of Manchester, Manchester, M13 9PL, UK]{} ]{}
[ address=[School of Physics and Astronomy, Schuster Building, The University of Manchester, Manchester, M13 9PL, UK]{} ]{}
[ address=[School of Physics and Astronomy, University of Minnesota, 116 Church Street SE, Minneapolis, Minnesota 55455, USA]{} ]{}
INTRODUCTION
============
The coupled cluster method (CCM) [@Bishop:1998_QMBT_coll] is one of the most pervasive, most powerful, and most successful of all [ *ab initio*]{} formalisms of quantum many-body theory. It has probably been applied to more systems in quantum field theory, quantum chemistry, nuclear, subnuclear, condensed matter, and other areas of physics than any other competing method. The CCM has yielded numerical results which are among the most accurate available for an incredibly wide range of both finite and extended physical systems defined on a spatial continuum. These range from atoms and molecules of interest in quantum chemistry, where the method has long been the recognized “gold standard”, to atomic nuclei; from the electron gas to dense nuclear and baryonic matter; and from models in quantum optics, quantum electronics, and solid-state optoelectronics to field theories of strongly interacting nucleons and pions.
This widespread success for both finite [@Bartlett:1989_ccm] and extended [@Bishop:1991_TheorChimActa_QMBT] physical systems has led to recent applications to corresponding quantum-mechanical systems defined on an extended regular spatial lattice. Such lattice systems are nowadays the subject of intense theoretical study. They include many examples of systems characterized by novel ground states which display [*quantum order*]{} in some region of the Hamiltonian parameter space, delimited by critical values or [*quantum critical points*]{} (QCPs), which mark the corresponding [*quantum phase transitions*]{}. The quantum critical phenomena often differ profoundly from their classical counterparts, and the subtle correlations present usually cannot easily be treated by standard many-body techniques such as perturbation theory or mean-field approximations.
A key challenge for modern quantum many-body theory has been to develop microscopic techniques capable of handling both these novel and more traditional systems. Our recent work, in the field of quantum magnetism, for example, shows that the CCM is clearly able to bridge this divide. We have shown how the systematic inclusion of multispin correlations for a wide variety of quantum spin-lattice problems can be efficiently implemented with the CCM [@Fa:2004_QM-coll]. The method is not restricted to bipartite lattices or to non-frustrated systems, and can thus deal with problems where many alternative techniques, such as the exact diagonalization (ED) of small lattices or quantum Monte Carlo (QMC) simulations, are faced with specific difficulties.
In this paper we illustrate the current power of the CCM to describe accurately the properties of strongly interacting and highly frustrated spin-lattice models of interest in quantum magnetism, especially in two spatial dimensions. The method itself is first briefly reviewed in Sec. \[model\_sec\], where we demonstrate how it may readily be implemented to high orders in a specific, systematically improvable, hierarchy ([*viz*]{}., a localized lattice-animal-based subsystem, LSUB$m$, scheme) of approximations, by the use of computer-algebraic techniques. In order to demonstrate how values for ground-state (GS) properties are obtained, using the CCM, which are fully competitive with those from other state-of-the-art methods, including the much more computationally intensive QMC techniques in the relatively rare (unfrustrated) cases where the latter can readily be applied, we apply it to a specific model of current interest. The model itself, which is a frustrated spin-half ($s=\frac{1}{2}$) antiferromagnet with nearest-neighbor (NN) $J_{1}>0$ and competing next-nearest-neighbor (NNN) $J_{2}>0$ exchange couplings on the honeycomb lattice, both of the anisotropic $XXZ$ type, is described in Sec. \[ccm\_sec\]. Results for the model are presented in Sec. \[results\_sec\], where we demonstrate the ability of the CCM to give an accurate description of the zero-temperature ($T=0$) GS phase diagram of this model, which contains two independent control parameter, [*viz*]{}., the frustration parameter $\kappa \equiv
J_{2}/J_{1}$, and the spin anisotropy parameter $\Delta$. The raw LSUB$m$ results themselves are shown to be generally excellent, and we demonstrate explicitly both how they converge rapidly and can also be accurately extrapolated in the truncation index to the exact limit, $m
\rightarrow \infty$. We show in Sec. \[phase\_sec\] how the results so obtained may be used to construct an accurate $T=0$ GS phase diagram for this model. Finally, in Sec. \[conclusion\] we present our conclusions.
A HONEYCOMB LATTICE MODEL {#model_sec}
=========================
Low-dimensional spin-lattice models of magnets exhibiting frustration, due either to the underlying lattice geometry or to competing interactions, have been the subject of intense study in recent years, both at the theoretical level and via their experimental realizations either in real materials or in ultracold atoms trapped in optical lattices. Their $T=0$ GS phase diagrams often differ profoundly from their classical ($s \rightarrow \infty$) counterparts, exhibiting, for example, such states without magnetic order as various valence-bond crystalline (VBC) phases or quantum spin-liquid (QSL) states.
Since quantum fluctuations of the order parameter destroy long-range order and hence prevent most types of continuous symmetry breaking in one-dimensional (1D) systems, even at $T=0$, 2D systems occupy a special role for studying QPTs. Since quantum fluctuations are generally weaker for higher values of the spin quantum number $s$, systems with $s=\frac{1}{2}$ typically exhibit the biggest differences from classical behavior. Furthermore, of all regular 2D lattices, one with the lowest coordination number, $z=3$, is the honeycomb lattice. Thus, good [*prima facie*]{} candidate systems for exhibiting novel behavior are spin-half models on the honeycomb lattice, and as a specific model that exhibits both frustration and anisotropy (in spin space), we consider here the so-called $J^{XXZ}_{1}$–$J^{XXZ}_{2}$ model [@Li:2014_honey_XXZ]. It is shown schematically in Fig.\[model\](a) and its Hamiltonian is given by
$$H = J_{1}\sum_{\langle i,j \rangle}(s^{x}_{i}s^{x}_{j}+s^{y}_{i}s^{y}_{j}+\Delta s^{z}_{i}s^{z}_{j}) + J_{2}\sum_{\langle\langle i,k \rangle\rangle}(s^{x}_{i}s^{x}_{k}+s^{y}_{i}s^{y}_{k}+\Delta s^{z}_{i}s^{z}_{k})\,, \label{H_honey_XXZ}$$
where $\langle i,j \rangle$ and $\langle\langle i,k \rangle\rangle$ denote NN and NNN pairs of spins respectively, and the respective sums count each bond once and once only; and ${\bf
s}_{i}=(s^{x}_{i},s^{y}_{i},s^{z}_{i}$) is the $s=\frac{1}{2}$ spin operator on the $i$th site of the honeycomb lattice. We shall be interested in the thermodynamic limit of an infinite lattice ($N
\rightarrow \infty$, where $N$ is the number of lattice sites).
The model of Eq. (\[H\_honey\_XXZ\]) interpolates continuously between the two cases where both NN and NNN exchange couplings have either an isotropic Heisenberg ($XXX$) form when $\Delta=1$ or an isotropic $XY$ ($XX$) form when $\Delta=0$. We shall be interested in the case where both bonds are antiferromagnetic in nature (i.e., when $J_{1}>0$ and $J_{2}>0$), so that they act to frustrate one another. With no further loss of generality we henceforth put $J_{1} \equiv 1$ to set the overall energy scale, and we study the model in the range $0 \leq \kappa \leq 1$ of the frustration parameter $\kappa \equiv
J_{2}/J_{1}$, and $0 \leq \Delta \leq 1$ of the spin anisotropy parameter. Although both limiting isotropic $s=\frac{1}{2}$ models on the honeycomb lattice have been well studied in the past (see, Refs.[@Varney:2011_honey_XY; @Varney:2012_honey_XY; @Zhu:2013_honey_XY; @Carrasquilla:2013_honey_XY; @Ciolo:2014_honey_XY; @Oitmaa:2014_honey_XY; @Bishop:2014_honey_XY] for the $\Delta=0$ $XX$ model and Refs.[@Rastelli:1979_honey; @Fouet:2001_honey; @Mulder:2010_honey; @Ganesh:2011_honey; @Clark:2011_honey; @Albuquerque:2011_honey; @Mosadeq:2011_honey; @Oitmaa:2011_honey; @Mezzacapo:2012_honey; @Li:2012_honey_full; @Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC; @Ganesh:2013_honey_J1J2mod-XXX; @Zhu:2013_honey_J1J2mod-XXZ; @Gong:2013_J1J2mod-XXX; @Yu:2014_honey_J1J2mod-XXZ] for the $\Delta=1$ $XXX$ model, there is still no overall consensus for either model for its respective complete $T=0$ GS phase diagram in the range of values of $\kappa$ and $\Delta$ under study. What is agreed, however, is that although the two limiting models share exactly the same $T=0$ GS phase diagram in the classical ($s
\rightarrow \infty$) case [@Rastelli:1979_honey; @Fouet:2001_honey], their $s=\frac{1}{2}$ counterparts differ significantly. For this reason alone, a complete study of the $T=0$ GS phase diagram of the $s=\frac{1}{2}$ model of Eq. (\[H\_honey\_XXZ\]) on the honeycomb lattice is of clear interest.
There is broad agreement from various theoretical studies that whereas both classical ($s \rightarrow \infty$) $XX$ and $XXX$ models have Néel ordering for $\kappa < \kappa_{{\rm cl}} =\frac{1}{6}$, their $s=\frac{1}{2}$ counterparts both retain Néel order out to larger values $\kappa_{c_{1}} \approx 0.2$. This finding is completely consistent with the general observation that quantum fluctuations tend to favor collinear forms of magnetic order over noncollinear ones since, in the classical cases, for $\kappa > \kappa_{{\rm cl}}$ the GS phase comprises an infinitely degenerate family of states with spiral magnetic order (and see Refs.[@Rastelli:1979_honey; @Fouet:2001_honey]). These spirally-ordered noncollinear states are very fragile against quantum fluctuations, and there is by now a broad consensus in the literature that neither the $s=\frac{1}{2}$ $XX$ or $XXX$ model has a stable $T=0$ GS phase with noncollinear spiral ordering for any value of $\kappa$ in the range $0
\leq \kappa \leq 1$ under study. On the other hand, as $\kappa
\rightarrow \infty$, both models reduce to Heisenberg antiferromagnets (HAFs) on two independent triangular lattices, for each of which one knows that the stable GS phase is one where the spins are arranged on three sublattices with relative 120$^{\circ}$ ordering. Whether such a state is stable against the imposition of NN $J_{1}$ exchange coupling for large but finite values of $\kappa$, or whether it then transforms continuously to a spiral state with a given pitch angle for a specific finite value of $\kappa$, is still unknown. What is broadly agreed, on the other hand, is that any such state only exists for values $\kappa > 1$.
The most interesting region for both the $s=\frac{1}{2}$ $XX$ and $XXX$ models is when $\kappa \gtrsim 0.2$. Thus, we know that novel quantum phases often emerge from classical models which have an infinitely degenerate family of GS phases in some region of phase space, as is the case here for the classical $XX$ and $XXX$ models for $\kappa > \kappa_{{\rm cl}}=\frac{1}{6}$. What is typically then found is that quantum fluctuations lift this (accidental) GS degeneracy, either wholly or partially, by the well-known [*order by disorder*]{} mechanism [@Villain:1977_ordByDisord; @Villain:1980_ordByDisord]. Either one or several members, respectively, of the classical family are then favored as the quantum GS phase. For the present $XXX$ model on the honeycomb lattice, for example, it has been shown [@Mulder:2010_honey] that to leading order, $O(1/s)$, spin-wave fluctuations lift the degeneracy in favor of specific wave vectors, leading to spiral order by disorder.
On the other hand, we know too that quantum fluctuations generally favor collinear ordering over noncollinear ordering, as mentioned above. Hence, one may easily intuit that the strong quantum fluctuations present in the $s=\frac{1}{2}$ models might melt the spiral order for a wide range of values of $\kappa$ in favor of some collinear state. One such clear collinear candidate state is actually among the infinitely degenerate family of ground states at the classical critical point $\kappa=\frac{1}{2}$, at which the closed contours of values of the spiral wave vector, all of which minimize the classical GS energy for a given value of $\kappa$, change character [@Mulder:2010_honey]. This special collinear state among the infinite family of $\kappa=\frac{1}{2}$ ground states is the so-called Néel-II state. It is characterized by having all NN bonds along any one of the three equivalent honeycomb lattice directions as being ferromagnetic (i.e., with spins parallel), while those along the remaining two directions are antiferromagnetic (i.e., with spins antiparallel), as illustrated in Fig. \[model\](d), for example.
In the extreme $s=\frac{1}{2}$ quantum limit one may also expect quantum fluctuations to destroy completely the magnetic order in any (collinear or noncollinear) quasiclassical state in some region or other of the $T=0$ GS phase space. Just such paramagnetic states have been found by using various theoretical techniques, for both the $s=\frac{1}{2}$ $XX$ and $XXX$ models on the honeycomb lattice, in the interesting regime $0.2 \lesssim \kappa \lesssim 0.4$ where, however, the least consensus exists for either model. For the $s=\frac{1}{2}$ $XX$ model, for example, the Néel $xy$ planar \[N(p)\] ordering that exists for $\kappa < \kappa_{c_{1}} \approx 0.2$ is predicted by different techniques to give way either to a GS phase with Néel $z$-aligned \[N($z$)\] order [@Zhu:2013_honey_XY; @Bishop:2014_honey_XY] or to one with a QSL nature [@Varney:2011_honey_XY; @Carrasquilla:2013_honey_XY] in a range $\kappa_{c_{1}} < \kappa < \kappa_{c_{2}} \approx 0.4$. By contrast, for the $s=\frac{1}{2}$ $XXX$ model, the Néel order that exists for $\kappa < \kappa_{c_{1}}$ is variously predicted to give way either to a GS phase with plaquette valence-bond crystalline (PVBC) order [@Albuquerque:2011_honey; @Mosadeq:2011_honey; @Li:2012_honey_full; @Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC; @Ganesh:2013_honey_J1J2mod-XXX; @Zhu:2013_honey_J1J2mod-XXZ] or to a QSL state [@Clark:2011_honey; @Mezzacapo:2012_honey; @Gong:2013_J1J2mod-XXX; @Yu:2014_honey_J1J2mod-XXZ] in the corresponding range $\kappa_{c_{1}} < \kappa < \kappa_{c_{2}}$.
In the range ($1 >$) $\kappa > \kappa_{c_{2}}$ there is broad agreement that for both models there is a strong competition to form the GS phase between states with collinear Néel-II $xy$ planar \[N-II(p)\] order and staggered-dimer valence-bond crystalline (SDVBC) order, which lie very close in energy to one another. Both of these states break the lattice rotational symmetry in the same way, and are correspondingly threefold-degenerate. Some theoretical treatments also favor a further QCP at $\kappa_{c_{3}} > \kappa_{c_{2}}$, at which a transition occurs between a GS phase with SDVBC ordering for $\kappa_{c_{2}} < \kappa < \kappa_{c_{3}}$, possibility mixed in some or all of this regime with N-II(p) ordering, to one with N-II(p) ordering alone for $\kappa > \kappa_{c_{3}}$. It is interesting to note in this context that alternative techniques such as the ED and density-matrix renormalization group (DMRG) methods, both of which are restricted to lattices with a finite number $N$ of lattice sites, find it particularly difficult to distinguish between the N-II(p) and SDVBC phases in the regime $\kappa > \kappa_{c_{2}}$ in the thermodynamic limit $N
\rightarrow \infty$ in which we are interested, for which finite-size scaling is required, especially for the $XX$ model. It is thus particularly valuable to use a size-extensive method such as the CCM used here, which works from the outset in the $N \rightarrow \infty$ limit at every level of LSUB$m$ approximation. Since such LSUB$m$ approximations form well-defined hierarchies, as explained in Sec.\[ccm\_sec\], the only final extrapolation needed by us is to the exact ($m \rightarrow \infty$) limit in the truncation index $m$. Furthermore, at the highest level of approximation feasible with available computational resources, results for physical quantities are often already very well converged, as our specific results in Sec.\[results\_sec\] for the $s=\frac{1}{2}$ $J^{XXZ}_{1}$–$J^{XXZ}_{2}$ model of Eq. (\[H\_honey\_XXZ\]) on the honeycomb lattice will show.
THE COUPLED CLUSTER METHOD {#ccm_sec}
==========================
Since the CCM is well documented in the literature (see, e.g., Refs.[@Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC; @Bishop:2014_honey_XY; @Bishop:1987_ccm; @Arponen:1991_ccm; @Bishop:1991_TheorChimActa_QMBT; @Bishop:1998_QMBT_coll; @Bishop:1991_XXZ_PRB44; @Zeng:1998_SqLatt_TrianLatt; @Farnell:2002_1D; @Fa:2004_QM-coll; @Bi:2008_PRB_J1xxzJ2xxz]) we present only a brief overview of its key features here. Any CCM calculation starts with the choice of a suitable model state (or reference state), $|\Phi \rangle$, on top of which the quantum correlations present in the exact GS phase under study can be systematically incorporated later, as we describe below. For the present model we use each of the N(p), N($z$), and N-II(p) states shown schematically in Figs.\[model\](b)–\[model\](d).
Once a model state $|\Phi \rangle$ is chosen, the exact GS ket- and bra-state wave functions that satisfy the corresponding Schrödinger equations, $$H|\Psi\rangle = E|\Psi\rangle\,; \quad \langle \tilde{\Psi}|H = E\langle \tilde{\Psi}|\,, \quad \label{ket_bra_eqs}$$ are parametrized as $$|\Psi \rangle = {\rm e}^{S} |\Phi \rangle\,; \quad \langle \tilde{\Psi}| = \langle\Phi|\tilde{S}{\rm e}^{-S}\,, \label{para_ket_bra_eqs}$$ where we use the intermediate normalization scheme for $|\Psi\rangle$, such that $\langle\Phi|\Psi\rangle = \langle\Phi|\Phi\rangle \equiv 1$, and then for $\langle \tilde{\Psi}|$ choose its normalization such that $\langle\tilde{\Psi}|\Psi\rangle = 1$. The correlation operators $S$ and $\tilde{S}$ are decomposed in terms of exact sets of multiparticle, multiconfigurational creation and destruction operators, $C^{+}_{I}$ and $C^{-}_{I} \equiv
(C^{+}_{I})^{\dagger}$, respectively, as $$S=\sum_{I \neq 0}{\cal S}_{I}C^{+}_{I}\,; \quad \tilde{S} = 1 + \sum_{I \neq 0}\tilde{{\cal S}}_{I}C^{-}_{I}\,, \label{corr_operators}$$ where $C^{+}_{0} \equiv 1$, the identity operator, and $I$ is a set index describing a complete set of single-particle configurations for all of the particles. The reference state $|\Phi \rangle$ thus acts as a fiducial (or cyclic) vector, or generalized vacuum state, with respect to the complete set of creation operators $\{C^{+}_{I}\}$, which are hence required to satisfy the conditions $\langle \Phi|C^{+}_{I} = 0 =
C^{-}_{I}|\Phi\rangle, \forall I \neq 0$.
In order to consider each site on the spin lattice to be equivalent to all others, whatever the choice of state $|\Phi \rangle$, it is convenient to form a passive rotation of each spin so that in its own local spin-coordinate frame it points in the downward, (i.e., negative $z$) direction. Clearly, such choices of local spin-coordinate frames leave the basic SU(2) spin commutation relations unchanged, but have the beneficial effect that the $C^{+}_{I}$ operators can be expressed as products of single-spin raising operators $s^{+}_{k} \equiv s^{x}_{k}
+ is^{y}_{k}$, such that $C^{+}_{I} \equiv s^{+}_{k_{1}}s^{+}_{k_{2}}\cdots
s^{+}_{k_{n}};\;n=1,2,\cdots,2sN$.
The complete set of multiparticle correlation coefficients $\{{\cal S}_{I},{\tilde{\cal S}}_{I}\}$ may now be evaluated by extremizing the energy expectation value $\bar{H}\equiv\langle\tilde{\Psi}|H|\Psi\rangle=\langle\Phi|{\tilde{S}}{\rm e}^{-S}H{\rm e}^{S}|\Phi\rangle$, with respect to each of them, $\forall I \neq 0$. Variation with respect to each coefficient ${\tilde{\cal S}}_{I}$ yields the coupled set of nonlinear equations, $$\langle\Phi|C^{-}_{I}{\rm e}^{-S}H{\rm e}^{S}|\Phi\rangle=0\,, \quad \forall I \neq 0\,, \label{nonlinear_eq}$$ for the coefficients $\{{\cal S}_{I}\}$, while variation with respect to each coefficient ${\cal S}_{I}$ yields the corresponding set of linear equations, $$\langle\Phi|\tilde{S}({\rm e}^{-S}H{\rm e}^{S} -
E)C^{+}_{I}|\Phi\rangle=0\,, \quad \forall I \neq 0\,, \label{ket_linearEqs}$$ for the coefficients $\{{\tilde{\cal S}}_{I}\}$, once the coefficients $\{{\cal S}_{I}\}$ have been calculated from Eq. (\[nonlinear\_eq\]), and where in Eq. (\[ket\_linearEqs\]) we have used Eqs. (\[ket\_bra\_eqs\]) and (\[para\_ket\_bra\_eqs\]) to introduce the GS energy $E$.
Up till now everything has been exact. In practice, of course, approximations need to be introduced, and these are made within the CCM by restricting the set of indices $\{I\}$ retained in the expansions of Eq. (\[corr\_operators\]) for the otherwise exact correlation operators $S$ and $\tilde{S}$. One such specific hierarchical scheme, [*viz*]{}., the LSUB$m$ scheme, is described below. It is important to realize, however, that no further approximations are made. In particular, the method is guaranteed by the use of the exponential parametrizations in Eq. (\[para\_ket\_bra\_eqs\]) to be size-extensive at every level of truncation, and hence we work from the outset in the $N \rightarrow
\infty$ limit. Similarly, the important Hellmann-Feynman theorem is also exactly obeyed at every level of truncation. Lastly, when the similarity-transformed Hamiltonian ${\rm e}^{-S}H{\rm e}^{S}$ in Eqs. (\[nonlinear\_eq\]) and (\[ket\_linearEqs\]) is expanded in powers of $S$ using the well-known nested commutator expansion, the fact that $S$ contains only spin-raising operators not only guarantees that all terms are linked, but also that the otherwise infinite expansion actually terminates at a finite order, so that no further approximations are needed.
Once an approximation has been chosen and the retained coefficients $\{{\cal S}_{I},{\tilde{\cal S}}_{I}\}$ have been calculated from Eqs. (\[nonlinear\_eq\]) and (\[ket\_linearEqs\]), any GS quantity can, in principle, be calculated. For example, the GS energy $E$ can be calculated in terms of the coefficients $\{{\cal S}_{I}\}$ alone, as $E=\langle\Phi|{\rm e}^{-S}H{\rm e}^{S}|\Phi\rangle$, while the average on-site GS magnetization (or magnetic order parameter) $M$ needs both sets $\{{\cal S}_{I}\}$ and $\{{\tilde{\cal S}}_{I}\}$ for its evaluation as $M = -\frac{1}{N}\langle\Phi|\tilde{S}{\rm
e}^{-S}\sum^{N}_{k=1}s^{z}_{k}{\rm e}^{S}|\Phi\rangle$, in terms of the rotated local spin-coordinate frames defined above.
Thus, the [*only*]{} approximation made in the CCM is to truncate the set of indices $\{I\}$ in the expansions of the correlation operators $S$ and $\tilde{S}$. We use here the well-studied LSUB$m$ scheme [@Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC; @Bishop:2014_honey_XY; @Bishop:1991_XXZ_PRB44; @Zeng:1998_SqLatt_TrianLatt; @Farnell:2002_1D; @Fa:2004_QM-coll; @Bi:2008_PRB_J1xxzJ2xxz] in which, at the $m$th level of approximation, one retains all multispin-flip configurations $\{I\}$ defined over no more than $m$ contiguous lattice sites. Such cluster configurations are defined to be contiguous if every site is NN to at least one other. The number, $N_{f}$, of such fundamental configurations is reduced by exploiting the space- and point-group symmetries and any conservation laws that pertain to the Hamiltonian and the model state being used. Even so, $N_{f}$ increases rapidly with increasing LSUB$m$ truncation index $m$, and it becomes necessary to use massive parallelization together with supercomputing resources [@Zeng:1998_SqLatt_TrianLatt],[^1] to derive and solve the corresponding coupled sets of CCM equations (\[nonlinear\_eq\]) and (\[ket\_linearEqs\]). For example, we have finally $N_{f}=818\,300$ for the N-II(p) reference state at the LSUB12 level.
Finally, as a last step, we need to extrapolate the approximate LSUB$m$ results to the limit $m \rightarrow \infty$ where the CCM becomes exact. For the GS energy per spin, $e \equiv E/N$, we use the well-tested extrapolation scheme [@Farnell:2002_1D; @Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC; @Bishop:2014_honey_XY; @Bi:2008_PRB_J1xxzJ2xxz; @Fa:2004_QM-coll], $$e(m) = e_{0}+e_{1}m^{-2}+e_{2}m^{-4}\,, \label{E_extrapo}$$ where results with $m=\{6,8,10,12\}$ are employed for the N(p) and N-II(p) states used as model state, and with $m=\{4,6,8,10\}$ for the N($z$) state. For the magnetic order parameter of systems near a QCP an appropriate extrapolation rule is the “leading power-law” scheme [@RFB:2013_hcomb_SDVBC; @Bishop:2014_honey_XY], $$M(m)=c_{0}+c_{1}(1/m)^{c_{2}}\,, \label{M_extrapo_nu}$$ which we use here for the LSUB$m$ results based on the N($z$) state with $m=\{4,6,8,10\}$. An alternative well-tested scheme for systems with strong frustration or where the order in question is zero or close to zero [@Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC; @Bishop:2014_honey_XY] is $$M(m)=d_{0}+d_{1}m^{-1/2}+d_{2}m^{-3/2}\,, \label{M_extrapo_frustrated}$$ when the leading exponent $c_{2}$ in Eq. (\[M\_extrapo\_nu\]) has been empirically found to be close to 0.5, as is the case here for results based on both the N(p) and N-II(p) model states with $m=\{6,8,10,12\}$.
RESULTS {#results_sec}
=======
We now firstly present our CCM extrapolated (LSUB$\infty$) results for the GS energy per spin, $E/N$, and magnetic order parameter, $M$, using the extrapolation schemes described above in Sec.\[ccm\_sec\]. For both quantities we present three different curves for each value of the anisotropy parameter $\Delta$ shown, corresponding respectively to calculations based on the N(p), N($z$), and N-II(p) states as our chosen CCM model state.
Results for the GS energy obtained in this way are shown in Fig. \[E\].
![(Color online) The GS energy per spin $E/N$ versus the frustration parameter $\kappa \equiv J_{2}/J_{1}$ for the spin-$\frac{1}{2}$ $J^{XXZ}_{1}$–$J^{XXZ}_{2}$ model on the honeycomb lattice (with $J_{1}=1$), for various values of the anisotropy parameter $\Delta = 0.0, 0.2, 0.5, 0.7, 0.8, 0.9, 1.0$ (from top to bottom, respectively). We show extrapolated CCM LSUB$\infty$ results (see text for details) based on the Néel planar, Néel $z$-aligned, and Néel-II planar model states, respectively. The times ($\times$) symbols mark the points where the respective extrapolations for the order parameter have $M \rightarrow 0$, and the unphysical portions of the solutions are shown by thinner lines (see text for details).[]{data-label="E"}](fig2.eps){width="9cm"}
A particularly noteworthy feature of the curves shown is that they all exhibit [*termination points*]{}. Thus, the N(p) curves all end at corresponding upper termination points, while the N-II(p) curves end at corresponding lower termination points. The intermediate N($z$) curves end at both corresponding lower and upper termination points. In each case the respective termination points relate to those points beyond which real solutions for the CCM multiconfigurational correlation coefficients $\{{\cal S}_{I}\}$ cease to exist in the LSUB$m$ approximation with the highest value of the truncation index $m$ used, for the particular extrapolated curve shown. Such termination points of LSUB$m$ solutions are both well understood and well documented in the literature (see, e.g., Refs.[@Fa:2004_QM-coll; @Bishop:2014_honey_XY; @Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC]). They are simply approximate manifestations of a corresponding QCP in the system, beyond which the order associated with the model state being employed melts. As would then be expected, we find for a given value of $\Delta$ that as the index $m$ is increased the range of values of $\kappa$ for which the LSUB$m$ equations have real solutions becomes narrower. Eventually, as $m \rightarrow \infty$, each termination point then becomes the respective exact QCP. Clearly, from what has just been explained, real LSUB$m$ solutions with a fixed finite value of $m$ can hence also exist in regions where the corresponding magnetic order is destroyed (i.e., where $M<0$).
Corresponding sets of curves to those shown in Fig. \[E\] for the GS energy per spin, $E/N$, are shown in Fig. \[M\] for the magnetic order parameter, $M$.
![(Color online) The GS magnetic order parameter $M$ versus the frustration parameter $\kappa \equiv J_{2}/J_{1}$ for the spin-$\frac{1}{2}$ $J^{XXZ}_{1}$–$J^{XXZ}_{2}$ model on the honeycomb lattice (with $J_{1}>0$) for various values of the anisotropy parameter $\Delta=0.0,0.2,0.5,0.7,0.8,0.9,1.0$ (from top to bottom, respectively). We show extrapolated CCM LSUB$\infty$ results (see text for details) based on the Néel planar, Néel $z$-aligned, and Néel-II planar states as CCM model states, respectively.[]{data-label="M"}](fig3.eps){width="9cm"}
In Fig. \[E\] we show by times ($\times$) symbols those points on each curve where $M=0$, as determined from the corresponding extrapolated LSUB$\infty$ curve in Fig. \[M\]. In Fig. \[E\] we also denote by thinner lines those portions of the curves which are “unphysical” in the sense that $M<0$, by contrast with the corresponding “physical” regions where $M>0$, which pertain to those portions of the curves denoted by thicker lines.
We can immediately draw several conclusions from the results shown in Figs. \[E\] and \[M\]. Firstly, it is clear that N(p) order is present, for all values of $\Delta$ shown, below a lower critical value, $0 < \kappa < \kappa_{c_{l}}(\Delta)$. Furthermore, $\kappa_{c_{l}}$ depends only very weakly on $\Delta$, taking the value $\kappa_{c_{l}}(\Delta) \approx 0.21$. Secondly, we observe both that N($z$) order is present within a rather narrow range of values around $\kappa \approx 0.3$ for $\Delta \lesssim 0.66$, but that it becomes unstable for $\Delta \gtrsim 0.66$. Thirdly, it is also clear that N-II(p) order is present, for all values of $\Delta$ shown, above some upper critical value, $\kappa_{c_{u}}(\Delta) < \kappa$ $(< 1)$, where $\kappa_{c_{u}}(\Delta)$ increases monotonically with $\Delta$. Fourthly, it is particularly clear from Fig. \[M\] that the GS phases with N(p) and N($z$) order melt at (or very close to) the [ *same*]{} value $\kappa_{c_{l}}(0)$ for $\Delta=0$, but as $\Delta$ is increased a very narrow region (in $\kappa$) opens up between these two phases in which the GS phase has neither of these orderings. Finally, Fig. \[M\] similarly shows that although the two GS phases with N($z$) and N-II(p) order also melt at (or very close to) the [ *same*]{} value $\kappa_{c_{u}}(0)$ for $\Delta=0$, as $\Delta$ is increased a GS phase with neither of these forms of order opens up between them. The range (in $\kappa$) of stability of this intermediate phase increases monotonically with $\Delta$.
We now turn to the issue of what might be the nature of the remaining GS phases outside the regimes of stability of the quasiclassical N(p), N($z$), and N-II(p) phases, as discussed above. Once we have identified any possible candidate phase with a specific form of ordering, described by a suitable operator $\hat{O}$, a very convenient way to test for the relative stability of a GS phase built on a given CCM model state against that new form of ordering is to consider its linear response to an imposed perturbation with a corresponding field operator, $F = \delta\,\hat{O}$, added to the original system Hamiltonian \[i.e., of Eq. (\[H\_honey\_XXZ\]) for the present case\], where $\delta$ is a (positive) infinitesimal. The perturbed energy per spin, $e(\delta) \equiv E(\delta)/N$, is then calculated at various LSUB$m$ levels of approximation based on the CCM model state whose stability is being investigated, for the infinitesimally perturbed Hamiltonian $H + F$. The corresponding susceptibility of the system to this perturbation is then defined, as usual, (and see, e.g., Refs.[@Bishop:2014_honey_XY; @Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC]) as $$\chi
\equiv - \left. \frac{\partial^2{e(\delta)}}{\partial {\delta}^2}
\right|_{\delta=0}.$$ The GS order of the CCM model state will thus become unstable against formation of the imposed form of order when $\chi \rightarrow \infty$ or, equivalently, when $1/\chi \rightarrow 0$. The corresponding LSUB$m$ results for the susceptibility of the given CCM model state against the imposed form of order are then extrapolated to the LSUB$\infty$ limit using the unbiased “leading power-law” scheme, $$\chi^{-1}(m) = x_{0}+x_{1}(1/m)^{\nu}\,, \label{Extrapo_inv-chi}$$ similar to that in Eq. (\[M\_extrapo\_nu\]) for the order parameter.
Previous results using the CCM for the current model of Eq.(\[H\_honey\_XXZ\]) in the limiting cases of the $XX$ model [@Bishop:2014_honey_XY] at $\Delta=0$ and the $XXX$ model [@Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC] at $\Delta=1$, as well as those using alternative techniques, suggest that N-II(p) ordering strongly competes with SDVBC ordering to form the stable GS phase in the relevant part of phase space. Hence, we now perform CCM calculations based on the N-II(p) state as model state where the perturbing field promotes SDVBC order, $\hat{O}
\rightarrow \hat{O}_{d}$, as illustrated schematically in the right-hand frame of Fig. \[X\_SDVBC\].
The results presented in Fig. \[X\_SDVBC\] for the corresponding inverse staggered dimer susceptibility, $1/\chi_{d}$, are LSUB$\infty$ extrapolations based on Eq. (\[Extrapo\_inv-chi\]), with LSUB$m$ results $m=\{4,6,8\}$ used as input, for each of the values of $\Delta$ shown. They show clearly that the lower critical value of the frustration parameter $\kappa$ at which SDVBC order appears is rather insensitive to the value of the spin anisotropy parameter $\Delta$ for all $\Delta \gtrsim 0.1$, where it takes the almost constant value $\kappa \approx 0.38$. However, the locus of such SDVBC critical points meets the corresponding locus of critical points $\kappa_{c_{u}}(\Delta)$ above which N-II(p) order appears, as taken from Fig. \[M\], at a value $\Delta \approx 0.1$. Hence, for values $\Delta \lesssim 0.1$, a “mixed” region opens up in the $T=0$ GS phase diagram in which both SDVBC and N-II(p) forms of order appear to coexist over a fairly narrow range of values of $\kappa$, above which N-II(p) order then reasserts itself as the sole form of ordering in the GS phase.
We turn finally to the remaining, and especially interesting, region in the $\kappa$–$\Delta$ phase space, which is outside the region of N(z) stability but between the two curves $\kappa =
\kappa_{c_{l}}(\Delta) \approx 0.21$ (below which N(p) order is stable) and $\kappa \approx 0.38$ (above which SDVBC and/or N-II(p) order is stable). For the limiting case of the $XXX$ model (at $\Delta=1$) some methods (including the CCM) favor the GS phase to have PVBC order over all or part of this region [@Albuquerque:2011_honey; @Mosadeq:2011_honey; @Li:2012_honey_full; @Bishop:2012_honeyJ1-J2; @RFB:2013_hcomb_SDVBC; @Ganesh:2013_honey_J1J2mod-XXX; @Zhu:2013_honey_J1J2mod-XXZ; @Gong:2013_J1J2mod-XXX], while others favor a QSL state [@Clark:2011_honey; @Mezzacapo:2012_honey; @Gong:2013_J1J2mod-XXX; @Yu:2014_honey_J1J2mod-XXZ], again over all or part of the region. Hence, we now perform CCM calculations based on the N(p) state as model state, in the presence of a perturbing field that now promotes PVBC order, $\hat{O}
\rightarrow \hat{O}_{p}$, as shown schematically in the right-hand frame of Fig. \[PVBC\].
The results presented in Fig. \[PVBC\] for the corresponding inverse plaquette susceptibility, $1/\chi_{p}$, are again LSUB$\infty$ extrapolations based on Eq. (\[Extrapo\_inv-chi\]), with LSUB$m$ results $m=\{4,6,8\}$ used as input, for each of the values of $\Delta$ shown. Once again, they show clear evidence for corresponding regions of stability of a GS phase with PVBC order.
$T=0$ GS PHASE DIAGRAM {#phase_sec}
======================
On the basis of the results presented so far in Sec.\[results\_sec\] it is now straightforward to construct the $T=0$ GS phase digram for the model, as shown in Fig. \[phase\].
![(Color online) Phase diagram for the spin-$\frac{1}{2}$ $J^{XXZ}_{1}$–$J^{XXZ}_{2}$ model on the honeycomb lattice (with $J_{1}>0$ and $\kappa \equiv J_{2}/J_{1}>0$) in the window $0 \leq \kappa \leq 1$ and $0 \leq \Delta \leq 1$, as obtained by a CCM analysis. The phase in the region marked “M” has both SDVBC and Néel-II planar order. See text for details.[]{data-label="phase"}](fig6.eps){width="9cm"}
Clearly, the regions of stability of the N(p), N($z$), and N-II(p) phases may be taken from Fig. \[M\] as those in which the respective magnetic order parameters $M$ take positive values. The corresponding points at which $M=0$ are shown in Fig. \[phase\] by open square ($\square$), times ($\times$), and open circle ($\bigcirc$) symbols, respectively. Similarly, the points at which $\chi^{-1}_{d} \rightarrow 0$ and $\chi^{-1}_{p} \rightarrow 0$, taken from Figs. \[X\_SDVBC\] and \[PVBC\], are shown in Fig.\[phase\] by open triangle ($\triangle$) and plus ($+$) symbols, respectively. The small region of mixed SDVBC and N-II(p) order, described in Sec. \[results\_sec\], is denoted in Fig. \[phase\] by “M”.
Based on the results for $1/\chi_{p}$ from Fig. \[PVBC\] we now tentatively identify the region denoted by “PVBC” in Fig.\[phase\] as having stable PVBC order. The remaining region denoted by “QSL(?)” is a clear candidate for a QSL phase, since we find no evidence for any form of magnetic (spin) ordering, nor of either form of VBC ordering, for which we have tested. In this context we also mention that a recent DMRG study [@Gong:2013_J1J2mod-XXX] of the limiting $XXX$ case (i.e., $\Delta=1$) of the present model found solid evidence of (weak) PVBC order in the thermodynamic ($N
\rightarrow \infty$) limit, in the range $0.26 \lesssim \kappa
\lesssim 0.35$ of the frustration parameter, in good agreement with our own estimate for this limiting $XXX$ case that PVBC order exists in the range $0.28 \lesssim \kappa \lesssim 0.38$. Very interestingly, the same DMRG study [@Gong:2013_J1J2mod-XXX] excluded, in the same thermodynamic limit, any form of either magnetic (spin) or VBC ordering in the range $0.22 \lesssim \kappa \lesssim 0.26$ immediately above the Néel-ordered regime for the $XXX$ model, which was identified as being the stable GS phase for $\kappa \lesssim 0.22$. These DMRG findings were thus consistent with a QSL phase in the region $0.22 \lesssim \kappa \lesssim 0.26$, again in broad agreement with our own tentative conclusion of a QSL phase in the region $0.21
\lesssim \kappa \lesssim 0.28$ for the $XXX$ limiting case of the model. Indeed, these results are backed up by our earlier CCM analysis [@Bishop:2012_honeyJ1-J2] of the $s=\frac{1}{2}$ $J_{1}$–$J_{2}$ $XXX$ model on the honeycomb lattice.
Thus, it was noted already in Ref. [@Bishop:2012_honeyJ1-J2] that the transition from the N(p) phase to the PVBC phase in the $XXX$ model might be via an intermediate phase. Any such intermediate phase was estimated to be restricted to a region $\kappa_{c_{1}} < \kappa <
\kappa_{c_{1}}'$. The value of $\kappa_{c_{1}}$ was accurately obtained from the point where Néel order vanishes as $\kappa_{c_{1}} = 0.207(3)$, and it is identical to that now shown in Fig. \[phase\] by the open square ($\square$) symbol at $\Delta=1$. The high accuracy obtained for $\kappa_{c_{1}}$ essentially stems from the shape of the N(p) order curve shown in Fig. \[M\], with its very steep (or infinite) slope at the point $\kappa_{c_{1}}$ where $M \rightarrow 0$. By contrast, the point $\kappa_{c_{1}}'$ was determined as in Fig. \[PVBC\] from the point where $1/\chi_{p} \rightarrow 0$. The relative inaccuracy in this value stems, conversely from the very shallow (or zero) slope in the $1/\chi_{p}$ curve at the point $\kappa_{c_{1}}'$ where it becomes zero. In the earlier CCM analysis [@Bishop:2012_honeyJ1-J2] a value $\kappa_{c_{1}}' \approx 0.24$ was quoted, without an error estimation. In the current analysis we have specifically examined the lower phase boundary of the PVBC phase in greater detail, and our best estimate for the limiting $XXX$ model is now $\kappa_{c_{1}}' \approx
0.28(2)$ from Fig. \[PVBC\], and as shown in Fig. \[phase\] by the plus ($+$) symbol at $\Delta=1$. Nevertheless, it is still the case that of all the phase boundaries shown in Fig. \[phase\], the one between the PVBC and putative QSL phases probably has the largest uncertainty, with a similar error along its whole length to that quoted above at the point $\Delta=1$. In this context we note too that Fig. \[phase\] shows that the plus ($+$) symbols denoting the lower boundary of PVBC stability do not fall precisely on top of the times ($\times$) symbols that denote the lower boundary of stability of the N($z$) phase, in the region $\Delta \lesssim 0.66$ where the latter phase exists as a stable GS phase. This difference is probably also another independent indication of the error bars associated with the lower PVBC boundary points.
These error bars could certainly be reduced by including higher-order LSUB$m$ results in the extrapolations. The entire PVBC and SDVBC regions of stability would also more definitively be confirmed by performing calculations of $1/\chi_{p}$ and $1/\chi_{d}$ based on other CCM model states to confirm their respective boundaries. For example, for the PVBC phase one might also use the N-II(p) state as CCM model state to confirm the upper boundary of the phase. In any case, more definitive evidence awaits higher-order LSUB$m$ calculations. Without them, for example, the possibility of a stable QSL phase also existing in the very narrow region between the N($z$) and SDVBC phases for $\Delta \lesssim 0.66$ also cannot be ruled out.
CONCLUSIONS {#conclusion}
===========
In this paper we have outlined how the well-known CCM technique, which has been very widely and very successfully applied to diverse (both finite and macroscopically extended) physical systems that exist in a spatial continuum, can be adapted for use with spin-lattice models of interest in quantum magnetism, in which the spins are confined to the sites of a regular periodic spatial lattice. In particular, we have explained how it may be applied, with comparable success, to high orders in a systematically improvable hierarchy of approximations. The method acts at every level of truncation in the thermodynamic limit ($N \rightarrow \infty$), and the [*only*]{} approximation made in practice is to a given $m$th level in the approximation hierarchy. Thus, unlike in such alternative techniques as ED and QMC methods, no finite-size scaling is ever needed within the CCM. We have also shown how GS quantities may readily be extrapolated to the exact $m
\rightarrow \infty$ limit of the truncation scheme, by the use of well-tested heuristic schemes.
As an illustration of the CCM technique we applied it here to the two-dimensional, frustrated, spin-half $J^{XXZ}_{1}$–$J^{XXZ}_{2}$ model on the honeycomb lattice. We demonstrated explicitly how a CCM analysis of the model could yield a fully coherent and accurate picture of its full $T=0$ GS phase diagram. We identified, in particular, a specific region in the phase space in which we positively excluded magnetic and VBC forms of order, and which is hence a strong candidate for a QSL phase. Clearly, it would be of value to apply other techniques to this model in order to check our findings.
We note finally that the CCM has been applied with comparable success in recent years to many other spin-lattice problems. Particular strengths of the method are that at every level of approximation it obeys both the Goldstone linked-cluster theorem (in the sense that it is manifestly size-extensive) and, perhaps even more importantly, the Hellmann-Feynman theorem.
In conclusion, we hope that we have convinced the reader that the CCM is extremely versatile, requiring only the choice of a suitable model state (or set of such states) as input, on top of which the method incorporates the multispin correlations systematically. Although we have demonstrated its use here for the case of a spin-half system, it is quite straightforward to generalize the CCM for use with spins of arbitrary quantum number $s$ [@Farnell:2002_1D].
We thank the University of Minnesota Supercomputing Institute for the grant of supercomputing facilities.
[^1]: We use the program package CCCM of D. J. J. Farnell and J. Schulenburg, see http://www-e.uni-magdeburg.de/jschulen/ccm/index.html.
|
---
abstract: 'Block diagonalization is a linear precoding technique for the multiple antenna broadcast (downlink) channel that involves transmission of multiple data streams to each receiver such that no multi-user interference is experienced at any of the receivers. This low-complexity scheme operates only a few dB away from capacity but requires very accurate channel knowledge at the transmitter. We consider a limited feedback system where each receiver knows its channel perfectly, but the transmitter is only provided with a finite number of channel feedback bits from each receiver. Using a random quantization argument, we quantify the throughput loss due to imperfect channel knowledge as a function of the feedback level. The quality of channel knowledge must improve proportional to the SNR in order to prevent interference-limitations, and we show that scaling the number of feedback bits linearly with the system SNR is sufficient to maintain a bounded rate loss. Finally, we compare our quantization strategy to an analog feedback scheme and show the superiority of quantized feedback.'
author:
-
bibliography:
- 'BD\_JSAC.bib'
title: 'Limited Feedback-based Block Diagonalization for the MIMO Broadcast Channel'
---
Introduction {#sec:intro}
============
In multiple antenna broadcast (downlink) channels, transmit antenna arrays can be used to simultaneously transmit data streams to receivers and thereby significantly increase throughput. Dirty paper coding (DPC) is capacity achieving for the MIMO broadcast channel [@WEIN04], but this technique has a very high level of complexity. Zero Forcing (ZF) and Block Diagonalization (BD) [@CHOI04] [@SPEN04] are alternative low-complexity transmission techniques. Although not optimal, these linear precoding techniques utilize all available spatial degrees of freedom and perform measurably close to DPC in many scenarios [@LEE06].
If the transmitter is equipped with $M$ antennas and there are at least $M$ aggregate receive antennas, zero-forcing involves transmission of $M$ spatial beams such that independent, de-coupled data channels are created from the transmit antenna array to $M$ receive antennas distributed amongst a number of receivers. Block diagonalization similarly involves transmission of $M$ spatial beams, but the beams are selected such that the signals received at different receivers, but not necessarily at the different antenna elements of a particular receiver, are de-coupled. For example, if there are $M/2$ receivers with two antennas each, then two beams are aimed at each of the receivers. If ZF is used, an independent and de-coupled data stream is received on each of the $M$ antennas. If BD is used, the streams for different receivers do not interfere, but the two streams intended for a single receiver are generally not aligned with its two antennas and thus post-multiplication by a rotation matrix (to align the streams) is generally required before decoding.
In order to correctly aim the transmit beams, both schemes require perfect Channel State Information at the Transmitter (CSIT). Imperfect CSIT leads to incorrect beam selection and therefore multiuser interference, which ultimately leads to a throughput loss. Unlike point to point MIMO systems where imperfect CSIT causes only an SNR offset in the capacity vs. SNR curve, the level of CSIT affects the slope of the curve and hence the *multiplexing* gain in broadcast MIMO systems. We consider the case when the CSI is known perfectly at the receiver and is communicated to the transmitter through a limited feedback channel and quantify the maximum rate loss due to limited feedback with BD.
MISO systems and ZF with limited feedback are analyzed in [@JIND05]. Similar to the results in [@JIND05], we show that scaling the number of feedback bits approximately linearly with the system SNR is sufficient to maintain the slope of the capacity vs. SNR curve and hence a constant gap from the capacity of BD with perfect CSIT. The scaling factor for BD offers an advantage over ZF in terms of the number of bits required to achieve the same sum capacity.
Rather than quantizing the CSIT into a finite number of bits and feeding this information back, the channel coefficients can also be explicitly transmitted over the feedback link. We compare this scheme to quantized feedback for an AWGN feedback channel, and show the superiority of quantized feedback.
System Model {#sec:sysmodel}
============
We consider a MIMO broadcast (downlink) system with a single transmitter or base station and $K$ receivers or users. Each user has $N$ antennas and the transmitter has $M$ antennas. The broadcast channel is described as: $${{\bf y}}_k = {{\bf H}}_k^{{\sf H}}{{\bf x}}+ {{\bf n}}_k,\quad k = 1, \dots, K$$ where ${{\bf H}}_k \in {\mbox{\bb C}}^{M \times N}$ is the channel matrix from the transmitter to the $k^\text{th}$ user ($1 \leq k \leq K$) and the vector ${{\bf x}}\in {\mbox{\bb C}}^{M \times 1}$ is the transmitted signal. ${{\bf n}}_k \in {\mbox{\bb C}}^{N \times 1}$ are independent complex Gaussian noise vectors of unit variance and ${{\bf y}}_k \in {\mbox{\bb C}}^{N \times 1}$ is the received signal vector at the $k^\text{th}$ user. We assume a transmit power constraint so that $E[||{{\bf x}}||^2] \leq P$ $(P > 0)$. We also assume that $K = \frac{M}{N}$ (with $K \geq 2$), which implies that the aggregate number of receive antennas equals the number of transmit antennas; as a result it is not necessary to select a subset of users for transmission.
The entries of ${{\bf H}}_k$ are assumed to be i.i.d. unit variance complex Gaussian random variables, and the channel is assumed to be block fading with independent fading from block to block. Each of the users are assumed to have perfect and instantaneous knowledge of their own channel matrix. The channel matrix is quantized by each user and fed back to the transmitter (which has no other knowledge of the instantaneous CSI) over a zero delay, error free, limited feedback channel.
It is assumed that a uniform power allocation policy is adopted (i.e., we do not perform waterfilling across streams), which is known to be asymptotically optimal for large SNR. Hence, in order to perform Block Diagonalization, it is only necessary to know the spatial direction of each user’s channel, i.e., the subspace spanned by the columns of ${{\bf H}}_k$, and the feedback only needs to convey this information.
The quantization codebook used by each user is fixed beforehand and is known to the transmitter. A quantization codebook ${{\cal C}}$ consists of $2^B$ matrices in ${\mbox{\bb C}}^{M \times N}$ i.e. $({{\bf W}}_1, \dots, {{\bf W}}_{2^B})$, where $B$ is the number of feedback bits allocated per user. The quantization of a channel matrix ${{\bf H}}_k$, say ${{\bf \widehat{ H}}}_k$, is chosen from the codebook ${{\cal C}}$ according to the following rule: $$\label{quantproc}
{{\bf \widehat{ H}}}_k = \mathop{{{\hbox{arg}}}\min}\limits_{{{\bf W}}\ \in\ {{\cal C}}}\ d^2\left( {{\bf H}}_k, {{\bf W}}\right)$$ where $d\left( {{\bf H}}_k, {{\bf W}}\right)$ is the distance metric. Here, we consider the *chordal distance* [@CONW96]: $$d\left( {{\bf H}}_k, {{\bf W}}\right) = \sqrt{\sum\limits_{j=1}^N \sin^2\theta_j}$$ where the $\theta_j$’s are the principal angles between the two subspaces spanned by the columns of the matrices ${{\bf H}}_k$ and ${{\bf W}}$ [@CONW96]. As the principal angles depend only on the subspaces spanned by the columns of the matrices, it can be assumed that the elements of ${{\cal C}}$ are unitary matrices (i.e. ${{\bf W}}^{{\sf H}}{{\bf W}}= {\bf I}_N\ \forall\ {{\bf W}}\in {{\cal C}}$), without loss of generality. An alternate form for the chordal distance is $d^2\left( {{\bf H}}_k, {{\bf W}}\right) = N - {{\hbox{tr}}}\left( \widetilde{{{\bf H}}}_k^{{\sf H}}{{\bf W}}{{\bf W}}^{{\sf H}}\widetilde{{{\bf H}}}_k \right)$, where $\widetilde{{{\bf H}}}_k$ forms an orthonormal basis for the subspace spanned by ${{\bf H}}_k$. Note that other distance metrics may also be considered, but we do not investigate this further in this work. No channel magnitude information is fed back to the transmitter.
Background {#sec:background}
==========
Block Diagonalization {#ssec:bdiag}
---------------------
The Block Diagonalization strategy, when perfect CSI is available at the transmitter, involves linear precoding that suppresses the interference at each user due to all other users (but does not suppress interference due to different antennas for the same user). If ${{\bf u}}_k \in {\mbox{\bb C}}^{N \times 1}$ contains the $N$ complex (data) symbols intended for the $k^\text{th}$ ($1 \leq k \leq K$) user and ${{\bf V}}_k \in {\mbox{\bb C}}^{M \times N}$ is the precoding matrix, then the transmitted vector is given by: $${{\bf x}}= \sum\limits_{k = 1}^K {{\bf V}}_k{{\bf u}}_k$$ and the received signal at the $k^\text{th}$ user is given by: $${{\bf y}}_k = {{\bf H}}_k^{{\sf H}}{{\bf V}}_k{{\bf u}}_k + \sum\limits_{j = 1, j \neq k}^K {{\bf H}}_k^{{\sf H}}{{\bf V}}_j{{\bf u}}_j + {{\bf n}}_k$$
The $\sum\limits_{j = 1, j \neq k}^K {{\bf H}}_k^{{\sf H}}{{\bf V}}_j{{\bf u}}_j$ term represents the multi-user interference at user $k$. In order to maintain the power constraint, it is assumed that ${{\bf V}}_k^{{\sf H}}{{\bf V}}_k = {\bf I}_N$ and $E\left[||{{\bf u}}_k||^2\right] \leq \frac{P}{M}$, for $k = 1, \dots, K$.
Following the BD procedure, each ${{\bf V}}_k$ is chosen such that ${{\bf H}}_j^{{\sf H}}{{\bf V}}_k$ is ${\bf 0},\ \forall k \neq j$. This amounts to determining an orthonormal basis for the left null space of the matrix formed by stacking all $\{{{\bf H}}_j\}_{j \neq k}$ matrices together. This reduces the interference terms in equation (\[eqn:rxbd\]) to zero at each user. This is different from Zero Forcing where each complex symbol to be transmitted to the $m^\text{th}$ antenna (among the $N$ antennas, i.e., $m = 1, \dots, N$) of the $k^\text{th}$ user is precoded by a vector that is orthogonal to all the columns of ${{\bf H}}_j, j \neq k$, as well as orthogonal to all but the $m^\text{th}$ column of ${{\bf H}}_k$.
However, zero interference can only be achieved with perfect knowledge of $\{{{\bf H}}_k\}_{k=1}^K$ at the transmitter. In the case of limited feedback, when only a quantized version of the subspace spanned by the columns of each ${{\bf H}}_k$ is available at the transmitter, namely ${{\bf \widehat{ H}}}_k$, we use a naive strategy where the precoding matrices are selected by treating ${{\bf \widehat{ H}}}_1, \dots, {{\bf \widehat{ H}}}_K$ as the true channels while performing BD. To distinguish these precoding matrices from those selected with perfect CSIT, we denote these matrices as ${{\bf \widehat{ V}}}_1, \dots, {{\bf \widehat{ V}}}_K$, where each ${{\bf \widehat{ V}}}_k$ is chosen such that ${{\bf \widehat{ H}}}_j^{{\sf H}}{{\bf \widehat{ V}}}_k = {\bf 0}\ \forall k \neq j$. Thus, ${{\bf H}}_j^H{{\bf \widehat{ V}}}_k \neq 0$ in general, which leads to residual interference terms and a loss in throughput. The received signal in the case of limited feedback is thus written as: $$\label{eqn:rxbd}
{{\bf y}}_k = {{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_k{{\bf u}}_k + \sum\limits_{j = 1, j \neq k}^K {{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf u}}_j + {{\bf n}}_k$$
Random Quantization Codebooks {#ssec:randcode}
-----------------------------
Since the design of optimal quantization codebooks for the given distance metric is a very difficult problem, we instead study performance averaged over *random* quantization codebooks. The Grassmann manifold is the set of all $N$ dimensional subspaces (or planes) passing through the origin, in an $M$ dimensional space. This is denoted by ${{\cal G}}_{M, N}$. We consider complex Euclidean subspaces in this work. Each of the $2^B$ unitary matrices making up the random quantization codebook are chosen independently and are uniformly distributed over ${{\cal G}}_{M, N}$ [@DAI06] [@marzetta1999cmm]. We alternatively refer to this uniform distribution as the isotropic distribution in the respective space. A random element drawn from this distribution (over ${{\cal G}}_{M, N}$) can be generated by generating an $M \times N$ matrix with i.i.d. complex Gaussian elements and then forming a specific orthonormal basis for the $N$ dimensional subspace spanned by the matrix (e.g., through a QR decomposition).
We analyze the performance averaged over all possible random codebooks. The distortion or error associated with a given codebook ${{\cal C}}$ for the quantization of ${{\bf H}}_k \in {\mbox{\bb C}}^{M \times N}$ is defined as: $$D \mathop{=}\limits^\Delta {\mbox{\bb E}}\left[ d^2({{\bf H}}_k,{{\bf \widehat{ H}}}_k) \right] = {\mbox{\bb E}}\left[ \mathop{\min\limits_{{{\bf W}}\in {{\cal C}}}}\ d^2({{\bf H}}_k,{{\bf W}}) \right],$$ where ${{\bf \widehat{ H}}}_k$ is the quantization of ${{\bf H}}_k$. It is shown in [@DAI06] that $D \leq \overline{D}$ where, $$\label{eqn:D}
\overline{D} = \frac{\Gamma(\frac{1}{T})}{T} (C_{MN})^{-\frac{1}{T}} 2^{-\frac{B}{T}} + N \exp\left[ -(2^BC_{MN})^{1-a} \right] ,$$ for a codebook of size $2^B$. Here, $T = N (M - N)$ and $a \in (0, 1)$ is a real number between $0$ and $1$ chosen such that $\left( C_{MN}2^B \right)^{-\frac{a}{T}} \leq 1$. $C_{MN}$ is given by $\frac{1}{T!}\ \prod\limits_{i = 1}^N\ \frac{(M - i)!}{(N - i)!}$. The second (exponential) term in (\[eqn:D\]) can be neglected for large $B$. For systems where $N = 2$ or $3$, the exponential term may be neglected for most practical cases.
Analysis and Results {#sec:tanal}
====================
In this section, we analyze the achievable throughput of the limited feedback-based system described so far. We first describe some preliminary mathematical results.
Preliminary Calculations {#ssec:prelim}
------------------------
\[lemma1\] The quantization ${{\bf \widehat{ H}}}_k$ of the channel ${{\bf H}}_k$ admits the following decomposition: $$\label{lem1}
\widetilde{{{\bf H}}}_k = {{\bf \widehat{ H}}}_k {{\bf X}}_k {{\bf Y}}_k + {{\bf S}}_k {{\bf Z}}_k$$ where
- $\widetilde{{{\bf H}}}_k \in {\mbox{\bb C}}^{M \times N}$ is an orthonormal basis for the subspace spanned by the columns of ${{\bf H}}_k$,
- ${{\bf X}}_k \in {\mbox{\bb C}}^{N \times N}$ is unitary and distributed uniformly over ${{\cal G}}_{N, N}$,
- ${{\bf Z}}_k \in {\mbox{\bb C}}^{N \times N}$ is upper triangular with positive diagonal elements, satisfying ${{\hbox{tr}}}({{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k) = d^2\left( {{\bf H}}_k, {{\bf \widehat{ H}}}_k \right)$,
- ${{\bf Y}}_k \in {\mbox{\bb C}}^{N \times N}$ is upper triangular with positive diagonal elements and satisfies ${{\bf Y}}_k^{{\sf H}}{{\bf Y}}_k = {\bf I}_N - {{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k$, and
- ${{\bf S}}_k \in {\mbox{\bb C}}^{M \times N}$ is an orthonormal basis for an isotropically distributed (complex) $N$ dimensional plane in the $M-N$ dimensional left nullspace of ${{\bf \widehat{ H}}}_k$.
Moreover, the quantities ${{\bf Y}}_k$, ${{\bf \widehat{ H}}}_k$ and ${{\bf X}}_k$ are distributed independent of each other, as are the pair ${{\bf S}}_k$ and ${{\bf Z}}_k$. This decomposition is a generalization of the decomposition in [@JIND05], which was for the specific case of $N = 1$. Similar to [@JIND05], the matrix ${{\bf Z}}_k$ represents the quantization error.
See Appendix \[lem1proof\].
A direct application of Lemma \[lemma1\] allows us to bound the rate loss due to limited feedback. This decomposition also allows us to perform low complexity Monte-Carlo simulations for evaluating the performance of random quantization codebooks, even for very large $B$, as described in detail in Section \[sec:simmethod\].
Throughput analysis for quantized feedback {#ssec:ffqual}
------------------------------------------
In the case of perfect CSIT and BD, the transmitter has the ability to suppress all interference terms giving a [*per user*]{} ergodic rate of: $$\label{eqn:R_CSIT}
R_\textsc{CSIT-BD}(P) = {\mbox{\bb E}}\left[ \ \log_2 \left| {\bf I}_N + \frac{P}{M}{{\bf H}}_k^{{\sf H}}{{\bf V}}_k{{\bf V}}_k^{{\sf H}}{{\bf H}}_k \right|\ \right]$$ where $k$ is any user from $1, \dots, K$. The expectation is carried out over the distribution of ${{\bf H}}_k$.
For limited feedback of $B$ bits per user, multiuser interference cannot be completely canceled and this leads to residual interference power. The per-user rate (throughput) is given by: $$\begin{aligned}
R_\textsc{Quant}(P) & = & {\mbox{\bb E}}\left[ I({{\bf u}}_k; {{\bf y}}_k \vert {{\bf H}}_k) \right]\\
& = & {\mbox{\bb E}}\left[ \log_2\left| {\bf I}_N + \frac{P}{M} \left( {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1, j \neq k}^K {{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf H}}_k \right)^{-1} {{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_k{{\bf \widehat{ V}}}_k^{{\sf H}}{{\bf H}}_k \right| \right]\\
& = & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1}^K {{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf H}}_k \right| \right] - \nonumber\\ & & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1, j \neq k}^K {{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf H}}_k \right| \right] \label{eqn:R_FB}\end{aligned}$$ where $k$ is any user between $1$ and $K$ and the expectation is carried out over the channel distribution as well as random codebooks ${{\cal C}}$.
\[thm:1\] The rate loss incurred per user due to limited feedback with respect to perfect CSIT using Block Diagonalization can be bounded from above by: $$\begin{aligned}
\Delta R_\textsc{Quant}(P) & = & \left[ R_\textsc{CSIT-BD}(P) - R_\textsc{Quant}(P) \right] \\
& \leq & N\ \log_2 \left(1 + \frac{P}{N} D \right)\end{aligned}$$
See Appendix \[thm1proof\].
This provides a bound on the rate loss per user[^1]. Furthermore, $D$ can be upper bounded tightly by $\overline{D}$ from (\[eqn:D\]).
Controlling feedback quality {#ssec:incfqual}
----------------------------
![Sufficient number of bits for a gap of 3 dB relative to BD with perfect CSIT, for $N = 2$ and $M = 4, 6$ and $8$[]{data-label="fig0"}](fig0){width="\ImgWidth"}
If $B$ is kept fixed and the SNR is taken to $\infty$, it is easy to see that residual interference will eventually overwhelm signal power, and this leads to a bounded throughput (i.e., zero multiplexing gain). Therefore, it is of interest to determine how fast $B$ must grow with SNR in order to prevent this behavior and to maintain a bounded rate loss relative to a perfect CSIT system.
\[thm:2\] In order to bound the per-user rate loss $\Delta R_\textsc{Quant}(P)$ from above by $\log_2(b) > 0$, it is sufficient for the number of feedback bits per user to be scaled with SNR as: $$\begin{aligned}
\label{eqn:B}
B \approx & \frac{N (M - N)}{3}P_{dB} - N (M - N) \log_2(N(b^{\frac{1}{N}}-1))\ + \nonumber\\
& N (M - N) \log_2 \left[ \frac{\Gamma(\frac{1}{N(M-N)})}{N(M-N)} \right] - \log_2(C_{MN})\end{aligned}$$
This expression can be found by equating the upper bound from Theorem \[thm:1\] with $\log_2 b$ and solving for $B$ as a function of $P$. Solving this numerically will yield the number of bits sufficient for a maximum rate loss of $\log_2 b$. We assume that $B$ is large enough to neglect the exponential term in the expression for $\overline{D}$ from (\[eqn:D\]), which yields the above approximation.
The total contribution of the term containing the logarithm of the gamma function is very small and can usually be neglected. To maintain a system throughput loss of $M$ bps/Hz, which corresponds to an SNR gap of no more than $3$ dB with respect to BD with perfect CSIT, it is sufficient to scale the bits as: $$\label{eqn:BDSca}
B \approx \frac{N (M - N)}{3}P_{dB} - \log_2(C'_{MN})$$ where $C'_{MN} = N^{N(M-N)} C_{MN}$. Figure \[fig0\] shows the sufficient number of bits required to maintain this level of performance, when $N = 2$ and $M = 4, 6$ and $8$.
The pre-log factor (i.e. the factor that multiplies the SNR in dB) is $N (M-N)$ rather than $MN$, which is intuitively because the space of $N$ dimensional subspaces in an $M$ dimensional space has a dimensionality of $N( M-N)$
Performance Comparison and Numerical Results {#sec:perfcomp}
============================================
Zero forcing vs. Block diagonalization {#ssec:ZF}
--------------------------------------
Zero forcing is simple low-complexity linear precoding strategy, and it is important to compare the performance of these two schemes under the presence of limited feedback. Zero forcing for a MIMO broadcast system with $K$ users and $N$ antennas per user is equivalent to a $KN = M$ user system with a single antenna per user. The feedback scaling law for such a system is derived in [@JIND05] to be: $$\label{eqn:ZFSca}
B_{ZF} \approx \frac{(M - 1)}{3}P_{dB}$$ to maintain an SNR gap of no more than $3$ dB with respect to ZF under perfect CSIT conditions. In this system, each user with $N$ antennas quantizes the direction of the channel vector (i.e. the channel vector normalized to have norm unity) of each of the $N$ antennas separately, and feeds this back to the transmitter.
In general, if BD with perfect CSIT achieves a sum rate of $R_{CSIT-BD}(P)$ with $M$, $N$ antennas at the transmitter and each of the $\frac{M}{N}$ users respectively, and ZF achieves $R_{CSIT-ZF}(P)$ for the same system, $R_{CSIT-BD}(P)$ will eventually dominate $R_{CSIT-ZF}(P)$ by a constant amount. Thus, we see an immediate advantage of BD with respect to ZF from (\[eqn:BDSca\]), where the pre-log factor for BD is $N(M-N)$ for $N$ antennas, or $M-N$ per user antenna. This is compared to the factor $M-1$ in (\[eqn:ZFSca\]), which is for a lower target rate. This difference between $M-1$ and $M-N$ is perhaps due to the fact that the space of $N$ dimensional subspaces in an $M$ dimensional space has a dimensionality of $N(M-N)$ while the space of $N$ one-dimensional subspaces in an $M$ dimensional space has dimensionality $N(M-1)$.
The rate gap between BD and ZF with perfect CSIT is given by [@LEE06]: $$R_g(P) = K\log_2(e) \sum_{j=1}^N\frac{N-j}{j}$$ at high SNR. For fair comparison of the number of bits required for BD and ZF under imperfect CSIT and limited feedback, it is necessary to fix a common target rate. By setting $b = 2^{R_g(P) + R}$ in (\[eqn:B\]) where $R_g(P)$ is the (per-user) rate gap between BD and ZF (with perfect CSIT) and $R$ the target (per-user) rate loss for the ZF system, we can compare the *sufficient* number of bits required to achieve the same sum rate for both strategies. For example, $R = 1$ for a $3$ dB target offset in SNR, relative to rate achievable with ZF and perfect CSIT. This suggests a bit savings of 48% for an $M = 6, N = 2$ system at 15 dB, and 63% for an $M = 9, N = 3$ system with BD. The scaling law in Theorem \[thm:2\] is slightly conservative for large $b$, and the advantage of BD is somewhat underestimated. Numerical results show that the bit savings possible with BD are even higher.
An alternative antenna combining method (when the users have multiple antennas) is proposed in [@jindal2007ant], where each user receives only a single stream of data (as opposed to $N$ streams of data with BD), but uses the extra antennas to obtain a very accurate quantization of the effective channel. This effectively allows for a reduction in feedback load, and produces the same pre-log factor as BD, i.e., $N(M-N)$, but needs $N$ times the number of users in the system (i.e. $K = M$ where each user as $N$ antennas, rather than the $K = \frac{M}{N}$ for BD). Table \[tab1\] compares the [*sufficient*]{} number of bits required to achieve the same target rate, i.e., 3 dB (in SNR) away from ZF with perfect CSIT, when using BD, ZF and Antenna combining for an $M = 6, N = 2$ system. ZF and BD have $K = 3$, while antenna combining has $K = 6$.
-- -- -- --
-- -- -- --
Analog Feedback {#ssec:analogfb}
---------------
We consider here the case when each user $k$ feeds back its channel ${{\bf H}}_k$ by explicitly transmitting the $MN$ complex coefficients $\left({{\bf H}}_k\right)_{mn}, m = 1, \dots M, n = 1, \dots, N$ over the feedback channel. We assume that the uplink feedback channel is unfaded AWGN with the same SNR as the downlink (i.e., $P$). Each user may transmit each coefficient effectively ‘$\beta$’ times on the uplink, resulting in the following matrix being received at the transmitter: $$\begin{aligned}
{{\bf G}}_k &=& \sqrt{\beta P} {{\bf H}}_k + {{\bf N}}_k .\end{aligned}$$ Here, ${{\bf N}}_k$ represents the feedback (additive white Gaussian) noise, whose entries are independent and complex Gaussian with unit variance. As the coefficients of ${{\bf H}}_k$ are also independent and complex Gaussian with unit variance, the optimal estimator is the MMSE estimator: $$\begin{aligned}
\breve{{{\bf H}}}_k &=& \frac{\sqrt{\beta P}}{1 + \beta P} {{\bf G}}_k, \end{aligned}$$ where $\breve{{{\bf H}}}_k$ is the estimate of ${{\bf H}}_k$ formed at the transmitter. It is convenient to express ${{\bf H}}_k$ in terms of the estimate $\breve{{{\bf H}}}_k$ and estimation noise as follows: $$\begin{aligned}
{{\bf H}}_k &=& \breve{{{\bf H}}}_k + \frac{1}{\sqrt{1 + \beta P}} {{\bf F}}_k,\end{aligned}$$ where the entries of ${{\bf F}}_k$ are also independent and complex Gaussian with unit variance, and independent of the estimator.
The beamformers $\{\breve{{{\bf V}}}_k\}_{k=1}^K$ are selected by treating $\{\breve{{{\bf H}}}_k\}_{k=1}^K$ as the ‘true’ set of channels, and following the BD procedure. Note that the marginal distribution of the beamformers are the same as in the quantized feedback case, as the addition of independent white Gaussian noise does not affect the isotropic property. As in the case for quantized (digital) feedback, we compute the quantity: $$\begin{aligned}
\label{int_ana}
{{\bf H}}_k^{{\sf H}}\breve{{{\bf V}}}_j &=& \frac{1}{\sqrt{1 + \beta P}} {{\bf F}}_k^{{\sf H}}\breve{{{\bf V}}}_j\end{aligned}$$ for $k \neq j$, which follows from the fact that $\breve{{{\bf H}}}_k^{{\sf H}}\breve{{{\bf V}}}_j = {\bf 0}$ for $k \neq j$. Similar to (\[eqn:R\_FB\]), we write the rate with ‘analog’ feedback as follows: $$\begin{aligned}
R_\textsc{Analog}(P) = {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1}^K {{\bf H}}_k^{{\sf H}}\breve{{{\bf V}}}_j\breve{{{\bf V}}}_j^{{\sf H}}{{\bf H}}_k \right| \right] - {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1, j \neq k}^K {{\bf H}}_k^{{\sf H}}\breve{{{\bf V}}}_j\breve{{{\bf V}}}_j^{{\sf H}}{{\bf H}}_k \right| \right]\end{aligned}$$
Similar to the proof of Theorem \[thm:1\] and using techniques similar to those in [@FastCSI], we compute a bound on the rate gap relative to BD with perfect CSIT to be: $$\begin{aligned}
\Delta R_\textsc{Analog}(P) & = & \left[ R_\textsc{CSIT-BD}(P) - R_\textsc{Analog}(P) \right]\\
& \leq & N\ \log_2 \left(1 + \frac{M-N}{M} \frac{P}{1 + \beta P} \right) \label{analogbound}\\
& < & N\ \log_2 \left(1 + \frac{M-N}{M} \frac{1}{\beta} \right) \label{analogbound2}\end{aligned}$$
The proof (\[analogbound\]) bound is given in Appendix \[analogproof\]. (\[analogbound2\]) is obtained by letting $P \rightarrow \infty$ in (\[analogbound\]).
In order to compare analog and quantized feedback, we measure the feedback quantity in terms of ‘feedback symbols’ rather than bits. Although analog feedback involves effectively $\beta MN$ channel uses per user (assuming that the users have orthogonal feedback channels), it also conveys more information that the quantized case, specifically information regarding the eigenvalues and eigenvector structure, which the ‘subspace’ information does not capture.
Hence, for fair comparison, we equate the $\beta MN$ analog channel uses to $\beta N(M-N)$ channel symbols in the quantized case (the ‘subspace’ information may be specified by $N(M-N)$ complex numbers). Under the simplifying assumption that error-free communication at capacity is possible, we set $B = \beta N(M-N) \log_2(1 + P)$ for $\beta N(M-N)$ channel uses of the AWGN feedback channel with SNR $P$. From Theorem \[thm:1\], we have: $$\begin{aligned}
\Delta R_\textsc{Quant}(P) & \leq & N \log_2\left(1 + \frac{P}{N} \frac{\Gamma\left((N(M-N))^{-1}\right)}{N(M-N)}C_{MN}^{\left(N(M-N)\right)^{-1}} 2^{-\frac{B}{N(M-N)}}\right)\\
& = & N \log_2\left(1 + \frac{P}{(1+P)^\beta} C''_{MN}\right)\end{aligned}$$ where $D$ has been bounded from (\[eqn:D\]) (neglecting the exponential term), and $$C''_{MN} = \frac{\Gamma\left((N(M-N))^{-1}\right)}{N^2(M-N)} C_{MN}^{\left(N(M-N)\right)^{-1}} .$$
Our conclusions are similar to the $N = 1$ case, which was considered in [@caire2007]. For $\beta \approx 1$, both bounds on the rate gap (i.e. for analog and quantized feedback) behave similarly, and the gap does not vanish as $P \rightarrow \infty$. For $\beta > 1$, the rate gap bound decreases rapidly (exponentially fast) for quantized feedback, and vanishes entirely as $P \rightarrow \infty$. However, for analog feedback, the decrease is relatively slow (i.e. only polynomially fast) and does not vanish as $P \rightarrow \infty$. The analysis may also be extended to the case when errors occur with quantized feedback, using techniques similar to those in [@caire2007].
Generation of Numerical Results {#sec:simmethod}
-------------------------------
The number of bits given by (\[eqn:B\]) can be very large and numerical simulation becomes a computationally complex task, as the chordal distance will have to be calculated for each of the $2^B$ matrices in the codebook. However, utilizing the statistics of random codebooks, the quantization procedure can be precisely [*emulated*]{} without having to do actual quantization. From Lemma \[lemma1\], we can repeat the argument by interchanging $\widetilde{{{\bf H}}}_k$ and ${{\bf \widehat{ H}}}_k$, to yield the following equivalent decomposition: $${{\bf \widehat{ H}}}_k = \widetilde{{{\bf H}}}_k {{\bf X}}_k {{\bf Y}}_k + {{\bf S}}_k {{\bf Z}}_k$$ which can be used to generate ${{\bf \widehat{ H}}}_k$, given $\widetilde{{{\bf H}}}_k$ and a codebook size. ${{\bf X}}_k$ is isotropic and independent of the codebook size, as is ${{\bf S}}_k$ which (in this decomposition) is isotropically distributed in the left nullspace of $\widetilde{{{\bf H}}}_k$. Samples drawn from the distribution of these matrices can thus be generated as samples from the isotropic distribution in their respective spaces.
Moreover, $d^2\left( \widetilde{{{\bf H}}}_k, {{\bf \widehat{ H}}}_k \right) = {{\hbox{tr}}}\left( {{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k \right)$ is the $1^\text{st}$ order statistic from $2^B$ samples. Here, each sample is drawn from the distribution of the trace of a matrix-variate beta distribution (as described in Appendix \[lem1proof\]). Thus, a sample drawn from the distribution of ${{\hbox{tr}}}\left( {{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k \right)$ can be generated by the ‘CDF inversion’ method, by computing the CDF for a specific $M$ and $N$. A general expression for the CDF has been computed in closed form in [@DAI06], for the case when $d^2\left( \widetilde{{{\bf H}}}_k, {{\bf \widehat{ H}}}_k \right) \leq 1$. For moderate to large $B$ and practical values of $M$, $N$, this event occurs with extremely high probability, allowing for low complexity CDF inversion. For very small values of $B$, $d^2\left( \widetilde{{{\bf H}}}_k, {{\bf \widehat{ H}}}_k \right)$ may be greater than 1 with appreciable probability, but an exhaustive searching among $2^B$ possibilities is not a problem in these cases.
From the eigen decomposition ${{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k = {{\bf E}}_k{{\bf D}}_k{{\bf E}}_k^{{\sf H}}$, as described in Appendix \[thm1proof\], ${{\bf E}}_k$ can be generated as the eigenvectors of any (complex) Beta$(N, M-N)$ distributed matrix. Further, the distribution of the eigenvalues (i.e., the entries of ${{\bf D}}_k$) [*conditioned*]{} on their sum (which is equal to $d^2(\widetilde{{{\bf H}}}_k, {{\bf \widehat{ H}}}_k)$), can be computed from their joint distribution [@muirhead1982ams] ([@DAI06] for the complex case). The conditional distribution can be easily computed for small values of $N$.
In particular, for $N = 2$, if $D_1, D_2$ are the diagonal elements of ${{\bf D}}_k$ with joint density $f_{D_1, D_2}(d_1, d_2)$, the distribution of $D_1$ conditioned on $Z = D_1 + D_2 \leq 1$ is given as: $$\begin{aligned}
F_{D_1|Z}(d_1|z) & = & \frac{\int\limits_{0}^z f_{D_1, D_2}(d_1, z-d_1)\ {d}(d_1)}{f_Z(z)}\\
& = & \frac{\int\limits_{0}^z V_{M} (z-2d_1)^2(1-d_1)^{M-4}(1-z+d_1)^{M-4}\ {d}(d_1)}{f_Z(z)}\end{aligned}$$ where $f_Z(z)$ is the pdf of $Z$ computed to be: $$f_Z(z) = \frac{z^{2M-5} (\Gamma(M))^2}{(M-1)\Gamma(2M-4)}$$ for $z \leq 1$. $V_M$ is a normalizing constant and is given by $V_M = \frac{1}{2} (M-1)(M-2)^2(M-3)$. For efficient CDF inversion, $F_{D_1|Z}(d_1|z)$ can be computed in closed form for specific values of $M$.
As ${{\bf Y}}_k^{{\sf H}}{{\bf Y}}_k = {\bf I}_N - {{\bf Z}}_k{{\bf Z}}_k^{{\sf H}}$, ${{\bf Y}}_k$ can be obtained as well. Putting all this together, one is able to randomly generate a realization of the quantized version of $\widetilde{{{\bf H}}}_k$, when random codebooks are used. This prevents the computational complexity from growing with $B$. However, for extremely large $B$, numerical errors may dominate and care must be taken to maintain numerical precision.
Numerical Results {#sec:simresults}
-----------------
![MIMO Broadcast Channel with $M = 4, N = 2, K = 4$[]{data-label="fig1"}](fig1){width="\ImgWidth"}
![MIMO Broadcast Channel with $M = 6, N = 2, K = 4$[]{data-label="fig2"}](fig2){width="\ImgWidth"}
![MIMO Broadcast Channel with $M = 8, N = 2, K = 4$[]{data-label="fig3"}](fig3){width="\ImgWidth"}
We present numerical results for $N=2$ and $M = 4, 6, 8$ in Figures \[fig1\], \[fig2\], \[fig3\] respectively, while scaling the bits as per (\[eqn:BDSca\]), i.e. with a target of staying at most 3 dB away (in SNR) from BD with perfect CSIT. As Theorem \[thm:2\] only provides the sufficient number of bits, this is a conservative strategy and the actual SNR gaps are found to be $2.65$ dB, $2.72$ dB and $2.84$ dB for $M = 4, 6$ and $8$ respectively, instead of $3$ dB. The results also show that keeping the number of bits fixed will result in a rate gap that increases unbounded with SNR.
Conclusion {#sec:concl}
==========
Accurate CSIT is clearly important for MIMO broadcast systems in order to achieve maximum throughput. When the receiver knows the channel perfectly and instantaneously feeds this information back to the transmitter using a finite number of bits, we have quantified the rate loss and have shown that increasing the number of bits linearly with the system SNR is sufficient to maintain a constant SNR loss with respect to perfect CSIT. Further, we have established the advantage of BD relative to ZF in terms of feedback load, and the advantage of using quantized feedback as opposed to using analog feedback. Note that BD is just one of many linear precoding techniques that can be used on the MIMO broadcast channel with multiple user antennas (for e.g., see coordinated beamforming [@chae2006cbm] and Multiuser Eigenmode Transmission [@boccardi2007not]). It remains to be seen which of these perform best in a limited feedback setting and also when multiuser diversity/user selection is considered.
Proof of Lemma \[lemma1\] {#lem1proof}
=========================
Let ${{\bf W}}$ be any arbitrary matrix in the codebook ${{\cal C}}$. Note that ${{\bf W}}$ is independent of $\widetilde{{{\bf H}}}_k$. We then decompose $\widetilde{{{\bf H}}}_k$ into components that lie in the column space of ${{\bf W}}$ and the left nullspace of ${{\bf W}}$ as follows: $$\begin{aligned}
\widetilde{{{\bf H}}}_k & = & {{\bf W}}{{\bf W}}^{{\sf H}}\widetilde{{{\bf H}}}_k + \left( {\bf I}_M - {{\bf W}}{{\bf W}}^{{\sf H}}\right) \widetilde{{{\bf H}}}_k \\
& = & {{\bf W}}{{\bf W}}^{{\sf H}}\widetilde{{{\bf H}}}_k + {{\bf W}}^\bot({{\bf W}}^\bot)^{{\sf H}}\widetilde{{{\bf H}}}_k\end{aligned}$$ where ${{\bf W}}{{\bf W}}^{{\sf H}}$ and ${{\bf W}}^\bot({{\bf W}}^\bot)^{{\sf H}}= {\bf I}_M - {{\bf W}}{{\bf W}}^{{\sf H}}$ are the projection matrices for the column space and left nullspace of ${{\bf W}}$ respectively. ${{\bf W}}^\bot \in {\mbox{\bb C}}^{M \times (M-N)}$ is chosen such that it forms an orthonormal basis for the left nullsapce of ${{\bf W}}$.
Let the (thin) QR decomposition of ${{\bf W}}{{\bf W}}^{{\sf H}}\widetilde{{{\bf H}}}_k$ be ${{\bf Q}}_k {{{\bf A}}}_k$ where ${{\bf Q}}_k \in {\mbox{\bb C}}^{M \times N}$ forms an orthonormal basis for the same space as ${{\bf W}}$, and ${{\bf A}}_k \in {\mbox{\bb C}}^{N \times N}$ is upper triangular with positive diagonal elements. Further, ${{\bf Q}}_k$ and ${{\bf A}}_k$ are independent, from [@gupta2000mvd Theorem 2.3.18] (after verification for the complex case). As ${{\bf Q}}_k$ and ${{\bf W}}$ describe the same subspace, ${{\bf Q}}_k$ may be represented as a rotation of ${{\bf W}}$, i.e., ${{\bf Q}}_k = {{\bf W}}{{\bf X}}_k$ for some unitary matrix ${{\bf X}}_k \in {\mbox{\bb C}}^{N \times N}$.
By isotropy and independence of ${{\bf W}}$ and $\widetilde{{{\bf H}}}_k$, ${{\bf X}}_k$ is also isotropically distributed and is independent of ${{\bf W}}$, which is an arbitrary orthonormal basis. Also note that ${{\bf W}}{{\bf W}}^{{\sf H}}= {{\bf Q}}_k{{\bf Q}}_k^{{\sf H}}$ and hence ${{\bf A}}_k^{{\sf H}}{{\bf A}}_k = \widetilde{{{\bf H}}}_k^{{\sf H}}{{\bf W}}{{\bf W}}^{{\sf H}}\widetilde{{{\bf H}}}_k$. Thus ${{\hbox{tr}}}\left({{\bf A}}_k^{{\sf H}}{{\bf A}}_k\right) = N - d^2\left({{\bf W}}, \widetilde{{{\bf H}}}_k\right)$.
Note that ${{\bf W}}^\bot({{\bf W}}^\bot)^{{\sf H}}\widetilde{{{\bf H}}}_k$ is the projection of $\widetilde{{{\bf H}}}_k$ onto the left nullspace of ${{\bf W}}$. As $\widetilde{{{\bf H}}}_k$ is isotropically distributed, the projection is also isotropically distributed in the corresponding $M-N$ dimensional nullspace. Let the (thin) QR decomposition of ${{\bf W}}^\bot({{\bf W}}^\bot)^{{\sf H}}\widetilde{{{\bf H}}}_k$ be ${{\bf S}}_k {{\bf B}}_k$, where ${{\bf S}}_k \in {\mbox{\bb C}}^{M \times N}$ is an orthonormal basis for an isotropically distributed (complex) $N$ dimensional plane in the $M-N$ dimensional left nullspace of ${{\bf W}}$ and ${{\bf B}}_k \in {\mbox{\bb C}}^{N \times N}$ is upper triangular with positive diagonal elements. Similar to the previous case, ${{\bf S}}_k$ and ${{\bf B}}_k$ are independently distributed. It is also straightforward to see that ${{\bf B}}_k^{{\sf H}}{{\bf B}}_k = {\bf I}_N - {{\bf A}}_k^{{\sf H}}{{\bf A}}_k$ and ${{\hbox{tr}}}\left({{\bf B}}_k^{{\sf H}}{{\bf B}}_k\right) = d^2\left({{\bf W}}, \widetilde{{{\bf H}}}_k\right)$.
As $\widetilde{{{\bf H}}}_k$ and ${{\bf W}}$ are independent, which has been our assumption thus far in the proof, ${{\bf B}}_k^{{\sf H}}{{\bf B}}_k$ is matrix-variate (complex) Beta$(N, M-N)$ distributed [@muirhead1982ams]. We will now argue that most of the above conclusions remain unchanged, even when the quantization procedure (\[quantproc\]) is followed.
The quantization procedure amounts to choosing a ${{\bf B}}_k^{{\sf H}}{{\bf B}}_k$ such that its trace is the minimum among $2^B$ choices. Thus, it follows that the quantization procedure only affects ${{\bf B}}_k$ (and ${{\bf A}}_k$, which is the ‘inverse’ quantization error and is related to ${{\bf B}}_k$ by ${{\bf A}}_k^{{\sf H}}{{\bf A}}_k = {\bf I}_N - {{\bf B}}_k^{{\sf H}}{{\bf B}}_k$). We use ${{\bf Y}}_k$ and ${{\bf Z}}_k$ to denote the matrices ${{\bf A}}_k$ and ${{\bf B}}_k$ after following the quantization procedure. Hence, even though ${{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k$ is not beta distributed, the distribution of the quantities ${{\bf X}}_k$, ${{\bf S}}_k$ and ${{\bf W}}$ remain the same, and are independent of ${{\bf Z}}_k$ (and ${{\bf Y}}_k$). We now use ${{\bf \widehat{ H}}}_k$ to denote ${{\bf W}}$ after following the quantization procedure, according to the convention in (\[quantproc\]).
Proof of Theorem \[thm:1\] {#thm1proof}
==========================
Theorem \[thm:1\] is proved as follows: $$\begin{aligned}
\Delta R_\textsc{Quant}(P) & = & \left[ R_\textsc{CSIT-BD}(P) - R_\textsc{Quant}(P) \right]\\
& \mathop{\leq}\limits^{\text{\tiny{(a)}}} & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M}{{\bf H}}_k^{{\sf H}}{{\bf V}}_k{{\bf V}}_k^{{\sf H}}{{\bf H}}_k \right| \right] - \nonumber\\
& & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M}\ {{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_k{{\bf \widehat{ V}}}_k^{{\sf H}}{{\bf H}}_k \right| \right] + \nonumber\\
& & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1, j \neq k}^K {{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf H}}_k \right| \right] \\
& \mathop{=}\limits^{\text{\tiny{(b)}}} & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1, j \neq k}^K{{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf H}}_k \right| \right]\\
& \mathop{=}\limits^{\text{\tiny{(c)}}} & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \widetilde{{{\bf H}}}_k^{{\sf H}}\left( \sum\limits_{j \neq k} {{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}\right) \widetilde{{{\bf H}}}_k {\bf \Lambda}_k \right| \right]\\
& \mathop{\leq}\limits^{\text{\tiny{(d)}}} & \log_2\ \left| {\bf I}_N + \frac{P(K-1)}{M} {\mbox{\bb E}}\left[ \widetilde{{{\bf H}}}_k^{{\sf H}}\left( {{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}\right)\widetilde{{{\bf H}}}_k \right] M\right|\\
& \mathop{=}\limits^{\text{\tiny{(e)}}} & \log_2\ \left| {\bf I}_N + P(K-1) {\mbox{\bb E}}\left[ {{\bf Z}}_k^{{\sf H}}\left( {{\bf S}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j {{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf S}}_k\right) {{\bf Z}}_k \right] \right|\\
& \leq & N\ \log_2 \left(1 + \frac{P}{N} D \right)\end{aligned}$$
Here, (a) follows by neglecting the positive semi-definite interference terms in the quantity: $${\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1}^K {{\bf H}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf H}}_k \right| \right].$$
By the BD procedure, both ${{\bf V}}_k$ and ${{\bf \widehat{ V}}}_k$ are distributed isotropically, and are chosen independent of ${{\bf H}}_k$, which results in (b). We write ${{\bf H}}_k{{\bf H}}_k^{{\sf H}}= \widetilde{{{\bf H}}}_k {\bf \Lambda}_k \widetilde{{{\bf H}}}_k^H$, where $\widetilde{{{\bf H}}}_k \in {\mbox{\bb C}}^{M \times N}$ forms an orthonormal basis for the subspace spanned be the columns of ${{\bf H}}_k$ and ${\bf \Lambda}_k = \text{diag}[\lambda_1, \dots, \lambda_N]$ are the $N$ non-zero, unordered eigenvalues of ${{\bf H}}_k{{\bf H}}_k^{{\sf H}}$ (${{\bf H}}_k$ is of rank $N$ and diagonalizable with probability 1). Both the density function of ${{\bf H}}_k$ (which is matrix-variate complex Normal distributed) [@gupta2000mvd] and the Jacobian of the singular value decomposition transformation of a matrix [@edelman2005rmt] can be separated into a product of functions of $\widetilde{{{\bf H}}}_k$ and ${\bf \Lambda}_k$ alone. Thus, $\widetilde{{{\bf H}}_k}$ and ${\bf \Lambda}_k$ are independent and ${\mbox{\bb E}}\left[ {\bf \Lambda}_k \right] = M{\bf I}_N$. Step (c) follows using this and the fact that $\left|{\bf I} + {{\bf A}}{{\bf B}}\right|$ = $\left|{\bf I} + {{\bf B}}{{\bf A}}\right|$, for matrices ${{\bf A}}$ and ${{\bf B}}$. Next, (d) follows from Jensen’s inequality due to the concavity of $\log|\cdot|$. Step (e) is proved as follows. First, we compute $$\begin{aligned}
\widetilde{{{\bf H}}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j & = & {{\bf Y}}_k^{{\sf H}}{{\bf X}}_k^{{\sf H}}{{\bf \widehat{ H}}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j + {{\bf Z}}_k^{{\sf H}}{{\bf S}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j\\
& = & {{\bf Z}}_k^{{\sf H}}{{\bf S}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j\end{aligned}$$ for $k \neq j$, which follows from Lemma \[lemma1\] and the fact that ${{\bf \widehat{ H}}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j = {\bf 0}\ \forall k \neq j$, by the BD procedure. Therefore, $$\begin{aligned}
\log_2\ \left| {\bf I}_N + P(K-1) {\mbox{\bb E}}\left[ \widetilde{{{\bf H}}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}\widetilde{{{\bf H}}}_k \right] \right| & = & \log_2\ \left| {\bf I}_N + P(K-1) {\mbox{\bb E}}\left[ {{\bf Z}}_k^{{\sf H}}\left( {{\bf S}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf S}}_k\right) {{\bf Z}}_k \right] \right|\nonumber\\
& \mathop{=}\limits^{\text{\tiny{(f)}}} & \log_2\ \left| {\bf I}_N + P(K-1) \frac{N}{M-N} {\mbox{\bb E}}\left[ {{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k \right] \right|\\
& \mathop{=}\limits^{\text{\tiny{(g)}}} & \log_2\ \left| {\bf I}_N + P(K-1) \frac{D}{N} \frac{N}{M-N} \right|\end{aligned}$$ Here, (f) follows from the fact that ${{\bf \widehat{ V}}}_j$ (which is just isotropically distributed in the left nullspace of ${{\bf \widehat{ H}}}_k$) and ${{\bf Z}}_k$ are independent, as are ${{\bf S}}_k$ and ${{\bf Z}}_k$ from Lemma \[lemma1\]. Further, ${{\bf S}}_k$ is also isotropically and distributed in the left nullspace of ${{\bf \widehat{ H}}}_k$, and is independent of ${{\bf \widehat{ V}}}_k$. Thus ${{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf S}}_k {{\bf S}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j$ is matrix-variate Beta$(N, M-2N)$ distributed [@gupta2000mvd], and ${\mbox{\bb E}}\left[ {{\bf Z}}_k^{{\sf H}}\left( {{\bf S}}_k^{{\sf H}}{{\bf \widehat{ V}}}_j{{\bf \widehat{ V}}}_j^{{\sf H}}{{\bf S}}_k\right) {{\bf Z}}_k \right] = \frac{N}{M-N} {\mbox{\bb E}}\left[ {{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k \right]$, by [@gupta2000mvd Theorem 5.3.12] and [@gupta2000mvd Theorem 5.3.19] (after verification for the complex case).
Let ${{\bf E}}_k{{\bf D}}_k{{\bf E}}_k^{{\sf H}}$ be the eigen decomposition of ${{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k$, where ${{\bf E}}_k \in {\mbox{\bb C}}^{N \times N}$ is orthonormal and ${{\bf D}}_k \in {\mbox{\bb C}}^{N \times N}$ is diagonal, with strictly positive elements along the diagonal. If an arbitrary matrix in the codebook ${{\cal C}}$ is selected as the quantization, ${{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k$ is matrix-variate (complex) Beta$(N, M-N)$ distributed (as described in Appendix \[lem1proof\]), and ${\mbox{\bb E}}\left[{{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k \right]$ is a multiple of the identity matrix. Both the density function of this distribution [@gupta2000mvd] and the Jacobian of the eigen decomposition transformation for a matrix [@edelman2005rmt] can be separated into a product of functions of ${{\bf E}}_k$ and ${{\bf D}}_k$ alone, and these are hence independently distributed.
For the actual quantization matrix, after following the procedure in (\[quantproc\]), only the distribution of the diagonal matrix ${{\bf D}}_k$ is affected, and the distribution of ${{\bf E}}_k$ remains unchanged and independent of ${{\bf D}}_k$. Thus, we have that ${\mbox{\bb E}}\left[ {{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k \right] = \rho {\bf I}_N$ for some constant $\rho$, even after following the quantization procedure. This can also be concluded by observing that ${{\bf Z}}_k^{{\sf H}}{{\bf Z}}_k$ is invariant to unitary rotations. In terms of the trace of the matrix, we have $\rho = \frac{{\mbox{\bb E}}\left[{{\hbox{tr}}}\left({{\bf Z}}_k{{\bf Z}}_k^{{\sf H}}\right)\right]}{N} = \frac{D}{N}$, and (g) follows.
Proof of equation (\[analogbound\]) {#analogproof}
===================================
$\Delta R_\textsc{Analog}(P) = \left[ R_\textsc{CSIT-BD}(P) - R_\textsc{Analog}(P) \right]$ $$\begin{aligned}
& \mathop{\leq}\limits^{\text{\tiny{(a)}}} & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M}{{\bf H}}_k^{{\sf H}}{{\bf V}}_k{{\bf V}}_k^{{\sf H}}{{\bf H}}_k \right| \right] - \nonumber\\
& & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M}\ {{\bf H}}_k^{{\sf H}}\breve{{{\bf V}}}_k\breve{{{\bf V}}}_k^{{\sf H}}{{\bf H}}_k \right| \right] + \nonumber\\
& & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1, j \neq k}^K {{\bf H}}_k^{{\sf H}}\breve{{{\bf V}}}_j\breve{{{\bf V}}}_j^{{\sf H}}{{\bf H}}_k \right| \right] \\
& \mathop{=}\limits^{\text{\tiny{(b)}}} & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \sum\limits_{j = 1, j \neq k}^K{{\bf H}}_k^{{\sf H}}\breve{{{\bf V}}}_j\breve{{{\bf V}}}_j^{{\sf H}}{{\bf H}}_k \right| \right]\\
& \mathop{=}\limits^{\text{\tiny{(c)}}} & {\mbox{\bb E}}\left[ \log_2 \left| {\bf I}_N + \frac{P}{M} \frac{1}{1 + \beta P} \sum\limits_{j = 1, j \neq k}^K{{\bf F}}_k^{{\sf H}}\breve{{{\bf V}}}_j\breve{{{\bf V}}}_j^{{\sf H}}{{\bf F}}_k \right| \right]\\
& \mathop{\leq}\limits^{\text{\tiny{(d)}}} & \log_2\ \left| {\bf I}_N + \frac{P(K-1)}{M} \frac{1}{1 + \beta P} {\mbox{\bb E}}\left[ {{\bf F}}_k^{{\sf H}}\breve{{{\bf V}}}_j\breve{{{\bf V}}}_j^{{\sf H}}{{\bf F}}_k \right] \right|\\
& \mathop{=}\limits^{\text{\tiny{(e)}}} & \log_2\ \left| {\bf I}_N + \frac{P(K-1)}{M} \frac{1}{1 + \beta P} N {\bf I}_N \right|\\
& = & N\ \log_2 \left(1 + \frac{M-N}{M} \frac{P}{1 + \beta P} \right)\end{aligned}$$ Here, (a) and (b) have the same justification as in the proof of Theorem \[thm:1\] (in Appendix \[thm1proof\]), (c) follows from (\[int\_ana\]), and (d) is obtained by applying Jensens inequality. By Gaussianity of ${{\bf F}}_k$ and independence of ${{\bf F}}_k$ and $\breve{{{\bf V}}}_j$, ${{\bf F}}_k^{{\sf H}}\breve{{{\bf V}}}_j$ is matrix-variate complex Gaussian distributed with i.i.d. elements, and ${\mbox{\bb E}}\left[ {{\bf F}}_k^{{\sf H}}\breve{{{\bf V}}}_j\breve{{{\bf V}}}_j^{{\sf H}}{{\bf F}}_k \right] = N {\bf I}_N$, which results in (e).
[^1]: Note that a factor of $N$ was erroneously omitted from this bound when this result was stated in [@ravindran2007mbc].
|
---
abstract: 'We study the longitudinal instabilities of two interpenetrating fluids interacting only through gravity. When one of the constituents is of relatively low density, it is possible to have a band of unstable wave numbers well separated from those involved in the usual Jeans instability. If the initial streaming is large enough, and there is no linear instability, the indefinite sign of the free energy has the possible consequence of explosive interactions between positive and negative energy modes in the nonlinear regime. The effect of dissipation on the negative energy modes is also examined.'
---
epsf [](\hspace{.2cm})
[**NEGATIVE ENERGY MODES AND GRAVITATIONAL**]{}
[**INSTABILITY OF INTERPENETRATING FLUIDS**]{}
A. R. R. CASTI$,^{a}$ P. J. MORRISON$,^{b}$ AND E. A. SPIEGEL$^{c}$
${ }^{a}$[*Department of Applied Physics/*]{}
[*Division of Applied Mathematics*]{}
${ }^{b}$[*Department of Physics and Institute for Fusion Studies*]{}
${ }^{c}$[*Department of Astronomy*]{}
—— P. A. M. Dirac, 1930
BACKGROUND {#sec-gravback}
==========
It used to be thought that the stellar component of two spiral galaxies would pass right through each other in the event of a collision and that only the gaseous components would merge. However, simulations over the past twenty years or so$^{1,2}$ have shown that the macroscopic energy of such a collision is quickly converted into internal energy and that merger of the stellar systems is a common natural outcome of a collision. How is this conversion effected?
An answer to this may lie in the physics of streaming instabilities. In the context of plasma physics, interpenetrating electron beams produce the two-stream instability$^{3}$ whose gravitational analog has long been recognized, beginning with the investigations of Sweet$^{4}$ and Lynden-Bell.$^{5}$ Of course, in the case of galaxy collisions, which occur quickly, the conventional two-stream instability may operate too slowly to be effective. However, it is known that even when two streams of interacting particles are not linearly unstable, they may collectively produce negative energy modes that lead to an explosive nonlinear growth of perturbations for arbitrarily small disturbances. There are well-developed criteria for the occurrence of this explosive growth in plasma physics$^{6}$ and, as Lovelace [*et al.*]{}$^{7}$ have suggested in their analysis of counter-rotating galaxies, we may expect something analogous in the gravitational setting. Our aim here is to briefly develop this topic of negative energy modes for the case of gravitational interaction in the expectation that the phenomena involved will be found significant in a variety of astrophysical processes.
Even in the event of linear instability, the case of counterstreaming populations is significantly different from the standard gravitational instability, which occurs only for perturbation scales greater than the Jeans length.$^{8}$ When there are two interpenetrating fluids such as stars and gas, modes of arbitrary wavelength can be rendered unstable. Numerous investigators have reported on these issues. Most of them (such as Ikeuchi et al.,$^{9}$ Fridman & Polyachenko,$^{10}$ and Araki$^{11}$) focused primarily on the symmetric situation of identical fluids in counterstreaming motion. In that case one finds that the spectrum of any instabilities arising from the relative motion is wholly contained within the Jeans instability band, and this blurs the distinction between the two processes. This need not be true when this symmetry is broken, and indeed not all authors restricted themselves entirely to the symmetric case. The present venture into the asymmetric problem is intended to focus on the possibility of well-separated instability bands, which has not been elucidated in the gravitational context, as far as we are aware.
EQUATIONS OF MOTION {#sec-graveqns}
===================
To see the problem in its simplest version, it is useful to have a uniform medium as the unperturbed state. Rather than formulate the problem inconsistently to achieve this end, as Jeans did, we prefer the Einstein device of introducing a cosmological repulsion term. In the Newtonian setting we readily see how to redefine the gravitational potential so that, instead of introducing such a repulsion term, we fill space with a fluid of [*negative*]{} gravitational mass of density $\rho_{_\Lambda}$. As in the one-fluid plasma model, we treat this density as a constant since its purpose is to allow a gravitationally neutral background state. One may also contemplate the analog of the two-fluid plasma model in which this background antigravitational fluid has its own dynamics, but we do not do that here. The two dynamically active fluids we consider are gravitationally ordinary and polytropic. Thus, the Poisson equation is written here as, \^2 V=4G(\_1+\_2-\_[\_]{}), \[eq:3Dgrav\] where $V$ is the gravitational potential, $\rho_1$ and $\rho_2$ are the source densities of the conventional fluids, and $-\rho_{_\Lambda}$ is the cosmological background density.
The equations of motion for the two fluids ($j=1,2$) are, \_j(\_t+[**u**]{}\_j)[**u**]{}\_j&=& -p\_j-\_jV \[eq:3Dmom\]\
\_t \_j+(\_j[**u**]{}\_j)&=&0, \[eq:3Dmass\] where we do [*not*]{} sum over repeated indices. Each fluid has a sound speed, $c_{j}^{2}=$, and a Jeans wavenumber, $k_{{Jj}}^{2}=$[${\frac{8\pi G{\widehat\rho_j}}{c_{j}^{2}}}$]{}, where the hat signifies the equilibrium value and an uncustomary factor of $2$ appears in the definition of the Jeans wavenumbers.
We shall use natural units with $k_{_{{J1}}}^{-1}$ as the length scale and $(k_{_{{J1}}}c_1)^{-1}$ as the time scale. We further simplify the description by considering only longitudinal motions in one-dimension, so each single-component velocity field may be expressed as the gradient of a velocity potential: $u_j=\partial_x\phi_j$. The fundamental equations (\[eq:3Dgrav\])-(\[eq:3Dmass\]) take the dimensionless form, M\_[1]{}\_t\_1+M\_[1]{}\^[2]{}(\_x\_1)\^2+ +V&=&B\_[\_1]{} \[eq:bern1\]\
cM\_[2]{}\_t\_2+c\^2 M\_[2]{}\^[2]{}(\_x\_2)\^2+ +V&=&B\_[\_2]{} \[eq:bern2\]\
\_t\_1+M\_[1]{}\_x(\_1\_x\_1)&=&0 \[eq:mass1\]\
\_t\_2+cM\_[2]{}\_x(\_2\_x\_2)&=&0 \[eq:mass2\]\
\_[x]{}\^[2]{}V-\_1- \_2+ &=&0,\[eq:grav\] where the $M_{j}=$[${\frac{{\cal U}_j}{c_j}}$]{} are Mach numbers, ${\cal U}_j$ measures the initial streaming velocities, $\beta=$[${\frac{\widehat\rho_2}{{\widehat\rho_1}}}$]{}, $c=$[${\frac{c_2}{c_1}}$]{}, and the Bernoulli constants $B_{_j}$ are chosen to balance the basic state.
LINEAR THEORY {#sec-gravlin}
=============
We perturb from the state of uniform densities and constant velocities by setting $\phi_j=(-1)^{j+1}x+\delta\phi_j$, $\rho_j = 1+ \delta \rho_j$, and $V={\widehat V}+\delta V$. The density terms of the Poisson equation (\[eq:grav\]) combine to vanish and ${\widehat V}$ is a constant. Since the linearized equations are separable we may decompose the perturbations into normal modes proportional to $\exp(i\omega t-ikx)$ to find the dispersion relation, (,k) = 1 ++ 1+\_1+\_2=0. \[eq:diagravic\] The quantity $\Gamma$, which we call the diagravic function by analogy with the dielectric function of electrodynamics, measures the collective response of the fluid to disturbances in the gravitational field and will serve to indicate the energy signature of any normal mode (section \[sec-gravnonlin\]).
For real $k$ the solutions of (\[eq:diagravic\]) with complex $\omega$ correspond to instability; if $\omega$ is real, then solutions of (\[eq:diagravic\]) with complex $k$ can give rise to wave amplification instability. Here we analyze only the case of real $k$. However, we have to deal with both the traditional Jeans instability as well as the two-stream instability, the latter of which involves a sympathetic bunching of particles and is effective for creating instability when the phase speed of the disturbance conspires to create a resonance between different modes.
**Symmetric Case** {#sec-symm}
------------------
If both fluids have the same basic properties ($c=1$ and $\beta=1$), a frame exists in which $M_{_1}=M_{_2}=M$. The dispersion relation then simplifies into a manageable biquadratic with solutions, \_\^[2]{}= -+k\^2(M\^2+1). \[eq:symroots\]
For $M=0$, we recover a simple version of the previously studied two-fluid Jeans problem.$^{10,12,13}$ We find $\omega_{+}^{2}=k^2$, corresponding to sound waves at all $k$, and $\omega_{-}^{2}=k^2-1$, which is the conventional Jeans dispersion relation. The new acoustic modes arise because the aggregate fluid now allows motions unaffected by the gravitational field; for these modes the perturbed gravitational potential is zero.
With relative velocity in the subsonic regime ($0<M<1$), there is only a single unstable mode that branches continuously from the Jeans mode at $M=0$. This mode is unstable for all wavenumbers below a critical value that approaches infinity as $M$ tends to unity from below (see Table 1). To study this limit, we let $M^2=1-\alpha/k^2$ with $0<\alpha<1$. As $k\rightarrow\infty$ we find the approximate solution $\omega_{-}^{2}\sim -$[${\frac{\alpha(1-\alpha)}{4k^2}}$]{}, which reveals a weak instability at large $k$. Thus, weak relative streaming allows gravitational instability at arbitrarily small wavelengths. These large-$k$ instabilities do not arise for Maxwellian velocity distributions within the context of the Vlasov equation.$^{9}$
For supersonic motion ($M>1$) the large-$k$ gravitational instability is no longer present, but a new instability emerges that we call a two-stream instability since it owes its presence to the energy contained in the initial streaming motion. As $M$ ranges from $1$ to $\infty$, the critical wavenumber for instability increases from $k_{{crit}}=1/2$ to $k_{{crit}}=\sqrt{2}/2$.
The upper half of figure \[f:quarticM2\] shows that near $k=0$ the two-stream modes are wholly contained within the Jeans band. This fact coupled with the larger growth rates of the Jeans modes has led some to believe that the two-stream instability is swamped by the Jeans instability and is essentially unimportant.$^{11}$ As $k$ increases the two-stream and Jeans modes collapse upon each other and together bifurcate into growing and damped oscillations. At still larger wavenumbers all motions are stable, propagating waves. The critical wavenumbers below which growth is possible at any Mach number are shown in Table 1.
[**TABLE 1**]{}[^1]
---------------------------------------------------------------------------------------------------------------------------------------------------------
Mach Range Mode Type $k_{{crit}}^{2}$ $\lim_{M\rightarrow 1} $\lim_{M\rightarrow\infty} k_{{crit}}^{2}$
k_{{crit}}^{2}$
-------------- ------------ ------------------------------------------------------- ------------------------ --------------------------------------------
$0\leq M< 1$ Jeans [${\frac{1}{1-M^2}}$]{} $\infty$ Not Applicable
$1\leq M$ Two-Stream [ ${\frac{\sqrt{M^2-1}}{4M(M^2-1+M\sqrt{M^2-1})}}$]{} ${\frac{1}{4}}$ $0$
$1\leq M$ Jeans [ ${\frac{\sqrt{M^2-1}}{4M(1-M^2+M\sqrt{M^2-1})}}$]{} ${\frac{1}{4}}$ ${\frac{1}{2}}$
---------------------------------------------------------------------------------------------------------------------------------------------------------
**Asymmetric Case** {#sec-asymm}
-------------------
When we relax the constraint of identical conditions in the two fluids, one of the more interesting consequences is the possibility of large wavenumber two-stream instability bands well-separated from the Jeans instabilities clustered at small $k$. For illustration we consider the effect of changing the initial relative streaming $M_1+cM_2$ for fixed $\beta$ and $c$. In fact, $\beta$ turns out to be the crucial parameter in achieving the spectral separation; variations in the sound speed ratio $c$ widens both bands together. The distancing of a bubble of two-stream modes from the Jeans band is illustrated in figure \[f:wispyM1.45\].
The large-$k$ two-stream modes can be explained qualitatively by examining the separate pieces of the dispersion relation (\[eq:diagravic\]). Since the streaming instability is related to resonant motions, it is revealing to examine the solutions to $1+\Gamma_1(\omega,k)=
1+\Gamma_2(\omega,k)=0$ in isolation and see where the curves intersect. The frequencies of these non-interacting modes are given by, \_1&=&kM\_1 \[eq:sepmode1\]\
\_2&=&-ckM\_2 \[eq:sepmode2\] When $\omega_1=\omega_2$, the assumed independent frequencies match one another for wavenumbers satisfying, k(M\_1+cM\_2)={
[l]{} (+)\
(-)
. \[eq:rescon\]
In the symmetric case where $M_1=M_2\equiv M$, $c=\beta=1$, we see that two of the resonances are lost except in the irrelevant cases $M=0$ and $k=0$. The other pair of possibilities, $kM=\pm\sqrt{k^2-{\frac{1}{2}}}$, just restate the critical wavenumber condition for what we know to be the modified Jeans instability when $M<1$. In the general case, we can expect another pair of intersections that account for the two-stream bubbles of figure \[f:wispyM1.45\].
NONLINEAR THEORY {#sec-gravnonlin}
================
**Hamiltonian Formulation and Energy Signature** {#sec-ham}
------------------------------------------------
The dynamical equations (\[eq:bern1\])-(\[eq:grav\]) derive from a variational principle and a conserved Hamiltonian functional. The variational formulation has the advantage of shedding light on the relation between the energy content of the disturbances and nonlinear stability. Here we present results for the symmetric case, though the formalism follows through for the asymmetric case as well.
The Hamiltonian and associated equations are, H&=&\_[j=1]{}\^[2]{}\_[0]{}\^[L]{} dx(\_j \_[jx]{}\^[2]{} +\_j[U]{}\_j -V\_[x]{}\^[2]{} ), \[eq:ham\]\
\_t\_j&=&{\_j,H}=-\[eq:bernpoisson\]\
\_t\_j&=&{\_j,H}=,\[eq:masspoisson\] where ${U}_j (\rho_j)=$[${\frac{\rho_{j}^{\gamma-1}}
{\gamma(\gamma-1)}}$]{} is the internal energy for the $j^{\rm th}$ fluid and the Poisson bracket is defined by, {F,G}=\_[j=1]{}\^[2]{}\_[0]{}\^[L]{}dx\^( - ). \[eq:poissonbracket\] For definiteness we have chosen a box-geometry of length $L$.
The energy content of a particular mode when the perturbation amplitude is small is given by the second variation of $H$ evaluated at equilibrium (the free energy). After some calculation this is seen to be, \^2 H=\_[0]{}\^[L]{}dx(M\^2+2M\^2+\_[1]{}\^[2]{} +\_2\^2-2V\_[x]{}\^[2]{}). \[eq:freeenergy\] It may be verified that this functional is conserved by the equations of motion. Suppose we now insert into (\[eq:freeenergy\]) an eigenfunction corresponding to a single stable mode with Im($\omega$)$=0$. Employing overbars to denote eigenvector components and $*$ for complex conjugation, we write, V=e\^[i(t-kx)]{}+[\_j\^\*]{} e\^[-i(\^[\*]{} t-kx)]{}, \[eq:rhoefunk\] and similarly for the other perturbation variables. Upon effecting the integrations, we can make use of the dispersion relation, (,k)=1++ =0, \[eq:nondimdia\] to express the modal free energy in the compact form, \^2 H=-2Lk\^2\^2 . \[eq:freediagravic\]
Wherever [$\omega{\frac{\partial\Gamma}{\partial\omega}}$]{}$<0$ a positive energy mode (PEM) is implied by (\[eq:freediagravic\]), while the condition [$\omega{
\frac{\partial\Gamma}{\partial\omega}}$]{}$>0$ defines a negative energy mode (NEM). This possibility of modes of either signature has been elucidated in the plasma physics literature$^{14}$ and a gravitational analog was suggested by Lovelace et al.$^{7}$ in the context of thin, counter-rotating stellar disks.
Figure \[f:diagravic\_SymmetricCase\] shows the diagravic function for both a subsonic and a supersonic case of stable modes with $k=3$. In the subsonic regime, we find the Hamiltonian to be positive definite near equilibrium since [$\omega{\frac{\partial\Gamma}{\partial\omega}}$]{}$<0$ at every crossing of $\Gamma$ on the $\omega$-axis. Right at the border of supersonic streaming ($M=1$), concomitantly with the appearance of the two-stream instability, the $\Gamma$ curves undergo a topological transition that allows the coexistence of positive and negative energy modes. The NEMs are [*slow modes*]{} in that they have smaller frequencies than their PEM counterparts. This is the typical situation; $\omega$ must pass through zero if the energy signature changes.$^{15}$ We expect from the precedents of plasma physics$^{6}$ that the simultaneous presence of positive and negative energy modes has dramatic consequences on the nonlinear stability of the system.
**Reduction to Action Angle Variables** {#sec-canon}
---------------------------------------
In the rest of this section we will concentrate on the nonlinear interactions between linearly stable modes in the symmetric problem (see figure \[f:diagravic\_SymmetricCase\]). From a physical standpoint, attention is focused on situations where the disturbances are of sufficiently small scale so that the Jeans instability can be ignored, though there are no compelling reasons why this ought to be the case. We will further assume supersonic motion in order to examine the interaction of positive and negative energy modes, a situation we expect to be the most interesting. Under these assumptions the equations of motion achieve their simplest form in action-angle coordinates that we now develop.
First we Fourier transform the field variables: \_j=\_[m=-]{}\^\_m\^[(j)]{}(t)e\^[ik\_m x]{} ,\_[[-m]{}]{}\^[(j)]{}=[\_[[m]{}]{}\^[(j)]{}]{}\^\*,k\_m=.\[eq:fourier\] We can then write the free energy in terms of real variables as, \^2 H=\_[m=1]{}\^([**q**]{}\^[T]{} [**A**]{}[**q**]{}+[**p**]{}\^[T]{}[**B**]{}[**p**]{}), \[eq:freeenergypq\] where ${\bf q}\equiv(q_1,q_2,q_3,q_4)^{T}$, ${\bf p}\equiv(p_1,p_2,p_3,p_4)^{T}$, are linear combinations of the complex modal amplitudes, ${\bf A}$ and ${\bf B}$ are symmetric matrices (given in Casti$^{16}$), and the “$T$” indicates transpose.
Defining the configuration variables ${\bf z}=(q_1,\ldots,p_4)^{T}$, we recast the linearized equations in the form, =[**J**]{}\_[[**z**]{}]{}[\^2 H]{},=(
[clcr]{} 0 & [**B**]{}\
-[**A**]{} & 0
) ,=(
[clcr]{} 0 &[**I**]{}\
-[**I**]{} & 0
), \[eq:Hameqns\] where ${\bf J}$ is the canonical $8\times 8$ cosymplectic form. The next order of business is to construct a symplectic transformation that puts $\delta^2 H$ in its normal form.$^{17,18}$ This can be achieved by writing ${\bf z}={\bf S}{\bf Z}$, where the matrix ${\bf S}$ consists of suitably ordered eigenvectors of ${\bf L}$ satisfying the symplectic condition, ${\bf S}^{T}{\bf J}{\bf S}={\bf J}$. After a final transformation to action-angle coordinates, the free energy expression (\[eq:freeenergypq\]) becomes a superposition of harmonic oscillators, \^2 H=\_[m=1]{}\^(\_+ J\_1+\_+ J\_2 -\_- J\_3-\_- J\_4). \[eq:freeaa\] The free energy is thus manifestly composed of two pairs each of positive and negative energy modes.
**Three-Wave Resonance and Explosive Growth** {#sec-3wave}
---------------------------------------------
Energy conservation forbids nonlinear runaway growth if $H$ is definite (Dirichlet’s theorem), as is the case here for subsonic motion. When the relative streaming is supersonic, however, interacting PEMs and NEMs can circumvent this restriction since they contribute energy of opposite sign.
We demonstrate the possibility of explosive growth with a three-wave resonant interaction between two NEMs and one PEM. Since the energy signature of a mode is not Galilean invariant, the existence of a reference frame in which all three modes have the same signature implies nonlinear stability. It may be shown that there is no reference frame in which all three modes have the same energy signature if the highest frequency wave has opposite signature to that of the other two.$^{19}$ This provides a criterion for three-wave interactions leading to instability.
The third-order resonance conditions for a triplet of modes are, m\_[\_1]{}k\_[\_1]{}+m\_[\_2]{}k\_[\_2]{}+m\_[\_3]{}k\_[\_3]{}&=&0\
m\_[\_1]{}\_[\_1]{}m\_[\_2]{}\_[\_2]{}m\_[\_3]{}\_[\_3]{}&=&0\[eq:rescon3\]\
m\_[\_1]{}+m\_[\_2]{}+m\_[\_3]{}&=&3(m\_[\_1]{},m\_[\_2]{},m\_[\_3]{}).which here may be satisfied by $(m_{_1},m_{_2},m_{_3})=(1,1,-1)$, $(k_{_1},k_{_2},k_{_3})=(k_m,k_m,2k_m)$, and $(\omega_{_1},\omega_{_2},\omega_{_3})=(\omega_+,\omega_-,\omega_-)$, where the $\omega_{_j}$ are taken to be positive. One can see from figure \[f:quarticM2\] that $\omega_{_1}>\omega_{_2},\omega_{_3}$, so the relative signatures of this triplet are immune to a Galilean shift. Note from figure \[f:quarticM2\] that a resonant triplet involving two PEMs and one NEM would not have robust relative signatures under a frame shift since the PEMs have larger frequencies.
The lowest order nonlinear terms come from the third variation of $H$ expanded about the dynamical equilibrium, \^3 H=\_[j=1]{}\^[2]{}\_[0]{}\^[L]{} dx(\_[\_j]{}\_[\_[jx]{}]{}\^[2]{}+ \_[\_j]{}\^[3]{}). \[eq:hamnon\] In terms of the Fourier amplitudes this expression is, \^3 H=\_[j=1]{}\^[2]{}\_[[m,n=1]{}]{}\^ \[-Mk\_[\_m]{}k\_[\_n]{}(\_[\_[m+n]{}]{}\^[(j)]{}\_[\_[-m]{}]{}\^[(j)]{} \_[\_[-n]{}]{}\^[(j)]{}-\_[\_[m-n]{}]{}\^[(j)]{}\_[\_[-m]{}]{}\^[(j)]{} \_[\_n]{}\^[(j)]{}+[c.c.]{})\[eq:fourierham3\]\
+(\_[\_m]{}\^[(j)]{}\_[\_n]{}\^[(j)]{} \_[\_[-m-n]{}]{}\^[(j)]{}+\_[\_m]{}\^[(j)]{}\_[\_[-n]{}]{}\^[(j)]{} \_[\_[n-m]{}]{}\^[(j)]{}+[c.c.]{})\].We then effect the same transformations on (\[eq:fourierham3\]) that led to the diagonalized free energy (\[eq:freeaa\]). This spawns a myriad of nonlinear terms, only some of which survive an averaging process that leads to the Birkhoff normal form.$^{20}$
For a three-wave resonance, one finds after near-identity transformations that the only higher order terms contributing to the normal form are of the type,$^{14}$ (3)[Terms]{}\~J\_[\_1]{}\^[|l|/2]{} J\_[\_2]{}\^[|m|/2]{} J\_[\_3]{}\^[|n|/2]{} ( l\_[\_1]{}+m\_[\_2]{}+n\_[\_3]{}), \[eq:cubicterm\] with $|l|+|m|+|n|=3$. The Hamiltonian up to $3^{\rm{rd}}$ order terms for the resonant NEM/PEM triplets can be written as, H=\_1 J\_[\_1]{}-\_2 J\_[\_2]{}-\_3 J\_[\_3]{} +(\_[\_1]{}+\_[\_2]{}+\_[\_3]{}),\[eq:3waveham\] where $\alpha=\alpha(k_m,M,L)$ is a nonlinear coupling constant that is neither especially large or small in the parameter regime of the three-wave resonance considered here.
Since the angles $(\theta_{_1},\theta_{_2},\theta_{_3})$ appear in only one combination in $H$, further simplification of (\[eq:3waveham\]) is possible via the generating function, F\_[\_2]{}([**I**]{},[****]{})=I\_[\_1]{}( \_[\_1]{}+\_[\_2]{})+I\_[\_2]{}\_[\_2]{}+I\_[\_3]{}\_[\_3]{}, \[eq:genfunk\] with ${\psi_{_j}}=$ and ${J_{_j}}=$. This canonical transformation yields, H=[\_[\_1]{}]{}I\_[\_1]{}-\_[\_3]{}I\_[\_3]{}+ (2\_[\_1]{}+\_[\_3]{}), \[eq:3waveham2\] with $2{\tilde\omega_{_1}}\equiv\omega_{_1}-\omega_{_2}$. If one chooses initial conditions satisfying $I_{_2}\equiv J_{_2}-J_{_1}=0$, then $H$ is identical to the normal form of a two-wave interaction originally presented by Cherry.$^{21,22}$ In terms of the $({\bf q},{\bf p})$ variables, Cherry’s Hamiltonian is H=[\_[\_1]{}]{}(p\_[\_1]{}\^2+q\_[\_1]{}\^2)- [\_[\_3]{}]{}(p\_[\_3]{}\^2+q\_[\_3]{}\^2)+ (2q\_[\_1]{} p\_[\_1]{} p\_[\_3]{}-q\_[\_3]{}), \[eq:cherry\] where $\epsilon=$[${\frac{\sqrt{2}\alpha}{4}}$]{}.
The dynamical system generated by the Cherry Hamiltonian is integrable. In the special case of a third-order resonance with $\omega_{_3}=2{\tilde\omega_{_1}}$, there exists a family of two-parameter solutions, q\_[\_1]{}=([\_[\_1]{}]{}t+) &,&p\_[\_1]{}=-([\_[\_1]{}]{}t+) \[eq:rescherrysolns\]\
q\_[\_3]{}=-(2[\_[\_1]{}]{}t+2) &,&p\_[\_3]{}=-(2[\_[\_1]{}]{}t+2),where $\xi$ and $\eta$ are constants depending on the initial conditions.
The solutions (\[eq:rescherrysolns\]) show the possibility of [*finite-time density singularities*]{} when two negative energy modes interact resonantly with a positive energy mode. A system exhibiting this behavior is said to undergo [*explosive growth*]{}, and it could be an important mechanism for structure formation in galactic and cosmological settings when relative motion between different fluid species is involved. If the resonance is detuned, separatrices bounding stable orbits emerge in phase space, but the dynamics are still prone to finite-amplitude instability.
DISSIPATIVE INSTABILITY {#sec-dissipation}
=======================
We close our investigation of the consequences of negative energy modes by examining the effects of dissipation on the linear stability of the system. With negative energy modes propagating through a dissipative medium, we may expect new instabilities since the damping can pump more negative energy into the wave. This somewhat counterintuitive effect of frictional forces in other contexts was first pointed out by Kelvin and Tait$^{23}$ (see also Zajac$^{24}$).
Suppose that collisions are important at some stage in the development of a gravitationally bound structure. A simple model of this effect incorporates a dynamical friction term $(-1)^j\nu\left({\bf u}_1-{\bf u}_2\right)$ on the right hand side of the momentum equations (\[eq:3Dmom\]), where $\nu$ is a positive damping coefficient.
In the dimensionless symmetric case the dispersion relation becomes, \^4-2i\^3 +\^2 &+&2i\[eq:dampeddis\]\
& &+k\^2(M\^2-1)=0. If we assume the damping is weak, $\nu\ll 1$, we may develop (\[eq:dampeddis\]) in a regular perturbation series ($\omega=\omega_{_0}+\nu\omega_{_1}+\ldots$) to find the lowest order corrections to the frequencies (\[eq:symroots\]), \_[\_1]{}\^=(1), \[eq:freqcorrection\] where we assume $k^2\gg{\frac{M\pm\sqrt{M^2-1}}{4M}}$ to avoid the singularity accompanying the vanishing denominator in (\[eq:freqcorrection\]) (see Casti$^{16}$ for the details). If we assume $M>1$ so that negative energy modes are present, then a close examination of the corrections reveals that the dissipation promotes instability in the wavenumber band ${\frac{M\pm\sqrt{M^2-1}}{4M}}\ll k<{\frac{\sqrt{2}}{2}}$ for any $M\ge1$ [*no matter how weak the damping*]{}. Since the instability as $k^2\rightarrow{\frac{1}{2}}$ is realized only in the $M\rightarrow\infty$ limit of the undamped problem, we see that the dissipation indeed has the effect of destabilizing modes that were stable in the conservative case. A numerical investigation revealed that this result holds for any $\nu>0$.
The modal bands destabilized by the damping become more significant in the asymmetric case. As remarked in section \[sec-asymm\], bubbles of unstable two-stream modes can pinch off from the Jeans-unstable bubble and result in well-separated instability bands. The inclusion of dissipation can destabilize the entire band of modes separating the bubbles, as well as some higher-$k$ modes beyond the undamped two-stream bubble. This is illustrated in figure \[f:damwispyM1.45\].
One should not assume that any form of dissipation will destabilize negative energy modes. For instance, if each fluid feels only a drag proportional to its own velocity, there are no new instabilities even with relative motion. In other words, the dissipation must in some sense project onto the eigenspace spanned by the negative energy modes in a way that decreases their energies. This depends not only on the nature of the dissipation, but also upon the initial equilibrium about which one perturbs.
The effect of damping can be understood by examining the time evolution of the free energy. If the dissipation acts to increase the energy of a positive energy mode or decrease the energy of a negative energy mode, then one can show that pure imaginary eigenvalues take on a positive real part.$^{25}$ To see that this is possible here, consider the temporal change in the free energy, which in the symmetric case can be written, =-M\_[0]{}\^[L]{} dx( \_[[1x]{}]{}-\_[[2x]{}]{})\^2 -M\_[0]{}\^[L]{} dx(\_1-\_2) (\_[[1x]{}]{}+\_[[2x]{}]{}). \[eq:hdot2\] Since the first term of ${\delta^2{\dot H}}$ is negative definite, the conditions for which the free energy decays or grows in time is determined by the relative phasings of the velocity and density perturbations comprising the second term. One may deduce the effect of the dissipation on any particular mode of the conservative problem by inserting the undamped modes into (\[eq:hdot2\]), which yields a formula for ${\delta^2{\dot H}}$ valid up to ${\cal O}(\nu^2)$. For subsonic relative motion, $M<1$, the expression (\[eq:hdot2\]) is negative definite and the damping lives up to its name and causes the PEMs to decay in time. When $M>1$, ${\delta^2{\dot H}}$ can be either positive or negative for an NEM depending on the value of $k$, which explains why some NEMs are destabilized and others are damped in the usual sense.
DISCUSSION {#sec-discussion}
==========
The formation of structures through the action of gravity is much analyzed in cosmology, galactic structure and cosmogony. Most of this analysis is centered on the operation of gravitational instability, though streaming fluids can resonantly interact via the gravitational field to cause linear instability in spectral ranges inaccessible to the traditional Jeans instability. As we have brought out here, the distinction between the two types of unstable modes, the Jeans and the two-stream, becomes sharper when one constituent is far denser than the other. Much of the previous work on the subject failed to take advantage of this crucial feature by focusing attention on situations where each component exists in equal abundance. Even when the two-stream instability does not occur, if the total energy of the gravitational two-stream interaction is indefinite, the positive and negative energy modes that are [*stable*]{} in the linear theory can interact to produce explosive development of disturbances of arbitrarily small amplitude. This can be a significant aspect of the theory of structure formation.
There are many clear instances where the dynamics of interpenetrating fluids may play a role in developing structures, but we close here by suggesting that even when the streaming is not apparent, two-stream dynamics may be relevant. An interesting example is provided by the coexistence of dark matter and luminous (baryonic) matter that is generally believed to occur throughout the cosmos. The locations of the two kinds of matter seem to be well correlated, which would not be the case if they were now streaming through each other. On the other hand, it might be reasonable to ask why there is this apparent correlation (or anticorrelation in the case of negative gravitational density) of the two kinds of material. Even if they had once been in relative motion this situation would not long persist, as we have seen. But the outcome, as far as large-scale structure is concerned, could be quite different if the kinematic history of the interaction of the two matters had been richer than has been supposed hitherto. Given the indefiniteness of the free energy if the initial streaming is large enough, waves of short length scale could have interacted in an explosive manner to quickly produce highly nonlinear density fluctuations. This is a feature of gravitational structure development that could be profitably studied, particularly in situations where the background Hubble expansion cannot be ignored. The dynamics of a two-fluid system with initially [*time-dependent*]{} relative motion is currently under investigation.
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9\. Ikeuchi, S., T. Nakamura & F. Takahara. 1974. Prog. Theor. Phys. [**52:**]{} 1807.
10\. Fridman, A. M. & V. L. Polyachenko. 1984. Physics of Gravitating Systems II.
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11\. Araki, S. 1987. Astron. J. [**94:**]{} 99.
12\. Spiegel, E. A. 1972, in Symposium on the Origin of the Solar System: 165. H. Reeves,
ed. Editions du Centre National de la Recherche Scientifique, Paris.
13\. de Carvalho, J. P. M. & P. G. Macedo. 1995. Astron. Astrophys. [**299:**]{} 326.
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16\. Casti, A. R. R. 1998. Ph. D. Thesis. Columbia University.
17\. Moser, J. K. 1958. Comm. Pure Appl. Math. [**11:**]{} 81.
18\. Meyer, K. R. & G. R. Hall. 1991. Introduction to Hamiltonian Dynamical Systems
and the N-Body Problem. Springer-Verlag New York Inc.
19\. Weiland, J. & H. Wilhelmsson. 1977. Coherent Nonlinear Interaction of Waves in
Plasmas. Pergamon, Oxford.
20\. Ozorio de Almeida, A. M. 1988. Hamiltonian Systems: Chaos and Quantization.
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22\. Whittaker, E. T. 1937. Analytical Dynamics: 142. Cambridge, London.
23\. Thompson, W. (Lord Kelvin) & P. G. Tait. 1921. Treatise on Natural Philosophy,
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[^1]: The value of $k_{{crit}}^{2}$ for the two-stream modes with $M\geq 1$ more accurately refers to the $k$ value at which the unstable two-stream and Jeans branches merge.
|
---
abstract: |
We discuss nuclear modification of fragmentation functions in the context of the so-called “rescaling models”. These models implement partial deconfinement inside nuclei by modifying the fragmentation functions perturbatively. We apply these models to the analysis of nuclear hadron production in deep inelastic scattering processes at the HERMES and EMC experiments.\
address:
- 'Institut für Theoretische Physik der Universität Heidelberg, Germany'
- 'Max-Planck-Institut für Kernphysik, Heidelberg, Germany'
author:
- 'Alberto Accardi [^1] and Hans J. Pirner [^2]'
title: 'Nuclear modifications of fragmentation functions and rescaling models [^3] [^4]'
---
In deep inelastic scattering a projectile lepton $\ell$ emits a virtual photon $\gamma^*$, which scatters on a quark $q$ from the target $T$, in our case a deuteron $D$ or a heavier nucleus with atomic number $A$. The struck quark fragments into the observed hadron $h$, see [Fig. \[fig:DIS\]]{}. We use kinematic variables as summarized in the same figure: $x$ is the Bjorken’s scaling variable, $\nu$ is the energy of the virtual photon in the target rest frame, and $z$ is the fraction of the energy transferred to the produced hadron.
-.5cm
-.7cm
The experimental data on nuclear effects in hadron production are usually presented in terms of the [*multiplicity ratio*]{} $$\begin{aligned}
R_M^h(z) =
\frac{{\displaystyle}1}{{\displaystyle}N_A^\ell} \frac{{\displaystyle}dN_A^h}{{\displaystyle}dz}
\bigg/ \frac{{\displaystyle}1}{{\displaystyle}N_D^\ell} \frac{{\displaystyle}dN_D^h}{{\displaystyle}dz} \ ,
\label{multratio}\end{aligned}$$ where $N_A^\ell$ is the number of outgoing leptons in DIS processes on a nuclear target of atomic number $A$ and $dN_A^h/dz$ is the $z$-distribution of produced hadrons in the same processes; the subscript $D$ refers to the same quantities when the target is a deuteron. A similar definition for the multiplicity ratio as a function of $\nu$ may be written. In absence of nuclear effects the ratio $R^h_M$ would be equal to 1. The experimental observation that $R^h_M \neq 1$ [@SLAC; @EMC; @HERMES] has been explained theoretically in many ways: as an effect of nuclear absorption of the produced hadrons [@BC83; @BG87], in a gluon-bremsstrahlung model for leading hadron production [@KNP96], as an higher-twist effect [@hightwist]. In this short note we will not discuss these models in detail (see [Ref. [@Muccifora02]]{}) and, instead, will concentrate on the so-called [*rescaling models*]{} [@NP84; @CJRR84; @DDD86].
The starting point of rescaling models is to assume a change in the confinement scale $\lambda_A$ in nuclei, compared to the confinement scale $\lambda_0$ in free nucleons: $$\begin{aligned}
\lambda_A > \lambda_0 \ .\end{aligned}$$ This assumed partial deconfinement in nuclei affects both the parton distribution functions (PDF) and the fragmentation functions (FF). For what concerns PDF’s [@NP84; @CJRR84], consider a valence quark which carries a momentum $Q_0$ when it is confined on a scale $\lambda_0$. If the scale changes to $\lambda_A$ it carries a corresponding momentum $Q_A$. Since there is no other dimensionful scale, the product $Q\lambda$ must remain constant, so that $$\begin{aligned}
Q_0 \lambda_0 = Q_A \lambda_A \ .
\label{deconf}\end{aligned}$$ Therefore, if we take $Q_0$ to be the initial scale for the DGLAP evolution of distribution functions, we may set $$\begin{aligned}
q_f^A \big( x, Q_A=\frac{{\displaystyle}\lambda_0}{\lambda_A} \, Q_0 \big)
= q_f\big(x,Q_0\big) \ ,
\label{PDF0}\end{aligned}$$ where $q_f$ is the distribution function of a quark of flavour $f$ in a free nucleon and $q_f^A$ is the same quantity when the nucleon is inside a nucleus. For FF’s [@DDD86] a similar argument holds because constituent quark and effective hadron masses are sensitive to the confinement scale. Therefore if we take $Q_0$ to be the physical threshold for hadron production we may set $$\begin{aligned}
D_f^{h|A} \big(x, Q_A=\frac{{\displaystyle}\lambda_0}{\lambda_A} \, Q_0 \big)
= D_f^h\big(x,Q_0\big) \ ,
\label{FF0}\end{aligned}$$ where $D_f^h$ is the fragmentation function of a quark of flavour $f$ into a hadron $h$ and $D_f^{h|A}$ is the nuclear modified fragmentation function. To extend [Eqs. ]{} and to an arbitrary scale $Q$ we apply pQCD evolution. The nuclear structure and fragmentation functions evolve over larger interval in momentum compared to the corresponding functions at the same scale Q, since the starting scale is smaller, see [Eq. ]{}. The final result [@NP84; @CJRR84; @DDD86] is $$\begin{aligned}
q^A_f(x,Q) & = q_f(x,\xi_A(Q) Q)
\label{rescPDF} \\
D^{h|A}_f(z,Q) & = D^h_f(x,\xi_A(Q) Q) \ ,
\label{rescFF}\end{aligned}$$ where the [*scale factor*]{} $\xi_A(Q)$ is greater than one. These two equations are the main tool which we will exploit. There are two models for the computation of the scale factor:
a) the (NP) [@NP84] assumes the onset of “colour conductivity” in nuclei and takes $\lambda_A=R_A$, where $R_A$ is the nuclear radius, so that $$\xi_A(Q) = R_A/R_p \ ;$$
b) -.2cm the (CJRR) [@CJRR84] assumes the deconfinement scale $\lambda_A$ to be proportional to the degree of overlap of the nucleons inside the given nucleus, and $$\xi_A(Q) = \big(\lambda_A/\lambda_0\big)
^{\frac12 \frac{\alpha_s(Q_0)}{\alpha_s(Q)}} \ .$$ -1cm
Note that the maximal deconfinement model assumes a much larger scale factor, which results in a larger nuclear modification of hadron production at high $z$.
We calculate the multiplicity ratio using the [*rescaled*]{} PDF and the [*rescaled*]{} FF in the leading order pQCD computation of $N_A^\ell$ and $dN_A^h/dz$: $$\begin{aligned}
\frac{1}{N_A^\ell}\frac{dN_A^h}{dz} & = \frac{1}{\sigma^{\gamma^*A}}
\int_{\rm exp.\ cuts}\hspace*{-1.0cm}
dx\, d\nu \sum_f e^2_f\, q_f(x,\xi_A\,Q) \,
\frac{d\sigma^{\gamma^* q}}{dx\,d\nu} \, D_f^h(z,\xi_A\,Q)
\label{dnhdz} \\
\sigma^{\gamma^*A} & = \int_{\rm exp.\ cuts}\hspace*{-1.0cm}
dx\, d\nu \sum_f e^2_f\, q_f(x,\xi_A\,Q) \,
\frac{d\sigma^{\gamma^* q}}{dx\,d\nu} \ , \nonumber\end{aligned}$$ where $e_f$ is the electric charge of a quark of flavour $f$, $d\sigma^{\gamma^* q}/dx\,d\nu$ is the differential cross-section for a $\gamma^* q$ scattering computed in pQCD at leading order, and $Q^2=2Mx\nu$, with $M$ the nucleon mass. In the numerical computations we used GRV98 parton distribution funtions [@GRV98] and Kretzer’s parametrization of FF’s at leading order [@K].
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-.9cm
The results for the $\nu$-dependence of the multiplicity ratio for charged hadrons are shown in [Fig. \[fig:res1\]]{}a and compared to EMC data [@EMC]. The maximal deconfinement model underestimates the data over nearly the whole range. On the contrary the partial deconfinement model fit the data at high $\nu$, but overestimates them at smaller $\nu$. Because of Lorentz dilatation, at high $\nu$ the hadron is formed mainly outside the nucleus and it is affected by rescaling effects only. On the contrary, at small $\nu$ the hadron is formed inside the nucleus and starts interacting with its nucleons, with some probability of being absorbed.
To take into account nuclear absorption, we follow the analysis of [Ref. [@BC83; @BG87]]{}: the quark propagates in the nucleus with a cross-section $\sigma_q$ for inelastic scattering on the nucleons. Subsequently, it creates a pre-hadronic states, which has a cross-section $\sigma_*$, and finally the observed hadron is formed, which has a cross-section $\sigma_h$. Following the experimental indications of EMC and HERMES, we take $\sigma_q=0$ and $\sigma_*=\sigma_h=20$ mbarn for charged hadrons. With the latter assumption we need to consider effectively only the [*formation time*]{} $\tau_F$ for the pre-hadronic state [@BG87]: $$\begin{aligned}
\tau_F = \left( \frac{-\ln(z^2) -1 + z^2}{1-z^2} \right)
\frac{z\nu}{\kappa} \ ,\end{aligned}$$ where $\kappa$ is a parameter which we fix to $\kappa = 0.25$ GeV/fm. Note that $\tau_F {{\rightarrow}}\frac{1}{\kappa} (1-z)\nu$ as $z{{\rightarrow}}1$, giving the formation time suggested by the gluon bremsstrahlung model of [Ref. [@KNP96]]{}. Finally, nuclear absorption effects are included in the computations by multiplying the integrand in by the [*nuclear absorption factor*]{} ${\cal N}_A$, which represent the fraction of hadrons which escape from the nucleus: $$\begin{aligned}
{\cal N}_A = \int d^2b \int_{-\infty}^{\infty}
dy \, \rho_A(b,y)\left[ S_A(b,y) \right]^{A-1} \ ,\end{aligned}$$ where $\rho_A(b,y)$ is the nuclear density normalized to 1 at transverse and longitudinal coordinates $(b,y)$, and $S_A(b,y)$ is the survival probability of a hadron produced in $(b,y)$: $$\begin{aligned}
S_A(b,y) = 1 - \sigma_h \int_y^\infty \hspace*{-.2cm}dy\,'
\rho_A(b,y\,')\left( 1 - e^{\,-(y\,'-y)/\tau_F} \right) \ . \end{aligned}$$ For the deuteron we used as a density the sum of the Reid’s soft-core S- and D-wave functions squared [@ReidSC]. For heavier nuclei we used a Woods-Saxon density with radius $R_A=1.12 A^{1/3} - 0.86 A^{-1/3}$ fm.
The $\nu$- and $z$-dependence of the multiplicity ratio at EMC and HERMES after the inclusion of nuclear absorption are shown in [Fig. \[fig:res1\]]{}b and [Fig. \[fig:res1\]]{}c. It is clearly seen that while the CJRR model gives a nice description of the data, the NP model is ruled out.
-.8cm
-.9cm
The average values of ${\langle \nu \rangle}$ available at HERMES are smaller than at EMC and the absorption effects larger. In [Fig. \[fig:res2\]]{} we compare the $z$-distributions in the partial deconfinement models at EMC and HERMES before and after inclusion of absorption. As expected, at EMC with a $^{63}$Cu target nuclear absorption is marginal and rescaling alone gives a satisfactory description of the data. At HERMES with a $^{84}$Kr target, which is comparable to copper, both effects are larger and absorption is dominant, tending to mask rescaling effects. With a $^{14}$N target, which is smaller than krypton, both effects are smaller.
In summary, rescaling models (supplemented by nuclear absorption) are shown to be able to describe both HERMES and EMC data on the nuclear modification of hadron production in DIS processes. While the maximal deconfinement model is ruled out by the data - it assumes a too large deconfinement - the partial deconfinement model is shown to be a good one. Further precise data at moderate and high $\nu$’s and for light and heavy targets are needed to disentangle rescaling and formation time effects.
.3cm [**Acknowledgments**]{} .15cm
[ We are grateful to N. Bianchi, P. di Nezza, B. Kopeliovich and V. Muccifora for stimulating discussions and to R. Fabbri for technical support during the conference. ]{}
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[^1]: E-mail address: accardi@tphys.uni-heidelberg.de
[^2]: E-mail address: pir@tphys.uni-heidelberg.de
[^3]: Talk presented at nthe “European workshop on the QCD structure of the nucleon” (QCD-N’02), Ferrara (ITA), April 3rd-6th, 2002.
[^4]: This work is (partially) funded by the European Commission IHP program under contract HPRN-CT-2000-00130.
|
---
abstract: 'We have studied a system consisted of two coupled quantum dots containing two electrons subjected by a laser field. The effect of the laser is described by the dressed-band approach involving the concept of the conduction/valence effective mass, valid far from resonance. The interaction between the electrons and the quantum dots is described by a phenomenological tridimensional potential, which simulates quantum dots in GaAs heterostructure. In this study we have employed the approach already presented in a previous work \[Olavo [*et al.*]{}, J. Phys. B: At. Mol. Opt. Phys. [**49**]{}, 145004 (2016)\]. We have used a code based on the full interaction configuration method. We have employed as basis set the [*Cartesian anisotropic Gaussian-type*]{} orbitals which allows one to explore the confining characteristics of a potential due to their flexibility of using different exponents for each direction space. We present an analysis based on the energy levels of the singlet and triplet as function of the confinement parameters.'
author:
- 'A. M. Maniero'
- 'C. R. de Carvalho'
- 'F. V. Prudente'
- Ginette Jalbert
title: Effect of a laser field in the confinement potential of two electrons in a double quantum dot
---
INTRODUCTION
============
The advances of the experimental techniques used in semiconductor structures of nanoscopic scale [@heinzel-2010] has increased the interest in the study of the physical properties of confined quantum systems. A consequence of this improvement on the manufacturing of semiconductor quantum dots (QDs) is the increase in the control of their size; this has attracted a great interest in the area of optoelectronics [@H.Liu15] and optical communications [@Schmeckebier17]. As long as the QD dimensions become of the order of nanometer, it has been noticed that the its physical properties are greatly affected by changes in its size [@Alivisatos96; @ScienceSpecialIssue96]. Frequently the size and geometric form of the quantum dot has been treated in terms of confinement profile and strength [@Diercksen-JPB34-1987-01; @Diercksen-JPB36-1681-03]. The influence of external fields on QDs has also attracted attention, in particular on double quantum dots (DQDs) aiming at quantum computation and general process in nanotechnology [@DiVincenzo-PRB59-2070-99; @CRC-GJ-JAP94-2579-03; @Szafran-PRB70-205318-04; @Dybalski-PRB72-205432-05; @Leburton-COSSMS10-114-06; @Yamamoto-RPP76-092501-13; @Prati-JPA48-065304-15]. The behavior of the exchange coupling ($J$), or exchange energy, has been one of the main subjects in the study of the properties of few-electron DQDs. In this context, different profiles of confining potential has been tried out such as quartic [@DiVincenzo-PRB59-2070-99; @CRC-GJ-JAP94-2579-03], gaussian [@Leburton-COSSMS10-114-06; @Leburton-JPCM21-095502-09] and few others [@Kwasniowski-JPCM20-215208-08; @Pedersen-PRB81-193406-10]. In all these cases a two-dimension geometry has always been considered.
In a previous work [@CRC-GJ-JAP94-2579-03] we have analyzed the exchange coupling ($J$) in the effective Heisenberg model within the Heitler-London approximation, so that it can be analytically calculated. We have discussed it as a function of the laser field and its detuning, as well as of the magnetic field. We have found that, due to the electronic confinement, the laser may play a role similar to the external magnetic field in the qualitative behavior of the exchange parameter ($J$). On the other hand, it has also been reported analytic expressions for the exchange coupling in 2D coupled quantum dots computed within the Heitler-London and the Hund-Mulliken approximations using different confining potentials under different regimes of magnetic field intensity [@Pedersen-PRB81-193406-10].
Aiming more precise results, one finds a variety of numerical methods or techniques employed for calculations of the electronic structure of quantum systems such as atoms, ions and molecules confined by an external potential [@Diercksen-JPB36-1681-03; @Diercksen-CPL349-215-01; @Klobukowski-MP103-2599-05; @Fred-JCP123-224701-05; @LeSech-JPB45-205101-12; @Bartkowiak-CP428-19-14; @Sen14; @JPB48-055002-15; @Cruz-JPB50-135002-17]. The interest on this type of problem arose from the wide range of issues found in many branches of chemistry and physics [@Sabin-AQC57-58-09]. Naturally these methods are also suitable for the study of QDs since they can be seen as artificial atoms or molecules [@Sen14].
In view of all these issues, we have developed a code [@olavo2016] which it allows to study arbitrary systems submitted to different confining potentials and external conditions, such as a laser field. This code allows one to lead with a set of anisotropic functions with different exponents for each space direction.
In the present work we shall use our code to study the energy spectrum of two electrons confined by a 3D anisotropic potential representing a 3D-DQD. We have adopted as confining potential a combination of the quartic potential $V(x,y)$ [@DiVincenzo-PRB59-2070-99; @CRC-GJ-JAP94-2579-03], for the $xy$ plane, with a parabolic potential on the $z-$direction [@Tarucha-Sci278-1788-97; @Manninen-RMP74-1283-02]. We shall discuss the confinement of the electrons in the $xy$ plane as a function of the characteristic parameters of the system: the laser intensity, the inter-dot distance, and the strengths of the potential along the $z-$ direction.
Throughout the paper the computations were done in atomic units (au), more common in atomic-molecular calculations, whereas the results were expressed in meV and nm which are more tangible in nanoscale.
The theoretical approach
========================
We want to solve the time independent Schrödinger equation for a system of $N$ electrons submitted to an arbitrary potential $\hat{V}(x,y,z)$ whose Hamiltonian is written as: $$\begin{aligned}
\hat H =\sum_i^N\hat{O}_1(\vec r_i)+\sum_i^N\sum_{j<i}^N\hat{O}_2(\vec r_i,\vec r_j),
\label{hamiltoniano}\end{aligned}$$ where $$\begin{aligned}
\hat{O}_1(\vec r_i) = -\frac{1}{2m^*_c}\vec\nabla^2_i+ \hat{V}(x_i,y_i,z_i) ,\end{aligned}$$ and $$\begin{aligned}
\hat{O}_2(\vec r_i,\vec r_j)=\frac{1}{\kappa\vert\vec r_i-\vec r_j\vert}.\end{aligned}$$ The parameters $\kappa$ and $m^*_c$ are respectively the static dielectric constant and the electron renormalized effective mass, allowing us to considered general conditions not necessarily in the vacuum.
In the present work we are interested in studying the electronic structure of a system composed of two electrons confined in a 3D CQD, whose potential is expressed as $$\begin{aligned}
\hat{V}(x,y,z) = \frac{m^*_c}{2}\left[\frac{\omega_x^2}{4a^2}\left(x^2-a^2\right)^2+\omega^2_y y^2+\omega^2_z z^2\right].
\label{Vq}\end{aligned}$$ The $xy$ dependence is modeled by a quartic potential $V(x,y)$[@DiVincenzo-PRB59-2070-99; @CRC-GJ-JAP94-2579-03]. Observe that the advantage of using the quartic potential, in modeling the double quantum dot, consists in controling the size of the inter-dot barrier with the laser intensity without the necessity of changing any other parameter, see Fig. \[Quartic potential\]. In the limit of inter-dot distance, $a\gg a^*_B$ where $a^*_B=\sqrt{1/(m^*_c\omega_x)}$, the potential splits into two harmonic wells of frequency $\omega_x$ and $\omega_y$ along $x$ and $y$, respectively. In the direction $z$ we assume an harmonic potential with frequency $\omega_z$, which can be chosen for instance to simulate a 2D double quantum dot by setting $\omega_z \gg \omega_x \mbox{ and } \omega_y$.
The electronic properties of free systems or confining potential in the study of quantum dots can be obtained due to the flexibility of our program which can take into account the anisotropy of the potential on the basis employed [@olavo2016]. In addition, one can use a different effective electronic mass $m^*_c$, once the laser is present through the electron renormalized effective mass, and/or change the environment in which they evolve via the $\kappa$ parameter (see Ref. [@CRC-GJ-JAP94-2579-03]).
The validity of the renormalized effective mass is discussed in detail in several works [@brandi1; @brandi2; @brandi3]. Briefly, the electronic band structure of the semiconductor is modeled by a two-parabolic, isotropic band in the $\vec{k}\cdot\vec{p}$ approximation [@callaway]. To incorporate the laser field into an effective mass formalism (renormalized effective mass approximation) the dressed atom [@Cohen] approach is extended to include a dispersion relation through the two-band model (dressed band approximation), the eigenvalue problem for the dressed bands is solved analytically and a $k$ expansion is performed. According to this model the renormalized effective mass of the conduction band ($m^*_c$) is given by $$\frac {1}{m^*_c}=\frac {1}{2M}\left[ 1+ \frac{M}{\mu} \frac{\left(
\frac{2\Lambda_0^2+\delta\Lambda_1}{\Lambda_1} \right) \left[
1-\frac{2\Lambda_0^2}{\Lambda_1^2} \left(1+\frac{2\Lambda_1}{E_g}
\right)\right] - \frac{4\Lambda _0^2}{E_g} }{\sqrt{4\Lambda_0^2+\left(
\frac{2\Lambda_0^2+\delta\Lambda_1}{\Lambda_1} \right)
^2}}\right] \label{massa}$$ where $E_g$ is the energy gap, $ 1/M= 1/{m_c}+1/{m_v}$, $1/{\mu} = 1/{m_c}-1/{m_v}$, and $m_c(m_v)$ is the undressed effective mass associated to the conduction (valence) band: $$\frac{1}{m_c}=1 + \frac{2p^2}{Eg} \mbox{ and } \frac{1}{m_v}=1 - \frac{2p^2}{Eg},$$ which leads to $m_c \approx 0.067$ and $m_v \approx -0,077$ for GaAs. We have also defined the laser detuning parameter $\delta =E_g-\hbar\Omega$ and $\Lambda_1=2E_g-\delta$ and $\Lambda_0 = \left[ \left( 2I/I_c
\right) 7.02 \times E_g^2\right] ^{\frac 12}$. In the expression of $\Lambda_0$, $I_c$ is a critical intensity defined in Ref. [@brandi1], whose value for GaAs is $I_c\approx 5\times 10^{13}W/cm^2$. We have only considered the case of $\delta/E_g = 0.05$ (see Fig.1 of Ref.[@CRC-GJ-JAP94-2579-03]), and we have taken the range of intensity from $I/I_c = 0$ to $10 \times 10^{-5}$. For this range the electron effective mass is displayed in the table \[m\_x\_I\].
$(I/I_c)\times 10^{-5}$ $m^*_c/m_c$
------------------------- -------------
0 1
1. 1.11025
2. 1.21163
3. 1.30639
4. 1.39599
5. 1.48149
6. 1.56364
7. 1.64304
8. 1.72013
9. 1.79527
10. 1.86877
: Electron effective mass as function of the laser field intensity. For details see the text.[]{data-label="m_x_I"}
The solution of Eq. (\[hamiltoniano\]), $\Phi$, was obtained by a Full-CI method and is written as $$\Phi=\sum_{i=1}^{N_\textrm{CSF}}C_i^{\textrm{CSF}}\Psi_i^{\textrm{CSF}}$$ where $N_\textrm{CSF}$ is the number of configuration state functions (CSF) and $C_i^\textrm{CSF}$ represent the coefficient of a given CSF. On the other hand, a CSF is constitute of Slater determinants, [*i.e.*]{}, $$\begin{aligned}
\Psi_i^\textrm{CSF}=\sum_{i_1=1}^{\textrm{Ndet}_i}C_{i_1}^\textrm{det}\textrm{det}(i,i_1),\end{aligned}$$ where $\textrm{det}(i,i_1)$ is the $i_1^{th}$ determinant of the $i^{th}$ CSF. As $[\hat{H},\hat{S}^2]=0$ and $[\hat{H},\hat{S}_z]=0$, $\Phi$ should be eigenfunction of $\hat{S}^2$ e $\hat{S}_z$.
Setting the bases
=================
As mentioned in the introduction, we use a computational code to study the energy spectrum of two electrons confined in a 3D anisotropic potential. In order to employ it, we have to establish anisotropic orbitals as the atomic basis set. Similar to what was done in [@olavo2016], we have chosen a basis set composed of the Cartesian anisotropic Gaussian-type orbitals (c-aniGTO) centred in the position $\vec R=(X, Y, Z)$ which, apart a normalization constant, are given by: $$\begin{aligned}
&&g_\mu(\vec r-\vec R ,\zeta)=
(x-X )^{n_x}
(y-Y )^{n_y}
(z-Z )^{n_z}\times\nonumber\\
&&\exp\left[
-\zeta_x(x-X )^2
-\zeta_y(y-Y )^2
-\zeta_z(z-Z )^2
\right]
\label{gaussianxyz}\end{aligned}$$ where one has the possibility of providing different exponents $\zeta_x$, $\zeta_y$ and $\zeta_z$ according to the problem analyzed and $\mu$ stands for $(n_x,n_y,n_z)$. In addition, in analogy to the standard convention for the atomic case, we shall classify the orbitals as $s$-, $p$-, $d$-,... type according to $n=n_x + n_y + n_z = 0, 1, 2,...$, respectively.
Since the potential $V(x,y,z)$ along the $y$ and $z$ direction has the same form of the potential used in previous work [@olavo2016], the same two types of exponents have been considered in those directions: $$\zeta_i^{(1)}=\frac{m^*_c \omega_i}{2} \mbox{ and } \zeta_i^{(2)} = \frac{3}{2}\zeta_i^{(1)},$$ where $i$ stands for $y$ and $z$.
On the other hand, the first type of exponent in the $x$ direction, $\zeta_x^{(1)}$, has been obtained by a variational method minimizing the following functional, $$\begin{aligned}
E(\zeta_x^{(1)})=\frac{\displaystyle\int_{-\infty}^{\infty}dx\phi_{\pm}^\ast(x,\zeta_x^{(1)})\hat O(x) \phi_{\pm}(x,\zeta_x^{(1)})}
{\displaystyle\int_{-\infty}^{\infty}dx\phi_{\pm}^\ast(x,\zeta_x^{(1)})\phi_{\pm}(x,\zeta_x^{(1)})},
\label{pvar_x}\end{aligned}$$ where $\hat O(x) = \left[-\frac{1}{2m^*_c}\frac{d^2}{dx^2}+\frac{m^*_c\omega_x^2}{8a^2}(x^2-a^2)^2\right]$.
The procedure to obtain this exponent was also employed in [@olavo2016] and is explained in its section 3. Once the potential displayed in the operator $\hat O(x)$ has minima in $x=\pm a$ and $y=z=0$, the functions $\phi_{\pm}(x,\zeta_x^{(1)})$ are taken as linear combination of the functions $g(\vec r-\vec R)$ centered at the same points. This means that they correspond to $x$-direction molecular orbitals given by $$\begin{aligned}
\phi_{\pm}(x,\zeta_x^{(1)})\hspace{-.1cm}&=& (x-a)^{n_x}e^{-\zeta_x^{(1)}\left(x-a\right)^2} \nonumber \\
&& \pm (x+a)^{n_x}e^{-\zeta_x^{(1)}\left(x+a\right)^2}
\label{sax}\end{aligned}$$ However, we have observed that the function $\phi_{+}(x,\zeta_x^{(1)})$ provides lower values for the energy than the one obtained with $\phi_{-}(x,\zeta_x^{(1)})$.
Finally, the second type of the exponent was chosen similarly as the second type of the $y$ and $z$ exponents, namely $\zeta_x^{(2)} = 3\zeta_x^{(1)}/2$. Observe that due to the minimizing process the $\zeta_x^{(1,2)}$ exponents will depend on $n_x$.
Now, considering the excitations levels, given by $(n_x,n_y,n_z)$, as we are interested in confining only along the $z$ direction, we shall use larger values for the $\omega_z$. Consequently we expect few excitation in this direction, [*i.e.*]{}, we shall take only $n_z = 0, 1$, whereas in the plane $xy$ we will consider larger values: $n_x, n_y = 0, 1, 2,...$
The following results were obtained with a basis of 40 functions (2s2p2d) in each center, with 820 (780) CSF’s and 1600 (780) determinants for the singlet (triplet) states.
Results and Discussion
======================
In the following we take $\omega_x=\omega_y=0.000111$ according to Ref.[@CRC-GJ-JAP94-2579-03] corresponding to a confinement potential of 3 meV. As a typical value for the static dielectric constant in GaAs, we consider $\kappa=13.6$. Besides, placing the coordinates origin in the middle of the dots, we consider two different values of the inter-dot distance $d=2a$: $a=270 a_0$ (14.3 nm) and 400$a_0$ (21.2 nm). We analyse three different confinement regimes, along the $z-$direction, whose strength is given by $\omega_z$.
In Fig. \[J=(T-S)\] it is displayed the exchange coupling parameter ($J$) as function of the laser intensity. The parameter $J$ is defined as the energy difference between the first triplet and singlet states ($J=E_T - E_S$).
The confinement or compression in the $z-$direction is characterized when $\omega _z \gg \omega_x, \omega_y$ in Eq. (\[Vq\]). In Ref. [@Diercksen10], the value $\omega _z = 100\times\omega_x$ was sufficient to consider the electrons strongly compressed along the $z-$direction. In the present work, we have used as a confinement criterion in the $z-$direction the behavior of the root-mean-square of $z$ ($\Delta _z$) of the wave function defined as: $$\Delta _z=\sqrt{\langle z^2\rangle-\langle z\rangle^2}
\label{Dz}$$ as a function of $\omega_z$. Indeed, Fig. \[Var\_z\] confirm the confinement condition of Diercksen [*et al*]{} [@Diercksen10] for the first singlet state. By observing the behavior of $J$ (Fig. \[J=(T-S)\]), one sees that there is a clear difference from $\omega_z=0.000111$ to $0.0111$, whereas from $\omega_z=0.0111$ to $0.111$ barely has any difference.
Next, we present the behavior of the electrons localization along the inter-dot direction ($x-$axis) by analysing the double-occupation probability in one of the dots. To do so we look at density function $\rho(x_1,x_2)$ defined as: $$\begin{aligned}
\rho(x_1,x_2)=\int d\omega_1d\omega_2 dy_1dy_2 dz_1dz_2 \vert\Phi\vert^2,
\label{1}\end{aligned}$$ where $\Phi=\Phi(\vec r_1, \vec r_2,\omega_1,\omega_2)$, with $\omega_1$ e $\omega_2$ representing the spin coordinates of the two electrons, and $\vec r_1=(x_1,y_1,z_1)$ and $\vec r_2=(x_2,y_2,z_2)$ their spatial coordinates.
In Figs. \[dens\_mc-1\_a-270\_wz-000111\] – \[dens\_mc-18\_a-400\_wz-111\] are displayed the contour plots of $\rho(x_1,x_2)$ for different conditions. We have only considered the system in its fundamental state, the singlet, to analyse the electrons spatial positioning along the $x-$axis. This choice is based in what is observed in Fig. \[J=(T-S)\], where $J=E_T-E_S$ is always positive. The graphic horizontal and vertical axes, $x_1$ and $x_2$, respectively, correspond to the position of electron 1 and 2 along the $x-$axis; once the electrons are undistinguishable, we expect to have reflection symmetry in respect to the diagonal line $x_1=x_2$.
We analyze the double-occupation as a function of the laser field intensity, via the effective mass $m^*_c/m_c$, the distance $a$ and the $z-$axis confinement parameter $\omega_z$.
Thus, in Fig. \[dens\_mc-1\_a-270\_wz-000111\] it is shown $\rho(x_1,x_2)$ for $m^*_c/m_c=1$, $a=270 a_0$ and $\omega_z=0.000111$. One observes that the probability of finding both electrons in the middle of the two dots is maximum; for small values of $x_1$ and $x_2$ simultaneously, one obtains the largest values of $\rho(x_1,x_2)$. At the same time, there is a considerable chance of finding them in the same dot $\rho(270,270)=\rho(-270,-270) \approx \frac{1}{2}\rho(\sim 0, \sim0)$.
Fig. \[dens\_mc-1\_a270\_wz-111\] shows the behaviour of $\rho(x_1,x_2)$ for the same parameters $m^*_c/m_c$ and $a$, but under an extreme large confinement in the $z-$direction ($\omega_z=1.11$). Now, one observes that the maximum probability occurs at $\sim (270, -270)$, and at the corresponding symmetrical place $\sim (-270, 270)$. This means that under strong $z-$confinement the electrons drain from the inter-dots region to the dots; consequently the probability of finding both electrons at the same dot becomes very low.
Although we have analysed the effect of confinement up to a strength $\omega_z=0.111$ in Figs. \[J=(T-S)\], here we have chosen an extreme confinement condition, corresponding to $\omega_z=1.11$, in order to compare with the regime of intense laser field, which is displayed in Fig. \[dens\_mc-18\_a270\_wz-000111\], where one can see the confinement property of the laser field. Now one can observe that the behaviour of $\rho(x_1,x_2)$, for the same parameters $a$ and $\omega_z$ as in Fig \[dens\_mc-1\_a-270\_wz-000111\] but with a higher mass $m^*_c/m_c=1.86877$, is similar to the one of Fig. \[dens\_mc-1\_a270\_wz-111\].
Now, let us look at the confinement property of the distance as in Fig. \[dens\_mc-1\_a-400\_wz-000111\]. We observe that the behaviour of $\rho(x_1,x_2)$ for $a=400 a_0$ and $\omega_z=0.000111$ is similar to the one observe in Fig. \[dens\_mc-18\_a270\_wz-000111\].
Finally, the confinement properties of the distance and laser field intensity are observed in Fig. \[dens\_mc-18\_a-400\_wz-111\]. We observe that the behaviour of $\rho(x_1,x_2)$ for $a=400 a_0$ and $\omega_z=1.11$ and $m^*_c/m_c=1.86877$ characterizes a situation where the electrons are localized in the opposite dots.
It is worth mentioning that the behaviour of the density for the first triplet state was also calculated. It indicates that the electrons tend to stay away from each other, each in a dot, for all considered conditions, as it was expected. Thus they are not presented.
CONCLUSIONS
===========
In this work we have analysed the confinement of the electrons in a coupled quantum dot. We have confirmed a criterion established in the literature concerning the confinement in the $z-$direction, analysing the exchange coupling $J$ and the dispersion of the electrons along the $z$ axis through the electrons position variance $\Delta_z$. In addition, we have presented another way of confining the electrons by applying a laser field. The advantage of using laser field is that one can vary the confinement in a simple manner, in contrast to others manners which involve the parameters $a$ (the inter-dot distance) and $\omega_z$ (connected to the potential profile along the $z-$direction) both constant or, at least, difficult to manage or vary. In order to establish that, we have performed calculations using a Full-CI wave functions to obtain information about the double-occupation of the electrons.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by the Brazilian agencies CNPq, CAPES, FAPESB and FAPERJ.\
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---
abstract: 'This article continues and completes [@QuantizacionMunozAlonso]. We present two methods of quantization associetd with a linear connection given on a differentiable manifold. One of these is that presented in [@QuantizacionMunozAlonso]. In the final section the equivalence between both methods is demonstrated, as a consequence of a remarkable property of the Riemannian exponential.'
address: 'Departamento de Matemáticas, Universidad de Salamanca, Plaza de la Merced 1-4, E-37008 Salamanca, Spain.'
author:
- 'J. Muñoz-D[í]{}az and R. J. Alonso-Blanco'
title: A note on the quantization of tensor fields and quantization of mechanical systems
---
Introduction
============
The purpose of this article is, as that of [@QuantizacionMunozAlonso], the research of bridges that link Classical Mechanics with Quantum Mechanics.
In the primitive rules of quantization for conservative mechanical systems, on each solution of the Hamilton-Jacobi equation, the Hamiltonian action supplies us with the phase for the waves. Without leaving the classical path, we can arrive to a wave equation which, by adding some hypotheses, is of “Schrödinger” ([@Holland; @RM]). But the Schrödinger equation itself is inaccessible by purely classical means. In Section \[uno\] of the paper we will give a quick review of the notions of Classical Mechanics that we need in the subsequent sections. We will also indicate how, even before any quantization rule, the Broglie waves and the aforementioned classical cases of the Schrödinger equation appear.
In Section \[tres\] we will present a quantization rule of contravariant tensors on a manifold $M$, which is canonically determined by the datum of a symmetric linear connection $\nabla$ on $M$. Such a rule establishes a biunivocal correspondence between contravariant tensors on $M$ (not necessarily homogeneous) and linear differential operators acting on $\A$. The passage *differential operator* $\to$ *contravariant tensor*, is the “dequantization” determined by $\nabla$. Since a contravariant tensor on $M$ is directly interpretable as a Hamiltonian function $F\in\C^\infty(T^*M)$, we can continue the dequantization with the passage $F\to$ *Hamiltonian field associated with $F$* by the symplectic structure of $T^*M$. In this way, an infinitesimal canonical transformation on the symplectic manifold $T^*M$ corresponds to each linear differential operator on $M$. When a Riemannian metric $T_2$ (of arbitrary signature) is given on $M$, makes sense to say whether a tangent field $D$ on $T^*M$ is or is not a second order differential equation (that is to say, the field that governs the evolution of a mechanical system with configuration space $(M,T_2)$). For the Levi-Civita connection of the metric $T_2$ it happens that the necessary and sufficient condition for the field Hamiltonian corresponding to a differential operator $P$ to be a second order differential equation is that $P$ is a differential operator “of Schrödinger” $P=-\frac{\hbar^2} 2\Delta+U$ ($U\in\A$, $\hbar$ a constant which is fixed when the rule of quantization is given). This fact proves that our rule cannot give quantization of non-conservative mechanical systems (There is such a rule?).
The content of Section \[tres\] is related with [@LychaginQ]. The quantization rule used is so natural that, quite probably, it is known; but we have not found a reference in which it is clearly formulated [@AE; @Woodhouse; @LychaginQ; @Bongaarts].
In Section \[dos\] we retake the quantization rule presented in [@QuantizacionMunozAlonso]. Here it is used the geodesic field $D$ of the connection $\nabla$, instead of directly use $\nabla$. The flow of $D$ allows us to inject in a non trivial way the ring $\A$ into the ring $\mathcal{O}(M)$ of germs of smooth functions on neighborhoods of 0-section of $TM$. On the other hand, each contravariant tensor $\Phi$ given on $M$ canonically determines a “vertical” differential operator on $\AT$; such an operator when applied on $\A$ (injected into $\mathcal{O}(M)$) gives, essentially, the quantization of $\Phi$ (it remains to add factors $i\hbar$). In [@QuantizacionMunozAlonso] we used the trajectories of $D$ parameterized by $[0,1]$. If we replace the final value of the parameter, the trajectories parameterized by $[0,s]$ give another quantization rule, that is the same given by $[0,1]$, except that $\hbar$ changes to $s\hbar$. The flow of $D$ produces a uniparametric family of quantizations in which the natural parameter is the Planck “constant”. Perhaps this purely formal result may have some physical meaning.
In Section \[cuatro\] the equivalence of both quantization rules used is demonstrated. Such an equivalence is derived from a property of the Riemannian exponential: it transforms symmetrized covariant differentials of arbitrary order on $M$ into “ordinary” differentials on the fibres of $TM$. This beautiful mathematical result seems to be new in the literature.
Notes on Classical Mechanics and Undulatory Mechanics {#uno}
=====================================================
Structures previous to the metric {#previas}
---------------------------------
Let $M$ be a smooth manifold of dimension $n$. Let $TM$, $T^*M$ be the tangent and cotangent bundles of $M$, respectively. Let $\C^\infty(M)$ be the ring of differentiable functions on $M$ with complex values. We will consider $\A$ as a subring of $\AT$ by means of the injection derived from the canonical projection $\pi\colon TM\to M$.
The vector fields tangent to $TM$ which (as derivations of the ring $\AT$) kill the subring $\A$, are the *vertical* tangent fields. The differentiable 1-forms on $TM$ which, by interior product, kill the vertical tangent fields are the *horizontal* 1-forms on $TM$. The lifting of 1-forms from $M$ to $TM$ by means of $\pi^*$ are horizontal and, locally, any horizontal 1-form on $TM$ is a linear combination of such 1-forms with coefficients in $\AT$.
Each horizontal 1-form $\alpha$ on $TM$ defines on $TM$ a function $\dot\alpha$ given by $\dot\alpha(u_x)=\langle\alpha,u_x\rangle$ (inner product), for each $u_x\in TM$. In particular, for each fuction $f\in\A$, the function $\dot{({df})}$ will be denoted, for short, $\dot f$. Essentially, $\dot f$ is $df$: $\dot f(u_x)=\langle df,u_x\rangle=u_x(f)$ (derivative of $f$ by $u_x\in TM$). If $(x^1,\dots,x^n)$ is a system of coordinates on an open subset of $M$, the $(x^1,\dots,x^n,\dot x^1,\dots,\dot x^n)$ are coordinates on the corresponding open subset of $TM$.
Each covariant tensor field $a$ of degree $r$ on $M$ canonically defines a function $\dot a$ on $TM$, polynomial along the fibres: $\dot a(u_x)=\langle a,\overset{\text{$r$ times}}{\overbrace{u_x\otimes\cdots\otimes u_x}}\rangle$. In local coordinates, $\dot a$ is obtained by substituting $dx^j$ by $\dot x^j$ in the expression of the tensor $a$.
The linear structure of each fibre $T_xM$, allows us to identify the tangent space to $T_xM$ at each one of its points $u_x$ with the very vector space $T_xM$: to the vector $v_x\in T_xM$ it corresponds the vector $V_{u_x}\in T_{u_x}(T_xM)$ that is the “derivative along $v_x$”. We will say that $V_{u_x}$ is the *vertical representative* of $v_x$ at $u_x$ and that $v_x$ is the *geometric representative* of $V_{u_x}$.
By going to the definitions it is checked that, for each $f\in\A$, we have $V_{u_x}(\dot f)=v_x(f)$. In this way, each tangent vector $v_x\in T_xM$ determines on its fibre $T_xM$ a tangent field which is “constant” (= parallel).
Each tangent field on $M$ determines a vertical tangent field on $TM$, constant along each fibre.
As a consequence, each contravariant tensor field $\Phi$ on $M$ determines a *vertical* contravariant tensor field $\PHI$ on $TM$, constant (=parallel) along each fibre.
In local coordinates, $\PHI$ is obtained from $\Phi$ by substituting each field $\partial/\partial x^j$ by its vertical representative $\partial/\partial\dot x^j$.
A symmetric contravariant tensor field $\Phi$ on $M$ determines on $T^*M$ a function $F$, polynomial on the fibres, defined by $$F(\alpha_x)=\langle\Phi,\overset{\text{$r$ times}}{\overbrace{\alpha_x\otimes\cdots\otimes\alpha_x}}\rangle$$ (tensor contraction). We will say that $F$ is the *Hamiltonian associated* with $\Phi$.
If $(x^1,\dots,x^n)$ are coordinates on an open subset of $M$, the Hamiltonian associated with $\partial/\partial x^j$ is usually denoted by $p_j$: $$p_j(\alpha_x)=\langle\alpha_x,\partial/\partial x^j\rangle.$$ The functions $(x^1,\dots,x^n,p_1,\dots,p_n)$ are local coordinates on $T^*M$. For a given contravariant tensor field $\Phi$ on $M$, its associated Hamiltonian is obtained by substituting in the expression of $\Phi$ each $\partial/\partial x^j$ by $p_j$.
Symmetric contravariant tensor fields on $M$, homogeneous or not, canonically corresponds to the functions $\in\C^\infty(T^*M)$ that are polynomials along the fibres. We will refer to this particular type of functions as *Hamiltonians*.
In $T^*M$ it is defined the *Liouville 1-form* $\theta$ by $\theta_{\alpha_x}=\pi^*\alpha_x$, for each $\alpha_x\in T^*M$ ($\pi\colon T^*M\to M$ is the canonical projection). We will simplify the notation by putting $\theta_{\alpha_x}=\alpha_x$, understanding that covariant tensors in general rise from $M$ to $T^*M$ by “pull-back” through $\pi^*$. In local coordinates $(x^1,\dots,x^n,p_1,\dots,p_n)$, we have $\theta=p_j dx^j$.
The 2-form $\omega_2:=d\theta$ is the *symplectic form* on $T^*M$. In local coordinates, $\omega_2=dp_j\wedge dx^j$.
The 2-form $\omega_2$ has no kernel, so establishes an isomorphism between the $\C^\infty(T^*M)$-module of tangent fields on $T^*M$ and that of the 1-forms on $T^*M$: $$D\mapsto \alpha:=D\,\lrcorner\,\omega_2.$$
The structure of Lie algebra (given by the commutator) in the module of tangent fields is translated to the module of 1-forms defining an structure of Lie algebra given by the *Poisson bracket*. The Poisson bracket of two closed 1-forms is an exact 1-form. However, in order not to leave arbitrary constants, a Poisson bracket of functions must be defined: for each function $F\in\C^\infty(T^*M)$ the *Hamiltonian field* of $F$ is defined by the condition $D\,\lrcorner\,\omega_2=dF$.
The Poisson bracket of two functions $F$, $G$, is defined by $$\{F,G\}:=D_FG$$ (which equals $-D_GF=2\omega_2(D_F,D_G)$).
The relationship between the Poisson bracket of functions and that of 1-forms is $$d\{F,G\}=\{dF,dG\}.$$ For the local coordinates $(x^j,p_j)$, the Hamiltonian fields are $D_{x^j}=\partial/\partial p_j$, $D_{p_j}=-\partial/\partial x^j$, so that $$\{x^j,x^k\}=0,\quad \{p_j,p_k\}=0,\quad\{p_j,x^k\}=-\delta_j^k.$$
In order to avoid confusions with the terminology, let us observe that the hamiltonian field associated with the tensor $\partial/\partial x^j$ is $p_j$, while $\partial/\partial x^j$ is the Hamiltonian field of function $-p_j$.
We have seen that to each covariant tensor field $a$ of order $r$ on $M$ it corresponds a function $\dot a$ on $TM$ polynomial on the fibres. On $T^*M$, the symplectic structure $\omega_2$ makes a vertical tangent field $\widetilde\alpha$ on $TM$ correspond to each 1-form $\alpha$ on $M$, by the rule $\widetilde\alpha\,\lrcorner\,\omega_2=\alpha$. In local coordinates, the field $\widetilde\alpha$ which corresponds to $dx^j$ is $\partial/\partial p_j$. For arbitrary order $r$, the correspondence established by the symplectic structure assigns to each symmetric covariant tensor field $a$ of order $r$, a symmetric sontravariant tensor field $\PHI_a$ of order $r$ and “vertical” (its contraction with any “horizontal” tensor vanishes); in local coordinates, $\PHI_a$ is obtained by substituting each $dx^j$ by $\partial/\partial p_j$ in the expression of $a$. This tensor field $\PHI_a$ gives on each fibre of $T^*M$ a differential operator of order $r$ that does not depend on the coordinates $(x^1,\dots,x^n)$, because changes of local coordinates on $M$ give always linear changes of coordinates in the fibres of $T^*M$ (and also of $TM$). By acting fibrewise it is obtained a differential operator $\widetilde\PHI_a$, on $\C^\infty(T^*M)$, that kills the subring $\A$.
Therefore, we have the correspondences $$\dot a(x,\dot x)\quad\longleftrightarrow\quad a(x,dx)\quad\longleftrightarrow\quad \PHI_a\quad\longleftrightarrow\quad\widetilde\PHI_a(x,\partial/\partial p).$$ $\PHI_a$ is the polynomial in the $\partial/\partial p_j$ that results by the substitution in the expression of $a$ each $dx^j$ by $\partial/\partial p_j$.
The correspondence $\dot a\to\widetilde\PHI_a$ is modified by a constant factor $\dot a\to k^{-r}\widetilde\PHI_a$ (for tensors of order $r$) if the symplectic form $\omega_2$ is changed to $k\omega_2$ (where $k\in\mathbb{C}$ is arbitrary). In Quantum Mechanics, it is taken $k=i/\hbar$.
The same association {tensor} $\to$ {vertical differential operator} is obtained from the Fourier transform, by using the linear duality between $TM$ and $T^*M$. Let us denote by $\mathcal{S}(TM)$ the space of complex functions on $TM$ which, when restricted to each fibre $T_xM$, are of class $\C^\infty$ and rapidly decreasing they and all of their derivatives. Analogous meaning for $\mathcal{S}(T^*M)$. On each fibre $T_xM$ (being a $\R$-linear space) there is a measure invariant by translation $\mu$, univocally determined up to a multiplicative constant (“Haar measure”). Once fixed that factor for each $T_xM$, it is defined the Fourier transform $\mathcal{S}(TM)\to\mathcal{S}(T^*M)$ by $$(\mathcal{F}f)(\alpha_x):=\int_{T_xM} f(v_x)e^{\frac i\hbar\langle v_x,\alpha_x\rangle}\,d\mu(v_x).$$ In local coordinates: $$(\mathcal{F}f)(x,p):=\int_{T_xM} f(x,\dot x)e^{\frac i\hbar p_j\dot x^j}\,d\dot x^1\cdots d\dot x^n.$$ (the constant factor that affects the integral is irrelevant for the following).
By differentiation under the integral sign we get the classical formula $$\F(\dot a(x,\dot x)f(x,\dot x))=a(x,-i\hbar\partial/\partial p)(\F f)(x,p),$$ that is, $\F\circ\dot a=\widehat\PHI_a\circ\F$, where $\widehat\PHI_a$ is the vertical differential operator which results of substituting in the tensor $a$ each $dx^j$ by $-i\hbar\partial/\partial p_j$.
This is the correspondence given by the symplectic structure $(i/\hbar)\omega_2$·
For later references, let us write the correspondence between symmetric covariant tensor fields on $M$ and vertical differential operators on $T^*M$, once $\omega_2$ is substituted by $(i/\hbar)\omega_2$: $$\label{corr1}
\dot a=a(x,\dot x)\quad\longleftrightarrow\quad a=a(x,dx)\quad\longleftrightarrow\quad \widehat\PHI_a(x,\partial/\partial p)=a(x,-i\hbar \partial/\partial p).$$
Introduction of a metric. Classical mechanical systems {#intrometrica}
------------------------------------------------------
Let $T_2$ be a Riemannian metric (non degenerate of arbitrary signature) on the manifold $M$. Such a metric determines an isomorphism of fibre bundles $TM\simeq T^*M$, that allows us to transport from one to each other all the structures that we have considered. Hence, the Liouville form $\theta$ and the symplectic form $\omega_2$ passe from $T^*M$ to $TM$, where we will denote them in the same way.
If the expression of the metric in local coordinates is $T_2=g_{jk}(x)dx^j\,dx^k$, the isomorphism $TM\simeq T^*M$ is expressed by the equations $p_j=g_{jk}\dot x^k$. The differential operators $\partial /\partial p_j$ transported to $TM$ become $g^{jk}\partial/\partial \dot x^k$, and in the correspondence (\[corr1\]), the operator $\widehat\PHI_a$ is $a(x,-i\hbar g^{jk}\partial/\partial\dot x^k)$. For the 1-form $\alpha_j=g_{jk} dx^k$, it holds $\dot\alpha_j=p_j$, and the corresponding operator $\widehat\PHI_a$ is $g_{jk}(-i\hbar g^{k\ell}\partial/\partial \dot x^\ell)=-i\hbar\partial/\partial\dot x^j$: $$\label{corr2}
p_j\mapsto -i\hbar\frac{\partial}{\partial\dot x^j}$$ in the correspondence of functions on $TM$ linear along the fibres with vertical tangent fields.
In coordinates of $TM$, $\theta=g_{jk}\dot x^jdx^k$ and the function associated with $\theta$ on $TM$ is $\dot\theta=g_{jk}\dot x^j\dot x^k=2T$ where $T$ is the *kinetic energy* function.
On $TM$ the Hamiltonian tangent field for the function $-T$ is the *geodesic field* of $(M,T_2)$; according its very definition, it holds $$\label{geodesico1}
{D_G}\lrcorner\,\omega_2+dT=0$$
For later references, the well known expression of the geodesic field is $$\label{geodesico2}
D_G=\dot x^j\frac\partial{\partial x^j}-\Gamma_{k\ell}^j(x)\dot x^k\dot x^\ell\frac\partial{\partial \dot x^j}$$ where the $\Gamma$’ are the Christoffel symbols of the metric (within our convention, $\Gamma_{k\ell}^j=\left\{{\substack{j\\ k\ell}}\right\}$).
Let us recall that a *second order differential equation* on $M$ is, by definition, a tangent field $D$ on $TM$ such that, as a derivation, takes each $f\in\A$ to $Df=\dot f$. Thereby, $D_G$ in (\[geodesico2\]) is a second order differential equation.
Two second order differential equations on $M$ derive in the same way the subring $\A$ of $\AT$. Thus, any second order differential equation $D$ on $M$ is of the form $D=D_G+V$, where $V$ is a vertical tangent field on $TM$.
The vertical tangent fields are the *forces* of the Classical Mechanics. A classical-mechanical system is a set comprised by three data $(M,T_2,V)$ and the *Newton law* says that the evolution of the space of states $TM$ is the flow of the field $D=D_G+V$. In particular, when $V=0$, the system evolves according the geodesic flow (*inertial law*).
In order to “visualize” a force $V$ in an state $u_x\in TM$ (“position-velocity state”) we must translate the vertical vector $V_{u_x}$ to its geometrical representative $v_x$. Once this is done, the Newton law can be stated in the original form “force = mass $\times$ acceleration”: the trajectory of the field $D$ that passes through the point $u_x$ is projected onto $M$ as a curve whose tangent field $u$ (defined along the curve) holds $v_x=\nabla_{u_x}u$. The left member is the “force” and the right member is the “mass times acceleration”, understood that masses and inertial moments are incorporated as factors in $T_2$. For further details see ([@MecanicaMunoz], Section 1).
By looking at the coordinate expression of the symplectic form, we immediately see that, in the correspondence established by $\omega_2$ between tangent fields and 1-forms on $T^*M$, the vertical fields correspond exactly with horizontal 1-forms: $i_V\omega_2=-\alpha$, horizontal. By applying this equality to the field $D=D_G+V$ it results \[Newton\] D\_2+dT+=0. Equation \[Newton\] expresses the biunivocal correspondence between second order differential equations on $M$ and horizontal 1-forms on $TM$. The form $\alpha$ is the *work form* of the mechanical system.
A mechanical system $(M,T_2,\alpha)$ is said to be *conservative* when the $\alpha$ is an exact differential form, $dU$. By taking into account that $\alpha$ is horizontal, $U$ have to be a function $\in\A$. The sum $H=T+U$ is the *Hamiltonian* of the system, and (\[Newton\]) is $$\label{hamiltoniano1}
D\lrcorner\,\omega_2+dH=0$$
$D$ is the hamiltonian field of the function $-H$ in the terminology of Section \[previas\].
Let us highlight the following consequence of (\[Newton\]), which will be important in Section \[tres\]:
The Hamiltonian fields on $T^*M$ that, by means of the metric $T_2$, passe into $TM$ as second order differential equations are exactly those that govern the evolution of conservative mechanical systems on $(M,T_2)$ through (\[hamiltoniano1\]).
No other infinitesimal canonical transformation on $T^*M$ is the law of evolution of a mechanical system on $ (M, T_2) $.
Equation (\[hamiltoniano1\]), when is written in coordinates of $T^*M$, is the system of Hamilton canonical equations.
A tangent field $u$ in $M$ is an *intermediate integral* of the field $D$ when the solution-curves of $u$ in $M$, lifted as curves to $TM$ (each point $x$ of the curve goes to the point $(x,u_x)$ of $TM$) is also a solution of $D$. We can think about a given vector field $u$ as a section of the fibre bundle $TM\to M$; when passing to $T^*M$, the section $u$ corresponds to a section $\alpha$ of $T^*M\to M$ where $\alpha=u\lrcorner\, T_2$ or, in other words, $u=\textrm{grad}\,\alpha$. If the section $\alpha$ is a lagrangian submanifold of $T^*M$, locally $\alpha=dS$ for a certain function $S$ on $M$ (or on some open subset). In such a way, the necessary and sufficient condition for $u=\textrm{grad}\,S$ to be an intermediate integral of the Hamiltonian field $D$ in (\[hamiltoniano1\]) is that $$\label{HJ}
H(\textrm{grad}\,S)=E,\quad\text{constant,}$$ where $H(\textrm{grad}\,S)$ is the specialization oh $H\in\AT$ to the section $u=\textrm{grad}\,S$. (\[HJ\]) is the *Hamilton-Jacobi equation* (see [@RM; @RMarxiv]).
De Broglie waves and Schrödinger equation
-----------------------------------------
Let us consider a conservative mechanical system with configuration space $(M,T_2)$ and Hamiltonian $H=T+U$. Let $S\in\C^\infty(M)$ be a solution of the Hamilton-Jacobi equation (\[HJ\]); $\textrm{grad}\,S$ is an intermediate integral of the equations of motion. Let $\textrm{Grad}\,S$ be the vertical field on $TM$ whose geometric representative is $\textrm{grad}\,S$; we have $\textrm{Grad}\,S\lrcorner\,\omega_2=\textrm{grad}\,S\lrcorner\, T_2=dS$, whereby $\textrm{Grad}\,S$ is the Hamiltonian field whose Hamiltonian function is $S$.
The correspondence (\[corr1\]) applied to the tensor $dS$ is $$\dot S\quad\leftrightarrow \quad dS=\frac{\partial S}{\partial x^j}dx^j\quad\leftrightarrow\quad -i\hbar\frac{\partial S}{\partial x^j}\frac{\partial }{\partial p_j}= -i\hbar\,\textrm{Grad}\,S.$$
For first order differential operators we have a rule, *previous to any quantization rule*, which assigns a field on $M$ to each vertical vector field constant along the fibres of $TM$: to go from a vertical field to its geometric representative. In this case, $-i\hbar\,\textrm{Grad}\,S\mapsto-i\hbar\,\textrm{grad}\,S$.
The correspondence: $$\text{\emph{Classical magnitude (function on $TM$)}, $\dot S$}\,\,\to\,\,
\text{\emph{Differential operator on $\C^\infty(M)$}, $-i\hbar\,\textrm{grad}\,S$}$$ must remain valid in any quantization law.
On the section $\textrm{grad}\,S$ of $TM$, the function $\dot S$ takes the value $\dot S\mid_{\textrm{grad}\,S}=\langle dS,\textrm{grad}\,S\rangle=\|\textrm{grad S}\|^2=2(E-U)$. The functions $\varphi$ on $M$ on which the classical magnitude $\dot S\mid_{\textrm{grad}\,S}$ and its associated differential operator, $-i\hbar\,\textrm{grad}\,S$, act (the first one by means of multiplying) giving the same result, are those which hold the differential equation: $$\label{Broglie1}
-i\hbar\,\textrm{grad}\,S(\varphi)=2(E-U)\cdot\varphi.$$
The parameter $t$ proper for the trajectories of the vector field $\textrm{grad}S$ holds on each trajectory \[tyS\] dt==.
By changing the parameter $t$ by the parameter $S$ on each trajectory, Equation (\[Broglie1\]) is $$-i\hbar\frac{d\varphi}{dS}=\varphi,$$ which gives $$\label{Broglie2}
\varphi=\varphi_0e^{i\frac S\hbar}=\varphi_0e^{2\pi i\frac Eh\cdot\frac SE},$$ where $\varphi_0$ is an arbitrary first integral of the vector field $\textrm{grad}\,S$.
The wave function $\varphi$ is derived form the condition of that, on it, give the same result the action of the differential operator $-i\hbar\,\textrm{grad}\,S$ and (by multiplication) the classical magnitude $\dot S$ (from which that operator proceeds) restricted to the section $\textrm{grad}\,S$. The generalization of this principle of formation of wave equations is that used in [@QuantizacionMunozAlonso].
Note that the advance rate of the wavefronts for $\varphi$ in (\[Broglie2\]) is uniform if the quotient $S/E$ is used as time (or a constant multiple), while the time $t$ that measures the motion of the virtual particles in the mechanical system holds (\[tyS\]). These two times are different, except for the geodesic field ($U=0$).
Going back to (\[Broglie2\]) and taking constant $\varphi_0$ an straightforward computation gives the identity $$\label{Broglie4}
\left(-\frac{\hbar^2} 2\Delta+U\right)\varphi=\left(\frac{\hbar}{2i}\Delta S+H(\textrm{grad} S)\right)\varphi.$$ From that identity it is derived the
Let $(M,T_2,dU)$ be a conservative mechanical system. Let $S\in\C^\infty(M)$. From the three conditions
1. $S$ holds the Hamilton-Jacobi equation (\[HJ\])
2. $S$ is harmonic: $\Delta S=0$
3. $\varphi=e^{iS/\hbar}$ holds the Schrödinger equation $\left(-\frac\hbar 2\Delta+U\right)\varphi=E\varphi$
each couple of them implies the third one.
When $M$ is oriented, the metric $T_2$ gives a volume form and condition B) can restated as
1. $-i\hbar\textrm{grad}\,S$ is self-adjoint.
By admitting a factor $\varphi_0$ a first integral of $\textrm{grad}\,S$, not necessarily constant, we obtained conditions on $\varphi_0$ allowing to generalize the above proposition, but it is not possible to reach a true general Schrödinger equation [@Holland; @RM]. There is not a continuous path classical-quantic. The pass requires rules of quantization for tensors of order higher than 1.
Quantization of contravariant tensors. Dequantization of differential operators {#tres}
===============================================================================
In this section we will study the way in which a symmetric linear connection $\nabla$ on the configuration space $M$ determines a canonical biunivocal correspondence (up to the concrete value of $h$) between contravariant tensor fields on $M$ and linear differential operators acting on $\A$. The passage $\text{\emph{tensor}}\to\text{\emph{differential operator}}$ is the rule of quantization defined by $\nabla$ and the reverse step $\text{\emph{differential operator}}\to\text{\emph{tensor}}$ is the rule of dequantization defined by $\nabla$; this second passage, once given, can be continued with another one $\text{\emph{tensor}}\to\text{\emph{infinitesimal contact transformation on $T^*M$}}$, which already only depends on the structure of $T^*M$.
The quantization rule established with the data $(M,\nabla)$ is an almost obvious generalization of the usual rule of quantization on the flat space $(\R^n,d)$, where we denote by $d$ the connection canonically associated with the vector structure of $\R^n$ (the “parallel transport” for $d$ is the transport by linear parallelism). Let us recall such a rule.
Let $E$ be a real $n$-dimensional vector space. Once fixed a system of vector coordinates $(x^1,\dots,x^n)$ on $E$, each real symmetric contravariant tensor of order $r$ at the origin of $E$ is written in the form: $$\Phi_0=a^{j_1\cdots j_r}\left(\frac\partial{\partial x^{j_1}}\right)_0\cdots\left(\frac\partial{\partial x^{j_r}}\right)_0,$$ where the $a$ are real numbers and by $\cdots$ we denote the symmetrized tensor product.
The linear structure of $E$ allows us to propagate “by parallelism” the tensor $\Phi_0$ to a tensor field $\Phi$ on the whole of $E$, whose expression is the same as that of $\Phi_0$, by deleting the subindex $0$. This tensor field $\Phi$ defines on $\CE$ a differential operator $$\widehat\Phi:=(-i\hbar)^ra^{j_1\cdots j_r}\frac{\partial^r}{\partial x^{j_1}\cdots\partial x^{j_r}}.$$ The assignation $\Phi\to\widehat\Phi$ is independent of changes of *vector* coordinates on $E$. When the numbers $a^{j_1\cdots j_r}$ are substituted by functions in $\CE$, the same formula assigns to the tensor field $\Phi$ a differential operator $\widehat\Phi$ independently of the concrete choice of *vector* coordinates.
In we wish that $\widehat\Phi$ to be self-adjoint (for the measure translation invariant of $E$) it is sufficient to replace it by $\frac 12(\widehat\Phi+\widehat\Phi^+)$.
When we work on a concrete problem in curvilinear coordinates, the quantization rule is applied by passing the tensors to vector coordinates, quantizing them according to the above rule and, then, coming back to the given curvilinear coordinates.
This recipe for quantization is *intrinsically determined* by the vector structure of $E$. On each $f\in\CE$ we have $$\widehat\Phi(f)=(-i\hbar)^r\langle\Phi,d^rf\rangle,$$ where $\langle\,,\,\rangle$ denotes tensor contraction, and $d^rf$ is the $r$-th iterated differential of $f$, that has an intrinsic sense on $E$ because of its vector structure.
The generalization to any smooth manifold $M$ endowed with a symmetric (=torsionless) linear connection $\nabla$ is immediate:
For each symmetric covariant tensor field of order $r$, $\Phi$, on $(M,\nabla)$, the *quantized of $\Phi$* is the differential operator $\widehat\Phi$ which, for each $f\in\A$ gives \[quantizado\] (f):=(-i)\^r,\^r\_fwhere $\langle\,,\,\rangle$ denotes tensor contraction and $\nabla^r_{\textrm{sym}}f$ is the symmetrized tensor of the $r$-th covariant iterated differential of $f$ with respect to the connection $\nabla$.
The quantized of a non-homogeneous tensor is the sum of the quantized of its homogeneous components.
This definition can be generalized giving a differential operator between sections of fibre bundles for each contravariant tensor $\Phi$ on $M$, once a linear connection is fixed in the first fibre bundle. This generalization does not affect what follows, and we leave it aside.
Let us recall that a differential operator of order $r$ on $M$ (= differential operator of order $r$ on $\A$) is an $\Com$-linear map $P\colon\A\to\A$ which holds the following condition: for each point $x\in M$, $P$ takes the ideal $\m_x^{r+1}$ into $\m_x$ ($\m_x$ is the ideal of the functions of $\A$ vanishing at $x$).
It is derived that $P$ takes the quotient $\m_x^{r}/\m_x^{r+1}$ into $\A/\m_x=\Com$. By taking into account that $\m_x^{r}/\m_x^{r+1}$ is the space of symmetric covariant tensors of order $r$ at the point $x$ (homogeneous polynomials of degree $r$, with coefficients in $\Com$, in the $d_xx^1,\dots,d_xx^n$, once taken local coordinates), we see that $P$ determines a symmetric contravariant tensor of order $r$ called *symbol of order $r$ of $P$ at $x$*, denoted by $\sigma_x^r(P)$, \[simbolo\] \_x\^r(P)\_x\^[r]{}/\_x\^[r+1]{}=T\^[\*r]{}\_xM, that is the map canonically associated with $P$ by pass to the quotient.
Given $f\in\m_x^r$, the differential operator $P$ of order $r$ gives $(Pf)(x)$, depending only on the class $[f]_{\textrm{mod}\,\m_x^{r+1}}$. But the identification of $\m_x^r/\m_x^{r+1}$ with the space of symmetric covariant tensors of order $r$ at the point $x$ is not unique. In order to fix the tensor $\sigma_x^r(P)$ in such a way that its contraction with the symmetric covariant tensor that represents $[f]_{\textrm{mod}\,\m_x^{r+1}}$, to be $(Pf)(x)$, we take such a covariant tensor as $d_x^rf$, computed in any local system of coordinates; the covariant tensor $d_x^rf$ so calculated for $f\in\m_x^r$, does not depends on the choice of coordinates.
When $x$ runs over $M$, we get the tensor field $\sigma^r(P)$ on $M$ called *symbol of order $r$ of $P$*. If $\sigma^r(P)=0$, $P$ is of order $r-1$.
In the case $M=\R^n$, with vector coordinates $x^1,\dots,x^n$, let us denote $\partial^\alpha$ the tensor $\partial^\alpha:=(\partial/\partial x^1)^{\alpha_1}\cdots(\partial/\partial x^n)^{\alpha_n}$. Its quantized by the rule (\[quantizado\]) (with the vector connection of $\R^n$) is $\widehat{\partial^\alpha}:=(-i\hbar)^{|\alpha|}D^\alpha$, where $D^\alpha$ is the differential operator ${\partial^{|\alpha|}}/{(\partial x^1)^{\alpha_1}\dots(\partial x^n)^{\alpha_n}}$. It is directly seen that $\sigma^{|\alpha|}(D^\alpha)=\partial^\alpha$, so that for any tensor field of order $r=|\sigma|$ on $\R^n$ is obtained, by adding terms, \[simboloplano\] \^r()=(-i)\^r
Going from $\R^n$ to the general case $(M,\nabla)$ let us observe that, when the iterated differentials of a function $f$ are calculated in local coordinates, the derivatives of order $r$ of $f$ appear in terms which does not contain Christoffel symbols (as in the case of $\R^n$). Since the symbol of an operator of order $r$ depends only on these terms, Formula (\[simboloplano\]) is still valid in general for the quantization rule (\[quantizado\]) on $(M,\nabla)$.
\[tquantizado1\] The rule of quantization (\[quantizado\]) establishes a biunivocal correspondence between linear differential operators $P$ and symmetric contravariant tensor fields (not necessarily homogeneous) on $M$. To the operator $P$ of order $r$ corresponds the tensor $\Phi=\Phi_{r}+\Phi_{r-1}+\cdots\Phi_{0}$ (each $\Phi_j$ denotes the homogeneous component of degree $j$) such that $$\sigma^r(P)=(-i\hbar)^r\Phi_r$$ and, for $k=1,\dots,r$: $$\sigma^{r-k}(P-\widehat\Phi_{r}-\cdots-\widehat\Phi_{r-k+1})=(-i\hbar)^{r-k}\Phi_{r-k}$$ and \[quantizado3\] P==\_[r]{}+\_[r-1]{}++\_[0]{}.
The contravariant tensor $\Phi$ in (\[quantizado3\]) is the *dequantized of the differential operator $P$ by the connection $\nabla$*.
We have seen in Section \[previas\] that symmetric contravariant tensor fields (homogeneous or not) on $M$ canonically correspond with functions $f\in \AC$ polynomials along the fibres.
The function $F\in\AC$ corresponding to the tensor $\Phi$ dequantized of the differential operator $P$ will be called *Hamiltonian of $P$* with respect to the connection $\nabla$.
The symplectic structure $\omega_2$ of $T^*M$ assign to each $F\in\AC$ a Hamiltonian vector field $D_F$, as we have already remembered in section \[previas\], by the rule $D_F\lrcorner\,\omega_2=dF$. These hamiltonian fields are the *infinitesimal contact transformations* of Lie [@Lie1; @Lie2]; they are the infinitesimal generators of the 1-parametric groups of automorphisms of $T^*M$ which preserve its symplectic structure.
We will call *infinitesimal contact transformation associated with the differential operator $P$ or Hamiltonian field associated with $P$* to the tangent field $D_P$ on $T^*M$ such that $${D_P}\lrcorner\,\omega_2+dF=0,$$ where $F$ is the Hamiltonian of $P$.
The path $P\to F\to D_P$ is univocal. The reverse path $D_P\to F$ determines $F$ up to a additive constant; then, $F\to P$ is univocal. Thus, up to an additive constant for $P$, the correspondence $P\leftrightarrow D_P$ is biunivocal.
\[tquantizado2\] The symmetric linear connection $\nabla$ on $M$ canonically establishes a biunivocal correspondence between linear differential operators $P$ on $\A$ (up to additive constants) and infinitesimal canonical transformations of the simplectic manifold $T^*M$ corresponding to functions polynomial along fibres (Hamiltonians).
Let us assume that $M$ is endowed with a Riemannian metric $T_2$ (of arbitrary signature = pseudoriemannian metric) and $\nabla$ the associated Levi-Civita connection. Under these conditions, it makes sense to say whether or not a tangent field $D$ on $T^*M$ is a second-order differential equation; that is, the tangent field that governs a mechanical system with the configuration space $(M,T_2)$. We have,
\[unicosbuenos\] The necessary and sufficient condition for a linear differential operator $P$ on $(M,T_2)$ to have as associated infinitesimal contact transformation $D_P$ a second order differential equation is that $P$ is of the form $$P=-\frac{\hbar^2}2\Delta+U$$ where $\Delta$ is the Laplacian operator of the metric and $U\in\A$.
Let us begin by checking that the tensor $\Phi$, contravariant form of the metric tensor, has as quantized operator $\widehat\Phi=-\hbar^2\Delta$. In local coordinates, with $T_2=g_{jk}dx^jdx^k$, is $\Phi=g^{rs}\frac{\partial}{\partial x^r}\otimes\frac{\partial}{\partial x^s}$. The expression for the second iterated covariant differential is $$\nabla^2f=\left(\frac{\partial^2f}{\partial x^k\partial x^j}-\Gamma_{jk}^\ell\frac{\partial f}{\partial x^\ell}\right)dx^j\otimes dx^k;$$ by contracting with $\Phi$, $$\langle\Phi,\nabla^2f\rangle=g^{jk}\left(\frac{\partial^2f}{\partial x^k\partial x^j}-\Gamma_{jk}^\ell\frac{\partial f}{\partial x^\ell}\right)=\Delta f.$$ By incorporating to $\phi$ the factor $(-i\hbar)^2$ we see that the quantized of $\Phi$ is $-\hbar^2\Delta$.
The Hamiltonian function corresponding to the tensor $\Phi$ is $g^{rs}p_rp_s=2T$ (where $T$ is the kinetic energy function). Finally, for the Hamiltonian $H=T+U$, the corresponding quantum operator is $(-\hbar^2/2)\Delta+U$.
When dequantizing, we go from the operator $(-\hbar^2/2)\Delta+U$ to the Hamiltonian $T+U=H$, and, then to the Hamiltonian field $D_P$ such that $D_P\lrcorner\,\omega_2+dH=0$; $D_P$ is the field of the canonical equations for the mechanical system $(M,T,dU)$.
Conversely, let us assume that $D_P$ is a second order differential equation. Equation \[Newton\] gives that it holds $D_P\lrcorner\,\omega_2+dT+\alpha=0$, where $\alpha$ is horizontal; since $D_P$ is a contact infinitesimal transformation, $\alpha$ has to be exact, so that of the form $dU$ for some $U\in\A$: $D_P\lrcorner\,\omega_2+dH=0$, for $H=T+U$. Since $T$ is the Hamiltonian function associted with the operator $\frac 12\Phi$ as before, when quantizing it turns that $P=-\frac{\hbar^2}2\Delta+U$.
**Problem.** There is something similar to a Schrödinger equation for non-conservative mechanical systems?
\(1) Let $\Phi_r$ be an homogeneous tensor of order $r$ that corresponds to $P$ by (\[quantizado3\]). Considered as a function on $T^*M$, $\Phi_r$ is $F_r$, homogeneous of degree $r$ on the fibres. The first order partial differential equation $F_r((dS)^r)=0$ has as solutions the hypersurfaces $S=\text{const.}$ *characteristic* for the differential operator $P$; they are the hypersurfaces of $M$ where the problem of initial conditions cannot be treated by the Cauchy-Kowalevsi method (for instance, for $\Delta$, the equation of characteristics is $\|dS\|^2=0$, the “Eikonal equation”). The Hamiltonian field of $F_r$ has as solutions the *bicharacteristics* of $P$. This field does not coincide, in general, with $D_P$. The field which propagates the singularities of $P$ is the hamiltonian field of $F_r$, not the one of the total Hamiltonian of $P$, $D_P$.
\(2) The relationship between the Poisson bracket of two Hamiltonians and the commutator of the corresponding quantum operators is: \[conmutadosimbolos\] \^[r+s-1]{}\[,\]=-{\^r(),\^s()} where $\Phi$ is a tensor of order $r$, $\Psi$ of order $s$, $[\,,\,]$ is the commutator of quantized tensors and $\{\,,\,\}$ is the Poisson bracket of $\sigma^r(\widehat\Phi)$, $\sigma^s(\widehat\Psi)$, by identified with the functions they define on $T^*M$ (the minus sign proceeds from the convention taken in \[previas\] for the Poisson bracket).
Formula (\[conmutadosimbolos\]) is valid for every connection $\nabla$, and its proof can be done as in the case of $\R^n$ with the vector connection, since only highest order terms of the operator intervene and we can use the symplectic form $\widehat\omega_2=dp_j\wedge dx^j$ as in vector coordinates. The checking of (\[conmutadosimbolos\]) is a simple calculation.
Quantization by means of Riemannian exponential. Families of quantizations parameterized by $h$ {#dos}
===============================================================================================
In [@QuantizacionMunozAlonso] we have presented a quantization rule for the classical system $(M,T_2)$ by means of a linear symmetric connection $\nabla$ on $M$. That rule is defined from the geodesic field $D$ associated with the connection. Instead of use directly $\nabla$, we use the geodesic field $D$ of $\nabla$. The flow of the field $D$ on $TM$ allows us to establish an isomorphism of manifolds between a certain neighborhood $\mathcal U_x$ of vector $0$ in $T_xM$ and a neighborhood $U_x$ of $x$ in $M$, by associating with the vector $v_x\in\mathcal U_x$ the final point of the geodesic (curve solution of $D$) parameterized by $[0,1]$ that starts from the point $x$ with initial velocity $v_x$. When $x$ runs over $M$, the union of all the $\mathcal U_x$ is a neighborhood $\mathcal U$ of the 0-section in $TM$, and the flow of $D$ defines, in the above described way, a differentiable map $\textrm{exp}\colon\mathcal U\to M$, in which the 0-section of $TM$ is identified (as a part of $\mathcal U$) with $M$.
For each $f\in\A$, let $$\label{exp1}
\widehat f:=\textrm{exp}^*(f)\in\C^\infty(\mathcal U).$$ Of $\widehat f$ the only thing we are interested in is its germ at the section $0$ of $TM$. If we denote by $\Ocal(M)$ the ring of germs of differentiable functions in neighborhoods of the section $0$ of $TM$, we identify $\widehat f$ with its germ $\in\Ocal(M)$. Thus, we have an injection of rings $\A\hookrightarrow\Ocal(M)$, $f\mapsto\widehat f$.
In the injection of rings $\A\hookrightarrow\AT$ produced by the natural projection $TM\to M$, the $f\in\A$ give functions constant along the fibres, annihilated by each vertical differential operator on $TM$ (except these of order 0). But in the injection $f\mapsto\widehat f$, such operators no longer annihilate the $\widehat f$.
In Section \[intrometrica\] we have seen how, with each covariant tensor field $a$ there is a vertical contravariant tensor field $\widehat\PHI_a$ on $TM$ associated by means of a rule determined by the metric $T_2$ and the symplectic form of $T^*M$ (or, alternatively, the Fourier transform). In local coordinates, $\widehat\PHI_a$ is obtained by substituting into the expression $a(x,dx)$, each $dx^j$ by the vertical vector field $-i\hbar g^{jk}\partial/\partial\dot x^k$. Or, by considering the coordinates $p_j$ as 1-forms, by substituting each $p_j$ by $-i\hbar\partial/\partial\dot x^j$ (\[corr2\]).
The quantization rule given in [@QuantizacionMunozAlonso] is
Let $(M,T_2)$ a configuration space, $\nabla$ a symmetric linear connection on $M$, $\A\hookrightarrow\Ocal(M)$ ($f\mapsto\widehat f$) the injection determined by the exponential defined by the geodesic field $D$ of $\nabla$. For each symmetric covariant tensor field $a$ of order $r$, the differential operator $\widehat a$ quantized of the function $\dot a$ by $\nabla$ gives, for each $f\in\A$ the value \[agorro\] a(f):=\_a,d\_0\^rf where $\langle\,,\,\rangle$ is the tensor contraction and $d_0^r\widehat f$ is the $r$-th differential of $\widehat f$ along each fibre of $TM$, and taking the value at the 0 section.
The “vertical differential” $d^r\left(\widehat f|_{T_xM}\right)$ makes sense due to $T_xM$ is a vector space.
So as not to get lost in technicalities in the discussion that follows, let us suppose that the geodesic field $D$ of $\nabla$ is complete. Let $\{\tau_s\}_{s\in\R}$ the 1-parametric group of automorphisms of the manifold $TM$ generated by $D$. The exponential map is, in this case, the composition $$\xymatrix{TM\ar[r]^-{\tau_1}\ar[dr]_-{\textrm{exp}} & TM\ar[d]^-{\pi\text{ (canonical projection)}}\\
& M}$$
From the mathematical point of view, the restriction of the parameter $s$ to the value $1$ is artificial. The natural thing is to consider an arbitrary segment $[0,s]$, $\tau_s$ instead of $\tau_1$, $\textrm{exp}_s=\pi\circ\tau_s$ and $\widehat f_s=\textrm{exp}_s^*(f)$. The classical magnitude $\dot a$ will be quantized as the differential operator $\widehat a_s$: \[agorros\] a\_s(f):=\_a,d\_0\^rf\_sThe interesting thing is that the quantization rule $\dot a\to\widehat a_s$ changes in such a way that $\widehat a_s=s^r\widehat a$ for tensors of order $r$. Indeed, since $D$ is a field of the form (\[geodesico2\]) (it does not matter how are the Christoffel symbols), it holds that $\pi\circ\tau_s(x,v_x)=\pi\circ\tau_1(x,sv_x)$ (the final point of the geodesic parameterized by $[0,s]$ with initial tangent vector $v_x$ at $x$, is the same that the final point of the geodesic parameterized by $[0,1]$ with initial tangent vector $sv_x$). It follows that $\widehat f_s(x,v_x)=\widehat f(x,sv_x)$, that is to say: $$\widehat f_s=\widehat f\circ (\text{Homothetie of ratio $s$ along each fibre of $TM$}).$$ It is derived that $d_0^r\widehat f_s=s^rd_0^r\widehat f$, then $\widehat a_s=s^r\widehat a$. This means that quantization $\dot a\to\widehat a_s$ is deduced from quantization $\dot a\to\widehat a$ by replacing $h$ by $sh$. The field $D$ canonically produces a 1-parametric family of quantizations whose parameter is the Planck “constant”.
Identity of the two considered rules of quantization {#cuatro}
====================================================
In this section all the functions are real.
Maintaining the above notation, $M$ is an smooth manifold of dimension $n$, $\nabla$ is a symmetric linear connection on $M$. The exponential map associated with $\nabla$ (defined on a neighborhood of $0$ on each fibre $T_{x_0}M$) assigns to each vector $v_{x_0}\in T_{x_0}M$ the point $\textrm{exp}(v_{x_0})\in M$ that is the final point of the geodesic of $\nabla$ parameterized by $[0,1]$ which starts from $x_0$ with tangent vector $v_{x_0}$. The local isomorphism $\exp\colon T_{x_0}M\to M$ assigns to each function $f\in\A$ a differentiable function defined in an neighborhood of $0$ in $T_{x_0}M$; when $x_0$ runs over $M$, $f$ gives a function $\widehat f$ defined in a neighborhood of the 0 section of $TM$; the map $f\mapsto\widehat f$ injects $\A$ into the ring $\mathcal O(M)$ comprised by germs of smooth functions on neighborhoods of the 0-section of $TM$.
Whatever the local coordinates $x^1,\dots,x^n$ in an neighborhood of $x_0$, the corresponding $\dot x^1,\dots,\dot x^n$ are linear coordinates on $T_{x_0}M$. Thus, for each $g\in\C^\infty(T_{x_0}M)$ the following tensor is intrinsically defined \[identidad1\] d\_0\^rg=\_[j\_1,…,j\_r=1]{}\^n(0)dx\^[j\_1]{}dx\^[j\_r]{}
The local isomorphism $\exp\colon T_{x_0}M\to M$ gives an isomorphism of tangent spaces $T_0(T_{x_0}M)\simeq T_{x_0}M$ (the already known) makes to correspond to each vertical vector in $TM$ its geometric representative: $(\partial/\partial\dot x^j)_0\to(\partial/\partial x^j)_{x_0}$. The dual morphism makes to correspond $d_{x_0}x^j$ to $d_0\dot x^j$. This isomorphism transforms the $d_0^rg$ of (\[identidad1\]) into a tensor at the point $x_0$ of $M$. In particular, for each $f\in\A$ we define the tensor \[identidad2\] d\_[x\_0]{}\^rf:=\_[j\_1,…,j\_r=1]{}\^n(0)d\_[x\_0]{} x\^[j\_1]{}d\_[x\_0]{} x\^[j\_r]{} which is the tensor at $x_0\in M$ that corresponds to $d_0^r\widehat f$ in (\[agorro\]) by means the isomorphism $T_{x_0}M\simeq T_0(T_{x_0}M)$.
\[tidentidad1\] For each $f\in\A$ and each $r$ is \[identidad3\] d\_[x\_0]{}\^rf=\^r\_[x\_0,]{}f
We begin by considering the case in which $f$ is one of the coordinates, $f=x^1$. The equation of the geodesics of $\nabla$ gives \[identidad4\] =-\_[jk]{}\^1x\^jx\^k=(dx\^1),xx where $\langle\quad,\quad\rangle$ is the coupling given by duality and $\dot x$ is the vector tangent to the geodesic in the point corresponding to the value $s$ of the parameter: $\dot x^j=dx^j/ds$.
Differentiating (\[identidad4\]) and applying the equation of the geodesics, we obtain $$\begin{aligned}
\frac{d^3x^1}{ds^3}=&-\left(\frac{\partial \Gamma_{jk}^1}{\partial x^\ell}-\Gamma_{rk}^1\Gamma_{j\ell}^r-\Gamma_{rj}^1\Gamma_{k\ell}^r\right)
\dot x^j\dot x^k\dot x^\ell=\langle \nabla\nabla dx^1,\dot x\otimes\dot x\otimes\dot x\rangle\\
&\cdots
\end{aligned}$$
By applying at each step the equation of geodesic, we obtain by induction
$$\begin{aligned}
\label{identidad5}
\frac{d^rx^1}{ds^r}&=\langle\overset{r-1}{\overbrace{\nabla\,\cdots\,\nabla}} dx^1,\overset{r}{\overbrace{\dot x\otimes\cdots\otimes\dot x}}\rangle
=\langle\nabla^rx^1,\overset{r}{\overbrace{\dot x\otimes\cdots\otimes\dot x}}\rangle\\
&=\langle\nabla_{\textrm{sym}}^rx^1,\overset{r}{\overbrace{\dot x\otimes\cdots\otimes\dot x}}\rangle\notag\end{aligned}$$
For the geodesic starting from $x_0$ with vector velocity $\xi$ of coordinates $\dot x^j(0)=\xi^j$, we will write its equations in the form $x^j=x^j(s;\xi^1,\dots,\xi^n)$. According with definitions, it will be $\widehat x^j(\xi^1,\dots,\xi^n)=x^j(1;\xi^1,\dots,\xi^n)$. By applying the well known property of the geodesic (used in Section \[dos\]) we will have, for $|s|<1$: $$\widehat x^j(s\xi^1,\dots,s\xi^n)=x^j(s;\xi^1,\dots,\xi^n).$$ Differentiating $r$ times and, then, taking the value for $s=0$, we obtain, since (\[identidad5\]) and (\[identidad2\]), $$\langle d_0^rx^1,\overset{r}{\overbrace{\xi\otimes\cdots\otimes\xi}}\rangle
=\langle\nabla^r_{x_0,\textrm{sym}}x^1,\overset{r}{\overbrace{\xi\otimes\cdots\otimes\xi}}\rangle$$ from which it turns the validity of (\[identidad3\]) for $f=x^1$.
Now, let $f\in\A$ be such that $d_{x_0}f\ne 0$. Such an $f$ can be taken as coordinate $x^1$ in a system of local coordinates $(x^1,x^2,\dots,x^n)$ around $x_0$. Thus, (\[identidad3\]) holds for $f$.
Finally, let $f\in\A$ arbitrary. We can express $f$ as the sum $f=f_1+f_2$ where $d_{x_0}f_1\ne 0$ and $d_{x_0}f_2\ne 0$. As (\[identidad3\]) is linear in $f$, being true for $f_1$ and $f_2$, also it is true for $f$.
**Conclusion.** The quantization defined in Section \[dos\] (by means of the Riemannian exponential) and the one defined in Section \[tres\] (by direct pairing between tensors) are identical. Indeed, the first one is obtained (in addition to the factors $-i\hbar$) for each tensor by applying on each $T_{x_0}M$ the direct quantization described in Section \[tres\] for $\R^n$, and applying to each $\widehat f|_{T_xM}$; by (\[identidad1\]), (\[identidad2\]), (\[identidad3\]), this is equivalent to directly coupling each $\nabla^r_{x_0,\textrm{sym}}f$ with the given tensor.
The exponential map reduces the symmetric covariant differential of arbitrary order to the corresponding linear differentials along each fibre.
[10]{}
<span style="font-variant:small-caps;">Ali, S. T.; Englisš, M.</span>, Quantization methods: a guide for physicists and analysts. Rev. Math. Phys. 17 (2005), no. 4, pp. 391-490.
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<span style="font-variant:small-caps;">Alonso-Blanco, R.J. and Muñoz Díaz, J.</span>, A note on the foundation of mechanics, arXiv:1404.1321 \[math-ph\].
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<span style="font-variant:small-caps;">Holland, P.R.</span>, [*The quantum theory of motion: an account of the de Broglie-Bohm causal interpretation of quantum mechanics*]{}, Cambridge University Press, Cambridge, 1993. <span style="font-variant:small-caps;">Lie, S.</span>, [*Theorie der Transformationsgruppen*]{} (german). Written with the help of Friedrich Engel. Teubner, Leipzig, 1888.
<span style="font-variant:small-caps;">Lie, S.</span>, [*Geometrie der Berührungstransformationen*]{} (german). Written with the help of Georg Scheffers. B. G. Teubner, Leipzig, 1896.
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<span style="font-variant:small-caps;">Muñoz Díaz, J., and Alonso-Blanco, R.J.</span>, Quantization of mechanical systems, J. Phys. Commun. 2 (2018) 025007, https://doi.org/10.1088/2399-6528/aaa850.
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UCB-PTH-04/28\
LBNL-56523\
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[**Supersymmetric Unification in Warped Space**]{}
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[Yasunori Nomura ]{}
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[*Department of Physics, University of California, Berkeley, CA 94720*]{}\
[*Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720*]{}
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Introduction {#sec:intro}
============
Supersymmetry has long been the leading candidate for physics beyond the standard model. It stabilizes the Higgs potential against potentially huge radiative corrections, giving a consistent theory of electroweak symmetry breaking. The minimal construction of the supersymmetric standard model, which contains supersymmetric multiplets for the standard-model gauge and matter fields as well as two Higgs doublets, also provides an elegant picture of gauge coupling unification at a scale of $M_X \simeq 10^{16}~{\rm GeV}$. A generic prediction of supersymmetric theories – the presence of a light Higgs boson – also seems to be supported by precision electroweak data.
Despite all these successes, the construction of a supersymmetric extension of the standard model is not yet complete. The origin of supersymmetry breaking is still a mystery and the communication of supersymmetry breaking to the supersymmetric standard model (SSM) sector is not yet fully understood. Some good news, though, is that in supersymmetric theories, the non-renormalization theorem guarantees that supersymmetry can be broken only by non-perturbative effects if it is not broken at tree level. Supersymmetry, then, can be broken at a dynamical scale $\Lambda$ of some gauge interaction $G$ responsible for dynamical supersymmetry breaking: $$\Lambda \sim M_{\rm Pl}\, e^{-\frac{8\pi^2}{|\tilde{b}|\tilde{g}^2}},
\label{eq:dim-trans}$$ where $\tilde{g}$ is the gauge coupling of $G$ renormalized at the Planck scale, and $\tilde{b}$ ($<0$) the beta-function coefficient for $\tilde{g}$. This provides a natural understanding of the smallness of the weak scale, as $\Lambda$ is exponentially smaller than the Planck scale for $|\tilde{b}|
\tilde{g}^2 \ll 8\pi^2$ [@Witten:1981nf].
What is the scale $\Lambda$? The answer depends on the mechanism by which supersymmetry breaking is mediated to the SSM sector, the sector that contains our quarks and leptons and their superpartners. If the mediation occurs through gravitational interactions, $\Lambda \simeq
10^{10}-10^{13}~{\rm GeV}$, while if it occurs through standard-model gauge interactions, $\Lambda$ can be much lower. These two interactions are selected as natural ways of mediation both because they are already known to exist and because they give “flavor universal” squark and slepton masses, which are needed to evade strong experimental constraints on the amount of flavor violation beyond that in the standard model [@Dimopoulos:1981zb]. Although it is possible to consider a scenario based on mediation through gravity ([*e.g.*]{} anomaly mediation), mediation by standard-model gauge interactions seems simpler to me, as it does not require a detailed understanding of Planck scale physics.
Now, suppose that the sector responsible for dynamical supersymmetry breaking (DSB) is charged under standard-model gauge interactions. Then it is possible that the gauginos obtain masses directly through their interaction with the DSB sector. The squarks and sleptons in the SSM sector then obtain flavor universal masses through standard-model gauge interactions. If this is the case, we do not need any other sector than the SSM and DSB sectors, which are in any case needed in any supersymmetric theory. We just have to assume that the DSB sector is charged under standard-model gauge interactions, giving masses to the gauginos. What can be simpler than this?
Constructing an explicit theory along the lines described above, however, is not an easy task. Typically what happens is that if we want to make the DSB sector charged under standard-model gauge interactions, $SU(3)_C \times SU(2)_L \times U(1)_Y$, the gauge group $G$ of the DSB sector becomes large, making $SU(3)_C$ strongly asymptotically non-free, and the successful prediction for gauge coupling unification is lost [@Dine:1993yw]. In general, it is not at all easy to find an explicit gauge group and matter content for the DSB sector that does the required job and to construct a fully realistic theory. One way out from this difficulty is to further separate the DSB sector from the SSM sector by introducing fields called messenger fields, which are charged both under standard-model gauge interactions and under interactions that mediate supersymmetry breaking from the DSB sector to the messenger fields [@Dine:1994vc]. This, however, loses a certain beauty that the original picture had.
In this talk I want to present explicit theories in which the picture described above is realized in a simple way. An important new ingredient is the correspondence between 4D gauge theories and their higher-dimensional dual gravitational descriptions, especially the AdS/CFT correspondence [@Maldacena:1997re]. This allows us to formulate our theories in higher dimensional spacetime, which does not require us to find the explicit gauge group and matter content for the DSB sector to construct the theories in a consistent effective field theory framework. In the construction, we require our theories to be fully realistic. In particular, we require that the successful prediction for gauge coupling unification is preserved. The theories are also free from problems of the simplest supersymmetric unified theories, such as the doublet-triplet splitting problem and the problem of overly rapid proton decay, and accommodate the successes of the conventional unification picture, such as the understanding of small neutrino masses by the see-saw mechanism. In much of the parameter space we find that the gauginos $\lambda$ and sfermions $\tilde{f}$ obtain masses of order $m_\lambda \sim (\alpha/4\pi)
\Lambda$ and $m_{\tilde{f}}^2 \sim (\alpha/4\pi)^2 \Lambda^2$, respectively. This implies that the scale for supersymmetry breaking is rather low $$\Lambda \approx 10\!\sim\!100~{\rm TeV}.$$ This may be the lowest possible scale for supersymmetry breaking we can attain in realistic supersymmetric theories.
This talk is mainly based on the works with Walter Goldberger, David Tucker-Smith, and Brock Tweedie, presented in Refs. .
Supersymmetry in Warped Space {#sec:susy-warped}
=============================
The theories we consider have the following basic structure. We have a sector, the DSB sector, that breaks supersymmetry dynamically at a scale $\Lambda \approx 10\!\sim\!100~{\rm TeV}$. We denote the gauge group of this sector as $G$. This sector is also charged under the standard-model gauge group, $SU(3)_C \times SU(2)_L \times U(1)_Y$ (321). Gaugino masses and flavor universal sfermion masses are then generated through standard-model gauge interactions. This basic picture is depicted in Fig. \[fig:structure\].
(350,95)(0,0) (140,50)(205,50)[3]{}[5]{} (172,65)\[b\][321]{} (100,50)(40,0,360) (100,50)\[\][$\tilde{q},\,\tilde{l}$]{} (250,50)(45,0,360) (251,71)\[\][Group $G$]{} (250,50)\[\][$\Lambda$]{} (251,28)\[\][SUSY]{} (233,23)(267,33)
In general, it is not easy to find an explicit calculable theory realizing the above basic structure, since it necessarily involves a sector that is strongly coupled at the scale $\Lambda \approx 10\!\sim\!100~{\rm TeV}$. Suppose now that the gauge coupling $\tilde{g}$ and the size (the number of “colors”) $\tilde{N}$ of the group $G$ satisfy the following relation: $\tilde{g}^2 \tilde{N}/16\pi^2 \gg 1$ and $\tilde{N} \gg 1$. In this case it is possible that the theory admits a dual higher-dimensional description that is weakly coupled, and allows explicit calculations of various quantities.
The way this duality works is the following. Let us first consider the 4D theory described in Fig. \[fig:structure\]. In this theory the DSB sector exhibits non-trivial infrared dynamics at the scale $\Lambda$. Besides dynamically breaking supersymmetry, this infrared dynamics produces a series of bound states, whose typical mass scale is the dynamical scale $\Lambda$. Since the DSB sector is charged under 321, these bound states are also charged under 321. For $\tilde{N} \gg 1$, there are a large number of such bound states which are weakly coupled, as suggested by the analysis of large-$N$ QCD [@'tHooft:1973jz]. We now consider another theory formulated in higher dimensions, [*e.g.*]{} in 5D, and assume that the extra dimension is compactified with the characteristic mass scale for the Kaluza-Klein (KK) towers $M_c$. Now, suppose that the spectrum of bound states obtained in the 4D theory of Fig. \[fig:structure\] and the KK spectrum of this 5D theory are exactly the same, $M_c \approx \Lambda$, and so are any physical quantities such as the scattering amplitudes among various states. If this is the case, we can never distinguish the two theories experimentally, implying that the two theories just correspond to two different descriptions of the same physics. This is the meaning of the duality, schematically depicted in Fig. \[fig:duality\]. Because the bound states of the 4D theory are charged under 321, the KK towers of the 5D theory should also be charged under 321. This implies that the 321 gauge fields must propagate in the 5D bulk in the “dual” 5D picture of the theory.
(470,235)(37,-5) (75,170)(170,170) (170,165)\[t\][$E$]{} (80,165)(80,225) (75,225)\[r\][$\tilde{g}$]{} (115,190)(165,190) (115,200)(10,190,270) (105,198)(101,222) (101,168)(101,172) (101,164)\[t\][$\Lambda$]{} (110,139)(103,126) (110,139)(105,126) (100,126)(103,126) (105,126)(108,126) (100,126)(102,120) (102,120)(108,126) (123,132)\[l\][$G$ confines]{} (45,10)(125,10) (55,0)(55,115) (48,115)\[r\][$E$]{} (52,35)(58,35) (48,37)\[r\][$\Lambda$]{} (80,30)(115,30) (80,50)(115,50) (80,70)(115,70) (80,90)(115,90) (80,60)\[r\][$\left\{ \matrix{ \cr\cr\cr\cr\cr\cr } \right.$]{} (130,95)\[l\][Resonances]{} (143,55)\[r\][$\left\{ \matrix{ \cr\cr\cr } \right.$]{} (140,70)\[l\][weakly coupled]{} (168,57)\[l\][for large $\tilde{N}$]{} (140,40)\[l\][charged under 321]{} (235,130)(270,130) (235,134)(270,134) (233,132)(239,138) (233,132)(239,126) (272,132)(266,138) (272,132)(266,126) (253,147)\[b\][dual]{} (325,165)(325,215) (350,175)(350,225) (325,165)(350,175) (325,215)(350,225) (395,165)(395,215) (420,175)(420,225) (395,165)(420,175) (395,215)(420,225) (340,185)(395,185) (395,185)(405,185)[2]{} (358,230)\[l\][extra]{} (369,218)\[l\][dim.]{} (360,139)(353,126) (360,139)(355,126) (350,126)(353,126) (355,126)(358,126) (350,126)(352,120) (352,120)(358,126) (295,10)(375,10) (305,0)(305,115) (298,115)\[r\][$E$]{} (302,35)(308,35) (298,37)\[r\][$M_c$]{} (330,30)(365,30) (330,50)(365,50) (330,70)(365,70) (330,90)(365,90) (330,60)\[r\][$\left\{ \matrix{ \cr\cr\cr\cr\cr\cr } \right.$]{} (380,95)\[l\][KK towers]{} (383,81)(383,52) (390,70)\[l\][charged under 321]{} (380,43)\[l\][321 gauge fields]{} (412,31)\[l\][in the bulk]{}
What the “dual” 5D theory looks like more explicitly? Let us now assume that the gauge coupling $\tilde{g}$ evolves very slowly above $\Lambda$, so that the $G$ sector is nearly conformal in a wide energy interval between $\Lambda$ and a high scale of order $M_X \simeq 10^{16}~{\rm GeV}$. The AdS/CFT correspondence then implies that the 5D theory is formulated in anti-de Sitter (AdS) space truncated by two branes, an ultraviolet (UV) brane and an infrared (IR) brane [@Arkani-Hamed:2000ds]. The metric of this spacetime is then given by $$ds^2 = e^{-2ky} \eta_{\mu\nu} dx^\mu dx^\nu + dy^2,
\label{eq:metric}$$ where $y$ is the coordinate for the extra dimension and $k$ denotes the inverse curvature radius of the AdS space. The two branes are located at $y=0$ (the UV brane) and $y=\pi R$ (the IR brane). This is the spacetime considered in Ref. [@Randall:1999ee], in which the large hierarchy between the weak and the Planck scales are generated by the AdS warp factor. We here choose the scales such that the scales on the UV and IR branes are roughly the 4D Planck scale and the scale $\Lambda$, respectively: $k \sim M_5 \sim M_* \sim M_{\rm Pl}$ and $kR \sim 10$ (the 4D Planck scale is given by $M_{\rm Pl}^2 \simeq M_5^3/k$). Here, $M_5$ is the 5D Planck scale, and $M_*$ the 5D cutoff scale, which is taken to be somewhat (typically a factor of a few) larger than $k$. With this choice of scales, the characteristic mass scale for the KK towers is given by $\pi k e^{-\pi kR}
\sim \Lambda \approx (10\!\sim\!100)~{\rm TeV}$.
The theories described below are thus formulated in 5D supersymmetric warped space truncated by two branes. The structure of Fig. \[fig:structure\] then corresponds to breaking supersymmetry on the IR brane (also called the TeV brane) and localizing quark and lepton superfields, $Q$, $U$, $D$, $L$ and $E$, to the UV brane (also called the Planck brane). The standard-model gauge fields propagate in the bulk. The overall picture is depicted in Fig. \[fig:overall\] (we can even see the similarity between the two pictures in Fig. \[fig:structure\] and Fig. \[fig:overall\]). Supersymmetry breaking on the TeV brane in this picture does not have to be suppressed — it can be an $O(1)$ breaking when measured in terms of the 5D metric of Eq. (\[eq:metric\]). Although supersymmetry breaking is directly transmitted to the 321 gauginos, the generated gaugino masses are of order TeV, because of the exponential warp factor. The squark and slepton masses are also generated through 321 gauge loops, which are flavor universal and thus do not introduce the supersymmetric flavor problem. This setup was first considered in Ref. [@Gherghetta:2000qt]. We will see that in our theories this picture coexists with most of the successes of the conventional weak-scale supersymmetry paradigm.
(250,192)(0,-20) (60,165)\[br\][“Planck” brane]{} (0,-15)(0,125) (60,15)(60,155) (0,-15)(60,15) (0,125)(60,155) (17,126)\[l\][$Q,U,D$]{} (25,113)\[l\][$L,E$]{} (30,20)(190,20) (190,20)(220,20)[3]{} (30,20)[1]{} (85,-325)(405,73,97) (85,-325)(405,73,97)[3]{}[15]{} (85,449)(405,263,287) (85,449)(405,263,287)[3]{}[15]{} (112,88)\[b\][$\lambda$]{} (36,77)(20,87)[3]{} (16,89)\[r\][$\tilde{q}$]{} (36,47)(20,37)[3]{} (16,36)\[r\][$\tilde{q}$]{} (83,62)(50,162,198) (29,62)\[r\][$q$]{} (125,128)\[b\][(321) gauge field]{} (190,165)\[bl\][“TeV” brane]{} (190,-15)(190,125) (250,15)(250,155) (190,-15)(250,15) (190,125)(250,155) (221,118)\[\][SUSY]{} (203,113)(237,123)
Here I want to emphasize that we should not take the view that our theory has solved the hierarchy problem [*twice*]{} by introducing both supersymmetry and a warped extra dimension. Rather, the picture of a supersymmetric warped extra dimension arises if the DSB sector, which is necessarily present in any supersymmetric theory, satisfies certain conditions, [*e.g.*]{} $\tilde{g}^2 \tilde{N}/16\pi^2 \gg 1$ and $\tilde{N} \gg 1$. A virtue of the higher dimensional construction is then that we do not need to know the gauge group or the matter content of the DSB sector explicitly. In fact, once we have the picture of higher dimensional warped space and construct a theory on this space, we can forget about the “original” 4D picture for all practical purposes, although such a picture is useful for estimating various physical quantities and for obtaining insight on physical properties of the theory. Although, strictly speaking, the presence of a higher dimensional theory does not necessarily guarantee the presence of the “corresponding 4D theory”, this need not concern us. Our higher dimensional warped supersymmetric theory is a consistent effective field theory, with which we can calculate various physical quantities and compare with experiments — the theory is even weakly coupled if the cutoff scale of the theory is sufficiently larger than the AdS curvature scale.
In the next section we present a complete theory built on this picture, which accommodates the successes of the conventional weak-scale supersymmetry paradigm. Alternative, related theories will be discussed in later sections.
Warped Supersymmetric Grand Unification {#sec:warped-sgut}
=======================================
We start by recalling that our DSB sector is charged under the standard-model gauge group so that it contributes to the evolution of the 321 gauge couplings. On the other hand, the successful prediction for gauge coupling unification in the minimal supersymmetric standard model (MSSM) implies that any additional contributions to the evolution of the 321 gauge couplings beyond those from the MSSM gauge, matter and Higgs fields must be universal for $SU(3)_C$, $SU(2)_L$ and $U(1)_Y$. In our context, this implies that the contribution from the DSB sector should be universal. This is most naturally attained if the DSB sector possesses a global $SU(5)$ symmetry, of which the 321 gauge group of the standard model is a subgroup. In the 5D picture, this corresponds to having a gauge group (at least) $SU(5)$ in the bulk, since the gauge symmetry in the 5D bulk corresponds to a global symmetry of the strong interacting sector in the 4D theory.
The higher dimensional unified gauge group must be broken to the 321 subgroup, as the gauge invariance in our low-energy world is 321. Because our theory looks higher dimensional, we can employ a higher-dimensional mechanism to break a gauge symmetry. In particular, we can break 5D unified gauge invariance, taken as $SU(5)$ here, by imposing non-trivial boundary conditions on the fields at a boundary of the spacetime. This way of breaking a unified gauge symmetry has many desirable features over the conventional Higgs mechanism; for example, it allows an elegant, simultaneous solution to the problems of doublet-triplet splitting, overly rapid proton decay and unwanted fermion mass relations [@Hall:2001pg]. Here we adopt this mechanism to construct our explicit theory.
The structure of our minimal warped supersymmetric unified theory is then given as follows [@Goldberger:2002pc]. The theory is formulated in 5D warped space with the metric given by Eq. (\[eq:metric\]) and the extra dimension compactified on an interval $0 \leq y \leq \pi R$. The bulk gauge group is taken to be $SU(5)$, which is broken by boundary conditions at $y=0$ (the Planck brane). The two Higgs hypermultiplets, which are ${\bf 5}$ and ${\bf 5}^*$ representations under $SU(5)$, are introduced in the bulk. Depending on the values of the bulk masses for the Higgs multiplets, which are conveniently parameterized by two dimensionless numbers $c_H$ and $c_{\bar{H}}$ (for notation see [@Goldberger:2002pc]), the wavefunctions for the zero modes arising from these multiplets can have varying shapes. The matter fields are localized to the Planck brane — they could either be located on the Planck brane or be introduced in the bulk but with the zero modes strongly localized to the Planck brane by bulk mass terms. The Yukawa couplings are also located on the Planck brane. Supersymmetry is broken on the TeV brane by a vacuum expectation value (VEV) for the auxiliary field of a chiral superfield $Z$. The overall picture of the theory is depicted in Fig. \[fig:theory\].
(270,175)(0,-33) (0,5)(270,5) (135,120)\[b\][$SU(5)$]{} (30,-10)(30,120) (30,130)\[b\][$321$]{} (30,-18)\[t\][$y=0$ (Planck)]{} (12,111)\[r\][$Q,U,D$]{} (22,97)\[r\][$L,E$ $(N)$]{} (66,20)(68,127,146) (26,75)(22,69) (26,75)(19,72) (-5,63)\[r\][Yukawa]{} (10,49)\[r\][couplings]{} (240,-10)(240,120) (240,130)\[b\][$SU(5)$]{} (240,-18)\[t\][$y=\pi R$ (TeV)]{} (298,36)(67,127,143) (244,76)(247,83) (244,76)(250,80) (278,98)\[\][SUSY]{} (260,93)(294,103) (273,86)\[tl\][$F_Z \neq 0$]{} (135,63)\[b\][$A_\mu^{321}, A_\mu^{\rm XY}, H_{\bf 5}, H_{{\bf 5}^*}$]{} (72,38)(127,38)[1]{}[4]{} (127,38)(122,41) (127,38)(122,35) (143,27)(198,27)[1]{}[4]{} (143,27)(148,30) (143,27)(148,24)
The boundary conditions for various fields are given more explicitly in Table \[table:bc\]. Here we represent the 5D gauge multiplet ${\cal V} \equiv \{V, \Sigma\}$ in terms of a 4D $N=1$ vector superfield $V$ and a chiral superfield $\Sigma$. The subscripts 321 and XY represent $321$ and $SU(5)/321$ components, respectively. A bulk hypermultiplet is represented by two 4D $N=1$ chiral superfields $\Phi$ and $\Phi^c$. Therefore, the two Higgs hypermultiplets are denoted as $\{ H, H^c \}$ and $\{ \bar{H}, \bar{H}^c \}$, with the subscripts $D$ and $T$ representing the doublet and triplet components, respectively. The boundary conditions are written in the language of orbifolding procedures. In the table we also give the boundary conditions for bulk matter fields. For bulk matter, we need two hypermultiplets $\{ T, T^c \} + \{ T', T'^c \}$ in the ${\bf 10}$ representation and two hypermultiplets $\{ F, F^c \} + \{ F', F'^c \}$ in the ${\bf 5}^*$ representation to complete a single generation. The subscripts for these fields denote the irreducible components under the 321 decompositions.[^1]
$(p,p')$ gauge and Higgs fields bulk matter fields
---------- ------------------------------------------- ------------------------------------------
$(+,+)$ $V_{321}$, $H_D$, $\bar{H}_D$ $T_{U,E}$, $T'_Q$, $F_D$, $F'_L$
$(-,-)$ $\Sigma_{321}$, $H^c_D$, $\bar{H}^c_D$ $T^c_{U,E}$, $T'^c_Q$, $F^c_D$, $F'^c_L$
$(-,+)$ $V_{\rm XY}$, $H_T$, $\bar{H}_T$ $T_Q$, $T'_{U,E}$, $F_L$, $F'_D$
$(+,-)$ $\Sigma_{\rm XY}$, $H^c_T$, $\bar{H}^c_T$ $T^c_Q$, $T'^c_{U,E}$, $F^c_L$, $F'^c_D$
: The boundary conditions for the bulk fields. The fields written in the $(p,p')$ column, $\varphi$, obey the boundary condition $\varphi(-y) = p\, \varphi(y)$ and $\varphi(-y') = p'\, \varphi(y')$ when we construct our space, $0 \leq y \leq 2\pi$, by the orbifolding procedure. Here, $y' \equiv y-\pi R$.[]{data-label="table:bc"}
The spectrum of the theory is obtained by KK decomposing the fields. We then find that the zero-mode sector contains only the MSSM fields: the 321 gauge field, $V_{321}$, two Higgs doublets, $H_D$ and $\bar{H}_D$, and three generations of matter fields $Q$, $U$, $D$, $L$ and $E$. The characteristic mass scale for the KK towers is of order TeV, $M_c \equiv
\pi k e^{-\pi kR}$. They are almost $N=2$ supersymmetric and $SU(5)$ symmetric. For example, for the gauge sector, each KK level contains $V_{321}$, $\Sigma_{321}$, $V_{\rm XY}$ and $\Sigma_{\rm XY}$. The spectrum for a characteristic case (for characteristic values of $c_H$ and $c_{\bar{H}}$) is depicted schematically in Fig. \[fig:KK\]. Because the spectrum at the TeV scale has a radical departure from that of the MSSM, one might wonder to what extent the successes of the conventional supersymmetric desert scenario are preserved. Below we will see that our theory preserves most of the successes of the conventional desert scenario and, moreover, is free from the problems which the minimal supersymmetric unified theory suffers from.
(385,165)(-10,-18) (5,0)(355,0) (15,-10)(15,135) (8,135)\[r\][mass]{} (8,75)\[r\][$\sim {\rm TeV}$]{} (45,133)\[b\][$V_{321}$]{} (30,0)(60,0) (45,0)[3]{} (30,45)(60,45) (45,45)[3]{} (30,105)(60,105) (45,105)[3]{} (85,133)\[b\][$\Sigma_{321}$]{} (70,45)(100,45) (85,45)[3]{} (70,105)(100,105) (85,105)[3]{} (125,133)\[b\][$V_{\rm XY}$]{} (110,45)(140,45) (125,45)[3]{} (110,105)(140,105) (125,105)[3]{} (165,133)\[b\][$\Sigma_{\rm XY}$]{} (150,45)(180,45) (165,45)[3]{} (150,105)(180,105) (165,105)[3]{} (205,133)\[b\][$H_{\!D},\!\bar{H}_{\!D}$]{} (190,0)(220,0) (205,0)[3]{} (190,80)(220,80) (205,80)[3]{} (245,133)\[b\][$H_{\!D}^c,\!\bar{H}_{\!D}^c$]{} (230,80)(260,80) (245,80)[3]{} (285,133)\[b\][$H_{\!T},\!\bar{H}_{\!T}$]{} (270,80)(300,80) (285,80)[3]{} (325,133)\[b\][$H_{\!T}^c,\!\bar{H}_{\!T}^c$]{} (310,80)(340,80) (325,80)[3]{}
Gauge coupling unification {#subsec:gcu}
--------------------------
As is suggested from the correspondence between the 4D and 5D pictures, the evolution of the gauge couplings in our theory is logarithmic. The fact that the gauge couplings for bulk gauge fields evolve logarithmically in warped space was first noticed in Ref. [@Pomarol:2000hp], in which the successful prediction was also anticipated based on a heuristic argument. There have been some debates on whether theories in warped space actually allow calculations of gauge coupling unification; in particular, whether threshold corrections at an IR scale are under control (see [*e.g.*]{} [@Arkani-Hamed:2000ds]). Subsequent theoretical works, however, have clarified that these corrections are in fact under control, and that theories on warped space retain calculability [@Randall:2001gc]. For an observer sitting at $y=y_*$, physics is essentially four dimensional up to an energy $E \sim k e^{-k y_*}$, so for the Planck-brane observer physics is four dimensional all the way up to $k \sim M_{\rm Pl}$. Now, the gauge couplings are measured, for example, by scattering two quarks — a process that occurs on the Planck brane. The evolution of the gauge couplings are then given by calculating diagrams as given in Fig \[fig:diag\] and summing up logarithms arising from them. At energies higher than the TeV scale $E \gg k' \sim {\rm TeV}$, the gauge propagator in the bulk cannot probe the region close to the TeV brane, as the propagation of a gauge field from $y=0$ to $y=\pi R$ receives a large suppression, $\propto \exp(-E/k')$, for $E \simgt k'$. In warped space all the KK modes are strongly localized to the TeV brane except for a single mode, which is often the zero mode. This implies that the contribution to the evolution of the gauge couplings at $E \simgt {\rm TeV}$ is dominated by the single mode and thus is logarithmic. For the case of a non-Abelian gauge field, the situation is somewhat more complicated due to the mass mixing between the different modes, but the essential physics is still the same and the evolution is still dominated by “a single mode” and is four dimensional.
(210,185)(0,-35) (0,-16)\[t\][Planck ($y=0$)]{} (0,-7)(0,120) (50,23)(50,150) (0,-7)(50,23) (0,120)(50,150) (26,46)(10,11)[3]{} (26,46)(40,31)[3]{} (26,98)(40,129)[3]{} (26,98)(10,111)[3]{} (90,72)(15,30)(0) (26,-2)(100,60,90)[2]{}[7]{} (26,146)(100,270,300)[2]{}[7]{} (160,-16)\[t\][TeV ($y=\pi R$)]{} (160,-7)(160,120) (210,23)(210,150) (160,-7)(210,23) (160,120)(210,150)
In Ref. [@Goldberger:2002pc] we showed that the successful prediction for gauge coupling unification (the same prediction as the MSSM) is automatically obtained if the following two conditions are obeyed:
- The bulk gauge group is $SU(5)$ (or a larger group containing $SU(5)$) that is broken on the Planck brane (at the scale $k$ or larger).
- The bulk mass parameters for the matter and Higgs fields are all larger than or equal to $1/2$: $c_{\rm matter}, c_{\rm Higgs} \geq 1/2$. This implies that zero modes for these fields have wavefunctions either conformally flat or localized to the Planck brane.
If the breaking scale of $SU(5)$ is somewhat larger than $k$, such as the case of boundary condition breaking (breaking by the Planck-brane boundary conditions corresponds to breaking $SU(5)$ at a scale much larger than $k$), tree-level operators on the Planck brane could potentially give incalculable non-universal corrections. These corrections, however, are naturally suppressed if the volume of the bulk is large, which is necessarily the case in warped space theories explaining the hierarchy between the Planck and the TeV scales. In our theory matter is localized to the Planck brane, corresponding to $c_{\rm Matter} \gg 1/2$. The only remaining condition is then that the two Higgs multiplets must have mass parameters larger than or equal to $1/2$: $c_H, c_{\bar{H}} \geq 1/2$. Under this condition, we find that the prediction for the low-energy 321 gauge couplings in our theory is given by $$\pmatrix{1/g_1^2 \cr 1/g_2^2 \cr 1/g_3^2}_{\! \mu = M_Z}
\simeq \:\: (SU(5)\,\,\, {\rm symmetric})
+ \frac{1}{8 \pi^2} \pmatrix{33/5 \cr 1 \cr -3}
\ln\left(\frac{k}{M_Z}\right),
\label{eq:gc-low}$$ which is identical to the MSSM prediction, with the AdS curvature $k$ identified as the conventional unification scale $M_X \simeq
10^{16}~{\rm GeV}$. This determines the scales in the theory to be $k \simeq 10^{16-17}~{\rm GeV}$ and $M_* \simeq M_5 \simeq
10^{17-18}~{\rm GeV}$. It is fortunate that we obtain these numbers, as we obtain roughly the correct size for the 4D Planck scale $M_{\rm Pl} \simeq (M_5^3/k)^{1/2}$ without introducing a new scale.
It is useful to consider gauge coupling unification in the 4D picture. In the 4D picture our theory appears as follows [@Goldberger:2002pc]. We have the DSB sector with the gauge group $G$, whose coupling $\tilde{g}$ evolves very slowly over a wide energy interval between $k \approx M_X
\sim 10^{16}~{\rm GeV}$ and $k' \approx \Lambda \sim {\rm TeV}$. The value of the coupling is $\tilde{g} \simeq 4\pi$ in this energy interval, and the size of the gauge group $\tilde{N}$ is sufficiently larger than $1$ so that $\tilde{g}^2\tilde{N}/16\pi^2 \gg 1$. The DSB sector possesses a global $SU(5)$ symmetry, of which the 321 subgroup is gauged and identified as the standard-model gauge group. Quark, lepton and two Higgs-doublet superfields are introduced as elementary fields, which interact with the DSB sector through 321 gauge interactions. The Higgs fields may also have direct interactions with the DSB sector through couplings of the form ${\cal L} \sim H {\cal O}_H + \bar{H} {\cal O}_{\bar{H}}$, where ${\cal O}_H$ and ${\cal O}_{\bar{H}}$ are operators of the DSB sector. The strengths of these couplings in the IR depend on the parameters $c_H$ and $c_{\bar{H}}$ in the 5D picture. Once supersymmetry is broken at the scale $\Lambda$ by the non-trivial IR dynamics of $G$, the 321 gauginos, squarks and sleptons (and the Higgs fields) receive masses through 321 gauge interactions.
Since the DSB sector is charged under 321, the evolution of the 321 gauge couplings receives a contribution from this sector as well as that from the elementary states. At low energies $q \sim {\rm TeV}$, the 321 gauge couplings are thus given by $$\frac{1}{g_a^2(q)} = \frac{1}{g_a^2(k)}
+ {b_{\rm DSB} \over 8\pi^2} \ln\left({k \over q}\right)
+ {b_a \over 8\pi^2} \ln\left({k \over q}\right),
\label{eq:gc-4D}$$ where $b_{\rm DSB}$ $(>0)$ represents the contribution from the DSB sector, which is universal due to the global $SU(5)$ symmetry, and $b_a$ the contribution from the elementary states: $(b_1, b_2, b_3) = (33/5, 1, -3)$. Now, in the 4D theory dual to the 5D theory with boundary condition $SU(5)$ breaking, the value of $b_{\rm DSB}$ is given such that the UV values of the 321 gauge couplings are strong, $g_a(k) \sim 4\pi$, in which case the first and second terms of the right-hand-side of Eq. (\[eq:gc-4D\]) are of $O(1/16\pi^2)$ and $O(1)$, respectively (the actual value is $b_{\rm DSB} \approx 5$). It is then clear that the contributions from these terms are approximately $SU(5)$ symmetric, so that the differences of the three couplings at low energies are essentially given by the last term. This gives the same prediction for gauge coupling unification as that of the MSSM. The schematic picture for the evolution of the gauge couplings are given in Fig. \[fig:couplings\]. It is interesting to note that in our theory the hierarchy between the Planck and the weak scales are generated by $$|\tilde{b}| \ll 1, \qquad \tilde{g} \sim 4\pi,$$ where $\tilde{b}$ is the beta-function coefficient for the evolution of $\tilde{g}$, while in the conventional picture it is generated by $|\tilde{b}| \sim 1$ and $\tilde{g} \ll 4\pi$ (see Eq. (\[eq:dim-trans\])).
(300,148)(-15,-23) (254,120)(220,180,193)[3]{} (50,78)(12,180,270)[3]{} (50,66)(230,66) (238,66)\[l\][$\tilde{g}$]{} (-2,657)(650,271.1,291) (10,5)\[r\][$g_1$]{} (-3,815)(800,271,287) (10,15)\[r\][$g_2$]{} (1,1126)(1100,270.6,282) (10,27)\[r\][$g_3$]{} (-10,50)(260,50)[2]{} (-15,50)\[r\][$1$]{} (50,0)(50,120)[2]{} (50,-5)\[t\][$\Lambda \sim k'$]{} (230,66)(230,0)[2]{} (230,66)[2]{} (230,-5)\[t\][$k$]{} (-10,-8)(-10,120) (-15,120)\[r\][$\frac{g^2 N}{16\pi^2}$]{} (-18,0)(260,0) (259,-6)\[t\][$E$]{}
Proton decay {#subsec:p-decay}
------------
Since the spectrum of the theory contains the XY gauge bosons and colored Higgs triplets at the TeV scale, one might worry that proton decay occurs at a disastrous rate in our theory. However, this should not be the case if quark and lepton fields are localized to the Planck brane. The scales on this brane do not receive a suppression by a warp factor under the dimensional reduction, so that any proton decay operator generated by integrating out baryon-number violating physics should be suppressed by a large mass scale of order $k$ in the low-energy 4D theory. This must always be the case as long as the scale of $SU(5)$ breaking is at or larger than $k$ (the boundary condition breaking can be regarded as the breaking at the scale much higher than $k$).
One might still wonder how the suppression of proton decay is explicitly realized in the KK decomposed 4D picture. To see this, note that the wavefunctions of the XY gauge bosons and the colored Higgs triplets in our theory are strongly localized to the TeV brane (this is exactly the reason why these states have TeV-scale masses — the masses for these states arise from the curvatures of the wavefunctions, which are localized to the TeV brane, so that they receive strong suppressions from a large warp factor). Therefore, the wavefunction overlaps of the XY-gauge or colored-Higgs states to the quark and lepton fields are exponentially small of $O({\rm TeV}/M_X)$.[^2] This leads to tiny couplings for the baryon-number-violating vertices such as the (quark)-(lepton)-(XY gauge bosons) vertex (see Fig. \[fig:pd\]a), and thus suppresses any proton decay process to the level of conventional unified theories, given by ${\cal L}_{\rm eff} \sim qqql/M_X^2$ or $qq\tilde{q}\tilde{l}/M_X$. Incidentally, the fact that the coupling in Fig. \[fig:pd\]a is tiny does not preclude the possibility of producing the XY gauge states at colliders, as they can be produced through the coupling to the gluon, which is the QCD coupling and $O(1)$ (see Fig. \[fig:pd\]b). This coupling, of course, does not lead to proton decay because it conserves baryon number.
(255,100)(0,-50) (25,-30)\[t\][(a)]{} (0,0)(20,25) (-5,0)\[r\][$l$]{} (0,50)(20,25) (-5,50)\[r\][$q$]{} (20,25)(50,25)[3]{}[3]{} (55,25)\[l\][$X_\mu$]{} (22,21)(30,5) (22,21)(23,13) (22,21)(27,16) (30,3)\[tl\][$\epsilon \sim \frac{\rm TeV}{M_X} \ll 1$]{} (227,-30)\[t\][(b)]{} (200,25)(230,25)[3]{}[3.5]{} (195,25)\[r\][$g_\mu$]{} (230,25)(255,0)[3]{}[3.5]{} (260,0)\[l\][$X_\mu$]{} (230,25)(255,50)[3]{}[3.5]{} (260,50)\[l\][$X_\mu$]{} (228,21)(220,5) (228,21)(227,13) (228,21)(223,16) (220,3)\[tr\][$g_3 = O(1)$]{}
In supersymmetric theories, it is in general not sufficient to ensure that proton decay operators are suppressed by the unified mass scale $M_X \simeq 10^{16}~{\rm GeV}$, because the dimension-five operators such as $W_{\rm eff} \sim QQQL/M_X$ could cause proton decay at a level contradicting to the experiments. In our theory, however, dimension-five proton decay operators are simply absent because of a $U(1)_R$ symmetry the original 5D theory possesses. The charges of various 4D $N=1$ superfields under $U(1)_R$ are given in Table \[table:U1R\]. This symmetry clearly forbids the dimension-five proton decay operators $W \sim QQQL/M_X$ and $UUDE/M_X$, as well as other phenomenologically dangerous operators such as $W \sim M_X H \bar{H}$. Diagrammatically, the absence of dimension-five proton decay due to the Higgs triplet exchange is understood by the structure of the mass terms for the Higgs triplets: $W \sim H_T H_T^c + \bar{H}_T \bar{H}_T^c$, which is different from that in the minimal supersymmetric grand unified theory $W \sim H_T \bar{H}_T$. After supersymmetry is broken, $U(1)_R$ is broken to its $Z_{2,R}$ subgroup, which is exactly $R$ parity. Thus, dangerous dimension-four proton decay operators are strictly forbidden, and the supersymmetric mass term for the Higgs doublets of order the weak scale can be generated. The breaking $U(1)_R \rightarrow Z_{2,R}$ does not reintroduce dimension-five proton decay at a dangerous level.
$V$ $\Sigma$ $H$ $H^c$ $\bar{H}$ $\bar{H}^c$ $T$ $T^c$ $F$ $F^c$ $N$ $N^c$
---------- ----- ---------- ----- ------- ----------- ------------- ----- ------- ----- ------- ----- -------
$U(1)_R$ 0 0 0 2 0 2 1 1 1 1 1 1
: $U(1)_R$ charges for 4D vector and chiral superfields. The charges of the primed matter fields, $T', T'^c, F', F'^c, N'$ and $N'^c$, are the same as the non-primed fields.[]{data-label="table:U1R"}
We finally mention that unwanted fermion mass relations such as $m_s/m_d
= m_\mu/m_e$ does not arise in our theory. This is because the Yukawa couplings are located on the Planck brane, on which the gauge group is reduced to 321. An extension of the theory leading to successful $b/\tau$ unification will be discussed in section \[subsec:SO10\].
Other issues {#subsec:other}
------------
We have seen that in our theory two scales coexist in an intriguing way. On one hand, we have an “extra dimension” at the TeV scale, which is characterized by the appearance of the KK towers (including those for the grand-unified states) at the TeV scale. The scale of supersymmetry breaking is also naturally set by this scale. On the other hand, we have a very high scale $k \approx M_X \simeq 10^{16}~{\rm GeV}$ at which the 321 gauge couplings “unify”. This is also the scale that suppresses effective proton decay operators. The reason why the scales for gauge coupling unification and proton decay are high is that we have broken $SU(5)$ at the Planck brane, where the warp factor is $1$ and does not give any suppression of the scales. We can now go further in using these two coexisting mass scales. For example, if we introduce right-handed neutrinos $N$ and introduce the Majorana masses and Yukawa couplings on the Planck brane, [*i.e.*]{} $\delta(y) \int d^2 \theta ((M_N/2)N^2+y_\nu LHN)$, we obtain small observed neutrino masses naturally through the conventional see-saw mechanism, because the Majorana masses, $M_N$, do not receive any suppressions from the warp factor.
In fact, the correspondence between the 4D and 5D pictures suggests that our theory can be interpreted as a purely 4D theory, in which physics between the weak and unification scales is simply 4D $N=1$ supersymmetric field theory. This implies, for example, that the cosmological evolution in the early universe is purely four dimensional in our theory. It is interesting to note that the theory is free from dangerous relics such as the gravitino and moduli. Because the supersymmetry-breaking scale is very low, $\Lambda
\approx (10\!\sim\!100)~{\rm TeV}$, we expect that the gravitino (and moduli, if any) is very light $$m_{3/2} \simeq \frac{\Lambda^2}{M_{\rm Pl}}
\approx (0.1\!\sim\!10)~{\rm eV},$$ in our theory. Such a light gravitino does not produce the “gravitino problem”, as its thermal relic abundance is small. We also note that in warped theories, the radion does not cause any cosmological problem, as its mass and interaction strengths are both dictated by the TeV scale so that it decays before the big-bang nucleosynthesis. Dark matter in our theory may come from conventional candidates such as axion, or may arise from a particle localized on the TeV brane, which has naturally TeV-scale mass and interactions and whose decay is protected by some discrete symmetry.[^3]
Phenomenology {#sec:pheno}
=============
The phenomenology of our theory is, naturally, quite rich, as it predicts a plethora of new particles at the TeV scale — superparticles and $SU(5)$ states as well as their KK towers (superparticles even form $N=2$ multiplets at higher KK level). In this section we study various aspects of the phenomenology associated with these particles.
Spectrum {#subsec:spectrum}
--------
In the minimal warped unified theory described in section \[sec:warped-sgut\], the spectrum of the TeV states are determined essentially by only two free parameters (up to parameters in the Higgs sector). This is because supersymmetry breaking occurs on the TeV brane, on which the gauge group is effectively $SU(5)$: supersymmetry breaking is transmitted to the SSM sector through the operator $${\cal L} = \delta(y-\pi R)\int d^2\theta
\frac{\zeta Z}{M_*}{\cal W}_a^\alpha {\cal W}_{a,\alpha}
\:\: \rightarrow \:\:
\delta(y-\pi R) M_\lambda\, \lambda_a^\alpha \lambda_{a,\alpha}.
\label{eq:gaugino-TeV}$$ which has only a single coupling for $a=SU(3)_C, SU(2)_L$ and $U(1)_Y$ ($M_\lambda$ does not depend on $a$). This allows us to calculate soft supersymmetry breaking parameters in terms of two parameters $x \equiv
M_\lambda/k$ and $k' = ke^{-\pi kR}$ [@Nomura:2003qb]. The result of this calculation is shown in Fig. \[fig:loglog\]. In the figure, we have normalized all the masses in units of $10 \, m_{\tilde{e}}$.
In the supersymmetric limit ($x=0$), the spectrum consists of the MSSM states, which are massless, and the KK states, which are $SU(5)$ symmetric and $N=2$ supersymmetric. Once supersymmetry is broken ($x \neq 0$), the 321 gauginos, squarks and sleptons obtain masses. In the meantime, the masses for the KK states are also shifted; in particular, one of the two 321 gauginos at each level becomes lighter and the other heavier, and similarly for the XY gauginos. In the limit of large supersymmetry breaking ($x \gg 1$), the 321 gauginos become pseudo-Dirac states by pairing up with the states that were previously the first KK excited 321 gauginos. On the other hand, the XY gaugino states become very light in this limit, $m_{\lambda_{\rm XY}} \propto 1/x$ — it becomes even lighter than the MSSM superparticles. These features are explained in more detail in Ref. [@Nomura:2003qb].
4D interpretation {#subsec:4D-interp}
-----------------
The characteristic features of the spectrum described above can also be understood from the 4D picture. In the 4D picture, the 321 gauginos and sfermions obtain masses, for small $x$, from the diagrams as shown in Fig. \[fig:gauge-med\]. Here, the gray discs represent contributions from the DSB sector. Using the scaling argument based on the large-$N$ expansion, the masses for the gauginos, $M_a \equiv m_{\lambda^{321}_a}$ ($a=1,2,3$), are estimated as $M_a \simeq g_a^2 (\tilde{N}/16\pi^2)
\hat{\zeta} m_\rho$, where $g_a$ are the 4D 321 gauge couplings, $\hat{\zeta}$ is a dimensionless parameters of $O(1)$, $\tilde{N}$ is the size of the DSB gauge group $G$, and $m_\rho$ is the typical mass scale for the resonances in the DSB sector.[^4] Similarly, the squared masses for the scalars, $m_{\tilde{f}}^2$, are estimated as $m_{\tilde{f}}^2 \simeq \sum_{a=1,2,3} (g_a^4 C_a^{\tilde{f}}/16\pi^2)
(\tilde{N}/16\pi^2) \hat{\zeta}^2 m_\rho^2$, where $\tilde{f} = \tilde{q},
\tilde{u}, \tilde{d}, \tilde{l}, \tilde{e}$ represents the MSSM squarks and sleptons, and $C_a^{\tilde{f}}$ are group theoretical factors given by $(C_1^{\tilde{f}}, C_2^{\tilde{f}}, C_3^{\tilde{f}}) = (1/60,3/4,4/3)$, $(4/15,0,4/3)$, $(1/15,0,4/3)$, $(3/20,3/4,0)$ and $(3/5,0,0)$ for $\tilde{f}
= \tilde{q}, \tilde{u}, \tilde{d}, \tilde{l}$ and $\tilde{e}$, respectively.
(290,95)(5,125) (60,145)\[t\][(a)]{} (5,190)(32,190)[3.5]{}[3]{} (5,190)(32,190) (88,190)(115,190)[3.5]{}[3]{} (88,190)(115,190) (5,198)\[b\][$\lambda$]{} (115,198)\[b\][$\lambda$]{} (60,190)(23,28)(0)[0.85]{} (61,190)\[\][SUSY]{} (40,185)(80,195) (240,145)\[t\][(b)]{} (185,188)(213,188)[3]{} (185,196)\[b\][$\tilde{f}$]{} (267,188)(295,188)[3]{} (295,196)\[b\][$\tilde{f}$]{} (238,188)(25,130,180)[3]{}[2.5]{} (238,188)(25,130,180) (242,188)(25,0,50)[3]{}[2.5]{} (242,188)(25,0,50) (207,203)\[bl\][$\lambda$]{} (274,203)\[br\][$\lambda$]{} (240,201)(30,204,336) (240,167)\[t\][$f$]{} (240,208)(13,18)(0)[0.85]{} (241,208)\[\][SUSY]{} (225,204)(255,212)
To represent the gaugino and scalar masses in terms of the 5D quantities, we use the correspondence relation between the 4D and 5D theories, which are given in the present context as $\tilde{N}/16\pi^2 \approx 1/g_B^2 k$ and $m_\rho \approx \pi k'$, where $g_B$ represents the $SU(5)$-invariant 5D gauge coupling. The parameter $\hat{\zeta}$ can be read off by matching the gaugino mass expressions of 4D and 5D theories as $\hat{\zeta} \approx
(\zeta g_B^2 F_Z/\pi M_*)$, where the parameter $\zeta$ appears in Eq. (\[eq:gaugino-TeV\]) and $F_Z$ is the VEV of the highest component of the chiral superfield $Z$.[^5] Using these relations, we obtain the following simple expressions for the gaugino and scalar masses: $$M_a = g_a^2\, \frac{\zeta F_Z}{M_*} \frac{k'}{k},
\label{eq:gaugino-masses}$$ and $$m_{\tilde{f}}^2
= \gamma\!\! \sum_{a=1,2,3} \frac{g_a^4 C_a^{\tilde{f}}}{16 \pi^2}\,
(g_B^2 k) \Biggl( \frac{\zeta F_Z}{M_*} \frac{k'}{k} \Biggr)^2,
\label{eq:scalar-masses}$$ where $g_a$ are the 4D gauge couplings given by $1/g_a^2 = \pi R/g_B^2
+ 1/\tilde{g}_{0,a}^2$ ($1/\tilde{g}_{0,a}^2$ are the renormalized coefficients for the Planck-brane gauge kinetic terms), and $\gamma$ is a numerical coefficient of $O(1)$. Note that the quantity $(\zeta F_Z/M_*)(k'/k)$, appearing in Eqs. (\[eq:gaugino-masses\], \[eq:scalar-masses\]) and setting the overall mass scale, is of $O(M_*
e^{-\pi kR}/16\pi^2)$, which is naturally of $O(100~{\rm GeV}\!\sim\!1~{\rm
TeV})$.
For the case of strong supersymmetry breaking, i.e. $\zeta F_Z/k^2 \gg 1$, the 321 gauginos become (pseudo-)Dirac states, where the extra degrees of freedom that pair up with the MSSM gauginos arise from the strong $G$ dynamics. In this case, the diagram giving the gaugino masses is the one that mixes elementary and composite states, instead of Fig. \[fig:gauge-med\]a, so that the gaugino masses are given by $M_a \simeq g_a (\sqrt{\tilde{N}}/4\pi)\,
m_\rho \simeq (g_a/\sqrt{g_B^2 k}) (\pi k')$. The scalar masses are still given by the diagram of Fig. \[fig:gauge-med\]b, but now with the insertion parameter $\hat{\zeta}$ replaced by $1$, as the physics does not depend much on the strength of brane supersymmetry breaking in the limit $\zeta F_Z/k^2
\gg 1$. This gives $m_{\tilde{f}}^2 \simeq \sum_{a=1,2,3} (g_a^4
C_a^{\tilde{f}}/16\pi^2) (\tilde{N}/16\pi^2) m_\rho^2 \simeq \sum_{a=1,2,3}
(g_a^4 C_a^{\tilde{f}}/16\pi^2) (1/g_B^2 k) (\pi k')^2$. These results, together with the formulae in Eqs. (\[eq:gaugino-masses\], \[eq:scalar-masses\]), explain almost all the features observed in Refs. [@Goldberger:2002pc; @Nomura:2003qb; @Chacko:2003tf] for the superparticle mass spectrum in warped unified theories.
Grand unified particles at colliders {#eq:collider}
------------------------------------
In our theory, grand unified particles such as the XY gauge bosons and color-triplet Higgs bosons are present at the TeV scale. What are experimental signatures of these particles? Studying the bulk Lagrangian of the theory, we find that it possesses the $Z_2$ parity under which all the MSSM states and their KK towers are even while the other “grand unified theory (GUT) states” are odd: $$\begin{aligned}
&& {\rm MSSM\:\: fields}\:\: (+):\:\:\: V_{321}, H_D, Q, L, \cdots,
\nonumber \\
&& {\rm GUT\:\: fields}\:\: (-):\:\:\: V_{\rm XY}, H_T, \cdots,
\nonumber\end{aligned}$$ which we call the GUT parity. This parity is not broken by the couplings present in the theory such as the Yukawa couplings and supersymmetry breaking operators. It can thus be an unbroken symmetry of the theory. If this is the case (at least approximately), the lightest GUT particle (LGP) is stable at colliders, leading to characteristic experimental signatures.[^6]
In the theory discussed in section \[sec:warped-sgut\], the LGP is expected to be the lightest of the XY gauginos, $\tilde{X}$, and leads to the following signatures [@Goldberger:2002pc]. Because $\tilde{X}$ is colored, it will hadronize after production by forming a bound state with a quark (or anti-quark). There are four mesons with almost degenerate masses: $$T^0 \equiv \tilde{X}_{\uparrow}\bar{d}, \quad
T^{-} \equiv \tilde{X}_{\uparrow}\bar{u}, \quad
T^{\prime -} \equiv \tilde{X}_{\downarrow}\bar{d}, \quad
T^{--} \equiv \tilde{X}_{\downarrow}\bar{u},$$ where $\tilde{X}_{\uparrow}$ and $\tilde{X}_{\downarrow}$ are the isospin up and down components of the XY gauginos, respectively. The mass splittings among these states are of order MeV, so that they are all sufficiently long-lived to traverse the entire detector without decaying. This yields distinctive signals; in particular, the charged states will easily be seen by highly ionizing tracks. These states can also cause intermittent highly ionizing tracks, generated through charge/isospin exchanges with the detector materials. The reach of the LHC in the masses of these states is estimated to be roughly $2~{\rm TeV}$. A detailed analysis for the case of the colored Higgs LGP can be found in [@Cheung:2003um].
Alternative Theories {#sec:alternative}
====================
In this section we present a variety of theories constructed along the lines presented in the previous two sections. The diversity of models presented here is an indication of how powerful the framework of warped supersymmetric grand unification is, and of the wide variety of phenomena we can obtain in this class of theories.
321-321 model {#subsec:321-321}
-------------
In the model of the previous sections, the bulk $SU(5)$ gauge group is broken to 321 on the Planck brane while it is unbroken on the TeV brane. We could, however, consider the case where the bulk $SU(5)$ is broken to 321 both at the Planck and TeV branes. This class of theories, called 321-321 theories, was considered in Ref. [@Nomura:2004is], where it was shown that the successful MSSM prediction for gauge coupling unification is preserved in such theories. Specifically, the boundary conditions for the bulk fields are given as $V_{321}(+,+)$, $\Sigma_{321}(-,-)$, $V_{\rm XY}(-,-)$ and $\Sigma_{\rm XY}(+,+)$ for the gauge sector and $H_D,\bar{H}_D(+,+)$, $H_D^c,\bar{H}_D^c(-,-)$, $H_T,\bar{H}_T(-,-)$ and $H_T^c,\bar{H}_T^c(+,+)$ for the Higgs sector (realistic theories could also be constructed with $H_D,\bar{H}_D(+,-)$, $H_D^c,\bar{H}_D^c(-,+)$, $H_T,\bar{H}_T(-,+)$ and $H_T^c,\bar{H}_T^c(+,-)$). The quark and lepton superfields are localized on the Planck brane, and proton decay is adequately suppressed. In the 4D picture, these theories have a similar structure to that of the theory in section \[sec:susy-warped\] (e.g. that given in Fig. \[fig:structure\]), but now the global $SU(5)$ symmetry of the DSB sector is spontaneously broken at $\Lambda$ by the IR dynamics of $G$.
As discussed in [@Nomura:2004is], this class of theories has the following distinctive features.
- Before supersymmetry breaking, the massless sector of the model contains not only the MSSM states but also exotic grand unified states $\Sigma_{\rm XY}$, $H_T^c$ and $\bar{H}_T^c$ (see Fig. \[fig:KK-321-321\]). Despite the presence of these exotic states, the successful MSSM prediction for gauge coupling unification is preserved. This is because in the 5D picture the wavefunctions of the exotic states are strongly localized to the TeV brane so that they do not contribute to the running defined through the Planck-brane correlators, and in the 4D picture the exotic states are composite and so do not contribute to the running above TeV.
(385,165)(-10,-18) (5,0)(355,0) (15,-10)(15,135) (8,135)\[r\][mass]{} (8,75)\[r\][$\sim {\rm TeV}$]{} (45,133)\[b\][$V_{321}$]{} (30,0)(60,0) (45,0)[3]{} (30,45)(60,45) (45,45)[3]{} (30,105)(60,105) (45,105)[3]{} (85,133)\[b\][$\Sigma_{321}$]{} (70,45)(100,45) (85,45)[3]{} (70,105)(100,105) (85,105)[3]{} (125,133)\[b\][$V_{\rm XY}$]{} (110,75)(140,75) (125,75)[3]{} (165,133)\[b\][$\Sigma_{\rm XY}$]{} (150,0)(180,0) (165,0)[3]{} (150,75)(180,75) (165,75)[3]{} (205,133)\[b\][$H_{\!D},\!\bar{H}_{\!D}$]{} (190,0)(220,0) (205,0)[3]{} (190,60)(220,60) (205,60)[3]{} (245,133)\[b\][$H_{\!D}^c,\!\bar{H}_{\!D}^c$]{} (230,60)(260,60) (245,60)[3]{} (285,133)\[b\][$H_{\!T},\!\bar{H}_{\!T}$]{} (270,90)(300,90) (285,90)[3]{} (325,133)\[b\][$H_{\!T}^c,\!\bar{H}_{\!T}^c$]{} (310,0)(340,0) (325,0)[3]{} (310,90)(340,90) (325,90)[3]{}
- After supersymmetry breaking, the exotic states, as well as the MSSM superparticles, obtain TeV-scale masses through operators localized on the TeV brane. Because the wavefunctions of the exotic states are strongly localized to the TeV brane, while the MSSM states are not, the masses for the exotic states are an order of magnitude larger than those of the MSSM superparticles. An exception is the $A_5^{\rm XY}$ state, the fifth component of the XY gauge bosons (the imaginary part of the lowest component of $\Sigma_{\rm XY}$), whose mass is forbidden at tree level by higher dimensional gauge invariance. The mass of this state is generated at loop level so that it could be as light as the MSSM superparticles. Thus the LGP is $A_5^{\rm XY}$, which is expected to be stable at colliders. In fact, we can understand the lightness of $A_5^{\rm XY}$ in the 4D picture, as it is the pseudo-Goldstone boson for the breaking $SU(5) \rightarrow 321$ caused by the $G$ dynamics.
- Because supersymmetry is broken at the TeV brane where the gauge group is only 321, the generated superparticle masses are non-unified. In particular, the masses for the three MSSM gauginos are completely free parameters in these theories (because the coupling $\zeta$ in Eq. (\[eq:gaugino-TeV\]) can take different values for $SU(3)_C$, $SU(2)_L$ and $U(1)_Y$). Squark and slepton masses are also non-unified, although they are still flavor universal. In fact, the superparticle masses are given by Eqs. (\[eq:gaugino-masses\], \[eq:scalar-masses\]) with $\zeta$ replaced by $\zeta_a$ (i.e. $\zeta$ now depends on the gauge group). In the 4D picture, we can understand the non-unified feature of the spectrum as the result of the spontaneous breakdown of the global $SU(5)$ symmetry in the DSB sector.
Warped supersymmetric SO(10) {#subsec:SO10}
----------------------------
It is possible to extend the bulk gauge group to a larger unified group in warped supersymmetric grand unification. In particular, we can extend the bulk group from $SU(5)$ to $SO(10)$. The bulk $SO(10)$ is then broken to 321 at the Planck brane by a combination of boundary condition and Higgs breaking, while at the TeV brane it can either be unbroken or broken to the $SU(4)_C \times SU(2)_L \times SU(2)_R$ (422) subgroup. These theories were considered in Ref. [@Nomura:2004it], and have the following (desirable) features.
- The theories provide an elegant understanding of the matter quantum numbers in terms of 422. In particular, the quantization of hypercharges can always be understood no matter how matter fields are introduced. Despite the presence of 422, realistic quark and lepton masses and mixings are easily obtained through higher dimension operators.
- Small neutrino masses are obtained quite naturally, as the seesaw mechanism arises as an automatic consequence of the theories.
- The successful prediction for $b/\tau$ Yukawa unification can be reproduced. The ratio of the VEVs for the two Higgs doublets, $\tan\beta \equiv H_D/\bar{H}_D$, is naturally predicted to be large, $\tan\beta \approx 50$.
- In the case where the gauge group on the TeV brane is 422 and the breaking of left-right symmetry is spontaneous, we have a non-trivial relation among the three MSSM gaugino masses: $M_1/g_1^2 = (2/5)(M_3/g_3^2)
+ (3/5)(M_2/g_2^2)$. This relation still leaves room for the gaugino masses to differ from the ones expected from the conventional grand-unified mass relations for the gauginos.
Model with heavy Higgs boson {#subsec:heavy-Higgs}
----------------------------
The construction of warped supersymmetric unification can also be used to construct supersymmetric theories in which the mass of the lightest Higgs boson is much larger than the conventional upper bound of $\approx 130~{\rm GeV}$ [@Birkedal:2004zx]. The basic idea is to introduce two sets of Higgs doublets — one localized on the TeV brane receiving a large quartic coupling from the TeV-brane superpotential term $W = \lambda S H \bar{H}$, and the other propagating the bulk having the Yukawa couplings to the quarks and leptons located on the Planck brane. The Higgs doublets responsible for electroweak symmetry breaking are linear combinations of these two sets, thus having both the Yukawa couplings and a large quartic coupling. We can then obtain the mass of the lightest Higgs boson as large as $\approx 200~{\rm GeV}$, keeping the successful MSSM prediction for gauge coupling unification. This class of theories allows the possibility of a significant reduction in the fine-tuning needed for correct electroweak symmetry breaking.
Conclusions {#sec:concl}
===========
Supersymmetric unification in warped space provides new possibilities for model building. The picture of warped supersymmetric unification arises naturally through the AdS/CFT correspondence from the assumption that supersymmetry is dynamically broken at $\Lambda \approx (10\!\sim\!100)~{\rm TeV}$ by gauge dynamics $G$ having certain special properties. In the minimal model [@Goldberger:2002pc], the bulk gauge group is $SU(5)$ broken to the 321 subgroup at the Planck brane. The theory leaves many of the most attractive features of conventional unification intact. In particular, the successful MSSM prediction for gauge coupling unification is preserved, and small neutrino masses are naturally obtained from the seesaw mechanism. Proton decay is also naturally suppressed at a level consistent with experiments. Yet physics at accessible energies could be quite different than in the conventional scenario. The model reveals its higher-dimensional nature near the TeV scale, through the appearance of KK towers and an $N=2$ supermultiplet structure. The spectrum of these particles are tightly constrained, so that several definite predictions can be drawn with distinct experimental signatures.
I have also presented several variations of the minimal model [@Nomura:2004is; @Nomura:2004it; @Birkedal:2004zx], which give distinct phenomenologies. These theories differ in the gauge groups of the bulk and branes and/or in locations of the Higgs (and matter) fields. Taking together, these theories, including the minimal one, provide a basis for phenomenological studies of dynamical supersymmetry breaking at low energies.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank David Tucker-Smith for reading the manuscript. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contract DE-AC03-76SF00098 and DE-FG03-91ER-40676, by the National Science Foundation under grant PHY-0403380, and by a DOE Outstanding Junior Investigator award.
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[^1]: Our theory can also be viewed as one with $SU(5)$ broken by a large VEV of a Higgs field on the Planck brane (see [*e.g.*]{} [@Nomura:2001mf]), although in that case an understanding of the doublet-triplet splitting should be attributed to unknown cutoff scale physics.
[^2]: In the theory with boundary condition $SU(5)$ breaking, the minimal couplings of the XY gauge bosons to quarks and leptons vanish, but the couplings of the XY-gauge or colored-Higgs fields to matter can still arise in 4D from the 5D couplings that involve a derivative with respect to the fifth coordinate.
[^3]: An alternative possibility is that there is an additional source of supersymmetry breaking on the Planck brane (such a breaking may in fact help to stabilize the radius of the extra dimension). As long as the scale $\Lambda'$ of this additional breaking is $\Lambda' \simlt 10^9~{\rm GeV}$, this does not reintroduce the supersymmetric flavor problem, even in the presence of generic interactions between the supersymmetry-breaking field on the Planck brane and the quark-lepton supermultiplets. This allows a wide range of possibilities for the gravitino mass, $0.1~{\rm eV} \simlt
m_{3/2} \simlt 1~{\rm GeV}$, some of which is consistent with the scenario that the gravitino is the super-WIMP dark matter of the universe.
[^4]: In a theory where $G$ is almost conformal above the dynamical scale $\Lambda$, the parameter $\tilde{N}$ may actually represent the [*square*]{} of the number of “colors” of $G$, and not the number of “colors” itself. Discussions on this and related issues in the AdS/CFT correspondence can be found, for example, in Ref. [@Burdman:2003ya].
[^5]: The definition of $F_Z$ here is that, in the normalization where the kinetic term of $Z$ is canonically normalized in 4D, $F_Z$ is defined by $F_Z = e^{\pi kR} \partial Z/\partial \theta^2|_{\theta = \bar{\theta} = 0}$. The natural size for $F_Z$ is then of order $k^2 \sim M_*^2$ (no exponential suppression factor).
[^6]: The GUT parity can in principle be broken by the presence of certain brane operators. In the present case of matter strongly localized to the Planck brane, however, the effect of the breaking is suppressed in the low-energy 4D theory so that the LGP is still effectively stable for collider purposes. It is possible, however, that the breaking leads to the lifetime of the LGP shorter than $\sim 1~{\rm s}$, which may be important for cosmology.
|
---
abstract: 'In recent years, it has become clear that large quantities of gas reside in the halos of many spiral galaxies. Whether the presence of this gas is ultimately a consequence of star formation activity in the disk, or accretion from outside of the galaxy, is not yet understood. We present new, deep H<span style="font-variant:small-caps;">i</span> observations of NGC 4395 as part of a continuing observational program to investigate this issue. We have detected a number of gas clouds with masses and sizes similar to Milky Way HVCs. Some of these are in regions without currently ongoing star formation, possibly indicating ongoing gas accretion.'
author:
- 'George Heald (1) and Tom Oosterloo (1,2)'
title: 'Anomalous HI Gas in NGC 4395: Signs of Gas Accretion'
---
Observations and Data Reduction
===============================
In the study of gaseous halos, most of the recent observational focus has been on nearby normal spiral galaxies [e.g., @o07; @b07]. Relatively little attention has so far been paid to low-luminosity spirals, but an H<span style="font-variant:small-caps;">i</span> halo has been detected by @mw03 in the edge-on UGC 7321. In order to determine the origin of gaseous halos, it is important to pursue the line of investigation into galaxies with very low rates of star formation, where the effects of star formation and accretion may be more easily distinguished.
To address this issue, we have recently performed deep H<span style="font-variant:small-caps;">i</span> observations of the nearby dwarf spiral NGC 4395. The data were obtained using the Westerbork Synthesis Radio Telescope (WSRT) for a total of $8\times12\,\mathrm{hr}$. The target was chosen because it is nearby \[$D\,\sim\,3.5\,\mathrm{Mpc}$; @s99\]; viewed at a moderately face-on inclination \[$i\,\sim\,46^{\circ}$; @s99\]; isolated; and has a low optical luminosity, with localized star formation activity (see Fig. \[fig:anomgas\]b).
The data were reduced using standard techniques in the `MIRIAD` software package. Subsequent analysis was performed using `GIPSY`.
Data Analysis & Conclusions
===========================
Inspection of the H<span style="font-variant:small-caps;">i</span> data cube reveals the presence of a large population of gas clouds at velocities which place them outside of the normal disk rotation. In total, approximately $5\,\times\,10^7\,M_{\odot}$, equivalent to 5% of the total H<span style="font-variant:small-caps;">i</span> mass, is found to be in this component. Here, we restrict ourselves to presenting a brief overview of the anomalous gas population; we defer a more detailed analysis of the individual clouds to a forthcoming paper.
In order to distinguish the gas at anomalous velocities from the gas participating in normal disk rotation, a velocity field was constructed by tracing the peaks of the individual velocity profiles. Next, the velocity profiles in each line of sight were shifted by the corresponding value in the velocity field, yielding a ‘derotated’ data cube. In this cube, the peak H<span style="font-variant:small-caps;">i</span> emission is located (by construction) at $V_{\mathrm{dr}}\,=\,0\,\mathrm{km\,s}^{-1}$. Following a visual inspection of the derotated cube, the gas outside of the somewhat conservative velocity range $|V_{\mathrm{dr}}|\,\leq\,30\,\mathrm{km\,s}^{-1}$ was called anomalous (cf. the typical H<span style="font-variant:small-caps;">i</span> velocity dispersion, $7\,\mathrm{km\,s}^{-1}$).
The global distribution of the anomalous velocity gas identified using the technique described above is displayed in Fig. \[fig:anomgas\]b. Four of the larger gas complexes have been labeled (A–D). Complexes B ($M_{\mathrm{HI}}\,\approx\,1.9\,\times\,10^6\,M_{\odot}$), C ($M_{\mathrm{HI}}\,\approx\,2.3\,\times\,10^6\,M_{\odot}$), and D ($M_{\mathrm{HI}}\,\approx\,1.1\,\times\,10^6\,M_{\odot}$) appear to be colocated with regions of active star formation, as suggested by the emission in the underlying GALEX FUV map. Complex A ($M_{\mathrm{HI}}\,\approx\,4.0\,\times\,10^6\,M_{\odot}$) on the other hand, appears to be unassociated with any of the actively star forming regions. It may be a signature of ongoing gas accretion onto the disk of NGC 4395.
The Westerbork Synthesis Radio Telescope is operated by ASTRON (Netherlands Foundation for Research in Astronomy) with support from the Netherlands Foundation for Scientific Research (NWO).
Boomsma, R. 2007, Ph.D. Thesis Matthews, L. D. & Wood, K. 2003, ApJ, 593, 721 Oosterloo, T., Fraternali, F., & Sancisi, R. 2007, AJ, 134, 1019 Swaters, R. A. 1999, Ph.D. Thesis
|
---
address: |
${}^{1}$Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa,\
${}^{2}$ Department of Chemistry and Physics, University of KwaZulu-Natal, Durban, South Africa
author:
- 'Amanda Weltman$^{1}$ and Anthony Walters$^{2}$'
title: Fast Radio Burst Cosmology and HIRAX
---
Introduction
============
Fast Radio Burst discovery is transitioning from a slow drip of data to a veritable deluge. In preparation for the expectation of a huge catalogue of FRBs and their dispersion measures, we consider the possible ways that they may be used as standard measures in cosmology as well as ways in which we can use FRB observations, cross correlated with cosmological surveys to answer other unsolved problems about our cosmos. In this short article based on a talk at the 2019 Moriond Gravity conference, we remind the reader about some of what is known about FRBs to date, as well as provide a brief introduction to the HIRAX experiment. We then go on to consider one such application for FRBs, the possibility of combining FRB results from HIRAX and Planck results to resolve the missing baryon problem.
HIRAX
=====
The Hydrogen Intensity and Real-time Analysis experiment (HIRAX) is an array of 6m parabolic radio telescopes currently under construction in South Africa [@HIRAX]. The primary array will be situated in the Karoo desert area in South Africa, an area that has particularly low Radio Frequency Interference (RFI) due to an act of parliament, the Astronomy Advantages act of 2008 that protects the Karoo area from RFI from the usual manmade sources. The central HIRAX array will be based in the Karoo with outriggers planned most likely in other parts of the country - and in various other countries in Africa, most likely Rwanda and Namibia at the very least. HIRAX will be able detect the effects of dark energy on the distribution of galaxies by using Hydrogen intensity mapping in the 400 to 800 MHz range, which corresponds to a redshift range of $0.8 < z < 2.5$. We expect that there are thousands, if not tens of thousands, of FRBs going off in some part of the observable sky every day. HIRAX will be able to observe a large fraction of these within its observing range, a 15 000 ${\rm deg}^2$ of the sky and will thus produce a huge catalogue of several thousand FRBs. In its full design, HIRAX will be made up of 1024 dishes with several sets of outriggers, each made up of 8 dishes. An advantage of the Southern location of HIRAX allows for cross correlations with several ongoing surveys; see [@HIRAX] for further discussions on HIRAX setup and strengths.
Fast Radio Bursts
=================
Since their discovery a dozen years ago, Fast Radio Bursts have rapidly become an exciting novel area of astrophysics research, in part due to their relatively scarcity in the data so far and partly because we cannot explain their cause. If there is anything we love in astrophysics research, it is unexplained phenomena. In the case of Fast Radio Bursts, we have been using the sparse data we have so far to try to put together a picture of how they are formed, their expected rates and properties and whether they are all one type of object or fall in multiple classes. All of this is set to change in the coming months and years as globally we shift from searching for signals of FRBs in archival data, to observing them live, and in large numbers across the sky. At this stage, what we know is that they are most likely extragalactic and found within host galaxies. We observe them at relatively low redshifts up to about $z =0.5$ which is likely the a limitation of our ability to observe them. They are of the order of 1 Jansky in brightness and so far are very short lived, with lifespans ranging from $\mu {\rm s}$ to 50 ms and observed in the radio range as low as 400 MHz. There is a public catalogue of all bursts observed so far at [www.frbcat.org]{} and a catalogue of theories at [www.frbtheorycat.org]{} with a companion paper expanding on each so far [@FRBcatalogue]. They are observed at a range of polarisations, with none observed so far that are unpolarised, and they have a range of rotation measures. All of these clues do not yet point to a clear picture but we await more data.
Fast Radio Burst Cosmology
==========================
Early indication that Fast Radio Bursts appeared to be extragalactic sources led to speculation on some possible cosmological applications of the bursts. And since the association of a repeating FRB with a host galaxy at $z=0.19$, many more applications have been proposed. Even without redshift information some cosmological information an be extracted, for example; a single FRB can constrain violations of the Einstein Equivalence Principal [@2015PhRvL.115z1101W; @tingay], or constrain the mass of the photon [@2016ApJ...822L..15W]. Strongly lensed FRBs could be used to probe dark matter [@2016PhRvL.117i1301M], or measure the value Hubble constant and cosmic curvature [@2017arXiv170806357L]. And, dispersion space distortions could be used to probe the clustering of matter [@2015PhRvL.115l1301M]. In the future, should more FRB events be associated with a host galaxies, for which redshifts can be acquired, this would give access to the Dispersion Measure (DM) redshift relation, $\mathrm{DM}(z)$, which can be used as a probe of the background cosmological parameters [@2014PhRvD..89j7303Z; @2014ApJ...783L..35D; @2014ApJ...788..189G]. However, the strength of this approach will strongly depend on the intrinsic scatter in the observed DM data caused by intervening matter inhomogeneities along the line of sight, host galaxy contamination to the observed DM, and knowledge of the cosmic mean gas fraction in the IGM [@ourFRBpaper]. Some other cosmological applicants include constraining the growth rate by cross-correlating FRBs with kSZ data [@newJonPaper], and testing the Copernican principal by testing the isotropy in the $\mathrm{DM}(z)$ relation [@newIsoPaper].
The Missing Baryon Problem
==========================
Despite our current era of concordance cosmology, a number of puzzles remain. Included in these is the missing baryon problem [@Fukugita]; despite our best efforts and most sophisticated analyses, it appears that we have lost 30% of the baryons we expect to see in the recent universe [@2004ApJ...616..643F; @2012ApJ...759...23S]. While we believe they are likely in the warm-hot intergalactic medium (WHIM), gaining direct observational evidence of this is challenging. Attempts at detecting the WHIM include large-scale structure cross-correlations, such as the cross-correlation between thermal Sunyaev-Zeldovich effect and galaxy weak lensing [@2015JCAP...09..046M; @Hojjati15; @Hojjati17], stacking luminous red galaxy pairs with thermal SZ map [@Tanimura19], detecting the temperature dispersion of kinetic SZ effect within the X-ray selected galaxy clusters [@Planck-dispersion2018], and the detection of the cross-correlation between kinetic SZ effect with velocity field [@Planck-unbound16; @Carlos15]. Since an FRBs DM is due to its propagation through regions with free electrons, they are directly sensitive to the location of baryons in the Universe, and thus may help to constrain the mass fraction of the WHIM. One such proposal considers cross-correlating FRB maps with the thermal Sunyaev-Zeldovich effect to find missing baryons [@2018PhRvD..98j3518M]. Another approach would be to use the $\mathrm{DM}(z)$ relation to constrain the mean cosmic gas (diffuse baryons) in the IGM, $f_\mathrm{IGM}$ [@newFRBpaper].
Future Constraints from FRB observations
========================================
A forecast using the $\mathrm{DM}(z)$ relation can be seen in Figure \[fig:ellipses\], which shows constraints in the $\Omega_k-\Omega_\mathrm{b}h^2$ plane, from a sample of $10^4$ simulated FRBs with redshifts, combined with the Planck 2015 constraints [@newFRBpaper]. Grey contours show the CMB + BAO + SNe + $H_0$ (CBSH) constraint from the Planck 2015 results. Magenta lines indicate the 1- and 2-$\sigma$ constraint contours for FRB+CBSH assuming one has perfect knowledge of the mean diffuse baryon fraction in the IGM, $f_\mathrm{IGM}$. Clearly, having perfect knowledge of $f_\mathrm{IGM}$ would allow for a dramatic improvement over the current CBSH constraints, however this function in poorly constrained by observations. Black lines show the same, but with no prior knowledge of $f_\mathrm{IGM}$. While the cosmological constraints do not shown any improvement over the CBSH priors, the priors allow for a measurement of $f_\mathrm{IGM}$ at the percent level (shown by the colorbar on the right). Since around $50-80\%$ of the baryons are believed to reside in the IGM, a detection of $f_\mathrm{IGM}$ in this range would provide further evidence that the most of the Universe’s baryons are located in diffuse gas in the IGM, and are in-fact not missing.
![Constraint forecast in the $\Omega_k-\Omega_\mathrm{b}h^2$ plane, from $10^4$ FRBs with redshifts, using the DM(z) relation and Planck priors, and.... Grey contours show the CMB + BAO + SNe + $H_0$ (CBSH) constraint from the Planck 2015 results. Magenta lines indicate the 1- and 2-$\sigma$ constraint contours for FRB+CBSH assuming one has perfect knowledge of the mean diffuse baryon fraction in the IGM, $f_\mathrm{IGM}$. Black lines show the same, but with no prior knowledge of $f_\mathrm{IGM}$. Coloured points correspond to the value of $f_\mathrm{IGM}$. []{data-label="fig:ellipses"}](omegabh2_omegak_wwof_v2.pdf){width=".65\textwidth"}
Conclusion
==========
From only a few FRBs observed in the last few years, we expect to soon have a catalogue of tens of thousands (if not far more) FRBs from radio telescope arrays around the world, including CHIME, ASKAP, and HIRAX as we have considered here. In this short article we have considered some novel cosmology and astrophysics questions we can probe using the incoming barrage of FRB data in the near future. We have highlighted the potential for localising these FRBs in particular with HIRAX, due to the comprehensive outrigger programme planned.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is based in part on a talk given by Weltman at the 2019 Moriond Gravity meeting, for which she would like to thank the Organisers. We gratefully acknowledge our collaborators on the projects referenced in this work, [@Amadeuspaper] and [@ourFRBpaper], and [@newFRBpaper]. Weltman is supported by the South African Research Chairs Initiative of the Department of Science and Technology and the National Research Foundation of South Africa (NRF). Any opinion, finding and conclusion or recommendation expressed in this material is that of the authors and the NRF does not accept any liability in this regard. Walters acknowledges support from the NRF (Grant Number 105925, 110984, 109577).
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|
---
abstract: 'We study a simple modification to the conventional time of flight mass spectrometry (TOFMS) where a *variable* and (pseudo)-*random* pulsing rate is used which allows for traces from different pulses to overlap. This modification requires little alteration to the currently employed hardware. However, it requires a reconstruction method to recover the spectrum from highly aliased traces. We propose and demonstrate an efficient algorithm that can process massive TOFMS data using computational resources that can be considered modest with today’s standards. This approach can be used to improve duty cycle, speed, and mass resolving power of TOFMS at the same time. We expect this to extend the applicability of TOFMS to new domains.'
author:
- |
Morteza Ibrahimi, Andrea Montanari, \
and George Moore, [^1][^2]
bibliography:
- 'lad\_references.bib'
title: 'Accelerated Time-of-Flight Mass Spectrometry'
---
Time of flight mass spectrometry.
Introduction
============
Mass spectrometry (MS) refers to a family of techniques used to analyze the constituent chemical species in a sample. The applications abound in science and technology and new fields of scientific investigations have evolved around these techniques. An example is proteomics which refers to the science of analyzing peptides and proteins. Proteins are the workhorse of many biological mechanism. Of great interest to biological sciences, medical research and drug discovery and developments is identifying and analyzing the composition and structure of proteins and other large chemical compounds. It has become possible only recently to analyze the composition of proteins with high throughput and accuracy through mass spectrometry techniques [@hillenkamp1991matrix] [@fenn1989electrospray]. Other applications include measuring isotopic ratio, space exploration, testing for illegal substances etc. Mass spectrometers are usually accompanied with gas or liquid chromatography techniques and are used in different configurations in tandem with other mass spectrometer of the same or different types. These configurations provide a wide range of utility and performance criteria making mass spectrometry relevant for many different applications.
A typical mass spectrometer consists of three main modules: an ionizer, a mass analyzer, and a detector. The ionizer converts the species of interest, and possibly other compounds in the sample to ions in gas phase. Recent advances in ionization techniques, namely matrix assisted laser desorption ionization [@doi:10.1021/ac00024a002], and electrospray ionization [@fenn1989electrospray] has made it possible to ionize and transform into gas phase large intact molecules like proteins. These techniques provided new applications for mass spectrometry and open new avenues for analyzing the composition and structure of proteins [@ragoussis2006matrix].
The purpose of the mass analyzer module is to separate the ions according to their mass to charge ratio (MCR). Today’s common mass analyzers separate the ions by subjecting them to electromagnetic fields. These fields exert different forces to different ions. One class of instruments, broadly referred to as sector instruments cause the ions with different MCR to take different trajectories, effectively beamforming a stream of flying ions of particular MCR toward a detector [@cross1951two] Another technique is to have all the ions travel a common trajectory but with different velocities. This is the basis for time of flight (TOF) mass spectrometers which we shall describe throughly in the sequel. A third approach is to *guide* only a particular MCR to have a stable trajectory. This is the basis for the Quadrupole and ion trap mass analyzers [@schwartz2002two]. These instruments can act as a MCR filter or scan a wider range by sweeping the filter pass band.
The detector module senses the ions by detecting the impact of charged compounds with the detector surface or the charge or current they induce by their particular motion.
![Different parts of a TOFMS.[]{data-label="fig:TOFMS_schematic"}](TOFMS_schematic_v2){width=".5\textwidth"}
In this paper we are concerned with time of flight mass spectrometry (TOFMS) which is a simple yet powerful MS technique. TOFMS was introduced in the 1940s by Stephens [@stephens1946pulsed]. TOFMS offers two major benefits over alternative techniques. It has essentially unlimited mass range and high repetition rate. These properties along with the recent advances in available hardware and ionization techniques have made TOFMS an appealing choice for the analysis of samples with wide mass range [@roberts2004application], biological macromolecules [@fenn2003electrospray] and in combination with other mass spectrometers [@shevchenko1997rapid].
A basic TOF mass spectrometer consists of four parts: an ionization chamber, an acceleration chamber, a drift region, and a detector (c.f., Fig \[fig:TOFMS\_schematic\]). The sample is ionized in the ionization chamber. These ions are then subjected to a very strong electrical field in the acceleration chamber, effectively firing them into the drift region. Ideally, the ions entering the drift region have kinetic energy, $K$, proportional to their charge $z$, i.e., if the potential difference in the acceleration chamber is $U$ then the following holds $K = Uz$. This means that an ion with mass $m$ has velocity $v = \sqrt{\frac{2Uz}{m}}$. Therefore, assuming that the length of the drift region to be $L$, the time to reach the detector is $$t = \sqrt{\frac{m}{2z}}\frac{L}{2U}.$$ In other words, the time that takes for an ion to reach the detector is proportional to $\sqrt{m/z}$ where $m$ is the mass of the ion and $z$ is its net charge.
TOFMS is a pulsed technique, i.e., ions are formed in an ionization stage and subsequently accelerated as a *packet* into the drift region with ions with different MCR traveling at different speeds. As the ions impact the detector, they generate a continuous electrical signal which is then sampled resulting in a discrete signal. The result of this process, which we call a *scan*, is a noisy sample of the $\sqrt{m/z}$ spectrum. A single scan is often too noisy and this process is repeated from hundreds to a few thousands times and averaged to obtain an accurate estimate of the $\sqrt{m/z}$ spectrum. We will call an estimate of $\sqrt{m/z}$, which is the result of processing many scans, one *acquisition*. In many applications multiple consecutive acquisitions are obtained: to construct a movie of an evolving sample like an ongoing chemical reaction; to analyze the output of a preceding chromatography stage; or to mass analyze samples through an automated system where TOFMS instrument in being fed automatically, e.g., in pharmaceutical applications [@janiszewski2008perspectives].
There are several metrics that describe the performance of a mass spectrometer. Some of the widely used metrics are: mass resolving power, mass accuracy, dynamic range, sensitivity and speed. Mass resolving power is the minimum difference in mass to charge ratio for two present species to be distinguishable by the instrument. Mass accuracy is the normalized precision with which the instrument can report the MCR of present species, measured in MCR error divided by MCR. Mass range refers to the range of MCRs the instrument can detect. Sensitivity is the minimum concentration of an specie to be detectable by the instruments. Finally, speed is the number of acquisition the instrument can acquire per unit time.
These metrics are not independent. Several trade offs exist among these metrics based on how a TOFMS instrument is designed and operated. For example, speed can be increased, at the cost of mass resolving power, mass accuracy and sensitivity, by decreasing the number of scans collected for each acquisition. Another trade off exists between speed in one hand and mass resolving power and accuracy on the other hand, which is the focus of this paper and is described in details below.
In conventional TOFMS, the time between consecutive pulses is set to be long enough to avoid overlap between different scans, i.e., for the slowest ion in an scan to arrive at the detector before the fastest ion of the next scan. Hence, acquisition time is lower bounded by, $$\begin{aligned}
\label{eq:acquisition_time_lbdd}
T_{\text{acquisition}} &\ge N \times T^*_{\text{scan}} \nonumber \\
& \ge N \times \frac{L}{\sqrt{2U}} \,(\sqrt{(m/z)_{\max}} - \sqrt{(m/z)_{\min}}),\end{aligned}$$ where $N$ is the number of scans collected for each acquisition. Furthermore, the difference in time of arrival for two ions is $$t_2 - t_1 = \frac{L}{\sqrt{2U}} \, (\sqrt{m_2/z_2} - \sqrt{m_1/z_1}).$$ Hence, increasing the length of the drift region $L$ $(i)$ increases the acquisition time and therefore decreases the speed $(ii)$ spreads the ions further apart and therefore increases the mass accuracy and resoling power. This issue is of fundamental importance because of the following.
Other factors that can improve the mass accuracy and resolving power of a TOFMS, e.g., detector characteristics and the speed of the electronics, have reached their limits while new applications demand even better performance in terms of mass accuracy and resolving power. One option that remains available for improving the mass accuracy and resolving power is to increase the length of the drift region.
However, first, there is the obvious desire for higher speed and accuracy at the same time. Second, some applications have stringent requirements in terms of speed, mass accuracy, resolving power and sensitivity. This could be due to exogenous time restrictions, e.g., when monitoring a chemical reaction or experimentation choice, e.g., when TOFMS is preceded by a chromatography stage or used in tandem, with another mass spectrometry stage. There is also significant economical implications, a high end instrument costs at the order of hundreds of thousands of dollars and improving the speed and throughput while keeping or improving the accuracy can result in significant savings. This is most clear in the case of large scale automated experiments used in drug discovery and development activities.
Therefore, simultaneous improvement of the speed and mass accuracy and resolving power is of fundamental interest in TOFMS [@trapp2004continuous].
Conventional TOFMS works by repeating the same experiment multiple times and averaging the results. The choice of averaging was mainly due to its simplicity.
In particular, the volume and rate of data generated by TOFMS instruments prohibited the use of more sophisticated techniques. Our ability to commit more computational resources has increased significantly since the introduction of TOFMS. At the same time, the data rate of these instruments generate has also dramatically increased. In this paper, we present an efficient, highly parallelizable algorithm that in conjunction with a simple modification to the conventional TOFMS can improve mass accuracy, mass resolving power and speed at the same time.
There has been previous work trying to alleviate this problem. One approach, called Fourier transform TOF, is to modulate a continuous ion beam at the source using a periodic waveform and subsequently accelerate it into the drift region [@knorr1986fourier]. The detected signal is then demodulated to obtain the spectrum. Another approach, called Hadamard transform TOF (HT-TOF) [@brock1998hadamard], is based on modulation (gating) of a continuous ion source by a $0/1$ pulse. In this approach, an ion beam is deflected according to a pseudorandom sequence of pulses. If the pulse is $1$, the beam is undeflected and will reach the detector. In contrast, if the pulse is $0$ the beam is deflected away from the detector. The pulse sequence has the same frequency as the detector. In an ideal case where there is neither shot noise nor additive noise, the output can be described as $$\label{eq:HTTOF_input_output}
\boldy = H \boldx,$$ where $\boldy \in \reals^n$ is the observed signal at the detector and $\boldx \in \reals^n$ is the TOF spectrum. $H \in \reals^{n \times n}$ is a $0/1$ matrix where each column is the pseudorandom sequence shifted by the index of the column. As long as $H$ is full rank and the model is accurate the TOF spectrum can be recovered by applying the inverse of $H$ to $\boldy$. The spectrum is obtained by a deconvolution that can be implemented efficiently using the fast Hadamard transform.
One drawback of these methods is that they treat the reconstruction process as a deterministic inversion problem and ignore the noisy nature of the observations. Furthermore, they require substantial modification to the hardware of a conventional TOFMS. In this paper, we describe a different method, called accelerated time of flight mass spectrometry (), which simultaneously achieves mass resolving power, duty cycle, and speed improvement using essentially the same hardware as a conventional TOFMS. Our reconstruction scheme acknowledges the stochastic nature of the observation. Simulation results using real data confirm the performance improvement of this scheme.
[**Notations and Terminology:**]{}\[sec:problem\_formulation\] Let $[n] = \{1, 2, \dots, n\}$ and $\{\boldx[i]\}_{i= [n]}$ be the output of the detector for a single scan. With a slight abuse of notation we also refer to $\boldx$ as a [*scan*]{}. Typically a TOFMS experiment consists of many scans which are later processed (simply averaged) to obtain a more accurate estimate of the spectrum. Let $\boldx^{(l)}[i]$ be the $l^{th}$ scan. Define the [*true spectrum*]{}, $\bar{\boldx}[i]$, as the average of infinitely many scans, i.e., $\bar{\boldx}[i] = \lim_{N \to \infty} \frac{1}{N}\sum_{l=1}^{N}\boldx^{(l)}[i]$. Each scan $\boldx^{(l)}[i]$ is therefore a noisy version of $\bar{\boldx}[i]$. We define the [*trace*]{}, $\boldy[t], \; t = 1, \dots T$, to be the observed detector response for multiple, possibly overlapping scan. Given an observed trace $\boldy$, the goal is to find a *good* estimate $\widehat{\boldx}$ of $\bar{\boldx}$.
In what follows we treat the discrete signals as column vectors. For $u$ and $v$ two vector of the same dimension, let $v^*$ denote the transpose of $v$ and $\langle u,v \rangle$ the scaler product of $u$ and $v$.
As a matter of convention we refer to each element of the vectors that represent the spectrum ($\boldx^{(l)}$ and $\bar{\boldx}$) as a bin and to that of the trace as a sample, e.g., $\boldy[1]$ represents the first sample of the trace. When an ion impacts the detector it generates a bell-shaped pulse in the output of the detector. We refer to an observed pulse in the trace as an *impact event*, or event for short. Usually the sampling rate of the detector response is such that an event spans multiple samples.
For pedagogical reasons, we first describe the algorithm *as if* each event could occupy only one sample and there was no time jitter, i.e., all the ions of the same species are associated with the same bin. We then describe the algorithm without this assumptions with some minor modifications. All the results presented in this paper are obtained using real data from a conventional TOFMS instrument which is used to simulate the output of an ATOFMS. The algorithm used to obtain these results is the generalized version of the algorithm.
Measurement Scheme and the Data Model {#sec:data_model}
=====================================
A TOF measurement from a single scan is commonly very sparse (after removing the additive electrical noise through preprocessing, c.f. Section \[sec:simulation\]). Furthermore, a single measurement of the whole spectrum is not expensive and it can be viewed as being performed in parallel as all ions are flying in the drift region at the same time. However, the observed signal from a single scan is too noisy and many repetitions of the same measurement are necessary to obtain an accurate estimate of the spectrum. In a conventional TOFMS setting, the observed trace can be expressed as $\boldy[t] = \sum_{l=1}^{N} \boldx^{(l)}[t-ln] $ where $\boldx^{(l)}$ is the detector response to the $l^{th}$ scan and $\boldx[i]$ is understood to be zero for $i\le 0$ or $i > n$.
We incorporate a simple, yet powerful, modification to this scheme [@moore2012statistical] (c.f. Fig. \[fig:ctof\_vs\_atof\_concept\]). Conventional TOF () requires collection of many scans, each scan collected [*independently*]{} with no overlap. ATOF idea is to increase the repetition rate and allow the subsequent scans to overlap.
![Difference between TOFMS and ATOFMS. In ATOFMS different scans can overlap resulting in shorter acquisition time for the same number of scans but also a [*convoluted*]{} observed trace.[]{data-label="fig:ctof_vs_atof_concept"}](CTOF_vs_ATOF){width=".48\textwidth"}
Define $\tau_l$, the *firing time*, to be the starting time of the $l^{th}$ scan, i.e., the time when the $l^{th}$ ion packet is accelerated into the drift region. Define $\Delta \tau_l \equiv \tau_{l+1} - \tau_l$. In $\Delta \tau_l = \Delta \tau \ge n$ to avoid overlapping between consecutive scans. We relax this condition and let $\Delta \tau_l$ be a random variable with $\expect[\Delta \tau_l] = \alpha n$, for some $\alpha < 1$. As is the case with the HT-TOF we assume that the detector response to overlapping scans is the superposition of the individual responses, $$\label{eq:trace_definition}
\boldy[t] = \sum_{l=1}^{N} \boldx^{(l)}[t-\tau_l].$$ In this case at each time $t$, $\boldy[t]$ is the superposition of multiple overlapping scans. Assume there are a total of $N$ scans and let $0 = \tau_1 < \tau_2 < \dots < \tau_N$ be the firing times. For a given $\boldtau = (\tau_1, \tau_2, \dots, \tau_N)$, define the matrix $A \in \reals^{T \times n}$ as $$\label{eq:adjacency_matrix}
A(t,i)= \left\{
\begin{array}{l l}
1 \quad &\text{if} \; \exists \; l\in [N] \; \mbox{s.t.} \; i = t-\tau_l \\
0 & \text{Otherwise.}
\end{array}
\right.$$ The matrix $A$ can be considered as the adjacency matrix of a bipartite graph (c.f., Fig \[fig:bipartite\_graph\]), with rows of $A$ corresponding to the samples in the trace $\boldy$ and columns of $A$ corresponding to the bins on the spectrum $\boldx$. Sample $t$ on the spectrum is connected to bin $i$ on the spectrum, i.e., $A_{ti} = 1$, if and only if for some scan $l \in [T]$ the ions from bin $i$ of the scan $\boldx^{(l)}$ arrive at time $t$ in the trace $\boldy$. In what follows, we will refer to the neighbors of sample $t$ as $\partial t = \left\{i\in [n] \;|\; A(t,i) > 0\right\} $, and similarly to the neighbors of bin $i$ as $\partial i$.
$\boldy[t]$ can be considered as a noisy version of linear measurements of $\bar{\boldx}$, $\langle A_t, \bar{\boldx} \rangle$, with $A_t$ the $t^{th}$ row of $A$ as a column vector. In this notation, the is a special case where each row of $A$ has only one nonzero element, i.e., measurement $\boldy[t]$ corresponds to a noisy observation of $\bar{\boldx}[i]$ for some bin $i$. The structure of matrix $A$ reveals the difference between and . $$A_{\scriptscriptstyle{\ctof{}}}=
{
\begin{pmatrix}
\boldsymbol{1}& 0& 0& 0 \\
0& \boldsymbol{1}& 0& 0 \\
0& 0& \boldsymbol{1}& 0 \\
0& 0& 0& \boldsymbol{1} \\
\hline
\boldsymbol{1}& 0& 0& 0 \\
0& \boldsymbol{1}& 0& 0 \\
0& 0& \boldsymbol{1}& 0 \\
0& 0& 0& \boldsymbol{1} \\
\hline
\boldsymbol{1}& 0& 0& 0 \\
0& \boldsymbol{1}& 0& 0 \\
0& 0& \boldsymbol{1}& 0 \\
0& 0& 0& \boldsymbol{1}
\end{pmatrix}
}
, \; A_{\scriptscriptstyle{ATOF}}=
\begin{pmatrix}
\boldsymbol{1}& 0& 0& 0 \\
0& \boldsymbol{1}& 0& 0 \\
0& 0& \boldsymbol{1}& 0 \\
\boldsymbol{1}& 0& 0& \boldsymbol{1} \\
0& \boldsymbol{1}& 0& 0 \\
\boldsymbol{1}& 0& \boldsymbol{1}& 0 \\
0& \boldsymbol{1}& 0& \boldsymbol{1} \\
\boldsymbol{1}& 0& \boldsymbol{1}& \boldsymbol{1} \\
0& \boldsymbol{1}& 0& 0 \\
0& 0& \boldsymbol{1}& 0 \\
0& 0& 0& \boldsymbol{1}
\end{pmatrix}$$
![Adjacency matrix $A$ and the corresponding bipartite graph. The signal on the top represents the spectrum and the bottom signal represents the trace. The trace is the overlapped concatenation of noisy copies of the spectrum. Nodes are color coded where blue correspond to the first scan, purple to the second, and green to the third. Neighbors of sample $t$ on the trace are those bins on the spectrum who could potentially contribute to an event at sample $t$ (again color coded). []{data-label="fig:bipartite_graph"}](trace_graph_ter){width=".48\textwidth"}
Given the trace $\boldy$ and adjacency matrix $A$ one can attempt to solve for $\bar{\boldx}$ using an ordinary least squares, $\hat{\boldx}_{{\scriptscriptstyle LS}} = {\argmin} \|A \boldx - y\|_2$ or $\ell_1$-regularized least squares $\hat{\boldx}_{{\scriptscriptstyle LASSO}} = {\argmin} \|A \boldx - y\|_2 + \lambda \|\boldx\|_1$ [@tibshirani1996regression]. However, simulation results demonstrate poor performance for both these methods. The reason lies in the choice of the objective function. Sum of square residuals approximates the negative log likelihood when the measurement noise is additive Gaussian. However, TOFMS is dominated by shot-noise which is signal dependent and non-additive. Similar issues arises in applications like photon-limited imaging where the observations are again shot-noise limited. Regularized maximum likelihood approaches proved effective in these settings [@harmany2010spiral]. Here we propose a stochastic model for the observation $\boldy$ and present an algorithm that optimizes the $\ell_1$-regularized log likelihood.
![A sample event from the trace[]{data-label="fig:sample_event"}](sample_event_bw_v2){width=".5\textwidth"}
Figure \[fig:sample\_event\] shows a sample observed event plotted as the output of ADC vs bin number. As in this figure, each such event can span multiple samples. However, to simplify the presentation of the algorithm we first assume that each ion impact can occupy only one sample on the trace. The algorithm is extended to the realistic case of events spanning multiple samples in the next section. All the experimental results presented in this paper also corresponds to this general case. However, we have not extended the theoretical results of our paper to the general case.
Define $\boldw \in \reals^n$ such that $\boldw[i]$ is the average number of ions that impact the detector at bin $i$ for a single scan. In each scan a large number of molecules of each species enter the instrument. However, each molecule has a slight chance of passing all the stages of the instrument and reaching the detector. Therefore, it is a natural choice to assume that the number of ions that impact the detector at time $i$ follows a Poisson distribution with mean $\boldw[i]$. Figure \[fig:epmf\_num\_impacts\] shows the estimated empirical probability mass function (EPMF) for the number of ions that impact the detector. Note that here we mentioned *estimated* EPMF since we do not directly observe the number of ion impacts. What we observe instead is the current at the output of the detector which has an arbitrary scaling. We estimate the number of ion impacts as follows. We start with ten thousands acquisition of the same sample and identify a set of *rare* ions as ions that are observed in more than $0.1\%$ but less than $1\%$ of acquisitions. These ions correspond to chemical species with low concentrations and have small probability of having multiple ion impacts in any acquisition. We take the median area under the pulse (weight) for these ions as the estimate for the weight of a single ion impact. We then normalize the weight of all events by this estimate and round it to the closest integer. Figure \[fig:epmf\_num\_impacts\] shows the estimated empirical probability mass function (EPMF) for the number of ions that impact the detector and its maximum likelihood Poisson fit.
This result indicates that a Poisson model for $\boldy$ is inadequate. In particular, a Poisson random variable, having its mean and variance tied together, cannot explain the observed variation in the tail. We assume that the detector response is additive for multiple concurrent impacts [@brock1998hadamard]. Furthermore, as suggested in [@wiza1979microchannel] we assume each ion impact generates a cumulative ADC count which itself is an exponentially distributed random variable. Therefore, conditioning on the event that $k$ ions impact the detector at a certain time the cumulative ADC count has an Erlang distribution with the shape parameter $k$. Figure \[fig:epdf\_event\_weight\] shows the empirical probability density function of normalized event weight and its maximum likelihood fit of the Poisson+Erlang model. Comparing Figs \[fig:epdf\_event\_weight\] and \[fig:epmf\_num\_impacts\] reveals that the Poisson+Erlang is a much more satisfactory model for this data.
![Estimated empirical probability mass function (EPMF) for the number of ions that impact the detector and the best (ML) Poisson fit for this data. See Section \[sec:generalization\] for details of estimating EPMF. This figure shows that a Poisson random variable is not adequate for modeling the number of impacts.[]{data-label="fig:epmf_num_impacts"}](EPMF_num_impacts_poiss_fit){width=".5\textwidth"}
![Empirical pdf of the normalized observed event weight (area under pulse) and a Poisson+Erlang model fitted to this data.[]{data-label="fig:epdf_event_weight"}](event_weight_density_model_v2){width=".5\textwidth"}
Let $\mu$ be the mean of the exponential random variable describing the detector response. Given $\boldw$ and $A$ the probability density function of $\boldy[t]$ is $$\begin{aligned}
\label{eq:density}
\bP(\boldy[t]|\boldw, A) &= e^{-\langle A_t, \boldw\rangle - w^0} \delta(\boldy[t]) \\
& \quad + \sum_{k=1}^{\infty}{E_{k,\mu}(\boldy[t]) P_{\langle A_t, \boldw\rangle + w^0}(k) },\end{aligned}$$ where $\delta(\cdot)$ is the Dirac delta representing a probability mass at zero and $E_{k,\mu}(\cdot)$ and $P_{\lambda}(\cdot)$ are the Erlang PDF and Poisson PMF defined as $$\begin{aligned}
E_{k,\mu}(y) = \frac{y^{k-1} \exp(-\frac{y}{\mu})}{\mu^{k}(k-1)\!\,!}, \\
P_{\lambda}(k) = \frac{\exp(-\lambda)\lambda^{k}}{k\!\,!}.\end{aligned}$$ $w^0 > 0 $ is a constant accounting for the spurious ion impacts observed at the detector.
Assuming that each ion impact is observed in only one sample of the trace, different trace samples are the result of different ion impacts. Hence, given $\boldw$ and $A$ the observed responses in different samples can be considered independent, namely $$\begin{aligned}
\bP(\boldy|\boldw, A) = \prod_{t=1}^T \bP(\boldy[t]|\boldw, A)
$$ Let $\mathbb{I}(\cdot)$ denote the indicator function, $\mathbf{1} \in \bR^T$ the vector of all ones, and define $\log(0)\mathbb{I}(\text{\small\sc FALSE}) = 0$. Hence the negative log-likelihood function takes the form $$\begin{aligned}
\label{eq:neg_log_likelihood}
&\ell(\boldw; \, \boldy, A) = \langle \mathbf{1}, A \boldw \rangle + N w^0 \nonumber \\
&- \sum_{t=1}^{T} \log\left(\sum_{k=1}^{\infty}
\frac{\boldy[t]^{k}(\langle A_t,\boldw\rangle + w^0)^{k}}{(k-1)\!\,! \, k\!\,! \,\mu^{k}} \right) \mathbb{I}(\boldy[t] > 0) \end{aligned}$$ for $\boldw \ge 0$, and infinity otherwise. Define $F \subseteq [T]$ as $F = \{t \in [T] |\, \boldy[t] > 0\}$. Alternatively, for $\boldw \ge 0$ the log-likelihood function can be written as $$\begin{aligned}
\label{eq:neg_log_likelihood_2}
& \ell(\boldw; \, \boldy, A) = N \| \boldw\|_1 \nonumber \\
& - \sum_{t\in F}\log\left\{\sqrt{\langle A_t,\boldw\rangle + w^0}\;
\bessel_1 (\frac{2}{\sqrt{\mu}}\sqrt{\boldy[t]\langle A_t,\boldw\rangle + w^0}) \right\},\end{aligned}$$ where $\bessel_1$ is the modified Bessel Function of the first kind, and we dropped the terms $\frac{1}{2} (\log(\boldy[t])- \log(\mu))$ and $N w^0$ which do not depend on $\boldw$.
We note that the negative log-likelihood function is strictly convex in $\boldw$ which is remarkable given the existence of the *hidden* variable $k$.
\[lem:neg\_log\_liklihood\_convex\] The function $\ell_{y, A}(\boldw)$ is strictly convex for $\boldw \ge 0$.
The proof is immediate using the following theorem due to Findling [@findling1995family].
[(Findling 95)]{} The function $x \bessel(x)$ is strictly log-concave on $\{x \in \reals \;|\; x >0 \}$
A simple transformation of the negative log-likelihood function is insightful. Let $ \lambda = 2N\mu, \quad \boldtw = \boldw/\mu$, then for $\boldtw \ge 0$, $$\begin{aligned}
\label{eq:regularized_NLL_decomposition}
\ell(\boldw; \, \boldy, A) = \widetilde{\ell}(\boldtw; y, A) + \lambda \|\boldtw\|_1,\end{aligned}$$ where $$\widetilde{\ell}(\boldtw; y, A) = - \underset{t \in F}{\sum}
{\log\left(\sqrt{y[t] \underset{i \in \partial t}{\sum} \boldtw_i} \;
\bessel_1 \left(\sqrt{y[t] \underset{i \in \partial t}{\sum} \boldtw_i}\right)\right) } .$$ A few remarks are in order. First, note that the scaling of the variable $\boldtw$ is irrelevant for our purpose and only the relative values are important. Hence, this is a single-parameter representation of the NLL function. This is of great importance for practical systems where optimal tuning of multiple parameters in different operational scenarios can be complicated and require additional expertise. Second, the single tuning parameter appears as the multiplicative factor in front of the $\ell_1$ regularization term. Given our intuition about the effect of $\ell_1$ regularization [@tibshirani1996regression; @zhao2006model] the parameter governs the sparsity of the estimate $\boldtw$, i.e., the number of species that appear in the output. In what follows we free the parameter $\lambda$ from our original interpretation of it as the product $2 N \mu$ and refer to it as the regularization parameter. Further, we define the regularized negative log-likelihood cost function $\mathcal{C}_{\lambda}(\boldw; \, \boldy, A)$ as $$\mathcal{C}_{\lambda}(\boldw; \, \boldy, A) = \tell(\boldw; y, A) + \lambda \|\boldw\|_1$$
Algorithm {#sec:algorithm}
=========
Given the regularized negative log-likelihood cost function $ \mathcal{C}_{\lambda}(\boldw; \, \boldy, A)$ our algorithm attempts to solve the following optimization problem. $$\label{eq:optimization_problem}
\underset{\boldw}{\text{minimize}} \quad \tell(\boldw; y, A) + \lambda \|\boldw\|_1 \, ,\;\;\;\;\;
\text{s.t.}\;\;\; \boldw \ge 0,
$$ We use the now standard method of iterative soft thresholding to solve this convex but non-differentiable optimization problem. For a doubly differentiable function $f: \mathcal{D} \subset \bR^n \rightarrow \bR$ let $\nabla f$ and $\nabla^2 f$ be the gradient and Hessian of $f$ respectively. Let $\gamma > 0$ be such that $\|\nabla^2 \tell(\boldw; \, \boldy, A)\|_2 < \gamma^{-1}$ for $\boldw \ge 0$. It is easy to see that $\gamma$ exists because of the presence of the chemical noise term $w^0$. We call the parameter $\gamma$ the step size as we use it to scale the steps the algorithm takes in each iteration. Let $\boldw^{(k)}$ be our estimate of $\boldw$ at step $k$. Then we can compute an upper bound for the cost function $\mathcal{C}_{\boldy, A, \lambda}(\boldw)$ as follows. $$\begin{aligned}
\mathcal{C}_{\lambda}(\boldw; \, \boldy, A) &\le
\tell(\boldw^{(k)}; \, \boldy, A) + \boldw^* \, \nabla
\tell(\boldw^{(k)}; \, \boldy, A) \nonumber\\
& + \gamma^{-1} \|\boldw - \boldw^{(k)}\|_2^2 + \|\boldw\|_1. \label{eq:cost_func_upper_bound}\end{aligned}$$ Equation provides an approximation for $\mathcal{C}_{\boldy, A, \lambda}(\boldw)$ when $\boldw$ is in a small neighborhood of $\boldw^{(k)}$. Minimizing the right-hand-side of Eq. with respect to $\boldw$ as a surrogate for the actual cost function results in $$\boldw^{(k+1)} = \eta_{\theta}\left( \boldw^{(k)} - \gamma \nabla \tell(\boldw^{(k)}; \, \boldy, A) \right).$$ where $\eta_{\theta}(\cdot)$ is the soft thresholding function, $\eta_{\theta}(x) = (|x|-\theta)_+$ with $(\cdot)_+$ being the positive part and $\theta \propto \lambda^{-1}$. Note that this is the *one-sided* soft thresholding function which differs from the two-sided soft thresholding function by mapping all negative values to zero. From Eq. , $\nabla \tell_{\lambda}(\boldw; \, \boldy, A)$ can be calculated as $$\begin{aligned}
& \nabla \tell_{\lambda}^*(\boldw; \, \boldy, A) = - \sum_{t\in F}\bigg( \frac{1}{2\langle A_t,\boldw\rangle} \nonumber \\
& + \frac{\bessel_0 \left( 2 \sqrt{\boldy[t] \langle A_t,\boldw\rangle}\right) +\bessel_2 \left( 2 \sqrt{ \boldy[t] \langle A_t,\boldw\rangle}\right)}{2 \sqrt{\boldy[t] \langle A_t,\boldw\rangle} \bessel_1 \left( 2 \sqrt{\boldy[t] \langle A_t,\boldw\rangle}\right)} \bigg) A_t.\end{aligned}$$ Given the gradient of the log-likelihood the algorithm is as follows.
**Algorithm**
-----------------------------------------------------------------------------------------------------------------------------------
**Input:** trace $\boldy$, firing times $\boldtau$, and constants ($\theta_0$, $\theta_1$, $\mu$)
**Output:** estimated spectrum $\widehat{\boldx}$
1:Calculate the adjacency matrix $A$ as in Eq. .
2:Set $\boldw^{(0)} = 0$, $\theta = \theta_0 + \theta_1$.
2:Repeat until stopping criterion is met
$\theta \leftarrow \theta_0 + \frac{1}{k^2} \theta_1$
$\boldw^{(k+1)} \leftarrow \eta_{\theta}\left( \boldw^{(k)} - \gamma \nabla \tell_{\lambda}(\boldw^{(k)}; \, \boldy, A) \right).$
3:Set $\widehat{\boldx} = 0$
4:For $t \in F$
$i_* = \underset{i \in \partial t}{\arg\max} \; \boldw[i]$
$\widehat{\boldx}[i_*] = \widehat{\boldx}[i_*] + \boldy[t]$
5:Return $\widehat{\boldx}$.
Step $4$ in the algorithm is worth noting. It was mentioned that each observed event $t$ has a set of possible bins on the spectrum it can be caused by, $\partial t$ (c.f., Fig. \[fig:bipartite\_graph\]). The problem of estimating the spectrum from the observed trace can be thought of as assigning each observed event to one of its neighbors which is the true cause of the event. This framing of the problem enables us to terminate the slow first order optimization method as soon as we are confident about the likely assignment of an event. In the generalized algorithm which is concerned with the case of multi-sample events this technique proves instrumental in decreasing the bias in the estimated spectrum.
Experimental Evaluation {#sec:simulation}
=======================
In this section we present performance evaluation results for the algorithm. We use a commercial TOFMS instrument for data collection and obtain $10,000$ scans for a high concentration multimode chemical sample using conventional TOFMS technique. Each scan is in length sampled at intervals. Therefore, in our notation $n=4\times 10^5$. The average of these ten thousand scans is considered the ground truth.
For evaluation, we use a random subset of $1,000$ scans and simulate using these scans as follows. First, we construct a vector of firing times, $\boldtau = (\tau_0, \tau_1, \dots, \tau_{N-1})$ by setting $\tau_0=0$ and choosing $\Delta \tau_i \equiv \tau_i-\tau_{i-1}$ uniformly at random from the interval $[\Delta\tau_{\min}, \Delta\tau_{\max}]$. Given $\boldtau$, the trace is constructed using unaliased scans as prescribed by Eq. .
![Preprocessing the data. There are four pulses that exceed the threshold $h_{w}$ from which three pass the minimum width condition $d_i \ge d_{\min}$ ($1$, $2$, and $4$). The markers at the $h_0$ level indicate the start and end of each marked event.[]{data-label="fig:thresholding_example"}](thresholding08-May-2012-15-54){width=".5\textwidth"}
![Illustration of true positive (TP), false positive (FP), and false negative (FN). []{data-label="fig:tp_fp_fn_illustration"}](TP_FP_FN){width=".4\textwidth"}
![False negative rate (dashed) and false discovery rate (solid) vs. iteration.[]{data-label="fig:fnr_fdr_vs_iteration"}](fnr_fdr_vs_iteration){width=".5\textwidth"}
![False negative rate vs. false discovery rate.[]{data-label="fig:fdr_fnr_iteration"}](fnr_fdr_iteration){width=".5\textwidth"}
In addition to the aliasing effect, the trace is corrupted by noise. Henceforth, we preprocess the trace before applying the reconstruction algorithm by setting the trace to zero unless it is likely to be the result of an ion impact. As mentioned before, the detector response to each ion impact is a bell shaped pulse which spreads across multiple samples. These pulses are corrupted by electrical noise that can be modeled as additive noise. However, the electrical noise level is significantly smaller in magnitude compared to the detector response to an ion impact and henceforth ion impacts can be marked with high confidence.
We select the potential ion impacts through the following procedure. Define three constants $h_0$, $h_w$, and $d_{\min}$ and label a pulse as a potential ion impact event if the width of the pulse at hight $h_{w}$ is greater than $d_{\min}$. If an event satisfies this criterion the support interval of the event is determined by thresholding the signal at level $h_0$. Figure \[fig:thresholding\_example\] demonstrates this procedure through an example. In this figure there are four pulses that exceed the threshold $h_{w}$ in peak magnitude. From these pulses $d_1$, $d_2$, and $d_4$ satisfy the minimum width condition of $d_i \ge d_{\min}$ at height $h_w$. Henceforth, after the preprocessing there are three valid events with the start and end times marked at level $h_0$. We set the trace equal to zero wherever it does not support a valid event. After the preprocessing step, the trace can be represented as a list of events whereby each event describes a single pulse in the trace. Note that observed traces are outputs of an ADC and have an arbitrary scaling. We keep this scaling but note that the absolute value of the amplitude of the trace and the corresponding parameters like $h_w$ and $h_0$ are irrelevant for our purposes.
Our procedure to identify valid events also enables us to define metrics for quantitative evaluation of different techniques. We take the true spectrum $\bar{\boldx}$ to be the average of all $10,000$ scans. Let $\hx$ be an estimate of $\bar{\boldx}$. For some constants $h_0$, $h_w$, and $d_{\min}$ define $\bcE = \{\bar{e}_1, \bar{e}_2, \dots, \bar{e}_{\bar{m}}\}$, and $\hcE = \{\he_1, \he_2, \dots, \he_{\widehat{m}}\}$ to be the set of events in $\bar{\boldx}$ and $\hx$ respectively, obtained through the procedure described above. For two events $\bar{e}_i$ and $\he_j$ we say $\he_j$ matches $\bar{e}_i$ if $\bar{e}_i$ overlap with at least $50\%$ of the width of $\he_j$. For $\bar{e}_i \in \bcE$ we say $\bar{e}_i$ is a false negative if none of the events in $\hcE$ matches $\bar{e}_i$. For $\he_j \in \hcE$ we say $\he_j$ is a true positive if there exist $\bar{e}_i \in \bcE$ such that $\he_j$ matches $\bar{e}_i$ and we say $\he_j$ is a false positive if it does not match any event in $\bcE$. See Fig \[fig:tp\_fp\_fn\_illustration\] for an illustration of these concepts.
Let [TP]{}, [FP]{}, and [FN]{} be the number of true positives, false positives and false negatives respectively. We consider false negative rate ([FNR = FN/(TP+FN)]{}), true positive rate ([TPR = TP/(TP+FN)]{}), and false discovery rate ([FDR = FP/(FP+TP)]{}) as the metrics of interest. Note that the notion of a true negative is ill-defined in this problem and hence we cannot use the false positive rate metric. In particular, observed pulses are of different width and shapes and they can overlap. Therefore, given an estimated spectrum the question *how many pulses are not observed?* is not well-posed.
![Sample reconstructed spectrum for the acceleration factor of $10$.[]{data-label="fig:sample_spectrum_atof_ctof_1"}](sample_spectrum_atof_vs_ctof_scale0_reducedsize){width=".5\textwidth"}
![Sample reconstructed spectrum for the acceleration factor of $10$. (different scale of Fig \[fig:sample\_spectrum\_atof\_ctof\_1\])[]{data-label="fig:sample_spectrum_atof_ctof_2"}](sample_spectrum_atof_vs_ctof_scale2){width=".5\textwidth"}
Unless otherwise stated, the parameters used to obtain the results of this section are as follows. $\mu = 225$, $\gamma = 2.5\times 10^{-3}$, $\theta_1 = 2 \times 10^{-2}$, $\theta_0 = 5 \times 10^{-4}$, $h_w^{(\text{trace})} = 2$, $h_w^{(\text{spectrum})} = 0.2$, $n = 4 \times 10^5$, $N = 10^3$, $ \Delta \tau_{\min} = 0$, $ \Delta \tau_{\max} = 2 \times 10^5$.
We also define the acceleration factor as the ratio $\equiv \frac{n}{\expect[\Delta \tau]}$. For example, $\mathbb{E}\Delta\tau = \frac{1}{4} n$ results in an acceleration factor of $4$, which means that is four times faster compared to conventional TOFMS in terms of the time it takes to collect the same number of scans. At the same time, each event has on average $4$ different positions on the spectrum it can be assigned to and the algorithm should be able to infer the correct position with satisfactory accuracy.
Figure \[fig:fnr\_fdr\_vs\_iteration\] shows the false negative rate and false discovery rate as a function of the iteration for the with acceleration factor ten. The algorithm starts with the all-zero spectrum. Hence, the false negative rate is equal to $1$ at iteration $0$ and as the algorithm proceeds the false negative rate decreases, converging to a final value of $0.47$. The false discovery rate on the other hand increases as the algorithm proceeds settling at a final value of about $0.085$. Inspecting Fig. \[fig:fnr\_fdr\_vs\_iteration\] suggest that the algorithm converges, in the sense of establishing the existence of ions, in about $15$ iterations.
Note that the large value of is by design. Firstly, by setting the $h_w$ threshold (c.f. Fig \[fig:thresholding\_example\]) very low we are requiring the algorithm to discover ions with diminutive abundance in the solutions that are observable in the ten thousand scans ground truth but very rarely appear in a random one thousand sample. Secondly, in a typical application of TOFMS declaring the presence of an ion that does not exist is considered a more costly mistake than missing an ion that is present. Hence, in line with these type of applications of TOFMS instruments we choose to operate in a high and low regime. Figure \[fig:fdr\_fnr\_iteration\] shows the same plot on the vs. plane. This figure shows how the algorithm converges as the number of iteration increases. Figures \[fig:sample\_spectrum\_atof\_ctof\_1\] and \[fig:sample\_spectrum\_atof\_ctof\_2\] show the ground truth (solid) and reconstructed (dashed) spectrums at two different scale. Visual inspection of the graphs indicates substantial match between the reconstructed spectrum and the ground truth.
[.48]{} {width="\textwidth"}\
[.48]{} {width="\textwidth"}\
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[.48]{} {width="\textwidth"}
The output of a TOFMS is usually used to generate a list of *peaks*, i.e., the list of MCRs that are deemed present by the instrument. Peak picking is an important task for a TOFMS instrument that can have significant effect on the instrument performance and the practice involves both publicly available methods as well as patents and trade secretes. These estimated peaks can be a few order of magnitudes more precise than the width of the pulses at the estimated spectrum.
Another more practically important but less transparent method for defining TP, FP, and FN is to use the list of peaks generated from the estimated spectrum. We use the peak picking software that ships with Agilent Technologies TOFMS. Since the estimated peaks are real-valued variables we also need to consider a precision. With a slight abuse of notation we define two peaks to match if they are within $\Delta m$ distance of each other on the MCR scale, i.e., $\sqrt{m_1/z_1} - \sqrt{m_2/z_2} \le \Delta m$.
Figure \[fig:tpr\_fdr\] shows the TPR vs FDR for different values of $\Delta m$ and number of peaks in the ground truth. Each plot contains four curves corresponding to with one hundred scans (-100), with one thousand scans and acceleration factor 10 (-1K), with one thousand scans (-1000), and with ten thousand scans and acceleration factor 10 (-10K). Note that (-100) and (-1K) have the same acquisition time. Similarly, (-1K) and (-10K) have the same acquisition time. For these plots we run the peack picking algorithm on the ground truth spectrum and obtain a list of picks. These picks are sorted based on their amplitude and we choose the $k$ most significant picks where $k \in \{400, 1000\}$. We then run the peak picking algorithm on the reconstructed spectrums and obtain the list of peaks. Similar to the process for the ground truth we keep the $k$ most significant peaks for each of these spectrums. A peak in an estimated spectrum is considered to match a peak in the ground truth if and only if their estimated MCRs are within $\Delta m$ of each other. For the top $400$ peaks and $\Delta m = 0.1$ (Fig. \[fig:tpr\_fdr\_top\_400\_deltam\_0100\], -$10$K achieves a nearly perfect reconstruction. TOF with $1$K scans (TOF-$1$K) demonstrates acceptable but significantly inferior performance compared to -$10$K This is while -10K and TOF-1K have the same acquisition time. -1K and TOF-1k have comparable performance for low TPR but TOF-1k outperforms -1K for high TPR. Note that -1K is ten times faster than -1K. -$100$ scans do not have enough information to achieve any significant accuracy in this regime. Figure \[fig:tpr\_fdr\_top\_400\_deltam\_0010\] shows the same curves when we decrease $\Delta m$ to $0.01$, i.e., when we adopt a more restrictive definition of two peaks matching. The overall trend is similar to that of Fig. \[fig:tpr\_fdr\_top\_400\_deltam\_0010\]. Figs \[fig:tpr\_fdr\_top\_1000\_deltam\_0100\] and \[fig:tpr\_fdr\_top\_1000\_deltam\_0010\] are similar but for the top one thousand significant peaks. The TPR and FDR degrade for all the cases since now we are expecting many more smaller peaks to be detected. However, the relative performance of different methods and configurations remain unchanged.
![Empirical CDF of the width (full width at half maximum) to intensity ratio for the output of (dashed) and ATOF (solid). ATOF results in little pulse broadening compared with with the same number of scans but $10$ times longer acquisition time.[]{data-label="fig:width_intesity_ratio"}](width_intensity_ratio){width=".5\textwidth"}
![True positive rate vs false discovery rate[]{data-label="fig:tpr_fdr_theta"}](agg_tpr_vs_fdr_atof_tof){width=".5\textwidth"}
The goal of a TOFMS instrument is to precisely measure the MCR of the present ions. Hence, for the instrument to be able to perform accurate peak picking it is important that does not significantly distort the shape of the pulses at the estimated spectrum. Figure \[fig:width\_intesity\_ratio\] shows, for and and for different number of scans and acceleration factor ten, the empirical CDF of the width to intensity (hight) ratio of the peaks (a measure of peak spikiness). As can be seen introduces little broadening in the peaks.
To better understand the achieved accuracy, we compare and three other cases in Fig. \[fig:tpr\_fdr\_theta\]. We run multiple randomized sample reconstruction and calculate bootstrap confidence intervals. The figure shows the true positive rate vs. false discovery rate, as parameterized by $\theta_0$ for the estimated spectrum $\widehat{\boldx}_{\newtof{}}$ obtained by the with one thousand scans and acceleration factor 4. Each data point is obtained dividing the ten thousand scans into ten buckets of one thousand scans each. The trace is constructed using one thousand scans from one bucket. The average of the other nine buckets is considered as the ground truth. Error bars indicate $2\hat{\sigma}$ confidence intervals where $\hat{\sigma}$ is the standard error estimate.
We Also plot the corresponding curves for three other cases. The red curve corresponds to the *naïve* , obtained by mapping each event to all possible positions on the spectrum, $$\widehat{\boldx}_{N}[i] = \sum_{t \in F}\frac{1}{\textrm{deg}_t}A_{tj} \boldy[t],$$ where $A_{tj}$ is the adjacency matrix defined in Eq. and $\textrm{deg}_t = \sum_{j=1}^n A_{tj}$ is the number of positions on the spectrum that event $t$ can be mapped to. The naïve uses the simplest way of processing the trace where one assumes that each event is equally likely to be caused by an ion from any of its potential locations on the spectrum.
The green (TOF-1K) and blue (TOF-250) curves correspond to conventional TOFMS, i.e., the spectrum obtained by averaging the scans when there is no overlapping, $$\widehat{\boldx}_{\rm ave} = \frac{1}{N} \sum_{l=1}^{N} \boldx^{(l)}.$$ In TOF-250, the number of scans is chosen such that the time which takes to perform the TOFMS and are the same. TOF-1K corresponds to TOFMS with the same number of scans as , which has an acquisition time $4$ times longer compared to the . Equivalently, one can think of TOF-1K corresponding to the case where an oracle is available for experiment that to each event assigns its true position on the spectrum.
The naïve and conventional TOFMS curves are parameterized by the thresholding parameter $h_w$ which is used to identify the events in the estimated spectrum. Similar to $\theta_0$, $h_w$ enables us to obtain a trade off between true positive rate and false discovery rate.
This comparison shows that significantly outperforms the conventional TOFMS. One the one hand, it allows for a four-fold speed-up with essentially unchanged accuracy (comparison with TOF-1K). On the other, it allows a two-fold increase in true positive rate for ${\rm FDR}= 0.2$ if the experiment duration is unchanged (comparison with TOF-250).
Events that span more than one sample {#sec:generalization}
=====================================
In order to simplify the presentation of the main ideas in the previous sections we assumed that all ion impacts are confined to one sample in the observed trace. However, as was evident from Fig. \[fig:sample\_event\] this assumption is far from being true. In this section we show how the algorithm presented in Section \[sec:algorithm\] can be extended to the case where each ion impact event can spread across multiple samples in the trace.
Consider the conditional probability of an observation $\boldy[t]$ from Eq. which is repeated below for convenience. $$\begin{aligned}
\label{eq:density2}
\bP(\boldy[a]|\boldw, A) &= e^{-\langle A_a, \boldw\rangle - w^0} \delta(\boldy[a]) \nonumber\\
& \quad + \sum_{k=1}^{\infty}{E_{k,\mu}(\boldy[a]) P_{\langle A_a, \boldw\rangle + w^0}(k) }.\end{aligned}$$ Assume instead of being confined to one sample the ion impact event $a$ is spread from time $\ut_a$ to $\ot_a$. Define the weight of ion impact $a$, $z_a$, as $$\label{eq:impact_weight}
z_a = \sum_{t = \ut_a}^{\ot_a} \boldy[t].$$ Recall that $\boldw[i] + w^0$ represents the arrival rate of ions at bin $i$ for a single scan, i.e., $\boldw[i] + w^0$ is the expected number of ions that arrive at bin $i$ in a single scan. Henceforth, assuming that ions arrive independent of each other $\langle A_t, \boldw + w^0 \mathbf{1}\rangle$ is the cumulative mean of the number of ions that arrive at time $t$ where $\mathbf{1}$ is the vector of all ones. Given the assumption that an ion impact being confined to only one sample we used $\langle A_t, \boldw + w^0 \mathbf{1}\rangle$ as the expected number of ions that caused the observation $\boldy[t]$. Here, we make the alternative assumption that for an ion impact observation spanning the interval $[\ut_a, \ot_a]$ the expected number of ions responsible, $K$, is given by $$\label{eq:cumulative_rate}
\expect[K] = \sum_{t = \ut_a}^{\ot_a} \langle A_t, \boldw + w^0\rangle$$ Let the generalized neighborhood of an event be $$\begin{aligned}
\label{eq: gen_neighbor_def}
\partial a = \{i \in [n] \,|\, \exists \, t \in [\ut_a, \ot_a] \; \text{s.t.} \; A_{ti} > 0 \}.\end{aligned}$$ Then, Eq. \[eq:cumulative\_rate\] can be written as $$\label{eq:cumulative_rate_2}
\expect[K] = \sum_{i \in \partial a}(\boldw[i] + w^0).$$ In other words, we are assuming that an observed event can be caused by one or multiple ions impacting at any time during the span of the event.
Similar to Section \[sec:data\_model\] we assume that $z$, the weight of an ion impact, is distributed as an Erlang random variable with shape parameter $K$ where $K$ is a Poisson random variable with mean given in Eq. . Let $a$ be an ion impact event with weight $z_a$ and expected number of ions $\sum_{i \in \partial a}\boldw[i]$. Then, similar to Eq. the probability of observing $a$ given $\boldw$ and $A$ can be written as $$\begin{aligned}
&\bP(a|\boldw, A) = \exp\left\{-\sum_{i \in \partial a}(\boldw[i] +w^0) \right\} \delta(z_a) \nonumber\\
& \quad + \exp\left\{-\frac{z_a}{\mu}-\sum_{i \in \partial a}(\boldw[i] + w^0)\right\} \, \nonumber \\
& \qquad \qquad \quad \sum_{k=1}^{\infty}{\frac{z_a^{k-1}}{\mu^{k}(k-1)\! \,!k\!\,!} \, (\sum_{i \in \partial a}\boldw[i] + w^0 )^{k} }.
\label{eq:density_impact_weight}\end{aligned}$$ For a given trace $\boldy$, let $F$ be the set of observed events. Similar to section \[sec:data\_model\] assume that given $\boldw$ and $A$ distinct events (ion impacts) are independent. Then the probability of observing the set of non-zero-weight events $F$ can be written as $$\begin{aligned}
\label{eq:density_impact_weight_joint}
\bP(F|\boldw, A) &= \prod_{a \in F} \bigg[ \exp\left\{-\frac{z_a}{\mu}-\sum_{i \in \partial a} (\boldw[i]+w^0) \right\} \nonumber \\
& \qquad \qquad \sum_{k=1}^{\infty}{\frac{z_a^{k-1} }{\mu^{k} {(k-1)\!\,!} {k\!\,!} } \, \, (\sum_{i \in \partial a}\boldw[i]+w^0)^{k} } \bigg].\end{aligned}$$ Note that here we dropped the first term in Eq since for an observed ion impact event the weight $z_a$ is always positive and the first term vanishes.
The next step is to write the negative log-likelihood function. There is a subtle point to be noted here. $F$ is the set of ion impact events with non-zero weights. However, $z_a$ can be zero while $\sum_{i \in \partial a} \boldw[i] > 0$. We refer to such observations as zero-weight observations. Zero-weight observations are informative and should be included in the log-likelihood function. In our model the conditional probability of observing a zero-weight event $a$ in the interval $[\ut_a, \ot_a]$ is given by $$\begin{aligned}
\label{eq:density_zero_weight}
\bP(a|\boldw, A) &= \exp\left\{- \sum_{i \in \partial a} (\boldw[i]+w^0) \right\} \nonumber \\
& = \exp\left\{- \sum_{t = \ut_a}^{\ot_a} \langle A_t, (\boldw+w^0)\rangle \right\}.\end{aligned}$$ The difficulty that seems to exist here is how to identify the zero-weight ion impact events when they can span more than one sample. However, as we shall see shortly, the particular form of the log-likelihood function for the zero-weight events eliminates the need to distinguish between adjacent zero-weight events for calculating the log-likelihood function.
Let $F_0$ be the set of zero-weight observations and define $U_0 = \cup_{a \in F_0} [\ut_a, \ot_a]$, i.e., the union of all time intervals we observed zero-weight events at. Again, making the assumption that distinct zero-weight ion impacts are independent events given $\boldw$ and $A$ we can write the joint probability of observing $F_0$ as $$\begin{aligned}
\label{eq:density_zero_weight_joint}
\bP(F_0|\boldw, A) &= \prod_{a \in F_0} \exp\left\{- \sum_{i \in \partial a} \boldw[i] \right\} \nonumber \\
&= \prod_{a \in F_0} \exp\left\{- \sum_{t = \ut_a}^{\ot_a} \langle A_t, \boldw\rangle \right\} \nonumber \\
&= \exp(- \sum_{t \in U_0} \langle A_t, \boldw\rangle).\end{aligned}$$ Note that $\bP(F_0|\boldw, A)$ does not depend on the number of zero-weight events or the beginning or end time of a particular event. We need only to identify all the samples that are part of a zero-weight event, i.e., not part of any observed ion impact event.
Putting Equations and together and assuming that all the events are independent we have the probability of observing a trace $\boldy$ as $$\begin{aligned}
\label{eq:generalized_prob_trace}
& \bP(\boldy|\boldw, A) = \bP(F|\boldw, A) \bP(F_0|\boldw, A) \nonumber \\
& \quad = \exp\left\{- \sum_{a \in F_0} \sum_{i \in \partial a} (\boldw[i] + w^0) \right\} \nonumber \\
& \qquad \exp\left\{ - \sum_{a \in F}\sum_{i \in \partial a}(\boldw[i]+w^0) - \frac{z_a}{\mu} \right\} \nonumber \\
& \qquad \prod_{a \in F} \bigg[ \sum_{k=1}^{\infty}{\frac{z_a^{k-1} }{\mu^{k} {(k-1)\!\,!} {k\!\,!} } \, \, \left(\sum_{i \in \partial a}(\boldw[i] + w^0)\right)^{k} } \bigg] \nonumber \\
& \quad = \exp\left\{- N \|\boldw\|_1 -\sum_{a \in F} \frac{z_a}{\mu} \right\} \nonumber \\
& \qquad\qquad \prod_{a \in F} \left[ \, \sum_{k=1}^{\infty}{\frac{z_a^{k-1} }{\mu^{k} {(k-1)\!\,!} {k\!\,!} } \, \left(\sum_{i \in \partial a}(\boldw[i] + w^0)\right)^{k} } \right],\end{aligned}$$ where the last equality is obtained since each sample on the trace is presented in either $F$ or $F_0$. The negative log-likelihood function then can be written as $$\begin{aligned}
\label{eq:generalized_neg_log_likelihood}
& \ell(\boldw| \boldy, A) = - N \|\boldw\|_1 \nonumber \\
& \qquad - \sum_{a \in F} \log \left[ \sum_{k=1}^{\infty}{\frac{z_a^{k} }{\mu^{k} {(k-1)\!\,!} {k\!\,!} } \, \left( \sum_{i \in \partial a}(\boldw[i] + w^0)\right)^{k} } \right]\end{aligned}$$ After some algebra, and keeping only the terms that depend on $\boldw$, $$\begin{aligned}
\ell(\boldw| \boldy, A) = & - N \|\boldw\|_1 \nonumber \\
& - \sum_{a \in F} \bigg[ \frac{1}{2} \log\left(\sum_{i \in \partial a}\frac{(\boldw[i] + w^0)}{\mu} \right) \nonumber \\
&+ \log \bessel_1 \left( 2 \sqrt{z_a \sum_{i \in \partial a} \frac{(\boldw[i] + w^0)}{\mu }} \right) \bigg],\end{aligned}$$ Using the same change of variable as before, namely $\boldtw = \frac{1}{\mu}\boldw$, $\lambda = N \mu$ and $\tell(\boldtw| \boldy, A) = \ell(\boldw| \boldy, A) - N \|\boldw\|_1$ we have $\ell(\boldw| \boldy, A) = \tell(\boldtw| \boldy, A) + \lambda \|\boldtw\|_1$ where $$\begin{aligned}
\tell(\boldtw| \boldy, A) = & - \sum_{a \in F} \bigg[ \frac{1}{2} \log\left(\sum_{i \in \partial a}(\boldtw[i] + w^0) \right) \nonumber \\
&+ \log \bessel_1 \left( 2 \sqrt{z_a \sum_{i \in \partial a} (\boldtw[i] + w^0)} \right) \bigg],\end{aligned}$$ And the gradient of the function $ \tell(\boldw| \boldy, A) $ is $$\begin{aligned}
\label{eq:generalized_neg_log_likelihood_gradient}
&\nabla \tell(\boldw| \boldy, A) = - \sum_{a \in F} \sum_{t = \ut_a}^{\ot_a} A_t \Bigg[ \frac{1}{2\sum_{i \in \partial a}(\boldw[i] + w^0)} \nonumber \\
& \quad + \left(\bessel_0 ( 2 \sqrt{z_a \sum_{i \in \partial a}(\boldw[i] + w^0) } ) + \bessel_2 ( 2 \sqrt{z_a\sum_{i \in \partial a}(\boldw[i] + w^0) } )\right) \nonumber \\
& \quad \left(2 \sqrt{z_a \sum_{i \in \partial a}(\boldw[i] + w^0)} \bessel_1 ( 2 \sqrt{z_a \sum_{i \in \partial a}(\boldw[i] + w^0) } )\right)^{-1} \Bigg] \end{aligned}$$ Having the gradient of the log-likelihood function the algorithm is similar to the algorithm of section \[sec:algorithm\] with the gradient of the log-likelihood calculated using Eq. . However, we need some additional notations to represent the generalized algorithm. Let $\deg(a)$ be the number of neighbors of event $a$, i.e., $\deg(a) = \sum_{i = 1}^{n} A_{\ut_a,i}$ [^3]. For $j \in [\deg(a)]$ let $\ui_a^j$ be the index of the $j^{\rm th}$ non-zero element of $A_{\ut_a}$. Similarly, let $\oi_a^j$ be the index of the $j^{\rm th}$ non-zero element of $A_{\ot_a}$. Then, $[\ui_a^j, \oi_a^j]$ is the true position of event $a$ on the spectrum if its $j^{th}$ neighbor corresponds to the true scan that caused event $a$.
The peak picking algorithms are usually sensitive to the shape of the pulses. Furthermore, the time of arrival of the ions are noisier than the observation error of the instrument. Observing many scans enables the instrument to measure the MCR of the ions with precision significantly better than the arrival noise level. To overcome the issue of a possible bias in the estimated MCR in our model, we employ one last trick. The algorithm constructs an estimate of the spectrum $\widehat{\boldx}$ by assigning each observed event to its most likely neighbor (c.f. Fig. \[fig:bipartite\_graph\]). In other words, let $$j_* = \underset{j \in [\deg(a)]}{\arg\max} \; \sum_{i = \ui_a^j}^{\oi_a^j} \boldw[i].$$ Then, given $\boldw$ we reconstruct the estimated spectrum as $$\widehat{\boldx}[\ui_a^{j^*} + \Delta] = \widehat{\boldx}[\ui_a^{j^*} + \Delta] + \boldy[t + \Delta].$$ Using this notation the algorithm as as follows.
**Algorithm**
-------------------------------------------------------------------------------------------------------------------------
**Input:** trace $\boldy$, firing times $\boldtau$, and constants ($\theta_0$, $\theta_1$, $\lambda$, $\gamma$)
**Output:** estimated spectrum $\widehat{\boldx}$
1:Calculate the adjacency matrix $A$ as in Eq. .
2:Set $\boldw^{(0)} = 0$, $\theta = \theta_0 + \theta_1$.
2:Repeat until stopping criterion is met:
$\theta \leftarrow \theta_0 + \frac{1}{k^2} \theta_1$
$\boldw^{(k+1)} \leftarrow \eta_{\theta}\left( \boldw^{(k)} - \gamma \nabla \tell(\boldw^{(k)}; \, \boldy, A) \right).$
3:Set $\widehat{\boldx} = 0$
4:For $a \in F$:
$j^* = \underset{j \in [\deg(a)]}{\arg\max} \; \sum_{i = \ui_a^j}^{\oi_a^j} \boldw[i]$
For $\Delta = 0, \dots, (\ot_a - \ut_a)$:
$\widehat{\boldx}[\ui_a^{j^*} + \Delta] = \widehat{\boldx}[\ui_a^{j^*} + \Delta] + \boldy[t + \Delta]$
5:Return $\widehat{\boldx}$.
[Morteza Ibrahimi]{} Morteza Ibrahimi received his PhD in Electrical Engineering from Stanford University in 2013, working with professor Andrea Montanari. His research interests are in high dimensional statistical signal processing, learning graphical models, optimization through message passing algorithms, and statistical inference with high dimensional data, specially through fast iterative algorithms. He received his BS from Sharif University of Technology in 2006, and MSc from University of Toronto in 2007.
[Andrea Montanari]{} Andrea Montanari received a Laurea degree in Physics in 1997, and a Ph. D. in Theoretical Physics in 2001 (both from Scuola Normale Superiore in Pisa, Italy). He has been post-doctoral fellow at Laboratoire de Physique Théorique de l’Ecole Normale Supérieure (LPTENS), Paris, France, and the Mathematical Sciences Research Institute, Berkeley, USA. Since 2002 he is Chargé de Recherche (with Centre National de la Recherche Scientifique, CNRS) at LPTENS. In September 2006 he joined Stanford University as a faculty, and since 2010 he is Associate Professor in the Departments of Electrical Engineering and Statistics. He was co-awarded the ACM SIGMETRICS best paper award in 2008. He received the CNRS bronze medal for theoretical physics in 2006 and the National Science Foundation CAREER award in 2008.
[George Moore]{} Goerge S. Moore is a research fellow at Agilent technology. He received his PhD in 1980 from Purdue University and his Bsc/Msc from Mississippi State University in 1974 all in Electrical Engineering. He has over 40 years of experience in the industry.
[^1]: M. Ibrahimi is with the Department of Electrical Engineering, Stanford University, Stanford, CA, 94305 USA email: ibrahimi@stanford.edu.
[^2]: A. Montanari is with the Department of Electrical Engineering and Department of Statistics, Stanford University, Stanford, CA, 94305 USA email: montanari@stanford.edu.
[^3]: This definition is slightly inaccurate since an event can potentially fall on the boundary of an scan resulting in $\sum_{i = 1}^{n} A_{\ut_a,i} \neq \sum_{i = 1}^{n} A_{\ot_a,i}$ but this is rare and the discrepancy is negligible.
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abstract: 'Selective control of qubits in a quantum register for the purposes of quantum information processing represents a critical challenge for dense spin ensembles in solid state systems. Here we present a protocol that achieves a complete set of selective electron-nuclear gates and single nuclear rotations in such an ensemble in diamond facilitated by a nearby NV center. The protocol suppresses internuclear interactions as well as unwanted coupling between the NV center and other spins of the ensemble to achieve quantum gate fidelities well exceeding 99%. Notably, our method can be applied to weakly coupled, distant, spins representing a scalable procedure that exploits the exceptional properties of nuclear spins in diamond as robust quantum memories.'
author:
- 'J. Casanova'
- 'Z.-Y. Wang'
- 'M. B. Plenio'
title: 'Noise-resilient Quantum Computing with a Nitrogen-Vacancy Center and Nuclear Spins'
---
[*Introduction –*]{} Quantum computing and quantum simulation hold the promise for tackling computational problems that are currently out of the reach of classical devices [@Feynman82; @Lloyd96]. With these applications in mind a wide variety of possible platforms have been proposed and realized which include trapped ions [@LeibfriedEtAl], superconducting circuits [@Devoret13], optical lattices [@Bloch05], coupled cavity arrays [@HartmannBP08], integrated photonics [@Obrien09], and hybrid systems involving nuclear spins and nitrogen vacancy (NV) centers [@Doherty13]. However, while the storage and processing of information by using quantum degrees of freedom promises to enhance our computational capabilities, the available quantum-bits (qubits) are fragile and strongly sensitive to environmental fluctuations.
Nuclear spin clusters in materials such as diamond have been identified as promising candidates for robust solid-state quantum memories because of their long coherence times and the potentially large number of available spins [@Doherty13]. Nuclear spins can be initialized, controlled, and read out for quantum information processing and sensing purposes with an NV center driven by optical fields and microwave radiation [@GruberDT+1997; @GaebelDP+2006; @Gurudev07; @Neuman10; @Robledo11; @vanderSar12; @Kolkowitz12; @Taminiau12; @Liu13; @Taminiau14; @WaldherrWZ+2014]. However quantum computing requires the precise manipulation of the information encoded in each qubit which becomes a delicate issue in samples with dense resonance spectra. Additionally, while nuclear spins can be effectively isolated from other spins [@Maurer12], to combine this protection with a sequential generation of a complete set of quantum gates on specific nuclei remains as a challenging task. Overcoming these issues would enable us to realize circuit-based algorithms in highly polarized nuclear registers [@Nielsen] as well as alternative models such as DQC1 computing that do not require initial nuclear polarization [@KnillL1998; @ParkerP2000].
In this Letter we present a protocol that combines the advantages of recently developed dynamical decoupling protocols [@Casanova15; @Wang15] for the suppression of both electronic decoherence and internuclear interactions with the robust and selective implementation of quantum gates. Following a description of the technical details of our method that incorporates the combined action of microwave and radio frequency fields, we show the existence of a low-energy branch that is useful for individual qubit coherent control. Finally we proceed to demonstrate with detailed numerical simulations that our scheme allows for protected single-qubit rotations and two-qubit gates between an NV center and weakly coupled $^{13}$C-nuclei and can achieve fidelities above 99% [@Fowler12]. Additionally, and although we use NV centers on diamond as the model system, our method is equally applicable to other platforms as the case of phosphorus in silicon, or silicon carbide.
[*Formalism –*]{} Let us consider an NV center in a nuclear spin bath where a static magnetic field $B_z$ is applied along the NV axis (the $\hat{z}$ axis). Microwave and rf fields are used for external control over the electron and nuclear spins as well as for achieving internuclear decoupling. The Hamiltonian that describes this situation reads ($\hbar = 1$) $$\label{model}
H = DS_{z}^{2} - \gamma_{e}B_{z}S_{z} - \sum_{j} \gamma_{n} B_{z} \ I_{j}^{z} +
S_{z}\sum_{j}\vec{A}_{j}\cdot\vec{I}_j + H_{\rm nn} + H_{\rm c}.$$ Here $\gamma_{e}$ ($\gamma_{n}$) is the electronic (nuclear) gyromagnetic ratio, $H_c$ describes the action of microwave and rf control fields (detailed below), and $H_{\rm nn}$ accounts for the internuclear coupling [@Supplemental]. The interaction between the NV center and the $j$-th nucleus is mediated by the hyperfine vector $\vec{A}_{j}$. Note that due to the large zero field splitting $D=2\pi\times2.87$ GHz, we have eliminated non-secular components in Eq. (\[model\]). The microwave field is tuned to address specific nuclear spins thanks to a tailored sequence of $\pi$-pulses, the AXY-$8$ sequence [@Casanova15; @Wang15]. The rf-field contains the decoupling field $\vec{B}_{\rm d} = B_{\rm d} \cos{(\omega_{\rm rf} t)} \ \hat{n}$ and the control field $\vec{B}_{\rm c} = B_{\rm c} \cos{(\omega_{\rm c} t + \phi_c)} \ \hat{n}_c$. Additionally and because of recently developed $3$D positioning methods [@Wang15] we can assume $\vec{A}_j$, $\hat{n}$, and $\hat{n}_c$ to be known quantities.
The decoupling field $\vec{B}_{\rm d}$ gives rise to the appearance of different branches of resonance frequencies of the nuclear spins that can be exploited both, to obtain entangling interactions between the electron spin and some specific nuclei as well as single qubit operations, see Eq. $(21)$ of [@Supplemental] for a view of the resonances map. These branches are $\omega_j$, $\omega_{\rm rf} \pm \omega_j$, $3\omega_{\rm rf} \pm \omega_j$, and $\omega_{\rm rf}$ where $j$ labels each nucleus and $$\label{resonances}
\omega_j = |\Delta| \sqrt{ \bigg[\delta + \frac{m_s}{2 \Delta} \tilde{n}_z \ \tilde{n}_x \ A_j^z \bigg]^2
+ \bigg[\frac{\delta}{\sqrt{2}} + \frac{m_s}{2 \Delta} \ \tilde{n}^2_z \ A_j^z \bigg]^2},$$ with $A_j^z = \vec{A}_j\cdot\hat{z}$, $\delta = \tilde{n}_x + \frac{\Omega_x}{2 \Delta} (1 - \tilde{n}_x^2 - \tilde{n}_z)$, $\Omega_x = \Omega \ n_x$, $n_x$ being the $x$-component of the decoupling-field vector $\hat{n}$, $\Omega = \frac{\gamma_n B_d}{2}$, and $m_s = \pm 1$. The quantities $\tilde{n}_x$ and $\tilde{n}_z$ accounts for the $x$- and $z$-components of $\tilde{n} = (\frac{\Omega_x}{\tilde\omega}, 0, -\frac{\omega
+ \omega_{\rm rf}}{\tilde\omega})$, while $\omega = \gamma _n B_z$, $\tilde{\omega} = \sqrt{(\omega + \omega_{\rm rf})^2 + \Omega_x^2 }$, and $\Delta = \tilde{\omega} - 2 \omega_{\rm rf}$. We consider $\omega_{\rm rf} \gg \max_j
\omega_j$, a condition that can be achieved with the application of moderate magnetic fields which, for the parameter regime considered in this work, need to satisfy $B_z > 0.1$ T.
Restricted to the low-energy resonance branch, i.e. to the set of frequencies $\{\omega_j\}$, the effective Hamiltonian after eliminating fast rotating terms reads [@Supplemental] $$\begin{aligned}
\label{effective}
H &=& \frac{m_s}{2} F(t) \ \sigma_z \sum_j g_j \bigg[I_j^x \cos{(\omega_j t)} - I_j^y \sin{(\omega_j t)} \bigg]\nonumber\\
&+& \Omega(t) \sum_j \vec{I}_j \cdot \bigg[ |\vec{\alpha}_j| \ \hat{\alpha}_j \cos{(\omega_j t)} - |\vec{\beta}_j| \
\hat{\beta}_j \sin{(\omega_j t)} \bigg].\end{aligned}$$ $F(t) = \pm 1$ is the modulation function, $\Omega(t) = 2\lambda\cos{(\omega_c t +\phi_c)}$ and $g_j = \big| A^z_j \ \tilde{n}_z \sin{(\theta_{\tilde{n}, \hat{n}_j})} \big|$ with $\theta_{\tilde{n},
\hat{n}_j}$ being the angle between the vectors $\tilde{n}$ and $\hat{n}_j$ with $\hat{n}_j =
\left( \Delta\delta + m_s/2 \tilde{n}_z \ \tilde{n}_x \ A_j^z, 0, \Delta\delta/\sqrt{2} + m_s/2 \
\tilde{n}^2_z \ A_j^z \right)/\omega_j.$ The nuclear quantization axes are $I_j^x = \vec{I}_j \cdot \hat{x}_j,
\ I_j^y = \vec{I}_j \cdot \hat{y}_j, \ I_j^z = \vec{I}_j \cdot \hat{z}_j,$ where $\vec{x}_j = (\vec{\gamma}_{3,j}
- \vec{\gamma}_{3,j}\cdot \hat{n}_j \ \hat{n}_j)$, $\vec{y}_j=(\hat{n}_j \times \vec{\gamma}_{3,j})$, $\vec{z}_j=\hat{n}_j $, $\vec{\gamma}_{3,j} = (\vec{A}_j \cdot \hat{z})(\hat{z} \cdot \tilde{n})\tilde{n}$, and the vectors $\vec{\alpha}_j$, $\vec{\beta}_j$ depend on the specific orientation of $\hat{n}_c$ with respect to the basis $\hat{x}_j$, $\hat{y}_j$, $\hat{z}_j$ [@Supplemental].
 Coherence ($L$) evolution as a consequence of the entangling gates mediated by $H^{s}$ a), and by $H^{a}$ b) for two different nuclei under the action of a decoupling field $\vec{B}_{\rm d}$ with $\Delta = 2\pi\times100$ kHz. $\rho \propto |+\rangle\langle +| \otimes I$ is the initial electronic-nuclear state, with $I$ the identity operator (we assume the nucleus in a thermal state). In all cases $6000$ imperfect decoupling pulses have been applied giving rise to an evolution time of a) $t \approx 3.4$ ms, and b) $t \approx 3.5$ ms.](tunning){width="1\columnwidth"}
The contribution of $H_{\rm nn}$ is suppressed in Eq. (\[effective\]) allowing to implement quantum operations between the electron spin and different nuclear spins without the interference of internuclear interactions. As shown in [@Supplemental] this suppression holds if $$\begin{aligned}
{\rm max}_j \ | A_j^z| &\ll& |2 \Delta|, \label{f} \\
\tilde{n}_x + \frac{\Omega_x}{2 \Delta} (1 - \tilde{n}_x^2 - \tilde{n}_z) &=& \sqrt{2} \
\big[ \tilde{n}_z + \frac{\Omega_x}{2 \Delta}\tilde{n}_x (1 - \tilde{n}_z) ],\label{s}\\
\big|\frac{\mu_{0}\gamma^2_{n}}{2|\vec{r}_{j,k}|^{3}}\big| &\ll& |\omega_j| \label{t},\end{aligned}$$ where $|\vec{r}_{j,k}|$ is the distance between nuclei $j$ and $k$. Eq. (\[f\]) accounts for the validity of the magic angle condition according to the Lee-Goldburg decoupling [@Lee65; @CaiRJ+2013] for all nuclei and, if satisfied, gives rise to Eq. (\[s\]). The latter corresponds to the magic angle relation in absence of hyperfine fields. Finally Eq. (\[t\]) assures that $H_{\rm nn}$ is eliminated up to a factor $ |\frac{A^z_j}{2\Delta} g_{j,k}|$ for each internuclear interaction [@Supplemental]. Here $g_{j,k}=\frac{\mu_{0}}{4}
\frac{\gamma_{n}^2}{r_{j,k}^{3}} \left[1 - 3 (n_{j,k}^z)^2 \right]$ are the internuclear coupling coefficients, $ r_{j k}$ the distance between the $j$-th and $k$-th nuclei, and $n^z_{j,k}$ the z-component of the unit vector $\vec{r}_{j,k}/r_{j,k}$. In our simulations we consider $\Delta=2\pi\times 100$ kHz which according to Eq. (\[s\]) result in $\Omega_x \approx \sqrt{2} \Delta$. Note that by using external coils rf fields of $\sim 0.1$ T have been demonstrated [@Michal08], and when applied to the case of $^{13}$C nuclei generate $\Omega_x > 2\pi\times 100$ kHz.
[*Gate performance–*]{} The application of AXY sequences provides us with a robust procedure to entangle the electron spin and each nuclei [@Casanova15]. However, in order to selectively couple the electron spin with the $I_l^x$ or $I_l^y$ operators we need to choose time-symmetric or antisymmetric pulse sequences, i.e. a modulation function $F(t)$ as $F^s(t) = \sum_{k >0} f^{s}_k \cos{(k\omega t)}$ ($F^a(t) = \sum_{k >0} f^{a}_k \sin{(k\omega t)}$) for the symmetric (anti-symmetric) case [@Supplemental], where, importantly for the following, $f^{s}_k$ and $f^{a}_k$ can be continuously tuned [@Casanova15; @Wang15]. In the absence of rf-control, $\lambda = 0$, once the symmetric (anti-symmetric) sequence is applied and fast rotating terms are eliminated, which requires that the condition $\left| \frac{f_{\tilde{k}}^{e/o} g_j}{4 (\omega_l - \omega_j)}\right| \ll 1$ $\forall j \neq l$ holds, see Eq. (\[effective\]) (note that this is always achievable because of the tunable $f_{\tilde{k}}^{e/o}$) we find $H^{s} = \frac{m_s}{4} f^{s}_{\tilde{k}} g_l
\sigma_z I_l^x$ ($H^{a} = -\frac{m_s}{4} f^{a}_{\tilde{k}} g_l \sigma_z I_l^y$) as the effective Hamiltonian from Eq. (\[effective\]). In both cases we assume $\tilde{k} \omega = \omega_l$ with $\tilde{k}\in\mathbb{N}$ and $\omega_l$ the resonance frequency of the $l$-th nucleus. Note that the resonances $\omega_l$ are independent of the magnitude of the $B_z$ field which allows to work in a wide variety of regimes including the case of high magnetic fields for better elimination of different rotating terms. When $\vec{B}_c \neq 0$ and $\omega_c = \omega_l$ we can add to $H^{s/a}$ a single-qubit term proportional to $I_l^x$ or $I_l^y$ by choosing the phase $\phi_c$ in $\Omega(t)$ as $\phi_c = 0$ and $\phi_c =
\frac{\pi}{2}$ respectively. Finally, when the NV center is driven outside of the resonance band $\{\omega_j\}$ we can eliminate the first line in Eq. (\[effective\]) achieving individual spin rotations. Hence our method provides a universal set of quantum gates.
The time-evolution operators associated to $H^{s/a}$ read $U^{s/a}_t = \exp(-i \frac{m_s}{4}
f^{s/a}_{\tilde{k}} g_l t \ \sigma_z I_l^{x/y} )=\exp(-i \varphi_{s/a} \sigma_z I_l^{x/y})$. Robustness of the protocol requires that the time $t$ is a multiple of the period $\tau$ of the pulse sequence, i.e $t = N \tau$ [@Supplemental]. Hence, we are restricted to a set of phases $\varphi_{s/a}\equiv
\varphi_{s/a}(N) = \frac{m_s}{4} f^{s/a}_{\tilde{k}} g_l N \tau$ depending on the number of applied periods $N$. The absence of tunable coefficients $f^{s/a}_{\tilde{k}}$ would limit the fidelity of each performed gate. For example the operation $\exp(-i \frac{\pi}{2}\sigma_z I_l^x )$ requires $\varphi_s(N) = \frac{\pi}{2}$ which, in general, does not hold for standard sequences as CPMG [@Carr54; @Meiboom58] or the XY family [@Maudsley86; @Gullion90] where $f^{s/a}_{\tilde{k}}
=\frac{4}{\pi k}$. However, in our case the coefficients $f^{s/a}_{\tilde{k}}$ can be arbitrarily selected giving access to any value for $\varphi_s(N)$, $\varphi_a(N)$ and to any quantum gate [@Casanova15; @Wang15].
This is shown in Fig. \[tunning\] where decoupling pulses are introduced according to $H_{\rm mw}=\Omega \cos{[(\omega_{\rm NV} + \Lambda) t}][ \cos{(\phi_i)}\ S_x + \sin{(\phi_i)}\ S_y]$ [@Loretz15], with $\Lambda$ a static detuning error with respect to the NV energy transition, $\omega_{\rm NV}$, caused, for example, by a change in temperature ($\Lambda \approx 2\pi\times 70$ kHz for a 1 K temperature shift [@Faraday15]) or by the imperfect polarization of the nitrogen spin of the NV center [@Doherty13] ($\Lambda \approx 2\pi\times2$ MHz for the $^{14}$N isotope). The Rabi frequency $\Omega$ is chosen to flip the electron spin in $12.5$ ns, and $S_{x,y}$ are spin-1 operators. In our numerics we consider $\vec{A}_{j}=\frac{\mu_{0}\gamma_{e}\gamma_{n}}{2|\vec{r}_{j}|^{3}}[\hat{z}
- 3\frac{(\hat{z}\cdot\vec{r}_{j})\vec{r}_{j}}{|\vec{r}_{j}|^{2}}]$, with $\vec{r}_j$ connecting the NV center and the nucleus, and the long relaxation time $(T_1)$ of the NV electron spin when operated at $T \sim 4$ K. Note that in these conditions $T_1$ measurements on the order of many seconds have been reported [@Jarmola12; @Cramer15].
The curves in a), b) correspond to the evolution of the signal $L = (1-2p_{|\psi_x\rangle} )$ with $p_{|\psi_x\rangle} = {\rm Tr}[\rho(t) |\psi_x\rangle\langle\psi_x|]$ [@Maze08; @Zhao12] due to the coherent interaction of the NV center with a nucleus for different values of the driving frequency ($\tilde{k}\omega$). $|\psi_x\rangle = \sigma_x |\psi_x\rangle$ is an eigenstate of the electronic Pauli operator $\sigma_x$. In a) we couple the NV to an spin (spin 1, $A^z = -2\pi\times16.59$ kHz) using the symmetric sequence tuned to obtain $\varphi_s = \pi, \frac{\pi}{2}, \frac{\pi}{4}$ ($f^s_1 = 0.045, 0.0225, 0.0112$) respectively. In b) we use a nucleus (spin 2, $A^z =
2\pi\times 9.63$ kHz) and the anti-symmetric sequence with $f^a_1 = 0.071, 0.0355, 0.01775$ such that $\varphi_a = \pi, \frac{\pi}{2}, \frac{\pi}{4}$. In all cases we consider a detuning error $\Lambda = 2\pi\times 70$ kHz and a Rabi frequency error (RFE) of $0.25 \%$ [@Cai12]. We note that these results are essentially indistinguishable from the case of ideal instantaneous pulses which confirms the robustness of both symmetric and anti-symmetric sequences. The vertical lines are located at the resonance positions predicted by Eq. (\[resonances\]) and their height can be calculated by using the expressions for $H^{s/a}$ as $L = -\cos{(\frac{m_s}{4}
f^{s/a}_{\tilde{k}} g_l t)}$, for $\frac{m_s}{4} f^{s/a}_{\tilde{k}} g_l t \equiv \varphi^{s/a} = \pi,
\frac{\pi}{2}, \frac{\pi}{4}$. Additionally to demonstrate the robustness of the method we chose $\tilde{k}= 1$ giving rise to the application of $6000$ imperfect decoupling pulses, however this number can be significantly reduced selecting $\tilde{k} > 1$.
[*Internuclear decoupling–* ]{} The decoupling efficiency of our method is shown in Fig. \[decoupling\] where we plot the evolution of $L$ for a nuclear cluster involving a dimer, i.e. a two-qubit register with the nuclei located at the minimum distance allowed in the diamond lattice $r_0 \approx 1.54 \ \mathring{\rm A}$, and an isolated nucleus. This gives rise to a internuclear coupling coefficient of $2\pi\times0.685$ kHz for the dimer, while the coupling of the latter with the additional qubit is smaller than $2\pi\times 1$ Hz. The hyperfine vectors for each nuclei have $A^z_1 =2\pi\times 4.35$ kHz, $A^z_2 = -2\pi\times7.49$ kHz, $A^z_3 = -2\pi\times11.82$ kHz, and we apply an rf decoupling field with $\Delta = 2\pi\times100$ kHz. $H_{\rm mw}$ accounts for imperfect pulses with $\Lambda =
2\pi\times70$ kHz, and a RFE of $0.25\%$.
In the absence of the decoupling field ${\vec B}_d$ we can observe the impact of the internuclear interactions due to $H_{\rm nn}$. The green curve in the background of Fig. \[decoupling\] a) ($H_{\rm nn} = 0$) exhibits three clearly identifiable peaks. In contrast, the grey curve in foreground of Fig. \[decoupling\] a) ($H_{\rm nn}\neq 0$) shows the distortion of the resonance peaks of the two nuclei of the dimer while the resonance curve of the isolated nucleus remains unchanged. Fig. \[decoupling\] b) demonstrates the effectiveness of the decoupling field, $\vec{B}_{\rm d}$, as now the response in the presence and in the absence of $H_{\rm nn} = 0$ (back dark blue curve and front light blue curve) produce the same evolutions for $L$. Again, vertical lines in Fig. \[decoupling\] a), b) account for the locations of the theoretically predicted resonances and for their height. For the case $\vec{B}_{\rm d} = 0$, Fig. \[decoupling\] a), we have resonances at $ |\omega \ \hat{z} - \frac{m_s}{2} \vec{A}_j|$, with $\omega =
\gamma _n B_z$ and $m_s=1$, see for example [@Casanova15], while the case $\vec{B}_{\rm d}
\neq 0$, Fig. \[decoupling\] b), resonates according to Eq. (\[resonances\]). The height of the peaks can be predicted theoretically when dealing with isolated nuclear spins, i.e. when $H_{\rm nn} = 0$, or $H_{\rm nn} \neq 0$ and the decoupling field is present. Note that in the $\vec{B}_{\rm d} = 0$ case we have $L = -\cos{(\frac{m_s}{4} f_{\tilde{k}} g_l t)}$ with $|g_l| = |\vec{A}_l - \vec{A}_l \cdot
\hat{\omega}_l \ \hat{\omega}_l|$ [@Casanova15].
[*Gate fidelities–* ]{} To check the fidelities of single- and two qubit-gates we simulate a sample such that $A^z_1 = 2\pi\times 23.10 $ kHz, $A^z_2 = 2\pi\times 9.63$ kHz, $A^z_3 = -2\pi\times 16.59$ kHz, the internuclear coupling coefficients are $g_{1, 2} = 2\pi\times 1.64$ Hz, $g_{1, 3} = 2\pi\times 1.40$ Hz, $g_{2, 3} = 2\pi\times -186.76$ Hz, and $\Delta =2\pi\times 100$ kHz. This situation involving a minimum distance between the $A^z_j$ components of $\approx 2\pi\times13$ kHz for, at least, three nuclear spins can be estimated to occur with a $0.06 \%$ of probability in diamond with $^{13}$C natural abundance. This means that 10 samples of this kind can be found in a diamond layer of dimensions 10 $\mu$m $\times$ 10 $\mu$m $\times$ 10 nm assuming a low NV concentration of 0.01ppm [@Jarmola12]. A relaxation of this condition by looking for an energy difference on the $A^z_j$ components of $2\pi\times 5$ kHz increases the rate of appearance to $\approx 1 \%$. These estimations have been performed by adding the restriction of discarding samples that present nuclei with $A_j^z > 2\pi\times45$ kHz, which assures a reduction of the internuclear coupling for moderate rf decoupling fields. If one does not include any condition on $A_j^z$, the percentage of suitable samples grows from $0.06\%$ to $5.7\%$.
We compute the fidelities of single- and two- qubit gates according to $F = \frac{|{\rm Tr}(A B^{\dag})|}{\sqrt{{\rm Tr}(A A^{\dag})
\ {\rm Tr}(B B^{\dag})}}$, with $A$ and $B$ two general quantum operations [@Wang08], by driving at the theoretically predicted resonance frequencies in Eq. (\[resonances\]) with imperfect pulses such that $\Lambda = 2\pi\times 70$ kHz and RFE $=0.25 \%$. The results are outlined in Table \[table1\] and an inspection confirms the high fidelity achieved by our method. See [@Supplemental] for further details.
Finally, to estimate the effect of environmental noise from other unused nuclei we have completed the previous sample with a bath containing $200$ $^{13}$C atoms and compare the value of $L$ of after the gate exp$(-i \pi \sigma_zI_1^x)$. This situation describes a diamond with a $0.27 \%$ abundance of $^{13}$C located within a radius of $\approx 4.7$ nm with the NV assumed to be located in the center. Because of computational restrictions, here we deal with instantaneous pulses. For an initial density matrix $\rho \propto |+\rangle\langle+| \otimes I_{2^N \times 2^N}$, $N$ being the total number of nuclear spins, our realisation of exp$(-i \pi \sigma_zI_1^x)$ gives rise to $L = 0.9985$ when only the three qubit register is consider and $L = 0.9936$ when the bath is included. Note that this value has been obtained by averaging the results for 10 different samples [@Supplemental]. This implies that thanks to our decoupling protocols the large number of bath spins has a small effect on the gate fidelity.
[[ |c | c | c | c | c| c|]{}]{} $F_{-,+}$ & exp$( \mp i\frac{\pi}{2} \sigma_z I_j^x)$& exp$(\mp i\frac{\pi}{2} \sigma_z I_j^y)$ & exp$( \mp i\frac{\pi}{2} I_j^x)$ & exp$( \mp i\frac{\pi}{2} I_j^y)$\
Spin$_1$ & $ 0.9984, 0.9980 $ &$0.9971, 0.9988$ & $0.9983, 0.9983$ &$0.9983, 0.9984$\
Spin$_2$ & $0.9918, 0.9918$ &$0.9935, 0.9930$ & $ 0.9986, 0.9986$ & $0.9987, 0.9986$\
Spin$_3$ & $0.9960, 0.9963$ &$0.9975, 0.9963$ & $0.9952, 0.9952$ & $ 0.9954, 0.9953$\
[*Applications–*]{} An algorithmic problem of interest in quantum chemistry and solid-state physics is the quantum simulation of fermionic systems. Through the Jordan-Wigner transformation [@Jordan28] any fermionic Hamiltonian $H_{f}$ of $N$ particles admits a form like $H_{f} = \sum_{(i, j, ...)} g_{(i, j, ...)} [\sigma^{\alpha}_{i} \otimes \sigma^{\beta}_{j}
\otimes ...]$, with $\alpha, \beta, ... = x, y, z,$ and $i, j, ... = 1,..., N$. By Trotter expanding the associated time-evolution operator $U_t \approx (\Pi_{(i, j, ...)}
U_{(i, j, ...)})^n$, where $U_{(i, j, ...)} = \exp(-i \frac{t}{n} g_{(i, j, ...)}
[\sigma^{\alpha}_{i} \otimes \sigma^{\beta}_{j} \otimes ...])$ we find that fermionic dynamics relies on the robust implementation of the gates $U_{(i, j, ...)}$. Note that the latter can be achieved by a single mediator [@Nielsen; @Casanova12; @Lamata14] which adapts to our protocol.
Any unitary gate between different nuclei can be implemented as $U_{q_i, \{ q_n \}}={\rm SWAP}_{q_e, q_i}
\ \tilde{U}_{q_e, \{ q_n\}} \ {\rm SWAP}_{q_e, q_i},$ where $q_e$ labels the electron spin while $q_j$ the nuclear spins, $\tilde{U}_{q_e, \{ q_n\}}$ represents a gate between the electron spin and some set of $n$ qubits $\{ q_n \}$ that does not contain the ${i}$-th qubit. When applied to some initial state we find $U_{q_i, \{ q_n \}} |\psi_e\rangle |\psi_i\rangle |\phi_{\{q_n\}}\rangle
=|\psi_e\rangle \tilde{U}_{q_i, \{ q_n\}} |\psi_i\rangle |\phi_{\{q_n\}}\rangle$. This procedure can be repeated to find $|\psi_e\rangle \tilde{U}_{q_{i_j}, \{ q_{n_j}\}} ... \tilde{U}_{q_{i_2}, \{ q_{n_2}\}}
\tilde{U}_{q_{i_1}, \{ q_{n_1}\}} |\phi_{\{q_N\}}\rangle$, that corresponds to a set of operations applied on the nuclei where $N=n+1$, and $n_j$, $q_{i_j}$ label different nuclear spin sets and spin targets respectively.
Quantum algorithms as outlined above assume a pure initial state of the entire quantum computer. Remarkably, however, there are models of quantum computation that achieve computational advantages with minimal coherence. Indeed, a single pure qubit (in our case the easily polarisable NV electron spin) executing a controlled unitary operation $|0\rangle\langle 0|\otimes {\mathbbm{1}}+ |1\rangle\langle 1|\otimes U$ on a maximally mixed register (the $^{13}$C nuclei) can be used to obtain the trace of the unitary $U$ efficiently after a single measurement of the control qubit. This setting realises the DQC1 protocol [@KnillL1998] which can be shown to achieve computational advantages over classical computation and allow for the execution of complex algorithms such as Shor’s factoring algorithm [@ParkerP2000; @ParkerP2002]. Furthermore, the DQC1 model can be adapted to obtain Heisenberg scaling in metrology tasks which ties in well with envisaged spin sensing applications [@BoixoS08].
[*Conclusions –*]{} We provide a robust scheme to perform highly-selective single- and two-qubit gates into different nuclear spins while efficiently eliminate the effect of internuclear interactions and the noise on the mediator. Our method does not requires strongly coupled spins representing an scalable procedure for general quantum computing purposes.
[*Acknowledgements –*]{} This work was supported by the Alexander von Humboldt Foundation, the ERC Synergy grant BioQ, the EU projects DIADEMS, SIQS, EQUAM and HYPERDIAMOND as well as the DFG via the SFB TRR/21 and the SPP 1601.
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**Supplemental Material:\
Noise-resilient Quantum Computing with a Nitrogen-Vacancy Center and Nuclear Spins**
Hamiltonian model under decoupling radio-frequency field
=========================================================
Here we show how to achieve the effective Hamiltonian in Eq. $(2)$ on the main text. In a rotating frame with respect to the electronic free energy terms $D S_z^2 - \gamma_e B_zS_z$ we have ($\hbar = 1$) $$\label{sup:root}
H = -\sum_j \omega \ I_j^z + \frac{m_s}{2} [F(t) \ \sigma_z + \mathbb{I}]\sum_j \vec{A}_j \cdot \vec{I}_j +2 \Omega \cos{(\omega_{\rm rf} t)} \sum_j \vec{I}_j \cdot \hat{n} + \tilde{H}_{\rm c} + H_{nn}.$$ Here $\omega = \gamma_n B_z$ with $\gamma_n = 2 \pi \times 10.705 \ \frac{\rm MHz}{\rm T}$ being the $^{13}$C gyromagnetic ratio, $m_s = \pm 1$ see [@Casanova15], $\vec{A}_{j}$ is the hyperfine vector for the $j$-th nucleus, $2\Omega = \gamma_n B_{\rm d}$, $\sigma_z = |m_s\rangle \langle m_s| - | 0 \rangle \langle 0|$ denotes the electron spin Pauli operator, $\tilde{H}_{\rm c}$ gives rise to the action of the rf control field, $F(t)=1$, or $F(t)=-1$ (depending on whether an even, $F(t)=1$, or odd, $F(t)=-1$, number of decoupling pulses have been applied) is the modulation function and $H_{\rm nn}$ is $$H_{\text{nn}}=\sum_{j>k}\frac{\mu_{0}}{2}\frac{\gamma_n^2}{r_{j,k}^{3}} \left[\vec{I}_{j}\cdot\vec{I}_{k}-\frac{3(\vec{I}_{j}\cdot\vec{r}_{j,k})
(\vec{r}_{j,k}\cdot\vec{I}_{k})}{r_{j,k}^{2}}\right],$$ where $|\vec{r}_{j,k}|$ is the distance between the $j$-, and $k$-th nuclei. Additionally and without loss of generality we consider $\hat{n} = (n^{z^{\perp}} , 0 , n^z)$ with $z^{\perp}\equiv \hat{x}$ being a direction orthogonal to $\hat{z}$.
Our analysis will be performed in a counter rotating frame that allows to a more detailed description of the resonances structure in Eq. (\[sup:root\]) than the one that can be obtained by merely applying the rotating wave approximation. More specifically, in order to take into account possible energy deviations coming from Bloch-Siegert shifts [@Allen] it is convenient to move into an interaction picture w.r.t. $\omega_{\rm rf} \sum_j I_j^z$. This yields $$\begin{array}{ccc}
H &=& \sum_j \bigg[ \Omega_x I_j^x - (\omega + \omega_{\rm rf}) I_j^z \ \bigg] + \Omega_x \sum_j \bigg[I_j^+ e^{i 2\omega_{\rm rf} t} + I_j^- e^{-i 2\omega_{\rm rf} t}\bigg] + 2 \Omega_z \cos{(\omega_{\rm rf} t)}\sum_j I_j^z \vspace{0.25 cm} \\ \nonumber
&&+ \frac{m_s}{2} [F(t) \ \sigma_z + \mathbb{I}]\sum_j e^{i \omega_{\rm rf} I_j^z t} \vec{A}_j \cdot \vec{I}_j e^{-i \omega_{\rm rf} I_j^z t} + e^{it \sum_j\omega_{\rm rf} I_j^z} \tilde{H}_{\rm c} e^{-it \sum_j\omega_{\rm rf} I_j^z} +\tilde{H}_{nn}.
\end{array}$$ Here, $\Omega_{x, z} = \Omega \ n^{z^{\perp}, z}$ and $\tilde{H}_{nn}$ reads $$\tilde{H}_{nn} = e^{it \sum_j \omega_{\rm rf} I_j^z} H_{nn} \ e^{-it \sum_j \omega_{\rm rf} I_j^z} \approx \sum_{j>k}\frac{\mu_{0}}{2}\frac{\gamma_{n}^2}{r_{j,k}^{3}} \left[1 - 3 (n_{j,k}^z)^2 \right] \left[I^z_{j} \ I^z_{k} - \frac{1}{2} (I_j^{x} I_k^{y} + I_j^{y} I_k^{x})\right].$$ By defining $ \bigg[ \Omega_x I_j^x - (\omega + \omega_{\rm rf}) I_j^z \ \bigg] = \tilde{\omega} \ \tilde{n} \cdot\vec{I}_j$, where $\tilde{\omega} = \sqrt{(\omega + \omega_{\rm rf})^2 + \Omega_x^2 }$ and $\tilde{n} = (\frac{\Omega_x}{\tilde\omega}, 0, -\frac{\omega + \omega_{\rm rf}}{\tilde\omega})$, we can write $$\begin{array}{ccc}
H &=& \tilde{\omega}\sum_j \tilde{n} \cdot \vec{I}_j + \Omega_x \sum_j \bigg[I_j^+ e^{i 2\omega_{\rm rf} t} + I_j^- e^{-i 2\omega_{\rm rf} t}\bigg] + 2 \Omega_z \cos{(\omega_{\rm rf} t)}\sum_j I_j^z\vspace{0.2cm}\\ \nonumber &&+ \frac{m_s}{2} [F(t) \ \sigma_z + \mathbb{I}]\sum_j e^{i \omega_{\rm rf} I_j^z t} \vec{A}_j \cdot \vec{I}_j e^{-i \omega_{\rm rf} I_j^z t} +e^{it \sum_j\omega_{\rm rf} I_j^z} \tilde{H}_{\rm c} e^{-it \sum_j\omega_{\rm rf} I_j^z} + \tilde{H}_{nn}.
\end{array}$$
In order to achieve internuclear decoupling the action of a magnetic field oriented into an specific angle is required. This condition is known as the magic angle condition [@Lee65] and it can be addressed in our formalism by expressing $\tilde{\omega} = \Delta + \xi$ and moving to a rotating frame w.r.t. $\xi \sum_j \tilde{n} \cdot \vec{I}_j$. By using the identity $$\label{sup:identity}
e^{i\vec{I}_{j}\cdot\hat{l}\phi}\vec{I}_{j}\cdot\vec{b}\ e^{-i\vec{I}_{j}\cdot\hat{l}\phi}=\vec{I}_{j}\cdot[(\vec{b}-\vec{b}\cdot\hat{l}\hat{l})\cos\phi-\hat{l}\times\vec{b}\sin\phi+\vec{b}\cdot\hat{l}\hat{l}],$$ and under the resonance condition $\xi = 2\omega_{\rm rf}$ we find $$\begin{aligned}
\label{sup:mesh}
H =&& \sum_j \bigg\{ \big[ \Delta \tilde{n}_x + \frac{\Omega_x}{2} (1 - \tilde{n}_x^2 - \tilde{n}_z)\big] \ I_j^x + \big[\Delta \tilde{n}_z + \frac{\Omega_x}{2}\tilde{n}_x (1 - \tilde{n}_z)\big] \ I_j^z \bigg\}\nonumber\\
&&+ \frac{m_s}{2} [F(t) \ \sigma_z + \mathbb{I}]\sum_j e^{i \xi \tilde{n} \cdot \vec{I}_j t} e^{i \omega_{\rm rf} I_j^z t} \vec{A}_j \cdot \vec{I}_j e^{-i \omega_{\rm rf} I_j^z t} e^{-i \xi \tilde{n} \cdot \vec{I}_j t}\nonumber\\
&&+ \ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} e^{it \sum_j\omega_{\rm rf} I_j^z} \tilde{H}_{\rm c} e^{-it \sum_j\omega_{\rm rf} I_j^z} e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} + e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \tilde{H}_{nn} e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t}.\end{aligned}$$ In the previous equation we have eliminated the term $2 \Omega_z \cos{(\omega_{\rm rf} t)}\sum_j I_j^z$ that it is not compensated with any other oscillating variable, see section \[secfidel\] for further considerations about the fidelity of this effective Hamiltonian. Note that the resonance condition $2\omega_{\rm rf} = \xi$ implies $$\Delta = \tilde{\omega} - 2\omega_{\rm rf} = \sqrt{(\omega + \omega_{\rm rf})^2 + \Omega^2_x} - 2\omega_{\rm rf}= (\omega + \omega_{\rm rf}) \sqrt{1+ \bigg(\frac{\Omega_x}{\omega + \omega_{\rm rf}} \bigg)^2} - 2\omega_{\rm rf} \approx \omega - \omega_{\rm rf} + \frac{1}{2} \frac{\Omega_x^2}{\omega + \omega_{\rm rf}},$$ where it can be explicitly seen an energy shift proportional to $\frac{\Omega_x^2}{\omega + \omega_{\rm rf}}$, see [@Allen].
The second line of Eq. (\[sup:mesh\]) can be simplified to $$\frac{m_s}{2} F(t) \ \sigma_z\sum_j e^{i \xi \tilde{n} \cdot \vec{I}_j t} e^{i \omega_{\rm rf} I_j^z t} \vec{A}_j \cdot \vec{I}_j e^{-i \omega_{\rm rf} I_j^z t} e^{-i \xi \tilde{n} \cdot \vec{I}_j t} + \frac{m_s}{2} \sum_j A_j^z (\hat{z} \cdot \tilde{n}) \ \vec{I}_j \cdot \tilde{n},$$ where we have eliminated fast rotating terms. This gives rise to
$$\begin{aligned}
\label{sup:demons}
H = &&\sum_j \bigg\{ \big[ \Delta \tilde{n}_x + \frac{\Omega_x}{2} (1 - \tilde{n}_x^2 - \tilde{n}_z) + \frac{m_s}{2} \tilde{n}_z \ \tilde{n}_x \ A_j^z \big] \ I_j^x + \big[\Delta \tilde{n}_z + \frac{\Omega_x}{2}\tilde{n}_x (1 - \tilde{n}_z)+ \frac{m_s}{2} \ \tilde{n}^2_z \ A_j^z \big] \ I_j^z \bigg\}\nonumber\\
&&+ \frac{m_s}{2} F(t) \ \sigma_z \sum_j e^{i \xi \tilde{n} \cdot \vec{I}_j t} e^{i \omega_{\rm rf} I_j^z t} \vec{A}_j \cdot \vec{I}_j e^{-i \omega_{\rm rf} I_j^z t} e^{-i \xi \tilde{n} \cdot \vec{I}_j t}\nonumber\\
&&+ \ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} e^{it \sum_j\omega_{\rm rf} I_j^z} \tilde{H}_{\rm c} e^{-it \sum_j\omega_{\rm rf} I_j^z} e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t}+ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \tilde{H}_{nn} e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t},\end{aligned}$$
where $\tilde{n}_x = \tilde{n} \cdot \hat{x}$, and $\tilde{n}_z = \tilde{n} \cdot \hat{z}$.
The elimination of the internuclear interactions happens if we match the magic-angle condition [@Lee65] which in our case corresponds to $$\big[ \Delta \tilde{n}_x + \frac{\Omega_x}{2} (1 - \tilde{n}_x^2 - \tilde{n}_z) + \frac{m_s}{2} \tilde{n}_z \ \tilde{n}_x \ A_j^z \big] = \sqrt{2} \ \big[\Delta \tilde{n}_z + \frac{\Omega_x}{2}\tilde{n}_x (1 - \tilde{n}_z) + \frac{m_s}{2} \ \tilde{n}^2_z \ A_j^z\big].$$ Because $A_j^z$ takes different values for each nuclei the above equality does not hold for any $j$, however it can be approximately fulfilled if $\Delta$ takes a value such that $$\label{sup:manglerestriction}
| A_j^z| \ll |2 \Delta|, \ \ \forall j,$$ This issue simplifies the magic-angle condition that now reads $$\label{sup:mangle}
\tilde{n}_x + \frac{\Omega_x}{2 \Delta} (1 - \tilde{n}_x^2 - \tilde{n}_z) = \sqrt{2} \ \big[ \tilde{n}_z + \frac{\Omega_x}{2 \Delta}\tilde{n}_x (1 - \tilde{n}_z) ].$$ Note that the above equation establishes a relation between the externally controllable parameters $\Omega$, $\hat{n}$, and $\Delta$.
In this manner the Hamiltonian is $$\begin{aligned}
\label{sup:tunable}
H = \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j + \frac{m_s}{2}F(t) \ \sigma_z \sum_j e^{i \xi \tilde{n} \cdot \vec{I}_j t} e^{i \omega_{\rm rf} I_j^z t} \vec{A}_j \cdot \vec{I}_j e^{-i \omega_{\rm rf} I_j^z t} e^{-i \xi \tilde{n} \cdot \vec{I}_j t}\nonumber\\
+ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} e^{it \sum_j\omega_{\rm rf} I_j^z} \tilde{H}_{\rm c} e^{-it \sum_j\omega_{\rm rf} I_j^z} e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t}+ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \tilde{H}_{nn} e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t}.\end{aligned}$$ The above expression contains a resonance frequency, $\omega_j$, depending on the nuclear register. We can exploit this issue to generate single-qubit addressing and implement entangling operations between the electron spin and some specific nucleus as well as single qubit rotations. More specifically that frequency is
$$\omega_j = |\Delta| \sqrt{ \bigg[\delta + \frac{m_s}{2 \Delta} \tilde{n}_z \ \tilde{n}_x \ A_j^z \bigg]^2 + \bigg[\frac{\delta}{\sqrt{2}} + \frac{m_s}{2 \Delta} \ \tilde{n}^2_z \ A_j^z \bigg]^2}$$
with $\delta = \tilde{n}_x + \frac{\Omega_x}{2 \Delta} (1 - \tilde{n}_x^2 - \tilde{n}_z)$. The $j$-th nuclear rotation axis reads $$\hat{n}_j = \bigg( \frac{\Delta\delta + \frac{m_s}{2} \tilde{n}_z \ \tilde{n}_x \ A_j^z}{\omega_j}, 0, \frac{\frac{\Delta\delta}{\sqrt{2}} + \frac{m_s}{2} \ \tilde{n}^2_z \ A_j^z}{\omega_j} \bigg)$$ An additional change of interaction picture w.r.t. $ \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j$ provides with
$$\begin{aligned}
H &= & \frac{m_s}{2}F(t) \ \sigma_z \sum_j e^{i \omega_j \ \hat{n}_j \cdot \vec{I}_j t} \ e^{i \xi \tilde{n} \cdot \vec{I}_j t} \ e^{i \omega_{\rm rf} I_j^z t} \ \vec{A}_j \cdot \vec{I}_j \ e^{-i \omega_{\rm rf} I_j^z t} \ e^{-i \xi \tilde{n} \cdot \vec{I}_j t} \ e^{-i \omega_j \ \hat{n}_j \cdot \vec{I}_j t} \nonumber\\
&+& e^{i \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j t} \ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \ e^{it \sum_j\omega_{\rm rf} I_j^z} \tilde{H}_{\rm c} \ e^{-it \sum_j\omega_{\rm rf} I_j^z} \ e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \ e^{-i \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j t} \nonumber\\
&+& e^{i \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j t} \ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \ \tilde{H}_{nn} \ e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \ e^{-i \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j t}.\end{aligned}$$
The last line of the previous equation, i.e. the one including the internuclear interactions, can be treated as follows. First, for large values of $B_z$ we have $\tilde{n} \approx \hat{z}$ and consequently $$\ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \ \tilde{H}_{nn} \ e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \approx \tilde{H}_{nn}.$$ Then we can average out the remaining expression, $e^{i \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j t} \ \tilde{H}_{nn} \ e^{-i \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j t}$, because of the magic angle condition in Eq. (\[sup:mangle\]) and the easily achievable requirement $$\big|\frac{\mu_{0}\gamma^2_{n}}{2|\vec{r}_{j,k}|^{3}}\big| \ll |\omega_j|$$ Note that for the diamond lattice the lowest Carbon-Carbon distance is $r_0 \approx 1.54 \ \mathring{\rm A}$, therefore nuclei located at that distance have the highest value of the coupling coefficient $\frac{\mu_{0}\gamma^2_{j}}{2|\vec{r}_{j,k}|^{3}}$. Consequently we need a set of frequencies $\omega_j$ such that $$2 \ \mbox{kHz} \ll |\omega_j|$$ In this manner internuclear interactions can be approximately neglected, note that in addition to the previous condition we are also restricted by the applicability of Eq. (\[sup:manglerestriction\]).
Hence we can simplify the Hamiltonian to the following expression
$$\begin{aligned}
\label{sup:prior}
H &= & \frac{m_s}{2}F(t) \ \sigma_z \sum_j e^{i \omega_j \ \hat{n}_j \cdot \vec{I}_j t} \ e^{i \xi \tilde{n} \cdot \vec{I}_j t} \ e^{i \omega_{\rm rf} I_j^z t} \ \vec{A}_j \cdot \vec{I}_j \ e^{-i \omega_{\rm rf} I_j^z t} \ e^{-i \xi \tilde{n} \cdot \vec{I}_j t} \ e^{-i \omega_j \ \hat{n}_j \cdot \vec{I}_j t} \nonumber\\
&+& e^{i \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j t} \ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \ e^{it \sum_j\omega_{\rm rf} I_j^z} \tilde{H}_{\rm c} \ e^{-it \sum_j\omega_{\rm rf} I_j^z} \ e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \ e^{-i \sum_j \omega_j \ \hat{n}_j \cdot \vec{I}_j t}.
\end{aligned}$$
With the help of the identity in Eq. (\[sup:identity\]) and writing the control Hamiltonian $H_{\rm c}$ as $$H_{\rm c} = 2\lambda\cos(\omega_{c} + \phi_c) \sum_j \vec{I}_j \cdot \hat{n}_c,$$ Eq. (\[sup:prior\]) can be fully developed providing a complete map to the resonances structure of the system under the action of the decoupling field. This is $$\begin{aligned}
\label{sup:super}
H = \frac{m_s}{2}F(t) \ \sigma_z \sum_j &\bigg\{& \vec{I}_j\cdot\bigg[(\vec{\alpha}_{1,j} - \vec{\alpha}_{1,j}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{\alpha}_{1,j}) \sin{(\omega_j t)} + \vec{\alpha}_{1,j}\cdot\hat{n}_j \ \hat{n}_j \bigg] \cos{(\xi t)} \cos{(\omega_{\rm rf } t)} \nonumber\\
&-& \vec{I}_j\cdot\bigg[(\vec{\alpha}_{2,j} - \vec{\alpha}_{2,j}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{\alpha}_{2,j}) \sin{(\omega_j t)} + \vec{\alpha}_{2,j}\cdot\hat{n}_j \ \hat{n}_j \bigg] \sin{(\xi t)} \cos{(\omega_{\rm rf } t)} \nonumber\\
&+& \vec{I}_j\cdot\bigg[(\vec{\alpha}_{3,j} - \vec{\alpha}_{3,j}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{\alpha}_{3,j}) \sin{(\omega_j t)} + \vec{\alpha}_{3,j}\cdot\hat{n}_j \ \hat{n}_j \bigg] \cos{(\omega_{\rm rf} t)} \nonumber\\
&-&\vec{I}_j\cdot\bigg[(\vec{\beta}_{1,j} - \vec{\beta}_{1,j}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{\beta}_{1,j}) \sin{(\omega_j t)} + \vec{\beta}_{1,j}\cdot\hat{n}_j \ \hat{n}_j \bigg] \cos{(\xi t)} \sin{(\omega_{\rm rf } t)}\nonumber\\
&+&\vec{I}_j\cdot\bigg[(\vec{\beta}_{2,j} - \vec{\beta}_{2,j}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{\beta}_{2,j}) \sin{(\omega_j t)} + \vec{\beta}_{2,j}\cdot\hat{n}_j \ \hat{n}_j \bigg] \sin{(\xi t)} \sin{(\omega_{\rm rf } t)}\nonumber\\
&-&\vec{I}_j\cdot\bigg[(\vec{\beta}_{3,j} - \vec{\beta}_{3,j}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{\beta}_{3,j}) \sin{(\omega_j t)} + \vec{\beta}_{3,j}\cdot\hat{n}_j \ \hat{n}_j \bigg] \sin{(\omega_{\rm rf} t)} \nonumber\\
&+&\vec{I}_j\cdot\bigg[(\vec{\gamma}_{1,j} - \vec{\gamma}_{1,j}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{\gamma}_{1,j}) \sin{(\omega_j t)} + \vec{\gamma}_{1,j}\cdot\hat{n}_j \ \hat{n}_j \bigg] \cos{(\xi t)} \nonumber\\
&-&\vec{I}_j\cdot\bigg[(\vec{\gamma}_{2,j} - \vec{\gamma}_{2,j}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{\gamma}_{2,j}) \sin{(\omega_j t)} + \vec{\gamma}_{2,j}\cdot\hat{n}_j \ \hat{n}_j \bigg] \sin{(\xi t)} \nonumber\\
&+&\vec{I}_j\cdot\bigg[(\vec{\gamma}_{3,j} - \vec{\gamma}_{3,j}\cdot \hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{\gamma}_{3,j}) \sin{(\omega_j t)} + \vec{\gamma}_{3,j}\cdot\hat{n}_j \ \hat{n}_j \bigg] \ \ \ \bigg\}\nonumber\\
+2\lambda\cos(\omega_{c} + \phi_c) \sum_j &\bigg\{& \vec{I}_j\cdot\bigg[(\vec{m}_{1,1} - \vec{m}_{1,1}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{m}_{1,1}) \sin{(\omega_j t)} + \vec{m}_{1,1}\cdot\hat{n}_j \ \hat{n}_j \bigg] \cos{(\xi t)} \cos{(\omega_{\rm rf } t)} \nonumber\\
&-& \vec{I}_j\cdot\bigg[(\vec{m}_{1,2} - \vec{m}_{1,2}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{m}_{1,2}) \sin{(\omega_j t)} + \vec{m}_{1,2}\cdot\hat{n}_j \ \hat{n}_j \bigg] \sin{(\xi t)} \cos{(\omega_{\rm rf } t)} \nonumber\\
&+& \vec{I}_j\cdot\bigg[(\vec{m}_{1,3} - \vec{m}_{1,3}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{m}_{1,3}) \sin{(\omega_j t)} + \vec{m}_{1,3}\cdot\hat{n}_j \ \hat{n}_j \bigg] \cos{(\omega_{\rm rf} t)} \nonumber\\
&-& \vec{I}_j\cdot\bigg[(\vec{m}_{2,1} - \vec{m}_{2,1}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{m}_{2,1}) \sin{(\omega_j t)} + \vec{m}_{2,1}\cdot\hat{n}_j \ \hat{n}_j \bigg] \cos{(\xi t)}\sin{(\omega_{\rm rf} t)} \nonumber\\
&+& \vec{I}_j\cdot\bigg[(\vec{m}_{2,2} - \vec{m}_{2,2}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{m}_{2,2}) \sin{(\omega_j t)} + \vec{m}_{2,2}\cdot\hat{n}_j \ \hat{n}_j \bigg] \cos{(\xi t)}\sin{(\omega_{\rm rf} t)} \nonumber\\
&-& \vec{I}_j\cdot\bigg[(\vec{m}_{2,3} - \vec{m}_{2,3}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{m}_{2,3}) \sin{(\omega_j t)} + \vec{m}_{2,3}\cdot\hat{n}_j \ \hat{n}_j \bigg] \sin{(\omega_{\rm rf} t)} \nonumber\\
&+& \vec{I}_j\cdot\bigg[(\vec{m}_{3,1} - \vec{m}_{3,1}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{m}_{3,1}) \sin{(\omega_j t)} + \vec{m}_{3,1}\cdot\hat{n}_j \ \hat{n}_j \bigg] \cos{(\xi t)} \nonumber\\
&-& \vec{I}_j\cdot\bigg[(\vec{m}_{3,2} - \vec{m}_{3,2}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{m}_{3,2}) \sin{(\omega_j t)} + \vec{m}_{3,2}\cdot\hat{n}_j \ \hat{n}_j \bigg] \sin{(\xi t)} \nonumber\\
&+& \vec{I}_j\cdot\bigg[(\vec{m}_{3,3} - \vec{m}_{3,3}\cdot\hat{n}_j \ \hat{n}_j) \cos{(\omega_j t)} - (\hat{n}_j \times \vec{m}_{3,3}) \sin{(\omega_j t)} + \vec{m}_{3,3}\cdot\hat{n}_j \ \hat{n}_j \bigg] \ \ \bigg\} \nonumber\\\end{aligned}$$
where $$\begin{aligned}
\vec{\alpha}_{1,j} &=& (\vec{A}_j - \vec{A}_j \cdot \hat{z} \ \hat{z}) - (\vec{A}_j - \vec{A}_j \cdot \hat{z} \ \hat{z}) \cdot \tilde{n} \ \tilde{n},\nonumber\\
\vec{\alpha}_{2,j} &=& \tilde{n} \ \times (\vec{A}_j - \vec{A}_j \cdot \hat{z} \ \hat{z}), \nonumber\\
\vec{\alpha}_{3,j} &=& (\vec{A}_j - \vec{A}_j \cdot \hat{z} \ \hat{z}) \cdot \tilde{n} \ \tilde{n},\nonumber\\
\vec{\beta}_{1,j} &=& (\hat{z} \times \vec{A}_j) - (\hat{z} \times \vec{A}_j) \cdot \tilde{n} \ \tilde{n},\nonumber\\
\vec{\beta}_{2,j} &=& \tilde{n} \ \times (\hat{z} \times \vec{A}_j), \nonumber\\
\vec{\beta}_{3,j} &=& (\hat{z} \times \vec{A}_j) \cdot \tilde{n} \ \tilde{n},\nonumber\\
\vec{\gamma}_{1,j} &=& (\vec{A}_j \cdot \hat{z} \ \hat{z}) - (\vec{A}_j \cdot \hat{z} \ \hat{z}) \cdot \tilde{n} \ \tilde{n},\nonumber\\
\vec{\gamma}_{2,j} &=& \tilde{n} \ \times (\vec{A}_j \cdot \hat{z} \ \hat{z}), \nonumber\\
\vec{\gamma}_{3,j} &=& (\vec{A}_j \cdot \hat{z}) \ (\hat{z} \cdot \tilde{n}) \ \tilde{n},\end{aligned}$$ and $$\begin{aligned}
\vec{m}_{j,1} &=& \vec{n}_{c, j} - \vec{n}_{c, j}\cdot\tilde{n} \ \tilde{n},\nonumber\\
\vec{m}_{j,2} &=& \tilde{n} \times \vec{n}_{c, j}, \nonumber\\
\vec{m}_{j,3} &=& \vec{n}_{c, j}\cdot\tilde{n} \ \tilde{n},\nonumber\\\end{aligned}$$ with $$\begin{aligned}
\vec{n}_{c, 1} &=& \hat{n}_c - \hat{n}_c \cdot \hat{z} \ \hat{z},\nonumber\\
\vec{n}_{c, 2} &=& \hat{z} \times \hat{n}_c, \nonumber\\
\vec{n}_{c, 3} &=& \hat{n}_c \cdot \hat{z} \ \hat{z}.\end{aligned}$$
An inspection of Eq. (\[sup:super\]) together with the condition $\xi = 2\omega_{\rm rf}$ reveals the existence of different resonance branches at $\omega_j$, $\omega_{\rm rf} \pm \omega_j$, and $3\omega_{\rm rf} \pm \omega_j$. We will restrict our analysis to the branch that resonates at $\omega_j$ frequencies, i.e. those going with the $\vec{\gamma}_{3,j}$, and $\vec{m}_{3, 3}$ in Eq. (\[sup:super\]). However the whole formalism can be applied to any other branch.
One can easily demonstrate that the set $\hat{x}_j$, $\hat{y}_j$, and $\hat{z}_j$, where $\vec{x}_j = (\vec{\gamma}_{3,j} - \vec{\gamma}_{3,j}\cdot \hat{n}_j \ \hat{n}_j)$, $\vec{y}_j=(\hat{n}_j \times \vec{\gamma}_{3,j})$, $\vec{z}_j=\hat{n}_j $ and $|\vec{x}_j| = |\vec{y}_j| =\big| (\vec{A}_j \cdot \hat{z}) (\hat{z} \cdot \tilde{n}) \sin{(\theta_{\tilde{n}, \hat{n}_j})} \big| = g_j$, $\theta_{\tilde{n}, \hat{n}_j}$ being the angle between the vectors $\tilde{n}, \hat{n}_j$, constitutes a set of orthogonal and unitary vectors that allows to define the computational basis of our problem as follows $$\begin{aligned}
I_j^x &=& \vec{I}_j \cdot \hat{x}_j,\nonumber\\
I_j^y &=& \vec{I}_j \cdot \hat{y}_j,\nonumber\\
I_j^z &=& \vec{I}_j \cdot \hat{z}_j.\nonumber\\\end{aligned}$$ Additionally we can express the set $(\vec{m}_{3,3} - \vec{m}_{3,3}\cdot\hat{n}_j \ \hat{n}_j)$, $(\hat{n}_j \times \vec{m}_{3,3})$, and $\hat{n}_j$ in terms of the previously defined vector basis.
Despite the complexity of the Hamiltonian (\[sup:super\]) we can excite only one resonance branch by means of the robust AXY-8 sequence [@Casanova15] where the function $F(t)$ can be modulated reading $F(t) = \sum_{k>0} f_k \cos{(k \omega t)}$ with $f_k$ and $\omega$ being fully tunable parameters (see section \[symantisym\] for a discussion of symmetric and antisymmetric pulse arrangements of the AXY sequence). Then, if we set $F(t)$ with a period $\tau$ and the control RF field with a frequency $\omega_c$ such that $\left(\frac{k}{\tau}\right)$ and $\omega_c$ are both on the range of the resonant $\omega_j$ frequencies, the Hamiltonian in Eq. (\[sup:super\]) is simplified to $$\label{sup:ya}
H = \frac{m_s}{2} F(t) \ \sigma_z \sum_j g_j \bigg[I_j^x \cos{(\omega_j t)} - I_j^y \sin{(\omega_j t)} \bigg] + 2\lambda\cos{(\omega_c t +\phi_c)} \sum_j \vec{I}_j \cdot \bigg[ |\vec{\alpha}_j| \ \hat{\alpha}_j \cos{(\omega_j t)} - |\vec{\beta}_j| \ \hat{\beta}_j \sin{(\omega_j t)} \bigg],$$ where $\vec{\alpha}_j$ and $\vec{\beta}_j$ depends on the specific orientation of $\hat{n}_c$ with respect to the basis $\hat{x}_j$, $\hat{y}_j$, and $\hat{z}_j$.
Decoupling field approximation {#secfidel}
==============================
In order to justify the decoupling field structure that gives rise to Eq. (\[sup:tunable\]) it is enough with considering the situation described by the expression $$\label{sup:real}
H = - \omega \ I^z + 2 \Omega \ \vec{I} \cdot \hat{n} \cos{(\omega_{\rm rf} t)}.$$
 fidelity values for the case $\Delta = 2\pi \times 200$ kHz and $\Omega $ calculated by using the magic-angle condition in Eq. (\[sup:mangle\]) for different values ($4$ and $5$ T) of the magnetic field in the $z$ direction. In both cases we assume an error in the radio frequency field alignment of $\pm 1.5^{\circ}$ in any direction considering $n_x =1$ as the ideal case. For these parameters the fidelity reaches values above $0.95$ for times up to $0.5$ seconds. In the inset we show the fidelity when the RWA is applied. Note that in the latter the approximation given by the Hamiltonian in Eq. (\[sup:RWA2\]) is only valid for some hundreds of microseconds. ](fidelity){width="0.6\columnwidth"}
In the case of tuning $\omega_{\rm rf} = \omega$, the RWA gives rise to the following Hamiltonian in the interaction picture of $-\omega \ I^z$ $$\label{sup:RWA}
H = \Omega_x I_x,$$ or when $\omega_{\rm rf} = \omega - \Delta$ $$\label{sup:RWA2}
H = -\Delta I_z + \Omega_x I_x.$$
However we can refine the RWA by proceeding as follows, in a rotating frame w.r.t. $\omega \ I^z$ (the counter rotating frame) we find $$H = \tilde{\omega} \ \tilde{n} \cdot \vec{I} + \Omega_x \ (I^{+} e^{i 2\omega t} + I^{-} e^{-i 2\omega t}) + 2\Omega_z \ I^z \cos{(\omega_{\rm rf} t)}.$$ By writing $\tilde{\omega} = \Delta + \xi$ and neglecting the term $2\Omega_z \ I^z \cos{(\omega_{\rm rf} t)}$ that will be not compensated with other oscillating variable we find that, in the rotating frame of $\xi \ \tilde{n} \cdot \vec{I}$ and under the resonance condition $\xi = 2\omega_{\rm rf}$, the Hamiltonian reads $$\label{sup:effectivo}
H = \bigg[ \Delta \tilde{n}_x + \frac{\Omega_x}{2} (1 - \tilde{n}_x^2 - \tilde{n}_z)\bigg] I^x + \bigg[\Delta \tilde{n}_z + \frac{\Omega_x}{2}\tilde{n}_x (1 - \tilde{n}_z)\bigg] I^z.$$
The above expression is only an approximation of the real Hamiltonian in Eq. (\[sup:real\]), however one can compute the fidelity between the propagators associated to Eqs. (\[sup:real\], \[sup:RWA\]) and Eqs. (\[sup:real\], \[sup:effectivo\]) as a function of time according to the definition $$F = \frac{|{\rm Tr}(A B^{\dag})|}{\sqrt{{\rm Tr}(A A^{\dag}) \ {\rm Tr}(B B^{\dag})}}$$ where $A$ are $B$ are two general quantum operations. In Fig. \[sup:fidelity\] we show the fidelity of the time evolution operators generated by the Hamiltonians in Eqs. (\[sup:real\], \[sup:effectivo\]) for different values of the static magnetic field. The inset shows the fidelity when the RWA is invoked. An inspection of the plots shows an improvement of almost three orders of magnitude in the fidelity when working in the counter-rotating picture.
Symmetric and anti-symmetric arrangement of AXY sequences {#symantisym}
=========================================================
 Different pulse arrangements giving rise to symmetric a), and anti-symmetric b) AXY sequences. Behind each pulse sequence the associated modulation function, $F(t)$, is plotted showing their even or odd character. The phases $\phi_j$, $\tilde{\phi}_j$ that determine the rotation axis in the X-Y plane are chosen such that $\phi_j = [\frac{\pi}{6}, 0, \frac{\pi}{2}, 0, \frac{\pi}{6}]$, and $\tilde{\phi}_j = [\frac{\pi}{6}+ \frac{\pi}{2}, \frac{\pi}{2}, \frac{\pi}{2}+ \frac{\pi}{2}, \frac{\pi}{2}, \frac{\pi}{6}+ \frac{\pi}{2}]$. Each pulse sequence is repeated $N$ times with the interpulse spacing governed by the time-variables $\delta_1$, and $\delta_2$ that allow to arbitrarily tune the $f^{e}_k$, $f^{o}_k$ coefficients. The highlighted $\pi$-pulse in a), b) represents the link between the even an odd pulse sequences, i.e. the odd sequence is displayed considering the rotation around the axis $\phi_4$ as the first pulse.](pulses){width="0.7\columnwidth"}
In order to couple the electron spin with the $I_j^x$, or $I_j^y$ components of each individual nuclei, we can choose between the symmetric and anti-symmetric versions of the AXY sequence, see Fig. \[sup:pulses\]. While the sequence in Fig. \[sup:pulses\] a) produces an even modulation function $F(t)$ such that $F(t) = \sum_{k>0} f^{s}_k \cos{(k \omega t)}$, the arrangement of pulses in Fig. \[sup:pulses\] b) gives rise to $F(t) = \sum_{k>0} f^{a}_k \sin{(k \omega t)}$ which presents an odd behavior. Note that the even an odd pulse sequences are simply related by a time-shift. More specifically, the $\pi$ pulse that rotates the electron spin along the axis defined by $\phi_4$ in Fig. \[sup:pulses\] a), i.e. in the even AXY sequence, is chosen as the first rotating pulse in the odd version of AXY. In both cases the coefficients $f^{s}_k$, $f^{a}_k$ can be arbitrarily tuned which provides with selective nuclear addressing and with high fidelity quantum gates (see discussion on the main text). In order to fit these pulse arrangements with the standard XY-$8$ sequence we can consider, in a), the composite X pulse as the pulse-set displayed from $0$ to $\frac{\tau}{2}$ while the composite Y pulse is conformed by the pulses from $\frac{\tau}{2}$ to $\tau$. In this manner the symmetric AXY-$8$ sequence corresponds to the repeated application of the XYXYYXYX pulse arrangement. The anty-simmetric AXY-$8$ sequence can be built in the same way but taking into account that the composite X and Y pulses have to be constructed according to Fig. \[sup:pulses\] b).
Decoupling Efficiency
=====================
The dynamics of interacting nuclear spins under the action of an static $B_z$ and a decoupling rf field is described by the Hamiltonian $$\begin{aligned}
H &=& -\sum_j \omega I_j^z + 2\Omega \cos{(\omega_{\rm rf} t)} \sum_j I^x_j + \sum_{j>k}\frac{\mu_{0}}{2}\frac{\gamma_n^2}{r_{j,k}^{3}} \left[\vec{I}_{j}\cdot\vec{I}_{k}-\frac{3(\vec{I}_{j}\cdot\vec{r}_{j,k})(\vec{r}_{j,k}\cdot\vec{I}_{k})}{r_{j,k}^{2}}\right]\nonumber\\
&\approx&-\sum_j \omega I_j^z + 2\Omega \cos{(\omega_{\rm rf} t)} \sum_j I^x_j + \sum_{j>k}\frac{\mu_{0}}{2}\frac{\gamma_n^2}{r_{j,k}^{3}} (1 - 3 \hat{r}_{j,k})
\left[I^z_j I^z_k -\frac{1}{2}(I^x_j I^x_k + I^y_j I^y_k ) \right].\end{aligned}$$ Then, in a rotating frame w.r.t. $\omega_{\rm rf} \sum_j I_j^z$ such that $\omega_{\rm rf} = -\omega + \Delta$ we have $$\label{sup:twonuclei}
H = -\Delta\sum_j I_j^z + \Omega \sum_j I^x_j + \sum_{j>k}\lambda_{j,k}
\left[I^z_j I^z_k -\frac{1}{2}(I^x_j I^x_k + I^y_j I^y_k ) \right],$$ where $\lambda_{j,k} = \frac{\mu_{0}}{2}\frac{\gamma_n^2}{r_{j,k}^{3}} (1 - 3 \hat{r}_{j,k})$. In order to simplify the explanation let us restrict to the simplest case of two interacting nuclei because the structure of the above Hamiltonian allows to straightforwardly extend the argument to many particles. In this manner we can consider the problem $$H = \omega_1 \tilde{I}^z_1 + \omega_2 \tilde{I}^z_2 + \lambda_{1,2} \left[I^z_1 I^z_2 -\frac{1}{2}(I^x_1 I^x_2 + I^y_1 I^y_2 ) \right],$$ where $\tilde{I}_{j}^z = \left[\frac{-\Delta}{\sqrt{\Delta^2 + \Omega^2 }}I_{j}^z + \frac{\Omega}{\sqrt{\Delta^2 + \Omega^2 }}I_{j}^x \right]$, for $j=1,2$ and $\omega_1 = \omega_2 = \sqrt{\Delta^2 + \Omega^2 }$. Then, one can define a new base in terms of the $\tilde{I}^\alpha_j$ operators as $$\begin{aligned}
\tilde{I}_j^z &=& e^{-i\theta I_j^y} \tilde{I}_j^z e^{i\theta I_j^y} =\cos{\theta} \ I_j^z + \sin{\theta} \ I_j^x,\nonumber\\
\tilde{I}_j^x &=& e^{-i\theta I_j^y} I_j^x e^{i\theta I_j^x} =\cos{\theta} \ I_j^x - \sin{\theta} \ I_j^z,\nonumber\\
\tilde{I}_j^y &=& I_j^y, \end{aligned}$$ or, conversely $$\begin{aligned}
I_j^z &=& e^{i\theta \tilde{I}_j^y} \tilde{I}_j^z e^{-i\theta \tilde{I}_j^y} = \cos{\theta} \ \tilde{I}_j^z - \sin{\theta} \ \tilde{I}_j^x,\nonumber\\
I_j^x &=& e^{i\theta \tilde{I}_j^y} \tilde{I}_j^x e^{-i\theta \tilde{I}_j^y} = \cos{\theta} \ \tilde{I}_j^x + \sin{\theta} \ \tilde{I}_j^z,\nonumber\\
I_j^y &=& \tilde{I}_j^y,\end{aligned}$$ where $\cos{\theta} = \frac{-\Delta}{\sqrt{\Delta^2 + \Omega^2 }}$ and $\sin{\theta} = \frac{\Omega}{\sqrt{\Delta^2 + \Omega^2 }}$.
Now it is easy to demonstrate that $$H = \omega_1 \tilde{I}^z_1 + \omega_2 \tilde{I}^z_2 + \lambda_{1,2} \left[I^z_1 I^z_2 -\frac{1}{2}(I^x_1 I^x_2 + I^y_1 I^y_2 ) \right] \approx \omega_1 \tilde{I}^z_1 + \omega_2 \tilde{I}^z_2$$ when $\theta = \theta_m=\arctan{( \sqrt{2} )} \approx 54.7^\circ$, i.e. equals to the magic angle [@Lee65].
In our case, and considering only the terms in Eq. (\[sup:demons\]) which are relevant for the demonstration, we have that the equivalent expression to Eq. (\[sup:twonuclei\]) is $$H = \sum_j \bigg\{ \big[ \Delta \tilde{n}_x + \frac{\Omega_x}{2} (1 - \tilde{n}_x^2 - \tilde{n}_z) + \frac{m_s}{2} \tilde{n}_z \ \tilde{n}_x \ A_j^z \big] \ I_j^x + \big[\Delta \tilde{n}_z + \frac{\Omega_x}{2}\tilde{n}_x (1 - \tilde{n}_z)+ \frac{m_s}{2} \ \tilde{n}^2_z \ A_j^z \big] \ I_j^z \bigg\}v+ e^{i \xi \sum_j\tilde{n} \cdot \vec{I}_j t} \tilde{H}_{nn} e^{-i \xi \sum_j\tilde{n} \cdot \vec{I}_j t}.$$ Now taking into account that $\tilde{n} = (\frac{\Omega_x}{\tilde\omega}, 0, -\frac{\omega + \omega_{\rm rf}}{\tilde\omega}) \approx (0, 0, -1)$, which is a reasonable assumption in the parameter regime we are considering, we can simplify the above expression to find
$$\label{generalsimplifyed}
H \approx \sum_j \Omega_x \ I_j^x + \sum_j \big[- \Delta + \frac{m_s}{2} \ A_j^z \big] \ I_j^z + \tilde{H}_{nn} = \sum_j \Omega_x \ I_j^x -\Delta \sum_j \big[1 - \frac{m_s A_j^z}{2 \Delta} \big] \ I_j^z + \tilde{H}_{nn}.$$
In this manner one can apply the same formalism to search for the magic angle condition but including a correction of the order $\frac{ A_j^z}{2 \Delta}$ (note that $|m_s| = 1$). Hence we can define a new basis such that $$\begin{aligned}
I_j^z &=& e^{i\theta_j \tilde{I}_j^y} \tilde{I}_j^z e^{-i\theta_j \tilde{I}_j^y} \approx e^{i\theta_m \tilde{I}_j^y} \tilde{I}_j^z e^{-i\theta_m \tilde{I}_j^y} (1 + O\left[\big| A_j^z /2 \Delta \big|\right] )= \cos{\theta_m} \ \tilde{I}_j^z - \sin{\theta_m} \ \tilde{I}_j^x + O\left[\big| A_j^z /2 \Delta \big|\right],\nonumber\\
I_j^x &=& e^{i\theta_j \tilde{I}_j^y} \tilde{I}_j^x e^{-i\theta_j \tilde{I}_j^y} \approx e^{i\theta_m \tilde{I}_j^y} \tilde{I}_j^x e^{-i\theta_m \tilde{I}_j^y} (1 + O\left[\big| A_j^z /2 \Delta \big|\right] )= \cos{\theta} \ \tilde{I}_j^x + \sin{\theta} \ \tilde{I}_j^z + O\left[\big| A_j^z /2 \Delta \big|\right] ,\nonumber\\
I_j^y &=& \tilde{I}_j^y,\end{aligned}$$ In this manner we have that the Hamiltonian in Eq. (\[generalsimplifyed\]) is $$H \approx \sum_j \Omega_x \ I_j^x + \sum_j \big[- \Delta + \frac{m_s}{2} \ A_j^z \big] \ I_j^z + O\left[\big| A_j^z /2 \Delta \big|\right].$$ Therefore the each nuclear-nuclear interaction is suppressed by a factor of $\big|A_j^z /2 \Delta\big|$.
Numerical simulations
=====================
Figure 2 sample
---------------
Figure $2$ of the main text simulate a sample consisting on an NV center and a nuclear spin cluster located at positions $$\begin{aligned}
\vec{r}_1 &=& [-0.1262, \ 0.8016, \ 0.2061 ] \ \mbox{nm},\\\nonumber
\vec{r}_2 &=& [0.2524, \ -0.5830, \ -0.3607 ] \ \mbox{nm},\\\nonumber
\vec{r}_3 &=& [0.2524, \ -0.7287, \ -0.4122 ] \ \mbox{nm}.\end{aligned}$$ Figure $2$ a) uses $B_z =0.1$ T, while in Fig. $1$ b) a high magnetic field of $B_{z} = 2$ T, and a value of $\Delta = 2\pi \times 100$ kHz have been employed.
Table 1 sample
--------------
In Table $1$ we used $$\begin{aligned}
\vec{r}_1 &=& [0.1262, \ 0.8016, \ 0.8245] \ \mbox{nm},\\\nonumber
\vec{r}_2 &=& [-0.6311, \ -0.2186, \ 0.6183 ] \ \mbox{nm},\\\nonumber
\vec{r}_3 &=& [-0.6311, \ -0.2644, \ 0.8760] \ \mbox{nm},\end{aligned}$$ with $B_z = 2$ T and $\Delta = 2\pi \times 100$ kHz.
Fidelities under strong error conditions
----------------------------------------
In Table \[sup:table1\] We show the fidelities for the sample used in the main text under strong error conditions $\Lambda = 2\pi \times 2$ MHz, and Rabi frequency error of $5 \%$.
[[ |c | c | c | c | c| c|]{}]{} $F_{-,+}$ & exp$( \mp i\frac{\pi}{2} \sigma_z I_j^x)$& exp$(\mp i\frac{\pi}{2} \sigma_z I_j^y)$ & exp$( \mp i\frac{\pi}{2} I_j^x)$ & exp$( \mp i\frac{\pi}{2} I_j^y)$\
Spin$_1$ & $ 0.9858, 0.9861 $ &$0.9766, 0.9831$ & $0.9861, 0.9862$ &$0.9839, 0.9849$\
Spin$_2$ & $0.9802, 0.9802$ &$0.9684, 0.9773$ & $ 0.9924, 0.9924$ & $0.9924, 0.9923$\
Spin$_3$ & $0.9893, 0.9852$ &$0.9738, 0.9778$ & $0.9917, 0.9915$ & $ 0.9911, 0.9913$\
Nuclear bath contribution
-------------------------
To estimate the effects of the nuclear environment we deal with $10$ different samples containing $200$ $^{13}$C nuclei, i.e. with $10$ different nuclear distributions that interfere with the NV center and the three qubit nuclear register, and obtain the results outlined in Table \[sup:table2\]. Note that because of machine restrictions we deal with instantaneous microwave pulses.
[[ |c | c | c | c| c|]{}]{} & $L$ & $d_{\rm min}$ (nm) & $d_{\rm max}$ (nm)\
Sample 1 & $0.9939$ & $1.4584$ &$4.7429$\
Sample 2 & $0.9949$ & $1.5433$ &$4.8623 $\
Sample 3 & $0.9921$ & $1.3913$ & $4.8492$\
Sample 4 & $0.9932$&$1.4140$ &$4.8096$\
Sample 5 & $0.9946$&$ 1.5840$ &$4.8096$\
Sample 6 & $0.9936$&$ 1.5840$ &$4.7319$\
Sample 7 & $0.9938$&$ 1.5433$ &$4.7387$\
Sample 8 & $0.9926$&$ 1.3117$ &$4.8360$\
Sample 9 & $0.9932$&$1.6934$ &$4.7722$\
Sample 10 & $0.9940$&$1.5225$ &$4.8492$\
[37]{} J. Casanova, Z.-Y. Wang, J. F. Haase, and M. B. Plenio, Phys. Rev. A [**92**]{}, 042304 (2015). L. Allen and J. H. Eberly, [*Optical Resonance and Two-Level Atoms*]{} (John Wiley & Sons, 1975). M. Lee and W. I. Goldburg, Phys. Rev. [**140**]{}, A1261 (1965).
|
---
abstract: 'In this paper, we strengthen the competitive analysis results obtained for a fundamental online streaming problem, the Frequent Items Problem. Additionally, we contribute with a more detailed analysis of this problem, using alternative performance measures, supplementing the insight gained from competitive analysis. The results also contribute to the general study of performance measures for online algorithms. It has long been known that competitive analysis suffers from drawbacks in certain situations, and many alternative measures have been proposed. However, more systematic comparative studies of performance measures have been initiated recently, and we continue this work, using competitive analysis, relative interval analysis, and relative worst order analysis on the Frequent Items Problem.'
author:
- |
Joan Boyar Kim S. Larsen Abyayananda Maiti\
University of Southern Denmark\
Odense, Denmark\
[{joan,kslarsen,abyaym}@imada.sdu.dk]{}
bibliography:
- 'ref\_stream.bib'
title: |
The Frequent Items Problem in\
Online Streaming under\
Various Performance Measures[^1]
---
Introduction
============
The analysis of problems and algorithms for streaming applications, treating them as online problems, was started in [@Becchetti09]. In online streaming, the items must be processed one at a time by the algorithm, making some irrevocable decision with each item. A fixed amount of resources is assumed. In the frequent items problem [@CH08], an algorithm must store an item, or more generally a number of items, in a buffer, and the objective is to store the items appearing most frequently in the entire stream. This problem has been studied in [@Giannakopoulos12]. In addition to probabilistic considerations, they analyzed deterministic algorithms using competitive analysis. We analyze the frequent items problem using relative interval analysis [@Dorrigiv09] and relative worst order analysis [@Boyar07]. In addition, we tighten the competitive analysis [@ST85; @KMRS88] results from [@Giannakopoulos12].
It has been known since the start of the area that competitive analysis does not always give good results [@ST85] and many alternatives have been proposed. However, as a general rule, these alternatives have been fairly problem specific and most have only been compared to competitive analysis. A more comprehensive study of a larger number of performance measures on the same problem scenarios was initiated in [@BIL09p] and this line of work has been continued in [@BLM12p; @BGL12p; @BGLtap]. With this in mind, we would like to produce complete and tight results, and for that reason, we focus on a fairly simple combinatorial problem and on simple algorithms for its solution, incorporating greediness and adaptability trade-offs to a varying extent.
Finally, we formalize a notion of competitive function, as opposed to competitive ratio, in a manner which allows us to focus on the constant in front of the high order term. These ideas are also used to generalize relative worst order analysis.
Preliminaries {#sec:prob}
=============
This is a streaming problem, but as usual in online algorithms we use the term sequence or input sequence to refer to a stream. We denote an *input sequence* by $I = a_1, a_2,\ldots, a_n$, where the items $a_i$ are from some universe $\mathcal{U}$, assumed to be much larger than $n$. We may refer to the index also as the *time step*. We consider online algorithms, which means that items are given one by one. We consider the simplest possible frequent items problem: An algorithm has a *buffer* with space for one item. When processing an item, the algorithm can either discard the item or replace the item in the buffer by the item being processed. The objective is to keep the most frequently occurring items in the buffer, where frequency is measured over the entire input, i.e., when an algorithm must make a decision, the quality of the decision also depends on items not yet revealed to the algorithm. We define this objective function formally:
Given an online algorithm $\mathcal{A}$ for this problem, we let $s^{\mathcal{A}}_t$ denote *the item in the buffer at time step $t$*. We may omit the superscript when it is clear from the context which algorithm we discuss.
Given an input sequence $I$ and an item $a\in \mathcal{U}$, the [*frequency*]{} of the item is defined as $f_I(a) = \frac{n_I(a)}{n}$, where $n_I(a) = | \{i\mid a_i=a\}|$ is the number of occurrences of $a$ in $I$. The objective is to maximize the *aggregate frequency* [@Giannakopoulos12], defined by $F_{{{\ensuremath{\mathcal{A}}}\xspace}}(I)=\sum_{t=1}^n f_I(s^{\mathcal{A}}_t)$, i.e., the sum of the frequencies of the items stored in the buffer over the time.
We compare the quality of the achieved aggregate frequencies of three different deterministic online algorithms from [@Giannakopoulos12]: the naive algorithm ([[$\textsc{Nai}$]{}]{}), the eager algorithm ([[$\textsc{Eag}$]{}]{}), and the majority algorithm ([[$\textsc{Maj}$]{}]{}). All three are practical streaming algorithms, being simple and using very little extra space.
\[def:nav\] [[$\textsc{Nai}$]{}]{}buffers every item as it arrives, i.e., $s^{{{\ensuremath{\textsc{Nai}}}\xspace}}_t = a_t$ for all $t=1,2,\ldots, n$.
The algorithm [[$\textsc{Eag}$]{}]{}switches mode upon detecting a [*repeated item*]{}, an item which occurs in two consecutive time steps.
\[def:eag\] Initially, [[$\textsc{Eag}$]{}]{}buffers every item as it arrives. If it finds a repeated item, then it keeps that item until the end, i.e., let $$t^* =\min_{1\leq t\leq n-1} \{t\mid a_t = a_{t+1}\},$$ if such a $t$ exists, and otherwise $t^* = n$. Then [[$\textsc{Eag}$]{}]{}is the algorithm with $s^{{{\ensuremath{\textsc{Eag}}}\xspace}}_t = a_t$ for all $t\leq t^*$ and $s^{{{\ensuremath{\textsc{Eag}}}\xspace}}_t = a_{t^*}$ for all $t>t^*$.
The algorithm [[$\textsc{Maj}$]{}]{}keeps a counter along with the buffer. Initially, the counter is set to zero.
\[def:maj\] If the counter is zero, then [[$\textsc{Maj}$]{}]{}buffers the arriving item and sets the counter to one. Otherwise, if the arriving item is the same as the one currently buffered, [[$\textsc{Maj}$]{}]{}increments the counter by one, and otherwise decrements it by one.
Finally, as usual in online algorithms, we let [[$\textsc{Opt}$]{}]{}denote an optimal offline algorithm. [[$\textsc{Opt}$]{}]{}is, among other things, used in competitive analysis as a reference point, since no online algorithm can do better. If ${{\ensuremath{\mathcal{A}}}\xspace}$ is an algorithm, we let ${{\ensuremath{\mathcal{A}}}\xspace}(I)$ denote the result (profit) of the algorithm, i.e., ${{\ensuremath{\mathcal{A}}}\xspace}(I)=F_{{{\ensuremath{\mathcal{A}}}\xspace}}(I)$.
In comparing these three algorithms, we repeatedly use the same two families of sequences; one where [[$\textsc{Eag}$]{}]{}performs particularly poorly and one where [[$\textsc{Maj}$]{}]{}performs particularly poorly.
\[sequences\] We define the sequences $$E_n = a,a,b,b,\ldots,b,$$ where there are $n-2$ copies of $b$, and $$W_n = \left\{ \begin{array}{ll}
a_1,a_0,a_2,a_0,\ldots, a_{\frac{n}{2}},a_0 & \mbox{for even $n$}\\
a_1,a_0,a_2,a_0,\ldots, a_{\lfloor\frac{n}{2}\rfloor},a_0,
a_{\lceil\frac{n}{2}\rceil} & \mbox{for odd $n$}.
\end{array} \right.$$
The four algorithms, including [[$\textsc{Opt}$]{}]{}, obtain the aggregate frequencies below on these two families of sequences. The arguments are simple, but fundamental, and also serve as an introduction to the algorithmic behavior of these algorithms.
\[prop\_sequences\] The algorithms’ results on $E_n$ and $W_n$ are as in Fig. \[fig-profit\].
$\begin{array}{|l||c|c|} \hline
& E_n & W_n \\ \hline\hline
\rule[-3ex]{0em}{7ex}{{\ensuremath{\textsc{Nai}}}\xspace}& n-4+\frac{8}{n} &
\left\{ \begin{array}{ll}
\frac{n}{4} + \frac{1}{2} & \mbox{for even $n$}\\[.5ex]
\frac{n}{4} + \frac{3}{4n} & \mbox{for odd $n$}
\end{array} \right. \\ \hline
\rule[-2ex]{0em}{5ex}{{\ensuremath{\textsc{Eag}}}\xspace}& 2 & \mbox{as {{\ensuremath{\textsc{Nai}}}\xspace}} \\ \hline
\rule[-2ex]{0em}{5ex}{{\ensuremath{\textsc{Maj}}}\xspace}& n-6+\frac{16}{n} & 1 \\ \hline
\rule[-3ex]{0em}{7ex}{{\ensuremath{\textsc{Opt}}}\xspace}& \mbox{as {{\ensuremath{\textsc{Nai}}}\xspace}} &
\left\{ \begin{array}{ll}
\frac{n}{2} - \frac{1}{2} + \frac{1}{n} & \mbox{for even $n$}\\[.5ex]
\frac{n}{2} -1 + \frac{3}{2n} & \mbox{for odd $n$}
\end{array} \right. \\ \hline
\end{array}$
In $E_n$, the frequency of $a$ is $\frac{2}{n}$ and the frequency of $b$ is $\frac{n-2}{n}$. Thus ${{\ensuremath{\textsc{Nai}}}\xspace}(E_n)= 2\frac{2}{n}+ (n-2)\frac{n-2}{n} = n-4+\frac{8}{n}$. In $W_n$, the frequency of $a_0$ is $\lfloor\frac{n}{2}\rfloor/n$, and the frequencies of all the other $a_i$, $1\leq i \leq \lceil \frac{n}{2}\rceil$, are $\frac{1}{n}$. Thus, ${{\ensuremath{\textsc{Nai}}}\xspace}(W_n)= \lceil\frac{n}{2}\rceil\frac{1}{n} + \lfloor \frac{n}{2} \rfloor \frac{\lfloor\frac{n}{2}\rfloor}{n}$. Considering both even and odd $n$ gives the required result.
When processing $E_n$, [[$\textsc{Eag}$]{}]{}keeps $a$ in its buffer. Hence, ${{\ensuremath{\textsc{Eag}}}\xspace}(E_n)= n\frac{2}{n} = 2$. Since $W_n$ has no repeated item, ${{\ensuremath{\textsc{Eag}}}\xspace}(W_n)={{\ensuremath{\textsc{Nai}}}\xspace}(W_n)$.
For $E_n$, [[$\textsc{Maj}$]{}]{}will have $a$ in its buffer for the first four time steps, so ${{\ensuremath{\textsc{Maj}}}\xspace}(E_n)$ is $4\frac{2}{n}+(n-4)\frac{n-2}{n} = n-6+\frac{16}{n}$. For $W_n$, [[$\textsc{Maj}$]{}]{}brings each $a_i$, $1\leq i\leq n$, into its buffer and never brings $a_0$ into its buffer. Thus, ${{\ensuremath{\textsc{Maj}}}\xspace}(W_n) = n\frac{1}{n} = 1$.
With $E_n$, [[$\textsc{Opt}$]{}]{}is forced to perform the same as [[$\textsc{Nai}$]{}]{}. In $W_n$, [[$\textsc{Opt}$]{}]{}must buffer $a_1$ in the first time step, but it buffers $a_0$ for the remainder of the sequence. Thus, ${{\ensuremath{\textsc{Opt}}}\xspace}(W_n) = \frac{1}{n}+(n-1)\frac{\lfloor \frac{n}{2}\rfloor}{n}$. Considering both even and odd $n$ gives the required result.
\[def:worstpermut\] Let ${\cal A}$ be any online algorithm. We denote the worst aggregate frequency of ${\cal A}$ over all the permutations $\sigma$ of $I$ by ${\cal A}_{W}(I) = \min_{\sigma} {\cal A} (\sigma(I))$.
It is convenient to be able to consider items in order of their frequencies. Let $D(I) = a_1', a_2',\ldots, a_n'$ be a sorted list of the item in $I$ in nondecreasing order of frequencies. For example, if $I = a, b, c, a, b, a$, then $D(I) = c,b,b,a,a,a$. We will use the notation $D(I)$ throughout the paper.
\[lem:wmaj\] For odd $n$, ${{\ensuremath{\textsc{Maj}}}\xspace}_{W}(I) = 2\sum_{i=1}^{\lfloor \frac{n}{2} \rfloor} f_I(a_i')
+ f_I(a_{\lceil \frac{n}{2}\rceil}')$, and for even $n$, ${{\ensuremath{\textsc{Maj}}}\xspace}_{W}(I) = 2\sum_{i=1}^{\frac{n}{2}} f_I(a_i')$, where the $a_i'$ are the items of $D(I)$.
Every time step where the counter is decremented can be paired with an earlier one where it is incremented and the same item is in the buffer. So, at least $\lceil\frac{n}{2}\rceil$ requests contribute to the aggregate frequency of the algorithm. One can order the items so that exactly the $\lceil \frac{n}{2}\rceil$ requests to that many least frequent items are buffered as follows: Assuming $n$ is even, then the worst permutation is $a_1', a_n', a_2',a_{n-1}',\ldots a_{\frac{n}{2}}', a_{\frac{n}{2}+1}'$. All (but the last request when $n$ is odd) of the requests which lead to an item entering the buffer contribute twice, since they are also in the buffer for the next step.
Competitive Analysis {#sec:comp}
====================
An online streaming problem was first studied from an online algorithms perspective using competitive analysis by Becchetti and Koutsoupias [@Becchetti09]. Competitive analysis[@ST85; @KMRS88] evaluates an online algorithm in comparison to an optimal offline algorithm. For a maximization problem, an algorithm, [[$\mathcal{A}$]{}]{}is called $c$-competitive, for some constant $c$, if there exists a constant $\alpha$ such that for all finite input sequences $I$, ${{\ensuremath{\textsc{Opt}}}\xspace}(I) \leq c\cdot {{\ensuremath{\mathcal{A}}}\xspace}(I) +\alpha$. The competitive ratio of [[$\mathcal{A}$]{}]{}is the infimum over all $c$ such that [[$\mathcal{A}$]{}]{}is $c$-competitive. Since, for the online frequent items problem, the relative performance of algorithms depends on the length of $I$, we define a modified and more general version of competitive analysis, providing a formal basis for our own claims as well as claims made in earlier related work. Functions have also been considered in [@DorrigivL05]. Here, we focus on the constant in front of the most significant term. Our definition can be adapted easily to minimization problems in the same way that the adaptations are handled for standard competitive analysis. In all these definitions, when $n$ is not otherwise defined, we use it to denote $|I|$, the length of the sequence $I$. As usual, when using asymptotic notation in inequalities, notation such as $f(n) \leq g(n) + o(g(n))$ means that there exists a function $h(n)\in o(g(n))$ such that $f(n) \leq g(n) + h(n)$. Thus, we focus on the multiplicative factors that relate the online algorithm’s result to the input length.
\[def:com\] An algorithm [[$\mathcal{A}$]{}]{}is $f(n)$-[*competitive*]{} if $$\forall I{\!:\;}{{\ensuremath{\textsc{Opt}}}\xspace}(I) \leq (f(n)+o(f(n)))\cdot{{\ensuremath{\mathcal{A}}}\xspace}(I).$$ [[$\mathcal{A}$]{}]{}has [*competitive function*]{} $f(n)$ if [[$\mathcal{A}$]{}]{}is $f(n)$-[*competitive*]{} and for any $g(n)$ such that [[$\mathcal{A}$]{}]{}is $g(n)$-[*competitive*]{}, $\lim_{n \rightarrow \infty}\frac{f(n)}{g(n)} \leq 1$.
If algorithm [[$\mathcal{A}$]{}]{}has [*competitive function*]{} $f(n)$ and algorithm [[$\mathcal{B}$]{}]{}has [*competitive function*]{} $f'(n)$, then [[$\mathcal{A}$]{}]{}is better than [[$\mathcal{B}$]{}]{}according to competitive analysis if $\lim_{n \rightarrow \infty}\frac{f(n)}{f'(n)} < 1$.
Thus, the concept of competitive function is an exact characterization up to the level of detail we focus on. It can be viewed as an equivalence relation, and if $\lim_{n\rightarrow\infty}\frac{f(n)}{g(n)}=1$ for two functions $f(n)$ and $g(n)$, then they belong to (and are representatives of) the same equivalence class. For example, $\frac{\sqrt{n}}{2}$ and $\frac{\sqrt{n}}{2-\frac{1}{\sqrt{n}}}$ are considered equivalent, whereas $\frac{\sqrt{n}}{2}$ and $\frac{\sqrt{n}}{4}$ are not.
All three algorithms discussed here are non-competitive according to the original definition. However, information regarding the relative quality of these algorithms can be obtained by considering the most significant constants from the corresponding functions. Giannakopoulos et al. has proved that no randomized algorithm for the online frequent items problem, where the buffer has room for one item, can have a competitive function better than $\frac{1}{3}\sqrt{n}$ [@Giannakopoulos12]. That result can be strengthened for the deterministic case:
\[thm:comp\_all\] No deterministic algorithm for the online frequent items problem can have a competitive function better than $\frac{\sqrt{n}}{2}$.
Consider any deterministic algorithm $\mathcal{A}$, and input of the form $$I_n = a_1, a_2, \ldots a_{n-\sqrt{n}}, x, x, \ldots, x$$ where the first $n-\sqrt{n}$ items are distinct and the last $\sqrt{n}$ items are identical. Since [[$\mathcal{A}$]{}]{}is deterministic, an adversary will know whether $a_1$ or $a_2$ is in the buffer upon completion of time step 2. The value of $x$ is based on this. If it is $a_2$, then the adversary sets $x=a_1$, and if it is $a_1$, then it sets $x=a_2$. As $x$ does not occur among the next $n-\sqrt{n}-2$ items, $\mathcal{A}$ has no chance of bringing $x$ into its buffer until the last $\sqrt{n}$ items arrive, so it stores $x$ in its buffer at most $\sqrt{n}+1$ times. ${{\ensuremath{\textsc{Opt}}}\xspace}$ stores $x$ at least $n-1$ times. That gives the ratio of $$\begin{aligned}
\frac{{{\ensuremath{\textsc{Opt}}}\xspace}(I_n)}{\mathcal{A}(I_n)} &\geq& \frac{\frac{1}{n} + (n-1)\frac{\sqrt{n} + 1}{n}}{(n-\sqrt{n} - 1)\frac{1}{n} +
(\sqrt{n} + 1)\frac{\sqrt{n} + 1}{n}} \nonumber
\\
&=& \frac{1 + (n-1)(\sqrt{n} +1)}{n- \sqrt{n} -1 + (\sqrt{n} + 1)^2} \nonumber
\\
&=& \frac{n + \sqrt{n} -1}{2\sqrt{n} + 1} \nonumber
\\
&\geq& \frac{\sqrt{n}}{2}, \nonumber \mbox{ for $n\geq 4$}\end{aligned}$$
In [@Giannakopoulos12], Giannakopoulos et al. proved that for all sequences $I$ of length $n$, ${{\ensuremath{\textsc{Opt}}}\xspace}(I)\leq \sqrt{n}\cdot{{\ensuremath{\textsc{Nai}}}\xspace}(I)$. Here we give a tighter result for [[$\textsc{Nai}$]{}]{}.
\[thm:comp\_nav\] [[$\textsc{Nai}$]{}]{}has competitive function $\frac{\sqrt{n}}{2}$. It is an optimal deterministic online algorithm for the frequent items problem.
Let $f$ be the frequency of the most frequent item in the input sequence $I$. Since the lowest possible frequency of an item is $\frac{1}{n}$, $${{\ensuremath{\textsc{Nai}}}\xspace}(I) \geq nf^2 + (n-nf)\frac{1}{n}
\mbox{ and }
{{\ensuremath{\textsc{Opt}}}\xspace}(I) \leq nf$$ Thus, $$\begin{aligned}
\frac{{{\ensuremath{\textsc{Opt}}}\xspace}(I)}{{{\ensuremath{\textsc{Nai}}}\xspace}(I)} &\leq& \frac{nf}{nf^2 + 1 -f} \label{eq:comp_nav}\end{aligned}$$ The right hand side of Ineq. \[eq:comp\_nav\] reaches its maximum when $f=\frac{1}{\sqrt{n}}$. Substituting this value into Ineq. \[eq:comp\_nav\], we get the result: $$\begin{aligned}
\frac{{{\ensuremath{\textsc{Opt}}}\xspace}(I)}{{{\ensuremath{\textsc{Nai}}}\xspace}(I)} &\leq& \frac{\sqrt{n}}{2-1/\sqrt{n}} = \frac{\sqrt{n}}{2} + \frac{1}{2(2-1/\sqrt{n})} \nonumber
$$ Thus, [[$\textsc{Nai}$]{}]{}is a $\frac{\sqrt{n}}{2}$-competitive algorithm and, by Theorem \[thm:comp\_all\], it is optimal.
For [[$\textsc{Maj}$]{}]{}Giannakopoulos et al. [@Giannakopoulos12] proved a competitive ratio of $\varTheta(n)$. We give the asymptotically tight bounds, including the multiplicative factor.
\[thm:comp\_maj\] [[$\textsc{Maj}$]{}]{}has competitive function $\frac{n}{2}$.
For the lower bound, consider the family of sequences, $W_n$, from Definition \[sequences\]. By Proposition \[prop\_sequences\], ${{\ensuremath{\textsc{Maj}}}\xspace}(W_n) = 1$, and $${{\ensuremath{\textsc{Opt}}}\xspace}(W_n) = \left\{ \begin{array}{ll}
\frac{n}{2} - \frac{1}{2} + \frac{1}{n} & \mbox{for even $n$}\\[1ex]
\frac{n}{2} -1 + \frac{3}{2n} & \mbox{for odd $n$}
\end{array} \right.$$ Consequently, $ {{\ensuremath{\textsc{Opt}}}\xspace}(W_n) \geq \frac{n}{2}{{\ensuremath{\textsc{Maj}}}\xspace}(W_n) - 1.$ Thus, the competitive function cannot be better than $\frac{n}{2}$.
For the upper bound, let $f$ be the largest frequency of any item in some input sequence $I$ of length $n$. [[$\textsc{Opt}$]{}]{}cannot have an aggregate frequency larger than $nf$.
If $f \leq \frac{1}{2}$, then, since no algorithm can have an aggregate frequency less than one in total, $\frac{{{\ensuremath{\textsc{Opt}}}\xspace}(I)}{{{\ensuremath{\textsc{Maj}}}\xspace}(I)}\leq nf\leq \frac{n}{2}$.
It remains to consider the range $\frac{1}{2} < f \leq 1$. Let $a_0$ denote the most frequent item in $I$. Note that $a_0$ must be in the buffer at some point since $f>\frac{1}{2}$. Since there are $n-fn$ items different from $a_0$, the total length of all subsequences where $a_0$ is not in the buffer is at most $2(n-fn)$. This means that $a_0$ [*is*]{} in the buffer at least $n - 2(n-fn) = 2fn - n$ times, collecting at least $(2fn - n)f=2nf^2-nf$. The remaining items collect at least $2(n-fn)\frac{1}{n}$. In total, this amounts to $2nf^2-nf+2-2f$. If we can prove that this quantity is at least $2f$ for large $n$, then asymptotically, $\frac{{{\ensuremath{\textsc{Opt}}}\xspace}(I)}{{{\ensuremath{\textsc{Maj}}}\xspace}(I)}\leq \frac{nf}{2nf^2-nf+2-2f}
\leq\frac{nf}{2f}=\frac{n}{2}$ and we will be done. Now, $ 2nf^2-nf+2-2f \geq 2f $ if and only if $ 2nf^2-(n+4)f+2\geq 0 $. Taking the derivative of the left side shows that the left side is an increasing function of $f$ for $n\geq 4$ and $f\geq \frac{1}{2}$. Thus, ${{\ensuremath{\textsc{Opt}}}\xspace}(I) \leq \frac{n}{2}{{\ensuremath{\textsc{Maj}}}\xspace}(I)$ holds for all $f$ and all $n\geq 4$. This implies that [[$\textsc{Maj}$]{}]{}is $\frac{n}{2}$-competitive and, combined with the lower bound result, that the competitive function of [[$\textsc{Maj}$]{}]{}is $\frac{n}{2}$.
\[thm:comp\_eag\] The competitive function of the algorithm [[$\textsc{Eag}$]{}]{}is $\frac{n}{2}$.
For the lower bound, consider the family of sequences, $E_n$, from Definition \[sequences\]. By Proposition \[prop\_sequences\], ${{\ensuremath{\textsc{Eag}}}\xspace}(E_n) =2$, and ${{\ensuremath{\textsc{Opt}}}\xspace}(E_n) = n-4+\frac{8}{n}$. Thus, ${{\ensuremath{\textsc{Opt}}}\xspace}(E_n) = \frac{n}{2}{{\ensuremath{\textsc{Eag}}}\xspace}(E_n) -4 + \frac{8}{n}$, and [[$\textsc{Eag}$]{}]{}’s competitive function cannot be better than $\frac{n}{2}$.
If there are no repeated items in $I$, then [[$\textsc{Eag}$]{}]{}behaves like [[$\textsc{Nai}$]{}]{}and that will give ${{\ensuremath{\textsc{Opt}}}\xspace}(I) \leq (\frac{\sqrt{n}}{2}+o(\sqrt{n})){{\ensuremath{\textsc{Eag}}}\xspace}(I)$ by Theorem \[thm:comp\_nav\]. It is evident from the lower bound result that the competitive function for [[$\textsc{Eag}$]{}]{}is worse than $\frac{\sqrt{n}}{2}$, so we assume that there is at least one repeated item in $I$. Let time steps $p+1$ and $p+2$ be the first occurrence of a repeated item in $I$. Let $b$ be the most frequent item in $I$. Note that $b$ is not necessarily the item which arrived at time steps $p+1$ and $p+2$. After $p$, all the items could conceivably be $b$, but among the first $p$ items, at most $\frac{p}{2}$ items can be $b$, because $p+1$ and $p+2$ are the indices of the first repeated item. So, an upper bound on the maximum frequency, $f_I(b)$, is $\frac{n-p + \frac{p}{2}}{n} = \frac{n-\frac{p}{2}}{n}$. This gives an upper bound of ${{\ensuremath{\textsc{Opt}}}\xspace}(I) \leq n\frac{n-\frac{p}{2}}{n} = n-\frac{p}{2}$. Now we consider a lower bound on ${{\ensuremath{\textsc{Eag}}}\xspace}(I)$. In the worst case for [[$\textsc{Eag}$]{}]{}, all the items before $p+1$ are distinct, so their contribution to ${{\ensuremath{\textsc{Eag}}}\xspace}(I)$ is at least $\frac{p}{n}$. In the worst case for [[$\textsc{Eag}$]{}]{}, the item that occurs at time steps $p+1$ and $p+2$ has frequency $\frac{2}{n}$, so the contribution to ${{\ensuremath{\textsc{Eag}}}\xspace}(I)$ from the items after $p$ is at least $(n-p)\frac{2}{n}$. Thus, ${{\ensuremath{\textsc{Eag}}}\xspace}(I) \geq \frac{p}{n} + (n-p)\frac{2}{n} = 2 - \frac{p}{n}$, and $$\frac{{{\ensuremath{\textsc{Opt}}}\xspace}(I)}{{{\ensuremath{\textsc{Eag}}}\xspace}(I)} \leq \frac{n-\frac{p}{2}}{2-\frac{p}{n}}=\frac{n}{2}.$$ Hence, [[$\textsc{Eag}$]{}]{}has competitive function $\frac{n}{2}$.
Relative Interval Analysis {#sec:realint}
==========================
Dorrigiv et al. [@Dorrigiv09] proposed another analysis method, relative interval analysis, in the context of paging. Relative interval analysis compares two online algorithms directly, i.e., it does not use the optimal offline algorithm as the baseline of the comparison. It compares two algorithms on the basis of the rate of the outcomes over the length of the input sequence rather than their worst case behavior. Here we define this analysis for maximization problems for two algorithms $\mathcal{A}$ and $\mathcal{B}$, following [@Dorrigiv09].
\[def:relint\] Define $$\operatorname{\textrm{Min}}_{\mathcal{A}, \mathcal{B}}(n) = \min_{|I| = n} \left \lbrace \mathcal{A}(I) - \mathcal{B}(I)\right\rbrace
\mbox{~and~}
\operatorname{\textrm{Max}}_{\mathcal{A}, \mathcal{B}}(n) = \max_{|I| = n} \left \lbrace \mathcal{A}(I) - \mathcal{B}(I)\right\rbrace,$$ and $$\operatorname{\textrm{Min}}(\mathcal{A}, \mathcal{B}) = \liminf _{n\to \infty} \frac{\operatorname{\textrm{Min}}_{\mathcal{A}, \mathcal{B}}(n)}{n} \mbox{ and } \operatorname{\textrm{Max}}(\mathcal{A}, \mathcal{B}) = \limsup _{n\to \infty} \frac{\operatorname{\textrm{Max}}_{\mathcal{A}, \mathcal{B}}(n)}{n}.$$ The [*relative interval*]{} of $\mathcal{A}$ and $\mathcal{B}$ is defined as $$l(\mathcal{A}, \mathcal{B}) = \left[ \operatorname{\textrm{Min}}(\mathcal{A}, \mathcal{B}), \operatorname{\textrm{Max}}(\mathcal{A}, \mathcal{B}) \right].$$ If $\operatorname{\textrm{Max}}(\mathcal{A}, \mathcal{B}) > |\operatorname{\textrm{Min}}(\mathcal{A}, \mathcal{B})| $, then $\mathcal{A}$ is said to have [*better performance*]{} than $\mathcal{B}$ in this model.
Note that $\operatorname{\textrm{Min}}(\mathcal{A}, \mathcal{B}) = - \operatorname{\textrm{Max}}(\mathcal{B}, \mathcal{A})$ and $\operatorname{\textrm{Max}}(\mathcal{A}, \mathcal{B}) = -\operatorname{\textrm{Min}}(\mathcal{B}, \mathcal{A})$.
For any pair of algorithms, [[$\mathcal{A}$]{}]{}and [[$\mathcal{B}$]{}]{}, for the frequent items problem, there is a trivial upper bound on $\operatorname{\textrm{Max}}({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace})$ and lower bound on $\operatorname{\textrm{Min}}({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace})$.
\[trivialMax\] For any pair of algorithms [[$\mathcal{A}$]{}]{} and [[$\mathcal{B}$]{}]{}, $\operatorname{\textrm{Max}}({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace})\leq 1$ and $\operatorname{\textrm{Min}}({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace})\geq -1$.
The maximum aggregate frequency any algorithm could have is for a sequence where all items are identical, giving the value $n$. The minimum is for a sequence where all items are different, giving the value $1$. The required bounds follow since $\limsup_{n\rightarrow\infty} \frac{n-1}{n} = 1$.
Naive vs. Eager
---------------
According to relative interval analysis, [[$\textsc{Nai}$]{}]{}has better performance than [[$\textsc{Eag}$]{}]{}.
\[thm:relint\_nav\_eag\] According to relative interval analysis $l({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace}) = [-\frac{1}{4},1]$.
By Proposition \[trivialMax\], $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace})\leq 1$.
We now consider a lower bound on $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace})$. By Proposition \[prop\_sequences\], we have that $ {{\ensuremath{\textsc{Nai}}}\xspace}(E_n) - {{\ensuremath{\textsc{Eag}}}\xspace}(E_n) = (n-4+\frac{8}{n})-2$, so $$\limsup _{n\to \infty} \frac{{{\ensuremath{\textsc{Nai}}}\xspace}(E_n) - {{\ensuremath{\textsc{Eag}}}\xspace}(E_n)}{n}
= \limsup _{n\to \infty} \frac{n-6+\frac{8}{n}}{n} = 1.$$ Thus, $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace})= 1$.
We now consider $\operatorname{\textrm{Min}}({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace})$. For the upper bound on the minimum value of ${{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Eag}}}\xspace}(I)$, let $I$ contain $r=\lceil \frac{n+1}{2} \rceil$ copies of $a$ and $\lfloor
\frac{n-1}{2}\rfloor$ distinct items $a_1, a_2, \ldots a_{n-r}$, and let $I$ start with $a a$. For this sequence, [[$\textsc{Nai}$]{}]{}’s aggregate frequency is $\lceil \frac{n+1}{2}\rceil \frac{\lceil\frac{n+1}{2}\rceil }{n} + \lfloor
\frac{n-1}{2}\rfloor \frac{1}{n}$, which is $\frac{n}{4} + \frac{3}{2}$ if $n$ is even and $\frac{n}{4} +1 -\frac{1}{4n}$ if $n$ is odd. [[$\textsc{Eag}$]{}]{}’s aggregate frequency is $n \frac{\lceil\frac{n+1}{2} \rceil}{n}$, which is $ \frac{n}{2} + 1$ if $n$ is even and $\frac{n+1}{2}$ if $n$ is odd. This gives an upper bound of $\operatorname{\textrm{Min}}_{{{\ensuremath{\textsc{Nai}}}\xspace},{{\ensuremath{\textsc{Eag}}}\xspace}}(n)\leq
{{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Eag}}}\xspace}(I)$, which is $-\frac{n}{4} +\frac{1}{2}$ if $n$ is even, and $-\frac{n}{4} +\frac{1}{2}-\frac{1}{4n}$ if $n$ is odd. Thus, $$\operatorname{\textrm{Min}}({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace})\leq \liminf _{n\to \infty} \frac{\frac{1}{2} - \frac{n}{4}}{n} = -\frac{1}{4}. \nonumber$$
Next we calculate a lower bound on $\operatorname{\textrm{Min}}({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace})$. Assume that among sequences of length $n$, $I$ gives the smallest possible value of ${{\ensuremath{\textsc{Nai}}}\xspace}(I)-{{\ensuremath{\textsc{Eag}}}\xspace}(I)$. From the definitions of [[$\textsc{Nai}$]{}]{}and [[$\textsc{Eag}$]{}]{}, it is evident that there must be a repeated item if ${{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Eag}}}\xspace}(I) <0$. Suppose the first repeated item is item $a$ at time steps $p+1$ and $p+2$. Before $p+1$, both [[$\textsc{Nai}$]{}]{}and [[$\textsc{Eag}$]{}]{}have the same items in the buffer, and both [[$\textsc{Nai}$]{}]{}and [[$\textsc{Eag}$]{}]{}have $a$ in their buffers every time it occurs. We show that we can assume that all items in $I$ different from $a$ each occur only once in $I$.
First, suppose that there is an item $b\not= a$ before $p+1$ with frequency greater than $1$ in $I$. Replace this occurrence of $b$ by a new item, $b'$, which does not occur in $I$ to obtain $I'$. The contribution to the aggregate frequency from $b'$ and any $b$s before $p+1$ is identical for [[$\textsc{Nai}$]{}]{}and [[$\textsc{Eag}$]{}]{}on $I'$. The contribution to [[$\textsc{Eag}$]{}]{}’s aggregate frequency from items after $p$ is unchanged, but if [[$\textsc{Nai}$]{}]{}has any $b$s after $p$, the contribution to [[$\textsc{Nai}$]{}]{}’s aggregate frequency from them is lower in $I'$ than in $I$. Thus ${{\ensuremath{\textsc{Nai}}}\xspace}(I')-{{\ensuremath{\textsc{Eag}}}\xspace}(I') < {{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Eag}}}\xspace}(I)$, contradicting the minimality for $I$.
Now we can assume that any repeated items other than $a$ occur only after $p$. Clearly, the same technique of replacing one of these repeated items by a new item which does not already occur will only affect [[$\textsc{Nai}$]{}]{}’s aggregate frequency and only decrease it, contradicting the minimality of $I$. Thus, we may assume that $a$ is the only repeated item.
We may also assume that the item $a$ does not occur before time $p+1$, since swapping such an occurrence with the item in location $p$ has no effect on either [[$\textsc{Nai}$]{}]{}’s or [[$\textsc{Eag}$]{}]{}’s aggregate frequency.
Consequently, if the number of occurrences of $a$ is denoted by $n_I(a)$, then ${{\ensuremath{\textsc{Nai}}}\xspace}(I)-{{\ensuremath{\textsc{Eag}}}\xspace}(I) = (n-n_I(a))\frac{1}{n} + n_I(a)\frac{n_I(a)}{n} -
(p\frac{1}{n} + (n-p)\frac{n_I(a)}{n})$. Since $n_I(a)>1$, this is clearly minimized at $p=0$, so the first two occurrences of $a$ are in the first two locations. Taking the derivative and setting it equal to zero gives that the minimum occurs when $n_I(a)=\frac{n+1}{2}$. This gives that ${{\ensuremath{\textsc{Nai}}}\xspace}(I)-{{\ensuremath{\textsc{Eag}}}\xspace}(I) \geq -\frac{n}{4} +\frac{1}{2}-\frac{1}{4n}$, and $\operatorname{\textrm{Min}}({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace})=-\frac{1}{4}$. Thus, $l({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace}) = [-\frac{1}{4}, 1]$.
Naive vs. Majority
------------------
[[$\textsc{Nai}$]{}]{}and [[$\textsc{Maj}$]{}]{}are equally good according to relative interval analysis.
\[thm:relint\_nav\_maj\] According to relative interval analysis $l({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Maj}}}\xspace}) = [-\frac{1}{4}, \frac{1}{4}]$.
For the maximum value of ${{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I)$, it is sufficient to consider the worst permutation of $I$ for [[$\textsc{Maj}$]{}]{}since [[$\textsc{Nai}$]{}]{}has the same output for all permutations of $I$. For the worst permutation, ${{\ensuremath{\textsc{Maj}}}\xspace}_W(I)$ will buffer only the first $\lceil \frac{n}{2} \rceil$ items of the distribution $D(I)$. The first $\lfloor \frac{n}{2} \rfloor$ items will be buffered twice and in case of odd $n$, the $\lceil \frac{n}{2} \rceil$th item will be stored once at the last time step. Let $D(I) = a_1', a_2', a_3',\ldots, a_n'$. Then $$\begin{aligned}
{{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}_W(I) &=& \sum_{i=1}^n f_I(a'_i) - 2\sum_{i=1}^{\lfloor \frac{n}{2} \rfloor}f_I(a'_i) - \left (\left\lceil \frac{n}{2} \right\rceil - \left\lfloor \frac{n}{2} \right\rfloor \right) f_I(a'_{\lceil \frac{n}{2} \rceil}) \nonumber\\
&=& \sum_{i=\lceil \frac{n+2}{2}\rceil}^n f_I(a'_i) - \sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}f_I(a'_i). \label{eq:dif_navmaj}\end{aligned}$$ Let $p$ be the number of occurrences of the most frequent item in $I$. Then $$\begin{aligned}
{{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}_W(I) &=&
\sum_{i=\lceil \frac{n+2}{2}\rceil}^n f_I(a'_i) - \sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}f_I(a'_i) \nonumber
\\
&\leq& \left\lfloor\frac{n}{2}\right\rfloor\frac{p}{n} - \left(p - \left\lceil\frac{n}{2}\right\rceil\right)\frac{p}{n} \nonumber
\\
&=& p - \frac{p^2}{n}. \nonumber\end{aligned}$$ If $n$ is even, an upper bound on the maximum difference will be achieved when $p = \frac{n}{2}$, and for odd $n$ when $p = \frac{n+1}{2}$. This gives an upper bound on the maximum of ${{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I)$ of $\frac{n}{4}$ for even $n$ and $\frac{n}{4} - \frac{1}{4n}$ for odd $n$. For a lower bound on the maximum value of ${{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I)$, we consider the family of sequences, $W_n$, from Definition \[sequences\]. By Proposition \[prop\_sequences\], for even $n$, ${{\ensuremath{\textsc{Nai}}}\xspace}(W_n) - {{\ensuremath{\textsc{Maj}}}\xspace}(W_n) = \frac{n}{4} - \frac{1}{2}$, and for odd $n$, ${{\ensuremath{\textsc{Nai}}}\xspace}(W_n) - {{\ensuremath{\textsc{Maj}}}\xspace}(W_n) = \frac{n}{4} - 1 + \frac{1}{4n}$. Thus, $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Maj}}}\xspace}) \geq \limsup _{n\to \infty} \frac{{{\ensuremath{\textsc{Nai}}}\xspace}(W_n) - {{\ensuremath{\textsc{Maj}}}\xspace}(W_n)}{n} = \frac{1}{4}$, matching the upper bound. To derive the minimum value of ${{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I)$, we calculate the maximum value of ${{\ensuremath{\textsc{Maj}}}\xspace}(I) - {{\ensuremath{\textsc{Nai}}}\xspace}(I)$. For an upper bound on this, we consider the best permutation, $I_B$, for [[$\textsc{Maj}$]{}]{}of an arbitrary sequence, $I$. For $I_B$, [[$\textsc{Maj}$]{}]{}would buffer the half of the requests in the sequence with the highest frequencies. The difference is $$\begin{aligned}
&& {{\ensuremath{\textsc{Maj}}}\xspace}(I_B) - {{\ensuremath{\textsc{Nai}}}\xspace}(I_B) \nonumber\\
&=& 2 \sum_{i=\lceil \frac{n+2}{2} \rceil}^{n} f_I(a'_i) + \left(\left\lceil \frac{n}{2} \right\rceil - \left\lfloor \frac{n}{2} \right\rfloor \right) f_I(a'_{\lceil \frac{n}{2} \rceil}) - \sum_{i=1}^n f_I(a'_i) \nonumber\\
&=& \sum_{i=\lceil \frac{n+2}{2}\rceil}^n f_I(a'_i) - \sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}f_I(a'_i). \nonumber $$ This expression is exactly the same as the expression for ${{\ensuremath{\textsc{Nai}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}_W(I)$ from Eq. \[eq:dif\_navmaj\], so we get the same upper bound of $\frac{1}{4}$. Now, for a lower bound on $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Maj}}}\xspace},{{\ensuremath{\textsc{Nai}}}\xspace})$, we use the family of sequences, $I_n$ defined as $$I_n = a_0,a_0,\ldots,a_0,a_1,a_2,\ldots,a_{\lfloor\frac{n}{2}\rfloor},$$ where there are $\lceil \frac{n}{2} \rceil$ copies of $a_0$. Then $${{\ensuremath{\textsc{Nai}}}\xspace}(I_n) = \left\lfloor\frac{n}{2}\right\rfloor\frac{1}{n} + \left\lceil \frac{n}{2} \right\rceil \frac{\lceil\frac{n}{2}\rceil}{n}
= \left\{ \begin{array}{ll}
\frac{n}{4} + \frac{1}{2} & \mbox{for even $n$}\\
\frac{n}{4} + 1 + \frac{1}{4n} & \mbox{for odd $n$}
\end{array} \right.$$ and $${{\ensuremath{\textsc{Maj}}}\xspace}(I_n) = n\frac{\lceil\frac{n}{2}\rceil}{n}=\left\lceil\frac{n}{2}\right\rceil.$$ $I_n$ gives a lower bound of $\frac{1}{4}$ on $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Maj}}}\xspace},{{\ensuremath{\textsc{Nai}}}\xspace})$, since ${{\ensuremath{\textsc{Maj}}}\xspace}(I_n)-{{\ensuremath{\textsc{Nai}}}\xspace}(I_n) \geq \left\lceil\frac{n}{2}\right\rceil-
\frac{n}{4} - 1 - \frac{1}{4n}$. It follows that, $\operatorname{\textrm{Min}}({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Maj}}}\xspace})= -\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Maj}}}\xspace}, {{\ensuremath{\textsc{Nai}}}\xspace}) = -\frac{1}{4}$, and $l({{\ensuremath{\textsc{Nai}}}\xspace}, {{\ensuremath{\textsc{Maj}}}\xspace}) = [-\frac{1}{4}, \frac{1}{4}]$.
Majority vs. Eager
------------------
According to relative interval analysis, [[$\textsc{Maj}$]{}]{}has better performance than [[$\textsc{Eag}$]{}]{}.
\[thm:relint\_maj\_eag\] According to relative interval analysis $l({{\ensuremath{\textsc{Maj}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace}) = [-\frac{1}{2}, 1]$.
By Proposition \[trivialMax\], $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Maj}}}\xspace},{{\ensuremath{\textsc{Eag}}}\xspace})\leq 1$. For the lower bound on $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Maj}}}\xspace},{{\ensuremath{\textsc{Eag}}}\xspace})$, we consider the family of sequences, $E_n$, from Definition \[sequences\]. By Proposition \[prop\_sequences\], ${{\ensuremath{\textsc{Maj}}}\xspace}(E_n) - {{\ensuremath{\textsc{Eag}}}\xspace}(E_n) = (n-6+\frac{16}{n}) -2
= n - 8 + \frac{16}{n}$, and $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Maj}}}\xspace},{{\ensuremath{\textsc{Eag}}}\xspace})\geq \limsup_{n\to \infty} \frac{n-8+\frac{16}{n}}{n} = 1$. Thus, $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Maj}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace}) = 1$.
For $\operatorname{\textrm{Min}}({{\ensuremath{\textsc{Maj}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace})$, we consider $\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Eag}}}\xspace}, {{\ensuremath{\textsc{Maj}}}\xspace})$. First we calculate an upper bound on ${{\ensuremath{\textsc{Eag}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I)$. Suppose the input sequence $I$ of length $n$ gives the maximum value of ${{\ensuremath{\textsc{Eag}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I)$ over all sequences of length $n$. Suppose $I$ has $k$ distinct items $a_1, a_2, a_3, \ldots , a_k$, and let $f_i=f_I(a_i)$ and $n_i = n_I(a_i)$ for all $i$. Assume that $f_1 \leq f_2 \leq f_3 \leq \ldots \leq f_k$, so $a_k$ is the most frequent item. First, assume $n_k \leq \lceil \frac{n}{2} \rceil$. $$\begin{aligned}
{{\ensuremath{\textsc{Eag}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I) &\leq& nf_k - 1
\leq n\frac{\lceil \frac{n}{2} \rceil}{n} - 1
\leq \frac{n}{2} - \frac{1}{2}
\label{eq:majcond}\end{aligned}$$ It remains to consider the range $\lceil \frac{n}{2} \rceil < n_k \leq n$. Assume for some positive integer $q$ that $n_k = \lceil \frac{n}{2} \rceil + q$. From Lemma \[lem:wmaj\], we know that [[$\textsc{Maj}$]{}]{}’s result has the lower bound ${{\ensuremath{\textsc{Maj}}}\xspace}_W(I) \geq 2(\sum_{i=1}^{k-1} n_i f_i + qf_k) + (\lceil \frac{n}{2} \rceil - \lfloor
\frac{n}{2} \rfloor)f_k$. The summation is minimized when the smallest $k-1$ frequencies are all equal to $\frac{1}{n}$. Since $k-1= \lfloor\frac{n}{2}
\rfloor-q$ in this case, ${{\ensuremath{\textsc{Maj}}}\xspace}(I)\geq 2\left((\lfloor\frac{n}{2}\rfloor -q)\frac{1}{n} + q\frac{\lceil \frac{n}{2}\rceil+q}{n}\right)$. Hence, $$\begin{aligned}
{{\ensuremath{\textsc{Eag}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I) &\leq& \left\lceil \frac{n}{2} \right\rceil + q - 2\left( \frac{1}{n}\left(\left\lfloor \frac{n}{2} \right\rfloor -q \right) + q \frac{\lceil \frac{n}{2} \rceil + q}{n} \right) \nonumber
\\
&=& \left\{ \begin{array}{ll}
\frac{n}{2} - 1 - \frac{2}{n}(q^2 - q) & \mbox{for even $n$}\vspace{1 mm} \\
\frac{n}{2} - \frac{1}{2} - \frac{1}{n}(2q^2 - q-1) & \mbox{for odd $n$}
\end{array} \right. \nonumber
\\
&\leq&
\frac{n}{2} - \frac{1}{2}
\label{eq:majcond2}\end{aligned}$$ Thus, the same upper bound holds both when $n_k\leq \lceil\frac{n}{2}\rceil$ and when $n_k > \lceil\frac{n}{2}\rceil$.
For a lower bound on the maximum value of ${{\ensuremath{\textsc{Eag}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I)$ for even $n$, we use the input sequence $I = a, a, a_1, a_2, a_3, a, a_4,a \ldots, a_{\frac{n}{2}}, a$ (an $a$ every second time after start-up). For this sequence $$\begin{aligned}
{{\ensuremath{\textsc{Eag}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I) &=& n\frac{1}{2} - \left( 4\frac{1}{2} + (n-4)\frac{1}{n}\right) \nonumber
\\
&=& \frac{n}{2} -3 + \frac{4}{n}. \nonumber\end{aligned}$$ For odd $n$, we add one $a$ at the end of the even length $I$ which gives ${{\ensuremath{\textsc{Eag}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I) = \frac{n}{2} -3 + \frac{5}{2n}$. These lower bounds and the upper bounds from Eq. \[eq:majcond\] and \[eq:majcond2\] are asymptotically all equal to $\frac{n}{2}$, so $$\operatorname{\textrm{Min}}({{\ensuremath{\textsc{Maj}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace}) = -\operatorname{\textrm{Max}}({{\ensuremath{\textsc{Eag}}}\xspace}, {{\ensuremath{\textsc{Maj}}}\xspace}) = -\limsup _{n\to \infty} \frac{{{\ensuremath{\textsc{Eag}}}\xspace}(I) - {{\ensuremath{\textsc{Maj}}}\xspace}(I)}{n} = -\frac{1}{2}.\nonumber$$ Therefore $l({{\ensuremath{\textsc{Maj}}}\xspace}, {{\ensuremath{\textsc{Eag}}}\xspace}) = [-\frac{1}{2}, 1]$.
Relative Worst Order Analysis {#sec:rel_worst}
=============================
Relative worst order analysis [@Boyar07] compares two online algorithms directly. It compares two algorithms on their worst orderings of sequences which have the same content, but possibly different order. The definition of this measure is somewhat more involved; see [@BFL07j] for more intuition on the various elements. As in the case of competitive analysis, here too the relative performance of the algorithms depend on the length of the input sequence $I$. As in Section \[sec:comp\], we define a modified and more general version of relative worst order analysis. The definition is given for a maximization problem, but trivially adaptable to be used for minimization problems as well; only the decision as to when which algorithm is better would change.
The following definition is parameterized by a total ordering, ${\sqsubseteq}$, since we will later use it for both $\leq$ and $\geq$.
\[def:wor\] $f$ is a *${\ensuremath{({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace},{\sqsubseteq})}\xspace}$-function* if $$\forall I{\!:\;}{{\ensuremath{\mathcal{A}}}\xspace}_W(I) {\sqsubseteq}(f(n)+o(f(n)))\cdot{{\ensuremath{\mathcal{B}}}\xspace}_W(I),$$ where [[$\mathcal{A}$]{}]{}and [[$\mathcal{B}$]{}]{}are algorithms and ${\sqsubseteq}$ is a total ordering. Recall from Definition \[def:worstpermut\] that the notation $\textsc{Alg}_W(I)$, where <span style="font-variant:small-caps;">Alg</span> is some algorithm, denotes the result of <span style="font-variant:small-caps;">Alg</span> on its worst permutation of $I$. $f$ is a *bounding function with respect to* ${\ensuremath{({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace},{\sqsubseteq})}\xspace}$ if $f$ is a ${\ensuremath{({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace},{\sqsubseteq})}\xspace}$-function and for any ${\ensuremath{({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace},{\sqsubseteq})}\xspace}$-function $g$, $\lim_{n\rightarrow\infty}\frac{f(n)}{g(n)}{\sqsubseteq}1$. If $f$ is a bounding function with respect to ${\ensuremath{({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace},\leq)}\xspace}$ and $g$ is a bounding function with respect to ${\ensuremath{({{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace},\geq)}\xspace}$, then [[$\mathcal{A}$]{}]{}and [[$\mathcal{B}$]{}]{}are said to be *comparable* if $\lim_{n\rightarrow\infty}f(n)\leq 1$ or $\lim_{n\rightarrow\infty}g(n)\geq 1$.
If $\lim_{n\rightarrow\infty}f(n)\leq 1$, then ${{\ensuremath{\mathcal{B}}}\xspace}$ is better than ${{\ensuremath{\mathcal{A}}}\xspace}$ and $g(n)$ is a *relative worst order function of ${{\ensuremath{\mathcal{A}}}\xspace}$ and ${{\ensuremath{\mathcal{B}}}\xspace}$*, and if $\lim_{n\rightarrow\infty}g(n)\geq 1$, then ${{\ensuremath{\mathcal{A}}}\xspace}$ is better than ${{\ensuremath{\mathcal{B}}}\xspace}$ and $f(n)$ is a *relative worst order function of ${{\ensuremath{\mathcal{A}}}\xspace}$ and ${{\ensuremath{\mathcal{B}}}\xspace}$*.
We use $\operatorname{\textrm{WR}}_{{{\ensuremath{\mathcal{A}}}\xspace},{{\ensuremath{\mathcal{B}}}\xspace}} = f(n)$ to indicate that $f(n)$ belongs to the equivalence class of *relative worst order functions of ${{\ensuremath{\mathcal{A}}}\xspace}$ and ${{\ensuremath{\mathcal{B}}}\xspace}$*.
The competitive function could also have been defined using this framework, but was defined separately as a gentle introduction to the idea.
Naive vs. Optimal {#subsec:relwor_nav_opt}
-----------------
Relative worst order analysis can show the strength of the simple, but adaptive, [[$\textsc{Nai}$]{}]{}algorithm by comparing it with the powerful [[$\textsc{Opt}$]{}]{}. [[$\textsc{Nai}$]{}]{}is an optimal algorithm according to relative worst order analysis, in the sense that it is equivalent to [[$\textsc{Opt}$]{}]{}.
\[thm:relwor\_nav\_opt\] According to relative worst order analysis $\operatorname{\textrm{WR}}_{{{\ensuremath{\textsc{Opt}}}\xspace},{{\ensuremath{\textsc{Nai}}}\xspace}}=1$, so [[$\textsc{Nai}$]{}]{}and [[$\textsc{Opt}$]{}]{}are equivalent.
In the aggregate frequency problem, even though [[$\textsc{Opt}$]{}]{}knows the whole sequence in advance, it cannot store an item before it first appears in the sequence. Thus, for any input sequence $I$, the worst permutation for [[$\textsc{Opt}$]{}]{}is the sorting of $I$ according to the increasing order of the frequencies of the items, i.e., $D(I)$. On this ordering, [[$\textsc{Opt}$]{}]{}is forced to behave like [[$\textsc{Nai}$]{}]{}. Therefore, the constant function $1$ is a bounding function with respect to both $({{\ensuremath{\textsc{Opt}}}\xspace},{{\ensuremath{\textsc{Nai}}}\xspace},\leq)$ and $({{\ensuremath{\textsc{Opt}}}\xspace},{{\ensuremath{\textsc{Nai}}}\xspace},\geq)$, so $\operatorname{\textrm{WR}}_{{{\ensuremath{\textsc{Opt}}}\xspace},{{\ensuremath{\textsc{Nai}}}\xspace}}=1$.
Naive vs. Eager
---------------
According to relative worst order analysis, [[$\textsc{Nai}$]{}]{}is better than [[$\textsc{Eag}$]{}]{}.
\[thm:relwor\_nav\_eag\] According to relative worst order analysis $\operatorname{\textrm{WR}}_{{{{\ensuremath{\textsc{Nai}}}\xspace}},{{{\ensuremath{\textsc{Eag}}}\xspace}}} = \frac{n}{2}$.
From Theorem \[thm:relwor\_nav\_opt\], we know that for [[$\textsc{Opt}$]{}]{}’s worst permutation, $I_W$, of any sequence $I$, ${{\ensuremath{\textsc{Opt}}}\xspace}(I_W) = {{\ensuremath{\textsc{Nai}}}\xspace}(I_W)$. Any arbitrary online algorithm ${{\ensuremath{\mathcal{A}}}\xspace}$ cannot be better than [[$\textsc{Opt}$]{}]{}on any sequence, so [[$\textsc{Nai}$]{}]{}and [[$\mathcal{A}$]{}]{}are comparable. For any arbitrary online algorithm ${{\ensuremath{\mathcal{A}}}\xspace}$ and a worst order, $I_W$, for [[$\mathcal{A}$]{}]{}of any sequence $I$, $\frac{{{\ensuremath{\textsc{Nai}}}\xspace}(I_W)}{{{\ensuremath{\mathcal{A}}}\xspace}(I_W)} = \frac{{{\ensuremath{\textsc{Opt}}}\xspace}(I_W)}{{{\ensuremath{\mathcal{A}}}\xspace}(I_W)}$, so a competitive function for [[$\mathcal{A}$]{}]{}is an upper bound on the relative worst order function of [[$\mathcal{A}$]{}]{}and [[$\mathcal{B}$]{}]{}. By Theorem \[thm:comp\_eag\], $\operatorname{\textrm{WR}}({{\ensuremath{\textsc{Nai}}}\xspace},{{\ensuremath{\textsc{Eag}}}\xspace})\leq \frac{n}{2}$. Consider the family of sequences, $E_n$, from Definition \[sequences\]. These sequences are in the worst ordering for both [[$\textsc{Eag}$]{}]{}and [[$\textsc{Opt}$]{}]{}. By Proposition \[prop\_sequences\], ${{\ensuremath{\textsc{Nai}}}\xspace}(E_n) = n-4+\frac{8}{n}$ and ${{\ensuremath{\textsc{Eag}}}\xspace}(E_n)=2$. Thus, ${{\ensuremath{\textsc{Nai}}}\xspace}(E_n) = \frac{n}{2}{{\ensuremath{\textsc{Eag}}}\xspace}(E_n) -4 + \frac{8}{n}$. Consequently, $\frac{n}{2}$ is a relative worst order function of [[$\textsc{Nai}$]{}]{}and [[$\textsc{Eag}$]{}]{}, and $\operatorname{\textrm{WR}}_{{{{\ensuremath{\textsc{Nai}}}\xspace}},{{{\ensuremath{\textsc{Eag}}}\xspace}}} = \frac{n}{2}$.
Naive vs. Majority
------------------
According to relative worst order analysis, [[$\textsc{Nai}$]{}]{}is better than [[$\textsc{Maj}$]{}]{}, though not quite as much better as compared to [[$\textsc{Eag}$]{}]{}.
\[thm:relwor\_nav\_maj\] According to relative worst order analysis, $\operatorname{\textrm{WR}}_{{{{\ensuremath{\textsc{Nai}}}\xspace}},{{{\ensuremath{\textsc{Maj}}}\xspace}}} = \frac{n}{4}$.
As in the proof of the previous theorem, since [[$\textsc{Nai}$]{}]{}and [[$\textsc{Opt}$]{}]{}perform the same on their worst orderings of any sequence, [[$\textsc{Nai}$]{}]{}and [[$\textsc{Maj}$]{}]{}are comparable.
Next we derive a bounding function with respect to $({{\ensuremath{\textsc{Nai}}}\xspace},{{\ensuremath{\textsc{Maj}}}\xspace},\leq)$. Since [[$\textsc{Nai}$]{}]{}’s aggregate frequency is the same on any ordering of that sequence, we can compare [[$\textsc{Nai}$]{}]{}and [[$\textsc{Maj}$]{}]{}on the same sequence, [[$\textsc{Maj}$]{}]{}’s worst ordering of it; that is also a worst ordering for [[$\textsc{Nai}$]{}]{}. Suppose the input sequence $I$ of length $n$ gives the largest ratio for $\frac{{{\ensuremath{\textsc{Nai}}}\xspace}_W(I)}{{{\ensuremath{\textsc{Maj}}}\xspace}_W(I)}$ for sequences of length $n$. Suppose $I$ has $k$ distinct items $a_1, a_2,\ldots, a_k$, and let $f_i=f_I(a_i)$ and $n_i = n_I(a_i)$ for all $i$. Assume that $f_1 \leq f_2 \leq f_3 \leq \ldots \leq f_k$, so $a_k$ is the most frequent item. If $n_k\leq \lfloor \frac{n}{2} \rfloor$ then $$\begin{aligned}
\frac{{{\ensuremath{\textsc{Nai}}}\xspace}_W(I)}{{{\ensuremath{\textsc{Maj}}}\xspace}_W(I)} &=& \frac{\sum_{i=1}^k n_if_i}{2(\sum_{i=1}^{j-1}n_if_i + pf_j) + \left (\lceil \frac{n}{2} \rceil - \lfloor \frac{n}{2} \rfloor \right) f_j} \nonumber
\\
&=& \frac{\sum_{i=1}^k n_i^2}{2(\sum_{i=1}^{j-1}n_i^2 + pn_j) + \left (\lceil \frac{n}{2} \rceil - \lfloor \frac{n}{2} \rfloor \right) n_j} \label{eq:majw_profit}\end{aligned}$$ where $j\leq k$ is the largest index such that $\sum_{i=1}^{j-1}n_i + p = \lfloor \frac{n}{2} \rfloor$ for some non-negative integer $p$. Create another sequence $I'$ from $I$ by replacing all the $a_i$’s where $j<i<k$ with $a_k$ and by replacing $n_j - p - \left (\lceil \frac{n}{2} \rceil - \lfloor \frac{n}{2} \rfloor \right)$ $a_j$’s with $a_k$. $I'$ will have $j+1$ distinct items and the most frequent item will have $\lfloor \frac{n}{2} \rfloor$ occurrences. Since all these changes will increase the numerator and not change the denominator in Eq. \[eq:majw\_profit\], $I'$ will give at least as large a ratio as $I$, so we consider the sequence $I'$ instead of $I$. Suppose the items of $I'$, in nondecreasing order of frequency, are $\hat{a}_1, \hat{a}_2,\ldots ,\hat{a}_{j+1}$ and the corresponding counts are $\hat{n}_1, \hat{n}_2,\ldots ,\hat{n}_{j+1}$. Then, $$\frac{{{\ensuremath{\textsc{Nai}}}\xspace}_W(I')}{{{\ensuremath{\textsc{Maj}}}\xspace}_W(I')} \leq \frac{\lfloor \frac{n}{2} \rfloor^2 + \sum_{i=1}^j \hat{n}_i^2}{2\sum_{i=1}^{j}\hat{n}_i^2 - \left (\lceil \frac{n}{2} \rceil - \lfloor \frac{n}{2} \rfloor \right) \hat{n}_j }\label{eq:majw_profit2}$$
Consider any item $\hat{a}_i$ where $i\leq j$. Suppose its count is $\hat{n}_i >1$. Replace the $\hat{n}_i$ copies of $\hat{a}_i$ by $\hat{n}_i$ distinct items which are different from all the other items in $I'$. In most cases, this replacement will decrease the numerator in Eq. \[eq:majw\_profit2\] by $\hat{n}_i^2 - \hat{n}_i$ and will decrease the denominator by $2(\hat{n}_i^2 - \hat{n}_i)$. The only exception is when $i=j$ and $n$ is odd, in which case the denominator will decrease by $2\hat{n}_i^2 - 3\hat{n}_i +1$. However, in either case, the decrease in the denominator is as large as that in the numerator. Since the lower bound on the ratio is $1$, this replacement will increase the ratio. Hence the maximum ratio will be achieved if all the items, except the most frequent item, have frequency $\frac{1}{n}$, so $I'$ has the same form as $W_n$. Using Proposition \[prop\_sequences\], $$\frac{{{\ensuremath{\textsc{Nai}}}\xspace}_W(I')}{{{\ensuremath{\textsc{Maj}}}\xspace}_W(I')} = \left\{ \begin{array}{ll}
\frac{n}{4} + \frac{1}{2} & \mbox{for even $n$}\vspace{1 mm}\\
\frac{n}{4} + \frac{3}{4n} & \mbox{for odd $n$}
\end{array} \right. \label{eq:majw_up}$$ It remains to consider the range $\lceil \frac{n}{2} \rceil \leq n_k \leq n$. In this case, $$\begin{aligned}
\frac{{{\ensuremath{\textsc{Nai}}}\xspace}_W(I)}{{{\ensuremath{\textsc{Maj}}}\xspace}_W(I)} &=& \frac{\sum_{i=1}^k n_if_i}{2(\sum_{i=1}^{k-1}n_if_i + qf_k) + \left (\lceil \frac{n}{2} \rceil - \lfloor \frac{n}{2} \rfloor \right) f_k} \nonumber
\\
&=& \frac{n_k^2 + \sum_{i=1}^{k-1} n_i^2}{2qn_k + 2\sum_{i=1}^{k-1}n_i^2 + \left (\lceil \frac{n}{2} \rceil - \lfloor \frac{n}{2} \rfloor \right) n_k} \label{eq:majw_profit3}\end{aligned}$$ where $\sum_{i=1}^{k-1}n_i + q = \lfloor \frac{n}{2} \rfloor$ for some non-negative integer $q$. As in the case of $n_k \leq \lfloor \frac{n}{2} \rfloor$, all the multiple instances of items other than $a_k$ can be replaced by distinct items with frequency $\frac{1}{n}$ without decreasing the ratio. Next, if $q>0$ and we replace one instance of $a_k$ with some an item with frequency $\frac{1}{n}$, i.e., decrease $q$ by one, then the numerator in Eq. \[eq:majw\_profit3\] will be decreased by $n_k^2 - (n_k-1)^2 - 1 = 2(n_k-1)$ and the denominator will be decreased by $$2qn_k - 2(q-1)(n_k-1)-2 + \left\lceil \frac{n}{2} \right\rceil - \left\lfloor \frac{n}{2} \right\rfloor = 2(n_k + q -2) + \left\lceil \frac{n}{2} \right\rceil - \left\lfloor \frac{n}{2} \right\rfloor$$ Since the lower bound of the ratio is $1$, this replacement will increase the ratio while decreasing value of $q$. Thus, the largest ratio will achieved when $q=0$, and $$\begin{aligned}
\frac{{{\ensuremath{\textsc{Nai}}}\xspace}_W(I)}{{{\ensuremath{\textsc{Maj}}}\xspace}_W(I)} &\leq& \frac{\lceil \frac{n}{2} \rceil^2 + \lfloor \frac{n}{2}\rfloor}{2\lfloor \frac{n}{2}\rfloor + \lceil \frac{n}{2} \rceil - \lfloor \frac{n}{2} \rfloor} \nonumber \\
& = & \left\{ \begin{array}{ll}
\frac{n}{4} + \frac{1}{2} & \mbox{for even $n$}\\
\frac{n}{4} + 1 - \frac{1}{4n} & \mbox{for odd $n$}
\end{array} \right. \label{eq:majw_up2}\end{aligned}$$ By Eqns. \[eq:majw\_up\] and \[eq:majw\_up2\], $\frac{n}{4}$ is a $({{\ensuremath{\textsc{Nai}}}\xspace},{{\ensuremath{\textsc{Maj}}}\xspace},\leq)$-function. Since the proof of the upper bounds above shows that $W_n$ gives the largest ratio among sequences of length $n$, we can use the same sequence for the lower bound, showing that $\frac{n}{4}$ is a bounding function with respect to $({{\ensuremath{\textsc{Nai}}}\xspace},{{\ensuremath{\textsc{Maj}}}\xspace},\leq)$, so $\operatorname{\textrm{WR}}_{{{\ensuremath{\textsc{Nai}}}\xspace},{{\ensuremath{\textsc{Maj}}}\xspace}} = \frac{n}{4}$.
Majority vs. Eager
------------------
\[thm:relwor\_maj\_eag\] According to relative worst order analysis, [[$\textsc{Maj}$]{}]{}and [[$\textsc{Eag}$]{}]{}are incomparable.
First, we show that [[$\textsc{Maj}$]{}]{}can be much better than [[$\textsc{Eag}$]{}]{}. Consider the family of sequences, $E_n$, from Definition \[sequences\]. These sequences are in their worst orderings for both [[$\textsc{Maj}$]{}]{}and [[$\textsc{Eag}$]{}]{}. By Proposition \[prop\_sequences\], ${{\ensuremath{\textsc{Eag}}}\xspace}(E_n) = 2$, so $${{\ensuremath{\textsc{Maj}}}\xspace}_W(E_n) = n-6+\frac{16}{n} \geq \left(\frac{n}{2}-3+\frac{8}{n}\right){{\ensuremath{\textsc{Eag}}}\xspace}_W(E_n).$$
Now, we show that [[$\textsc{Eag}$]{}]{}can be much better than [[$\textsc{Maj}$]{}]{}. Consider the family of sequences, $W_n$, from Definition \[sequences\]. These sequences are in their worst orderings for [[$\textsc{Maj}$]{}]{}, so by Proposition \[prop\_sequences\], ${{\ensuremath{\textsc{Maj}}}\xspace}_W(W_n) = 1$. A worst ordering for [[$\textsc{Eag}$]{}]{}is $$W'_n = a_1,a_2,\ldots,a_{\lceil\frac{n}{2}\rceil},a_0,a_0,\ldots,a_0,$$ where there are $\lfloor \frac{n}{2} \rfloor$ copies of $a_0$. ${{\ensuremath{\textsc{Eag}}}\xspace}(W'_n) = {{\ensuremath{\textsc{Nai}}}\xspace}(W_n)$, which by Proposition \[prop\_sequences\] is $\frac{n}{4} + \frac{1}{2}$ when $n$ is even and $\frac{n}{4} + \frac{3}{4n}$ when $n$ is odd. Thus, $${{\ensuremath{\textsc{Eag}}}\xspace}_W(W_n) \geq \frac{n}{4}{{\ensuremath{\textsc{Maj}}}\xspace}_W(W_n).$$
These two families of sequences show that [[$\textsc{Maj}$]{}]{} and [[$\textsc{Eag}$]{}]{} are incomparable under relative worst order analysis.
Conclusion and Future Work
==========================
The frequent items problem for streaming was considered as an online problem. Three deterministic algorithms, [[$\textsc{Nai}$]{}]{}, [[$\textsc{Maj}$]{}]{}, and [[$\textsc{Eag}$]{}]{}were compared using three different quality measures: competitive analysis, relative worst order analysis, and relative worst order ratio. According to competitive analysis, [[$\textsc{Nai}$]{}]{}is the better algorithm and [[$\textsc{Maj}$]{}]{}and [[$\textsc{Eag}$]{}]{}are equivalent. According to relative interval analysis, [[$\textsc{Nai}$]{}]{}and [[$\textsc{Maj}$]{}]{}are equally good and both are better than [[$\textsc{Eag}$]{}]{}. According to relative worst order analysis, [[$\textsc{Nai}$]{}]{}and [[$\textsc{Opt}$]{}]{}are equally good and better than [[$\textsc{Maj}$]{}]{}and [[$\textsc{Eag}$]{}]{}, which are incomparable.
All three analysis techniques studied here are worst case measures. According to both competitive analysis and relative worst order analysis, [[$\textsc{Nai}$]{}]{}is the best possible online algorithm, and according the relative worst order analysis, it is as good as [[$\textsc{Maj}$]{}]{}and better than [[$\textsc{Eag}$]{}]{}. This is a consequence of [[$\textsc{Nai}$]{}]{}being very adaptive and, as a result, good at avoiding the extreme poor performance cases. Both [[$\textsc{Maj}$]{}]{}and [[$\textsc{Eag}$]{}]{}attempt to keep the most frequent items in the buffer for longer than their frequency would warrant. The heuristic approaches hurt these algorithms in the worst case.
Relative interval analysis compares the algorithms on the same sequence in a manner which, in addition to the worst case scenarios, also takes the algorithms’ best performance into account to some extent. This makes [[$\textsc{Maj}$]{}]{}’s sometimes superior performance visible, whereas [[$\textsc{Eag}$]{}]{}, not being adaptive at all, does not benefit in the same way from its best performance. In some sense, [[$\textsc{Maj}$]{}]{}’s behavior can be seen as swinging around the behavior of [[$\textsc{Nai}$]{}]{}, with worse behavior on some sequences counter-acted by correspondingly better behavior on other sequences.
Our conclusion is that purely worst behavior measures do not give indicative results for this problem. Relative interval analysis does better, and should possibly be supplemented by some expected case analysis variant. To that end, natural performance measures to consider would be bijective and average analysis [@Angelopoulos07]. However, as the problem is stated in [@Giannakopoulos12] and studied here, the frequent items problem has an infinite universe from which the items are drawn. Thus, these analysis techniques cannot be applied directly to the problem in any meaningful way. Depending on applications, it could be realistic to assume a finite universe. This might give different results than those obtained here, and might allow the problem to be studied using other measures, giving results dependent on the size of the universe. Another natural extension of this work is to consider multiple buffers, which also allows for a richer collection of algorithms [@Berinde09], or more complicated, not necessarily discrete, objective functions [@Cohen06].
[^1]: Supported in part by the Danish Council for Independent Research. Part of this work was done while the authors were visitng the University of Waterloo.
|
---
abstract: |
A consistent theory of quantum gravity (QG) at Planck scale almost sure contains manifestations of Lorentz local symmetry violations (LV) which may be detected at observable scales. This can be effectively described and classified by models with nonlinear dispersions and related Finsler metrics and fundamental geometric objects (nonlinear and linear connections) depending on velocity/ momentum variables. We prove that the trapping brane mechanism provides an accurate description of gravitational and matter field phenomena with LV over a wide range of distance scales and recovering in a systematic way the general relativity (GR) and local Lorentz symmetries. In contrast to the models with extra spacetime dimensions, the Einstein–Finsler type gravity theories are positively with nontrivial nonlinear connection structure, nonholonomic constraints and torsion induced by generic off–diagonal coefficients of metrics, and determined by fundamental QG and/or LV effects.
0.1cm
**Keywords:** quantum gravity, Lorentz violation, nonlinear dispersion, Finsler geometry, brane physics.
PACS: 02.40.-k, 04.50.Kd, 04.60.Bc, 04.70.-s, 04.90.+e, 11.25.-w, 11.30.Cp, 98.80.Cq
author:
- |
**Sergiu I. Vacaru** [^1]\
*University “Al. I. Cuza” Iaşi, Science Department,*\
*54 Lascar Catargi street, Iaşi, Romania, 700107*
date: 'July 5, 2011'
title: Finsler Branes and Quantum Gravity Phenomenology with Lorentz Symmetry Violations
---
Introduction and Preliminaries
==============================
There are several reasons to study generalizations of the Einstein gravity theory to models with local anisotropy, extra dimensions, analogous gravitational interactions and Finsler geometries. The first one goes in relation to the so–called quantum gravity (QG) phenomenology being supported by a number of ideas and research on possible observable QG effects and induced violations of Lorentz invariance (LV), see recent reviews [@kost4; @xiao; @liberati; @mavromatos1].
There were also analyzed possible production QG scenarios of mini–black holes in TeV–scale at colliders [@dimopoulos], or in cosmic rays [anchor]{}, and Planck–scale fuziness of spacetime [@amelino]. For some special situations, the QG effects/objects may manifest as a non–commutative geometry [@carroll] or on some brane–world backgrounds [@burgess]. A series of tentative results in various approaches to QG and string theory, and computations of local quantum field theory, suggest the proposal that the Lorenz invariance may be only a low energy symmetry.
Let us consider two important arguments about nonlinear dispersion relations in QG and possible related modifications of the fundamental concepts on spacetime geometry:
1. **Nonlinear dispersions and LV in QG. **Generically, we can write for a particle of mass $m_{0}$ propagating in a ”slightly deformed” four dimensional (4–d) Minkowski spacetime $$E^{2}=p^{2}c^{2}+m_{0}^{2}c^{4}+\varphi (E,p;\mu ;M_{P}), \label{defep}$$where $c$ is the light velocity, $E$ and $p$ are respectively the energy and momentum of the particle; $\mu $ is some particle physics mass scale and (normally) assumes that the Planck mass $M_{P}\approx 1.22\times 1019GeV$ denotes the mass scale at which the QG corrections become appreciable. The nonlinear term $\varphi (...)$ encodes possible quantum matter and gravity effects and LV terms[^2]. For $\varphi =0,$ we get locally the standard mass/energy/momentum relation describing a point particle in the special theory of relativity (SR). Assuming $\ E\sim \frac{\partial }{\partial t},p_{i}\sim \frac{\partial }{\partial x^{i}}$ for some background bosonic media with ”effective light velocity” $c_{s}$ (see details in [@kost4; @xiao; @liberati]), the nonlinear energy–momentum relation (\[defep\]) results in $$\omega ^{2}=c_{s}^{2}k^{2}+c_{s}^{2}\left( \frac{\overline{h}}{2m_{0}c_{s}}\right) ^{2}\text{ }k^{4}+... \label{ndisp1}$$when the ”phonon” dispersion relation $\omega \approx c_{s}|k|$ violates the acoustic Lorentz invariance with the wave length $\lambda =2\pi /|k|,$ for $k^{2}=(k_{1})^{2}+(k_{2})^{2}+(k_{3})^{2}$ and $\overline{h}=const.$ It is possible to derive more ”sophisticate” dispersion relations with cubic on $k$ and higher order terms and different coefficients than those in (\[ndisp1\]) if more general models of ”effective media” with fermionic and/or bosonic fields are considered.
2. **Finsler generating functions from nonlinear dispersions.** Nonlinear dispersions of type (\[ndisp1\]) encode not only ”energy–momentum” properties of point particles for LV. They contain also a very fundamental information about possible metric elements defining more general spacetime geometries than those postulated in SR and GR. Here, we briefly present a setup for such constructions in terms of Finsler geometry [@lammer; @stavr0; @girelli; @vacpr]. A Minkowski metric $\eta
_{ij}=diag[-1,+1,+1,+1]$ (for $i=1,2,3,4)$ defines a quadratic line element in SR,$$ds^{2}=\eta
_{ij}dx^{i}dx^{j}=-(dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2}+(dx^{4})^{2},
\label{minkmetr}$$with space type, $(x^{2},x^{3},x^{4}),$ and time like, $x^{1}=ct,$ coordinates where $c$ is the light speed.[^3] We can write for some classes of coordinate systems (for simplicity, omitting priming of indices and considering that indices of type $\widehat{i},\widehat{j},...=2,3,4)$ $$c^{2}=g_{\widehat{i}\widehat{j}}(x^{i})y^{\widehat{i}}y^{\widehat{j}}/\tau
^{2}. \label{lightf}$$This formula can be used also in GR if we consider that $g_{\widehat{i}\widehat{j}}(x^{i})$ are solutions of Einstein equations. The above quadratic on $y^{\widehat{i}}$ expression can be generalized to an arbitrary nonlinear one, $\check{F}^{2}(y^{\widehat{j}}),$ in order to model propagation of light in anisotropic media and/or for modeling an (ether) spacetime geometry. We have to impose the the condition of homogeneity, $\check{F}(\beta y^{\widehat{j}})=\beta \check{F}(y^{\widehat{j}})$ for any $\beta >0,$ which is necessary for description of light propagation. The formula (\[lightf\]) transforms into $$c^{2}=\check{F}^{2}(y^{\widehat{j}})/\tau ^{2}. \label{lightfd}$$Using approximations of type $\check{F}^{2}(y^{\widehat{j}})\approx \left(
\eta _{\widehat{i}\widehat{j}}y^{\widehat{i}}y^{\widehat{j}}\right) ^{r}+q_{\widehat{i}_{1}\widehat{i}_{2}...\widehat{i}_{2r}}y^{\widehat{i}_{1}}...y^{\widehat{i}_{2r}},$ for $r=1,2,....$ and $\widehat{i}_{1},\widehat{i}_{2},...,\widehat{i}_{2r}=2,3,4,$ we can parametrize small deformations of (\[lightf\]) to (\[lightfd\]). For $r=1$ and $q_{\widehat{i}_{1}\widehat{i}_{2}...\widehat{i}_{2r}}\rightarrow 0,$ we get the propagation of light rays in SR. Instead of $\eta _{\widehat{i}\widehat{j}},$ we can introduce a metric $g_{\widehat{i}\widehat{j}}(x^{i})$ from GR and include it in $\check{F}^{2}$ for gravitational fields when $\check{F}^{2}(x^{i},y^{\widehat{j}})\approx \left( g_{\widehat{i}\widehat{j}}(x^{k})y^{\widehat{i}}y^{\widehat{j}}\right) ^{r}+q_{\widehat{i}_{1}\widehat{i}_{2}...\widehat{i}_{2r}}(x^{k})y^{\widehat{i}_{1}}...y^{\widehat{i}_{2r}}$. For such deformations (derived from (\[minkmetr\]) and (\[lightf\])), we get generalized nonlinear homogeneous quadratic elements, [$$ds^{2}=F^{2}(x^{i},y^{j})\approx -(cdt)^{2}+g_{\widehat{i}\widehat{j}}(x^{k})y^{\widehat{i}}y^{\widehat{j}}[1+\frac{1}{r}\frac{q_{\widehat{i}_{1}\widehat{i}_{2}...\widehat{i}_{2r}}(x^{k})y^{\widehat{i}_{1}}...y^{\widehat{i}_{2r}}}{\left( g_{\widehat{i}\widehat{j}}(x^{k})y^{\widehat{i}}y^{\widehat{j}}\right) ^{r}}] +O(q^{2}), \label{fbm}$$]{}when $F(x^{i},\beta y^{j})=\beta F(x^{i},y^{j}),$ for any $\beta >0.$ A value $F$ is called a fundamental (generating) Finsler function usually satisfying the condition that the Hessian $$\ ^{F}g_{ij}(x^{i},y^{j})=\frac{1}{2}\frac{\partial F^{2}}{\partial
y^{i}\partial y^{j}} \label{hess}$$is not degenerate, see details in [@cartan; @ma; @bcs; @vsgg]. For light rays, the nonlinear element (\[fbm\]) defines a nonlinear dispersion relation between the frequency $\omega $ and the wave vector $k_{i},$[^4]$$\omega ^{2}=c^{2}[g_{\widehat{i}\widehat{j}}k^{\widehat{i}}k^{\widehat{j}}]^{2}\ (1-\frac{1}{r}\frac{q_{\widehat{i}_{1}\widehat{i}_{2}...\widehat{i}_{2r}}k^{\widehat{i}_{1}}...k^{\widehat{i}_{2r}}}{[g_{\widehat{i}\widehat{j}}k^{\widehat{i}}k^{\widehat{j}}]^{2r}}). \label{disp}$$
The dispersion relations should be parametrized and computed differently for various classes of theories formulated in terms of Finsler geometry and generalizations. Here we cite a series of works on very special relativity [@gibbons; @stavr1], generalized (super) Finsler gravity and LV induced from string gravity [@vstr1; @vstr2; @mavromatos], double special relativity [@mignemi; @ghosh], Finsler–Higgs mechanism [@sindoni], Finsler black holes/ellipsoids induced by noncommutative variables [@vncfinsl]. In particular, we can chose such subsets of coefficients $q_{\widehat{i}_{1}\widehat{i}_{2}...\widehat{i}_{2r}}$ when (\[disp\]) transforms into ([ndisp1]{}).
The main conclusion we derive from above points 1 and 2 is that various classical and quantum gravity theories are with local nonlinear dispersions of type (\[ndisp1\]) and/or (\[disp\]). Such theories are positively with LV and can be characterized geometrically by nonlinear Finsler type quadratic elements (\[fbm\]) constructed as certain deformations of standard quadratic elements for Minkowski (\[minkmetr\]) and/or pseudo–Riemannian spacetimes. This results in geometric constructions on tangent, $TV$ (with local coordinated $u^{\alpha }=(x^{i},y^{a}),$ where $y^{a}$ label fiber coordinates; we shall write in brief $u=(x,y)$). We can elaborate physical models on cotangent, $T^{\ast }V$ (with local coordinates $\ \check{u}^{\alpha }=(x^{i},p_{a}),$ where $p_{a}$ label co–fiber coordinates), bundles to a curved spacetime manifold $V$ (with local coordinates $x^{i}=(x^{1},x^{2},x^{3},x^{4})$ of pseudo–Euclidean signature). Constructions on $TM$ and $T^{\ast }V$ are typical for Finsler–Lagrange, and/or Cartan–Hamilton geometries, and generalizations, see details and references in [@cartan; @ma; @vsgg; @vacpr]. In modern particle physics and cosmology, see [@stavr2; @lichang4; @lichang6], there is a renewed interest in Finsler geometry applications, see reviews of results and critical remarks in [@vcrit; @vrevflg; @vexsol; @vqgr1; @vqgr2].
Nonlinear dispersions and associated Finsler like generating functions suggest the idea that a self–consistent QG theory may be constructed not just for a 4–d pseudo–Riemannian spacetime $V$ but for certain Finsler type extensions on $TV$ and/or $T^{\ast }V.$ Following a nonholonomic generalization of Fedosov deformation quantization, such quantum gravity models were studied in [@vqgr1; @vqgr2]. Roughly speaking, a QG model with some generalized nonlinear dispersions, and associated fundamental Finsler structures, should replace GR at very short distances approaching the Planck length, $\mathit{l}_{P}\simeq \sqrt{\frac{\ ^{4}G\hbar }{c^{3}}}\simeq 1.6\times 10^{-33}cm,$ where $\ ^{4}G$ is the 4–d Newton constant and $\hbar =h/2\pi $ is the Planck constant.
Over short distances, we have certain modifications of GR which seem to be of Finsler type with additional depending ”velocity/momentum” type coordinates. A Finsler spacetime geometry/ gravity model is not completely determined only by its nonlinear quadratic element $F(x,y)$ (\[fbm\]), or Hessian (\[hess\]). It is completely stated after we choose (following certain physical arguments) what types of metric tensor, $\ ^{F}\mathbf{g},$ nonlinear connection (N–connection), $^{F}\mathbf{N},$ and linear connection, $^{F}\mathbf{D},$ are canonically induced by a generating Finsler function $F(x,y)$ on $\widetilde{TV}\equiv TV/\{0\}$ (we exclude the null sections $\{0\}$ over $TM).$[^5] The nature of QG and LV effects derived in certain theoretical construction is related to a series of assumptions on fundamental spacetime structure and considered classes of fundamental equations, conservation laws and symmetries. For instance, it depends on the fact if $^{F}\mathbf{D}$ and $\ ^{F}\mathbf{g}$ are compatible, or not; what type of torsion $\ ^{F}\mathcal{T}$ of $\ ^{F}\mathbf{D}$ is induced by $F$ and/or $\ ^{F}\mathbf{g}$ and $\ ^{F}\mathbf{N,}$ if there are considered compact and/or noncompact extra/velocity/momentum type dimensions etc (in section \[s2\], we present rigorous definitions of such geometric/physical objects).
In this work, our focus is on LV effects and QG phenomenology determined by mechanisms for trapping/locallyzing gravitational and matter fields from a Finsler spacetime on $TV$ to a 4–d observable pseudo–Riemannian spacetime $V.$ Such ideas were originally considered in brane gravity, see [gogb1,gogb1a,gogb2]{} and [@rs1; @rs2], with a non–compact extra dimension coordinate. In Finsler gravity theories on tangent bundles of 3–d and/or 4–d pseudo–Riemannian spacetimes, there are considered respectively 3+3 and/or 4+4 dimensional locally anisotropic spacetime models determined by data $\left[ F:\ ^{F}\mathbf{g,\ }^{F}\mathbf{N,\ }^{F}\mathbf{D}\right] $. This contains a more rich geometric structure than that for Einstein spaces determined by a metric $\mathbf{g,}$ and its unique torsionless and metric compatible Levi–Civita connection $\ ^{g}\nabla .$
Warped Finsler like configurations and related trapping ”isotropization” have to be adapted to the N–connection structure (as we studied for the case of locally anisotropic black holes and propagating solitonically black holes and wormholes [@vsing1]). We can consider solutions with both types of exponential and non–exponential factors by introducing non–gravitational interactions or considering a pure gravitational trapping mechanism for all types of spin fields similarly to 5-d and 6–d pseudo–Riemannian configurations in [@midod; @gm2; @gs2; @singlbr]. The physics of locally isotropic brane theories with extra dimensions and the Finsler–brane models are very different even the LV effects (see [gensol2,vexsol,grojean]{} for locally isotropic branes in 5–d) can be computed in both cases[^6].
The portion of this paper developing conceptual and theoretical issues of Finsler gravity and brane theories spans sections \[s2\] and \[s3\]. It begins in section \[s2\] with a review of the Einstein–Finsler gravity model and the anholonomic deformation method of constructing exact solutions for gravitational field equations. Section \[s3\] concerns explicit Finsler–brane solutions in 3+3 and 4+4 dimensional gravity on tangent bundles. Finally, in section \[s4\] we conclude the results. For convenience, in Appendices we provide two important Theorems and relevant computations on constructing exact solutions in Einstein and Finsler gravity theories. Throughout the paper, we follow the conventions of Refs. [vrevflg,vsgg]{} and [@vacpr] where possible.
Einstein–Finsler Gravity {#s2}
========================
In general, there are two different classes of Finsler gravity models which can be constructed on a $\mathbf{TV}=(TV,\pi ,V),$ where $TV$ is the total space, $\pi $ is a surjective projection and $V,\dim V=n$. In this work, we consider that $V$ is a pseudo–Riemannian/Einstein manifold of dimension $n=2,3,$ or 4), is the base manifold, see details and critical remarks in Refs [@vcrit; @vacpr; @vrevflg; @vsgg].
The first class of theories is with metric noncompatible linear Finsler connections $\ ^{F}\mathbf{D}$ when $\ ^{F}\mathbf{D}\ ^{F}\mathbf{g=\
^{F}Q\neq 0}$ (the typical example is that when $\ ^{F}\mathbf{D=}\ ^{Ch}\mathbf{D}$ is the Chern connection for which the total $\ ^{Ch}\mathbf{Q}$ is not zero but torsion vanishes, $\ ^{Ch}\mathcal{T}=0).$ Because of nonmetricity, it seems that there are a number of conceptual/theoretical and technical problems with definition of spinors and Dirac operators, conservations laws and performing quantization of such theories [vcrit,vacpr,vrevflg]{}. In our opinion, such geometries have less perspectives for applications in standard particle physics and ”simple” modifications, for instance, for purposes of modern cosmology.
The second class of Finsler gravity models is with such $\ ^{F}\mathbf{D}$ which are metric compatible, i.e. $\ ^{F}\mathbf{D}\ ^{F}\mathbf{g}=0.$ Such a locally anisotropic gravity theory is positively with nontrivial torsion, $\ ^{F}\mathcal{T}\neq 0.$ A very important property is that there are $\ ^{F}\mathbf{D}$ when $\ ^{F}\mathcal{T}$ is completely defined by the total Finsler metric structure $\ ^{F}\mathbf{g}$ and a prescribed nonlinear connection (N–connection) $\ ^{F}\mathbf{N}$. For instance, this is the case of canonical distinguished connection (d–connection) $\ ^{F}\mathbf{D=\widehat{\mathbf{D}},}$ see details in [@ma; @vrevflg; @vsgg], and the Cartan d–connection $\ ^{F}\mathbf{D}=\widetilde{\mathbf{D}},$ see formula (\[cdc\]) below. There are preferred constructions with $\widetilde{\mathbf{D}}$ because it defines canonically an almost Kähler structure (for instance, this is important for deformation/ A–brane quantization of gravity [@vqgr1; @vqgr2]).
An an **Einstein–Finsler gravity** theory (EFG), we consider a model of gravity on $TV$ defined by data $\left[ F:\ ^{F}\mathbf{g,\ }^{F}\mathbf{N,\ }^{F}\mathbf{D=}\widetilde{\mathbf{D}}\right] $ and corresponding gravitational field equations in such variables (see section \[ss2\]) following the same principles (postulates) as in GR stated by data $\left[
\mathbf{g,}\ ^{g}\nabla \right].$ Additionally, we suppose that there is a trapping/warped mechanism defined by explicit solutions of (Finsler type) gravitational field equations which in classical limits for $\mathit{l}_{P}\rightarrow 0,$ when EFG $\rightarrow $ GR, determining QG corrections to gravitational and matter field interactions at different scales depending on the class of considered models and solutions.
Fundamental objects in EFG
--------------------------
A (pseudo) $F^{n}=(V,F)$ corresponding, for instance, to a (pseudo) Riemannian manifold $V$ of signature $(-,+,+,...)$ consists of a Finsler metric (fundamental/generating function) $F(x,y)$ (\[fbm\]) defined as a real valued function $F:TV\rightarrow \mathbb{R}$ with the properties that the restriction of $F$ to $\widetilde{TM}$ is a function 1) positive; 2) of class $C^{\infty }$ and $F$ is only continuous on $\{0\};$ 3) positively homogeneous of degree 1 with respect to $y^{i},$ i.e. $F(x,\beta y)=|\beta |F(x,y),$ $\beta \in \mathbb{R};$ and 4) the Hessian $\
^{F}g_{ij}=(1/2)\partial ^{2}F/\partial y^{i}\partial y^{j}$ (\[hess\]) defined on $\widetilde{TV},$ is nondegenerate (for Finsler spaces, this condition is changed into that $\ ^{F}g_{ij}$ is positive definite. In brief, a Finsler space is a Lagrange space with effective Lagrangian $L=F^{2}.$[^7]
### The canonical (Finsler) N–connection
One of three fundamental geometric objects induced by a Finsler metric $F,$ and defining a Finsler space, is the nonlinear connection (N–connection). A N–connection $\mathbf{N}$ is by definition ($:=$) a Whitney sum $$TTV:=hTV\oplus vTV. \label{whitney}$$Geometrically, this an example of nonholonomic (equivalentl, anholonomic, or non–integrable) distribution with conventional horizontal (h) – vertical (v) decomposition/ splitting which can be considered for the module of vector fields $\chi (TTV)$ on $TV.$ For instance, $\mathbf{Y}=\ ^{h}\mathbf{Y}+\ ^{v}\mathbf{Y}$ for any vector $\mathbf{Y}\in \chi (TTV),$ where $\ ^{h}\mathbf{Y}\doteqdot h\mathbf{Y}\in \chi (hTV)$ and $\ ^{v}\mathbf{Y}\doteqdot v\mathbf{Y}\in \chi (vTV).$
There is a canonical N–connection structure $\mathbf{N=\ }^{c}\mathbf{N}$ $\ $which is defined by $F$ following such arguments. Considering that $L=F^{2}$ is a regular Lagrangian (i.e. with nondegenerate $^{F}g_{ij}$ ([hess]{})) and define the action integral $S(\tau
)=\int\limits_{0}^{1}L(x(\tau ),y(\tau ))d\tau $, with $y^{k}(\tau
)=dx^{k}(\tau )/d\tau ,$ for $x(\tau )$ parametrizing smooth curves on $V$ with $\tau \in \lbrack 0,1]$. By straightforward computations, we can prove that the Euler–Lagrange equations of $S(\tau ),$ i.e. $\frac{d}{d\tau
}\frac{\partial L}{\partial y^{i}}-\frac{\partial L}{\partial x^{i}}=0,$ are equivalent to the ”nonlinear geodesic” (equivalently, semi–spray) equations $\frac{d^{2}x^{k}}{d\tau ^{2}}+2G^{k}(x,y)=0,$ where $G^{k}=\frac{1}{4}g^{kj}\left( y^{i}\frac{\partial ^{2}L}{\partial y^{j}\partial x^{i}}-\frac{\partial L}{\partial x^{j}}\right) $ defines the canonical N–connection $\mathbf{\ }^{c}\mathbf{N=}\{\ ^{c}N_{j}^{a}\},$ where $\
^{c}N_{j}^{a}=\frac{\partial G^{a}(x,y)}{\partial y^{j}}.$
Under general (co) frame/coordinate transform, $\mathbf{e}^{\alpha
}\rightarrow \mathbf{e}^{\alpha ^{\prime }}=e_{\ \alpha }^{\alpha ^{\prime }}\mathbf{e}^{\alpha }$ and/or $u^{\alpha }\rightarrow u^{\alpha ^{\prime
}}=u^{\alpha ^{\prime }}(u^{\alpha }),$ preserving the splitting ([whitney]{}), we transform $\ ^{c}N_{j}^{a}\rightarrow N_{j^{\prime
}}^{a^{\prime }},$ when $\mathbf{N}=N_{i^{\prime }}^{a^{\prime
}}(u)dx^{i^{\prime }}\otimes \frac{\partial }{\partial y^{a^{\prime }}}$ is given locally by a set of coefficients $\{N_{j}^{a}\}$. Hereafter, we shall omit priming, underlying etc of indices if that will not result in ambiguities.
### Sasaki types lifts of metrics in Finsler spaces
For a fundamental Finsler function $F(x,y),$ we can construct a canonical (Sasaki type) metric structure $$\begin{aligned}
\ ^{F}\mathbf{g} &=&\ ^{F}g_{ij}(x,y)\ e^{i}\otimes e^{j}+\left( \mathit{l}_{P}\right) ^{2}\ ^{F}g_{ij}(x,y)\ \ ^{F}\mathbf{e}^{i}\otimes \ \ ^{F}\mathbf{e}^{j}, \label{slm} \\
e^{i} &=&dx^{i}\mbox{ and }\ \ ^{F}\mathbf{e}^{a}=dy^{a}+\
^{F}N_{i}^{a}(u)dx^{i}, \label{ddifl}\end{aligned}$$where $\ ^{F}\mathbf{e}^{\mu }=(e^{i},\ ^{F}\mathbf{e}^{a})$ (\[ddifl\]) is the dual to $\ ^{F}\mathbf{e}_{\alpha }=(\ ^{F}\mathbf{e}_{i},e_{a}),$ for $$\ ^{F}\mathbf{e}_{i}=\frac{\partial }{\partial x^{i}}-\ ^{F}N_{i}^{a}(u)\frac{\partial }{\partial y^{a}}\mbox{ and }e_{a}=\frac{\partial }{\partial
y^{a}}. \label{dder1}$$We shall put the square of an effective Planck length $\mathit{l}_{P}$ before the v–part of metric (\[slm\]) if we shall wont to have the same dimensions for the h– and v–components of metric when coordinates have the dimensions $[x^{i}]=cm$ and $[y^{i}\sim dx^{i}/ds]=cm/cm.$
Using frame transforms $e^{\alpha ^{\prime }}=e_{\ \alpha }^{\alpha ^{\prime
}}\mathbf{e}^{\alpha },$ any metric $$\mathbf{g}=g_{\alpha \beta }du^{\alpha }\otimes du^{\beta } \label{cm}$$on $TM,$[^8] including $\ ^{F}\mathbf{g}$ (\[slm\]), can be represented in N–adapted form $$\ \mathbf{g}=\ g_{ij}(x,y)\ e^{i}\otimes e^{j}+\left( \mathit{l}_{P}\right)
^{2}\ h_{ab}(x,y)\ \mathbf{e}^{a}\otimes \ \mathbf{e}^{b}, \label{dm}$$for an N–adapted base $\mathbf{e}_{\nu }=(\mathbf{e}_{i},e_{a}),$ where $$\mathbf{e}_{i}=\frac{\partial }{\partial x^{i}}-\ N_{i}^{a}(u)\frac{\partial
}{\partial y^{a}}\mbox{ and
}e_{a}=\frac{\partial }{\partial y^{a}}, \label{nader}$$and the dual frame (coframe) structure is $\mathbf{e}^{\mu }=(e^{i},\mathbf{e}^{a}),$ for $$e^{i}=dx^{i}\mbox{ and }\mathbf{e}^{a}=dy^{a}+\ N_{i}^{a}(u)dx^{i}.
\label{nadif}$$
The local bases induced by N–connection structure, for instance, ([nader]{}) satisfy nontrivial nonholonomy relations of type $$\lbrack \mathbf{e}_{\alpha },\mathbf{e}_{\beta }]=\mathbf{e}_{\alpha }\mathbf{e}_{\beta }-\mathbf{e}_{\beta }\mathbf{e}_{\alpha }=W_{\alpha \beta
}^{\gamma }\mathbf{e}_{\gamma }, \label{anhrel}$$with (antisymmetric) nontrivial anholonomy coefficients $W_{ia}^{b}=\partial
_{a}N_{i}^{b}$ and $W_{ji}^{a}=\Omega _{ij}^{a}$ determined by the coefficients of curvature of N–connection.
The above formulas define h– and v–splitting of metrics on $TM,$ respectively, $\ ^{h}\mathbf{g=}\{g_{ij}(u)\}$ and $\ ^{v}\mathbf{g=}\{h_{ab}(u)\}.$ Extending the principle of general covariance from $V$ to $TV,$ i.e. from GR to EFG, we can work equivalently with any parametrization of metrics in the form (\[slm\]), (\[cm\]), or (\[dm\]). The first parametrization show in explicit form that our gravity model is for a Finsler spacetime, the second one states the coefficients of metric with respect to local coordinate (co) bases and the third one will be convenient for constructing exact solutions in EFG.
### Canonical linear/distinguished connections
For any Finsler metric $\ ^{F}\mathbf{g}$ (\[slm\]), we can compute in standard form the Levi–Civita connection $\ ^{F}\nabla .$ But such a linear connection is not used in Finsler geometry because it is not adapted to the N–connection structure. We have to revise the concept of linear connection for nonholonomic bundles/manifolds enabled with splitting of type ([whitney]{}): A distinguished connection (d–connection) is a linear connection $\mathbf{D}$ preserving by parallelism the N–connection splitting (\[whitney\]).[^9]
To a d–connection $\mathbf{D}=(\ ^{h}D,\ ^{v}D)=(L_{\ jk}^{i},C_{jc}^{i}), $ for $L_{\ jk}^{i}=L_{\ bk}^{a}$ and $C_{jc}^{i}=C_{bc}^{a}$ (with a chosen contraction for h- and v–indices), we can associate a 1–form $\mathbf{\Gamma }_{\ \beta }^{\alpha }=[\mathbf{\Gamma }_{\ j}^{i},\mathbf{\Gamma }_{\ b}^{a}]$ with $$\mathbf{\Gamma }_{\ j}^{i}=\mathbf{\Gamma }_{\ j\gamma }^{i}\mathbf{e}^{\gamma }=L_{\ jk}^{i}e^{k}+C_{jc}^{i}\mathbf{e}^{c},\ \mathbf{\Gamma }_{\
b}^{a}=\mathbf{\Gamma }_{\ b\gamma }^{a}\mathbf{e}^{\gamma }=L_{\
bk}^{a}e^{k}+C_{bc}^{a}\mathbf{e}^{c}.$$The torsion, $\mathcal{T}$ $=\{\mathbf{T}_{\beta \gamma }^{\alpha }\},$ and curvature, $\mathcal{R}=\{\mathbf{R}_{\ \beta \gamma \tau }^{\alpha }\},$ tensors of a d–connection $\mathbf{D}$ are defined and computed in usual forms as for linear connections for any $\mathbf{X,Y,Z}\in \chi (TTV)$. Using Cartan’s structure equations$$\begin{aligned}
de^{i}-e^{k}\wedge \mathbf{\Gamma }_{\ k}^{i} &=&-\mathcal{T}^{i},\ d\mathbf{e}^{a}-\mathbf{e}^{b}\wedge \mathbf{\Gamma }_{\ b}^{a}=-\mathcal{T}^{a},
\notag \\
d\mathbf{\Gamma }_{\ j}^{i}-\mathbf{\Gamma }_{\ j}^{k}\wedge \mathbf{\Gamma }_{\ k}^{i} &=&-\mathcal{R}_{j}^{i}, \label{cartseq}\end{aligned}$$we can compute the N–adapted coefficients of torsion and curvature, see details in [@ma; @vrevflg; @vsgg]. For instance, an explicit computation results in $$\mathcal{T}^{i}=C_{\ jc}^{i}e^{i}\wedge \mathbf{e}^{c}\mbox{
and }\mathcal{T}^{a}=-\frac{1}{2}\Omega _{ij}^{a}e^{i}\wedge e^{j}+\left(
e_{b}N_{i}^{a}-L_{\ bi}^{a}\right) e^{i}\wedge \mathbf{e}^{b},
\label{nztors}$$with nontrivial values (anti–symmetric on lower indices) of $T_{\
jc}^{i}=-T_{\ cj}^{i}=C_{\ jc}^{i},$ $T_{\ ji}^{a}=-T_{\ ij}^{a}=\frac{1}{2}\Omega _{ij}^{a},$ $T_{\ bi}^{a}=-T_{\ ib}^{a}=e_{b}N_{i}^{a}-L_{\ bi}^{a}.$
For a metric structure $\mathbf{g}=[g_{ij},h_{ab}]$ (\[dm\]), there is a unique normal d–connection $\widetilde{\mathbf{D}}$ which is metric compatible, $\widetilde{\mathbf{D}}\ \mathbf{g=0,}$ and with vanishing $hhh$- and $vvv$–components ($\ ^{h}\widetilde{\mathcal{T}}(h\mathbf{X,}h\mathbf{Y})=0$ and $\ \ ^{v}\widetilde{\mathcal{T}}(v\mathbf{X,}v\mathbf{Y})=0,$ for any vectors $\mathbf{X}$ and $\mathbf{Y)}$ of torsion$\ \widetilde{\mathcal{T}}$ computed following formulas (\[nztors\]). If $\mathbf{g}=\ ^{F}\mathbf{g,}$ we get get the coefficients of the so–called Cartan d–connection in Finsler geometry [@cartan; @ma; @vrevflg; @vsgg]. We can verify that locally the normal d–connection $\widetilde{\mathbf{D}}=(\ ^{h}\widetilde{\mathbf{D}}\mathbf{,}\ ^{v}\widetilde{\mathbf{D}})$ is given respectively by coefficients $\widetilde{\mathbf{\Gamma }}_{\ \beta \gamma
}^{\alpha }=\left( \widetilde{L}_{\ bk}^{a},\widetilde{C}_{bc}^{a}\right) ,$ $$\widetilde{L}_{\ jk}^{i}=\frac{1}{2}g^{ih}(\mathbf{e}_{k}g_{jh}+\mathbf{e}_{j}g_{kh}-\mathbf{e}_{h}g_{jk}),\widetilde{C}_{\ bc}^{a}=\frac{1}{2}h^{ae}(e_{b}h_{ec}+e_{c}h_{eb}-e_{e}h_{bc}), \label{cdc}$$are computed with respect to N–adapted frames. The covariant h–derivative is $\ ^{h}\widetilde{\mathbf{D}}=\{\widetilde{L}_{\ jk}^{i}\}$ and v–derivative is $\ ^{v}\widetilde{\mathbf{D}}=\{\widetilde{C}_{bc}^{a}\}.$ The torsion coefficients $\widetilde{\mathbf{T}}_{\ \beta \gamma }^{\alpha }$ of $\widetilde{\mathbf{D}}$ are $\widetilde{T}_{jk}^{i}=0$ and $\widetilde{T}_{bc}^{a}=0$ but with non–zero cross coefficients, $\widetilde{T}_{ij}^{a}=\Omega _{ij}^{a},\widetilde{T}_{ib}^{a}=e_{b}N_{i}^{a}-\widetilde{L}_{\ bi}^{a}.$
### Finsler variables in (pseudo) Riemannian geometry
Finsler variables can be introduced not only on $TM$ but also, via corresponding nonholonomic distributions, on any pseudo–Riemannian manifold [@gensol2; @vexsol; @vqgr2; @vrevflg] $\mathbf{V},\dim \mathbf{V}=2n,n\geq 2,$ enabled with metric structure $\mathbf{g}$. On such a manifold, we can prescribe any type of nonholonomic frames/ distributions. For instance, we can choose a distribution defined by a regular generating function of necessary type homogeneity, $F(x,y),$ when coordinates $u=(x,y)$ are local ones on $\mathbf{V,}$ with nondegenerate Hessian $\ ^{F}g_{ij}$, and define $\mathbf{g}=\ ^{F}\mathbf{g}$. We model on $\mathbf{V}$ a Finsler geometry if we construct from $\ ^{F}\mathbf{g,}$ in a unique form, the Cartan d–connection $\ \widetilde{\mathbf{D}}.$
In ”standard” variables, a (pseudo) Riemannian geometry is characterized by the Levi–Civita connection $\nabla .$ [^10] We have $$\mathbf{\ }\widetilde{\mathbf{D}}=\mathbf{\ }^{F}\nabla +\mathbf{\ }\widetilde{\mathbf{Z}}, \label{dist}$$where the distortion tensor $\mathbf{\ }^{F}\widetilde{\mathbf{Z}}$ is determined by the torsion $\widetilde{\mathcal{T}},$ see explicit coefficients (\[nztors\]). All such geometric objects (i.e.$\ \widetilde{\mathbf{D}},\ ^{F}\nabla , \widetilde{\mathbf{Z}})$ are completely defined by the same metric structure $\mathbf{g.}$ Any geometric (pseudo) Riemannian data $(\mathbf{g,\nabla })$ can be transformed equivalently into $(\mathbf{g}=\ ^{F}\mathbf{g,}\widetilde{\mathbf{D}})$ and inversely.
The question of (at least formal) equivalence of two gravity theories given by data/ variables $[F:\ ^{F}\mathbf{g,\ }^{F}\mathbf{N,\ }^{F}\mathbf{D}=\widetilde{\mathbf{D}}]$ or $[\mathbf{g=\ ^{F}\mathbf{g},}\ ^{g}\nabla =\mathbf{\ }^{F}\nabla = \widetilde{\mathbf{D}}-\widetilde{\mathbf{Z}}]$ (on $TV,$ or $\mathbf{V)}$ depends on the type of gravitational field equations (for $\mathbf{\ }\widetilde{\mathbf{D}}$ or $\nabla $) and matter field sources are postulated for a model of relativity theory.
Field equations in EFG {#ss2}
----------------------
We can elaborate a Finsler gravity theory on $TM$ using the d–connection $\widetilde{\mathbf{D}}$ and following in general lines the same postulates as in GR. Such a model present a minimal metric compatible Finsler extension of the Einstein gravity but for the generating function $F$.
The curvature 2–form of $\widetilde{\mathbf{D}}=\{\widetilde{\mathbf{\Gamma
}}_{\beta \gamma }^{\alpha }\}$ is computed (see (\[cartseq\]))$$\widetilde{\mathcal{R}}_{\ \gamma }^{\tau }=\widetilde{\mathbf{R}}_{\ \gamma
\alpha \beta }^{\tau }\ \mathbf{e}^{\alpha }\wedge \ \mathbf{e}^{\beta }=\frac{1}{2}\widetilde{R}_{\ jkh}^{i}e^{k}\wedge e^{h}+\widetilde{P}_{\
jka}^{i}e^{k}\wedge \mathbf{e}^{a}+\frac{1}{2}\widetilde{S}_{\ jcd}^{i}\mathbf{e}^{c}\wedge \mathbf{e}^{d},$$when the nontrivial N–adapted coefficients of curvature $\ \widetilde{\mathbf{R}}_{\ \beta \gamma \tau }^{\alpha }$ are $$\begin{aligned}
\widetilde{R}_{\ hjk}^{i} &=&\mathbf{e}_{k}\widetilde{L}_{\ hj}^{i}-\mathbf{e}_{j}\widetilde{L}_{\ hk}^{i}+\widetilde{L}_{\ hj}^{m}\widetilde{L}_{\
mk}^{i}-\widetilde{L}_{\ hk}^{m}\widetilde{L}_{\ mj}^{i}-\widetilde{C}_{\
ha}^{i}\Omega _{\ kj}^{a}, \label{ncurv} \\
\widetilde{P}_{\ jka}^{i} &=&e_{a}\widetilde{L}_{\ jk}^{i}-\widetilde{\mathbf{D}}_{k}\widetilde{C}_{\ ja}^{i},\ \widetilde{S}_{\ bcd}^{a}=e_{d}\widetilde{C}_{\ bc}^{a}-e_{c}\widetilde{C}_{\ bd}^{a}+\widetilde{C}_{\
bc}^{e}\widetilde{C}_{\ ed}^{a}-\widetilde{C}_{\ bd}^{e}\widetilde{C}_{\
ec}^{a}. \notag\end{aligned}$$
The Ricci tensor $\widetilde{R}ic=\{\widetilde{\mathbf{R}}_{\alpha \beta }\}$ is defined by contracting respectively the components of curvature tensor, $\widetilde{\mathbf{R}}_{\alpha \beta }\doteqdot \widetilde{\mathbf{R}}_{\
\alpha \beta \tau }^{\tau }.$ The scalar curvature is $\ ^{s}\widetilde{\mathbf{R}}\doteqdot \mathbf{g}^{\alpha \beta }\widetilde{\mathbf{R}}_{\alpha \beta }=g^{ij}\widetilde{R}_{ij}+h^{ab}\widetilde{R}_{ab},$ where $\widetilde{R}=g^{ij}\widetilde{R}_{ij}$ and $\ \widetilde{S}=h^{ab}\widetilde{R}_{ab}$ are respectively the h– and v–components of scalar curvature.
The gravitational field equations for our Finsler gravity model with metric compatible d–connection$\ ^{F}\mathbf{D=}\widetilde{\mathbf{\mathbf{D}}},$ $$\widetilde{\mathbf{E}}_{\ \beta \delta }=\widetilde{\mathbf{R}}_{\ \beta
\delta }-\frac{1}{2}\mathbf{g}_{\beta \delta }\ ^{s}\widetilde{R}=\widetilde{\mathbf{\Upsilon }}_{\beta \delta } \label{ensteqcdc}$$ can be introduced in geometric and/or variational forms on $TM,$ similarly to Einstein equations in GR, $$\ _{\shortmid }E_{\ \beta \delta }=\ _{\shortmid }R_{\ \beta \delta }-\frac{1}{2}\mathbf{g}_{\beta \delta }\ \ _{\shortmid }^{s}R=\ _{\shortmid }\Upsilon
_{\beta \delta }, \label{einsteqlc}$$where all values (the Einstein and Ricci tensors, respectively, $\
_{\shortmid }E_{\ \beta \delta }$ and $\ _{\shortmid }R_{\ \beta \delta }$, scalar curvature, $\ _{\shortmid }^{s}R,$ and the energy–momentum tensor, $\ _{\shortmid }\Upsilon _{\beta \delta }$) are for the Levi–Civita connection $\ ^{F}\nabla $ computed for the same $\mathbf{g}_{\beta \delta
}=\ ^{F}\mathbf{g}_{\beta \delta }.$
A source $\widetilde{\mathbf{\Upsilon }}_{\beta \delta }$ can be defined following certain geometric and/or N–adapted variational principles for matter fields, see such examples in [@vsgg]. An important property of the equations (\[ensteqcdc\]) is that it can be integrated in very general forms. On exact solutions for such equations (related to black hole physics, locally anisotropic thermodynamics etc) see [gensol2,vexsol,vqgr2,vrevflg,vsgg]{} and references therein. Finsler modified Einstein equations of type (\[ensteqcdc\]) can be such way constructed that they would be equivalent to the Einstein equations for $\nabla$.[^11] Such an equivalence is important if we reformulate the GR theory in Finsler, or almost Kähler variables [@gensol2; @vexsol; @vqgr2], but there are not strong theoretical and/or experimental arguments to impose such conditions for Finsler gravity theories on $TM.$
Finally, we emphasize that the EFG theory is positively with nontrivial torsion structure $\widetilde{\mathbf{T}}_{\ \beta \gamma }^{\alpha }$ induced by fundamental generating function $F(x,y).$ This torsion is completely defined by certain off–diagonal coefficients of the metric structure $\ ^{F}\mathbf{g,}$ including $\ ^{F}\mathbf{N.}$
Finsler–Branes {#s3}
==============
Examples of Einstein–Finsler gravity model and QG phenomenology can be elaborated for metrics $\ ^{F}\mathbf{g},$ see parametrizations ([slm]{}) and (\[dm\]), transforming into Einstein metrics for $\mathit{l}_{P}\rightarrow 0.$ In the classical limit, the gravitational physics is satisfactory described by GR (perhaps with certain exceptions related to accelerating Universes and dark energy/matter problems (see [stavr2,vsgg,lichang4,lichang6,vcrit]{}). In this section, we study scenarios of QG phenomenology and LV when classical 4–d Einstein spacetimes are embedded into 8–d Finsler spaces with non–factorizable velocity type coordinates. Experimentally, the light velocity is finite and metrics in GR do not depend explicitly on velocity/momentum type variables which can be modelled via trapping/warping solutions in EFG.
General ansatz and integrable filed equations
---------------------------------------------
The system of equations (\[ensteqcdc\]) can be integrated in very general forms (following geometric methods reviewed in details in Refs. [gensol2,vexsol]{}). In this paper, we can use a simplified approach because our 8–d Finsler gravity models are with Killing symmetries and Finsler branes can be described by some off–diagonal ansatz for metrics and connections.[^12]
It possible to extend GR theory to holonomic 8–d models on tangent bundle considering a trivial N–connection/Finsler structure for the EFG when solutions with diagonal metrics play an important role. To select a more realistic model of velocity/momentum depending gravity, we have to solve the 8–d Einstein equations (\[einsteqlc\]) (defining a “velocity” depending type of scalar–tensor gravity theory, see discussion in Ref. [@vacpr]) and compare such classes of solutions with generic off–diagonal ones and nontrivial d–torsion and N–connection structures constructed for Finsler gravity.
We use an ansatz which via frame transform can be parametrized [$$\begin{aligned}
&&\mathbf{g} =\ \phi ^{2}(y^{5})[g_{1}(x^{k})\ e^{1}\otimes
e^{1}+g_{2}(x^{k})\ e^{2}\otimes e^{2} +h_{3}(x^{k},v)\ \mathbf{e}^{3}\otimes \mathbf{e}^{3}+ \notag \\
&& h_{4}(x^{k},v)\ \mathbf{e}^{4}\otimes \mathbf{e}^{4}] +\left( \mathit{l}_{P}\right) ^{2}\ [h_{5}(x^{k},v,y^{5})\ \mathbf{e}^{5}\otimes \ \mathbf{e}^{5}+h_{6}(x^{k},v,y^{5})\ \mathbf{e}^{6}\otimes \ \mathbf{e}^{6}] \notag \\
&&+\left( \mathit{l}_{P}\right) ^{2}\ [h_{7}(x^{k},v,y^{5},y^{7})\ \mathbf{e}^{7}\otimes \ \mathbf{e}^{7}+h_{8}(x^{k},v,y^{5},y^{7})\ \mathbf{e}^{8}\otimes \ \mathbf{e}^{8}], \label{ans8d} \\
&& \mathbf{e}^{3} =dv+w_{i}dx^{i},\ \mathbf{e}^{4}=dy^{4}+n_{i}dx^{i},
\mathbf{e}^{5} =dy^{5}+\ ^{1}w_{i}dx^{i}+\ ^{1}w_{3}dv+\ ^{1}w_{4}dy^{4},
\notag \\
&&\mathbf{e}^{6} =dy^{6}+\ ^{1}n_{i}dx^{i}+\ ^{1}n_{3}dv+\ ^{1}n_{4}dy^{4},
\notag \\
&&\mathbf{e}^{7}=dy^{7}+\ ^{2}w_{i}dx^{i}+\ ^{2}w_{3}dv+\ ^{2}w_{4}dy^{4}+\
^{2}w_{5}dy^{5}+\ ^{2}w_{6}dy^{6}, \notag \\
&&\mathbf{e}^{8} =dy^{8}+\ ^{2}n_{i}dx^{i}+\ ^{2}n_{3}dv+\ ^{2}n_{4}dy^{4}+\
^{2}n_{5}dy^{5}+\ ^{2}n_{6}dy^{6}, \notag\end{aligned}$$for nontrivial N–connection coefficients $$\begin{aligned}
N_{i}^{3} &=&w_{i}(x^{k},v),N_{i}^{4}=n_{i}(x^{k},v); \label{ncon8d} \\
N_{i}^{5} &=&\ ^{1}w_{i}(x^{k},v,y^{5}),N_{3}^{5}=\
^{1}w_{3}(x^{k},v,y^{5}),N_{4}^{5}=\ ^{1}w_{4}(x^{k},v,y^{5}); \notag \\
N_{i}^{6} &=&\ ^{1}n_{i}(x^{k},v,y^{5});N_{3}^{6}=\
^{1}n_{3}(x^{k},v,y^{5}),N_{4}^{6}=\ ^{1}n_{4}(x^{k},v,y^{5}); \notag \\
N_{i}^{7} &=&\ ^{2}w_{i}(x^{k},v,y^{7}),N_{3}^{7}=\
^{2}w_{3}(x^{k},v,y^{7}),N_{4}^{7}=\ ^{2}w_{4}(x^{k},v,y^{7}), \notag \\
&&N_{5}^{7}=\ ^{2}w_{3}(x^{k},v,y^{7}),N_{6}^{7}=\ ^{2}w_{4}(x^{k},v,y^{7});
\notag \\
N_{i}^{8} &=&\ ^{2}n_{i}(x^{k},v,y^{7}),N_{3}^{8}=\
^{2}n_{3}(x^{k},v,y^{7}),N_{4}^{8}=\ ^{2}n_{4}(x^{k},v,y^{7}), \notag \\
&&N_{5}^{8}=\ ^{2}n_{3}(x^{k},v,y^{7}),N_{6}^{8}=\ ^{2}n_{4}(x^{k},v,y^{7}).
\notag\end{aligned}$$]{} The local coordinates in the above ansatz (\[ans8d\]) are labeled in the form $x^{i}=(x^{1},x^{2}),$ for $i,j,...=1,2;$ $y^{3}=v.$
Our goal is to construct and analyze physical implications of solutions of equations (\[einsteqlc\]) and (\[ensteqcdc\]) defined by ansatz ([ans8d]{}) with, respectively, trivial and non–trivial N–connection coefficients (\[ncon8d\]).
Holonomic brane configurations
------------------------------
A trapping scenario with diagonal metric from QG with LV to GR can be constructed for an ansatz of type (\[ans8d\]) with zero N–connection coefficients (\[ncon8d\]) when $h_{5},h_{7},h_{8}=const$ and data $\left[g_{i},h_{a}\right] $ define a trivial solution in GR and the local signature for metrics of of type $(+,-,-,...-)$. Such metrics are written $$\begin{aligned}
\mathbf{g} &=&\ \phi ^{2}(y^{5})\eta _{\alpha \beta }du^{\alpha }\otimes
du^{\beta }- \label{ansdiag8d} \\
&&\left( \mathit{l}_{P}\right) ^{2}\overline{h}(y^{5})[\ dy^{5}\otimes \
dy^{5}+dy^{6}\otimes \ dy^{6}\pm dy^{7}\otimes \ dy^{7}\pm dy^{8}\otimes \
dy^{8}], \notag\end{aligned}$$where $\eta _{\alpha \beta }=diag[1,-1,-1-,1]$ and $\alpha ,\beta
,...=1,2,3,4.$ We shall use also generalized indices of type $\ ^{1}\alpha
=(\alpha ,5,6)$ and $\ ^{2}\alpha =(\ ^{1}\alpha ,7,8),$ respectively for 6–d and 8–d models. Indices of type $\ ^{2}\alpha ,\ ^{2}\beta ,...$ will run values $1,2,3,4,5,...,m$, where $m\geq 2.$
We consider sources for Einstein equations (\[einsteqlc\]) with nonzero components defined by cosmological constant $\Lambda $ and stress–energy tensor $$\ _{\shortmid }\Upsilon _{\ \delta }^{\beta }=\Lambda -M^{-(m+2)}\overline{K}_{1}(y^{5}),\ _{\shortmid }\Upsilon _{\ 5}^{5}=\ _{\shortmid }\Upsilon _{\
6}^{6}=\Lambda -M^{-(m+2)}\overline{K}_{2}(y^{5}), \label{source3}$$for a fundamental mass scale $M$ on $TV,$ $\dim TV=8.$ The fiber coordinates $y^{5},y^{6},y^{7},y^{8}$ are velocity/momentum type. Diagonal trivial Finsler brane solutions can be constructed following the methods elaborated (for extra dimensional gravity) in Refs. [@midod; @gm2; @gs2; @singlbr].[^13] A metric (\[ansdiag8d\]) is a solution of (\[einsteqlc\]) if $$\phi ^{2}(y^{5})=\frac{3\epsilon ^{2}+a(y^{5})^{2}}{3\epsilon
^{2}+(y^{5})^{2}}\mbox{ and }\mathit{l}_{P}\sqrt{|\overline{h}(y^{5})|}=\frac{9\epsilon ^{4}}{\left[ 3\epsilon ^{2}+(y^{5})^{2}\right] ^{2}},
\label{cond1}$$where $a$ is an integration constant and the width of brane is $\epsilon ,$ with some fixed integration parameters when $\frac{\partial ^{2}\phi }{\partial (y^{5})^{2}}\mid _{y^{5}=\epsilon }=0$ and $\mathit{l}_{P}\sqrt{|\overline{h}(y^{5})|}\mid _{y^{5}=0}=1;$ this states the conditions that on diagonal branes the Minkowski metric on $TV$ is 6–d or 8–d. We get compatible (with field equations) sources (\[source3\]) if [$$\begin{aligned}
&&\overline{K}_{1}(y^{5})M^{-(m+2)}=\Lambda +\left[ 3\epsilon
^{2}+(y^{5})^{2}\right] ^{-2}[\frac{2am(a(m+2)-3)}{3\epsilon ^{2}}(y^{5})^{4}+ \notag \\
&&2[-2a(m^{2}+2m+6)+3(m+3)(1+a^{2})](y^{5})^{2} -6\epsilon ^{2}m(m-3a+2)],
\label{cond2} \\
&&\overline{K}_{2}(y^{5})M^{-(m+2)} =\Lambda +\left[ 3\epsilon
^{2}+(y^{5})^{2}\right] ^{-2}[\frac{2a(m-1)(a(m+2)-4)}{3\epsilon ^{2}}(y^{5})^{4} + \notag \\
&&4[-a(m^{2}+m+10)+2(m+2)(1+a^{2})](y^{5})^{2} -6\epsilon ^{2}(m-1)(m-4a+2)].
\notag\end{aligned}$$ ]{} The above formulas for $m=2$ are similar to those for usual 6–d diagonal brane solutions with that difference that in our case the width $\epsilon
^{2}=40M^{4}/3\Lambda $ is with extra velocity/momentum coordinates and certain constants are related to $\mathit{l}_{P}.$ Here we also emphasize that $y^{5} $ has a finite maximal value $y^{5}_0 $ on $TM$ because the light velocity is finite.
The Einstein equations (\[einsteqlc\]) are for the Levi–Civita connection when [$$\nabla _{\ ^{2}\alpha }\ _{\shortmid }\Upsilon _{\ \quad }^{\ ^{2}\alpha \
^{2}\beta }=(\sqrt{|\ ^{F}\mathbf{g}|})^{-1}\mathbf{e}_{\ ^{2}\alpha }(\sqrt{|\ ^{F}\mathbf{g}|}\ _{\shortmid }\Upsilon _{\ \quad }^{\ ^{2}\alpha \
^{2}\beta })+\ _{\shortmid }\Gamma _{\ \ ^{2}\alpha \ ^{2}\gamma }^{\
^{2}\beta }\ _{\shortmid }\Upsilon _{\ \quad }^{\ ^{2}\alpha \ ^{2}\gamma
}=0. \label{cond3}$$]{} For our ansatz (\[ansdiag8d\]) and (\[source3\]) with coefficients (\[cond1\]) and (\[cond2\]), such a conservation law is satisfied if $$\frac{\partial \overline{K}_{1}}{\partial (y^{5})}=4\left( \overline{K}_{2}-\overline{K}_{1}\right) \frac{\partial \ln |\phi |}{\partial (y^{5})}.
\label{cond3a}$$
We conclude that a metric (\[ansdiag8d\]), when the coefficients are subjected to conditions (\[cond1\]) – (\[cond3a\]), defines trapping solutions containing “diagonal” extensions of GR to a 8–d $TM$ and/or possible restrictions to 6–d and 7–d. Such solutions provide also mechanisms of corresponding gravitational trapping for fields of spins $0,1/2,1,2$ (see similar proofs in Refs. [@midod; @gm2; @gs2; @singlbr]). The above results are in some sense expected since for diagonal configurations our model is similar to the 6–d and higher dimension ones constructed in the mentioned papers. There are two substantial differences that $m$ is fixed to have a maximal value $m=4$ and that $y^{5}\leq $ $y_{0}^{5}$ where $y_{0}^{5}$ is determined by the maximal speed of light propagation (in the supposition that it is the same as for propagation of gravitational interactions in QG).
The behavior of physical suitable sources determined by ansatz $\overline{K}_{1}(y^{5})$ and/or $\overline{K}_{2}(y^{5})$ depends (for this class of solutions) on four parameters, $m,\epsilon ,\Lambda $ and $a.$ This is quite surprising for QG and solutions with LV because usually it is expected that quantum effects and/or Lorenz violations may be important for distances $\sim \mathit{l}_{P}. $ It is possible to have either $\overline{K}_{1}(y^{5}) $ or $\overline{K}_{2}(y^{5}),$ or both, go to zero for corresponding choices of the mentioned four parameters. To see this we may use the analysis from the Conclusion section of Ref. [@singlbr] even on $TM$ with finite $y_{0}^{5}$ it is not necessary to consider $y^{5}\rightarrow \infty .$ For $m>4,$ which is not the case of Finsler geometry from QG dispersions, the function $\phi ^{2}(y^{5})$ may become singular at $y^{5}\rightarrow \infty .$ Such problems can be avoided because for Finsler configurations derived from GR we can take always $m=1,2,3$ and/or consider $y_{0}^{5}.$ We can consider that in Finsler gravity that $\overline{K}_{1}(y^{5})\rightarrow 0$ or $\overline{K}_{2}(y^{5})\rightarrow
0$ for $y^{5}\rightarrow y_{0}^{5}.$
We do not address the question of stability of Finsler brane solutions in this work. In general, stabile configurations can be constructed for diagonal solutions which survive for nonholonomically constrained off–diagonal ones (proofs are similar to those for extra dimensional brane solutions; we shall address the problem in details in our further works).
Finsler brane solutions
-----------------------
One of the main purposes of this work is to elaborate trapping scenarios for Finsler configurations with positively nontrivial N–connections as solutions of nonholonomic gravitational equations (\[ensteqcdc\]). The priority of such generic off–diagonal solutions is that they allow us to distinguish the QG phenomenology and effects with LV of (pseudo) Finsler type from that described, on $TV$ by (pseudo) Riemannian ones.
### Decoupling of equations in Einstein–Finsler gravity
We consider an ansatz (\[ans8d\]) multiplied to $\phi ^{2}(y^{5})$ and with non–trivial N–connection coefficients (\[ncon8d\]) and define the conditions when the coefficients generate exact solutions of (\[ensteqcdc\]) we get extending the solutions and sources (\[source3\]). The sources are parametrized in a form similar to (\[sourceans\]), $$\begin{aligned}
\widetilde{\mathbf{\Upsilon }}_{\ \delta }^{\beta } &=&diag[\widetilde{\mathbf{\Upsilon }}_{\ 1}^{1}=\widetilde{\mathbf{\Upsilon }}_{\ 2}^{2}=\widetilde{\mathbf{\Upsilon }}_{2}(u^{\ ^{2}\alpha }),\widetilde{\mathbf{\Upsilon }}_{\ 3}^{3}=\widetilde{\mathbf{\Upsilon }}_{\ 4}^{4}=\widetilde{\mathbf{\Upsilon }}_{4}(u^{\ ^{2}\alpha }), \notag \\
&&\widetilde{\mathbf{\Upsilon }}_{\ 5}^{5}=\widetilde{\mathbf{\Upsilon }}_{\
6}^{6}=\widetilde{\mathbf{\Upsilon }}_{6}(u^{\ ^{2}\alpha }),\widetilde{\mathbf{\Upsilon }}_{\ 7}^{7}=\widetilde{\mathbf{\Upsilon }}_{\ 8}^{8}=\widetilde{\mathbf{\Upsilon }}_{8}(u^{\ ^{2}\alpha })], \label{sourcb}\end{aligned}$$when the coefficients are subjected to algebraic conditions (for vanishing N—coefficients, containing respectively the functions (\[source3\]) determining sources in the gravitational field equations) $\ ^{h}\Lambda
(x^{i}) = \widetilde{\mathbf{\Upsilon }}_{4}+\widetilde{\mathbf{\Upsilon }}_{6}+\widetilde{\mathbf{\Upsilon }}_{8},\ ^{v}\Lambda (x^{i},v)=\widetilde{\mathbf{\Upsilon }}_{2}+\widetilde{\mathbf{\Upsilon }}_{6}+\widetilde{\mathbf{\Upsilon }}_{8}, \ ^{5}\Lambda (x^{i},y^{5}) = \widetilde{\mathbf{\Upsilon }}_{2}+\widetilde{\mathbf{\Upsilon }}_{4}+\widetilde{\mathbf{\Upsilon }}_{8},\ \ ^{7}\Lambda (x^{i},y^{5},y^{7})=\widetilde{\mathbf{\Upsilon }}_{2}+\widetilde{\mathbf{\Upsilon }}_{4}+\widetilde{\mathbf{\Upsilon }}_{6}$.
Using the above assumptions on metric ansatz and sources, the conditions of Theorem \[th3\] can be extended step by step for dimensions 2+2+2+2. We obtain a system of equations with decoupling (separation) of partial differential equations (generalizing respectively (\[4ep1a\]) and ([4ep4a]{})): [$$\begin{aligned}
\widetilde{R}_{1}^{1} &=&\widetilde{R}_{2}^{2} =\frac{1}{2g_{1}g_{2}}[\frac{g_{1}^{\bullet }g_{2}^{\bullet }}{2g_{1}}+\frac{(g_{2}^{\bullet })^{2}}{2g_{2}}-g_{2}^{\bullet \bullet }+\frac{g_{1}^{^{\prime }}g_{2}^{^{\prime }}}{2g_{2}}+\frac{(g_{1}^{^{\prime }})^{2}}{2g_{1}}-g_{1}^{^{\prime \prime
}}]=-\ ^{h}\Lambda (x^{i}), \label{4ep1b} \\
\widetilde{R}_{3}^{3} &=&\widetilde{R}_{4}^{4}=\frac{1}{2h_{3}h_{4}}[-h_{4}^{\ast \ast }+\frac{\left( h_{4}^{\ast }\right) ^{2}}{2h_{4}}+\frac{h_{3}^{\ast }\ h_{4}^{\ast }}{2h_{3}}]=\ -\ ^{v}\Lambda (x^{i},v),
\label{4ep2b} \\
\widetilde{R}_{5}^{5} &=&\widetilde{R}_{6}^{6}=\frac{1}{2h_{5}h_{6}}[-\partial _{y^{5}y^{5}}^{2}h_{6}+\frac{\left( \partial _{y^{5}}h_{6}\right)
^{2}}{2h_{6}}+\frac{(\partial _{y^{5}}h_{5})\ (\partial _{y^{5}}h_{6})}{2h_{5}}] =-\ ^{5}\Lambda (x^{i},y^{5}), \notag \\
\widetilde{R}_{7}^{7} &=&\widetilde{R}_{8}^{8}=\frac{1}{2h_{7}h_{8}}[-\partial _{y^{7}y^{7}}^{2}h_{8}+\frac{\left( \partial _{y^{7}}h_{8}\right)
^{2}}{2h_{8}}+\frac{(\partial _{y^{7}}h_{7})\ (\partial _{y^{7}}h_{8})}{2h_{7}}] = -\ ^{7}\Lambda (x^{i},y^{5},y^{7}), \notag\end{aligned}$$]{} with partial derivatives on velocity/momentum type coordinates taken on respective fibers, for instance, $\partial _{y^{5}}h_{6}=\partial
h_{6}/\partial y^{5}.$ The equations (\[4ep1b\]) are completely similar to (\[4ep1a\]) and the equations (\[4ep2b\]) reproduce three times (correspondingly, for couples of variables $\left( y^{3}=v,y^{4}\right)
,\left( y^{5},y^{6}\right) ,\left( y^{7},y^{8}\right) ,$ and ”anisotropic” coordinates $v,y^{5},y^{7\text{ }}$and Killing symmetries on vectors $\partial /\partial y^{4},$ $\partial /\partial y^{6}$ and $\partial
/\partial y^{8})$ the equations (\[4ep2a\]). The equations [$$\begin{aligned}
\ \widetilde{R}_{3j} &=&\frac{h_{3}^{\ast }}{2h_{3}}w_{j}^{\ast }+A^{\ast
}w_{j}+B_{j}=0, \notag \\
\ \widetilde{R}_{5j} &=& \frac{\partial _{y^{5}}h_{5}}{2h_{5}}\partial
_{y^{5}}\ ^{1}w_{j}+\left( \partial _{y^{5}}\ ^{1}A\right) \ ^{1}w_{j}+\
^{1}B_{j}=0, \notag \\
\widetilde{R}_{7j} &=&\frac{\partial _{y^{7}}h_{7}}{2h_{7}}\partial
_{y^{7}}\ ^{2}w_{j}+\left( \partial _{y^{7}}\ ^{2}A\right) \ ^{2}w_{j}+\
^{2}B_{j}=0, \label{4ep3b}\end{aligned}$$]{} generalize on 8–d $TM$ the equations (\[4ep3a\]). The system [$$\begin{aligned}
\widetilde{R}_{4i} &=&-\frac{h_{4}^{\ast }}{2h_{3}}n_{i}^{\ast }+\frac{h_{4}^{\ast }}{2}K_{i}=0,\ \widetilde{R}_{6i} = -\frac{\partial _{y^{5}}h_{6}}{2h_{5}}\partial _{y^{5}}\ ^{1}n_{i}+\frac{\partial _{y^{5}}h_{6}}{2}\
^{1}K_{i}=0, \notag \\
\widetilde{R}_{8i} &=&-\frac{\partial _{y^{7}}h_{8}}{2h_{7}}\partial
_{y^{7}}\ ^{2}n_{i}+\frac{\partial _{y^{7}}h_{8}}{2}\ ^{2}K_{i}=0,
\label{4ep4b}\end{aligned}$$]{} is an extension of (\[4ep4a\]).
In the above formulas (\[4ep3b\]) and (\[4ep4b\]), there are considered nontrivial N–connection coefficients (\[ncon8d\]) and and extensions of of (\[aux\]), [$$\begin{aligned}
&&A =(\frac{h_{3}^{\ast }}{2h_{3}}+\frac{h_{4}^{\ast }}{2h_{4}}) ,\ B_{k}=\frac{h_{4}^{\ast }}{2h_{4}}( \frac{\partial _{k}g_{1}}{2g_{1}}-\frac{\partial _{k}g_{2}}{2g_{2}}) -\partial _{k}A, K_{1} =-\frac{1}{2}(\frac{g_{1}^{\prime }}{g_{2}h_{3}}+\frac{g_{2}^{\bullet }}{g_{2}h_{4}}) , \\
&& K_{2}=\frac{1}{2}(\frac{g_{2}^{\bullet }}{g_{1}h_{3}}-\frac{g_{2}^{\prime
}}{g_{2}h_{4}}); \ ^{1}A =(\frac{\partial _{y^{5}}h_{5}}{2h_{5}}+\frac{\partial _{y^{5}}h_{6}}{2h_{6}}) , \ ^{1}B_{k}=\frac{\partial _{y^{5}}h_{6}}{2h_{6}}(\frac{\partial _{k}g_{1}}{2g_{1}}-\frac{\partial _{k}g_{2}}{2g_{2}})
\\
&&-\partial _{k}\ ^{1}A, \ ^{1}K_{1}= -\frac{1}{2}(\frac{g_{1}^{\prime }}{g_{2}h_{5}}+\frac{g_{2}^{\bullet }}{g_{2}h_{6}}) ,\ ^{1}K_{2}=\frac{1}{2}(\frac{g_{2}^{\bullet }}{g_{1}h_{5}}-\frac{g_{2}^{\prime }}{g_{2}h_{6}}); \\
&& \ ^{2}A =(\frac{\partial _{y^{7}}h_{7}}{2h_{7}}+\frac{\partial
_{y^{7}}h_{8}}{2h_{8}}),\ ^{2}B_{k}=\frac{\partial _{y^{7}}h_{8}}{2h_{8}}(\frac{\partial _{k}g_{1}}{2g_{1}}-\frac{\partial _{k}g_{2}}{2g_{2}})
-\partial _{k}\ ^{2}A, \\
&&\ ^{2}K_{1}=-\frac{1}{2}(\frac{g_{1}^{\prime }}{g_{2}h_{7}}+\frac{g_{2}^{\bullet }}{g_{2}h_{8}}),\ ^{2}K_{2}=\frac{1}{2}(\frac{g_{2}^{\bullet }}{g_{1}h_{7}}-\frac{g_{2}^{\prime }}{g_{2}h_{8}}).\end{aligned}$$ ]{}
### Integration of equations
The conditions of Theorem \[th4\] can be extended on 8–d $TM$ which allows us to integrate in general forms the system of gravitational field equations (see respectively the equations (\[4ep1b\]), (\[4ep2b\]), ([4ep3b]{}) and (\[4ep4b\]) in EFG. Such solutions can be parametrized additionally to the data (\[sol1\])–(\[sol4a\]) (for $g_{i}(x^{k}),h_{a}(x^{k},v),w_{i}(x^{k},v)$ and $n_{i}(x^{k},v))$ by coefficients [$$\begin{aligned}
h_{5}(x^{i},y^{5}) &=&\epsilon _{5}\ _{1}^{0}h(x^{i})\ [\partial _{y^{5}}\
^{1}f(x^{i},y^{5})]^{2}|\ ^{1}\varsigma (x^{i},y^{5})|,\ ^{1}\varsigma =\
_{1}^{0}\varsigma (x^{i})-\frac{\epsilon _{5}}{8}\ _{1}^{0} \notag \\
h(x^{i}) &&\int (dy^{5})\ ^{5}\Lambda (x^{i},y^{5})\ [\partial _{y^{5}}\
^{1}f(x^{i},y^{5})]\ [\ ^{1}f(x^{i},y^{5})-\ _{1}^{0}f(x^{i})], \notag \\
h_{6}(x^{i},y^{5}) &=&\epsilon _{6}[\ ^{1}f(x^{i},y^{5})-\
_{1}^{0}f(x^{i})]^{2}; \notag\end{aligned}$$ $$\begin{aligned}
\ ^{1}w_{j}(x^{i},y^{5}) &=&\ _{0}^{1}w_{j}(x^{i})\exp \left\{
-\int_{0}^{y^{5}}\left[ \frac{2h_{5}\partial _{y^{5}}(\ ^{1}A)}{\partial
_{y^{5}}h_{5}}\right] _{y^{5}\rightarrow v_{1}}dv_{1}\right\} \label{data2a}
\\
\int_{0}^{y^{5}}dv_{1} &&\left[ \frac{h_{5}\ ^{1}B_{j}}{\partial
_{y^{5}}h_{5}}\right] _{y^{5}\rightarrow v_{1}}\exp \left\{
-\int_{_{0}}^{v_{1}}\left[ \frac{2h_{5}\partial _{y^{5}}\ ^{1}A}{\partial
_{y^{5}}h_{5}}\right] _{y^{5}\rightarrow v_{1}}dv_{1}\right\} , \notag \\
\ ^{1}n_{j}(x^{i},y^{5}) &=&\ _{0}^{1}n_{j}(x^{k})+\int dy^{5}\ h_{5}\
^{1}K_{j}, \notag \\
h_{7}(x^{i},y^{5},y^{7}) &=&\epsilon _{7}\ _{2}^{0}h(x^{i})\ [\partial
_{y^{7}}\ ^{2}f(x^{i},y^{5},y^{7})]^{2}|\ ^{2}\varsigma
(x^{i},y^{5},y^{7})|,\ ^{2}\varsigma =\ _{2}^{0}\varsigma (x^{i}) \notag \\
-\frac{\epsilon _{7}}{8}\ _{2}^{0}h(x^{i}) &&\int (dy^{7})\ ^{7}\Lambda
\lbrack \partial _{y^{7}}\ ^{2}f(x^{i},y^{5},y^{7})]\ [\
^{2}f(x^{i},y^{5},y^{7})-\ _{2}^{0}f(x^{i})], \notag \\
h_{8}(x^{i},y^{5},y^{7}) &=&\epsilon _{8}[\ ^{2}f(x^{i},y^{5},y^{7})-\
_{2}^{0}f(x^{i})]^{2}; \notag \\
\ ^{2}w_{j}(x^{i},y^{5},y^{7}) &=&\ _{0}^{2}w_{j}(x^{i})\exp \left\{
-\int_{0}^{y^{7}}\left[ \frac{2h_{7}\partial _{y^{7}}(\ ^{2}A)}{\partial
_{y^{7}}h_{7}}\right] _{v\rightarrow v_{1}}dv_{1}\right\}
\int_{0}^{y^{7}}dv_{1} \notag \\
&&\left[ \frac{h_{7}\ ^{2}B_{j}}{\partial _{y^{7}}h_{7}}\right]
_{y^{7}\rightarrow v_{1}}\exp \left\{ -\int_{_{0}}^{v_{1}}\left[ \frac{2h_{7}\partial _{y^{7}}\ ^{2}A}{\partial _{y^{7}}h_{7}}\right]
_{y^{7}\rightarrow v_{1}}dv_{1}\right\} , \notag \\
\ ^{2}n_{j}(x^{i},y^{7}) &=&\ _{0}^{2}n_{j}(x^{k})+\int dy^{7}\ h_{7}\
^{2}K_{j} \notag\end{aligned}$$]{} Such solutions with nonzero $h_{3}^{\ast },$ $h_{4}^{\ast },\partial
_{y^{5}}h_{5},\partial _{y^{5}}h_{6},\partial _{y^{7}}h_{7},\partial
_{y^{7}}h_{8}$ are determined by generating functions $f(x^{i},v),f^{\ast
}\neq 0,$ $\ ^{1}f(x^{i},y^{5}),\partial _{y^{5}}\ ^{1}f\neq 0,$ $\ ^{2}f(x^{i},y^{5},y^{7}),$ $\partial _{y^{7}}\ ^{2}f\neq 0,$ and integration functions $\ ^{0}f(x^{i}),\ ^{0}h(x^{i}),$ $\ _{0}w_{j}(x^{i}),$ $\ _{0}n_{i}(x^{k}),\ _{1}^{0}f(x^{i}),\ _{1}^{0}h(x^{i}),\
_{0}^{1}w_{j}(x^{i}),\ _{0}^{1}n_{i}(x^{k}),\ _{2}^{0}f(x^{i}),\
_{2}^{0}h(x^{i}),\ _{0}^{2}w_{j}(x^{i}),\ _{0}^{2}n_{i}(x^{k}).$ We should chose and/or fix such functions following additional assumptions on symmetry of solutions, boundary conditions etc.
There are substantial differences between branes in Finsler gravity and in extra dimension theories. In the first case, the physical constants/ parameters are induced in quasi–classical limits from QG on (co) tangent bundles but in the second case the constructions are for high dimensional spacetime models. An important problem to be solved for such geometries is to show that there are trapping mechanisms for nonholonomic configurations to Finsler branes with finite widths (determined by the maximal value of light velocity) and possible warping on ”fiber” coordinates.
### On (non) diagonal brane solutions on $TM$
It is not clear what physical interpretation may have the above general solutions for Finsler gravity. We have to impose additional restrictions on some coefficients of metrics and sources in order to construct in explicit form certain Finsler brane configurations and model a trapping mechanism with generic off–diagonal metrics.
Let us consider a class of sources in EFG when for trivial N–connection coefficients (i.e. for zero values (\[ncon8d\])) the sources $\widetilde{\mathbf{\Upsilon }}_{\ \ ^{2}\delta }^{\ ^{2}\beta }$ (\[sourcb\]) transform into data $\ _{\shortmid }\Upsilon _{\ \ ^{2}\delta }^{\ ^{2}\beta
}$ (\[source3\]), with nontrivial limits for $\ _{\shortmid }\Upsilon _{\
\delta }^{\beta }=\Lambda -M^{-(m+2)}\overline{K}_{1}(y^{5})$ and $\
_{\shortmid }\Upsilon _{\ 5}^{5}=\ _{\shortmid }\Upsilon _{\ 6}^{6}=\Lambda
-M^{-(m+2)}\overline{K}_{2}(y^{5})$. The generating $f$–functions are taken in the form when $h_{5} =\mathit{l}_{P}\frac{\overline{h}(y^{5})}{\phi
^{2}(y^{5})}\ ^{q}h_{5}(x^{i},y^{5}),\ h_{6} =\mathit{l}_{P}\frac{\overline{h}(y^{5})}{\phi ^{2}(y^{5})}\ ^{q}h_{6}(x^{i},y^{5}), h_{7} = \mathit{l}_{P}\frac{\overline{h}(y^{5})}{\phi ^{2}(y^{5})}\ ^{q}h_{7}(x^{i},y^{5},y^{7}),$ $h_{8} =\mathit{l}_{P}\frac{\overline{h}(y^{5})}{\phi ^{2}(y^{5})}\
^{q}h_{8}(x^{i},y^{5},y^{7})$, where the generating functions are parametrized in such a form that $\phi ^{2}(y^{5})$ and $h_{5}(y^{5})$ are those for diagonal metrics, i.e. of type (\[cond1\]), and $\ ^{q}h_{5},\
^{q}h_{6},\ ^{q}h_{7},$ $\ ^{q}h_{8}$ are computed following formulas ([sol1]{})–(\[sol4a\]) and (\[data2a\]). The resulting off–diagonal solutions are [$$\begin{aligned}
\mathbf{g} &=&g_{1}dx^{1}\otimes dx^{1}+g_{2}dx^{2}\otimes dx^{2}+h_{3}\mathbf{e}^{3}{\otimes }\mathbf{e}^{3}\ +h_{4}\mathbf{e}^{4}{\otimes }\mathbf{e}^{4}\ + \label{fbr} \\
&& \left( \mathit{l}_{P}\right) ^{2}\frac{\overline{h}}{\phi ^{2}} [\
^{q}h_{5}\mathbf{e}^{5}\otimes \ \mathbf{e}^{5}+\ ^{q}h_{6}\mathbf{e}^{6}\otimes \ \mathbf{e}^{6}+\ ^{q}h_{7}\mathbf{e}^{7}\otimes \ \mathbf{e}^{7}+\ ^{q}h_{8}\mathbf{e}^{8}\otimes \ \mathbf{e}^{8}], \notag \\
\mathbf{e}^{3} &=&dy^{3}+w_{i}dx^{i},\mathbf{e}^{4}=dy^{4}+n_{i}dx^{i}, \
\mathbf{e}^{5} = dy^{5}+\ ^{1}w_{i}dx^{i}, \label{ncfbr} \\
\mathbf{e}^{6} &=& dy^{6}+\ ^{1}n_{i}dx^{i},\ \mathbf{e}^{7} = dy^{7}+\
^{2}w_{i}dx^{i},\mathbf{e}^{8}=dy^{8}+\ ^{2}n_{i}dx^{i}. \notag\end{aligned}$$ ]{} Any solution of type (\[fbr\]) describes an off–diagonal trapping for 8–d (respectively, for corresponding classes of generating and integration functions, 5–, 6–, 7–d) to 4–d modifications of GR with some corrections depending on QG ”fluctuations” and LV effects. There is a class of sources when for vanishing N–connection coefficients (\[ncfbr\]) we get diagonal metrics of type (\[ans8d\]) but multiplied to a conformal factor $\phi
^{2}(y^{5})$ when the $h$–coefficients are solutions of equations of type (\[4ep2b\]). Even for some diagonal limits, such metrics metrics are very different and can not be transformed, in general form, from one to another even asymptotically, when $\phi (y^{5})\rightarrow a$ for $y^{5}\rightarrow
\infty ,$ they may mimic some similar behavior and QG contributions.
With respect to a local coordinate cobase $du^{\ ^{2}\alpha
}=(dx^{i},dy^{a},dy^{\ ^{1}a},dy^{\ ^{2}a}),$ a solution (\[fbr\]) is parametrized by an off–diagonal matrix $g_{\ ^{2}\alpha \ ^{2}\beta }=$[$$\left[
\begin{array}{cccccccc}
A_{11} & A_{12} & w_{1}h_{3} & n_{1}h_{4}+ & \ ^{1}w_{1}h_{5} & \
^{1}n_{1}h_{6}+ & \ ^{2}w_{1}h_{7} & \ ^{2}n_{1}h_{8} \\
A_{21} & A_{22} & w_{2}h_{3} & n_{2}h_{4} & \ ^{1}w_{2}h_{5} & \
^{1}n_{2}h_{6} & \ ^{2}w_{2}h_{7} & \ ^{2}n_{2}h_{8} \\
w_{1}h_{3} & w_{2}h_{3} & h_{3} & 0 & 0 & 0 & 0 & 0 \\
n_{1}h_{4} & n_{2}h_{4} & 0 & h_{4} & 0 & 0 & 0 & 0 \\
\ ^{1}w_{1}h_{5} & ^{1}w_{2}h_{5} & 0 & 0 & h_{5} & 0 & 0 & 0 \\
\ ^{1}n_{1}h_{6} & \ ^{1}n_{2}h_{6} & 0 & 0 & 0 & h_{6} & 0 & 0 \\
\ ^{2}w_{1}h_{7} & \ ^{2}w_{2}h_{7} & 0 & 0 & 0 & 0 & h_{7} & 0 \\
\ ^{2}n_{1}h_{8} & \ ^{2}n_{2}h_{8} & 0 & 0 & 0 & 0 & 0 & h_{8}\end{array}\right]$$]{} where the possible observable QG and LV contributions (fluctuations in general form) are distinguished by terms proportional to $\left( \mathit{l}_{P}\right) ^{2}$ in $$\begin{aligned}
A_{11} &=&g_{1}+w_{1}^{2}h_{3}+n_{1}^{2}h_{4}+\left( \mathit{l}_{P}\right)
^{2}\frac{\overline{h}}{\phi ^{2}}\times \\
&&\left[ (\ ^{1}w_{1})^{2}\ ^{q}h_{5}+(\ ^{1}n_{1})^{2}\ ^{q}h_{6}+(\
^{2}w_{1})^{2}\ ^{q}h_{7}+(\ ^{2}n_{1})^{2}\ ^{q}h_{8}\right] , \\
A_{12} &=&A_{21}=w_{1}w_{2}h_{3}+n_{1}n_{2}h_{4}+\left( \mathit{l}_{P}\right) ^{2}\frac{\overline{h}}{\phi ^{2}}\times \\
&&\left[ \ ^{1}w_{1}\ ^{1}w_{2}\ ^{q}h_{5}+\ ^{1}n_{1}\ ^{1}n_{2}\
^{q}h_{6}+\ ^{2}w_{1}\ ^{2}w_{2}\ ^{q}h_{7}+\ ^{2}n_{1}\ ^{2}n_{2}\ ^{q}h_{8}\right] , \\
A_{22} &=&g_{2}+w_{2}^{2}h_{3}+n_{2}^{2}h_{4}+\left( \mathit{l}_{P}\right)
^{2}\frac{\overline{h}}{\phi ^{2}}\times \\
&&\left[ (\ ^{1}w_{2})^{2}\ ^{q}h_{5}+(\ ^{1}n_{2})^{2}\ ^{q}h_{6}+(\
^{2}w_{2})^{2}\ ^{q}h_{7}+(\ ^{2}n_{2})^{2}\ ^{q}h_{8}\right] .\end{aligned}$$It is possible to distinguish experimentally such off–diagonal metrics in Finsler geometry from diagonal configurations (\[ansdiag8d\]) with Levi–Civita connection on $TM.$
On Finsler branes determined by data (\[data2a\]), the gravitons are allowed to propagate in the bulk of a Finsler spacetime with dependence on velocity/ momentum coordinates. The reason to introduce warped Finsler geometries and consider various trapping mechanisms is that following modern experimental data there are not explicit observations for Finsler like metrics in gravity even such dependencies can be always derived in various QG models. There are expectations that brane trapping effects may allow us to detect QG and LV effects experimentally even at scales much large than the Planck one and for different scenarios than those considered in Refs. [@kost4; @liberati; @amelino; @dimopoulos; @anchor; @lammer; @stavr0; @girelli].
It is not surprising that two classes of solutions, of type (\[fbr\]) and (\[ans8d\]), are very different on structure and physical implications because such metrics were subjected to the conditions to solve two different classes of gravitational field equations, respectively, (\[ensteqcdc\]) and (\[einsteqlc\]). It should also emphasized here that conservation laws of type (\[cond3\]) are not satisfied for Finsler type solutions even the conditions (\[cond3a\]) can be imposed for some initial data for $\overline{K}_{1},\overline{K}_{2}$ and $\phi .$ In EFG with the Cartan d–connection, the conservation law $\nabla _{\ ^{2}\alpha }\ _{\shortmid
}\Upsilon _{\ \quad }^{\ ^{2}\alpha \ ^{2}\beta }=0$ is nonholonomically deformed into [$$(\sqrt{|\ ^{F}\mathbf{g}|})^{-1}\mathbf{e}_{\ ^{2}\alpha }(\sqrt{|\ ^{F}\mathbf{g}|}\ _{\shortmid }\Upsilon _{\ \quad }^{\ ^{2}\alpha \ ^{2}\beta
})+\ _{\shortmid }\Gamma _{\ \ ^{2}\alpha \ ^{2}\gamma }^{\ ^{2}\beta }\
_{\shortmid }\Upsilon _{\ \quad }^{\ ^{2}\alpha \ ^{2}\gamma }=\widetilde{\mathbf{Z}}_{\ \ ^{2}\alpha \ ^{2}\gamma }^{\ ^{2}\beta }\ _{\shortmid
}\Upsilon _{\ \quad }^{\ ^{2}\alpha \ ^{2}\gamma }, \label{const}$$]{} where the distortion term $\widetilde{\mathbf{Z}}_{\ \ ^{2}\alpha \
^{2}\gamma }^{\ ^{2}\beta }$ (\[dist\]) is determined by nontrivial torsion components (\[nztors\]) (in their turn completely defined by generic off–diagonal terms of $\mathbf{g}$ and respective N–connection coefficients).
Conservation laws of type (\[const\]) are typical for systems with some degrees of freedom subjected to anholonomic constraints, see (\[anhrel\]), which is the case of Finsler spaces (more than that, possible LV result in more sophisticate local spacetime symmetries). They are derived from generalized Bianchi equations for the normal/Cartan d–connection $\widetilde{\mathbf{D}} $ [@ma] (the problem of formulating conservation laws for gravity theories with local gravity is discussed in Refs. [vrevflg,vcrit,vsgg]{}). In our approach, with respect to N–adapted frames, we can compute constraints of type (\[cond3a\]) with some additional terms following from (\[const\]) reflecting some arbitrariness for fixing nonholonomic distributions and frames on $TM$ for EFG. In general, the anholonomic deformation method allows us to construct Finsler brane type solutions with generic off–diagonal QG and LV terms expressed in general form (not depending explicitly on the type of metric compatible d–connection we consider, type of fundamental Finsler function and generating/integration functions).
Discussion and Conclusions {#s4}
==========================
During the last decade, Finsler like gravity models were studied because they appeared to provide possible scenarios of Lorentz symmetry violations (LV) in quantum gravity (QG), new ideas for modified gravity theories with local anisotropy and in relation to dark matter and dark energy problems in modern cosmology [liberati,mavromatos1,stavr0,girelli,gibbons,stavr1,mavromatos,sindoni,stavr2,lichang4,lichang6]{}. The crux of the argument that QG can be related to Finsler geometry follows from three important physical results:
1. There are fundamental uncertainty relations for quantum physics, $$x_{k}p_{j}-p_{k}x_{j}\sim i\hbar , \label{uncert}$$where $x_{i}$ are operators associated to coordinates on a manifold $V$ and $p_{j}$ are momentum variables associated to $T^{\ast }V,$ being dual to certain ”velocities” $y_{k}$ on $TV;i^{2}=-1$ and $\hbar $ is the Planck constant.
2. The bulk of QG theories are with nonlinear dispersion relations (\[disp\]) which encode certain Finsler structure of type (\[fbm\]).
3. The general relativity (GR) theory can be written equivalently in so–called “formal” Finsler variables which can be defined on any (pseudo) Riemannian manifold with conventional horizontal (h) and (v) vertical splitting (for instance, via non–integrable 2+2 distributions/frame decompositions) [vexsol,gensol2,vrevflg,vsgg]{}.
Quantum theories are, at least in quasi–classical limits, some geometric models on (co) tangent bundles of certain manifolds endowed with geometric, dynamical and nonholonomic structure adapted to non–integrable distributions on $TV,$ or $T^{\ast }V$, determined by generating functions $F(x,y)$ (in particular, of uncertainty type ([uncert]{})). The principle of equivalence in GR imposes via nonlinear dispersion relations (\[fbm\]) and (\[disp\]) the condition that $F(x,y)$ is a homogeneous on $y$–variables Finsler metric, see details in Refs. [@vncfinsl; @vacpr; @stavr0; @girelli; @lammer]. Generalizations of the principle of general covariance and axiomatics of GR to $TV$ result in theories with arbitrary (nonhomogeneous) $F$ and more general metric structures $g_{\alpha \beta }(x,y)$ and frame transforms and deformations.
A number of papers on Finsler gravity and applications written by physicists are restricted only to models with ”nonlinear” quadratic Finsler elements $ds^{2}=F^{2}(x,y)$ without important studies of physical implications of nonlinear and distinguished connection structures. Non–experts in Finsler geometry consider that locally anisotropic theories are completely defined by $F$ in a form which is similar to (pseudo) Riemanian geometry which is completely determined by a quadratic $\left(
~^{0}F\right) ^{2}=g_{ij}(x)y^{i}y^{j}$, for $y^{i}\sim dx^{i}.$ Really, in GR a metric tensor field $g_{ij}(x)$ defines a unique metric compatible Levi–Civita connection $\nabla $ on $V,$ and $TV,$ and corresponding fundamental Riemann/Ricci/Einstein tensors when the torsion field is constrained to be zero. Nevertheless, this is not true for Finsler geometries and related gravity models because in such approaches the geometric constructions are based on three fundamental geometric objects: a total metric, $^{F}\mathbf{g,}$ a nonlinear connection, $^{F}\mathbf{N,}$ and a distinguished (adapted) linear connection, $^{F}\mathbf{D.}$ For certain well–defined geometric/physical principles, all such values are uniquely generated by $F$ and this means that a Finsler space[^14] is defined by a triple of geometric data $\left( F:~\mathbf{g,N,D}\right)$. Finsler theories are with more rich geometric structures than the (pseudo) Riemannian ones determined by data $( \mathbf{g,\nabla }).$
In order to elaborate a self–consistent geometric model of classical and quantum Finsler gravity theory we have to involve into constructions all fundamental geometric/physical objects. Such values must be included into certain gravitational and matter field gravitational field equations (derived on $TM$ following certain generalized variational/geometric principles). It is also necessary to try to perform a quantization program and then to analyze possible consequences/applications, for instance, in modern cosmology and astrophysics, or geometric mechanics, see details on such a series of works in Refs. [@vrevflg; @vsgg; @vacpr; @vqgr1; @vqgr2; @vsing1].
We must solve two important problems for quantum/noncommutative Finsler generalizations of GR:
- What type of Finsler nonlinear and linear connections, $(\ ^{F}\mathbf{N},\ ^{F}\mathbf{D})$, are chosen following certain geometric and physical arguments? For instance, mathematicians [@bcs] prefer to work with the Chern and Berwald connections which are metric noncompatible (certain cosmological models [@lichang4; @lichang6] were elaborated following such an approach). Nevertheless, constructions with metric noncompatible connections are less relevant to generalizations of standard theories of particle physics because does not allow to define in a usual form a particle classification, Dirac equations, conservation laws etc, see critical remarks in [@vcrit; @vrevflg; @vsgg]. In our works, we preferred to elaborate physical models when $\ ^{F}\mathbf{D}$ is chosen to be the canonical distinguished and/or normal/Cartan distinguished connections. Such constructions are metric compatible and allow ”more standard” theories of Finsler extension of GR (the so–called Einstein–Finsler gravity, EFG, models).
- Another problem is that if existing experimental data do not constrain ”too much” the perspectives of Finsler gravity for realistic QG and LV theories? For instance, in Ref. [@lammer], such an analysis is performed with the conclusion that coefficients $q_{\widehat{i}_{1}\widehat{i}_{2}...\widehat{i}_{2r}}$ in a Finsler metric (\[fbm\]) and a related dispersion relation (\[disp\]) seem to be very small and this sounds to be very pessimistic for detecting a respective QG phenomenology and LV. Here we note that a conclusion drown only using certain data for a Finsler metric $F(x,y)$ is not a final one because any parametrization (\[fbm\]) is “geometric gauge“ dependent. Really, using frame/coordinate transforms and nonholonomic deformations, $$\left( F:~\mathbf{g,N,D}\right) \rightarrow \left( ~^{0}F:~\mathbf{\check{g},\check{N},\check{D}}\right)$$when $\ ^{0}F$ is a typical quadratic form in GR, the LV effects are removed into data $\left( \mathbf{\check{N},\check{D}}\right) $ modeling nonlinear generic off–diagonal quantum, and quasi–classical, interactions in QG. An explicit example of such systems is that of noncommutative Finsler black holes [vncfinsl]{}. The black hole solutions and various gravitational–gauge–fermion interactions, in GR and EFG can not be studied experimentally only via Mikelson–Morley and possible related nonlinear dispersion effects determined only by $F$. Off–diagonal metrics and anholonomic frames (nonlinear connection) and induced torsion effects (distinguished connection) effects are of crucial importance in Finsler theories.
A rigorous mathematical and physically motivated approach should consider formulations of Finsler gravity theories for certain physically important classes of nonlinear and distinguished connections. We should try to find exact solutions and after that to analyze possible physical implications not restricting our approach only to $F$ but to complete theories with nontrivial $\mathbf{N}$ and $\mathbf{D.}$ Surprisingly, such solutions can be constructed in very general off–diagonal forms [@vexsol; @gensol2; @vrevflg; @vsgg] and we apply such methods in this paper. Nontrivial Finsler spaces on $TV$ are with generic off–diagonal metrics $\mathbf{g}$ which in certain coordinate bases contain contributions from $\mathbf{N.}$ Various limits from EFG to GR can be modeled by a corresponding nonholonomic and nonlinear dynamics when the coefficients of metrics depend anisotropically at least on 3–5–7 space, time and velocity type coordinates on 8–d $TV$ Finsler spacetimes. Such scenarios are more complex than the well–known compactification of extra dimensions in Kaluza–Klein gravity. Even diagonal metrics for Finsler–Kaluza–Klein gravity can be used for certain rough estimations, we need more sophisticate classes of generic off–diagonal exact solutions with warping and trapping of interactions in order to get a constant nonzero value for the speed of light and generic off–diagonal ”bulk” configurations with nontrivial $\mathbf{N}$.
In this article, we have constructed brane world solutions of gravitational field equations for metric compatible EFG theories of QG and possible LV. We found that using generic off–diagonal metrics, non–integrable constraints, Finsler connections and stress–energy ansatz functions it is possible to realize trapping gravitational configurations with physically resonable properties for a range of parameters (for instance, the extra dimension, $m\geq 1;$ bulk cosmological constant $\Lambda ,$ in general, with locally anisotropic polarizations; brane width $\epsilon ;$ a constant value $a$ of gravitational interactions for the maximal speed of light/ gravitational interactions etc). Such brane effects of QG with LV depend on the mechanism of Finsler type gravitational and matter fields interactions on tangent bundle $TV$ over a spacetime $V$ in general relativity (GR) and it is expected that they may be detected in TeV physics, or via modifications in modern cosmology and astrophysics (on locally anisotropic Finsler cosmological scenarios and exact solutions with black ellipsoids, wormholes etc see [@vsing1; @vsgg]).
The solutions with trapping from $TV$ to a GR spacetime $V$ are of two general forms: The first class consists from almost standard diagonal generalizations/ modifications of results from Refs. [@midod; @gm2] for 6-d (and higher) dimensions, when the Einstein equations for the Levi–Civita connection were extended to 8–d (pseudo) Riemannian spacetimes, with possible two time coordinates. If such diagonal brane effects of QG and/or LV origin can be detected experimentally, we can conclude that QG gravity is a (co) tangent bundle geometric theory for the Levi–Civita connection determined by special types of nonlinear dispersions (generating a trivial nonlinear connection, N–connection structure).
Nevertheless, very general assumptions on LV effects and nonlinear dispersions induced from QG seem to result in a second class of Finsler like nonlinear quadratic elements and canonically induced linear connections (for instance, the so–called normal/Cartan d–connection) which are different from the well known Levi–Civita connection. On $TV,$ it is naturally to work with metric compatible d–connections with effective torsion completely determined by a (Finsler) metric and N–connection structure as we discussed in details in Refs. [@vrevflg; @vcrit; @vacpr; @vsgg]. To construct brane solutions with nontrivial N–connection structure and generic off–diagonal metrics (the first attempts where considered in [@vsing1; @vsgg]) is a more difficult technical task which can be solved following the so–called anholonomic deformation/frame method [@vexsol; @gensol2]. We shall address possible applications of nonholonomic geometry methods and Finsler brane solutions in (non) commutative locally anisotropic cosmology and black holes physics [@vncfinsl; @vacpr; @vqgr1].
**Acknowledgements:** This paper contains some results presented at Spanish Relativity Meeting, ERE2010, in Granada, Spain.
Einsten–Finsler Spaces of Dimension 2+2
=======================================
In this Appendix, we study a toy model of Einstein–Finsler gravity on $TM$ over a 2–dimensional manifold $M.$ We prove that such a theory can be integrated in general form. Local coordinates are labeled $u^{\alpha
}=(x^{k},y^{a}),$ where indices run respectively the values: $i,j,k,...=1,2;\ a,b,c,...=3,4;\ $ and $y^{3}=v.$ Using frame transforms any (pseudo) Finsler/ Riemannian 4–d metric can parametrized in the form $$\begin{aligned}
\ \mathbf{g} &\mathbf{=}&g_{i}(x^{k})dx^{i}\otimes dx^{i}+\omega
^{2}(x^{j},y^{b})h_{a}(x^{k},v)\mathbf{e}^{a}{\otimes }\mathbf{e}^{a}
\label{killingdm} \\
&&\mbox{ for }\mathbf{e}^{3}=dy^{3}+w_{i}(x^{k},v)dx^{i},\mathbf{e}^{4}=dy^{4}+n_{i}(x^{k},v)dx^{i}, \notag\end{aligned}$$which is a particular case of (\[dm\]). We label in brief the partial derivatives in the form $g_{1}^{\bullet }=\partial g_{1}/\partial
x^{1},g_{1}^{\prime }=\partial g_{1}/\partial x^{2}$ and $h_{3}^{\ast
}=\partial h_{3}/\partial v.$
For Finsler configurations, the condition of homogeneity results in at least on Killing symmetry for metrics. We can always introduce such a N–adapted frame/coordinate parametrization when $\omega ^{2}=1$ and the above metric does not depend on variables $y^{4}.$ The coefficients of the normal/ Cartan d–connection $\widetilde{\mathbf{\Gamma }}_{\ \alpha \ \beta }^{\
\gamma }$ (\[cdc\]) can be computed for a metric (\[killingdm\]) with $\omega ^{2}=1,$ when $g_{\alpha \beta }=diag[g_{i}(x^{k}),h_{a}(x^{i},v)]$ and $N_{k}^{3}=w_{k}(x^{i},v),N_{k}^{4}=n_{k}(x^{i},v).$[^15] Using the Cartan structure equations (\[cartseq\]), it is possible to determine the $h$– and $v$–components of the Riemannian, torsion, Ricci d–tensors etc.
For the 2+2 dimensional EFG theory, there is a very important property of decoupling/separation of field equations with respect to a class of N–adapted frames which allows us to integrate the theory in very general forms (see Theorem \[th4\]) depending on the types of prescribed nonholonomic constraints and given sources parametrized by frame transform as [$$\widetilde{\mathbf{\Upsilon }}_{\beta }^{\ \delta }= diag[ \widetilde{\mathbf{\Upsilon }}_{1}^{\ 1}= \widetilde{\mathbf{\Upsilon }}_{2}^{\ 2}=\
^{v}\Lambda (x^{i},v),\widetilde{\mathbf{\Upsilon }}_{3}^{\ 3}= \widetilde{\mathbf{\Upsilon }}_{4}^{\ 4}=\ ^{h}\Lambda (x^{i})]. \label{sourceans}$$]{} As particular cases, such sources generalize contributions from nontrivial cosmological constants (for instance, if $\ ^{h}\Lambda =\ ^{v}\Lambda
=\Lambda =const$), their nonholonomic matrix polarizations, approximations for certain dust/radiation locally anisotropic states of matter etc.
\[th3\]The Finsler gravitational field equations (\[ensteqcdc\]) for a metric (\[killingdm\]) with $\omega ^{2}=1$ and source (\[sourceans\]) are equivalent to this system of partial differential equations: [$$\begin{aligned}
\widetilde{R}_{1}^{1} &=&\widetilde{R}_{2}^{2} = \frac{1}{2g_{1}g_{2}}[\frac{g_{1}^{\bullet }g_{2}^{\bullet }}{2g_{1}}+\frac{(g_{2}^{\bullet })^{2}}{2g_{2}}-g_{2}^{\bullet \bullet }+\frac{g_{1}^{^{\prime }}g_{2}^{^{\prime }}}{2g_{2}}+\frac{(g_{1}^{^{\prime }})^{2}}{2g_{1}}-g_{1}^{^{\prime \prime
}}]=-\ ^{h}\Lambda, \label{4ep1a} \\
\widetilde{R}_{3}^{3} &=&\widetilde{R}_{4}^{4}=\frac{1}{2h_{3}h_{4}}\left[
-h_{4}^{\ast \ast }+\frac{\left( h_{4}^{\ast }\right) ^{2}}{2h_{4}}+\frac{h_{3}^{\ast }h_{4}^{\ast }}{2h_{3}}\right] =\ -\ ^{v}\Lambda, \label{4ep2a}
\\
\ \widetilde{R}_{3j} &=&\frac{h_{3}^{\ast }}{2h_{3}}w_{j}^{\ast }+A^{\ast
}w_{j}+B_{j}=0, \label{4ep3a} \\
\widetilde{R}_{4i} &=&-\frac{h_{4}^{\ast }}{2h_{3}}n_{i}^{\ast }+\frac{h_{4}^{\ast }}{2}K_{i}=0, \label{4ep4a} \\
A&=&\left( \frac{h_{3}^{\ast }}{2h_{3}}+\frac{h_{4}^{\ast }}{2h_{4}}\right)
,\ B_{k}=\frac{h_{4}^{\ast }}{2h_{4}}\left( \frac{\partial _{k}g_{1}}{2g_{1}}-\frac{\partial _{k}g_{2}}{2g_{2}}\right) -\partial _{k}A, \label{aux} \\
K_{1} &=&-\frac{1}{2}\left( \frac{g_{1}^{\prime }}{g_{2}h_{3}}+\frac{g_{2}^{\bullet }}{g_{2}h_{4}}\right) ,\ K_{2}=\frac{1}{2}\left( \frac{g_{2}^{\bullet }}{g_{1}h_{3}}-\frac{g_{2}^{\prime }}{g_{2}h_{4}}\right) .
\notag\end{aligned}$$ ]{}
We apply the constructions for the canonical d–connection from [vexsol,gensol2,vsgg]{} to the case of normal/ Cartan d–connection on 4–d $TM.$ Following definition of coefficients $\widetilde{\mathbf{\Gamma }}_{\
\alpha \ \beta }^{\ \gamma }$ (\[cdc\]), the $h$– and $v$–components are similar to those for those for the canonical d–connection. So, the proofs for equations (\[4ep1a\]) and (\[4ep2a\]) are completely similar to those for presented in the mentioned works. For $\widetilde{\mathbf{\Gamma
}}_{\ \alpha \ \beta }^{\ \gamma },$ there are differences for (\[4ep3a\]) and (\[4ep4a\]) which are analyzed in this Appendix.
We perform a N–adapted differential calculus if instead of partial derivatives $\partial _{\ \alpha }=\partial /\partial u^{\ \alpha }$ there are considered operators (\[nader\]) parametrized in the form $\mathbf{e}\
_{i}=\partial _{\ i}-N_{\ i}^{\ a}\partial _{\ a}=\partial _{\ i}-w_{i\ }^{\
}\partial _{v}-n_{i}^{\ }\partial _{4\ }.$ For $N_{k}^{3}=w_{k}(x^{i},v),N_{k}^{4}=n_{k}(x^{i},v),$ the nontrivial coefficients of N–connection curvature are $$\Omega _{\ 12}^{\ 3}=w_{2}^{\ \bullet }-w_{\ 1}^{\prime \ }-w_{1\ }^{\
}w_{2}^{\ \ast }+w_{2\ }^{\ }w_{1}^{\ \ast },\Omega _{\ 12}^{\
4}=n_{2}^{\bullet \ }-n_{1}^{\prime \ }-w_{1\ }^{\ }n_{2}^{\ \ast }+w_{2\
}^{\ }n_{1}^{\ \ast }. \label{nccan}$$ There are nontrivial coefficients of $\ \widetilde{\mathbf{\Gamma }}_{\ \
\alpha \ \beta }^{\ \gamma }$, [$$\begin{aligned}
\widetilde{L}_{11}^{1} &=&\frac{g_{1}^{\bullet }}{2g_{1}},\ \widetilde{L}_{12}^{1}=\frac{g_{1}^{\prime }}{2g_{1}},\widetilde{L}_{22}^{1}=-\frac{g_{2}^{\bullet }}{2g_{1}},\widetilde{L}_{11}^{2}=-\frac{g_{1}^{\prime }}{2g_{2}},\ \widetilde{L}_{12}^{2}=\frac{g_{2}^{\bullet }}{2g_{2}},\ \notag
\\
\ \ \widetilde{L}_{22}^{2} &=&\frac{g_{2}^{\prime }}{2g_{2}},\widetilde{C}_{33}^{3}=\frac{h_{3}^{\ast }}{2h_{3}},\ \widetilde{C}_{44}^{3}=-\frac{h_{4}^{\ast }}{2h_{3}},\ \widetilde{C}_{34}^{4}=\frac{h_{4}^{\ast }}{2h_{4}}.
\label{auxcartc}\end{aligned}$$]{} The nontrivial coefficients of torsion (\[nztors\]) are $$\begin{aligned}
\widetilde{T}_{\ 12}^{\ 3} &=&\Omega _{\ 21}^{\ 3},\ \widetilde{T}_{\ 12}^{\
4}=\Omega _{\ 21}^{\ 4}, \widetilde{P}_{i3}^{3} = w_{i}^{\ast }-\frac{\partial _{i}g_{1}}{2g_{1}}; \widetilde{P}_{\ 14}^{3}=-\frac{g_{1}^{\prime }}{2g_{1}},\widetilde{P}_{\ 24}^{3}=\frac{g_{2}^{\bullet }}{2g_{1}}, \\
\widetilde{P}_{\ 13}^{4} &=& n_{1}^{\ast }+\frac{g_{1}^{\prime }}{2g_{2}},
\widetilde{P}_{\ 23}^{4}=n_{2}^{\ast }-\frac{g_{2}^{\bullet }}{2g_{1}};\widetilde{P}_{i4}^{4}=-\frac{\partial _{i}g_{2}}{2g_{2}}.\end{aligned}$$
The h–v components of the Ricci tensor are derived from $$\begin{aligned}
\widetilde{R}_{\ bka}^{c} &=&\frac{\partial \widetilde{L}_{.bk}^{c}}{\partial y^{a}}-\widetilde{C}_{.ba|k}^{c}+\widetilde{C}_{.bd}^{c}\widetilde{P}_{.ka}^{d} \\
&=&\frac{\partial \widetilde{L}_{.bk}^{c}}{\partial y^{a}}-(\frac{\partial
\widetilde{C}_{.ba}^{c}}{\partial x^{k}}+\widetilde{L}_{.dk}^{c\,}\widetilde{C}_{.ba}^{d}-\widetilde{L}_{.bk}^{d}\widetilde{C}_{.da}^{c}-\widetilde{L}_{.ak}^{d}\widetilde{C}_{.bd}^{c}) +\widetilde{C}_{.bd}^{c}\widetilde{P}_{ka}^{d}.\end{aligned}$$Contracting indices, we get $\widetilde{R}_{bk}=\widetilde{R}_{\ bka}^{a}=\frac{\partial \widetilde{L}_{.bk}^{a}}{\partial y^{a}}-\widetilde{C}_{b|k}+\widetilde{C}_{.bd}^{a}\widetilde{P}_{ka}^{d}$, where $\widetilde{C}_{b}=\widetilde{C}_{.ba}^{c}$ and $\partial \widetilde{L}_{.bk}^{a}/\partial
y^{a}=0$ for (\[killingdm\]) with $\omega ^{2}=1.$ We have $$\begin{aligned}
\widetilde{C}_{b|k} &=&\mathbf{e}_{k}\widetilde{C}_{b}-\widehat{L}_{\
bk}^{d\,}\widetilde{C}_{d}=\partial _{k}\widetilde{C}_{b}-N_{k}^{e}\partial
_{e}\widetilde{C}_{b}-\widetilde{L}_{\ bk}^{d}\widetilde{C}_{d} \\
&=&\partial _{k}\widetilde{C}_{b}-w_{k}\widetilde{C}_{b}^{\ast }-\widetilde{L}_{\ bk}^{d\,}\widetilde{C}_{d},\end{aligned}$$for $\widetilde{C}_{3}=\widetilde{C}_{33}^{3}+\widetilde{C}_{34}^{4}=\frac{h_{3}^{\ast }}{2h_{3}}+\frac{h_{4}^{\ast }}{2h_{4}},\ \widetilde{C}_{4}=\widetilde{C}_{43}^{3}+\widetilde{C}_{44}^{4}=0$, see (\[auxcartc\]).
We express $\widetilde{R}_{bk}=\ ^{1}\widetilde{R}_{bk}+\ ^{2}\widetilde{R}_{bk}+\ ^{3}\widetilde{R}_{bk},$ where$$\begin{aligned}
\ ^{1}\widetilde{R}_{bk} &=&\left( \widetilde{L}_{bk}^{4}\right) ^{\ast
}=0,\ ^{2}\widetilde{R}_{bk}=-\ \widetilde{C}_{b|k}=-\partial _{k}\
\widetilde{C}_{b}+w_{k}\ \widetilde{C}_{b}^{\ast }+\widetilde{L}_{\
bk}^{d\,}\ \widetilde{C}_{d}, \\
\ \ ^{3}\widetilde{R}_{bk} &=&\ \widetilde{C}_{bd}^{a}\widetilde{P}_{.ka}^{d}=\ \widetilde{C}_{b3}^{3}\widetilde{P}_{k3}^{3}+\ \widetilde{C}_{b4}^{3}\widetilde{P}_{k3}^{4}+\ \widetilde{C}_{b3}^{4}\widetilde{P}_{k4}^{3}+\ \widetilde{C}_{b4}^{4}\widetilde{P}_{k4}^{4}.\end{aligned}$$Then, it is possible to compute $\widetilde{R}_{3k}=\ ^{2}\widetilde{R}_{3k}+\ ^{3}\widetilde{R}_{3k}$ when, for instance, $\widehat{L}_{\
3k}^{3\,}\rightarrow \widehat{L}_{\ 1k}^{1\,}$ and $\widehat{L}_{\
4k}^{4\,}\rightarrow \widehat{L}_{\ 2k}^{2\,},\ $ with [$$\begin{aligned}
\ ^{2}\widetilde{R}_{3k} &=&-\partial _{k}\widetilde{C}_{3}+w_{k}\widetilde{C}_{3}^{\ast }+\widehat{L}_{\ 3k}^{3\,}\widetilde{C}_{3} \\
&=& -\partial _{k}\left( \frac{h_{3}^{\ast }}{2h_{3}}+\frac{h_{4}^{\ast }}{2h_{4}}\right) +\left( \frac{h_{3}^{\ast }}{2h_{3}}+\frac{h_{4}^{\ast }}{2h_{4}}\right) ^{\ast }w_{k}+\left( \frac{h_{3}^{\ast }}{2h_{3}}+\frac{h_{4}^{\ast }}{2h_{4}}\right) \frac{\partial _{k}g_{1}}{2g_{1}}, \\
\ ^{3}\widetilde{R}_{3k} &=&\ \ \widetilde{C}_{33}^{3}\widetilde{P}_{k3}^{3}+\ \widetilde{C}_{34}^{3}\widetilde{P}_{k3}^{4}+\ \widetilde{C}_{33}^{4}\widetilde{P}_{k4}^{3}+\ \widetilde{C}_{34}^{4}\widetilde{P}_{k4}^{4} \\
&=& \frac{h_{3}^{\ast }}{2h_{3}}\left( w_{i}^{\ast }-\frac{\partial _{i}g_{1}}{2g_{1}}\right) -\frac{h_{4}^{\ast }}{2h_{4}}\frac{\partial _{i}g_{2}}{2g_{2}}.\end{aligned}$$]{} (\[4ep3a\]) can be obtained summarizing above formulas, [$$\widetilde{R}_{3k}=\frac{h_{3}^{\ast }}{2h_{3}}w_{i}^{\ast }+\left( \frac{h_{3}^{\ast }}{2h_{3}}+\frac{h_{4}^{\ast }}{2h_{4}}\right) ^{\ast
}w_{k}-\partial _{k}\left( \frac{h_{3}^{\ast }}{2h_{3}}+\frac{h_{4}^{\ast }}{2h_{4}}\right) +\frac{h_{4}^{\ast }}{2h_{4}}\left( \frac{\partial _{k}g_{1}}{2g_{1}}-\frac{\partial _{k}g_{2}}{2g_{2}}\right).$$ ]{}
Similarly, we compute $\ \widetilde{R}_{4k}=\ ^{2}\widetilde{R}_{4k}+\ ^{3}\widetilde{R}_{4k},$ where [$$\begin{aligned}
\ \ ^{2}\widetilde{R}_{4k} &=&-\partial _{k}\ \widetilde{C}_{4}+w_{k}\widetilde{C}_{4}^{\ast }+\widetilde{L}_{\ 4k}^{3\,}\ \widetilde{C}_{4}=0; \\
\ \ ^{3}\widetilde{R}_{4k} &=&\ \widetilde{C}_{43}^{3}\widetilde{P}_{k3}^{3}+\ \widetilde{C}_{44}^{3}\widetilde{P}_{k3}^{4}+\ \widetilde{C}_{43}^{4}\widetilde{P}_{k4}^{3}+\ \widetilde{C}_{44}^{4}\widetilde{P}_{k4}^{4} = \widetilde{C}_{44}^{3}\widetilde{P}_{k3}^{4}+ \widetilde{C}_{43}^{4}\widetilde{P}_{k4}^{3}.\end{aligned}$$]{} Putting together, we obtain (\[4ep4a\]) \[which ends the proof of Theorem \[th3\]\], $$\ \widetilde{R}_{41}=-\frac{h_{4}^{\ast }}{2h_{3}}(n_{1}^{\ast }+\frac{g_{1}^{\prime }}{2g_{2}}) -\frac{g_{2}^{\bullet }}{2g_{2}}\frac{h_{4}^{\ast }}{2h_{4}},\ \widetilde{R}_{42}=-\frac{h_{4}^{\ast }}{2h_{3}}(n_{2}^{\ast }-\frac{g_{2}^{\bullet }}{2g_{1}}) -\frac{g_{2}^{\prime }}{2g_{2}}\frac{h_{4}^{\ast }}{2h_{4}}.$$
Integration of field equations
==============================
\[th4\]The general solutions of equations (\[4ep1a\])–(\[4ep4a\]) defining Einstein–Finsler spaces are parametrized by metrics of type ([killingdm]{}) with coefficients computed in the form [$$\begin{aligned}
g_{i} &=&\epsilon _{i}e^{\psi (x^{k})},\mbox{\ for }\epsilon _{1}\psi
^{\bullet \bullet }+\epsilon _{2}\psi ^{\prime \prime }=\ ^{h}\Lambda
(x^{k}); \label{sol1} \\
h_{3} &=&\epsilon _{3}\ ^{0}h(x^{i})\ [f^{\ast }(x^{i},v)]^{2}|\varsigma
(x^{i},v)|, \label{sol2} \\
\varsigma &=&\ ^{0}\varsigma (x^{i})-\frac{\epsilon _{3}}{8}\
^{0}h(x^{i})\int(dv)\ ^{v}\Lambda (x^{k},v) f^{\ast }(x^{i},v)\
[f(x^{i},v)-\ ^{0}f(x^{i})], \notag \\
h_{4} &=&\epsilon _{4}[f(x^{i},v)-\ ^{0}f(x^{i})]^{2}; \label{sol3} \\
w_{j} &=&\ _{0}w_{j}(x^{i})\exp \left\{ -\int_{0}^{v}[ {2h_{3}A^{\ast }}/{h_{3}^{\ast }}] _{v\rightarrow v_{1}}dv_{1}\right\} \times \label{sol4} \\
&&\int_{0}^{v}dv_{1}[{h_{3}B_{j}}/{h_{3}^{\ast }}] _{v\rightarrow v_{1}}\exp
\left\{ -\int_{_{0}}^{v_{1}}[{2h_{3}A^{\ast }}/{h_{3}^{\ast }}]
_{v\rightarrow v_{1}}dv_{1}\right\} , \notag \\
n_{i} &=&\ _{0}n_{i}(x^{k})+\int dv\ h_{3}K_{i}. \label{sol4a}\end{aligned}$$]{} Such solutions with $h_{3}^{\ast }, h_{4}^{\ast }\neq 0$ are determined by generating, $f(x^{i},v),f^{\ast }\neq 0,$ and integration, $\ ^{0}f(x^{i}),\
^{0}h(x^{i}),$ $\ _{0}w_{j}(x^{i}),\ _{0}n_{i}(x^{k}),$ functions.
We sketch a proof following two steps:
1. **Solutions with Killing symmetry for h– and v–components of metric:** The equation (\[4ep1a\]) is for a two dimensional (semi) Riemannian metric. Any such metric can be diagonalized and expressed as a conformally flat metric. Choosing $\epsilon _{i}e^{\psi (x^{k})},$ we get the Poisson equation in (\[sol1\]). The equation (\[4ep2a\]) is similar to that for the canonical d–connection configurations which was solved in general form [@vexsol; @gensol2; @vsgg]. Such equations relate two un–known functions. For instance, if we prescribe any $h_{3}(x^{i},v),$ we can construct (at least via some series decompositions) $h_{4}(x^{i},v),$ and inversely. By straightforward computations, we can verify that any $h_{3}$ and $h_{4}$ with nonzero $h_{3}^{\ast }$ and $h_{4}^{\ast }$ given by ([sol2]{}) define exact solutions for (\[4ep2a\]). Solutions with $h_{3}^{\ast }=0$ and/or $h_{4}^{\ast }=0$ should be re–considered as some particular degenerated cases.
2. **Solutions for the N–connection coefficients:** The main differences between our former results for the canonical d–connection and the normal/ Cartan d–connection (in this work) consist in equations ([4ep3a]{}) and (\[4ep4a\]) and coefficients (\[aux\]). We provide the proofs of formulas (\[sol3\]) and (\[sol4a\])) in Appendix \[assc\]. Taking together the solutions (\[sol1\])–(\[sol4a\]) for ansatz ([killingdm]{}) with $\omega ^{2}=1,$ we constrict the general class of exact solutions with Killing symmetry on $\partial /\partial y^{4}$ defining Einstein–Finsler spaces[^16]. Considering different types of frame transforms, with coordinates parametrized for tangent bundles, such metrics can be transformed into standard ones $\ ^{F}\mathbf{g}$ (\[slm\]) Finsler spaces.
**Some computations for Theorem \[th4\]:**
\[assc\]The solutions of (\[4ep3a\]) and (\[4ep4a\]) can be always considered for $|g_{1}|=|g_{2}|,$ when $B_{k}=\partial _{k}A.$ We construct them for three more special cases.
Case 1: $h_{3}^{\ast }=0,h_{4}^{\ast }\neq 0$ and $A=h_{4}^{\ast }/2h_{4}.$ We must solve the equation $h_{4}^{\ast \ast }-\frac{\left( h_{4}^{\ast
}\right) ^{2}}{2h_{4}}=2h_{3}h_{4}\ ^{v}\Lambda (x^{i},v),$ for any given $h_{3}=h_{3}(x^{i})$ and $\ ^{v}\Lambda (x^{i},v).$ We have $w_{j}^{\ast }=0$ and we obtain, from (\[4ep3a\]), $w_{j}=-B_{j}/A^{\ast }=-\partial
_{j}A/A^{\ast }$ and, from (\[4ep4a\]), $n_{i}^{\ast }=K_{i}h_{3}.$
Case 2: $h_{4}^{\ast }=0,$ any $h_{3}$ and $n_{i}$ for $\ ^{v}\Lambda =0.$ Let us consider in (\[4ep3a\]) that $h_{3}\neq 0.$ We have to solve $\frac{h_{3}^{\ast }}{2h_{3}}w_{j}^{\ast }+A^{\ast }w_{j}+B_{j}=0.$ Representing $w_{j}=\ ^{1}w_{j}\cdot \ ^{2}w_{j}$ and introducing $\ w_{j}^{\ast }=\
^{1}w_{j}^{\ast }\cdot \ ^{2}w_{j}+\ ^{1}w_{j}\cdot \ ^{2}w_{j}^{\ast }$ into above equation, we obtain $$\ ^{1}w_{j}^{\ast }\cdot \ ^{2}w_{j}+\ ^{1}w_{j}\cdot \ ^{2}w_{j}^{\ast }+\frac{2h_{3}A^{\ast }}{h_{3}^{\ast }}\ ^{1}w_{j}\cdot \ ^{2}w_{j}+\frac{2h_{3}B_{j}}{h_{3}^{\ast }}=0.$$We can chose $\ ^{1}w_{j}=-\ _{0}^{1}w_{j}(x^{i})\exp \left[ -\int \frac{2h_{3}A^{\ast }}{h_{3}^{\ast }}dv\right] ,$ for some integration functions $\ _{0}^{1}w_{j}(x^{i}),$ and transform the equation into$\ ^{2}w_{j}^{\ast
}=2\ _{0}^{1}w_{j}(x^{i})$$\int_{v_{2}}^{v}dv_{1}\frac{h_{3}B_{j}}{h_{3}^{\ast }}\exp \left[
-\int_{v_{0}}^{v_{1}}\frac{2h_{3}A^{\ast }}{h_{3}^{\ast }}dv_{1}\right] $, which can be integrated in general form. Finally, we get $$w_{j}=\ _{0}w_{j}(x^{i})\exp \left[ -\int \frac{2h_{3}A^{\ast }}{h_{3}^{\ast
}}dv\right] \int_{v_{2}}^{v}dv_{1}\frac{h_{3}B_{j}}{h_{3}^{\ast }}\exp \left[
-\int_{v_{0}}^{v_{1}}\frac{2h_{3}A^{\ast }}{h_{3}^{\ast }}dv_{1}\right] ,$$for some $v_{0},v_{2}=const.$ The solution of (\[4ep4a\]) is constructed by a direct integration on $v$ of values $K_{i}$ from (\[aux\]).
Case 3: $h_{3}^{\ast }\neq 0,h_{4}^{\ast }\neq 0,$ which is stated in Theorem \[th4\]. As a general solution of (\[4ep3a\]), we can consider (\[sol3\]) with the coefficients $A,B_{j}$ and $K_{i}$ and computed for arbitrary $h_{3}$ and $h_{4}$ depending on $v.$ Integrating (\[4ep4a\]), we get the formula (\[sol4a\]).
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[^1]: sergiu.vacaru@uaic.ro, Sergiu.Vacaru@gmail.com
[^2]: in explicit form, we have to consider additional dependencies and characteristic parametrizations on spacetime coordinates $x^{i},$ metric $g_{ij},$ spin of particle, chosen types of spacetime connections etc
[^3]: Light rays can be parametrized as $x^{i}(\varsigma )$ with a real smooth parameter $0\leq \varsigma \leq \varsigma _{0},$ when $ds^{2}/d\varsigma
^{2}=0;$ there is a ”null” tangent vector field $y^{i}(\varsigma
)=dx^{i}/d\varsigma ,$ with $d\tau =dt/d\varsigma .$ Under general coordinate transforms $x^{i^{\prime }}=x^{i^{\prime }}(x^{i}),$ we have $\eta _{ij}\rightarrow g_{i^{\prime }j^{\prime }}(x^{k});$ the condition $ds^{2}/d\varsigma ^{2}=0$ holds always for propagation of light, i.e. $g_{i^{\prime }j^{\prime }}y^{i^{\prime }}y^{j^{\prime }}=0.$
[^4]: for simplicity, we can consider such a relation in a fixed point $x^{k}=x_{(0)}^{k},$ when $g_{\widehat{i}\widehat{j}}(x_{0}^{k})=g_{\widehat{i}\widehat{j}}$ and $q_{\widehat{i}_{1}\widehat{i}_{2}...\widehat{i}_{2r}}=q_{\widehat{i}_{1}\widehat{i}_{2}...\widehat{i}_{2r}}$ $(x_{0}^{k})$
[^5]: Following our conventions [@vcrit; @vrevflg; @vsgg], we use ”boldface” symbols for spaces/ geometrical/physical objects endowed with /adapted to nonlinear connection structure, see definitions in next section; we also put left up/low labels in order to emphasize that a geometric/physical object is completely defined/induced by a corresponding fundamental generating function, for instance, that $^{F}\mathbf{g}$ is completely and uniquely determined by $F.$
[^6]: the Finsler configurations are or different nature when the generating functions are determined by certain general coefficients in a QG model, and related LV, which is not the case, for instance, of Rundall–Sundrum branes
[^7]: Similar theories can be elaborated for (pseudo) Lagrange spaces and generalizations as it is provided in [@ma; @vrevflg; @vsgg]. This way, we can construct different ”analogous gravity” and geometric mechanics models. Here we also note that Finsler–Lagrange variables can be introduced even in Einstein gravity which is very convenient for constructing exact solutions and developing certain models of QG.
[^8]: a dual local coordinate basis os $du^{\beta }=(dx^{j},dy^{b}),$ when $\partial _{\alpha }=\partial /\partial u^{\alpha }=(\partial _{i}=\partial
/\partial x^{i},\partial _{a}=\partial /\partial y^{a})$
[^9]: In Lagrange–Finsler geometry, there are used the terms distinguished tensor/ vector / spinor connection etc (d–tensor, d–vector, d–spinor, d–connection etc) [@ma; @vrevflg; @vsgg] for the corresponding geometric objects defined with respect to N–adapted (co) bases.
[^10]: By definition, it is metric compatible, $\nabla \mathbf{g}=0,$ and torsionless, $\ ^{\nabla }\mathcal{T}=0.$
[^11]: This is possible for the normal/ Cartan d–connection $\widetilde{\mathbf{D}}
$ being completely defined by $\mathbf{g}_{\beta \delta }$ (\[dm\]) and if $\ \widetilde{\mathbf{\Upsilon }}_{\beta \delta }=\ ^{matter}\mathbf{\Upsilon }_{\beta \delta }+\ ^{z}\mathbf{\Upsilon }_{\beta \delta }$ are derived in such a way that they contain contributions from 1) the N–adapted energy–momentum tensor (defined variationally following the same principles as in GR but on $TV$) and 2) the distortion of the Einstein tensor in terms of $\ \widehat{\mathbf{Z}}$ (\[dist\]), $\widetilde{\mathbf{Z}}_{\ \beta \delta }=\ _{\shortmid }E_{\alpha \beta }+\ ^{z}\widetilde{\mathbf{Z}}_{\ \beta \delta },$ for $\ ^{z}\mathbf{\ }\widetilde{\mathbf{Z}}_{\ \beta \delta }=\ ^{z}\mathbf{\Upsilon }_{\beta \delta }.$ The value $\ ^{z}\widetilde{\mathbf{Z}}_{\ \beta \delta }$ is computed by introducing $\widetilde{\mathbf{D}}=\ ^{F}\nabla +\mathbf{\ }\widetilde{\mathbf{Z}}$ into (\[ncurv\]) and corresponding contractions of indices in order to find the Ricci d–tensor and scalar curvature.
[^12]: For convenience, we provide in Appendix two theorems on constructing exact solutions for a 4–d Einstein–Finsler toy model which is exactly integrable. Various extensions of the outlined there anholonomic deformation method to 6–d and 8–d Finsler brane spacetimes with nontrivial N–connection structures are straightforward.
[^13]: In this paper we shall use some adapted classes of solutions from the just cited paper where the extra dimensions (2,3 etc) are analyzed in general form. Here we also note that our notations for Finsler gravity models on tangent bundles are different for those used in the above papers on 6-d, and other dimensions, brane gravity solutions.
[^14]: spacetime, if $^{F}\mathbf{g}
$ is related to a Minkowski metric in special relativity, or GR; for simplicity, we omit the left label $F$ is this does not result in ambiguities
[^15]: Following methods elaborated in Refs. [@gensol2; @vexsol], we can construct exact solutions with $\omega ^{2}\neq 1.$ For Finsler brane configurations, for simplicity, we do not consider such “very” general classes of solutions.
[^16]: such a symmetry exists if the coefficients of metrics do not depend on coordinate $y^{4}$
|
---
abstract: 'The Reynolds number dependency of intermittency for 2D turbulence is studied in a flowing soap film. The Reynolds number used here is the Taylor microscale Reynolds number $R_{\lambda}$, which ranges from 20 to 800. Strong intermittency is found for both the inverse energy and direct enstrophy cascades as measured by (a) the pdf of velocity differences $P( \delta u(r))$ at inertial scales $r$, (b) the kurtosis of $P(\partial_x u)$, and (c) the scaling of the so-called intermittency exponent $\mu$, which is zero if intermittency is absent. Measures (b) and (c) are quantitative, while (a) is qualitative. These measurements are in disagreement with some previous results but not all. The velocity derivatives are nongaussian at all $R_{\lambda}$ but show signs of becoming gaussian as $R_{\lambda}$ increases beyond the largest values that could be reached. The kurtosis of $P(\delta u(r))$ at various $r$ indicates that the intermittency is scale dependent. The structure function scaling exponents also deviate strongly from the Kraichnan prediction. For the enstrophy cascade, the intermittancy decreases as a power law in $R_{\lambda}$. This study suggests the need for a new look at the statistics of 2D turbulence.'
author:
- 'R.T. Cerbus'
- 'W.I. Goldburg'
title: Intermittency in 2D soap film turbulence
---
Introduction
============
Three-dimensional turbulence appears in bursts, a phenomenon called “intermittency”. Said in another way, the flow is characterized by velocity fluctuations of size $r$ that deviate strongly from their mean value. The issue addressed in this experimental study is the presence and strength of intermittency in flows that are very close to being two-dimensional.
There are several prior studies, both experimental and numerical, that have addressed this issue, but there is little agreement. These experiments support the existence of strong intermittency in two-dimensional (2D) turbulence. In fact they imply that it is even stronger in 2D than in three dimensions (3D).
Turbulent flow in 2D is decidedly different from its 3D counterpart. In 3D, eddies of size $r$ break up into smaller and smaller ones (say, of size $r/2, r/2^2, r/2^3$...). The total vorticity amplitude is amplified by local velocity gradients, a well-studied effect called “vortex stretching” [@frisch1995]. On the other hand, in 2D the total squared vorticity $\langle {\bf \omega}^2 \rangle$ (enstrophy) is a constant of the motion in the absence of viscosity, just like the total kinetic energy density $\langle u^2 \rangle$ in 3D. It was R. Kraichnan who first recognized that this new conservation law comes into play with the result that energy is transferred to large scales while enstrophy is handed down to small scales [@kraichnan1967a; @kraichnan1980; @boffetta2012; @kellay2002; @tabeling2002a].
Assume that the turbulence is forced at a scale $r = r_f$. In the steady state this forcing results in a mean rate of kinetic dissipation per unit mass $\epsilon = d \langle u^2 \rangle/dt$ and a mean rate of dissipation of the enstrophy, $\beta = d \langle \omega ^2 \rangle/dt$. The Kraichnan analysis gives the result for the longitudinal velocity fluctuations or second-order structure functions
$$\begin{aligned}
S_2(r) \equiv \langle \delta u(r)^2 \rangle \propto \epsilon ^{2/3} r^{2/3} , \,\,\, r>r_f \\
S_2(r) \equiv \langle \delta u(r)^2 \rangle \propto {\beta}^{2/3} r^2 , \,\,\,\, r < r_f
\label{mlett:2}\end{aligned}$$
where $\langle \delta u(r) \rangle \equiv \langle u(x+r) - u(x) \rangle_x$ is the velocity difference on a scale $r$ averaged over $x$ (flow direction). The two cascades do not extend to infinity in both directions. The energy cascade eventually reaches a scale $r = r_{\alpha}$ on the order of the system size where it is cut off by boundary friction or coupling to the 3D environment [@boffetta2012]. Likewise, the enstrophy cascade eventually reaches a scale $r = \eta \equiv \nu^{1/2}/\beta^{1/6}$ where viscosity takes over.
A simplified picture of the double cascade process is to view eddies of size $r > r_f$ combining to create the energy cascade. The interstices between these eddies contain most of the shear (and hence large vorticity) but little energy. The result is a cascade of enstrophy to small scales. Recently this picture has come into question [@chen2006; @chen2003], but there is full agreement on the strong difference between 3D and 2D turbulence. Both of these cascades are studied in these experiments, although they do not appear simultaneously. This is the case in nearly all 2D experiments and simulations [@boffetta2012; @kellay2002; @tabeling2002a]. Some high resolution simulations and a few experiments are able to achieve the classical dual cascade picture [@boffetta2007; @boffetta2010; @rutgers1998; @daniel2000; @bruneau2005].
Intermittency is not a sharply defined effect in 2D or 3D. The presence of strong, unlikely velocity fluctuations is manifest in many different types of analysis. Most measures of intermittency are defined in terms of a deviation from the predictions of the scaling theory of Kolmogorov (K41) in 3D [@kolmogorov1941] or Kraichnan (Kr67) in 2D [@kraichnan1967a], although intermittency has a larger meaning than this [@frisch1995]. Some of those measurements are made in the “inertial range” of velocity fluctuations, defined as those eddy sizes where damping effects are negligible. The K41 and Kr67 predictions are for this range of scales. On the other hand, intermittency is reported in measurements of the probability density function (pdf) of velocity gradients $\partial_x u$ or velocity differences $\delta u(r)$. The pdfs are expected to be nearly gaussian if there is no intermittency, with large fluctuations being manifested in the tails. For values of $r$ in the inertial range, the assumptions of K41 and Kr67 also require the shape of the pdfs to be independent of $r$, i.e. self-similar. The statistics of the velocity gradients provide a measure of the “smoothness" of the velocity field at very small scales outside the inertial range. For 3D energy and 2D enstrophy cascades, these scales lie in the dissipative (viscous) range. Although these measurements of $\partial_x u$ are perhaps not in contradiction to K41 or Kr67, it is still surprising [@sreenivasan1997; @kraichnan1967b].
Whereas intermittency is well-established for 3D turbulent flows, there is little agreement about its presence or origin (if present) in 2D turbulence. It is important to consider the two cascades separately. Numerical simulations by Boffetta [*et al.*]{} and experiments in salt layers by Paret and Tabeling (PT) suggest that the inverse energy cascade is intermittency-free [@boffetta2000; @paret1998]. An experiment by Jun and Wu (JW) involving a horizontal soap film at a large Reynolds number indicates the opposite [@jun2005]. The soap film experiments of Daniel and Rutgers (DR), which exhibit a dual cascade, indicate the presence of intermittency in both [@daniel2000].
The role of 3D effects (in this case air drag) is also not clear. In some simulations it is responsible for intermittent effects in the inertial scales of the enstrophy cascade [@boffetta2002; @nam2000; @tsang2005]. The same linear drag force produces no such effect in numerical simulations of the inverse cascade and is necessary to establish a steady state [@boffetta2000; @boffetta2012]. Moreover, there is negligible intermittency in salt layer experiments for either the energy or enstrophy cascade despite significant 3D effects from the floor of the container [@paret1998; @paret1999]. However, a systematic study of the effects of drag on the enstrophy cascade indicate the opposite [@boffetta2005]. Other researchers claim that the dissipative scales of the enstrophy cascade are intermittency-free but that coherent structures emerge to produce inertial range intermittency [@benzi1986; @jimenez2007]. The possible effect of Marangoni stresses, which are present in soap films, is also unknown [@chakraborty2011].
This study focuses on three measures of intermittency. They are (a) the shape of the pdf of velocity differences $\delta u(r)$ at inertial scales $r$ to determine if they are nongaussian and self-similar, (b) the kurtosis of the pdf of $\partial_x u$ and $\delta u(r)$ and (c) measurements of an exponent $\mu$ that characterizes the structure functions $S_n(r)$. Measures (b) and (c) are quantitative, while (a) is qualitative.
Assuming an appreciable range of inertial scales $r$ where $S_n(r)$ is of algebraic form $$S_n(r) \equiv \langle | \delta u(r) ^n | \rangle \sim r^{\zeta_n},
\label{zeta_ndefined}$$ one defines the intermittency exponent $\mu$ as $$\mu = \frac{2 \zeta_3 -\zeta_6}{\zeta_3}.
\label{mudefined}$$ In the absence of intermittency, $\zeta_n / \zeta_3 = n/3$ for both cascades and so $\mu = 0$. The parameter $\mu$ was first introduced as the fitting parameter for the lognormal model of intermittency and is also the codimension of the dissipation in 3D [@frisch1995]. Normalizing by $\zeta_3$ will enable a direct comparison of the degree of intermittency between the cascades. The same normalization choice was made by PT and JW [@paret1998; @jun2005]. In this work, as in other studies, the average is taken outside the absolute value of the velocity difference. This makes no difference for the even moments but it is not clear why this should be valid for the odd moments [@sreenivasan1997].
An important feature of the intermittency studied here is its strong Reynolds number dependence. Here we use the Taylor microscale Reynolds number $R_{\lambda} \equiv u' \lambda / \nu$, where $u'$ is the rms velocity and $\lambda = u'/\sqrt{\langle \partial_x u)^2\rangle}$ is the Taylor microscale. For unknown reasons the energy and enstrophy cascade are here observed to occur for low and high $R_{\lambda}$ respectively (see Fig. \[derivativeflatness\] below). Most previous studies considered only one $R_{\lambda}$ or do not quote it [@boffetta2000; @boffetta2002; @paret1998; @nam2000; @tsang2005; @jun2005; @benzi1986]. (A recent simulation claims intermittency in 2D turbulence is $R_{\lambda}$-independent, but the only measures considered are the pdfs of the vorticity and velocity but not their derivatives or structure functions [@bracco2010].) Intermittency measures are $R_{\lambda}$-dependent in 3D, so there is little surprise that the same should be true for 2D [@kahalerras1998; @sreenivasan1997; @jimenez2007]. This being the case, one should take care in comparing results for intermittency in 2D turbulence without reference to $R_{\lambda}$.
Experimental Setup
==================
The experimental setup is diagrammed in Fig. \[setup\]. A soap solution in reservoir $RT$ flows between two blades into reservoir $RB$, where it is pumped back to $RT$ by a small pump $P$, keeping the pressure head constant. The blades have a separation $w$ that is adjustable in the cm range, with a separation of 1-2 cm for most experiments. The blades, which are made of stainless steel, have a thickness of 0.7 mm. The soap solution is 2% Dawn$^\copyright$ dish-washing detergent in tap water. The valve $V$ regulates the soap solution flux $\Phi$, which is in the range of 0.2-1 ml/s. The film thickness $h$ is roughly 2-20 $\mu$m. The turbulence is generated by a comb usually located 10 cm below the point where thin nylon wires are joined at the top. The spacing of the comb teeth is 2 mm and their thickness is 1 mm. A laser Doppler velocimeter (LDV) permits monitoring the film velocity as function of time at a lateral distance $y$ from a blade. The soap solution contains 4 $\mu$m polystyrene particles that scatter light into the LDV’s photodetector. These seed particles are small enough to follow the local velocity fluctuations, their Stokes number being in the range of 0.1 [@merzkirch1987]. As discussed below, one of the smooth blades is sometimes replaced by a serrated blade having an indentation depth of 2 mm. In this case the comb is replaced by a single 1 mm rod thrust through the film.
The soap film typically flows with mean vertical velocity (averaged over the film width) of $U \simeq 2$ m/s, with the rms velocity fluctuations $u'$ roughly 0.1 $U$, so the turbulent intensity ${\cal I} = u'/U \simeq$ 0.1. Replacing one smooth blade with a serrated one, and by varying $\Phi$ and $w$, ${\cal I}$ becomes an adjustable parameter, as does $R_{\lambda}$. The ratio ${\cal I}$ is sufficiently small that one can use the Taylor frozen turbulence hypothesis, enabling translation of temporal shifts into spacial displacements [@belmonte2000]. In this work only longitudinal velocities are considered. However, a few sets of transverse data, which are of lesser quality, show similar behavior. This, along with similar rms values, suggests that the flow is reasonably isotropic.
The mean data rate is typically 5000 Hz, making the smallest scales of the inertial range easily resolvable. The data are interpolated for equal spacing. To ensure insensitivity to the interpolation method, several are used. Linear and nearest neighbor schemes (similar to sample and hold [@ramond2000]) are used in conjunction with the usual Taylor hypothesis: $x = tU$. The third method introduced by Kahalerras [*et al.*]{} seeks to correct for biases in the sampling stemming from the fluctuations in velocity [@kahalerras1998]. No significant difference is found between these different interpolation schemes.
Using the above methods, one extracts the structure functions $S_n(r)$ and the energy spectrum $E(k)$ at desired points ($x,y$) in the soap film [@pope2000]. The measurements are usually made 10 cm below the comb, but the distance is not found to be important. Taking the Fourier transform of $u$ yields the one-dimensional energy spectrum $E_1 (k)$ [@pope2000]. If the turbulence is isotropic, $E(k) = 10 E_1(k)$.
Energy spectra are measured using either two smooth walls or one smooth and one rough wall. The enstrophy cascade is observed when smooth walls are used, and the inverse energy cascade is usually present when one smooth wall is replaced by a serrated one. This presumably changes decaying turbulence to forced turbulence and is the method recently used to study friction in the energy cascade [@kellay2012].
The energy and enstrophy spectra are seen in Fig. \[spectra\]. Though the slopes $\gamma$ in the scaling relation $E(k) \sim k^{-\gamma}$ are only approximately equal to the canonical values $\gamma$ = 5/3 (energy) and $\gamma$ = 3 (enstrophy), they will be referred to as such. The reason for the departure from 5/3 and 3 is unclear, but it may be related to the intermittency studied here. Additional evidence that the $\gamma = 5/3$ spectra really correspond to the inverse energy cascade comes from the third order structure function $S_3(r)$ (where the absolute value has not been taken). See Appendix A for the details.
Results
=======
Probability Density Functions
-----------------------------
Figure \[pdf\] shows the normalized longitudinal velocity difference probability $P(\delta u(r) / \sqrt{\langle \delta u(r)^2 \rangle})$ as a function of the dimensionless velocity difference $\delta u(r)/\sqrt{\langle \delta u(r)^2\rangle}$ for several values of $r$ for both cascades. Data for the energy cascade (a) and the enstrophy cascade (b) are shown. Both pdfs must be skewed and hence not perfectly gaussian to assure that the energy or enstrophy flux is nonzero [@belmonte1999]. It is apparent from this qualitative measure of intermittency that the energy cascade pdf has wider flanks than the enstrophy cascade pdf and is notably nongaussian. On the other hand, the enstrophy cascade pdf is well-fitted by a gaussian function (dashed line). The relative difference between the $R_{\lambda}$-values is significant. Some enstrophy data with smaller $R_{\lambda}$ do not have a perfect gaussian distribution.
The pdfs at different $r$ have measurably different shapes. This indicates a lack of self-similarity, since the statistics of velocity fluctuations should be the same at all scales $r$ in the inertial range. The absence of self-similarity provides additional evidence that the flows are intermittent. The classical Kr67 theory of 2D turbulence (as well as the K41 theory of 3D turbulence) assumes self-similarity in the inertial range [@kraichnan1967a; @frisch1995].
Although these measures of intermittency are qualitative, they signal a disagreement with the most recent and prevalent view of 2D intermittency [@boffetta2012]. That is, it is held that the energy cascade should be non-intermittent while the enstrophy cascade is necessarily intermittent (for a variety of reasons). The quantitative measures will be investigated next.
Flatness
--------
### Velocity Derivative Flatness
The flatness is now considered as the first quantitative measure of intermittency. The velocity derivative $\partial_x u$ in turbulence plays a special role compared to inertial-scale velocity differences $\delta u(r)$. Experiments in 3D show that the mean viscous energy dissipation rate per unit mass $\epsilon_{\nu}$, which is given by $$\epsilon_{\nu} = \nu \langle { |\nabla {\bf u}|^2} \rangle,
\label{meandissipationrate}$$ approaches a constant as $R_{\lambda} \rightarrow \infty$ [@frisch1995]. Imagine increasing $R_{\lambda}$ to infinity by taking the limit $\nu \rightarrow 0$. It is then necessary for the velocity gradients to diverge in order to retain the constancy of $\epsilon_{\nu}$. This can occur if the pdf of velocity gradients develop increasingly wider skirts. Define the derivative flatness as $$F_{\eta} \equiv \frac{ \langle (\partial_x u) ^4 \rangle }{ \langle (\partial_x u)^2 \rangle ^2}.$$ A broader pdf for $\partial_x u$ means a larger $F_{\eta}$. Keep in mind that $F_{\eta}$ is a measure of small scale intermittency. A question posed by 3D experiments, and by these 2D experiments as well, concerns the dependence of $F_{\eta}$ on $R_{\lambda}$. In 3D it is found that $F_{\eta}$ increases monotonically with $R_{\lambda}$ in a self-similar fashion [@sreenivasan1997], eventually flattening out [@tabeling2002b]. In practice the velocity derivative must be approximated by a velocity difference on a very small scale. It should be smaller than the dissipative scale $\eta$ for the enstrophy cascade. Although the smallest scale for the energy cascade inertial range $r_f$ is not well-defined in these experiments, the smallest scales probed here are significantly smaller than the smallest scale (highest wavenumber $k$) in the power law region of $E(k)$. Moreover, $F_{\eta}$ may have no meaning for the energy cascade, since it probes intermittency in scales much smaller than inertial range scales.
The mean data rate to which the measurements are interpolated is high enough that velocity derivatives are calculated at a scale smaller than the inertial range scales for each case considered. The velocity derivative is estimated using the central difference method ($\partial_x u \simeq \frac{u_{i+1} - u_{i-1}}{x_{i+1} - x_{i-1}}$). (The simpler adjacent point method gives slightly higher values of $F_{\eta}$.)
Figure \[derivativeflatness\] shows $F_{\eta}$ vs. $\log R_{\lambda}$. The data span a wide range of $R_{\lambda}$ and include both energy and enstrophy data. There is a perceptible downward trend (note dashed line), but the scatter is indeed large, especially for the enstrophy data. In the energy range, $F_{\eta}$ is larger than the gaussian value of 3 and is comparable to 3D values at the same $R_{\lambda}$ [@tabeling2002b].
One might extrapolate this downward trend in $F_{\eta}(R_{\lambda})$ to infer that as $R_{\lambda} \rightarrow \infty$, 2D turbulence has no small-scale intermittency. There seem to be no previous studies of small-scale intermittency for 2D turbulence, but in 3D both experiments and simulations show that $F_{\eta}$ is an increasing function of $R_{\lambda}$ before possibly leveling off at a value larger than 3 [@tabeling2002b]. The $R_{\lambda}$-independence of $\epsilon_{\nu}$ is the likely explanation of this increase of $F_{\eta}$ in 3D turbulence, as discussed previously. In 2D turbulence this dissipative anomaly is absent: $\epsilon_\nu \rightarrow 0$ as $R_{\lambda} \rightarrow \infty$ (or as $\nu \rightarrow 0$) [@boffetta2012]. Hence, $F_{\eta}$ need not increase as $R_{\lambda}$ increases. Some claim that $\beta$ asymptotes to zero slower than logarithmically: $\beta \rightarrow 0$ as $(\log R_{\lambda})^{-1/2}$, which may be related to the apparently (sub-) logarithmic vanishing of $F_{\eta}$ with $R_{\lambda}$ (see Fig. \[derivativeflatness\]) [@tran2006; @dritschel2007].
### Inertial Range Flatness
Now consider the flatness of $P(\delta u(r))$ for $r$ at any scale. $$F(r) \equiv \frac{\langle \delta u(r) ^4 \rangle}{\langle \delta u(r)^2 \rangle ^2} = \frac{S_4(r)}{S_2(r)^2}.$$ This quantity allows one to measure the deviations from gaussianity, specifically the strength of rare fluctuations, at various scales $r$, whereas $F_{\eta}$ focuses on the small scales only. Figure \[flatness\_sfs\] shows $F(r)$ for several $R_\lambda$ for both the energy and enstrophy cascades. The values of $r$ span a wide range of scales that include the inertial range (indicated by arrows). The curves with low $R_{\lambda}$ show significant nongaussianity.
The two cascades show rather different behaviors as $r$ increases. The energy cascade data appear to approach 3, indicating that gaussianity is restored at large scales just as in 3D [@jimenez2007]. The enstrophy cascade data reach a value of 3 in the inertial range and then continue to decrease to slightly smaller values. For some of the lower $R_{\lambda}$ enstrophy data, there are small bumps at large $r$ outside the inertial range. Since $F(r)$ is larger for the energy cascade than the enstrophy cascade, even in the inertial range, this means that the large scale intermittency is not only present but also stronger in the energy cascade. Moreover, given the direction of the cascades, one may infer that the eddies become intermittency-free as they combine for the energy cascade but become more intermittent as they break up in the enstrophy cascade.
Intermittency Exponent
----------------------
One of the many measures of intermittency involves the structure functions $S _n(r)$. In the absence of intermittency, $S_n(r) \sim r^{\zeta_n}$, with $\zeta_n = n/3$ for the energy cascade (in 3D or 2D), according to K41 and Kr67 [@kolmogorov1941; @frisch1995; @kraichnan1967a]. For the 2D enstrophy cascade, $\zeta_n = n$ [@kraichnan1980]. Subsequent 3D experiments have shown that $\zeta_n/n$ is not a constant but decreases with increasing $n$. This is one identifier of intermittency [@frisch1995].
Just as with the velocity derivatives and indeed any measure of intermittency, there are sometimes delicate and controversial issues involved in estimations. The issue of estimating scaling exponents will be discussed below, but before this can be done the reliability of the higher order structure functions should be addressed. As in the experiment of JW [@jun2005], the method advanced by Dudok de Wit is used to indicate the highest reliable order [@wit2004]. This method has the advantage of being less subjective than other methods. See Appendix B for a more complete description. Also discussed in Appendix A is an additional technique for determining the reliability of the data. Using Dudok de Wit’s method, most data were reliable up to order 12 or 13.
The soap film experiments described here add evidence that intermittency is also present in 2D. The strongest prior evidence for its presence in 2D comes from a study of forced steady-state turbulence at $R_{\lambda}$ comparable to that used here. In the experiments of JW [@jun2005], the 2D system is an electrically conducting soap film driven by an array of small magnets placed below it.
They evaluated the moments $S_n(r)$ using extended self-similarity (ESS) and several other methods as a check [@benzi1993]. With ESS, one plots $S_n(r)$ as a function of $S_3(r)$ rather than $r$ itself. With this approach the range of self-similarity extends over a broad range of the independent variable $S_3(r)$, as compared with a plot of $S_n(r)$ vs. $r$ itself. The same has been done in this study. Figure \[ESS246\] shows an example with $S_2(r)$, $S_4(r)$, and $S_6(r)$ vs. $S_3(r)$ for enstrophy cascade data with $R_\lambda$ = 490. The dashed lines in the figure are least square fits to the data.
Figure \[zeta\_n\] provides a comparison of the various methods for estimating $\zeta_n$. One sees here the normalized structure function exponents $\zeta_n/\zeta_3$ for values of $n$ in the interval 1 through 10. The open squares ($\square$) designate direct measurements of $\zeta_n/\zeta_3$ from the enstrophy cascade data ($R_{\lambda} = 490$) without invoking ESS. This method is labeled “Simple". The open circles ($\bigcirc$) are obtained using ESS, as discussed above. The open triangles ($\triangle$) are calculated as $\langle \frac{d \log S_n(r)}{d \log r} / \frac{d \log S_3(r)}{d \log r} \rangle$ just as in JW [@jun2005]. The straight dashed line is the Kr67 prediction and has a slope of $\zeta_n/\zeta_3$ = $n/3$ (no intermittency). It is apparent that the various methods give similar results with any significant differences occurring for larger $n$. The main results for $\mu$ presented here were obtained using the ESS method.
In the present study $\mu$ is measured as a function of $R_{\lambda}$, which was not an adjustable parameter in prior measurements (JW’s experiment varied $R_{\lambda}$ but focused on only one value [@jun2005]). The results in Fig. \[mu\] show that the dependence of $\mu$ on $R_{\lambda}$ cannot be ignored. The observations are divided into two sets. The open circles ($\bigcirc$) denote measurements made for the energy cascade, where $10 < R_{\lambda} < 110$. The open squares ($\square$) denote measurements made for the enstrophy cascade, where $50 < R_{\lambda} < 700$. For the energy cascade, the maximum value of $\mu$ is $\simeq$ 0.5, which is more than twice its value in 3D flows [@anselmet1984; @sreenivasan1997]. It is notable that $\mu$ depends rather strongly on $R_{\lambda}$, at least for the enstrophy cascade. Just as with $F_{\eta}$, this decrease of $\mu$ with $R_{\lambda}$ suggests that as $R_{\lambda} \rightarrow \infty$, intermittency vanishes for 2D turbulence.
Although there is significant $R_{\lambda}$-dependence for the intermittency, the mean values and variances will roughly indicate the strength for each cascade (energy and enstrophy). Table \[tab:tabulation\], which contains the statistics for $F_{\eta}$ and $\mu$, shows that they are statistically reliable.
$\langle F_{\eta} \rangle $ $ \sigma_{F_{\eta}}$ $\langle \mu \rangle $ $ \sigma_{\mu} $
----------- ----------------------------- ---------------------- ------------------------ ------------------
energy 4.06 0.3 0.36 0.05
enstrophy 3.80 1 0.11 0.05
: Average ($\langle \cdot \rangle$) and standard deviation values ($\sigma$) of intermittency measures for energy and enstrophy data. All mean values show significant deviations from the non-intermittent standard. The enstrophy cascade standard deviations are large due to the $R_{\lambda}$-dependence. Recall that for 3D, $\mu \simeq 0.2$ [@frisch1995].[]{data-label="tab:tabulation"}
The values of $\mu$ measured here are significantly larger than that obtained by PT for the energy cascade [@paret1998]. For their magnetically driven salt layer experiment, they found (also using ESS) that $\mu \simeq 0.02$. The value of $\mu$ found by JW was $\mu \simeq 0.11$, which is comparable with the enstrophy cascade values measured here, although JW focused on the energy cascade [@jun2005]. The JW experiment quotes $R_{\lambda} \simeq 170$, which is significantly higher than the maximum value obtained in the present energy cascade experiments. This may explain the discrepancy in the measured values of $\mu$. The experiments of DR do not quote $\mu$, but there is a stronger deviation in the exponents in the enstrophy cascade range than the energy cascade [@daniel2000]. Their work has the novelty of looking at a dual cascade, which may be the reason for the disagreement with the present measurements. The results for $\mu$ for the enstrophy cascade with drag obtained by numerical simulations seem comparable to the energy cascade result here (see Table \[tab:tabulation\]). As mentioned earlier, it may not be meaningful to compare the intermittency between experiments when $R_{\lambda}$ is different. For most previous studies, no $R_{\lambda}$ (or any Reynolds number) is quoted [@boffetta2000; @boffetta2002; @tsang2005; @paret1998; @paret1999].
The argument that air drag is responsible for the intermittency in the enstrophy cascade does not deal with its $R_{\lambda}$-dependence. A term linearly proportional to the velocity (or vorticity) is usually added to the 2D Navier-Stokes equation to account for 3D drag effects: $$\frac{\partial {\bf u}}{\partial t} + ({\bf \nabla} \cdot {\bf u} ) {\bf u} = - \frac{1}{\rho} {\bf \nabla} p + \nu \nabla^2 {\bf u} - \alpha {\bf u} + {\bf f}$$ where ${\bf f}$ is some external forcing and $\alpha$ is the drag coefficient [@nam2000; @tsang2005; @boffetta2000; @boffetta2002; @rivera2000]. The value of $\alpha$ extracted from a horizontal soap film experiment is $\alpha \simeq 0.7 \pm 0.3$ Hz [@rivera2000], while estimates based on a boundary layer approximation suggest $\alpha \simeq 0.1$ Hz [@boffetta2012]. The corrections to the enstrophy cascade scaling exponents are on the order of $\alpha' = \alpha/\omega'$, where $\omega'$ is the rms vorticity [@boffetta2005; @boffetta2002; @tsang2005; @nam2000]. Namely, the exponents are modified to become $\zeta_n \simeq n(1+\alpha')$. This correction is estimated to be on the order of $10^{-3}$ for the system used here. This value of $\alpha'$ is in good agreement with that calculated in [@boffetta2012] for falling soap films. It is clearly too small to be detectable. Experiments with soap films in partial vacuums, where the air drag has been considerably reduced, indicate that the air drag has little effect on the spectral exponent [@martin1998]. The effects of air drag on the inverse energy cascade have not previously been considered since such 3D effects are necessary to maintain the steady state. With regard to coherent structures, their presence is always accompanied by energy spectra $E(k)$ that are much steeper than is observed in this work ($\gamma \ge 5$), so they cannot be the root of the intermittency here [@benzi1986].
Fractal dimension of turbulence and intermittency
=================================================
In an attempt to understand why intermittency decreases with increasing $R_{\lambda}$, the $\beta$ model is invoked ($\beta \neq \omega^2$). The $\beta$ model is based on a cartoonish description of the cascade where “mother" eddies do not transfer all of their energy or enstrophy to their “daughter" eddies. According to this model, turbulent fluctuations are not space-filling but rather occupy a spatial dimension $D$ (fractal) that is less than the embedding dimension $d$ (in this case 2) [@frisch1995]. Consider the structure functions $S_n(r) \propto r^{\zeta_n}$. The exponents for the energy and enstrophy cascades respectively are
$$\begin{aligned}
\zeta_n = \frac{n}{3} + (2-D)(1-\frac{n}{3}) \\
\zeta_n = n + (2-D)(1-\frac{n}{3}).
\label{exponent}\end{aligned}$$
The energy spectrum is similarly altered for both the energy and enstrophy cascades respectively
$$\begin{aligned}
E(k) \sim k^{-\frac{5}{3} - \frac{2-D}{3}} \\
E(k) \sim k^{-3 - \frac{2-D}{3}}.
\label{spectrumfix}\end{aligned}$$
In terms of the intermittency exponent, one has $\mu_{\epsilon} = 2-D$ for the energy cascade and $\mu_{\beta} = \frac{1}{3}(2-D)$ for the enstrophy cascade. This establishes a link between the intermittency measured here and the fractal dimension of the turbulence. Taking $D$ equal to the embedding dimension $d = 2$, gives $\mu_{\epsilon} = \mu_{\beta}$ = 0. Using the typical value of $\mu_{\beta} \simeq 0.1$ for the enstrophy cascade or $\mu_{\epsilon} \simeq 0.3$ for the energy cascade, one finds that $D \simeq 1.7$. On the other hand it is well-established that for the enstrophy cascade in soap films, $E(k) \propto k^{-3.3}$ [@boffetta2012; @kellay2002], which suggests that $D \simeq 1.1$ and $\mu \simeq 0.6$. These values do not seem realistic and point to the failure of the $\beta$ model to reproduce the numerical values of the intermittency, although the picture of a fractal turbulence is still attractive.
So far there are no measurements establishing that $D$ is a function $R_{\lambda}$, but some of the enstrophy-range measurements discussed above suggest that $D$ approaches 2 with increasing $R_{\lambda}$ (see Figs. \[derivativeflatness\], \[flatness\_sfs\] and \[mu\]). This is equivalent to the enstrophy cascade becoming more space-filling as $R_{\lambda} \rightarrow \infty$. This should be compared with the 3D energy cascade, where it appears that $D$ saturates to a value less than $d = 3$ as $R_{\lambda} \rightarrow \infty$ [@frisch1995].
Using Eq. \[exponent\], the flatness of the structure functions goes as $F(r) \propto r^{D-2}$, which indicates that this quantity blows up as $r \rightarrow 0$ if intermittency is present. This is in accord with the data shown in Fig. \[flatness\_sfs\]. However, as in 3D one expects that a finite value is reached as the limit $r \rightarrow 0$, $F(r) \rightarrow F_{\eta}$. With the assumption that the turbulence becomes space-filling as $R_{\lambda} \rightarrow 0$, a multi-fractal formalism then suggests that $F_{\eta} \rightarrow$ constant (see section 8.5 in [@frisch1995]), which is in reasonable agreement with the results in Fig. \[derivativeflatness\].
Summary
=======
Presented here is a study of intermittency in a flowing soap film. In accord with some prior studies and in disagreement with others, the observed intermittency is as strong in 2D as in 3D. The measurements, which span more than two decades in $R_{\lambda}$, explore both the energy and enstrophy cascades. The intermittency is strongest in the energy cascade and is nearly $R_{\lambda}$-independent. While there is intermittency in the enstrophy cascade, it decreases with increasing $R_{\lambda}$, suggesting that it may vanish in the limit $R_{\lambda} \rightarrow \infty$.
The origin of the intermittency is not clear. There have been several explanations proposed in the literature for the direct enstrophy cascade, but they do not account for its strength or deal with its $R_{\lambda}$-dependence. There also seems to be no fundamental reason to expect the inverse energy cascade to be intermittency-free. Some of the difficulties in coming up with a unified picture of intermittency in 2D may also be due to the failure of the Kr67 picture. The present results suggest the need for taking a fresh look at the statistics of 2D turbulence, as some are already doing [@chen2006; @chen2003; @tran2006; @dritschel2007].
Acknowledgments
===============
The authors are grateful to G. Boffetta for clarifying and explaining the physics of the linear drag. The authors also benefitted from discussions with C. C. Liu and P. Chakraborty. This work is supported by NSF Grant No. 1044105 and by the Okinawa Institute of Science and Technology (OIST). R.T.C. is supported by a Mellon Fellowship through the University of Pittsburgh.
Appendix A: Third-order Structure Function
==========================================
The sign of third order structure function $S_3(r)$ indicates the direction of energy transfer in the turbulent cascade. (Here the absolute value is not taken.) If $S_3(r) < 0$ for $r$ in the inertial range, then the energy is going to small scales, as in 3D [@frisch1995]. If $S_3(r) > 0$, then the energy is going to large scales, as is predicted for the inverse energy cascade of 2D turbulence [@kraichnan1980; @boffetta2012; @kellay2002]. The experimental data that exhibit the energy spectrum scaling $E(k) \propto k^{-5/3}$ also show this positive third moment, as indicated in Fig. \[thirdmoment\].
![The third order structure function $S_3(r)$ vs. $r$ for several $R_{\lambda}$. The sign of $S_3(r)$ is positive for $r$ in the inertial range of each case. This indicates that energy is being transferred to large scales, in agreement with the prediction for the inverse energy cascade of 2D turbulence.[]{data-label="thirdmoment"}](third_order.eps){width="60.00000%"}
The curves in this figure are positive for $r$ in the inertial range and even appear to behave linearly for small $r$ as the theory suggests. At larger $r$ outside the inertial range they approach zero.
Appendix B: Data Quality
========================
The method of Dudok de Wit for determining the quality of data for measuring structure functions involves an empirical observation [@wit2004]. Take all of the measured velocity differences $\delta u(r)$and sort them according to size in descending order $\delta u(r)_k$ ($k$ = 1, 2, 3,...). When plotted in a log-log plot against their ranking, one finds that the distribution is initially a power law $\delta u(r)_k = \alpha {\Big (} \frac{k}{N} {\Big )} ^{-\gamma}$ for roughly $M$ out of the first $N$ elements. This is shown in Fig. \[dudok\] and has been found to be the general empirical rule for turbulence data.
The structure function can then be broken up according to $$S_n(r) = \frac{1}{N} \sum_{k=1}^N (\delta u(r)_k)^n = \frac{1}{N} \sum_{k=1}^M (\delta u(r)_k)^n + \frac{1}{N} \sum_{k=M+1}^N (\delta u(r)_k)^n = \frac{1}{N} \sum_{k=1}^M \alpha^n {\Big ( } \frac{k}{N} {\Big )} ^{-n\gamma} + \frac{1}{N} \sum_{k = M+1}^N (\delta u(r)_k)^n$$ The first term is the power law contribution. This represents rare events, since empirically one finds that only a small fraction of the data follow this power law. A simple criterion for the data to have sufficient quality is for this sum to converge as $N$ increases. This leads to the requirement that $n\gamma < 1$. In other words, the highest order moment for which this converges is $n_{max} = \Big{ [} \frac{1}{\gamma} {\Big ]} -1$, where the brackets denote rounding to the nearest integer. We took an average $\gamma$ from all scales $r$ in the inertial range to find $n_{max}$. Using this method, most data were reliable up to order 12 or 13. The main results in this paper only require accuracy up to order 6.
![Rank-ordered log-log plot of velocity differences for two sets of experimental data. The curves initially behave as power laws. The power law exponent may be used to estimate the highest order of structure functions that can be accurately measured.[]{data-label="dudok"}](dudok.eps){width="60.00000%"}
As a check, we use a second method to test the quality of our data. Since the structure functions are defined as $$S^n(r) = \int_{-\infty}^{+\infty} \delta u(r)^n p(\delta u(r)) d (\delta u(r)),$$ the area underneath the integrand curve $\delta u(r)^n p(\delta u(r))$ must be finite. That is, the integrand plotted as a function of $\delta u(r)$ must not diverge. This is easily checked, although it has the failure of being a less objective method as compared to the method of Dudok de Wit. In Fig. \[tennekes\] we plot the integrand for several values of $n$ with $r$ chosen as small as possible. The results are the same for larger $r$ values.
The areas underneath the integrand curves are clearly finite. This indicates that the structure functions may be accurately measured up to order 6. Higher orders converged as well, but these plots are not shown here since only order 6 is necessary for the main results of this paper.
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abstract: 'We present a computer assisted method for proving the existence of globally attracting fixed points of dissipative PDEs. An application to the viscous Burgers equation with periodic boundary conditions and a forcing function constant in time is presented as a case study. We establish the existence of a locally attracting fixed point by using rigorous numerics techniques. To prove that the fixed point is, in fact, globally attracting we introduce a technique relying on a construction of an absorbing set, capturing any sufficiently regular initial condition after a finite time. Then the absorbing set is rigorously integrated forward in time to verify that any sufficiently regular initial condition is in the basin of attraction of the fixed point.'
author:
- |
Jacek Cyranka[^1]\
[Institute of Computer Science, Jagiellonian University]{}\
[prof. Stanisława Łojasiewicza 6, 30-348 Kraków, Poland]{}\
[jacek.cyranka@ii.uj.edu.pl]{}
title: 'Existence of globally attracting fixed points of viscous Burgers equation with constant forcing. A computer assisted proof'
---
#### Keywords:
[viscous Burgers equation, computer assisted proof, fixed point, dissipative PDE, rigorous numerics, Galerkin projection]{}
#### AMS classification:
[Primary: 65M99, 35B40. Secondary: 35B41]{}
Introduction
============
The field of computer assisted proofs for ordinary differential equations (ODEs) is a quite well established and analysed topic. Still, it seems to us that the development of methods for investigating the dynamics of PDEs by performing rigorous computer assisted proofs is at a pioneering stage.
In the present paper we develop a computer aided method which is interesting for two main reasons. First, it provides not only a local, but also a global perspective on the dynamics. Second, it allows to establish results which have not been achieved using known analytical techniques. As a case study we present the forced viscous Burgers equation, where the forcing is constant in time and periodic in space. More specifically, we consider the initial value problem with periodic boundary conditions for the equation $$\label{eq:vBEq}
u_t+u\cdot u_x-\nu u_{xx}=f(x).$$ In the present paper we deal with the case of non-zero forcing, which is not reducible to a linear PDE by the Hopf-Cole transform anymore.
To our knowledge, there exist two rigorous numerics methods for studying the non-stationary PDE problem using the Fourier basis. The method of self-consistent bounds, presented in the series of papers [@ZM], [@Z2], [@Z3], [@ZAKS], and the method presented in [@AK]. Both of them have been applied to the Kuramoto-Sivashinsky equation. In [@KKN] authors obtained some rigorous numerics prototype results for a non-stationary PDE problem using the Finite Element basis. Related work regarding a rigorous numerics study of the global dynamics of PDE includes [@DHMO], and [@MPMW]. In [@FTKS] the viscous Burgers equation with zero forcing was used as an illustration of a computer aided technique of proving existence of stationary solutions.
It has been shown that belongs to the class of dissipative PDEs (dPDEs) possessing inertial manifolds [@V]. Using our technique we demonstrate that the global attractor exhibited by is in fact a unique stable fixed point. In [@JKM] it was shown that for any viscosity and the time independent forcing the attractor of is a single point. This is a stronger than ours result, but the methods in present paper have also some advantages. Contrary to the approach from [@JKM] we are not invoking any unconstructive functional analysis techniques, thus the speed of convergence could be obtained from our construction. Moreover, we are not using the maximum principle, so our method should apply to a class of systems of PDEs.
To establish the existence of an attracting fixed point locally, we use the computer techniques from [@ZAKS]. We construct a small neighbourhood of a candidate for the fixed point and prove the existence and uniqueness of a fixed point within said neighbourhood by calculating an explicit upper bound for the logarithmic norm. In case of the negative logarithmic norm, we claim that there exists a locally attracting fixed point. On the other hand, we show the global existence of solutions by constructing trapping regions inspired by the analogical sets constructed for the Navier-Stokes equations [@MS], [@ES], see also [@ZNS].
We link those results by constructing an absorbing set, which captures any initial condition after a finite time. Then we integrate the obtained absorbing set forward in time rigorously until it is mapped into a small region with the established existence of an attracting fixed point within. By doing so, we verify that any initial condition is in the basin of attraction of the fixed point. The aforementioned elements applied together give an original technique that allows to extend the property of attractiveness obtained locally on a small region to a global fact. We would like to stress that our method concerns the evolution of dPDEs in time, not only the stationary problem. Moreover, it is worth pointing out that we do not restrict ourselves by assuming zero spatial average, i.e. $\int_Q{u(t,x)}\,dx=0$ on a domain $Q$, which was often assumed in related work, see for instance [@V] or [@FTKS]. Our theory can be applied when zero is replaced by any number. We remark that exclusively in the case of non-zero spatial average the equation admits travelling wave-like solutions.
An example result obtained with the presented method is the following
\[thm:main\] For any $\nu\in\paperExampleNu$ and\
$f\in \left\{x\mapsto\paperExampleForcing+\paperExampleBall\right\}$ there exists a steady state solution of , which is unique and attracts globally any initial data $u_0$ satisfying $u_0\in C^4$ and $\int_0^{2\pi}{u_0(x)\,dx}=\paperExampleAzero$.
Other examples are given in Section \[sec:exampleTheorem\]. The function $\paperExampleForcing$, added to the forcing was chosen as an example to show that our method is not limited to the simpler case of low energy forcings. Note that Theorem \[thm:main\] covers a whole set of forcing functions within a “ball” $\paperExampleBall$. To achieve this we used the interval arithmetic in a way to be explained later.
By using the presented algorithm we could prove a more general case, namely replace in Theorem \[thm:main\] $\beta_k,\ \gamma_k$ with arbitrary continuous functions $\beta_k(t),\ \gamma_k(t)$, such that $\beta_k(t),\ \gamma_k(t)\in[-0.03,0.03]$ for $t\geq 0$. This will be exploited in the next paper [@CZ] where we prove existence of globally attracting periodic orbits for viscous Burgers equation with nonautonomous forcing.
This paper is dependent on [@Z3] and [@ZAKS], we recall only crucial definitions and results from the previous works and focus on the new elements. Proper references are always provided whenever necessary. We are convinced that the presented techniques are applicable to higher dimensional dPDEs, including the Navier-Stokes equations, and we will address this problem in our forthcoming papers.
We organize the paper as follows: the first part comprises the theory and it is concluded by the proof of Theorem \[thm:main\] in Section \[sec:exampleTheorem\]. A presentation and discussions of the algorithms follows.
The viscous Burgers equation {#sec:burgers}
============================
As the viscous Burgers equation we consider the following PDE\
$$\frac{\partial u}{\partial t}+u\cdot\frac{\partial u}{\partial x}-\nu\bigtriangleup u=0\quad{\mbox{in}\ } \Omega,\quad t>0,$$ where $\nu$ is a positive *viscosity constant*. The equation was proposed by Burgers (1948) as a mathematical model of turbulence. Later on it was successfully showed that the Burgers equation models certain gas dynamics (Lighthill (1956)) and acoustic (Blackstock (1966)) phenomena, see e.g. [@Wh]. We consider the equation on the real line $\Omega:=\mathbb{R}$ with periodic boundary conditions and *a constant in time forcing $f$*, i.e. $$u\colon\mathbb{R}\times[0,\mathcal{T})\to\mathbb{R},$$ $$f\colon\mathbb{R}\to\mathbb{R},$$
\[eq:burgers\] $$\begin{aligned}
&u_t+u\cdot u_x-\nu u_{xx}=f(x),\quad x\in\mathbb{R},\ t\in[0, \mathcal{T}),\label{eq:burgers1}\\
&u(x,t)=u(x+2k\pi, t),\quad x\in\mathbb{R},\ t\in[0, \mathcal{T}),\ k\in\mathbb{Z},\label{eq:burgers3}\\
&f(x)=f(x+2k\pi),\quad x\in\mathbb{R},\ k\in\mathbb{Z},\label{eq:burgers4}\\
&u(x,0)=u_0(x),\quad x\in\mathbb{R}.\label{eq:burgers2}
\end{aligned}$$
The viscous Burgers equation in the Fourier basis
-------------------------------------------------
In this section we rewrite using *the Fourier basis* of $2\pi$ periodic functions $\{e^{ikx}\}_{k\in\mathbb{Z}}$. From now on we assume that all functions we use are sufficiently regular to be expanded in the Fourier basis and all necessary Fourier series converge.
\[def:FourierModes\] Let $u\colon\mathbb{R}\to\mathbb{R}$ be a $2\pi$ periodic function. We call $\{a_k\}_{k\in\mathbb{Z}}$ [the Fourier modes]{} of $u$, where $a_k\in\mathbb{C}$ satisfies $$\label{eq:FourierModes1}
a_k=\frac{1}{2\pi}\int_0^{2\pi}{u(x)e^{-ikx}}\,dx,$$moreover, the following equality holds $$\label{eq:FourierModes2}
u(x)=\sum_{k\in\mathbb{Z}}{a_ke^{ikx}},\quad x\in\mathbb{R}.$$
Let $\nmid\cdot\nmid\colon\mathbb{R}\to\mathbb{R}$ be given by $$\nmid a\nmid:=\left\{\begin{array}{ll}|a|&\text{ if }a\neq 0,\\1&\text{ if }a=0.\end{array}\right.$$
\[lem:fourierC\] Let $\gamma>1$. Assume that $|a_k|\leq\frac{M}{\nmid k\nmid^\gamma}$ for $k\in\mathbb{Z}$. If $n\in\mathbb{N}$ is such that $\gamma-n>1$, then the function $u(x)=\sum_{k\in\mathbb{Z}}{a_ke^{ikx}}$ belongs to $C^n$. The series $$\frac{\partial^s u}{\partial x^s}(x)=\sum_{k\in\mathbb{Z}}{a_k\frac{\partial^s}{\partial x^s}e^{ikx}}$$ converges uniformly for $0\leq s\leq n$.
Let $u_0$ be an initial value for the problem and $f$ be a forcing. Then rewritten in the Fourier basis becomes
\[eq:burgers\_infinite\] $$\begin{aligned}
&\frac{d a_k}{d t}=-i\frac{k}{2}\sum_{k_1\in\mathbb{Z}}{a_{k_1}\cdot a_{k-k_1}}+\lambda_k a_k+f_k,\quad k\in\mathbb{Z},\label{eq:burgers_infinite1}\\
&a_k(0)=\frac{1}{2\pi}\int_0^{2\pi}{u_0(x)e^{-ikx}}\,dx,\quad k\in\mathbb{Z},\label{eq:burgers_infinite2}\\
&f_k=\frac{1}{2\pi}\int_0^{2\pi}{f(x)e^{-ikx}}\,dx,\quad k\in\mathbb{Z},\label{eq:burgers_infinite3}\\
&\lambda_k=-\nu k^2.\label{eq:burgers_infinite4}
\end{aligned}$$
For the proof refer [@SuppMat].
\[def:symmetricGalerkinProjection\] For any given number $m>0$ [the $m$-th Galerkin projection]{} of is $$\label{eq:symmetricGalerkinProjection}
\frac{d a_k}{d t}=-i\frac{k}{2}\sum_{\substack{|k-k_1|\leq m\\|k_1|\leq m}}{a_{k_1}\cdot a_{k-k_1}}+\lambda_k a_k+f_k,\quad|k|\leq m.$$
Note that in our case $\{a_k\}_{k\in\mathbb{Z}}$ are not independent. The solution $u$ of is real valued, which implies that $$\label{eq:first}
a_k=\overline{a_{-k}}.$$ Note that condition is invariant under all Galerkin projections as long as $f_k=\overline{f_{-k}}$.
In Section \[sec:analytic\] and Section \[sec:global\] we will assume that the initial condition for satisfies $$\label{eq:fixedInt}
\frac{1}{2\pi}\int_{0}^{2\pi}{u_0(x)\,dx}=\alpha,\quad\text{for a fixed }\alpha\in\mathbb{R}.$$ We will require additionally that $f_0=0$, and then implies that $a_0(t)$ is constant in time, namely $$\label{eq:fixedA0}
a_0=\alpha.$$ Note that condition is invariant under all Galerkin projections as long as $f_0=0$.
Analytic arguments {#sec:analytic}
==================
In this section we provide some analytic arguments that we use in proving the global existence and regularity results for solutions of .
Energy as Lyapunov function
---------------------------
\[def:energy\] *Energy* of (\[eq:burgers\_infinite1\]) is given by the formula $$\label{eq:energy}
E(\{a_k\})=\sum_{k\in\mathbb{Z}}{|a_k|^2}.$$ Energy of (\[eq:burgers\_infinite1\]) with $a_0$ excluded is given by the formula $$\label{eq:energyWithoutZero}
\mathcal{E}(\{a_k\})=\sum_{k\in\mathbb{Z}\setminus\{0\}}{|a_k|^2}.$$
The following lemma provides an argument for the statement that *the energy* of is being absorbed by a ball whose radius depends on the forcing and the viscosity constant. Basing on this argument, later on, we will construct a trapping region for any . In particular, any trapping region constructed encloses the absorbing ball.
\[lem:energy\] For any solution of or a such that $a_{-k}=\overline{a_k}$ the following equality holds $$\label{eq:energyEquality}
\frac{d E(\{a_k\})}{d t}= -2\nu\sum_{k\in\mathbb{Z}}{k^2|a_k|^2}+\sum_{k\in\mathbb{Z}}{f_{-k}\cdot a_k}+\sum_{k\in\mathbb{Z}}{f_{k}\cdot a_{-k}}.$$
#### *Proof*
Using the symmetry of the index in we rewrite $$\begin{gathered}
\frac{d E}{d t}=\sum_{k\in\mathbb{Z}}{(\frac{d a_k}{d t}\cdot a_{-k})}+\sum_{k\in\mathbb{Z}}{(\frac{d a_{-k}}{d t}\cdot a_k)}=\sum_{k\in\mathbb{Z}}{-i\frac{k}{2}\sum_{k_1\in\mathbb{Z}}{a_{k_1}\cdot a_{k-k_1}\cdot a_{-k}}}\\
+\sum_{k\in\mathbb{Z}}{i\frac{k}{2}\sum_{k_1\in\mathbb{Z}}{a_{k_1}\cdot a_{-k-k_1}\cdot a_{k}}}-2\nu\sum_{k\in\mathbb{Z}}{k^2a_k\cdot a_{-k}}+\sum_{k\in\mathbb{Z}}{f_{-k}\cdot a_k}+\sum_{k\in\mathbb{Z}}{f_{k}\cdot a_{-k}}\\
=\sum_{k\in\mathbb{Z}}{-ik\sum_{k_1\in\mathbb{Z}}{a_{k_1}\cdot a_{k-k_1}\cdot a_{-k}}}-2\nu\sum_{k\in\mathbb{Z}}{k^2a_k\cdot a_{-k}}+\sum_{k\in\mathbb{Z}}{f_{-k}\cdot a_k}+\sum_{k\in\mathbb{Z}}{f_{k}\cdot a_{-k}}.
\end{gathered}$$ We want to show that $\sum_{|k|\leq N}{k\sum_{\substack{|k_1|\leq N\\|k-k_1|\leq N}}{a_{k_1}\cdot a_{k-k_1}\cdot a_{-k}}}=0$. In order to facilitate the proof explanation we denote $S_{N,k}:=\sum_{\substack{|l|\leq N\\|k-l|\leq N}}{a_{k-l}\cdot a_l}$ and $S_N:=\sum_{|k|\leq N}{k\sum_{\substack{|k_1|\leq N\\|k-k_1|\leq N}}{a_{k_1}\cdot a_{k-k_1}\cdot a_{-k}}}=\sum_{|k|\leq N}{kS_{N,k}a_{-k}}$.
####
We proceed by induction, firstly we check if for $N=1$ the thesis is fulfilled $$S_1=-1(a_{-1}\cdot a_0\cdot a_1+a_0\cdot a_{-1}\cdot a_1)+1(a_1\cdot a_0\cdot a_{-1}+a_0\cdot a_1\cdot a_{-1})=0.$$ We verify the induction step $S_{N-1}=0 \Rightarrow S_N=0$ $$S_N=S_{N-1}+\sum\left\{\begin{array}{lll}
a_N\cdot a_{-N+k}\cdot a_{-k}2k&, 0<k<N,&\ \left(S_{I}\right)\\
a_{-N}\cdot a_{k+N}\cdot a_{-k}2k&, -N<k<0,&\ \left(S_{II}\right)\\
S_{N,-N}\cdot a_N(-N)&, k=-N,&\ \left(S_{III}\right)\\
S_{N,N}\cdot a_{-N}N&, k=N,&\ \left(S_{IV}\right)
\end{array}\right.$$ we match elements with the same modes from $\left(S_I\right)$ and $\left(S_{III}\right)$. Let $e(N)=1$ for $N$ even and $e(N)=0$ for $N$ odd, $$\begin{gathered}
\sum_{0<k<N}{a_N\cdot a_{k-N}\cdot a_{-k}(2k-N)}\\
=\sum_{0<k<\frac{N}{2}}{a_N\cdot a_{k-N}\cdot a_{-k}(2k-N+N-2k)}+e(N)a_N\cdot a^2_{-\frac{N}{2}}(N-N)=0.
\end{gathered}$$ When elements with the same modes from $\left(S_{II}\right)$ and $\left(S_{IV}\right)$ are matched analogously as above the result is also zero. After substitution all that is left is $$S_N=S_{N-1}+2Na_N\cdot a_0\cdot a_{-N}-2Na_{-N}\cdot a_0\cdot a_N=0.\quad\qed$$
A trapping region for {#sec:analyticalTR}
----------------------
In this section we provide a forward invariant set for each , called *the trapping region* . If we consider an arbitrary initial condition that is inside a trapping region, then the corresponding trajectory remains in this set in the future. This is an argument for the existence of solutions of each within a trapping region. Moreover, due to the existence of a trapping region, the solution of , obtained by passing to the limit, conserves the initial regularity. We use this fact to argue that a solution of with sufficiently regular initial data exists for all times, is unique, and is a classical solution of . Calculations performed in this section were inspired by the trapping regions built for the Navier-Stokes equations, see [@MS] and [@ES].
#### *Notation*
Let $l^2(\mathbb{Z})=\left\{\left\{a_k\right\}_{k\in\mathbb{Z}}\colon\sum{|a_k|^2<\infty}\right\}$, where $a_k\in\mathbb{C}$ for $k\in\mathbb{Z}$. In the sequel the space $l^2(\mathbb{Z})$ will be denoted by $H$. We equip $H$ with the standard scalar product. Let $m>0$, we define $P_m(H)$ to be $\mathbb{C}^{2m+1}$.
####
Formally an element of a Galerkin projection is a finite sequence. In the sequel we will use the following embedding, and with some abuse of notation we will use the same symbol to denote the element of infinite dimensional space $H$ $$P_m(H)\ni\left\{a_{-m},\dots,a_0,\dots,a_m\right\}\equiv\left\{\dots,0,\dots,0,a_{-m},\dots,a_0,\dots,a_m,0,\dots,0,\dots\right\}\in H.$$ In consequence we assume the inclusion $P_m(W)\subset W$, for all $W\subset H$.
\[lem:estimateNk\] Let $\{a_k\}_{k\in\mathbb{Z}}\in H$, $N_k:=-i\frac{k}{2}\sum_{k_1\in\mathbb{Z}}{a_{k_1}\cdot a_{k-k_1}}$. Assume that there exists $C>0$ and $s>0.5$ such that $\{a_k\}_{k\in\mathbb{Z}}$ satisfy $|a_k|\leq\frac{C}{\nmid k\nmid^s}$, $k\in\mathbb{Z}$.\
Then $$ |N_k|\leq\frac{\sqrt{E(\{a_k\})}C\left(2^{s-{\frac{1}{2}}}+\frac{2^{s-1}}{\sqrt{2s-1}}\right)}{|k|^{s-\frac{3}{2}}},\quad k\in\mathbb{Z}\setminus\{0\}.$$
#### *Proof*
In order to prove the bound for $N_k$, we split $N_k=N_k^I+N_k^{II}$, and bound $N_k^I$ and $N_k^{II}$ separately
#### Case 1
First, we bound the following sum $N_k^{I}=-i\frac{k}{2}\sum_{k_1}{a_{k_1}\cdot a_{k-k_1}}$,where $|k_1|\leq\frac{1}{2}|k|$\
$$\begin{gathered}
|N_k^{I}|\leq\sum_{|k_1|\leq\frac{1}{2}|k|}{{\frac{1}{2}}|k||a_{k_1}||a_{k-k_1}|}\leq\sum_{|k_1|\leq{\frac{1}{2}}|k|}{{\frac{1}{2}}|k|\frac{C}{|k-k_1|^{s}}|a_{k_1}|}\\
\leq\frac{2^{s-1}C}{|k|^{s-1}}\sqrt{\sum_{|k_1|\leq\frac{1}{2}|k|}{|a_{k_1}|^2}}\sqrt{\sum_{|k_1|\leq\frac{1}{2}|k|}{1}}\leq\frac{2^{s-1}\sqrt{2}\sqrt{E(\{a_k\})}C}{|k|^{s-\frac{3}{2}}}.
\end{gathered}$$
#### Case 2
Second, we bound the remaining part $N_k^{II}=-i\frac{k}{2}\sum_{k_1}{a_{k_1}\cdot a_{k-k_1}}$,where $|k_1|>{\frac{1}{2}}|k|$\
$$\begin{gathered}
|N_k^{II}|\leq\sum_{|k_1|>{\frac{1}{2}}|k|}{\frac{1}{2}|k||a_{k_1}||a_{k-k_1}|}\leq\frac{1}{2}|k|C\sum_{|k_1|>{\frac{1}{2}}|k|}{\frac{1}{|k_1|^s}|a_{k-k_1}|}\\
\leq\frac{1}{2}|k|C\sqrt{\sum_{|k_1|>{\frac{1}{2}}|k|}\frac{1}{|k_1|^{2s}}}\sqrt{\sum_{|k_1|>{\frac{1}{2}}|k|}{|a_{k-k_1}|^2}}\\
\leq\frac{1}{2}|k|\sqrt{E(\{a_k\})}C\sqrt{\frac{2^{2s}}{(2s-1)|k|^{2s-1}}}=\frac{\sqrt{E(\{a_k\})}C\frac{2^{s-1}}{\sqrt{2s-1}}}{|k|^{s-\frac{3}{2}}}.
\end{gathered}$$ We used the following estimation due to the convexity $$\sum_{|k_1|>{\frac{1}{2}}|k|}{\frac{1}{|k_1|^{2s}}}<2\int_{{\frac{1}{2}}|k|}^{\infty}\frac{1}{r^{2s}}\,d r=2\left[-\frac{1}{(2s-1)r^{2s-1}}\right]_{{\frac{1}{2}}|k|}^{\infty}=\frac{2^{2s}}{(2s-1)|k|^{2s-1}}.$$ After summing together Case 1 and Case 2 $$|N_k|\leq|N_k^{I}|+|N_k^{II}|=\frac{\sqrt{E(\{a_k\})}C\left(2^{s-{\frac{1}{2}}}+\frac{2^{s-1}}{\sqrt{2s-1}}\right)}{|k|^{s-\frac{3}{2}}}.$$ holds for any $k\in\mathbb{Z}\setminus\{0\}$.
\[thm:analyticTrappingRegion\] Let $\{a_k\}_{k\in\mathbb{Z}}\in H$, $\alpha\in\mathbb{R}$, $J>0$, $s>0.5$, $E_0=\frac{\Ef}{\nu^2}$, $\energyBound>E_0$, $D=\D$, $C>\sqrt{\energyBound}N^s$, $N>\max{\left\{J,\left(\frac{\sqrt{\energyBound+\alpha^2}D}{\nu}\right)^{2}\right\}}$. Assume that . Then $$W_0(\energyBound, N, C, s,\alpha)=\{\{a_k\}\ |\ \Ezero(\{a_k\})\leq \energyBound,\ |a_k|\leq\frac{C}{|k|^s}\text{ for }|k|>N\}$$ is a trapping region for each .
#### *Proof*
We first show that $$\label{eq:energyDecreasing}
\text{if }\Ezero(\{a_k\})>E_0=\frac{\Ef}{\nu^2}\text{ then }\frac{d \Ezero(\{a_k\})}{dt}<0.$$ Under the assumption $f_0=0$ we have $\sum_{k\in\mathbb{Z}}{|f_{-k}||a_k|} + \sum_{k\in\mathbb{Z}}{|f_k||a_{-k}|}=\sum_{k\in\mathbb{Z}\setminus\{0\}}{|f_{-k}||a_k|} + \sum_{k\in\mathbb{Z}\setminus\{0\}}{|f_k||a_{-k}|}$, and $\frac{d a_0}{d t}=0$, the latter implies that $\frac{d E}{d t}=\frac{d\Ezero}{d t}$.
Taking the square root of $\Ezero(\{a_k\})>\frac{\Ef}{\nu^2}$ gives $$\nu\sqrt{\Ezero(\{a_k\})}>\sqrt{\Ef},$$ multiplying both of the sides by $2\sqrt{\sum_{k\in\mathbb{Z}\setminus\{0\}}{|a_k|^2}}$ gives $$2\nu\sum_{k\in\mathbb{Z}\setminus\{0\}}{|a_k|^2}>2\sqrt{\sum_{k\in\mathbb{Z}\setminus\{0\}}{|f_k|^2}}\sqrt{\sum_{k\in\mathbb{Z}\setminus\{0\}}{|a_k|^2}},$$ moreover the following inequalities are satisfied $$2\nu\sum_{k\in\mathbb{Z}}{k^2a_k\cdot a_{-k}}\geq 2\nu\sum_{k\in\mathbb{Z}\setminus\{0\}}{|a_k|^2}>2\sqrt{\sum_{k\in\mathbb{Z}\setminus\{0\}}{|f_k|^2}}\sqrt{\sum_{k\in\mathbb{Z}\setminus\{0\}}{|a_k|^2}}\geq\sum_{k\in\mathbb{Z}\setminus\{0\}}{|f_{-k}||a_k|} + \sum_{k\in\mathbb{Z}\setminus\{0\}}{|f_k||a_{-k}|}.$$ Simply, the linear term dominates the forcing term in , i.e. $$\label{eq:linearDominates}
2\nu\sum_{k\in\mathbb{Z}}{k^2a_k\cdot a_{-k}}>\sum_{k\in\mathbb{Z}\setminus\{0\}}{|f_{-k}||a_k|} + \sum_{k\in\mathbb{Z}\setminus\{0\}}{|f_k||a_{-k}|}.$$ The condition is sufficient to satisfy .
Next observe that the condition $|a_k|\leq\frac{C}{|k|^s}$ is satisfied for all $\{a_k\}\in W_0$ and $k\in\mathbb{Z}\setminus\{0\}$. Since $\Ezero(\{a_k\})\leq \energyBound$ and $|a_k|\leq\sqrt{\energyBound}$ for $k\neq 0$, $$|a_k|\leq \sqrt{\energyBound}\leq\frac{C}{|k|^s}\text{ because }C>\sqrt{\energyBound}N^s.$$
####
Now, we shall check if the vector field points inwards on $\partial W_0$. For $\{a_k\}\in\partial W_0$ such that $\Ezero(\{a_k\})=\energyBound$ and $\energyBound>\frac{\Ef}{\nu^2}$ vector field points inwards from . Let us pick a point $\{a_k\}_{k\in\mathbb{Z}}\in\partial W$ such that $|a_k|=\frac{C}{|k|^s}$ for some $|k|>N$, and perform calculations to check if the diminution condition $\frac{d |a_k|}{d t}<0$ holds. Observe that $E(\{a_k\}_{k\in\mathbb{Z}})\leq\energyBound+\alpha^2$ and we apply Lemma \[lem:estimateNk\] with $E(\{a_k\}_{k\in\mathbb{Z}})$ replaced by $\energyBound+\alpha^2$. $$\begin{gathered}
\frac{d |a_k|}{d t}<-\nu|k|^2\frac{C}{|k|^s}+\frac{D\sqrt{\energyBound+\alpha^2}C}{|k|^{s-\frac{3}{2}}}<0,\\
\nu|k|^2\frac{C}{|k|^s}>\frac{D\sqrt{\energyBound+\alpha^2}C}{|k|^{s-\frac{3}{2}}},\\
\nu\sqrt{|k|}>D\sqrt{\energyBound+\alpha^2},\\
|k|>\left(\frac{D\sqrt{\energyBound+\alpha^2}}{\nu}\right)^2,
\end{gathered}$$ $\frac{d |a_k|}{d t}<0$ holds if $|k|>\frac{D^2\left(\energyBound+\alpha^2\right)}{\nu^2}$. The proof is complete because $|k|>N>\frac{D^2\left(\energyBound+\alpha^2\right)}{\nu^2}$. $\qed$
Global results {#sec:global}
==============
The subspace ${\overline{H}}\subset H$ is defined by $${\overline{H}}:=\left\{\{a_k\}\in H\colon\text{there exists }0\leq C<\infty\text{ such that }|a_k|\leq\frac{C}{\nmid k\nmid^4}\text{ for }k\in\mathbb{Z}\right\}.$$
#### *Notation*
Let $l>0$, we define $P_l(H)$ to be $\mathbb{C}^{2l+1}$. From now on by $\varphi^l(t, x)$ we denote the solution of $l$-th at a time $t>0$, with an initial value $x\in P_l(H)$. By $\{a_k\}_{|k|\leq l}$ we denote an initial condition $x\in P_l(H)$. The operator $N_k$ is the nonlinear part of , and is defined by $N_k(\{a_k\}_{k\in\mathbb{Z}}):=-i\frac{k}{2}\sum_{k_1\in\mathbb{Z}}{a_{k_1}\cdot a_{k-k_1}}$ for $k\in\mathbb{Z}$. For a sequence of complex numbers $\{c_k\}_{k\in\mathbb{Z}}$ let $c_{k,j}$ denotes the $j$-th component of $c_k$ for $k\in\mathbb{Z}$ and $j=1,2$, complex numbers are considered as elements of $\mathbb{R}^2$ here.
Let $P_l(H)\ni\{a^l_k(t)\}_{|k|\leq l}:=\varphi^l(t, \{a_k\}_{|k|\leq l})$, $t>0$, $l>0$. Observe that $\{a^l_k(t)\}_{|k|\leq l}$ is well defined, as solutions for each Galerkin projection of exist for all times $t>0$ due to Theorem \[thm:analyticTrappingRegion\] (existence of a trapping region) and are unique due to the fact that is a finite system of ODEs with a locally Lipschitz right-hand side. We will drop the index $l$ when it is known either from the context or irrelevant in the context.
\[lem:energyDissipation\] Let $\alpha\in\mathbb{R}$, $J>0$, $M_1\geq 0$, $E_0=\frac{\Ef}{\nu^2}$, $\energyBound>E_0$. Assume that . Let [$H\supset W$ be a trapping region for $l$-th for all $l>M_1$]{}.\
There exists a finite time $t_1=t_1(W)\geq 0$ such that $\Ezero\left(\varphi^l(t_1,P_l(\{a_k\}_{k\in\mathbb{Z}}))\right)\leq \energyBound$ holds uniformly for all $\{a_k\}_{k\in\mathbb{Z}}\in W$ and $l>M_1$.
#### *Proof*
Let us take $\{\hat{a}_k\}_{k\in\mathbb{Z}}$ from the boundary of $W$, such that $\Ezero(\{a_k\}_{k\in\mathbb{Z}})\leq\Ezero(\{\hat{a}_k\}_{k\in\mathbb{Z}})$ for all $\{a_k\}_{k\in\mathbb{Z}}\in W$. Let $\Ezero\left(\left\{\hat{a}_k\right\}_{k\in\mathbb{Z}}\right)=\Ezero_I$ be the initial energy. It is enough to take either $t_1=0$ if $\Ezero_I\leq\energyBound$ or $t_1(W)=\frac{1}{2\nu\varepsilon}\ln{\frac{\Ezero_I}{\energyBound}}$ if $\Ezero_I>\energyBound$, where $\varepsilon=\left(1-\sqrt{\frac{E_0}{\energyBound}}\right)$.\
To see this, we calculate in a similar fashion as in the proof of Theorem \[thm:analyticTrappingRegion\]. Let $\Ezero_I>\energyBound$, by Lemma \[lem:energy\] and the assumption that $f_0=0$ (observe that in this case $\frac{d E}{d t}=\frac{d\Ezero}{d t}$, because $a_0$ is a constant) we have $$\frac{d\Ezero}{dt}\leq -2\nu \Ezero+2\sqrt{\Ezero}\sqrt{E(\{f\})}=-2\nu \Ezero\left(1-\frac{\sqrt{E(\{f\})}}{\nu\sqrt{\Ezero}}\right)\leq
-2\nu \Ezero\left(1-\sqrt{\frac{E_0}{\energyBound}}\right),$$ therefore by Gronwall’s inequality $$\Ezero(t)\leq e^{-2\nu t\left(1-\sqrt{\frac{E_0}{\energyBound}}\right)}\Ezero_I.$$ We set $t_1=t$, where $t$ satisfies $e^{-2\nu\varepsilon t}\Ezero_I=\energyBound$. The time $t_1$ is uniform for the trapping region $W$, because $\Ezero_I$ is the maximal energy within the trapping region $W$.
\[lem:akbk1\] Let $M_1\geq 0$, $k\in\mathbb{Z}\setminus\{0\}$, $j=1,2$, $\lambda_k$ be the $k$-th eigenvalue , [$H\supset W$ be a trapping region for $l$-th for all $l>M_1$]{}. .\
Assume that $N_{k,j}^\pm\in\mathbb{R}^2$ are bounds such that $$\left(N_{k,1}(\{a_k\}_{k\in\mathbb{Z}}), N_{k,2}(\{a_k\}_{k\in\mathbb{Z}})\right)\in[N_{k,1}^-, N_{k,1}^+]\times[N_{k,2}^-, N_{k,2}^+]\text{ for all }\{a_k\}_{k\in\mathbb{Z}}\in W.$$ Then for any $\varepsilon>0$ there exists a finite time $\hat{t}>0$ such that for all $l>\max{\{M_1, |k|\}}$ and $t\geq\hat{t}$ $a_k^l(t)$ with any initial condition in $P_l(W)$ satisfies $$\left(a_{k,1}^l(t), a_{k,2}^l(t)\right)\in \left[b_{k,1}^-,b_{k,1}^+\right]\times\left[b_{k,2}^-,b_{k,2}^+\right]+\left[-\varepsilon,\varepsilon\right]^2,$$ where $b_{k,j}^\pm=\frac{N_{k,j}^\pm+f_{k,j}}{-\lambda_k}$.
#### *Proof*
In the calculations we drop the index $l$ denoting the Galerkin projection dimension and the index $j$ denoting the coordinate for better clarification, for instance instead of $a_{k,j}^l$ we write $a_k$. We perform the calculations for the first and the second component simultaneously; thus, we finally obtain two values $t_{k, 1}>0$ and $t_{k,2}>0$. For any from $\frac{d a_k}{dt}\leq\lambda_k a_k+N_k^++f_k^+$, $\frac{d a_k}{dt}\geq\lambda_k a_k+N_k^-+f_k^-$ it follows that $$a_k(t)\geq\left(a_k^--b_k^-\right)e^{\lambda_k t}+b_k^-,\quad a_k(t)\leq\left(a_k^+-b_k^+\right)e^{\lambda_k t}+b_k^+,$$ where $a_k^\pm$ are bounds such that $\left(a_{k,1}, a_{k,2}\right)\in[a_{k,1}^-, a_{k,1}^+]\times[a_{k,2}^-, a_{k,2}^+]$, which exist as the initial condition is contained in a compact trapping region.\
Because $\lambda_kt=-\nu k^2t<0$ for any $t>0$ ($k\in\mathbb{Z}\setminus\{0\}$ by assumption) it follows that for a sufficiently large time $t_k>0$ we have $\left(\left|a_k^+-b_k^+\right|+\left|a_k^--b_k^-\right|\right)e^{\lambda_k t} \leq \varepsilon$ for any $t\geq t_k$. It is enough to take $$t_k=-\ln{\frac{\varepsilon}{\left(\left|a_k^+-b_k^+\right|+\left|a_k^--b_k^-\right|\right)}}/{\nu k^2}.$$ Finally $\hat{t}:=\max\left\{t_{k,1}, t_{k,2}\right\}$.
\[lem:akbk2\] Let $J>0$, $M_1\geq 0$, [$H\supset W$ be a trapping region for $l$-th for all $l>M_1$]{}. Assume that $C_a$, $s_a$ are numbers such that $$|a_k|\leq\frac{C_a}{|k|^{s_a}}\text{ for }|k|>M_1\text{, and for all }\{a_k\}\in W.$$ Assume that $\{f_k\}$ satisfies $f_k=0$ for $|k|>J$, $f_0=0$, and $C_N$, $s_N$ are numbers such that $$\left|N_k(\{a_k\}_{k\in\mathbb{Z}})\right|\leq\frac{C_N}{|k|^{s_N}}\text{ for }|k|>M_1.$$\
Then for any $\varepsilon>0$ there exists a finite time $\hat{t}\geq 0$ such that for all $l>M_1$ and $t\geq\hat{t}$ $\left\{a_k^l(t)\right\}_{|k|\leq l}$ with any initial condition in $P_l(W)$ satisfy $$|a_k^l(t)|\leq\frac{C_b+\varepsilon}{|k|^{s_b}}\text{ for }|k|>M_1,$$ where $C_b=\left(C_N+\max_{0<|k|\leq J}{\left\{|f_k||k|^{s_N}\right\}}\right)/\nu$, $s_b=s_N+2$.
#### *Proof*
We will use the same notation as in Lemma \[lem:akbk1\]. For any from the fact that $\frac{d a_k^l}{d t}\leq\lambda_k\left(a_k^l+\frac{N_k^++f_k^+}{\lambda_k}\right)$ and $\frac{d a_k^l}{d t}\geq\lambda_k\left(a_k^l+\frac{N_k^-+f_k^-}{\lambda_k}\right)$, it follows $$\begin{gathered}
a_k^l(t)\leq\left(\frac{C_a}{|k|^{s_a}}-\frac{C_b}{|k|^{s_b}}\right)e^{\lambda_{k}t}+\frac{C_b}{|k|^{s_b}},\\
a_k^l(t)\geq\left(-\frac{C_a}{|k|^{s_a}}+\frac{C_b}{|k|^{s_b}}\right)e^{\lambda_{k}t}-\frac{C_b}{|k|^{s_b}},
\end{gathered}$$ for $|k|>M_1$. Due to the fact that $s_b>s_a$ $$|a_k^l(t)|\leq\frac{C_a(k_{max}(t))^{s_b-s_a}e^{\lambda_{k_{max}(t)}t}+C_b}{|k|^{s_b}},\quad |k|>M_1$$ for all $l>M_1$ and $t>0$, where $C_b=\left(C_N+\max_{0<|k|\leq J}{\left\{|f_k||k|^{s_N}\right\}}\right)/\nu$, $s_b=s_N+2$, $k_{max}(t)$ is the value at which the maximum of $f_t(k)=C_ak^{s_b-s_a}e^{\lambda_k t}$ is attained. Analogically, for a sufficiently large time $t_\mathcal{F}>0$ $${C_a(k_{max}(t))^{s_b-s_a}e^{\lambda_{k_{max}(t)}t}\leq\varepsilon},\quad t\geq t_\mathcal{F},$$ therefore $$|a_k^l(t)|\leq\frac{C_b+\varepsilon}{|k|^{s_b}},\quad t\geq t_\mathcal{F},\ l>M_1.$$ Finally, the obtained time $t_\mathcal{F}$ is uniform with respect to the projection dimension $l$.
\[lem:energyEstimate\] Let $\hat{E}>0$. The following estimate holds $$\left|N_k(\{a_k\}_{k\in\mathbb{Z}})\right|\leq{\frac{1}{2}}|k|\hat{E}$$ for all $\{a_k\}_{k\in\mathbb{Z}}\in\left\{\{a_k\}_{k\in\mathbb{Z}}\in H\ |\ E(\{a_k\}_{k\in\mathbb{Z}})\leq \hat{E}\right\}$.
#### *Proof*
Let $\{a_k\}_{k\in\mathbb{Z}}\in \left\{\{a_k\}_{k\in\mathbb{Z}}\in H\ |\ E(\{a_k\}_{k\in\mathbb{Z}})\leq \hat{E}\right\}$. We start with the easy estimate $\left|N_k(\{a_k\}_{k\in\mathbb{Z}})\right|\leq{\frac{1}{2}}|k|\sum_{k_1\in\mathbb{Z}}{|a_{k-k_1}||a_{k_1}|}$, by the Cauchy-Schwarz inequality $\left|N_k(\{a_k\}_{k\in\mathbb{Z}})\right|\leq{{\frac{1}{2}}|k|\sqrt{\sum_{k_1\in\mathbb{Z}}{|a_{k_1}|^2}}\sqrt{\sum_{k_1\in\mathbb{Z}}{|a_{k-k_1}|^2}}}$, which is the following energy estimate $\left|N_k(\{a_k\}_{k\in\mathbb{Z}})\right|\leq{\frac{1}{2}}|k|\hat{E}$.
####
Now, we shall introduce *the absorbing sets*. For any initial condition there exists a finite time after which the solutions of Galerkin projections are trapped in an absorbing set. We use absorbing sets as a tool for studying the global dynamics of .
\[def:absorbingSet\] Let $M_1>0$. A set $A\subset H$ is called [the absorbing set]{} for large , if for any initial condition $\{a_k\}_{k\in\mathbb{Z}}\in H$ there exists a finite time $t_1\geq 0$ such that for all $l>M_1$ and $t\geq t_1$ $\varphi^l\left(t, P_l(\{a_k\}_{k\in\mathbb{Z}})\right)\in P_l(A)$.
In what follows we will often call the absorbing set for large simply *the absorbing set*.
In the next result, to show the existence of an absorbing set, we construct analytically an absorbing set. Furthermore, we construct absorbing sets with any [order of polynomial decay]{}. Later on, in the context of a computer assisted proof of the main theorem, we will construct an absorbing set using *the interval arithmetic*. Accomplishing this task requires the established existence of an absorbing set with a sufficiently large [order of polynomial decay]{}.
\[lem:absorbingSet\] Let $\alpha\in\mathbb{R}$, $\varepsilon>0$, $J>0$, $M_1\geq 0$, $E_0=\frac{\Ef}{\nu^2}$, $\energyBound>E_0$. Assume that . Put $$\begin{aligned}
s_i&=&i/2\text{ for }i\geq 2,\\
D_i&=&2^{s_i-\frac{1}{2}}+\frac{2^{s_i-1}}{\sqrt{2s_i-1}}\text{ for }i\geq 2,\\
C_2&=&\varepsilon+\left({\frac{1}{2}}\left(\energyBound+\alpha^2\right)+\max_{0<|k|\leq J}{\frac{|f_k|}{|k|}}\right)/\nu,\\
C_i&=&\varepsilon+\left(C_{i-1}\sqrt{\energyBound+\alpha^2}D_{i-1}+\max_{0<|k|\leq J}{|k|^{s_i-2}|f_k|}\right)/\nu\text{ for }i>2.
\end{aligned}$$\
Then for all $i\geq 2$ $$H\supset W_i\bigl(\energyBound,M_1,\varepsilon,\alpha\bigr):=\left\{\{a_k\}_{k\in\mathbb{Z}}\ |\ \Ezero(\{a_k\}_{k\in\mathbb{Z}})\leq\energyBound,\ |a_k|\leq\frac{C_i}{|k|^{s_i}}\text{ for }|k|>M_1\right\}$$ is an absorbing set for large .
#### *Proof*
Let $\energyBound>E_0$, $\{\widehat{a_k}\}_{k\in\mathbb{Z}}$ be an arbitrary initial condition for , $E^{max}:=\max\left\{\Ezero\left(\{\widehat{a_k}\}_{k\in\mathbb{Z}}\right), \energyBound\right\}$. Let $C_0\geq 0$ and $s_0> 0$ be constants such that $$\label{eq:W_0}
W_0:=\left\{\{a_k\}_{k\in\mathbb{Z}}\ |\ \Ezero(\{a_k\}_{k\in\mathbb{Z}})\leq E^{max},\ |a_k|\leq\frac{C_0}{\nmid k\nmid^{s_0}}\right\}$$ is a trapping region for each Galerkin projection of enclosing $\{\widehat{a_k}\}_{k\in\mathbb{Z}}$. This trapping region exists due to Theorem \[thm:analyticTrappingRegion\]. Note that a trapping region can be scaled to make it enclose an arbitrary sufficiently smooth initial condition. It follows from Lemma \[lem:energyDissipation\] that there exists a finite time $t_1\geq 0$ such that for all $\{a_k\}_{k\in\mathbb{Z}}\in W_0$ and $l>M_1$ $$\label{eq:energyW0}
\Ezero(\varphi^l(t_1,P_l(\{a_k\}_{k\in\mathbb{Z}})))\leq \energyBound.$$ We define $W_1:=W_0\cap\left\{\{a_k\}_{k\in\mathbb{Z}}\ |\ \Ezero(\{a_k\}_{k\in\mathbb{Z}})\leq \energyBound\right\}$. From and that $W_0$, $W_1$ are trapping regions we immediately have that $\varphi^l(t, P_l(\{a_k\}_{k\in\mathbb{Z}})\in W_1$ for all $\{a_k\}_{k\in\mathbb{Z}}\in W_0$, $t\geq t_1$ and $l>M_1$. Using Lemma \[lem:energyEstimate\] we bound the nonlinear part $$\label{eq:energyEstimate}
|N_k(\{a_k\}_{k\in\mathbb{Z}})|\leq{\frac{1}{2}}|k|\left(\energyBound+\alpha^2\right)\text{ for all }\{a_k\}_{k\in\mathbb{Z}}\in W_1.$$ It follows from Lemma \[lem:akbk2\] that there exists a finite time $t_2\geq t_1$ such that for all $t\geq t_2$ and $l>M_1$, $\{a_k^l(t)\}_{|k|\leq l}$ with any initial condition in $P_l(W_1)$ satisfy $$\label{eq:C_2}
|a_k^l(t)|\leq\frac{C_2}{|k|}\text{ for }|k|>M_1.$$ It is important to start with the energy estimate to bound the nonlinear part $N_k$ because the goal is to estimate $|a_k|$ uniformly with respect to $C_0$ and $s_0$ . We emphasize that $C_2$ from does not depend on $C_0$ and $s_0$. Having the bound , we construct the following absorbing set $$W_2:=\left\{\{a_k\}_{k\in\mathbb{Z}}\ |\ \Ezero(\{a_k\}_{k\in\mathbb{Z}})\leq\energyBound,\ |a_k|\leq\frac{C_2}{|k|}\text{, for }|k|>M_1\right\}.$$ Due to Lemma \[lem:estimateNk\] the following estimate holds $$\left|N_k(\{a_k\}_{k\in\mathbb{Z}})\right|\leq\frac{C_2\sqrt{\energyBound+\alpha^2}D_2}{|k|^{-{\frac{1}{2}}}},$$ for all $\{a_k\}_{k\in\mathbb{Z}}\in W_2$. Due to Lemma \[lem:akbk2\] again there exists a finite time $t_3\geq t_2$ such that for all $t\geq t_3$ and $l>M_1$, $\{a_k^l(t)\}_{|k|\leq l}$ with any initial condition in $P_l(W_2)$ satisfy $$|a_k^l(t)|\leq\frac{C_3}{|k|^{\frac{3}{2}}}\text{ for }|k|>M_1,$$ where $C_3=\varepsilon+\left(C_2\sqrt{\energyBound+\alpha^2}D_2+\max_{0<|k|\leq J}{\frac{|f_k|}{|k|^{{\frac{1}{2}}}}}\right)/\nu$. Having this bound, we construct the following absorbing set $$W_3:=\left\{\{a_k\}_{k\in\mathbb{Z}}\ |\ \Ezero(\{a_k\}_{k\in\mathbb{Z}})\leq \energyBound,\ |a_k|\leq\frac{C_3}{|k|^{\frac{3}{2}}}\text{ for }|k|>M_1\right\}.$$ Note the gain of ${\frac{1}{2}}$ in the [order of polynomial decay]{} of $\{a_k\}_{k\in\mathbb{Z}}$ in $W_3$ compared to $W_2$. From applying Lemma \[lem:estimateNk\] and Lemma \[lem:akbk2\] further we obtain a sequence of times $t_3<t_4<\dots<t_n<\dots$ such that $$|a_k^l(t_3)|\leq\frac{C_3}{|k|^{s_3}},\dots,\ |a_k^l(t_n)|\leq\frac{C_n}{|k|^{s_n}},\dots,\text{ for }|k|>M_1,$$ with $s_i=i/2$, $C_i=\varepsilon+\left(C_{i-1}\sqrt{\energyBound+\alpha^2}D_{i-1}+\max_{0<|k|\leq J}{|k|^{s_i-2}|f_k|}\right)/\nu$. The obtained $W_i$, $i\geq 2$ are absorbing sets for large , which follows from the construction and that $C_i$ for all $i\geq 2$ depend on the energy $\energyBound$ and $\alpha$ only.
Assume the same as in Lemma \[lem:absorbingSet\]. The inclusion $W_i\subset{\overline{H}}$ holds for all $i\geq 8$, where $W_i$ is an absorbing set proved to exist in Lemma \[lem:absorbingSet\].
\[lem:refinementOfAbsorbingSet\] Let $k\in\mathbb{Z}\setminus\{0\}$, $\varepsilon>0$, $\lambda_k$ denotes the $k$-th eigenvalue . Let $H\supset A$ be an absorbing set for large . . Assume that $N_k^\pm\in\mathbb{R}^2$ are bounds such that $$\left(N_{k,1}(\{a_k\}_{k\in\mathbb{Z}}), N_{k,2}(\{a_k\}_{k\in\mathbb{Z}})\right)\in[N_{k,1}^-, N_{k,1}^+]\times[N_{k,2}^-, N_{k,2}^+]\text{ for all }\{a_k\}_{k\in\mathbb{Z}}\in A.$$ Then $A\cap\left\{\{a_k\}_{k\in\mathbb{Z}}\ |\ a_k\in\left[b_{k,1}^-,b_{k,1}^+\right]\times\left[b_{k,2}^-,b_{k,2}^+\right]+\left[-\varepsilon,\varepsilon\right]^2\right\}$ is also an absorbing set for large , where $b_{k,j}^\pm=\frac{N_{k,j}^\pm+f_{k,j}}{-\lambda_k}$, $j=1,2$.
#### *Proof*
Immediate consequence of Lemma \[lem:akbk1\].
General method of self-consistent bounds. {#sec:generalMethod}
=========================================
The same symbols as in the preceding part are used in a more general context. For the purpose of the presented work we call a dissipative PDE a PDE of the following type $$\label{eq:dPDE}
\frac{du}{dt}=Lu+N(u,Du,\dots,D^r u)+f=F(u),$$ where $u(x,t)\in\mathbb{R}^n$, $x\in\mathbb{T}^d$, ($\mathbb{T}^d=(\mathbb{R}/2\pi)^d$ is a $d$-dimensional torus), $L$ is a linear operator, $N$ a polynomial and by $D^su$ we denote the collection of $s$-th order partial derivatives of $u$. The right-hand side contains a constant in time forcing function $f$. We require that $L$ is diagonal in the Fourier basis $\{e^{ikx}\}_{k\in\mathbb{Z}^d}$ $$Le^{ikx}=\lambda_k e^{ikx}$$ and the *eigenvalues* $\lambda_k$ satisfy
\[eq:dPDEassumptions\] $$\begin{gathered}
\lambda_k=-\nu(|k|)|k|^p,\\
0<\nu_0\leq\nu(|k|)\leq \nu_1,\quad\text{for }|k|>K_-,\\
p>r,
\end{gathered}$$ for some $v_0>0$, $v_1\geq v_0$ and $K_-\geq 0$, $r$ is the maximal order of derivatives appearing in the nonlinear part , $|\cdot|$ is the Euclidean norm.
Self-consistent bounds
----------------------
We recall, in the context of dPDEs, the definition of self-consistent bounds from [@Z3]. Let $H$ be a Hilbert space, actually $L^2$ or one of its subspaces in the context of dPDEs. We assume that a domain of $F$, the right hand side of , is dense in $H$. By a solution of we understand a function $u\colon[0,\mathcal{T})\to{\operatorname{dom}\left(F\right)}$ such that $u$ is differentiable and is satisfied for all $t\in[0,\mathcal{T})$ and $\mathcal{T}$ is a maximal time of the existence of solution. We assume that there is a set $I\subset\mathbb{Z}^d$ and a sequence of subspaces $H_k\subset H$ for $k\in I$ such that $\dim{H_k}=d_1 <\infty$, $H_k$ and $H_{k'}$ are mutually orthogonal for $k\neq k'$ and $H=\overline{\oplus_{k\in I}{H_k}}$. Let $A_k\colon H\to H_k$ be the orthogonal projection onto $H_k$, for each $u\in H$ holds $u=\sum_{k\in I}{u_k}=\sum_{k\in I}{A_k u}$. Analogously if $B$ is a function with the range in $H$, then $B_k(u)=A_kB(u).$
We assume that a a metric space $(T,\rho)$ is provided, for $X\subset T$ by $\overline{X}$ we denote the closure of $X$, by $\partial X$ we denote the boundary of $X$. For $n>0$ we set $X_n=\oplus_{|k|\leq n,k\in I}{H_k}$, $Y_n=X_n^\perp$. By $P_n\colon H\to X_n$ and $Q_n\colon H\to Y_n$ we denote the orthogonal projections onto $X_n$ and $Y_n$ respectively, $T\supset B(c,r)=\left\{x\in T\colon \rho(c,x)<r\right\}$ denotes a ball with the centre at $c$ and the radius $r$.
[[@Z3 Def. 2.1]]{} We say that $F\colon H\supset{\operatorname{dom}\left(F\right)}\to H$ is admissible if the following conditions are satisfied for any $n>0$ such that $\dim{X_n}>0$
- $X_n\subset{\operatorname{dom}\left(F\right)}$
- $P_nF\colon X_n\to X_n$ is a $C^1$ function
[[@Z3 Def. 2.3]]{} \[def:scs1\] Assume $F$ is an admissible function. Let $m,\ M\,\in\mathbb{R}$ with $m\leq M$. Consider an object consisting of: a compact set $W\subset X_m$ and a sequence of compact sets $B_k\subset H_k$ for $|k|>m,\ k\in I$. We define the conditions **C1**, **C2**, **C3**, **C4a** as follows:
#### **C1**
For $|k|>M,\ k\in I$ holds $0\in B_k$.
#### **C2**
Let $\hat{a}_k\colon=\max_{a\in B_k}{{\lVerta\rVert}}$ for $|k|>m,\ k\in I$ and then $\sum_{|k|>m,k\in I}{\hat{a}^2_k}<\infty$.\
In particular $$W\oplus\Pi_{|k|>m}{B_k}\subset H$$ and for every $u\in W\oplus\Pi_{k\in I,|k|>m}{B_k}$ holds , ${\lVertQ_n u\rVert}^2\leq\sum_{|k|>n,k\in I}{\hat{a}_k^2}$.
#### **C3**
The function $u\mapsto F(u)$ is continuous on $W\oplus\Pi_{k\in I,|k|>m}{B_k}\subset H$.\
Moreover, if we define for $k\in I$, $\hat{f}_k=\max_{u\in W\oplus\Pi_{k\in I,|k|>m}{B_k}}{\left|F_k(u)\right|}$, then $\sum{\hat{f}_k^2}<\infty$.
#### **C4a**
For $|k|>m,\ k\in I\ B_k$ is given by or $$\begin{aligned}
B_k &= \overline{B(c_k,r_k)},\quad r_k>0,\label{def:ball1}\\
B_k &= \Pi_{s=1}^{d_1}\left[a_{k,s}^-,a_{k,s}^+\right],\quad a_{k,s}^-<a_{k,s}^+\label{def:ball2}.
\end{aligned}$$ Let $u\in W\oplus\Pi_{|k|>m}{B_k}$, $F_{k,s}$ be the $s$-th component of $F_k$. Then for $|k|>m$ holds:
- if $B_k$ is given by then $$u_k\in\partial_{H_k}{B_k}\Rightarrow\left(u_k-c_k|F_k(u)\right)<0.$$
- if $B_k$ is given by then $$\begin{aligned}
u_{k,s}=a_{k,s}^-&\Rightarrow F_{k,s}(u)>0,\\
u_{k,s}=a_{k,s}^+&\Rightarrow F_{k,s}(u)<0.
\end{aligned}$$
[[@Z3 Def. 2.4]]{} \[def:scs2\] Assume $F$ is an admissible function. Let $m,M\in\mathbb{R}$ with $m\leq M$. Consider an object consisting of: a compact set $W\subset X_m$ and a sequence of compacts $B_k\subset H_k$ for $|k|>m,k\in I$. We say that set $W\oplus\Pi_{k\in I,|k|>m}{B_k}$ forms [self-consistent bounds for]{} $F$ if conditions **C1**, **C2**, **C3** are satisfied.\
If additionally condition **C4a** holds, then we say that $W\oplus\Pi_{k\in I,|k|>m}{B_k}$ forms [topologically self-consistent bounds for ]{}$F$.
####
We start our approach by replacing a sufficiently regular $u$ and $f$ in by the Fourier series, i.e. $u(x,t)=\sum_{k\in\mathbb{Z}^d}{a_k(t)e^{ikx}}$ and $f(x)=\sum_{k\in\mathbb{Z}^d}{f_ke^{ikx}}$. We obtain a system of ODEs describing the evolution of the coefficients $\{a_k\}_{k\in\mathbb{Z}^d}$, where $a_k$ is the coefficient corresponding to $e^{ikx}$ $$\label{eq:evolutionFourierModes}
\frac{d a_k}{d t}=F_k(a)=L_k(a)+N_k(a)+f_k=\lambda_k a_k+N_k(a)+f_k ,\quad k\in\mathbb{Z}^d.$$ The method works for dPDEs only. The Burgers equation on the real line with forcing, which is the subject of the case study given in this paper is in fact a dPDE.
\[lem:burgersDPDE\] Let $\nu$ be the viscosity constant in , then satisfies the conditions with $d=1$, $r=1$, $p=2$, $\nu(k)=\nu$, $\lambda_k=-\nu k^2$.
In our approach we solve the system of equations instead of . is defined on $l^2=\left\{\{a_k\}\colon\,\sum{|a_k|^2<\infty}\right\}$ space or one of its subspaces. We associate $a_k$ with the coefficient corresponding to $e^{ikx}$ in the Fourier expansion of $u$. Assuming that the initial condition $u_0\in H$ is sufficiently regular, then and are equivalent. In our approach we expand $u_0$ in the Fourier basis to get the initial value for all the variables $\left\{a_k(0)\right\}_{k\in\mathbb{Z^d}}$. We argue that the solution of is defined for all times $t>0$. Moreover, the solution conserves its initial regularity due to the existence of *trapping regions* and is, in fact, a classical solution of . For the details refer to Section \[sec:local\] and Section \[sec:exampleTheorem\].
To establish the notation in the next sections we provide
\[def:tail\] Given an object $W\oplus\Pi_{|k|>m}{B_k}$, $W\subset X_m$ and a sequence of compact sets $B_k\subset H_k$ for $|k|>m$, $m,M\in\mathbb{R}_{+}$, $m\leq M$
- $W$ is called [the finite part]{},
- $\Pi_{|k|>m}{B_k}$ is called [the tail]{} and denoted by $T$,
- $\Pi_{m<|k|\leq M}{B_k}$ is called [the near tail]{} and denoted by ${T_\mathcal{N}}$,
- $\Pi_{|k|> M}{B_k}$ is called [the far tail]{} and denoted by ${T_\mathcal{F}}$.
${T_\mathcal{N}}$ is the finite part of a tail, whereas ${T_\mathcal{F}}$ is the infinite part of a tail. In fact in our approach we use ${T_\mathcal{F}}$ of the form $$\label{eq:farTail}
{T_\mathcal{F}}:=\prod_{|k|>M}{\overline{B(0, C/|k|^s)}},\quad C\in\mathbb{R}_+,\ s\geq d+p+1.$$
####
First of all, any $F$ in is admissible, because any finite truncation of a $l^2$ series is in the domain of $F$, and the Galerkin projection of the right-hand side, being a smooth function, is a polynomial. $W\oplus T\subset H$ with ${T_\mathcal{F}}$ defined in satisfies conditions C1, C2 and C3 of Definition \[def:scs1\] with $I=\mathbb{Z}^d$, in particular $F$ in is a continuous function on $W\oplus T$. This property was proved in [@Z3 Theorem 3.6], i.e. $W\oplus T$ forms self-consistent bounds for and equivalently forms a self-consistent bounds for . It is allowable to associate the finite part $W$ with the near tail ${T_\mathcal{N}}$, but we keep the distinction because of the different treatment of both in the algorithm.
We do not address here the question if solutions of a general dPDE exist and are unique as it was thoroughly answered in [@Z3], see [@Z3 Theorem 3.7].
Local existence and uniqueness {#sec:local}
==============================
####
Regarding local existence and uniqueness we rely on results from [@ZAKS]. For the sake of completeness we recall the main theorems. The same symbols as in the preceding part are used in a more general context.
[[@ZAKS Def. 3.1]]{} A decomposition of $H$, into into a sum of subspaces is called a block decomposition of $H$ if the following conditions are satisfied.
1. $H=\oplus_i{H_i}$,
2. for every $i$ $h_i=dim\,H_i\leq h_{max}<\infty$,
3. for every $i$ $H_i=\langle e_{i_1},e_{i_2},\dots,e_{i_{h_i}}\rangle$,
4. if $dim\,H=\infty$, then there exists $k$ such that for $i>k$ $h_i=1$.
#### Notation
In this section we adopt the notation from [@ZAKS], namely, we make a distinction between blocks and one dimensional spaces spanned by $\langle e_i\rangle$. For the blocks we use $H_{(i)}=\langle e_{i_1},\dots,e_{i_k}\rangle$, where $(i)=(i_1,\dots,i_k)$. The symbol $H_i$ will always mean the subspace generated by $e_i$. For a block decomposition of $H$ and block $(i)$, we set $\dim{(i)}=\dim{H_{(i)}}$. For any $x\in H$ by $x_{(i)}$ we will denote a projection of $x$ onto $H_{(i)}$, by $P_{(i)}$ we will denote an orthogonal projection onto $H_{(i)}$. For $x\in\mathbb{R}^n$ we set $|x|$ to be the Euclidean norm. We define the norm (*the block-infinity norm*) by $|x|_{b,\infty}=\max_{(i)}{|P_{(i)}x|}$.
For any norm $||\cdot||$ on $\mathbb{R}^n$ we use the notion of *the logarithmic norm* of a matrix.
[[@ZAKS Definition 3.4]]{} Let , $Q$ be a square matrix, then we call $$\mu(Q)=\limsup_{h>0,h\to 0}{\frac{||I+hQ||-1}{h}}$$ *the logarithmic norm* of $Q$.
Let $R\subset H$, $R$ is convex, $l>0$, $x\in X_l$, $\varphi^l(t, x)$ be the local flow inducted by the $l$-th Galerkin projection of . We call $P_l(R)$ [a trapping region]{} for the $l$-th Galerkin projection of if $\varphi^l(t, P_l(R))\subset P_l(R)$ for all $t>0$ or equivalently the vector field on the boundary of $P_l(R)$ points inwards.
[[@ZAKS Thm. 3.7]]{} \[thm:7\] Assume that $R\subset H$, $R$ is convex and $F$ satisfies conditions C1, C2, C3. Assume that we have a block decomposition of $H$, such that condition **Db** holds
#### Db
there exists $l\in\mathbb{R}$ such that for any (i) and $x\in R$ $$\label{eq:logarithmic}
\mu\left(\frac{\partial F_{(i)}}{\partial x_{(i)}}\left(x\right)\right)+\sum_{(k),(k)\neq(i)}{\left|\frac{\partial F_{(i)}}{\partial x_{(k)}}\left(x\right)\right|}\leq l$$ Assume that $P_n(R)$ is a trapping region for the $n$-dimensional Galerkin projection of for all $n>M_1$. Then
1. **Uniform convergence and existence**. For a fixed $x_0\in R$, let $x_n\colon[0,\infty]\to P_n(R)$ be a solution of $x^{\prime}=P_n(F(x)),\ x(0)=P_nx_0$. Then $x_n$ converges uniformly in a max-infinity norm on compact intervals to a function $x^{*}\colon[0,\infty]\to R$, which is a solution of and $x^{*}(0)=x_0$. The convergence of $x_n$ on compact time intervals is uniform with respect to $x_0\in R$.
2. **Uniqueness within $R$**. There exists only one solution of the initial value problem , $x(0)=x_0$ for any $x_0\in R$ such that $x(t)\in R$ for $t>0$.
3. **Lipschitz constant**. Let $x\colon[0,\infty]\to R$ and $y\colon[0,\infty]\to R$ be solutions of , then $$ \left|y(t)-x(t)\right|_{b,\infty}\leq e^{lt}\left|x(0)-y(0)\right|_{b,\infty}$$
4. **Semidynamical system**. The map $\varphi\colon\mathbb{R}_+\times R\to R$, where $\varphi(\cdot,x_0)$ is a unique solution of the equation such that $\varphi(0,x_0)=x_0$ defines a semidynamical system on $R$, namely
- $\varphi$ is continuous,
- $\varphi(0,x)=x$,
- $\varphi(t, \varphi(s, x))=\varphi(t+s,x)$.
####
The following Theorem is the main tool used to prove the existence of a locally attracting fixed point.
[[@ZAKS Thm. 3.8]]{} \[thm:8\] The same assumptions on $R, F$ and a block decomposition $H$ as in Theorem \[thm:7\]. Assume that $l<0$.
####
Then there exists a fixed point for $x^*\in R$, unique in $R$, such that for every $y\in R$ $$\begin{split}
&\left|\varphi(t,y)-x^*\right|_{b,\infty}\leq e^{lt}\left|y-x^*\right|_{b,\infty},\quad\text{for }t\geq 0,\\
&\lim_{t\to\infty}{\varphi(t,y)}=x^*.
\end{split}$$
Proof of Theorem \[thm:main\] {#sec:exampleTheorem}
=============================
Now we are ready to prove Theorem \[thm:main\]. The complete algorithm that we used to prove Theorem \[thm:main\] and other results are demonstrated in Section \[sec:cap\]. This proof is a prototype for any other result that is obtained using the algorithm, however each case requires construction of different sets. The sets and all the relevant numbers used in the proof of Theorem \[thm:main\] are presented in Appendix \[sec:proofData\].
#### *Proof of Theorem \[thm:main\]* {#proof-of-theoremthmmain}
Let $u_0\in C^4$ be an arbitrary initial condition satisfying $\int_0^{2\pi}{u_0(x)\,dx}=\paperExampleAzero$, $\left\{a_k\right\}_{k\in\mathbb{Z}}$ be the Fourier coefficients of $u_0$, i.e. $u_0=\sum{a_ke^{ikx}}$. Let $A\subset{\overline{H}}$ be an absorbing set for large , which exists due to Lemma \[lem:absorbingSet\] (for instance $W_8$). Firstly, the existence of a locally attracting fixed point for is established by constructing a set $\widetilde{W\oplus T}\subset{\overline{H}}$ satisfying the assumptions of Theorem \[thm:8\], using the interval arithmetic. This is constructed in step \[step:trappingRegion\] of Algorithm from Section \[sec:cap\]. Observe that $\widetilde{W\oplus T}$ is a trapping region for $m$-th for all $m>\widehat{m}$ and the logarithmic norm is bounded from above by $l<0$. The purpose of the notation $\widetilde{W\oplus T}$ is to keep the consistency with the description of the algorithm used for proving this theorem in Section \[sec:cap\]. $\widetilde{W\oplus T}$ satisfies the conditions C1, C2 and C3 of Definition \[def:scs1\], i.e. forms self-consistent bounds for , the conditions C1 and C2 are satisfied trivially, because $\overline{H}\subset H$. The condition C3 is also satisfied, the right-hand side of , denoted here by $F$, is continuous on $\widetilde{W\oplus T}$. First, notice that $\widetilde{W\oplus T}\subset{\operatorname{dom}\left(F\right)}$ because of the following inequality, let $u\in\widetilde{W\oplus T}$ $$\label{eq:Fl2}
|F(u)_k|\leq\frac{D_1}{|k|^{5/2}}+\frac{\nu D_2}{|k|^2}\leq\frac{\tilde{D}}{|k|^2},\quad |k|>\widehat{m},$$ therefore $F(u)\in H$. The continuity of $F$ on $\widetilde{W\oplus T}$ follows from the general theorem [@Z3 Theorem 3.7]. All the assumptions of [@Z3 Theorem 3.7] are satisfied here, i.e. belongs to the proper class, see Lemma \[lem:burgersDPDE\], and the order of decay of $\widetilde{W\oplus T}$ is sufficient, see .
By Theorem \[thm:8\] within $\widetilde{W\oplus T}$ there exists a locally attracting fixed point for .
Then, $V\oplus\mathnormal{\Theta}\subset{\overline{H}}$, an absorbing set for large satisfying $$\label{eq:rigorousIntegration}
\varphi^m\left(t,P_m(V\oplus\mathnormal{\Theta})\right)\subset P_m\left(\widetilde{W\oplus T}\right),$$ for all $t\geq \widehat{t}$ and $m>\widehat{m}$, is constructed. This is constructed in Algorithm from Section \[sec:scb\]. The absorbing set $V\oplus\mathnormal{\Theta}$ forms self-consistent bounds for and thus is verified by rigorous integration of $V\oplus\mathnormal{\Theta}$ forward in time using Algorithm \[alg:main\] presented hereafter. From and the fact that $V\oplus\mathnormal{\Theta}$ is an absorbing set for large it follows that $$\varphi^m\left(t, P_m\left(\left\{a_k\right\}_{k\in\mathbb{Z}}\right)\right)\in P_m\left(\widetilde{W\oplus T}\right).$$ after a finite time and for all $m>\widehat{m}$. Therefore $\left\{a_k\right\}_{k\in\mathbb{Z}}$ is located in the basin of attraction of the fixed point for . The sets $\widetilde{W\oplus T}$ and $V\oplus\mathnormal{\Theta}$ are presented in Appendix \[sec:proofData\].
To close the proof we will argue that the fixed point for is the steady state solution of . There exists $C>0$ such that the Fourier coefficients $\left\{a_k\right\}_{k\in\mathbb{Z}}$ of $u_0$ satisfy $$\label{eq:regularity}
|a_k|\leq\frac{C}{\nmid k\nmid^4}.$$ Let $W_0\subset{\overline{H}}$ be a trapping region enclosing $\{a_k\}_{k\in\mathbb{Z}}$, and let $\{a_k(t)\}_{k\in\mathbb{Z}}$ be the unique solution of existing for all times $t>0$, $\{a_k\}_{k\in\mathbb{Z}}\in W_0$ due to Theorem \[thm:7\]. The solution is unique, as the logarithmic norm on $W_0$ is bounded, see e.g. [@ZNS]. Moreover, the solution conserves the initial regularity . The sequence $\{a_k(t)\}_{k\in\mathbb{Z}}$ for $t>0$ is a classical solution of , as from Lemma \[lem:fourierC\], the condition suffices to $\sum{a_k e^{ikx}}$ and every term that appears in converge uniformly. Therefore, the solution of within $\widetilde{W\oplus T}$ is in fact the classical solution of , in particular, the fixed point of is the steady state solution of .
####
In the table below we present example results which we obtained using our algorithm.
$$\label{eq:table}
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\mathbf{\nu} & \mathbf{\int_0^{2\pi}{u_0(x)\,dx}} & \mathbf{E_0} & \mathbf{\varepsilon} & \mathbf{m} &\eqref{eq:logarithmic}\ \mathbf{l< }& \textbf{1.} & \textbf{2.} & \textbf{3.} & \textbf{4.} & \textbf{5.}\\\hline\hline
[10,10.1] & 14\pi & 0.5 & 0.001 & 5 & -9.94489 & 20.06 & 1041 & \checkmark & \checkmark & \checkmark \\\hline
[4,4.1] & 4\pi & 0.5 & 0.001 & 7 & -2.65147 & 61.41 & 1305 & \checkmark & \checkmark & \checkmark \\\hline
\paperExampleNu & \paperExampleAzero & 0.82 & 0.03 & 3 &\paperExampleL &\paperExampleTotalTime &\paperExampleNrOfSteps& \checkmark & \checkmark & \checkmark \\\hline
1 & 0.4\pi & 0.25 & 0.0001 & 20 & -0.0442416 & 452.23 & 452 & \checkmark & \checkmark & \checkmark \\\hline
0.5 & 0.1\pi & 0.08 & 0.0001 & 20 & -0.0456 & 556.73 & 629 & \checkmark & \checkmark & \checkmark \\\hline
0.15 & 0 & 0.22 & 0 & 40 & 1340.95 & 26.19 & - & \checkmark & & \\\hline
\end{array}$$
The meaning of the labels in Table \[table\] is the following [**1.**]{} total execution time in seconds, [**2.**]{} number of integration steps, [**3.**]{} if existence of a fixed point was proved, [**4.**]{} if the fixed point is locally attracting, [**5.**]{} if the fixed point is attracting globally. Order of the Taylor method was $6$, time step length was $0.005$ in all cases.
For each case we fixed the radius of the energy absorbing ball $E_0$ and chose at random a forcing $f(x)$ which satisfies $\frac{E(\{f_k\})}{\nu^2}=E_0$. The forcing $f(x)$ was defined by a finite number of modes $\{f_k\}_{|k|\leq m}$. We added to each forcing mode $f_k$ the uniform perturbation $[f_\varepsilon]:=[-\varepsilon,\varepsilon]\times[-\varepsilon,\varepsilon]$ (the parameter $\varepsilon$ is also provided in Table \[table\]) in order to perform simultaneously a proof for a ball of functions.
####
We would like to stress the fact that the provided cases are only examples and our program can attempt to prove any case. The package with the program along with the instruction and all the data from the proofs is available [@Package].
Algorithm for constructing an absorbing set for large {#sec:scb}
======================================================
####
The goal of this section is to present an algorithm for constructing a set $V\oplus\mathnormal{\Theta}\subset\overline{H}$, forming self-consistent bounds for such that $V\oplus\mathnormal{\Theta}$ is an absorbing set for large . It is important to require that $V\oplus\mathnormal{\Theta}$ forms self-consistent bounds for because in Algorithm \[alg:main\] $V\oplus\mathnormal{\Theta}$ is integrated forward in time to verify that any solution in $V\oplus\mathnormal{\Theta}$ after a finite time enters a trapping region.
To support our claim that the constructed $V\oplus\mathnormal{\Theta}$ is in fact an absorbing set, in the following description we argue each estimate. We drop the indication of Galerkin projections and times. For the precise meaning, the reader is referred to the proof of Lemma \[lem:absorbingSet\].
#### Notation
${\operatorname{Sq}\left(r\right)}:=[-r,r]\times[-r,r]\subset\mathbb{R}^2,\quad {\operatorname{B}\left(r\right)}:=\overline{B(0,r)}\subset\mathbb{R}^2$.
#### Input data
- $\nu>0,\quad M>m>0$ defining the dimensions of self-consistent bounds as in Definition \[def:tail\], $\alpha\in\mathbb{R}$,
- $\left\{[f_k]\right\}_{0<|k|\leq m}$ set of forcing modes perturbed by a uniform and constant perturbation $[f_\varepsilon]$, i.e. $[f_k]=f_k+[f_\varepsilon]$ for $0<|k|\leq m$ and $[f_k]=0$ for $|k|>m$, $[f_0]=0$,
- $E_0$, where $E_0=\max_{\{f_k\}\in\left\{[f_k]\right\}}{\frac{E(\{f_k\})}{\nu^2}}$.
#### Output data
$V\oplus\mathnormal{\Theta}\subset{\overline{H}}$ forming self-consistent bounds for .
#### begin
#### Initialization
$\hat{E}:=1.01\cdot \left(E_0+\alpha^2\right)$, $\hat{\varepsilon}:=10^{-15}$.
#### I Step
- For $0<|k|\leq M$ set $$(V\oplus\mathnormal{\Theta})_k:={\operatorname{Sq}\left(\frac{\hat{\varepsilon}+\left({\frac{1}{2}}\hat{E}+\max_{0<|k|\leq m}{\frac{\left|[f_k]\right|}{|k|}}\right)/\nu}{|k|}\right)}.$$
- For $|k|>M$ set $$(V\oplus\mathnormal{\Theta})_k:={\operatorname{B}\left(\frac{\hat{\varepsilon}+\left({\frac{1}{2}}\hat{E}+\max_{0<|k|\leq m}{\frac{\left|[f_k]\right|}{|k|}}\right)/\nu}{|k|}\right)}.$$
*Initial data is the absorbing ball of radius* $\hat{E}$*, then by Lemma \[lem:akbk2\] combined with Lemma \[lem:energyEstimate\] after a finite time the coefficients* $\{a_k\}$ *satisfy* $$|a_k|\leq\frac{\hat{\varepsilon}+\left({\frac{1}{2}}\hat{E}+\max_{0<|k|\leq m}{\frac{|f_k|}{|k|}}\right)/\nu}{|k|}=:\frac{C}{|k|},\quad |k|>M.$$
#### II Step
- For $0<|k|\leq M$ calculate $$b_{k,j}^-:=\frac{\left(-C\sqrt{\hat{E}}D+\frac{f^-_{k,j}}{|k|^{\frac{1}{2}}}\right)/\nu}{|k|^\frac{3}{2}},\quad
b_{k,j}^+:=\frac{\left(C\sqrt{\hat{E}}D+\frac{f^+_{k,j}}{|k|^{\frac{1}{2}}}\right)/\nu}{|k|^\frac{3}{2}},\quad j=1,2.$$ *Initial data is the set* $V\oplus\Theta$ *from I Step, then the following estimate due to Lemma \[lem:estimateNk\] is used* $$|N_k(V\oplus\Theta)|\leq\frac{C\sqrt{\hat{E}}D}{|k|^{-\frac{1}{2}}},$$ *where $C$ is defined in I Step.*
- For $0<|k|\leq M$ set $$(V\oplus\mathnormal{\Theta})_k:=
[b_{k,1}^-,b_{k,1}^+]\times[b_{k,2}^-,b_{k,2}^+]+[-\hat{\varepsilon},\hat{\varepsilon}]^2.$$ *This is a refinement step. Using the data from I Step, a new value of* $V\oplus\Theta$ *is defined. By Lemma \[lem:akbk1\] and Lemma \[lem:refinementOfAbsorbingSet\] after a finite time the coefficients* $\{a_k\}$ *satisfy* $$a_k\in[b_{k,1}^-,b_{k,1}^+]\times[b_{k,2}^-,b_{k,2}^+]+[-\hat{\varepsilon},\hat{\varepsilon}]^2,
\quad 0<|k|\leq M.$$
- For $|k|>M$ set $$(V\oplus\mathnormal{\Theta})_k:={\operatorname{B}\left(\frac{\hat{\varepsilon}+\left(C\sqrt{\hat{E}}D\right)/\nu}{|k|^\frac{3}{2}}\right)}.$$ *This is a refinement step. Using the data from I Step, a new value of* $V\oplus\Theta$ *is defined. By Lemma \[lem:akbk2\] after a finite time the coefficients* $\{a_k\}$ *satisfy* $$|a_k|\leq\frac{\hat{\varepsilon}+\left(C\sqrt{\hat{E}}D\right)/\nu}{|k|^\frac{3}{2}},\quad |k|>M.$$ *Observe that $[f_k]=0$ for $|k|>M$ and Lemma \[lem:akbk2\] is used with $M_1=M$.*
#### III Step
Iterate the refinement, until $V\oplus\mathnormal{\Theta}$ forms self-consistent bounds for , as the stopping criterion use the condition $s(\mathnormal{\Theta})>d+p$, where $d$ and $p$ are from and $s(\mathnormal{\Theta})$ is the order of polynomial decay of the tail $\mathnormal{\Theta}=\Pi_{|k|>M}{\overline{B\left(0,\frac{C(\mathnormal{\Theta})}{|k|^{s(\mathnormal{\Theta})}}\right)}}$. To calculate $[b_{k,1}^-,b_{k,1}^+]\times[b_{k,2}^-,b_{k,2}^+]$ and $C(b)$ use the estimates derived in [@SuppMat], in principle giving much sharper bounds than energy-like estimates used in the previous steps.
*Every such refinement generates bounds that are reached by the solutions after a finite time. Moreover, to see that the procedure will stop, note that at each iteration the [order of polynomial decay]{} $s(\mathnormal{\Theta})$ is increased by $1$. Using the formulas derived in [@SuppMat] a bound such that $|N_k|\leq\frac{D}{|k|^{s(N)}}$ is received, where $s(N)=s(\mathnormal{\Theta})-1$ and, therefore, $s_{new}(\mathnormal{\Theta})=s(b)=s(N)+2=s(\mathnormal{\Theta})+1$. Finally, as soon as $s_{new}(\mathnormal{\Theta})>d+p$, stop.*
#### end
Rigorous integration forward in time
====================================
By *rigorous numerics* we mean algorithms for estimating solutions of differential equations that operate on sets and produce sets that always contain an exact solution. Rigorous numerics for ODEs is a well established and analysed topic. There are a few algorithms that offer reliable computations of the solution trajectories for ODEs which are based on interval arithmetic. The approach used in this paper is based on the Lohner algorithm, presented in [@Lo] , see also [@ZLo]. It has made possible to prove many facts concerning the dynamics of certain ODEs, e.g. the Rossler equation, the Lorenz equation or the restricted n-body problem (see [@ZLo], [@KZ] and references therein). In the context of rigorous integration of ODEs we consider an abstract Cauchy problem $$\left\{\begin{array}{ccc}
\dot{x}(t)&=&f(x(t)),\\
x(0)&=&x_0.
\end{array}\right.
\label{cauchy}$$ $x\colon[0, \mathcal{T})\to\mathbb{R}^n$, $f\colon\mathbb{R}^n\to\mathbb{R}^n$, $f\in C^\infty$. The goal of a rigorous ODEs solver is to find a set $\mathbf{x_k}\subset\mathbb{R}^n$ compact and connected such that $$\varphi(t_k, \mathbf{x_0})\subset \mathbf{x_k},$$ $t_k\in[0, \mathcal{T}),\quad \mathbf{x_0}\subset\mathbb{R}^n$. By $\varphi(t_k, x_0)$ we denote the solution of at the time $t_k$ with initial condition $x_0\in\mathbb{R}^n$, and therefore $\varphi(t_k, \mathbf{x_0})$ denotes the set of all the values which are attained at the time $t_k$ by any solution of with the initial condition in $\mathbf{x_0}$.
#### *Notation*
We denote by $[x]$ an *interval set* $[x]\subset\mathbb{R}^n$, $[x]=\Pi_{k=1}^n[x^-_k, x^+_k]$, $[x^-_k, x^+_k]\subset\mathbb{R},\ -\infty<x_k^-\leq x_k^+<\infty$, ${\operatorname{mid}\left([x]\right)}$ is the middle of an interval set $[x]$ and ${\operatorname{r}\left([x]\right)}$ is the rest, i.e. $[x]={\operatorname{mid}\left([x]\right)}+{\operatorname{r}\left([x]\right)}$.
There are some subtle issues regarding intervals and set representation in the Lohner algorithm, which are discussed e.g. in [@ZLo]. Let us only mention that it is highly ineffective to use the interval set representation explicitly $\Pi{[a^-_k, a^+_k]}$ because it leads to the so-called *wrapping effect* [@ZLo], large over-estimates appear and prevents us from integrating over a longer time interval. In order to avoid those problems we do not use interval sets explicitly, but to represent sets in a suitable coordinate system we use the *doubleton* representation of sets [@ZLo] $$\label{eq:doubletonRepresentation}
[x_k]+B_k\cdot [r_k]+C_k\cdot [r_0],$$ where $B_k$ and $C_k$ are matrices representing a coordinate systems, $[x_k]$ is an interval set, likely a single point, $[r_k]$ is an interval set that represents local errors that arise during integration, $[r_0]$ is an interval set that represents the error at the beginning (the diameter of a set at the beginning).
We stress the fact that we are interested in rigorous numerics for dPDEs, we develop main ideas in the following sections.
Algorithm for integrating rigorously dPDEs {#sec:rigorousIntegration}
------------------------------------------
In context of dPDEs we have to solve the following infinite system of ODEs $$\label{eq:projection}
\left\{
\begin{array}{l}
\frac{d x}{d t}=P_mF(x+y),\\
\frac{d y}{d t}=Q_mF(x+y),\\
\end{array}
\right.$$ $x\in X_m$, $y\in Y_m$.
Following [@KZ], [@Z3] we will get estimates for by considering the following differential inclusion $$\frac{d x}{d t}(t)\in P_mF(x(t))+\delta,
\label{eq:delta}$$ where $\delta\subset X_m$ describes influence of $y$ onto $P_mF(x+y)$. We call $$\label{eq:galerkinProjection}
\frac{d x}{d t}=P_mF(x)$$ the $m$ dimensional Galerkin projection of , where $m>0$.
We also consider a Cauchy problem, with $a\in X_m$, $x_0\in X_m$ $$\left\{\begin{array}{l}
\frac{d x}{d t}(t)= P_mF(x(t))+a,\\
x(0)=x_0.
\end{array}\right.
\label{eq:associatedCauchyProblem}$$
####
Let $d_{X_m}$, $d_{Y_m}$ dimensions associated with $X_m$ and $Y_m$ respectively. From now on we switch to a more concrete setting, which is $$X_m:=\mathbb{R}^{d_{X_m}}\text{ and }Y_m:=\mathbb{R}^{d_{Y_m}},\ d_{X_m}<\infty,\ d_{Y_m}=\infty.$$ In this section we assume that the solutions of problems , and are defined and unique and later we will prove it.
#### *Notation*
$T, T(0), T(t_1), T([0,h])\subset\spaceYm$ are tails satisfying , in the context of tails, for notational purposes, the symbol $T(\cdot)$ is not used to denote a function of time, but an enclosure for a tail at the provided time. By
- $\overline{\varphi}^{m}\left(t, x_0,a\right)$ we denote the solution of at a time $t>0$ with $a\in\spaceXm$ and an initial condition $x_0\in\spaceXm$,
- $\varphi^{X_m}\left(t,x_0,y_0\right)$ we denote the solution of at a time $t>0$, projected onto $X_m$ with an initial condition $x_0\in\spaceXm$ and $y_0\in\spaceYm$,
- $\varphi^{m}\left([0,h], x_0, T\right)$ denotes a collection of all possible values of the solution of the inclusion $\frac{d x}{d t}\in P_mF(x+T)$ on the time interval $[0,h]$ with $T\subset\spaceYm$ and an initial condition $x_0\in\spaceXm$.
Below we present all steps of the algorithm needed to rigorously integrate . Whereas [@KZ], [@Z3] algorithm is given in an abstract setting, here we provide a detailed description of an algorithm designed for dPDEs exclusively.
####
In Algorithm \[alg:main\] we present steps needed to calculate rigorous bounds for the solutions of at $t_1=h$. The main idea is to get estimates for the solutions of each Galerkin projection of simultaneously. For the correctness proof of Algorithm \[alg:main\] we refer the reader to [@KZ] or [@Z3]. Note that Algorithm \[alg:main\] is a subcase of a general algorithm, with the set $[W_y]\subset\spaceXm$ chosen to be the Galerkin projection error.
#### Input
- a time step $h$,
- $[f_\varepsilon]\subset\spaceXm$, a constant forcing perturbation,
- $[x_0]\subset\spaceXm$, an initial finite part,
- $T(0)\subset\spaceYm$, an initial tail,\
$[x_0]\oplus T(0)\subset H$ forms self-consistent bounds for .
#### Output
- $[x_{t_1}]\subset \spaceXm$ such that $\varphi^{X_m}(t_1, [x_0], T(0))\subset[x_{t_1}]$, enclosure for the finite part of the solutions at the time $t_1$.
- $T(t_1)\subset \spaceYm$, an enclosure for the tail at the time $t_1$,\
$[x_{t_1}]\oplus T(t_1)\subset H$ forms self-consistent bounds for .
#### begin
1. find \[step:first\] $T\subset \spaceYm$ such that $T([0, h])\subset T$ and $[W_2]\subset \spaceXm$ such that $\varphi^{m}([0, h], [x_0], T)\subset [W_2]$. Enclosure for the tail on the whole time interval $[0,h]$ and the enclosure for the collection of solutions of the differential inclusion respectively. $[W_2]\oplus T$ forms self-consistent bounds for ,
2. calculate the *Galerkin projection error* $\spaceXm\supset[W_y]:=\{P_mF(x+T)-P_mF(x)\,|\,x\in[W_2]\}_I$,
3. do the selection $[W_y]\ni y_c:={\operatorname{mid}\left([W_y]\right)}$,
4. \[step:deterministic\] apply the *$C^0$ Lohner algorithm* to solve the system of autonomous ODEs with $a=y_c$. The result is a rigorous enclosure for the solution $[\overline{x_{t_1}}]\subset\spaceXm\colon\overline{\varphi}^{m}(t_1, [x_0], y_c)\subset [\overline{x_{t_1}}]$. As a mid-step the enclosure $[W_1]$ such that $\overline{\varphi}^{m}\left([0,h], [x_0], y_c\right)\subset[W_1]$ is calculated and returned. Refer to [@ZLo] for the details,
5. calculate *the perturbations vector* $\spaceXm\supset[\delta]:=\left[y_c-[W_y]+[f_\varepsilon]\right]_I$,
6. initialize the single valued vector $\spaceXm\ni C_i:=\sup{\left|[\delta_i]\right|}$,
7. compute the “Jacobian” matrix $\mathbb{R}^{d_{X_m}\times d_{X_m}}\ni J\colon
J_{ij}\geq\left\{\begin{array}{ll}\sup{\frac{\partial F_i}{\partial x_j}([W_2], y_c)}&\text{if }i=j\\\left|\sup{\frac{\partial F_i}{\partial x_j}([W_2],y_c)}\right|&\text{if }i\neq j\end{array}\right.$,
8. perform component-wise estimates in order to calculate the set $[\Delta]\subset \spaceXm$, $D:=\int_0^h{e^{J(t_1-s)}C}\,ds$, $[\Delta_i]:=[-D_i, D_i]$ for $i=1,\dots,d_{X_m}$,
9. obtain the final rigorous bound $[x_{t_1}]\subset \spaceXm$ for the solution of a differential inclusion by combining results from the previous steps $\varphi^{X_m}(t_1, [x_0], T(0))\subset[x_{t_1}]=[\overline{x_{t_1}}]+[\Delta],\quad [x_0]\subset \spaceXm,\ T(0)\subset \spaceYm$,
10. perform rearrangements into the doubleton representation,
11. \[step:second\] compute $T(t_1)\subset\spaceYm$ such that $\varphi^{Y_m}(t_1, [x_0], T(0))\subset T(t_1)$.
#### end
####
\[alg:main\]
Basing on the framework of Algorithm \[alg:main\] we have developed an algorithm which apparently has been better in tests, the improvement concerns Step \[step:first\] and Step \[step:second\] of Algorithm \[alg:main\]. As the details are very technical we do not present them here. The interested reader can find a detailed presentation in Appendix \[sec:improvement\], whereas in Appendix \[sec:pseudoCode\] we included the pseudo-codes. We omitted all the remaining steps of Algorithm \[alg:main\] that have already been described in previous works. To realize some of the elements we used the [@CAPD] package.
Algorithm for proving Theorem \[thm:main\] {#sec:cap}
==========================================
#### *Notation*
By a capital letter we denote *a single valued matrix*, e.g. $A$, by $[A]$ we denote an *interval matrix*. The inverse matrix of $A$ is denoted by $A^{-1}$, we use the symbol $[A^{-1}]$ to denote an interval matrix such that $[A^{-1}]\ni A^{-1}$. $[M]_I$ denotes an interval hull of a matrix $M$, we also use this notation in the context of vectors.
![Flow diagram presenting steps of Algorithm for proving Theorem \[thm:main\]](diagram.pdf){width="\textwidth"}
#### Input
- $m>0$, an integer, the Galerkin projection dimension,
- $[\nu_1,\nu_2]>0$, an interval of the viscosity constant values,
- $\alpha\in\mathbb{R}$, a constant value, equal to $\frac{1}{2\pi}{\int_{0}^{2\pi}{u_0(x)\,dx}}$,
- $s$, the [order of polynomial decay]{} of coefficients that is required from the constructed bounds and trapping regions, have to be an integer satisfying $s\geq 4$,
- order and the time step of the Taylor method used by the $C^0$ Lohner algorithm,
- set of $2\pi$ periodic forcing functions $f(x)$ for , defined by a finite number of modes $\{f_k\}_{0<|k|\leq m}$ and a uniform and constant perturbation $[f_\varepsilon]=[-\varepsilon, \varepsilon]\times[-\varepsilon,\varepsilon]$.
#### Output
- $\overline{x}$, an approximate fixed point for ,
- $J\approx dP_mF(\overline{x})$, an approximate Jacobian matrix at $\overline{x}$,
- $[A]$ and $[A^{-1}]$ interval matrices reducing $[dP_mF(\overline{x})]_I$ to an almost diagonal matrix $[D]$ - with dominating blocks on the diagonal,
- $[D]=\left[\left[A\right]\cdot [dP_mF(\overline{x})]_I\cdot[A^{-1}]\right]_I$, almost diagonal form of the Jacobian matrix used to estimate the eigenvalues of $dP_mF(\overline{x})$,
- $W\oplus T\subset{\overline{H}}$ and $\widetilde{W\oplus T}\subset{\overline{H}}$, trapping regions for $W\oplus T\subset\widetilde{W\oplus T}$,
- $l$, upper bound of the logarithmic norm on $\widetilde{W\oplus T}$,
- $V\oplus \mathnormal{\Theta}\subset{\overline{H}}$, an absorbing set forming self-consistent bounds for ,
- a rigorous bounds for the fixed point location,
- total time and integration steps needed to complete the proof.
#### begin
1. find an approximate fixed point location $\overline{x}$ by non-rigorous integration of $\dot{x}=P_mF(x)$. Refine the provided candidate $\overline{x}$ using *the Newton method* iterations,
2. calculate non-rigorously the Jacobian matrix, $J\approx dP_mF(\overline{x})$ (use $\nu_1$ as the viscosity constant in both steps),
3. calculate non-rigorously an approximate orthogonal matrix $S$ used for reducing $J$ to an approximate upper triangular matrix $T$ (with 1x1 and 2x2 blocks on the diagonal). Use *the QR algorithm* with multiple shifts to find $S$. Then find a rigorous inverse $[S^{-1}]\colon S^{-1}\in[S^{-1}]$ using *the Krawczyk operator* [@N],
4. calculate *the eigenvectors* of $T$ to form a block upper triangular matrix $E$ that is used to further reduce $T$ to an almost diagonal matrix, then calculate a rigorous inverse matrix $[E^{-1}]\colon E^{-1}\in[E^{-1}]$ using the Krawczyk operator again,
5. calculate $[A]:=[E\cdot S]_I$, $[A^{-1}]:=[[S^{-1}]\cdot[E^{-1}]]_I$ and $[D]:=\left[\left[A\right]\cdot [dP_mF(\overline{x})]_I\cdot[A^{-1}]\right]_I$ , where $[D]$ is in an almost diagonal form, having blocks on the diagonal and negligible intervals as non-diagonal elements, suitable form to estimate the eigenvalues,
6. \[step:trappingRegion\] find $W\oplus T\subset{\overline{H}}$ a trapping region in block coordinates that encloses $\overline{x}$. This step requires $[A]$ and $[A^{-1}]$, the change of coordinates matrices calculated in the previous step. A detailed description of an algorithm performing this task is provided by [@ZAKS],
7. calculate $l$ an upper bound for the logarithmic norm on the set $[[A^{-1}]\cdot W]_I\oplus T$, for the details refer to [@ZAKS]. In case $l<0$ by Theorem \[thm:8\] claim that there exists a locally attracting fixed point. Observe that in this case $W\oplus T$ is the basin of attraction of the fixed point found. One may be tempted to use the “analytical” trapping region, calculated in Section \[sec:analyticalTR\] for that purpose, but this is an unfeasible goal in general as an analytical trapping region may simply be too large to include it into the calculation process,
8. enlarge $W\oplus T$ and return the largest calculated self-consistent bounds $\widetilde{W\oplus T}\subset{\overline{H}}$ such that $\widetilde{W\oplus T}$ is a trapping region, $l<0$ and $W\oplus T\subset\widetilde{W\oplus T}$. By Theorem \[thm:8\] claim that the basin of attraction of the fixed point found is $\widetilde{W\oplus T}$,
9. using the procedure from Section \[sec:scb\] calculate the absorbing set $V\oplus \mathnormal{\Theta}\subset\overline{H}$,
10. integrate $V\oplus \mathnormal{\Theta}$ rigorously forward in time until $\varphi\left(t, \left[\left[A\right]\cdot V\right]_I\oplus\mathnormal{\Theta}\right)\subset\widetilde{W\oplus T}$ . If this step finishes successfully conclude that $\widetilde{W\oplus T}$ after a finite time contains any solution of the problem with sufficiently smooth initial data and claim the existence of a globally attracting fixed point,
11. translate $[[A^{-1}]\cdot W]_I\oplus T$ into the doubleton representation and integrate it forward in time in order to estimate the fixed point location with a relatively high accuracy.
#### end
All the trapping regions constructed in the main algorithm presented above are expressed in block coordinates. Where the block decomposition of $H$ is given by $H=\oplus_{(i)}H_{(i)}$, where for $(i)>m$ blocks are given by $H_{(i)}=<e_i>$, and $(i)=i$ in this case. Whereas for $(i)\leq m$ each block $H_{(i)}$ is a two-dimensional eigenspace of $J$. In case of two dimensional blocks $(i)=(i_1,i_2)\in\mathbb{Z}^2$, the expression $(i)<m$ means that $i_j<m$ for $j=1,2$.
Therefore given a trapping region $W\oplus T\subset H$ the finite part $W$ has the following form $$W=\prod_{(i)}\left\{\begin{array}{ll}
\overline{B}\left(0, r_i\right)&\text{, for }(i)\in\mathcal{I},\\
{[a_i^-,\ a_i^+]}&\text{, for }(i)\notin\mathcal{I},
\end{array}\right.$$ where $\mathcal{I}=\left\{(i)\colon H_{(i)}\text{ is two dimensional eigenspace of }J\right\}$.
\[rem:statement\] In all the proofs presented in Table \[table\] from Section \[sec:exampleTheorem\] we have $$\mathcal{I}=\left\{\begin{array}{ll}
\emptyset&\text{ when $\int_0^{2\pi}{u_0(x)\,dx}=0$},\\
\{(i)\colon(i)\leq m\}&\text{ when $\int_0^{2\pi}{u_0(x)\,dx}\neq 0$}.
\end{array}\right.$$ We have not been able to prove this rigorously.
Conclusion
==========
A method of proving the existence of globally attracting fixed points for a class of dissipative PDEs has been presented. A detailed case study of the viscous Burgers equation with a constant in time forcing function has been provided. All the computer program sources used are available online [@Package]. There are several paths for the future development of the presented method we would like to suggest. An option is, for instance, to apply a technique for splitting of sets in order to see what the largest domain approachable by this technique is. One may also consider working on proving the statement given in Remark \[rem:statement\]. Another very interesting possibility is application of the presented method to higher dimensional PDEs, such as the Navier-Stokes equation, and we will address this topic in our forthcoming papers.
[00]{} Program package, http://ww2.ii.uj.edu.pl/`‘~`cyranka/Burgers. Supplementary material with detailed computations, http://ww2.ii.uj.edu.pl/`‘~`cyranka/Burgers. G. Arioli, H. Koch, *Integration of Dissipative Partial Differential Equations: A Case Study*, SIAM J. Appl. Dyn. Syst., vol. 9(2010), 1119-1133. CAPD - Computer Assisted Proofs in Dynamics, a package for rigorous numeric, http://capd.ii.uj.edu.pl. J. Cyranka, P. Zgliczyński, *Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing - a computer assisted proof*, in preparation. S. Day, Y. Hiraoka, K. Mischaikow, T. Ogawa, *Rigorous numerics for global dynamics: a study of Swift-Hohenberg equation*, SIAM J. Appl. Dyn. Syst. 4, 1-31 (2005). W. E and Y. Sinai, *New results in mathematical and statistical hydrodynamics*, Uspekhi Mat. Nauk 55 (2000), vol. 4(334), 25-58. O. Fogelklou, W. Tucker, G. Kreiss, M. Siklosi *A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition*, Commun. Nonlinear Sci., vol. 16(2011), 1227-1243. H.R. Jauslin, H.O. Kreiss, J. Moser, *On the Forced Burgers Equation with Periodic Boundary Condition*, Proceedings of Symposia in Pure Mathematics, vol. 65(1999), 133-153. T. Kinoshita, T. Kimura, M. T. Nakao, *On the a posteriori estimates for inverse operators of linear parabolic equations with applications to the numerical enclosure of solutions for nonlinear problems*, Numerische Mathematik, DOI 10.1007/s00211-013-0575-z. T. Kapela, P. Zgliczyński, *The existence of simple choreographies for the N-body problem - a computer assisted proof*, Nonlinearity, vol. 16(2003), 1899-1918. T. Kapela, P. Zgliczyński, *A Lohner-type algorithm for control systems and ordinary differential inclusions*, Discrete Cont. Dyn. Sys. B, vol. 11(2009), 365-385. R.J. Lohner, *Einschliessung der Losung gewonhnlicher Anfangs–and Randwertaufgaben und Anwendungen*, Universitat Karlsruhe (TH), 1988. R.J. Lohner, *Computation of Guaranteed Enclosures for the Solutions of Ordinary Initial and Boundary Value Problems*, Computational Ordinary Differential Equations, J.R. Cash, I. Gladwell Eds., Clarendon Press, Oxford, 1992. S. Maier-Paape, K. Mischaikow, T. Wanner, *Structure of the attractor of the Cahn-Hillard equation on a square*, International Journal of Bifurcation and Chaos, Vol. 17(2007), No. 4 1221-1263. J. Mattingly, Y. Sinai, *An Elementary Proof of the Existence and Uniqueness Theorem for Navier-Stokes Equations*, Comm. in Contemporary Mathematics, vol. 1(1999), 497-516. A. Neumeier, *Interval methods for system of equations*. Cambridge University Press, 1990. J. Vukadinovic, *Global dissipativity and inertial manifolds for diffusive Burgers equations with low-wavenumber instability*, Discrete Cont. Dyn. Sys., vol. 29(2011), 327-341. G. B. Whitham, *Linear and Nonlinear Waves*. John Wiley & Sons, 1975. P. Zgliczyński, *Attracting fixed points for the Kuramoto-Sivashinsky equation - a computer assisted proof*, SIAM Journal on Applied Dynamical Systems, vol. 1(2002), 215-288. P. Zgliczyński, *Rigorous numerics for dissipative Partial Differential Equations II. Periodic orbit for the Kuramoto-Sivashinsky PDE - a computer assisted proof*, Foundations of Computational Mathematics, vol. 4(2004), 157-185. P. Zgliczyński, *Rigorous Numerics for Dissipative PDEs III. An effective algorithm for rigorous integration of dissipative PDEs*, Topological Methods in Nonlinear Analysis, vol. 36(2010), 197-262. P. Zgliczyński, *$C^1$-Lohner algorithm*, Foundations of Computational Mathematics, vol. 2(2002), 429-465. P. Zgliczyński and K. Mischaikow, *Rigorous Numerics for Partial Differential Equations: the Kuramoto-Sivashinsky equation*, Foundations of Computational Mathematics, vol. 1(2001), 255-288. P. Zgliczyński, *Trapping regions and an ODE-type proof of an existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane*, Univ. Iag. Acta Math., vol. 41(2003), 89-113.
Data from the example proof {#sec:proofData}
===========================
The parameters were as follows $\nu\in\paperExampleNu$ (the whole interval was inserted), $\widehat{m}=3$, $a_0=0.5$. To present the following data all the numbers were truncated, for more precise data we refer the reader to the package with data from proofs available [@Package].
The change of coordinates $$\mbox{\scriptsize
${\operatorname{mid}\left([A]\right)}=
\left[\begin{array}{@{\,}c @{\,} c @{\,} c @{\,} c @{\,} c @{\,} c@{\,}}-0.998 & 0.0623 & -8.86\cdot 10^{-3} & -2.28\cdot 10^{-3} & -7.28\cdot 10^{-4} & -6.15\cdot 10^{-3}\\
0.0509 & 0.816 & 1.39\cdot 10^{-3} & -7.02\cdot 10^{-3} & -5.02\cdot 10^{-3} & 6.59\cdot 10^{-4}\\
6.35\cdot 10^{-3} & 0.0175 & 1.83\cdot 10^{-4} & 8.79\cdot 10^{-4} & -0.863 & 0.505\\
0.0175 & -6.12\cdot 10^{-3} & 6.8\cdot 10^{-4} & -1.86\cdot 10^{-4} & 0.505 & 0.863\\
0.0114 & 0.0153 & -0.288 & 0.957 & -6.9\cdot 10^{-4} & 4.41\cdot 10^{-5}\\
-0.0139 & 9.46\cdot 10^{-3} & 0.957 & 0.288 & -1.21\cdot 10^{-4} & 3.88\cdot 10^{-4}\\
\end{array}\right].
$}$$
The Jacobian matrix in almost diagonal form $$\mbox{\scriptsize
$\begin{split}
&{\operatorname{mid}\left(\left[[A]\cdot [dF(\overline{x})]_I\cdot[A^{-1}]\right]_I\right)}\\
=&
\left[\begin{array}{@{\,}c @{\,} c @{\,} c @{\,} c @{\,} c @{\,} c@{\,}}-2.06 & 0.402 & 0.0558 & -1.98\cdot 10^{-3} & 0.0123 & 0.0985\\
-0.598 & -2.04 & -7.17\cdot 10^{-3} & -0.0545 & -0.0985 & 0.0123\\
0.109 & -3.97\cdot 10^{-3} & -8.2 & 1 & 1.33\cdot 10^{-3} & 5.19\cdot 10^{-3}\\
-0.0143 & -0.112 & -1 & -8.2 & -5.19\cdot 10^{-3} & 1.33\cdot 10^{-3}\\
-0.0369 & 0.295 & -2\cdot 10^{-3} & 7.78\cdot 10^{-3} & -18.4 & 1.5\\
-0.295 & -0.0369 & -7.78\cdot 10^{-3} & -2\cdot 10^{-3} & -1.5 & -18.4\\
\end{array}\right].
\end{split}$}$$
Note that the matrix $\left[[A]\cdot [dF(\overline{x})]_I\cdot[A^{-1}]\right]_I$ does not have negligible elements beyond the diagonal blocks. This is because we have performed the calculations for all the values $\nu\in\paperExampleNu$ simultaneously. If we perform the same calculations for one particular value of $\nu$ we would get a thin matrix with intervals of diameter $\thicksim10^{-15}$.
The approximate eigenvalues $\operatorname{spect}(J)\approx$ $$\mbox{\scriptsize
$ \begin{split}
\approx\left(-2.00088+0.489685i, -2.00088-0.489685i, -17.9982+1.50012i, -17.9982-1.50012i, -8.00096+0.999759i, -8.00096-0.999759i\right).
\end{split}
$}$$ The logarithmic norm upper bound $l = \paperExampleL$.\
The trapping region expressed in canonical coordinates $[[A^{-1}]\cdot \widetilde{W]_I\oplus T}=$ $$\mbox{\scriptsize
$
\begin{array}{|c|c|c|}\hline\mathbf{k} & \mathbf{{\operatorname{Re}\left(a_k\right)}} & \mathbf{{\operatorname{Im}\left(a_k\right)}}\\\hline\hline
1 & 2.59365\cdot 10^{-3}+[-0.144158,0.144158] & -6.66462\cdot 10^{-4}+[-0.171969,0.171969]\\
2 & 0.0984977+[-9.55661,9.55661]10^{-2} & -0.0123068+[-9.64073,9.64073]10^{-2}\\
3 & 4.57814\cdot 10^{-3}+[-5.7441,5.7441]10^{-2} & 0.0551328+[-5.7053,5.7053]10^{-2}\\
4 & -2.88994\cdot 10^{-4}+[-5.90827,5.90827]10^{-3} & -1.14901\cdot 10^{-3}+[-5.57297,5.57297]10^{-3}\\
5 & 1.01516\cdot 10^{-3}+[-1.99265,1.99265]10^{-3} & -2.33225\cdot 10^{-4}+[-2.46885,2.46885]10^{-3}\\
6 & 2.22928\cdot 10^{-5}+[-7.83055,7.83055]10^{-4} & 2.53646\cdot 10^{-4}+[-6.63366,6.63366]10^{-4}\\
7 & -1.08421\cdot 10^{-5}+[-1.60583,1.60583]10^{-4} & -2.06754\cdot 10^{-5}+[-1.48048,1.48048]10^{-4}\\
8 & 9.3526\cdot 10^{-6}+[-5.09454,5.09454]10^{-5} & -3.26429\cdot 10^{-6}+[-5.69385,5.69385]10^{-5}\\
\geq 9 & \multicolumn{2}{|c|}{|a_k|\leq0.970056/k^{4}}\\\hline\end{array}
$}$$ The absorbing set $V\oplus \mathnormal{\Theta}=$ $$\mbox{\scriptsize
$
\begin{array}{|c|c|c|}\hline\mathbf{k} & \mathbf{{\operatorname{Re}\left(a_k\right)}} & \mathbf{{\operatorname{Im}\left(a_k\right)}}\\\hline\hline
1 & 4.96368\cdot 10^{-3}+[-0.142913,0.142913] & -2.33252\cdot 10^{-3}+[-0.144686,0.144686]\\
2 & 0.0971252+[-5.3667,5.3667]10^{-2} & -0.0125347+[-5.2554,5.2554]10^{-2}\\
3 & 4.25602\cdot 10^{-3}+[-2.65075,2.65075]10^{-2} & 0.0542654+[-2.71581,2.71581]10^{-2}\\
4 & -2.69437\cdot 10^{-4}+[-1.41625,1.41625]10^{-2} & -1.31697\cdot 10^{-3}+[-1.35799,1.35799]10^{-2}\\
5 & 1.06171\cdot 10^{-3}+[-7.40709,7.40709]10^{-3} & -2.2071\cdot 10^{-4}+[-7.931,7.931]10^{-3}\\
6 & -4.23977\cdot 10^{-6}+[-5.02386,5.02386]10^{-3} & 2.65357\cdot 10^{-4}+[-4.90303,4.90303]10^{-3}\\
7 & -2.23332\cdot 10^{-5}+[-3.434,3.434]10^{-3} & -3.58771\cdot 10^{-5}+[-3.42765,3.42765]10^{-3}\\
8 & 1.33681\cdot 10^{-5}+[-2.49967,2.49967]10^{-3} & -7.06016\cdot 10^{-6}+[-2.50257,2.50257]10^{-3}\\
\geq 9 & \multicolumn{2}{|c|}{|a_k|\leq147.297/k^{4}}\\\hline\end{array}
$}$$ The absorbing set is apparently larger than the trapping region, it has been necessary to integrate it rigorously forward in time. The Taylor method used in the $C^0$ Lohner algorithm was of order $6$ with time step $0.005$. Total execution time was $\paperExampleExecutionTime$ seconds, total number of integration steps needed to verify that $\varphi\left(V\oplus \mathnormal{\Theta}\right)\subset \widetilde{W\oplus T}$ (having in mind that the sets are expressed in different coordinates) was $\paperExampleNrOfSteps$, therefore $\widehat{t}=\paperExampleTotalTime$.
Improvement of Algorithm \[alg:main\] {#sec:improvement}
=====================================
### Step \[step:first\] of Algorithm \[alg:main\]. The main loop. {#sec:mainLoop}
\[def:polynomialBound\] Let $W\subset H$, $W$ convex. We call $W$ [the polynomial bound]{} if there exist numbers $M>0$, $C>0$, $s\geq 0$ such that $$\max_{x\in W_k}{||x||}\leq\frac{C}{|k|^s},\ |k|>M.$$ To denote the polynomial bound we use the quadruple $(W, M, C, s)$.
Basically, during step \[step:first\] of Algorithm \[alg:main\] we have to calculate $T\subset Y_m$ a good enclosure for the tail during the whole time interval $[0,h]$, i.e. $T$ has to satisfy $T([0,h])\subset T$. Apparently, the bounds for $T([0,h])$ can be calculated explicitly, due to the following monotonicity of the bounds formula $$\label{eq:monotonicityOfBounds}
T([0,h])_k\subset T(0)_k\cup g_k,\quad k\in\mathbb{Z}\setminus\{0\},$$ where $g_k$ is the linear approximation of the solution defined in Definition \[def:nonlinearTerms\], see [@Z3 Lemma 6.1]. $T(0)$ in the formula is known as it is the initial condition, and the polynomial bounds enclosing $g$ can be calculated in a finite number of steps. We describe an appropriate procedure in the following part.
\[def:nonlinearTerms\] Let $W\oplus T\subset H$ forms a self-consistent bounds for , $m>0$ be the Galerkin projection dimension, $N_k$, $f_k$ and $\lambda_k$ appear on the right-hand side of , $f_0=0$, $f_k=0$ for $|k|>m$. For $k\in\mathbb{Z}\setminus\{0\}$ and $i=1,\dots,d_1$ we define
$$\begin{gathered}
N_{k,i}^{\pm}\colon N_{k,i}^{-}\leq N_{k,i}(W\oplus T)\leq N_{k,i}^{+},\label{eq:boundsNk}\\
b_{k,i}^{\pm}:=\frac{N_{k,i}^{\pm}+f_{k,i}}{-\lambda_k},\label{eq:boundsbk}\\
g_{k,i}^{\pm}:=\left(T(0)_{k,i}^\pm-b_{k,i}^\pm\right)e^{\lambda_k h}+b_{k,i}^\pm\label{eq:boundsgk},\\
N_k:=\Pi_{i=1}^{d_1}{[N_{k,i}^-,N_{k,i}^+]},\quad b_k:=\Pi_{i=1}^{d_1}{[b_{k,i}^-,b_{k,i}^+]},\quad g_k:=\Pi_{i=1}^{d_1}{[g_{k,i}^-,g_{k,i}^+]}\nonumber.
\end{gathered}$$
Now the question is how to verify the relations in a finite number of steps. In general, it is impossible. Apparently, in the setting studied here, when sets are represented by polynomial bounds defined in Definition \[def:polynomialBound\] the relations can be verified in a finite number of steps. Observe that the self-consistent bounds introduced in Section \[sec:generalMethod\] are in particular polynomial bounds.
We present procedures dealing with ${T_\mathcal{N}}$ and ${T_\mathcal{F}}$ in Algorithm \[alg:validateNearTail\] and Algorithm \[alg:validateFarTail\], to be found in Appendix \[sec:pseudoCode\], separately for better clarification. For the exact meaning of the symbols refer to Definition \[def:tail\]. The crucial part in Step \[step:first\] of Algorithm \[alg:main\] is to verify if ${T_\mathcal{F}}([0,h])\subset{T_\mathcal{F}}$ in a finite number of steps, where ${T_\mathcal{F}}$ is a candidate for the far tail.
Now, let us present the procedure. Our goal is to enclose the interval sets $g_k$ by a uniform polynomial bound. Once we have a uniform polynomial bound, denoted by $g$, the verification of ${T_\mathcal{F}}([0,h])\subset{T_\mathcal{F}}$ is straightforward, because of the property . Firstly, given a polynomial bound $$\label{eq:polynomialBoundT}
(W\oplus T, {M_T}, {C_T}, {s_T})$$ a polynomial bound $$(N, {M_T}, {C_N}, {s_N})\text{ such that }\Pi_{k\in\mathbb{Z}}{N_k}\subset N$$ is found. This task requires performing some tedious estimates and we do not present them here. We derived the required estimates for a class of dPDEs including the viscous Burgers equation in [@SuppMat]. Generally, a polynomial bound satisfying ${s_N}={s_T}-r$ is found. Then we immediately obtain a polynomial bound $$\label{eq:polyBdBk}
(b, {M_T}, {C_b}, {s_b})\text{ such that }\Pi_{k\in\mathbb{Z}}{b_k}\subset b,$$ with ${C_b}=\frac{{C_N}}{V(M)}$, $V(M)=\inf{\left\{\nu(|k|)\colon|k|>M\right\}}$ and ${s_b}={s_N}+p$. Finally, a polynomial bound $$\label{eq:polyBdGk}
(g, {M_T}, {C_g}, {s_g})\text{ such that }\Pi_{k\in\mathbb{Z}}{g_k}\subset g$$ is obtained using the formulas as follows
If $|k|>M$ then \[lem:formulasG\] $$\label{eq:Gformula1}
|g_k|\leq\frac{{C_{T(0)}}e^{\lambda_{k}h}\cdot|k|^{{s_b}-{s_{T(0)}}}-{C_b}(e^{\lambda_{k}h}-1)}{|k|^{{s_b}}}$$ and $$\label{eq:Gformula2}
|g_k|\leq\frac{{C_{T(0)}}\cdot e^{\lambda_{k_{max}}h}(k_{max})^{{s_b}-{s_{T(0)}}}+{C_b}}{|k|^{{s_b}}}=\colon\frac{{C_g}}{|k|^{{s_g}}}$$ where $k_{max}$ is the $k$ for which function $e^{\lambda_kh}\cdot k^{r}$ attains its maximum.
#### *Proof*
Maximum of $f(k)=e^{\lambda_k h}k^r$, with ${\mbox{dom}\,{f}}=\{k\colon |k|>M\}$ is reached at $k_{max}$, therefore is estimated by for any $|k|\geq M$.
####
Note that $$\label{eq:powers}
{s_g}>{s_T}$$ because ${s_N}={s_T}-r$, ${s_g}={s_b}={s_T}-r+p$ and $p>r$.
#### The main loop
#### Input
$\left([x_0]\oplus T(0), {M_{T(0)}}, {C_{T(0)}}, {s_{T(0)}}\right)$ a polynomial bound, $[x_0]\oplus T(0)\subset H$ forms self-consistent bounds for .
#### Output
$\left([W_2]\oplus T, {M_T}, {C_T}, {s_T}\right)$ a polynomial bound such that $T([0,h])\subset T$, $\varphi^m\left([0,h],[x_0],T\right)\subset [W_2]$ and $[W_2]\oplus T\subset H$ forms self-consistent bounds for .
#### begin
1. Initialize $T:=T(0)$.
2. Update ${T_\mathcal{F}}$ using function.
3. *validated*
- $[W_2]:=$ ($[x_0]$, $T$), calculate a rough-enclosure $[W_2]$ for the differential inclusion using a current candidate for a tail enclosure $T$, after this step $\varphi^m\left([0,h],[x_0],T\right)\subset [W_2]$ holds,
- calculate the polynomial bounds $(b, {M_T}, {C_b}, {s_b})$ and $(g, {M_T}, {C_g}, {s_g})$,
- *validated* $:=$ ($T(0)$, $T$, $b$, $g$, $[W_2]$) (if $T$ was changed during this step *validated*=).
#### end
In our algorithm the number ${M_T}$ in is chosen adaptively in and changes from step to step.
####
is *the rough enclosure algorithm based on isolation*, designed for dPDEs presented in [@Z3].
We present a correctness proof of the and functions in the comments within the code listings from Appendix \[sec:pseudoCode\]. By a correctness proof we show that a polynomial bound $T$, such that the condition $T([0,h])_k\subset T_k$ holds for all $k\in\mathbb{Z}$, is returned by the algorithm whenever the algorithm stops.
Now, we shall focus on explaining the main idea behind and explain why we consider it an improvement of the existing algorithm. Basically, when a $-\lambda_k$ in is small, the nonlinear part $N_k$ dominates the linear term. However, there exists an index $\tilde{k}\in\mathbb{N}$ such that $-\lambda_k$ for $|k|>\tilde{k}$ becomes large enough to make the linear part overtake the nonlinear part. The position of the threshold $\tilde{k}$ depends on the maximum order of the “Laplacian” that appears in the linear part $L$ of , as well as on the order of the polynomial that appears in the nonlinear part. We remark that the solution of the $m$-th Galerkin projection of with $m<\tilde{k}$ greatly differs from the solution of the whole system .
The aforementioned effects show that a proper choice of the Galerkin projection dimension $m$ (in our algorithm taken only once at the beginning) and the number ${M_T}$ of the polynomial bound (in our algorithm taken at each time step) is of critical importance and has to be performed carefully. The application of a too small value may result in blow-ups and may prevent the completion of the calculations. In the original algorithm from [@Z3] the number ${M_T}$ was fixed in advance. Then heuristic formulas were derived for the KS equation in order to predict if the tail validation function would finish successfully for a given ${M_T}$, ${s_T}$ and to guess the initial values of ${C_T}$ and ${s_T}$ in , see [@Z3 Section 8]. We found the original approach insufficient for the purpose of rigorously integrating PDEs that are the subject of our research (for example the Burgers or the Navier-Stokes equations). A similar approach for the mentioned dPDEs is problematic and, especially in the case of lower viscosities, heuristic formulas cause performance issues and sometimes offer infeasible values, mainly due to the lower order of the “Laplacian” in the linear part.
### Step \[step:second\] of Algorithm \[alg:main\]
#### Input
$\left([W_2]\oplus T, {M_T}, {C_T}, {s_T}\right)$, a polynomial bound such that $T([0,h])\subset T$, $\varphi^m\left([0,h],[x_0],T\right)\subset [W_2]$ and $[W_2]\oplus T\subset H$ forms self-consistent bounds for .
#### Output
$\left(T(h), {M_{T(h)}}, {C_{T(h)}}, {s_{T(h)}}\right)$, a polynomial bound.
#### begin
1. ${M_{T(h)}}:={M_T}$, $T(h)$ inherits $M$ from the enclosure $T$,
2. calculate the polynomial bound $(g, {M_T}, {C_g}, {s_g})$,
3. $T(h):=g$, $C_{T(h)}:=C_g$, $s_{T(h)}:=s_g$.
#### end
Validate tail function in pseudo-code {#sec:pseudoCode}
=====================================
Here we present a pseudo-code of the functions and used in Section \[sec:mainLoop\]. First, we present the internal representation of sets that was used in actual program, written in C++ programming language and available at [@Package].
#### *Data representation*
- *double* is a floating point number of double precision in *C++ programming language*,
- *interval* is $[a^-, a^+]\subset\mathbb{R}$ where $a^-$, $a^+$ are *double* numbers. All arithmetic operations on such intervals are rigorous and are performed using implementation of the CAPD library [@CAPD]. It is verified that the interval arithmetic provides proper in mathematical sense results [@N],
- *Vector* represents an interval set, a vector composed of *intervals*,
- *PolyBd* is a structure used for representing a polynomial bound $(W, M, C, s)$. A given *PolyBd* $V$ contains a *Vector* representing the finite part of $W\subset H$, an *integer* representing $M$ denoted by $M(V)$ and two *intervals* representing $C$ and $s$ denoted by $C(V)$ and $s(V)$ respectively.
Below, in Algorithm \[alg:validateNearTail\] and Algorithm \[alg:validateFarTail\], we present functions and respectively along with correlated functions. Wherever $previous$ keyword appear the value from the previous step is used.\
$L:=\left(\frac{C(T)}{C(g)}\right)^{s(T)-s(g)}$ $L$
$PolyBd$ $g:=g(T(0), T, W_2)$ $currentM:=$ ($T$, $g$) $potentialM:=$ ($T$, $g$ with decreased $s$)
$vector<bool> inflatesRe$ $vector<bool> inflatesIm$
$L:=\left(\frac{C(T)}{C(g)}\right)^{\frac{1}{s(T)-s(g)}}$; $L_2:=\left\lceil\left(\frac{C(b)}{C(T(0))}\right)^{\frac{1}{s(b)-s(T0)}} \right\rceil$
#### Case 1
$s(b)>s(T(0))$\
()[$T(0)_{M+1}\subset b_{M+1}$]{}[ ]{}()[ ]{}
#### Case 2
$s(b)=s(T(0))$\
()[ ]{}
#### Case 3
$s(b)<s(T(0))$\
()[ ]{}
[^1]: Research has been supported by National Science Centre grant DEC-2011/01/N/ST6/00995.
|
---
abstract: 'Stochastic optimization (SO) considers the problem of optimizing an objective function in the presence of noise. Most of the solution techniques in SO estimate gradients from the noise corrupted observations of the objective and adjust parameters of the objective along the direction of the estimated gradients to obtain locally optimal solutions. Two prominent algorithms in SO namely Random Direction Kiefer-Wolfowitz (RDKW) and Simultaneous Perturbation Stochastic Approximation (SPSA) obtain noisy gradient estimate by randomly perturbing all the parameters simultaneously. This forces the search direction to be random in these algorithms and causes them to suffer additional noise on top of the noise incurred from the samples of the objective. Owing to this additional noise, the idea of using deterministic perturbations instead of random perturbations for gradient estimation has also been studied. Two specific constructions of the deterministic perturbation sequence using lexicographical ordering and Hadamard matrices have been explored and encouraging results have been reported in the literature. In this paper, we characterize the class of deterministic perturbation sequences that can be utilized in the RDKW algorithm. This class expands the set of known deterministic perturbation sequences available in the literature. Using our characterization, we propose construction of a deterministic perturbation sequence that has the least cycle length among all deterministic perturbations. Through simulations we illustrate the performance gain of the proposed deterministic perturbation sequence in the RDKW algorithm over the Hadamard and the random perturbation counterparts. We also establish the convergence of the RDKW algorithm for the generalized class of deterministic perturbations.'
author:
- 'Chandramouli K.$^1$, Prabuchandran K.J.$^1$$^2$, D. Sai Koti Reddy$^3$, and Shalabh Bhatnagar$^1$$^4$ [^1] [^2] [^3][^4]'
bibliography:
- 'reference.bib'
title: '**Generalized Deterministic Perturbations For Stochastic Gradient Search** '
---
Introduction
============
Stochastic optimization (SO) problems frequently arise in engineering disciplines such as transportation systems, machine learning, service systems, manufacturing etc. Practical limitations, lack of model information and the large dimensionality of these problems prohibit analytic solutions to these problems. Simulation is often employed to evaluate the performance of the current parameters of the system. Simulating and evaluating the system’s performance is generally expensive and one is typically constrained by a simulation budget. In such scenarios, owing to the simulation budget one aims to drive the system to optimal parameter settings using as few simulations as possible.
Under the SO framework, we have a system that gives noise-corrupted feedback of the performance for the currently set parameters, i.e., given the system parameter vector $\theta$, the feedback that is available is the noisy evaluation $h(\theta, \xi)$ of the performance $J(\theta)=\E_{\xi}[h(\theta, \xi)]$ where $\xi$ is the noise term inherent in the system and $J(\theta)$ denotes the expected performance of the system for the parameter $\theta$. The pictorial description of such a system is shown in Figure \[fig:so\]. The objective in the SO problem then is to determine a parameter $\theta^*$ that gives the optimal expected performance of the system, i.e., $$\begin{aligned}
\theta^* = \arg\min_{\theta \in \R^p} J(\theta). \label{eq:pb}\end{aligned}$$
= \[draw, fill=white, rectangle, minimum height=3em, minimum width=6em\] = \[draw, fill=white, circle, node distance=1cm\] = \[coordinate\] = \[coordinate\] = \[pin edge=[to-,thin,black]{}\]
Analogous to solutions for deterministic optimization problems where the explicit analytic gradient of the objective function is used to adjust the parameters along the negative gradient directions, many of the solution approaches in SO mimic the familiar gradient descent algorithm. However, unlike the deterministic setting, the SO setting only has access to noise corrupted samples of the objective. Thus, in the SO setting, one essentially aims at estimating the gradient of the objective function using noisy cost samples. In the pioneering work by Kiefer and Wolfowitz [@kiefer1952], the gradient is estimated by approximating each of the partial derivatives using either a two-sided or a one-sided finite difference approximation (FDSA) algorithm. This algorithm requires $2p$ objective function evaluations (or simulations) per iteration for the two-sided gradient approximation scheme and $p+1$ simulations per iteration for the one-sided scheme (for a $p$-dimensional parameter problem, see [@bhatnagar-book]). As the number of simulations per iteration required for gradient estimation scales linearly with the dimension of the problem, FDSA algorithm is expensive to deploy under high-dimensional parameter settings.
In [@kushcla], Random Direction Kiefer-Wolfowitz (RDKW) algorithm that uses only two simulations per iteration for obtaining gradient estimates has been proposed. In the RDKW algorithm, all the parameters are randomly perturbed simultaneously using two parallel simulations and function evaluations at those perturbed parameters are used to obtain the gradient estimate. In the RDKW algorithm, the random perturbation vector as well as the random direction vector involved in estimating the gradient have been kept the same. For the choice of random direction (or perturbation), various distributions like spherical uniform distribution [@kushcla], uniform distribution [@ermol1969method], normal and Cauchy distribution [@styblinski1990experiments], asymmetric Bernoulli [@prashanth2017adaptive] have been explored. The number of simulations required for estimating the gradients in the RDKW algorithm is significantly less compared to the FDSA algorithm and the algorithm is seen to perform empirically better than FDSA.
In a seminal work [@spall], the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm that uses two simulations similar to RDKW has been proposed. Unlike the RDKW algorithm, SPSA employs different choices for parameter perturbations and the random direction of movement, in particular, the random perturbation direction and the random direction of movement have been chosen to be inverses of each other. In [@spall], symmetric Bernoulli distribution has been shown to be the best choice for random perturbations among all the distributions and the proposed SPSA scheme has been proven to perform asymptotically better compared to FDSA. In [@chin1997comparative], a comprehensive comparative study of the stochastic optimization algorithms namely FDSA, RDKW and SPSA has been provided. Further, under a general third order cross derivative assumption on the loss function, RDKW with symmetric Bernoulli distribution has been shown to be the best choice for random directions. In [@theiler2006choice], an example of a loss function that does not satisfy the third order cross derivative condition in [@chin1997comparative] has been constructed. For such a loss function, it has been shown that the optimal distribution choice for random directions need not be symmetric Bernoulli.
In [@kushcla] and [@spall1997one], to further reduce simulation cost per iteration, extensions of the RDKW and SPSA algorithms that estimate the gradient with only one simulation or measurement of the objective have been considered. However, it is observed that the one-simulation gradient estimate has higher bias compared to the two-simulation gradient estimate. In [@sandilya1997deterministic] and [@wang1998deterministic], deterministic conditions for the perturbation and noise sequences required to obtain almost sure convergence of the iterates have been discussed. In [@bhatfumarcwang], to enhance the performance of one-sided SPSA scheme, deterministic perturbations based on lexicographical ordering and Hadamard matrices have been proposed. Further, the numerical results in [@bhatfumarcwang], illustrate the benefit of Hadamard matrix based perturbation sequences as it has been shown to improve the performance of SPSA empirically for the case of one sided measurements. In [@xiong2002randomized], a unified view of both RDKW and SPSA is presented and a binary deterministic perturbation sequence using orthogonal arrays [@hedayat1999orthogonal] for obtaining gradient estimate in both of the algorithms has been discussed.
In this paper, we generalize the class of deterministic perturbation sequences that can be utilized in the RDKW algorithm. Based on this characterization, we provide a construction of a deterministic perturbation sequence using a specially chosen circulant matrix. We empirically study the performance of the constructed sequence against the afore mentioned Hadamard matrix based deterministic perturbations and the randomized perturbations. We expect with our generalization the study of rate of convergence for the RDKW algorithm based on deterministic perturbation sequences would be possible. We now summarize our contributions:
- We generalize the class of deterministic perturbation sequences that can be applied in the RDKW algorithm.
- We provide a special construction of deterministic perturbation sequence with smaller cycle length compared to Hadamard perturbation sequence.
- We illustrate the performance gain of the proposed deterministic perturbations over the Hadamard matrix based perturbations as well as random perturbations.
- We prove the convergence of the RDKW algorithm for the class of deterministic perturbations.
Conditions on Deterministic Perturbations
=========================================
In this section, we describe the classical RDKW algorithm and motivate the necessary conditions that a deterministic perturbation sequence should satisfy for almost sure convergence of the iterates in the deterministic perturbation version of RDKW algorithm.
The standard RDKW algorithm iteratively updates the parameter vector along the direction of the negative estimated gradient, i.e., $$\begin{aligned}
\label{eq:grad-descent}
\theta_{n+1} = \theta_n - a_n \widehat{\nabla J}(\theta_n), \end{aligned}$$ where $a_n$ is the step-size that satisfies standard stochastic approximation conditions (see Assumption **A2** in section \[sec:convergenceresults\]) and $\widehat{\nabla J}$ is the estimate of the gradient of the objective function $J$ at the current parameter.
In the case of two-simulation RDKW algorithm, the gradient estimate at $\theta$ is obtained as $$\begin{aligned}
\label{gradEst1}
\widehat{\nabla J}(\theta) = \frac{J(\theta + \delta d)-J(\theta-\delta d)}{2\delta}d,\end{aligned}$$ where $d$ is the random perturbation direction chosen according to a specific probability distribution. The properties that the specific distribution on $d$ should satisfy can be obtained as explained below. The Taylor series expansion of $J(\theta \pm \delta d)$ around $\theta$ is given by $$\label{eq:pmTaylor}
J(\theta \pm \delta d)=
J(\theta) \pm \delta d^T \nabla J(\theta)+o(\delta^2).$$ From , the error between the estimate and the true gradient at $\theta$ can be obtained as $$\begin{aligned}
\label{eq:Taylor}
&\frac{J(\theta +\delta d )-J(\theta -\delta d )}{2\delta }d
- \nabla J(\theta ) \nonumber \\
&= (d d ^T-I)\nabla J(\theta )+o(\delta ).\end{aligned}$$ Note that the term $(dd ^T-I)\nabla J(\theta )$ constitutes the bias in the gradient estimate. For the error estimate in to be negligible, we require $$\label{eq:2sidedbias}
\mathbb{E}\Big{[}dd^T \Big{]}=I.$$ Here, the expectation $\mathbb{E}[\cdot]$ is taken over the random perturbation distribution.
In the one-simulation version of the RDKW algorithm, the gradient estimate at $\theta$ is obtained as $$\begin{aligned}
\label{gradEst2}
\widehat{\nabla J}(\theta) = \frac{J(\theta + \delta d)}{\delta}d.\end{aligned}$$ By analogous Taylor series argument, we obtain the error between the estimate and the true gradient as $$\begin{aligned}
\label{eq:1sided}
&\frac{J(\theta +\delta d )}{\delta }d
- \nabla J(\theta ) \nonumber \\
&=\frac{J(\theta )}{\delta }d
+(d d ^T-I)\nabla J(\theta )+O(\delta ).\end{aligned}$$ From , we require the following to hold in addition to in the case of random perturbations for the one simulation version of RDKW algorithm, i.e., $$\label{eq:1sidedbias}
\mathbb{E}[d]=0.$$ For the random perturbations, $d \sim F$, $F$ is any distribution that satisfies and , the noise in the gradient estimates gets averaged asymptotically. An example distribution for $F$ is symmetric Bernoulli where each component of the perturbation vector is $\pm 1$ with equal probability.
From and clearly one is motivated to look for perturbations that satisfy similar properties. In what follows, the sequence of deterministic perturbations (that will be used in either or ) will be denoted by $\{d_n\}_{n\geq 1}$ and we require the following two properties to hold for the perturbation sequence $d_n$ for the almost sure convergence of the iterates to a local minima.
1. Let $D_n:=d_nd_n^T-I_{p \times p}.$ For any $s \in \mathbb{N}$ there exists a $P \in \mathbb{N}$ such that $\sum\limits_{n=s+1}^{s+P}D_n=0$ and,
2. $\sum\limits_{n=s+1}^{s+P}d_n=0.$
\[rem1\] The properties $\textbf{P1}$ and $\textbf{P2}$ are the deterministic analogues of and . For the properties $\textbf{P1}$ and $\textbf{P2}$ to hold, it is sufficient to determine a finite sequence $\{d_1,d_2,\dots,d_{P}\}$ such that $\sum_{n=1}^{P} d_n d_n^T = PI$ and $\sum_{n=1}^{P} d_n=0$ and for $n \geq P+1$, periodically cycle through this sequence, i.e., set $d_n=d_{n \% P + 1 }$. We will refer the length of the deterministic perturbation sequence $P$ as the cycle length.
Construction Of Deterministic Perturbations {#sec3}
===========================================
In section \[detPerturb\], following Remark \[rem1\], we first characterize the finite sequences $\{d_1,d_2,\dots,d_{P}\}$ that satisfy properties $\textbf{P1}$ and $\textbf{P2}$ by providing a matrix equation whose solution gives the deterministic perturbations. In Section \[construct\], we then construct a specific sequence using a circulant matrix that has the least possible cycle length among all the deterministic perturbation sequences. Finally in section \[gradEstSec\], we completely describe the RDKW algorithm that uses the deterministic perturbation sequence constructed using the circulant matrix approach.
Matrix condition for Deterministic Perturbations {#detPerturb}
------------------------------------------------
The properties **P1** and **P2** can be satisfied individually. For example, to satisfy property **P1**, let $P=p$ and $d_{n}=\sqrt{p} e_{n}, ~n \in \{1,\ldots,P\}$, the scaled canonical basis vectors, then $\sum_{n=1}^{P} d_{n}d_{n}^T = \sum_{n=1}^{p} pe_{n}e_{n}^T = pI$. To satisfy property **P2**, consider any set of linearly dependent vectors $\{v_0,\cdots,v_P\}$. Then there exists scalars $\alpha_1,\cdots, \alpha_P$ such that $\sum_{n=1}^{P} \alpha_n v_n=0$. Now for the choice $d_n=\alpha_nv_n$ the property **P2**, $\sum_{n=1}^{P} d_n=\sum_{n=1}^{P} \alpha_nv_n=0$ is trivially satisfied. A natural question would be to determine sequences $\{d_n\}_{1 \leq n \leq P}$ that satisfy both the properties simultaneously.
To address this problem, let us consider a $p \times P$ matrix $Y$ as follows: $Y:=\left[\begin{array}{cccc}
\uparrow \ &\uparrow \ &\cdots \ &\uparrow \\
d_1 \ &d_2 \ &\cdots \ &d_{P} \\
\downarrow \ &\downarrow \ &\cdots \ & \downarrow \\
\end{array}\right].$ Let $u=[1,1,\cdots,1]^T$ be a $P \times 1$ dimension vector. The perturbations that satisfy properties **P1** and **P2** essentially solve the two matrix equations $Yu=0$ and $YY^{T}=PI$. These equations can be compactly written in a single matrix equation as $$\begin{aligned}
\label{single}
XX^{T}=PI_{(p+1)\times(p+1)},\end{aligned}$$ where $X=\left[\begin{array}{c}
u^{T}\\ Y \end{array}\right]$. Note that $Y_{p\times P}$ and $P$ are the unknowns here.
It can observed from that $\frac{X}{\sqrt{P}}$ could be treated as a $p \times P$ submatrix of a $P \times P$ orthogonal matrix with the first row being $\frac{u^T}{\sqrt{P}}$, a $1 \times P$ vector. It has been shown in [@bhatfumarcwang] that columns of Hadamard matrices satisfy properties **P1** and **P2** simultaneously with $\bar{P}=2^{\log_2 \ceil{p+1}}$, i.e., $X$ is chosen as a $(p+1) \times 2^{\log_2 \ceil{p+1}}$ submatrix of the Hadamard matrix. It is not in general clear if the equation can be solved for a smaller $P \leq \bar{P}$.
\[rem2\] We note that similar analysis for matrix condition for the construction of deterministic perturbations for SPSA estimates involves solving the following matrix system. $AB=PI$,$Au=0$ and $A \circ B^T=vu^T$ where $A$ is $p\times P$, $B$ is $P\times p$, $u$ is $P \times 1$ vector of ones, $v$ is $p \times 1$ vector of ones and $\circ$ denotes the Hadamard product of the matrices $A$ and $B$. It is not clear how to solve for $P,$ $A$ and $B$ due to the presence of Hadamard product in this system.
Specific Perturbation Sequence Construction {#construct}
-------------------------------------------
In this section, our goal is to obtain a sequence with least cycle length. Using a simple matrix rank argument it can be shown that $P$ is at least $p+1$. Thus, in what follows, we give a construction of deterministic perturbation sequence with cycle length $P=p+1$. We first write $$Y=\left[\begin{array}{ccc}
\uparrow \ &\cdots \ &\uparrow \\
Z \ & \ &-ZU \\
\downarrow \ &\cdots \ & \downarrow \\
\end{array}\right]$$ where $Z$ is a $p\times p$ matrix and $U$ is any $p \times (P-p)$ matrix with columns that sum to 1. Clearly $Yu=0$ satisfies property **P2**.
To satisfy property **P1**, i.e., $YY^{T}=I$ is equivalent to $$\begin{aligned}
\label{mainEq}
ZZ^{T}+ZUU^{T}Z^{T}=Z(I+UU^{T})Z^{T}=PI.\end{aligned}$$ Clearly construction of deterministic perturbations with smaller cycle length $P$ is equivalent to solving for $Z$ with an appropriate choice of $U$.
The simplest choice of $U$ with column sums being 1 is $U=u$, a $p \times 1$ vector, thus $P=p+1$. Let $C=I+UU^{T}=I+uu^{T}$ ($p \times p$ dimensional matrix) $$\label{eq:C}
C = \left[\begin{array}{cccc}
2 \ 1 \ 1 \cdots 1\\
1 \ 2 \ 1 \cdots 1 \\
\vdots \ \vdots \ \vdots \ \vdots\\
1 \ 1 \ 1 \cdots 2
\end{array}\right].$$ Observe that $C$ is a positive definite circulant matrix. Hence $C^{-1/2}$ is well defined and the choice $Z=C^{-1/2}$ satisfies and solves the system $YY^{T}=I$ with $P=p+1$, i.e., $$\begin{aligned}
\label{yEqn}
Y=\sqrt{p+1}[C^{-1/2},-C^{-1/2}u].\end{aligned}$$ The columns of $Y$ finally give us the deterministic perturbations. We note that in general the computation of $C^{-1/2}$ is $O(p^3)$ and can be very expensive for large $p$. However owing to the special structure of $C$, using a Sherman-Morrison type result (see Lemma \[lemma: gen Sherman-Morrison\], Section \[sec:convergenceresults\]), $C^{-1/2}$ can be computed in $O(p^2)$ time complexity.
Gradient estimation {#gradEstSec}
-------------------
In this section, we present the RDKW algorithms that use the deterministic perturbation sequence constructed above in two-simulation and one-simulation gradient estimates of the objective. We denote the corresponding algorithms by DSPKW-2C and DSPKW-1C respectively.
\[sec:algo\]
- $\theta_0 \in \mathbb{R}^p,$ initial parameter vector
- $\delta_n, n \geq 0,$ a sequence of sensitivity parameters to approximate gradient
- Matrix of perturbations $$Y=\sqrt{p+1}[C^{-1/2},-C^{-1/2}u],$$ with $u=[1,1,\cdots,1]^T;$
- noisy measurements of cost objective $J$
- $a_n, n \geq 0,$ step-size sequence satisfying assumption \[stepsizes\] (see section \[sec:convergenceresults\])
- $n_{end}$, the total number of iterations determined by simulation budget
$\theta_{n_{\text{end}}}$, approximate local optimal solution Let $d_n$ be the mod$(n,p+1)^{\text{th}}$ column of $Y$. Update the parameter as follows: $$\theta_{n+1}=\theta_n-a_n \widehat{\nabla J}(\theta_n)$$ $\widehat{\nabla J}(\theta_n)$ is chosen according to either or for DSPKW-2C and DSPKW-1C respectively. $\theta_{n_{\text{end}}}$
Let $\delta_n, n\geq 0$ denote a sequence of diminishing positive real numbers satisfying assumption \[stepsizes\] in section \[sec:convergenceresults\]. Let $y_{n}^{+}$, $y_{n}^{-}$ denote the noisy objective function evaluations at the perturbed parameters $\theta_n+\delta_n d_n$ and $\theta_n -\delta_n d_n$ respectively, i.e., $y_{n}^{+} = J(\theta_n+\delta_n d_n) + M_{n+1}^{+}$ and $y_{n}^{-} = J(\theta_n-\delta_n d_n) + M_{n+1}^{-}$. We assume the noise terms $M_{n}^{+}, M_{n}^{-}$ are martingale difference noise sequence, $\E\left[M_{n+1}^{+} | \F_n \right] = \E\left[ M_{n+1}^{-} | \F_n\right] = 0$ where $\F_n = \sigma(\theta_m, M^{+}_{m}, M^{-}_{m}, ~m\le n)$ is the information conditioned on the past parameter values and martingale difference terms.
The two-simulation and one-simulation estimates of the gradient $\nabla J(\theta_n)$ based on the observed noisy objective samples for the RDKW algorithm are respectively given by $$\begin{aligned}
\label{eq:grad-twosided}
\widehat{\nabla J}(\theta_n)=
\left[\dfrac{(y_{n}^{+} - y_{n}^{-})d_n}{2\delta_n}\right],\end{aligned}$$ $$\begin{aligned}
\label{eq:grad-onesided}
\widehat{\nabla J}(\theta_n)=
\left[\dfrac{(y_{n}^{+})d_n}{\delta_n}\right],\end{aligned}$$ respectively. Observe that in the two-sided estimate we use two function samples $y_{n}^{+}$ and $y_{n}^{-}$ and the estimate in uses only one function sample $y_{n}^{+}$.
Now we briefly describe the DSPKW algorithm. Inputs to the DSPKW algorithm are randomly chosen initial point $\theta_0$, diminishing sequences $\delta_n$ and $a_n$ satisfying assumption \[stepsizes\] and the matrix of deterministic perturbations $Y$ chosen according to . In our algorithms, we iteratively choose the perturbations by cycling through columns of $Y$ with period $p+1$ and in steps 2-4, we update the parameters along the direction of estimated gradient according to in the DSPKW-2C algorithm and according to in the DSPKW-1C algorithm. Note the choice of gradient estimate (or the algorithm) is dictated by the simulation budget given to us. The algorithms terminate by returning the parameter $\theta_{n_{end}}$ at the end of $n_{end}$ iterations.
Convergence Analysis {#sec:convergenceresults}
====================
In this section we first provide a few lemmas that assist in computing the proposed deterministic perturbation sequence (see in Section \[construct\]). In the latter part of the section, we prove the almost sure convergence of the iterates for the class of deterministic perturbations characterized in Section \[detPerturb\]. The following lemma is useful in obtaining the negative square root of $C$, i.e., $C^{-1/2}$ in a computationally efficient manner. Also note that it takes only $O(p^2)$ operations to compute $C^{-1/2}$ using the lemma and the circulant structure of $C^{-1/2}$. Note that the following lemma could also be utilized in an independent context for efficient computation.
\[lemma: gen Sherman-Morrison\] Let $I$ be a $p \times p$ identity matrix and\
$u=[1,1, \cdots 1]^{T}$ be a $p \times 1$ column vector of 1s, then $$(I+uu^T)^{-1/2}= I-\frac{uu^T}{p}+\frac{uu^T}{p\sqrt{(1+p)}}.$$
It is enough to show that $$(I+uu^T)\Bigg{[}I-\frac{uu^T}{p}+\frac{uu^T}{p\sqrt{(1+p)}}\Bigg{]}^2=I.$$ Using $\|u\|^2=u^Tu=p$ in the expansion of $\Big{[}I-\frac{uu^T}{p}+\frac{uu^T}{p\sqrt{(1+p)}}\Big{]}^2$ gives the result.
Let $C$ be defined as in and $Y=\sqrt{p+1}[C^{-1/2},-C^{-1/2}u].$ Let the perturbations $d_n$ be the columns of $Y.$
The perturbations $d_n$ chosen as columns of Y satisfy properties $\textbf{P1}$ and $\textbf{P2}$.
It easily follows from the discussion in section \[construct\] on the construction of this specific perturbation sequence.
In what follows, we prove the almost sure convergence of the iterates in the DSPKW algorithm (Section \[gradEstSec\]) under the following assumptions. Note that $\|.\|$ denotes the 2-norm.
1. The map $J:\mathbb{R}^p \rightarrow \mathbb{R}$ is Lipschitz continuous and is differentiable with bounded second order derivatives. Further, the map $L:\mathbb{R}^p \rightarrow \mathbb{R}^p$ defined as $L(\theta)=-\nabla J(\theta)$ is Lipschitz continuous.
2. The step-size sequences $a_n, \delta_n >0, \forall n $ satisfy $$\label{stepsizes}
a_n,\delta_n \rightarrow 0 , \sum_na_n=\infty,
\sum_n \Big{(}\frac{a_n}{\delta_n}\Big{)}^2<\infty.$$ Further, $\frac{a_j}{a_n}\rightarrow 1$ as $n\rightarrow \infty$, for all $j \in \{n,n+1,n+2\cdots,n+M\}$ for any given $M>0$ and $b_n=\frac{a_n}{\delta_n}$ is such that $\frac{b_j}{b_n}\rightarrow 1$ as $n\rightarrow \infty$, for all $j \in \{n,n+1,n+2,\cdots,n+M\}.$
3. $\max_n \|d_n\|= K_{0}, \max_n \|D_n\|= K_{1}$.
4. The iterates $ \theta_n$ remain uniformly bounded almost surely, i.e., $ \sup_n\|\theta_n\|<\infty, \text{ a.s.}$
5. The ODE $\dot{\theta}(t)=-\nabla J(\theta(t))$ has a compact set $G \subset \mathbb{R}^p$ as its set of asymptotically stable equilibria (i.e., the set of local minima of $J$ is compact).
6. The sequences $(M_{n}^{+},\F_n),(M_{n}^{-},\F_n), n\geq0 $ form martingale difference sequences. Further, $(M_{n}^{+},M_{n}^{-},n\geq0)$ are square integrable random variables satisfying $$\E[\|M_{n+1}^{\pm}\|^2|\F_n]\leq K(1+\|\theta_n\|^2) \text{ a.s., } \forall n\geq0,$$ for a given constant $K > 0.$
\[rem3\] Assumptions **A1**, **A2** and **A5** are standard stochastic approximation conditions. Assumption **A3** trivially follows from Remark \[rem1\]. Assumption **A4** is the stability condition on the iterates and holds in many applications [@spall] (see the discussion in pp 40-41 of [@kushcla]). This condition can also be enforced by projecting the iterates into a compact set, however, the iterates converge to a limiting set that contains all possible limit points (see pp.191 in [@kushcla]). Assumption **A6** gives the condition on the maximum strength of the martingale difference noise under which convergence of the iterates could be ensured and in many stochastic optimization settings this condition could be easily verified using Jensen’s inequality and Lipschitz continuity of $\nabla J$ .
The following two lemmas aid in the proof of almost sure convergence of the iterates in the DSPKW algorithm.
Given any fixed integer $P>0$, $\|\theta_{m+k}-\theta_{m}\| \rightarrow 0$ $w.p.1,$ as $m \rightarrow \infty,$ for all $k \in \{1,\cdots, P\}.$
Fix a $k \in \{1,\cdots,P \}.$ Now $$\begin{aligned}
\begin{split}
\theta_{n+k} = \theta_n & -\sum_{j=n}^{n+k-1}a_j\Bigg{(}\frac{J(\theta_j+\delta_j d_j)-J(\theta_j-\delta_j d_j)}{2\delta_j}\Bigg{)}d_j \\
&-\sum_{j=n}^{n+k-1}a_jM_{j+1},
\end{split}
\end{aligned}$$ where $M_{j+1}=\frac{(M_{j+1}^{+}-M_{j+1}^{-})d_{j}}{2\delta_j}$. Thus, $$\begin{aligned}
\begin{split}
\|\theta_{n+k}-\theta_n\| &\leq \sum_{j=n}^{n+k-1}a_j\Bigg{|}\frac{J(\theta_j+\delta_{j} d_j)-J(\theta_j-\delta_{j} d_j)}{2\delta_j}\Bigg{|}\|d_j\|\\
&+\sum_{j=n}^{n+k-1}a_j\|M_{j+1}\|.
\end{split}\end{aligned}$$ Now clearly, $N_n=\sum\limits_{j=0}^{n-1}a_jM_{j+1}, n\geq1,$ forms a martingale sequence with respect to the filtration $\{\F_n\}$. Further, from the assumption (A6) we have, $$\begin{aligned}
\sum_{m=0}^{n}\mathbb{E}[\|N_{m+1}-N_{m}\|^2|\mathcal{F}_{m}]& =\sum_{m=0}^{n}\mathbb{E}[a_{m}^2\|M_{m+1}\|^2|\mathcal{F}_{m}]\\
& \leq \sum_{m=0}^{n}a_{m}^2K(1+\|\theta_m\|^2).\end{aligned}$$ From the assumption (A4), the quadratic variation process of $N_n,n\geq0$ converges almost surely. Hence by the martingale convergence theorem, it follows that $N_n, n\geq0$ converges almost surely. Hence $\|\sum\limits_{j=n}^{n+k-1}a_jM_{j+1}\|\rightarrow 0$ almost surely as $n\rightarrow \infty.$ Moreover $$\begin{aligned}
&\Big{\|}\Big{(}J(\theta_j+\delta_j d_j)-J(\theta_j-\delta_j d_j)\Big{)}d_j\Big{\|} \\
& \leq \Big{|}\Big{(}J(\theta_j+\delta_j d_j)-J(\theta_j-\delta_j d_j)\Big{)}\Big{|}\|d_j\|\\
& \leq K_{0} \Big{(}|J(\theta_j+\delta_j d_j)|+|J(\theta_j-\delta_j d_j)|\Big{)},\end{aligned}$$ since $\|d_j\|\leq K_{0}, \forall j \geq0.$ Note that $$\begin{aligned}
|J(\theta_j+\delta_j d_j)|-|J(0)| & \leq|J(\theta_j+\delta_j d_j)-J(0)| \\
& \leq \hat{B} \|\theta_j+\delta_j d_j\|,\end{aligned}$$ where $\hat{B}$ is the Lipschitz constant of the function $J.$ Hence, $$|J(\theta_j+\delta_j d_j)|\leq \tilde{B}(1+\|\theta_j+\delta_j d_j\|),$$ for $\tilde{B}=$max$(|J(0)|,\hat{B}).$ Similarly, $$|J(\theta_j-\delta_j d_j)|\leq \tilde{B}(1+\|\theta_j-\delta_j d_j\|).$$ From assumption (A1), it follows that $$\sup_j\Big{\|}\Big{(}J(\theta_j+\delta_j d_j)-J(\theta_j-\delta_j d_j)\Big{)}d_j\Big{\|}\leq \tilde{K}<\infty,$$ for some $\tilde{K}>0.$ Thus, $\|\theta_{n+k}-\theta_n\| \leq \tilde{K}\sum\limits_{j=n}^{n+k-1}\frac{a_j}{2\delta_j}+\|\sum_{j=n}^{n+k-1}a_jM_{j+1}\|$ $\rightarrow 0 \text{ a.s. with } n \rightarrow \infty,$ proving the lemma.
$\text{ For any } m \geq0,$ $\Big{\|}\sum\limits_{n=m}^{m+P-1}\frac{a_n}{a_m}D_n\nabla J(\theta_n)\Big{\|} \text{ and }$ $\Big{\|}\sum\limits_{n=m}^{m+P-1}\frac{b_n}{b_m}d_nJ(\theta_n)\Big{\|}\rightarrow 0,$ $\text{almost surely, as } m \rightarrow \infty.$
From Lemma 3, it can be seen that $\|\theta_{m+s}-\theta_{m}\|\rightarrow 0$ as $m\rightarrow \infty,$ for all $s=1,\cdots,P.$ Also, from assumption (A1), we have $\|\nabla J(\theta_{m+s})-\nabla J(\theta_{m})\|\rightarrow 0$ as $m\rightarrow \infty,$ for all $s=1,\cdots,P.$ Now from Lemma 2, $\sum\limits_{n=m}^{m+P-1}D_n=0$ $\forall m\geq0.$ Hence $D_m=-\sum\limits_{n=m+1}^{m+P-1}D_n.$ Consider first $$\begin{aligned}
&\Big{\|}\sum_{n=m}^{m+P-1}\frac{a_n}{a_m}D_n\nabla J(\theta_n)\Big{\|}\\
& = \Big{\|}\sum_{n=m+1}^{m+P-1}\frac{a_n}{a_m}D_n\nabla J(\theta_n)
+D_{m}\nabla J(\theta_{m})\Big{\|}\\
&=\Big{\|}\sum_{n=m+1}^{m+P-1}\frac{a_n}{a_m}D_n \nabla J(\theta_n)
-\sum_{n=m+1}^{m+P-1}D_n\nabla J(\theta_{m})\Big{\|}\\
&=\Big{\|}\sum_{n=m+1}^{m+P-1}D_n\Big{(}\frac{a_n}{a_m}\nabla J(\theta_n)
-\nabla J(\theta_{m})\Big{)}\Big{\|}\\
&\leq\sum_{n=m+1}^{m+P-1}{\|}D_n{\|}\Big{\|}\Big{(}\frac{a_n}{a_m}\nabla J(\theta_n)
-\nabla J(\theta_{m})\Big{)}\Big{\|}\\
&\leq K_{1}\sum_{n=m+1}^{m+P-1}\Big{\|}\Big{(}\frac{a_n}{a_m}-1\Big{)}\nabla J(\theta_n)\Big{\|}
+\Big{\|}\nabla J(\theta_n)-\nabla J(\theta_{m})\Big{\|}
\end{aligned}$$ $\rightarrow 0 \text{ a.s. with } n \rightarrow \infty,$ from assumptions (A1) and (A2). Now observe that $\|J(\theta_{m+k})-J(\theta_{m})\|\rightarrow 0$ as $m\rightarrow \infty,$ for all $k \in \{1,\cdots,P\}$ as a consequence of (A1) and Lemma 3. Moreover from $d_m=-\sum\limits_{n=m+1}^{m+P-1}d_n$ we have $$\begin{aligned}
& \Big{\|}\sum_{n=m}^{m+P-1}\frac{b_n}{b_m}d_n J(\theta_n)\Big{\|}\\
& =\Big{\|}\sum_{n=m+1}^{m+P-1}\frac{b_n}{b_m}d_n J(\theta_n)+d_m J(\theta_{m})\Big{\|}\\
& =\Big{\|}\sum_{n=m+1}^{m+P-1}\frac{b_n}{b_m}d_n J(\theta_n)-\sum_{n=m+1}^{m+P-1}d_n J(\theta_{m})\Big{\|}\\
& =\Big{\|}\sum_{n=m+1}^{m+P-1}d_n\Big{(}\frac{b_n}{b_m}J(\theta_n)-J(\theta_{m})\Big{)}\Big{\|}\\
& \leq \sum_{n=m+1}^{m+P-1}\|d_n\|\Big{\|}\Big{(}\frac{b_n}{b_m}J(\theta_n)-J(\theta_{m})\Big{)}\Big{\|}\\
&\leq K_{0} \sum_{n=m+1}^{m+P-1}\Big{\|}\Big{(}\frac{b_n}{b_m}-1\Big{)} J(\theta_n)\Big{\|}
+\Big{\|}\Big{(} J(\theta_n)- J(\theta_{m})\Big{)}\Big{\|}\end{aligned}$$ The claim now follows as a consequence of assumptions (A1) and (A2).
Finally, using the following theorems, we conclude the analysis by proving the almost sure convergence of the iterates to the set of local minima $G$ of the function $J.$
$\theta_n, n\geq0$ obtained from DSPKW-2C satisfy $\theta_n \rightarrow G$ almost surely.
Note that $$\theta_{n+P} = \theta_n-\sum\limits_{l=n}^{n+P-1}a_l\Big{[}\frac{J(\theta_l+\delta_l d_l)-J(\theta_l-\delta_l d_l)}{2\delta_l}d_l+M_{l+1}\Big{]}.$$ It follows that $$\begin{aligned}
& \theta_{n+P} = \theta_n-\sum_{l=n}^{n+P-1}a_l\nabla J(\theta_l) -\sum_{l=n}^{n+P-1}a_l o(\delta_l) \\
& -\sum_{l=n}^{n+P-1}a_l(d_ld_l^T-I)\nabla J(\theta_l)-\sum_{l=n}^{n+P-1}a_lM_{l+1}.
\end{aligned}$$ Now the fourth term on the RHS above can be written as $$a_n\sum_{l=n}^{n+P-1}\frac{a_l}{a_n}D_{l}\nabla J(\theta_l)=a_n\xi_{n},$$ where $\xi_{n}=o(1)$ from Lemma 4. Thus, the algorithm is asymptotically analogous to $$\theta_{n+1}=\theta_n-a_n(\nabla J(\theta_n)+o(\delta)+M_{n+1}).$$ Hence, from Theorem 2 in chapter 2 of [@borkar2008stochastic], it follows that $\theta_n, n\geq0$ converge to a local minima of the function $J.$
$\theta_n, n\geq0$ obtained from DSPKW-1C satisfy $\theta_n \rightarrow G$ almost surely.
Note that $$\begin{aligned}
\theta_{n+P} = \theta_n-\sum_{l=n}^{n+P-1}a_l\Big{(}\frac{J(\theta_l+\delta_l d_l)}{2\delta_l}\Big{)}d_l-\sum_{l=n}^{n+P-1}a_lM_{l+1}.
\end{aligned}$$ It follows that $$\begin{aligned}
\begin{split}
& \theta_{n+P} = \theta_n- \sum_{l=n}^{n+P-1}a_l\nabla J(\theta_l)
-\sum_{l=n}^{n+P-1}a_l\frac{J(\theta_l)}{\delta_l}d_l\\
&-\sum_{l=n}^{n+P-1}a_l(d_ld_l^T-I)\nabla J(\theta_l)-\sum_{l=n}^{n+P-1}a_lO(\delta_l)\\
&-\sum_{l=n}^{n+P-1}a_lM_{l+1}.
\end{split}
\end{aligned}$$ Now we observe that the third term on the RHS above is $$\begin{aligned}
& \sum_{l=n}^{n+P-1}a_l\frac{J(\theta_l)}{\delta_l}d_l= \sum_{l=n}^{n+P-1} b_{l}J(\theta_l)d_l\\
& = b_n\sum_{l=n}^{n+P-1}\frac{b_l}{b_n}\frac{J(\theta_l)}{\delta_l}d_l=b_n\xi^{1}_{n},
\end{aligned}$$ where $\xi^{1}_{n}=o(1)$ by Lemma 4. Similarly $$\sum_{l=n}^{n+P-1}a_l(d_ld_l^T-I)\nabla J(\theta_l)=a_n\xi^{2}_{n},$$ with $\xi^{2}_{n}=o(1)$ by Lemma 4. The rest follows as in Theorem 5.
Simulation Experiments {#sec:expts}
======================
In this section, we compare the numerical performance of our DSPKW-2C algorithm against the RDKW algorithm that uses random Bernoulli perturbations and another variant of the RDKW algorithm that uses Hadamard matrix based deterministic perturbations. We refer them by the acronyms RDKW-2R and RDKW-2H respectively. In a similar manner, we also compare DSPKW-1C algorithm against the one-simulation variants RDKW-1R and RDKW-1H. Note that 2 or 1 in the acronyms of these algorithms denote the number of simulations utilized per iteration.[^5]
Experimental setup
------------------
For the empirical performance evaluation, we consider the following two loss functions:
#### Quadratic loss
$$\begin{aligned}
\label{eq:quadratic}
J(\theta) = \theta\tr A \theta + b\tr \theta. \end{aligned}$$
#### Fourth-order loss
$$\begin{aligned}
\label{eq:4thorder}
J(\theta) = \theta \tr A\tr A\theta+0.1 \sum_{j=1}^N (A\theta)^3_j+0.01 \sum_{j=1}^N (A\theta)^4_j.
\end{aligned}$$
In the loss functions considered above, we set the dimension $p=10$. We choose $A$ such that $pA$ is an upper triangular matrix with each nonzero entry equal to one and $b$ is a $p$-dimensional vector of ones. In our experiments, we follow the same noise assumptions considered in [@spall_adaptive], i.e., for any $\theta$, the additive noise in the objective is given by $[\theta \tr, 1]z$ where $z \sim \N(0,\sigma^2 I_{p+1 \times p+1})$. In all algorithms, we set the step-size schedule as $\delta_n = c/(n+1)^{\gamma}$ and $a_n = 1/(n+B+1)^{\alpha}$ with $\alpha=0.602$ and $\gamma=0.101$. Note that the chosen values for $\alpha$ and $\gamma$ have demonstrated good finite-sample performance empirically, while satisfying the theoretical requirements needed for asymptotic convergence (see [@spall_adaptive]. We set the same initial point $\theta_0$ for all the algorithms.
We consider two settings in our experiments. In the first noise-free setting, we do not add any noise to the objective function evaluations and in the second setting, we corrupt the function evaluations by adding noise (with variance parameter $\sigma=0.01$ as described above). We evaluate the performance of these algorithms based on Normalized Mean Square Error (NMSE) metric. NMSE is defined as the ratio $\l \theta_{n_\text{end}} - \theta^* \r^2 / \l \theta_0 - \theta^*\r^2$, where $\theta_{n_\text{end}}$ is the parameter returned by the algorithm.
Discussion of Results
---------------------
The performance comparisons of all the algorithms based on NMSE values are summarized in Tables \[tab:NMSE-quadratic\], \[tab:NMSE-fourthorder\], \[tab:NMSE-quadratic-1sim\] and \[tab:NMSE-fourthorder-1sim\]. In the tables, we have highlighted the algorithm that has the minimum NMSE. We summarize our findings:
- Even in the absence of noise, due to the random directions chosen by RDKW-2R and RDKW-1R algorithms, the standard deviation is significantly high compared to the corresponding deterministic counterparts.
- We would like to emphasize that the quality of the solution (characterized by standard deviation) is significantly better for the case of proposed deterministic perturbations compared to the existing Hadamard based deterministic perturbations and random perturbations. Note however that we do not make comparisons between two-simulation and one-simulation algorithms.
- In the case of two simulation algorithms (see Tables \[tab:NMSE-quadratic\] and \[tab:NMSE-fourthorder\]), DSPKW-2C performs marginally better than RDKW-2H, while both of them outperform RDKW-2R significantly.
- In the case of one simulation algorithms (see Tables \[tab:NMSE-quadratic-1sim\] and \[tab:NMSE-fourthorder-1sim\]), DSPKW-1C performs better than both RDKW-1H and RDKW-1R.
Conclusions {#sec:conclusions}
===========
We have generalized the deterministic perturbation sequences from lexicographical ordering and Hadamard matrix based constructions for the RDKW algorithm and presented a novel construction of deterministic perturbations that has least cycle length within the class of deterministic perturbation sequences. Further, we have proved the almost sure convergence of the iterates for the class of deterministic perturbation sequences. Now that we have a characterization of the class of deterministic perturbation sequences, it would be interesting as future work, to theoretically study and compare the rate of convergence of deterministic perturbation algorithms against their random perturbation counterparts. A challenging future direction would be to study the asymptotic normality or weak convergence of the iterates. It would also be interesting to similarly characterize the class of deterministic perturbation sequences for the SPSA algorithm.
[^1]: $^1$ Department of Computer Science and Automation, Indian Institute of Science (IISc)
[^2]: $^2$ Supported by Amazon-IISc Postdoctoral fellowship
[^3]: $^3$ IBM Research, Bangalore
[^4]: $^4$ Robert Bosch Centre for Cyber-Physical Systems, IISc
[^5]: The implementation is available at <https://github.com/cs1070166/1RDSA-2Cand1RDSA-1C/>
|
---
abstract: 'Graphical calculi for representing interacting quantum systems serve a number of purposes: compositionally, intuitive graphical reasoning, and a logical underpinning for automation. The power of these calculi stems from the fact that they embody generalized symmetries of the structure of quantum operations, which, for example, stretch well beyond the Choi-Jamiolkowski isomorphism. One such calculus takes the GHZ and W states as its basic generators. Here we show that this language allows one to encode standard rational calculus, with the GHZ state as multiplication, the W state as addition, the Pauli X gate as multiplicative inversion, and the Pauli Z gate as additive inversion.'
author:
- 'Bob Coecke[^1], Aleks Kissinger[^2], Alex Merry[^3]'
- 'Shibdas Roy[^4]'
bibliography:
- 'rationals.bib'
title: 'The GHZ/W-calculus contains rational arithmetic'
---
Introduction
============
*Categorical quantum mechanics *[@AC] aims to recast quantum mechanical notions in terms of symmetric monoidal categories with additional structure. One layer of extra structure, compactness [@KellyLaplaza], encompasses the well-known Choi-Jamiolkowski isomorphism. Compactness is itself subsumed by the much richer commutative Frobenius algebra structure [@CarboniWalters], which governs classical data, observables, and certain tripartite states [@CPav; @CD; @CES2; @CK]. In this symmetric monoidal form, quantum mechanics enjoys:**
an *operational interpretation *by making sequential and parallel composition of systems and processes the basic connectives of the language [@ContPhys];**
an intuitive *diagrammatic calculus *[@ContPhys] via the Penrose-Joyal-Street diagrammatic calculus for symmetric monoidal categories [@Penrose; @JS], augmented with Kelly and Laplaza’s coherence result for compact categories, and Lack’s work on distributive laws [@Lack];**
a *logical underpinning *[@RossThesis] via the closed structure resulting from compactness. The last allows the application of automated reasoning techniques to quantum mechanics [@DD; @quanto; @DK]. A prototype software implementation, [quantomatic]{}, already exists and is jointly developed in Edinburgh and Oxford.**
Categorical quantum mechanics has meanwhile been successful in solving problems in quantum information [@DP] and quantum foundations [@CES2], where other methods and structures failed to be adequate. Key to these results is the description of *interacting basis structures *in [@CD]. The language of that paper consists of a pair of abstract bases or *basis structures*, which are, again in abstract terms, mutually unbiased, and an abstract generalisation of phases relative to bases. This formalism has been implemented in [quantomatic]{}, and is expressive enough to universally model any linear map $f:\mathbb{Q}^{\otimes n}\to \mathbb{Q}^{\otimes m}$, where $\mathbb{Q}=\mathbb{C}^2$. On the other hand, if we restrict the language to the two basis structures only it becomes very poor, describing no more than 2 qubit states.****
This brings us to the subject of this paper. In [@CK] two of the authors introduced pairs of interacting commutative Frobenius algebras that do not model bases, but the tripartite GHZ and W states [@DVC]. Both these states can indeed be endowed with the structure of a commutative Frobenius algebra, yielding a *GHZ structure *and a *W structure *as we recall in Section \[sec:ghz-w\]. The main point of this paper is that the language consisting of the GHZ structure (which is essentially the same as a basis structure) and the W structure is already rich enough to encode rational arithmetic, with the exception of additive inverses. Now an infinite number of qubit states can be described, corresponding to the rational numbers of the arithmetic system. We demonstrate this in Section \[sec:arithmetic\]. In Section \[sec:additive-inverses\] we extend the GHZ/W-calculus with one basic graphical element which then allows additive inverses to be captured. Section \[sec:automation\] addresses the issue of how to implement the calculus within the [quantomatic]{} software.****
We assume that the reader is familiar with the diagrammatic calculus for symmetric monoidal categories [@JS; @SelingerSurvey], which is also reviewed in [@CK]. We also assume that the reader is familiar with the (very) basics of finite dimensional Hilbert spaces and Dirac notation as used in quantum computing.
Frobenius Algebras and the GHZ/W-calculus {#sec:ghz-w}
=========================================
Fix a symmetric monoidal category $({\bf V},\otimes,I,\sigma)$. Throughout this paper, we shall define morphisms in $\bf V$ using the graphical notation defined in [@SelingerSurvey]. In this notation, ‘wires’ correspond to objects and vertices, and ‘boxes’ correspond to morphisms. We shall express composition vertically, from top to bottom, and the monoidal product as (horizontal) juxtaposition of graphs. When wires are not labeled, they are assumed to represent a fixed object, $Q$.
A canonical example throughout will be ${\bf FHilb}$, the category of finite-dimensional Hilbert spaces and linear maps. In this case, $\otimes$ is the usual tensor product, $\sigma$ the swap map $v \otimes w \mapsto w \otimes v$, $I := \mathbb C$ and $Q := \mathbb C^2$, the space of qubits. We shall also refer the “projective” category of finite-dimensional Hilbert spaces, ${\bf FHilb}_p$, whose objects are the same as ${\bf FHilb}$ and whose arrows are linear maps, taken to be equivalent iff they differ only by a non-zero scalar.
Commutative Frobenius Algebras
------------------------------
A *commutative Frobenius algebra *(CFA) consists of an internal commutative monoid $(Q, \mult\ , \unit)$ and an internal cocommutative comonoid $(Q, \comult\ , \counit)$ that interact via the Frobenius law: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, -1.5) {};
\node [style=none] (1) at (1.5, -1.5) {};
\node [style=none] (2) at (3, -1.5) {};
\node [style=none] (3) at (4, -1.5) {};
\node [style=dot] (4) at (1.5, -2) {};
\node [style=dot] (5) at (3.5, -2) {};
\node [style=none] (6) at (2.5, -2.25) {=};
\node [style=dot] (7) at (1, -2.5) {};
\node [style=dot] (8) at (3.5, -2.5) {};
\node [style=none] (9) at (1, -3) {};
\node [style=none] (10) at (2, -3) {};
\node [style=none] (11) at (3, -3) {};
\node [style=none] (12) at (4, -3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (8) to (12.center);
\draw (9.center) to (7);
\draw (5) to (2.center);
\draw (4) to (7);
\draw (11.center) to (8);
\draw (4) to (1.center);
\draw (5) to (3.center);
\draw (5) to (8);
\draw[bend left=15] (7) to (0.center);
\draw[bend left=15, looseness=1.25] (4) to (10.center);
\end{pgfonlayer}
\end{tikzpicture}$$**
One can show that any connected graph consisting only of $\mult, \unit, \comult, \counit, \sigma$ and $1_Q$ depends only upon the number of inputs, outputs, and loops. As such, it can be reduced to a canonical normal form: $$\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-2, 4.25) {};
\node [style=none] (1) at (-1, 4.25) {};
\node [style=none] (2) at (0, 4.25) {};
\node [style=none] (3) at (2, 4.25) {};
\node [style=dot] (4) at (-1.5, 3.75) {};
\node [style=dot] (5) at (-1, 3.25) {};
\node [style=none] (6) at (-0.75, 3) {};
\node [style=none] (7) at (-0.5, 2.75) {\small ...};
\node [style=none] (8) at (-0.25, 2.5) {};
\node [style=dot] (9) at (0, 2.25) {};
\node [style=dot] (10) at (0, 1.5) {};
\node [style=dot] (11) at (0, 0.75) {};
\node [style=none] (12) at (0, 0.25) {};
\node [style=none] (13) at (0, 0) {\small ...};
\node [style=none] (14) at (0, -0.25) {};
\node [style=dot] (15) at (0, -0.75) {};
\node [style=dot] (16) at (0, -1.5) {};
\node [style=dot] (17) at (0, -2.25) {};
\node [style=none] (18) at (-0.25, -2.5) {};
\node [style=none] (19) at (-0.5, -2.75) {\small ...};
\node [style=none] (20) at (-0.75, -3) {};
\node [style=dot] (21) at (-1, -3.25) {};
\node [style=dot] (22) at (-1.5, -3.75) {};
\node [style=none] (23) at (-2, -4.25) {};
\node [style=none] (24) at (-1, -4.25) {};
\node [style=none] (25) at (0, -4.25) {};
\node [style=none] (26) at (2, -4.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw[bend left=60, looseness=1.25] (10) to (11);
\draw[bend right=60, looseness=1.25] (15) to (16);
\draw[bend right=60, looseness=1.25] (10) to (11);
\draw (16) to (17);
\draw (9) to (10);
\draw (4) to (5);
\draw (9) to (3.center);
\draw (1.center) to (4);
\draw (25.center) to (21);
\draw (21) to (20.center);
\draw (14.center) to (15);
\draw (8.center) to (9);
\draw (5) to (6.center);
\draw (23.center) to (22);
\draw (24.center) to (22);
\draw (0.center) to (4);
\draw (22) to (21);
\draw (2.center) to (5);
\draw (17) to (26.center);
\draw[bend left=60, looseness=1.25] (15) to (16);
\draw (18.center) to (17);
\draw (11) to (12.center);
\end{pgfonlayer}
\end{tikzpicture}$$
In any connected graph, loops are counted as the total number of edges that can be removed without disconnecting the graph. We shall use ‘spider’ notation to represent graphs of Frobenius algebras using vertices of any arity. We express any connected graph as above with $m$ inputs, $n$ outputs, and no loops as a single vertex of the same colour: $$\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (3, 3.25) {$\overbrace{\qquad\qquad\qquad\qquad}^m$};
\node [style=none] (1) at (1, 2.5) {};
\node [style=none] (2) at (2, 2.5) {};
\node [style=none] (3) at (3, 2.5) {};
\node [style=none] (4) at (5, 2.5) {};
\node [style=dot] (5) at (1.5, 2) {};
\node [style=none] (6) at (-1, 1.75) {$\overbrace{\qquad\qquad}^m$};
\node [style=dot] (7) at (2, 1.5) {};
\node [style=none] (8) at (2.25, 1.25) {};
\node [style=none] (9) at (-2, 1) {};
\node [style=none] (10) at (-1.5, 1) {};
\node [style=none] (11) at (0, 1) {};
\node [style=none] (12) at (2.5, 1) {\small ...};
\node [style=none] (13) at (-0.75, 0.75) {\small ...};
\node [style=none] (14) at (2.75, 0.75) {};
\node [style=dot] (15) at (3, 0.5) {};
\node [style=dot] (16) at (-1, 0) {};
\node [style=none] (17) at (1, 0) {=};
\node [style=dot] (18) at (3, -0.5) {};
\node [style=none] (19) at (-0.75, -0.75) {\small ...};
\node [style=none] (20) at (2.75, -0.75) {};
\node [style=none] (21) at (-2, -1) {};
\node [style=none] (22) at (-1.5, -1) {};
\node [style=none] (23) at (0, -1) {};
\node [style=none] (24) at (2.5, -1) {\small ...};
\node [style=none] (25) at (2.25, -1.25) {};
\node [style=dot] (26) at (2, -1.5) {};
\node [style=none] (27) at (-1, -1.75) {$\underbrace{\qquad\qquad}_n$};
\node [style=dot] (28) at (1.5, -2) {};
\node [style=none] (29) at (1, -2.5) {};
\node [style=none] (30) at (2, -2.5) {};
\node [style=none] (31) at (3, -2.5) {};
\node [style=none] (32) at (5, -2.5) {};
\node [style=none] (33) at (3, -3.25) {$\underbrace{\qquad\qquad\qquad\qquad}_n$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (14.center) to (15);
\draw[bend right=15] (10.center) to (16);
\draw (29.center) to (28);
\draw (28) to (26);
\draw (30.center) to (28);
\draw[bend right=15] (16) to (22.center);
\draw (5) to (7);
\draw (15) to (4.center);
\draw[bend left] (16) to (23.center);
\draw (3.center) to (7);
\draw (20.center) to (18);
\draw (15) to (18);
\draw (7) to (8.center);
\draw[bend right] (9.center) to (16);
\draw (2.center) to (5);
\draw[bend left=15] (11.center) to (16);
\draw (31.center) to (26);
\draw (18) to (32.center);
\draw[bend right] (16) to (21.center);
\draw (1.center) to (5);
\draw (26) to (25.center);
\end{pgfonlayer}
\end{tikzpicture}$$
We give two of these graphs special names. The *cup* is defined as $\blackcup$ and the *cap* is defined as $\blackcap$. These induce a compact structure, since $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (-0.75, 0.75) {};
\node [style=none] (1) at (1.75, 0.75) {};
\node [style=dot] (2) at (-4.25, 0.5) {};
\node [style=dot] (3) at (-0.75, 0.25) {};
\node [style=none] (4) at (0.25, 0.25) {};
\node [style=none] (5) at (-4.75, -0) {};
\node [style=none] (6) at (-2.75, -0) {};
\node [style=none] (7) at (-2, -0) {$=$};
\node [style=none] (8) at (1, -0) {$=$};
\node [style=none] (9) at (-1.25, -0.25) {};
\node [style=dot] (10) at (-0.25, -0.25) {};
\node [style=dot] (11) at (-3.25, -0.5) {};
\node [style=dot] (12) at (-0.25, -0.75) {};
\node [style=none] (13) at (1.75, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw[out=0, in=270] (11) to (6.center);
\draw[out=90, in=180] (5.center) to (2);
\draw (3) to (10);
\draw[out=0, in=180] (2) to (11);
\draw (0) to (3);
\draw (1.center) to (13.center);
\draw (9.center) to (3);
\draw (12) to (10);
\draw (10) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}$$
Phases
------
[[@CD]]{}\[Phasedef\] Given a CFA on an object $A$, a morphism $f:A\to A$ is a *phase *if we have \[eq:phases1\] = = **
Equivalently, phases can be described as module endomorphisms, where $\mult$ is considered as a left (or right) module over itself.
\[prop:phases-states\] A phase $f:A\to A$ can be equivalently defined as a morphism of the form: \[eq:phases2\] := for some element $\psi:\II\to A$.
Given eqs. (\[eq:phases1\]) we have $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=square box] (2) at (0, 0) {$f$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw (2.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=dot] (2) at (0, -0.5) {};
\node [style=square box] (3) at (0, 0.5) {$f$};
\node [style=dot] (4) at (1, 1.5) {};
\node [style=none] (5) at (1, 0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (3.center);
\draw (3.center) to (2.center);
\draw (2.center) to (1);
\draw [bend right=45] (2.center) to (5.center);
\draw (5.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=dot] (2) at (0, -0.5) {};
\node [style=square box] (3) at (1, 0.5) {$f$};
\node [style=dot] (4) at (1, 1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw (2.center) to (1);
\draw (3.center) to (4);
\draw [bend right=60] (2.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=small white dot,font=\footnotesize] (3) at (1, 1) {$f \circ \unit$};
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=dot] (2) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=60] (2.center) to (3.center);
\draw (2) to (1);
\draw (2) to (0);
\end{pgfonlayer}
\end{tikzpicture}}$$ where we used unitality of the CFA, and conversely, given eq. (\[eq:phases2\]), eqs. (\[eq:phases1\]) straightforwardly follow by associativity and commutativity of the CFA.
\[prop:invphase\] The inverse of a phase is a phase.
Setting $
\raisebox{-4mm}{
\begin{tikzpicture}[scale=0.5]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1) {};
\node [style=square box] (1) at (0, 0) {$f$};
\node [style=none] (2) at (0, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1);
\draw (1) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
}
:=
\left(\raisebox{-4mm}{
\begin{tikzpicture}[scale=0.5]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1) {};
\node [style=white dot] (1) at (0, 0) {$\psi$};
\node [style=none] (2) at (0, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1);
\draw (1) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
}\right)^{-1}
$ we have $$\begin{tikzpicture}[scale=0.8]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-5.75, 1.5) {};
\node [style=none] (1) at (-4.75, 1.5) {};
\node [style=none] (2) at (-3.25, 1.5) {};
\node [style=none] (3) at (-2.25, 1.5) {};
\node [style=none] (4) at (-0.75, 1.5) {};
\node [style=none] (5) at (0.25, 1.5) {};
\node [style=none] (6) at (1.75, 1.5) {};
\node [style=none] (7) at (2.75, 1.5) {};
\node [style=square box] (8) at (-5.75, 0.75) {$f$};
\node [style=none] (9) at (-4.75, 0.75) {};
\node [style=square box] (10) at (-3.25, 0.75) {$f$};
\node [style=none] (11) at (-2.25, 0.75) {};
\node [style=square box] (12) at (-0.75, 0.75) {$f$};
\node [style=dot] (13) at (-5.25, 0) {};
\node [style=none] (14) at (-4, 0) {$=$};
\node [style=dot] (15) at (-2.75, 0) {};
\node [style=none] (16) at (-1.5, 0) {$=$};
\node [style=none] (17) at (1, 0) {$=$};
\node [style=white dot] (18) at (-0.75, -0.25) {$\psi$};
\node [style=none] (19) at (0.25, -0.25) {};
\node [style=none] (20) at (1.75, -0.25) {};
\node [style=none] (21) at (2.75, -0.25) {};
\node [style=white dot] (22) at (-2.75, -0.75) {$\psi$};
\node [style=dot] (23) at (-0.25, -1) {};
\node [style=dot] (24) at (2.25, -1) {};
\node [style=square box] (25) at (-2.75, -1.75) {$f$};
\node [style=square box] (26) at (-0.25, -1.75) {$f$};
\node [style=square box] (27) at (2.25, -1.75) {$f$};
\node [style=none] (28) at (-5.25, -2.5) {};
\node [style=none] (29) at (-2.75, -2.5) {};
\node [style=none] (30) at (-0.25, -2.5) {};
\node [style=none] (31) at (2.25, -2.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (22) to (25);
\draw (9.center) to (1.center);
\draw[bend right] (20.center) to (24);
\draw (6.center) to (20.center);
\draw (26) to (30.center);
\draw[bend right] (24) to (21.center);
\draw[bend right] (13) to (9.center);
\draw (7.center) to (21.center);
\draw (27) to (31.center);
\draw[bend right] (15) to (11.center);
\draw (0.center) to (8);
\draw[bend right] (10) to (15);
\draw (4.center) to (12);
\draw[bend right] (8) to (13);
\draw[bend right] (23) to (19.center);
\draw (19.center) to (5.center);
\draw (23) to (26);
\draw (13) to (28.center);
\draw (24) to (27);
\draw[bend right] (18) to (23);
\draw (2.center) to (10);
\draw (25) to (29.center);
\draw (15) to (22);
\draw (11.center) to (3.center);
\draw (12) to (18);
\end{pgfonlayer}
\end{tikzpicture}$$
GHZ/W calculus
--------------
In this paper, we are concerned not only with general CFAs, but two specific cases, depending on the behaviour of the loops. We refer to $\mult \circ \comult$ as the *loop map* of a CFA.
[[@CK]]{} A *GHZ-structure* is a special commutative Frobenius algebra; that is, a commutative Frobenius algebra where the loop map is equal to the identity: $$\begin{tikzpicture}[dotpic,yshift=5mm]
\node [dot] (a) at (0,0) {};
\node [dot] (b) at (0,-1) {};
\draw [bend left] (a) to (b);
\draw [bend right] (a) to (b);
\draw (0,0.5) to (a) (b) to (0,-1.5);
\end{tikzpicture} = \
\begin{tikzpicture}[dotpic]
\draw (0,1) -- (0,-1);
\end{tikzpicture}$$
These GHZ-structures have also been referred to as *basis structures*, for example in [@CES2], because of their strong connection to bases in finite-dimensional vector spaces. See Theorem \[basisthm\] below.
[[@CK]]{} A *W-structure* is an anti-special commutative Frobenius algebra. This is commutative Frobenius algebra whose loop map obeys the following equation: $$\label{eq:antispec}
\circl\ \begin{tikzpicture}[dotpic]
\node [dot] (a) at (0,0.5) {};
\node [dot] (b) at (0,-0.5) {};
\draw [bend left] (a) to (b);
\draw [bend right] (a) to (b);
\draw (0,1) to (a) (b) to (0,-1);
\end{tikzpicture}\ \ =
\begin{tikzpicture}[dotpic]
\node [dot] (a) at (0,0.7) {};
\node [dot] (b) at (0,-0.7) {};
\draw (0,1.2) to (a) (b) to (0,-1.2);
\draw (a) to [downloop] ();
\draw (b) to [uploop] ();
\end{tikzpicture}$$ where we use the following short-hand notation: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (1, 0.75) {};
\node [style=none] (1) at (5.25, 0.75) {};
\node [style=none] (2) at (3.25, 0.5) {};
\node [style=dot] (3) at (1, 0.25) {};
\node [style=dot] (4) at (5.25, 0.25) {};
\node [style=dot] (5) at (-1, 0) {};
\node [style=none] (6) at (0, 0) {=};
\node [style=dot] (7) at (3.25, 0) {};
\node [style=none] (8) at (4.25, 0) {=};
\node [style=dot] (9) at (1, -0.25) {};
\node [style=dot] (10) at (5.25, -0.25) {};
\node [style=none] (11) at (-1, -0.5) {};
\node [style=none] (12) at (1, -0.75) {};
\node [style=dot] (13) at (5.25, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw[bend left=300, looseness=1.25] (10) to (4);
\draw[bend right=300, looseness=1.25] (3) to (9);
\draw[out=45, in=135, loop] (5) to ();
\draw (4) to (1.center);
\draw[bend left=60, looseness=1.25] (10) to (4);
\draw (9) to (12.center);
\draw (0) to (3);
\draw[bend right=60, looseness=1.25] (3) to (9);
\draw[out=-45, in=-135, loop] (7) to ();
\draw (5) to (11.center);
\draw (7) to (2.center);
\draw (13) to (10);
\end{pgfonlayer}
\end{tikzpicture}$$
This distinction essentially comes down to whether the loop map is singular or invertible.
\[lem:iso-scfa\] If the loop map of a CFA is an isomorphism, the CFA can be made special via a phase.
Consider a CFA $(\mult, \unit, \comult, \counit)$. Since its loop map is a phase, by Proposition \[prop:invphase\] so is the inverse of the loop map, which we denote $f$. Then $(\,\mult\, , \, \unit\,,
\raisebox{-4.5mm}{
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-8, 1.5) {};
\node [style=square box] (1) at (-8, 0.75) {$f$};
\node [style=dot] (2) at (-8, 0) {};
\node [style=none] (3) at (-8.5, -0.5) {};
\node [style=none] (4) at (-7.5, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (4.center);
\draw (1) to (2);
\draw (0.center) to (1);
\draw (2) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}}
,\,\cololli\,)$ is easily seen to be a special CFA.
If the loop of a CFA is disconnected, i.e. factors over the tensor unit, then it obeys eq. [(\[eq:antispec\])]{}, that is the CFA is necessarily anti-special.
The following is an example of a GHZ-structure in ${\bf FHilb}$: $$\label{GHZ-SCFA}
\begin{split}
\whitemult & = {|0\rangle}{\langle00|} + {|1\rangle}{\langle11|} \qquad\qquad
\whiteunit = \sqrt{2}\, {|+\rangle} := {|0\rangle}+{|1\rangle} \\
\whitecomult & = {|00\rangle}{\langle0|} + {|11\rangle}{\langle1|} \qquad\qquad
\whitecounit = \sqrt{2} {\langle+|} := {\langle0|}+{\langle1|}
\end{split}\vspace{-1.5mm}$$ and we also have an example of a W-structure in ${\bf FHilb}$: $$\label{W-ACFA}
\begin{split}
\mult & = {|1\rangle}{\langle11|} + {|0\rangle}{\langle01|} + {|0\rangle}{\langle10|}
\qquad\qquad\qquad
\unit = {|1\rangle}\qquad\qquad \\
\comult & = {|00\rangle}{\langle0|} + {|01\rangle}{\langle1|} + {|10\rangle}{\langle1|}
\qquad\qquad\qquad
\counit = {\langle0|}\qquad\qquad
\end{split}$$ Note that the cups for these CFAs do not coincide: $$\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (1.5, 0.75) {};
\node [style=white dot] (1) at (-1.5, 0.25) {};
\node [style=white dot] (2) at (1.5, 0.25) {};
\node [style=none] (3) at (0, 0) {$:=$};
\node [style=none] (4) at (-2, -0.25) {};
\node [style=none] (5) at (-1, -0.25) {};
\node [style=none] (6) at (1, -0.25) {};
\node [style=none] (7) at (2, -0.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2);
\draw (7.center) to (2);
\draw[bend right=45] (5.center) to (1);
\draw (6.center) to (2);
\draw[bend left=45] (4.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}\ = {|00\rangle}+{|11\rangle}
\qquad\qquad
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (1.5, 0.75) {};
\node [style=dot] (1) at (-1.5, 0.25) {};
\node [style=dot] (2) at (1.5, 0.25) {};
\node [style=none] (3) at (0, 0) {$:=$};
\node [style=none] (4) at (-2, -0.25) {};
\node [style=none] (5) at (-1, -0.25) {};
\node [style=none] (6) at (1, -0.25) {};
\node [style=none] (7) at (2, -0.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2);
\draw (7.center) to (2);
\draw[bend right=45] (5.center) to (1);
\draw (6.center) to (2);
\draw[bend left=45] (4.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}\ = {|01\rangle}+{|10\rangle}$$
However, the composition of a cap from one CFA with a cup from the other yields the Pauli X, or ‘NOT’, gate: $$\begin{tikzpicture}[dotpic,scale=0.5]
\node [bn] (0) at (0,1.5) {};
\node [bn] (1) at (0,-1.5) {};
\draw (0)-- node[tick]{-} (1);
\end{tikzpicture}
:=\
\begin{tikzpicture}[dotpic,yshift=-5mm,scale=0.5]
\node [bn] (b0) at (-1,2) {};
\node [dot] (0) at (0,0) {};
\node [white dot] (1) at (1.5,1) {};
\node [bn] (b1) at (2.5,-1) {};
\draw (b0) to [out=-90,in=180] (0) (0) to [out=0,in=180] (1) (1) to [out=0,in=90] (b1);
\end{tikzpicture} \ =\
\begin{tikzpicture}[dotpic,yshift=-5mm,scale=0.5]
\node [bn] (b0) at (-1,2) {};
\node [white dot] (0) at (0,0) {};
\node [dot] (1) at (1.5,1) {};
\node [bn] (b1) at (2.5,-1) {};
\draw (b0) to [out=-90,in=180] (0) (0) to [out=0,in=180] (1) (1) to [out=0,in=90] (b1);
\end{tikzpicture}
\ =\
\left(\begin{array}{cc}
0 & 1 \\
1 & 0
\end{array}\right)$$ These CFAs respectively induce the following tripartite states: $$\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.75) {};
\node [style=white dot] (1) at (0, 0) {};
\node [style=white dot] (2) at (-0.5, -0.5) {};
\node [style=none] (3) at (-1, -1) {};
\node [style=none] (4) at (0, -1) {};
\node [style=none] (5) at (1, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (2);
\draw (3.center) to (2);
\draw (2) to (1);
\draw (0) to (1);
\draw (5.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}
\ = {|000\rangle}+{|111\rangle} = {{|\textit{GHZ}\,\rangle}}\qquad
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 0.75) {};
\node [style=dot] (1) at (0, 0) {};
\node [style=dot] (2) at (-0.5, -0.5) {};
\node [style=none] (3) at (-1, -1) {};
\node [style=none] (4) at (0, -1) {};
\node [style=none] (5) at (1, -1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (2);
\draw (3.center) to (2);
\draw (2) to (1);
\draw (0) to (1);
\draw (5.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}
\ = {|100\rangle}+{|010\rangle}+{|001\rangle} = {{|\textit{W\,}\rangle}}$$ As the name suggests, the associated tripartite state of the above GHZ-structure is a GHZ state, and that of the W-structure is a W state. Furthermore, Theorem \[GHZ/Wthm\] asserts that for qubits, the associated tripartite state of *any* GHZ-structure (resp. W-structure) is a GHZ state (resp. W state), up to local operations.
[[@CK]]{}\[GHZ/Wthm\] For any special (respectively anti-special) CFA on a qubit in ${\bf FHilb}$, the induced tripartite state is SLOCC-equivalent to ${{|\textit{GHZ}\,\rangle}}$ (respectively ${{|\textit{W\,}\rangle}}$). Furthermore, any tripartite state ${|\Psi\rangle}$ either induces a special or anti-special CFA-structure, depending on whether it is SLOCC-equivalent to ${{|\textit{GHZ}\,\rangle}}$ or to ${{|\textit{W\,}\rangle}}$.
Theorem \[basisthm\] justifies the alternative name *basis structure *for GHZ-structures.**
[[@Aguiar]]{}\[basisthm\] Special commutative Frobenius algebras on a finite-dimensional Hilbert space ${\cal H}$ are in 1-to-1 correspondence with (possibly non-orthogonal) bases for ${\cal H}$.
For any special CFA, phases are matrices that are diagonal in the corresponding basis. The corresponding $|\psi\rangle$ (as in proposition \[prop:phases-states\]) lies on the equator of the Bloch sphere, justifying the name ‘phases’.
We can also consider interactions between a GHZ-structure and a W-structure.
[[@CK]]{} A GHZ- and a W-structure form a *GHZ/W-pair *if the following equations hold:**
[nodelayer]{} (0) at (2.5, 1.5) ; (1) at (5, 1.5) ; (2) at (-3, 1.25) ; (3) at (-0.75, 1) ; (4) at (0.25, 1) ; (5) at (-1.75, 0.75) [$\stackrel{\beta}{=}$]{}; (6) at (3.75, 0.75) [$\stackrel{\gamma}{=}$]{}; (7) at (2.5, 0.5) ; (8) at (5, 0.5) ; (9) at (-3, 0.25) ; (10) at (-0.75, 0) ; (11) at (0.25, 0) ; (12) at (-3.5, -0.25) ; (13) at (-2.5, -0.25) ; (14) at (1.75, -0.25) ; (15) at (3.25, -0.25) ; (16) at (4.25, -0.25) ; (17) at (5.75, -0.25) ;
[edgelayer]{} (7) to (15.center); (2) to (9); (12.center) to (9); (9) to (13.center); (8) to node\[tick\][-]{} (17.center); (16.center) to node\[tick\][-]{} (8); (0.center) to node\[tick\][-]{} (7); (3) to (10.center); (1.center) to (8); (14.center) to (7); (4) to (11.center);
By eqs. ($\beta$, $\gamma$) we also have:
[nodelayer]{} (0) at (-1.5, -0.75) ; (1) at (0, -1.25) [$\stackrel{\beta'}{=}$]{}; (2) at (1.25, -1.25) ; (3) at (2.25, -1.25) ; (4) at (-1.5, -1.75) ; (5) at (-2, -2.25) ; (6) at (-1, -2.25) ; (7) at (1.25, -2.25) ; (8) at (2.25, -2.25) ;
[edgelayer]{} (0) to node\[tick\][-]{} (4); (3) to node\[tick\][-]{} (8.center); (5.center) to (4); (2) to node\[tick\][-]{} (7.center); (4) to (6.center);
Plugging
--------
Since we are often concerned with objects in a monoidal category that are finitary in nature, we can deduce many new identities using a technique we call *plugging*.
\[def:plugging-set\] A set of points $\{ \psi_i : I \rightarrow Q \}$ form a *plugging set* for $Q$ if they suffice to distinguish maps from $Q$. That is, for all objects $A$ and maps $f,g : Q \rightarrow A$, $$\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-3.5, 1.5) {};
\node [style=none] (1) at (-1.5, 1.5) {};
\node [style=white dot, font=\footnotesize] (2) at (2.5, 1.5) {$\psi_i$};
\node [style=white dot, font=\footnotesize] (3) at (4.5, 1.5) {$\psi_i$};
\node [style=none, font=\footnotesize] (4) at (-3.75, 1.25) {$Q$};
\node [style=none, font=\footnotesize] (5) at (-1.75, 1.25) {$Q$};
\node [style=square box] (6) at (-3.5, 0) {$f$};
\node [style=none] (7) at (-2.5, 0) {=};
\node [style=square box] (8) at (-1.5, 0) {$g$};
\node [style=none] (9) at (0, 0) {$\Leftrightarrow$};
\node [style=none] (10) at (1.5, 0) {$\forall i .$};
\node [style=square box] (11) at (2.5, 0) {$f$};
\node [style=none] (12) at (3.5, 0) {=};
\node [style=square box] (13) at (4.5, 0) {$g$};
\node [style=none, font=\footnotesize] (14) at (-3.75, -1.25) {$A$};
\node [style=none, font=\footnotesize] (15) at (-1.75, -1.25) {$A$};
\node [style=none, font=\footnotesize] (16) at (2.25, -1.25) {$A$};
\node [style=none, font=\footnotesize] (17) at (4.25, -1.25) {$A$};
\node [style=none] (18) at (-3.5, -1.5) {};
\node [style=none] (19) at (-1.5, -1.5) {};
\node [style=none] (20) at (2.5, -1.5) {};
\node [style=none] (21) at (4.5, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (18.center);
\draw (1.center) to (19.center);
\draw (3) to (21.center);
\draw (2) to (20.center);
\end{pgfonlayer}
\end{tikzpicture}$$
When we prove a graphical identity by showing two maps are not distinguished by a plugging set, we call this ‘proof by plugging.’ Also note that we can extend such proofs to maps of the form $f : Q
\otimes A \rightarrow B$ or $f' : A \rightarrow Q \otimes B$ by using the Frobenius caps and cups when $Q$ has a CFA $(\mult, \unit, \comult, \counit)$ and $A$ a CFA $(\whitemult, \whiteunit,
\whitecomult, \whitecounit)$, $$\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-2.25, 1.75) {};
\node [style=white dot] (1) at (4.25, 1.75) {};
\node [style=none] (2) at (-4, 1.5) {};
\node [style=none] (3) at (1.5, 1.5) {};
\node [style=none, font=\footnotesize] (4) at (-4.25, 1.25) {$Q$};
\node [style=none, font=\footnotesize] (5) at (-3.25, 1.25) {$A$};
\node [style=none, font=\footnotesize] (6) at (1.75, 1.25) {$Q$};
\node [style=none, font=\footnotesize] (7) at (3.25, 1.25) {$A$};
\node [style=none] (8) at (-3, 1) {};
\node [style=none] (9) at (-1.5, 1) {};
\node [style=none] (10) at (3.5, 1) {};
\node [style=none] (11) at (5, 1) {};
\node [style=none] (12) at (-4, 0.5) {};
\node [style=none] (13) at (-3, 0.5) {};
\node [style=none] (14) at (3.5, 0.5) {};
\node [style=square box, minimum width=1 cm] (15) at (-3.5, 0) {$f$};
\node [style=square box, minimum width=1 cm] (16) at (3.5, 0) {$f'$};
\node [style=none] (17) at (-3.5, -0.5) {};
\node [style=none] (18) at (3, -0.5) {};
\node [style=none] (19) at (4, -0.5) {};
\node [style=none] (20) at (1.5, -0.75) {};
\node [style=none] (21) at (3, -0.75) {};
\node [style=none, font=\footnotesize] (22) at (-3.75, -1.25) {$B$};
\node [style=none, font=\footnotesize] (23) at (4.25, -1.25) {$B$};
\node [style=none] (24) at (-3.5, -1.5) {};
\node [style=none] (25) at (-1.5, -1.5) {};
\node [style=dot] (26) at (2.25, -1.5) {};
\node [style=none] (27) at (4, -1.5) {};
\node [style=none] (28) at (5, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (12.center);
\draw (8.center) to (13.center);
\draw[out=0, in=90] (0) to (9.center);
\draw (17.center) to (24.center);
\draw[out=180, in=90] (1) to (10.center);
\draw (11.center) to (28.center);
\draw (9.center) to (25.center);
\draw (10.center) to (14.center);
\draw (19.center) to (27.center);
\draw[out=180, in=90] (0) to (8.center);
\draw[out=0, in=90] (1) to (11.center);
\draw (21.center) to (18.center);
\draw[out=0, in=-90] (26) to (21.center);
\draw (20.center) to (3.center);
\draw[out=180, in=-90] (26) to (20.center);
\end{pgfonlayer}
\end{tikzpicture}$$
The axioms of a GHZ/W-pair suffice to prove the following lemma for Hilbert spaces.
\[lem:dot-lolli-2d\] For a GHZ/W-pair on $H$ in ${\bf FHilb}$ with $\dim(H) \geq 2$, the points $\unit$ and $\tickunit$ span a 2-dimensional space; hence for $H=\mathbb{C}^2$ the points $\unit$ and $\tickunit$ form a basis.
Motivated by this fact, we assume the that $\{ \unit, \tickunit \}$ forms a plugging set for $Q$. More explicitly: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (-6, 0.75) {};
\node [style=dot] (1) at (-4.5, 0.75) {};
\node [style=dot] (2) at (-2.5, 0.75) {};
\node [style=dot] (3) at (-1, 0.75) {};
\node [style=none] (4) at (2, 0.75) {};
\node [style=none] (5) at (3.5, 0.75) {};
\node [style=square box] (6) at (-6, 0) {$f$};
\node [style=none] (7) at (-5.25, 0) {$=$};
\node [style=square box] (8) at (-4.5, 0) {$g$};
\node [style=none] (9) at (-3.5, 0) {$\wedge$};
\node [style=square box] (10) at (-2.5, 0) {$f$};
\node [style=none] (11) at (-1.75, 0) {$=$};
\node [style=square box] (12) at (-1, 0) {$g$};
\node [style=none] (13) at (0.5, 0) {$\Leftrightarrow$};
\node [style=square box] (14) at (2, 0) {$f$};
\node [style=none] (15) at (2.75, 0) {$=$};
\node [style=square box] (16) at (3.5, 0) {$g$};
\node [style=none] (17) at (-6, -0.75) {};
\node [style=none] (18) at (-4.5, -0.75) {};
\node [style=none] (19) at (-2.5, -0.75) {};
\node [style=none] (20) at (-1, -0.75) {};
\node [style=none] (21) at (2, -0.75) {};
\node [style=none] (22) at (3.5, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (10) to (19.center);
\draw (0) to (6);
\draw (2) to node[tick]{-} (10);
\draw (16) to (22.center);
\draw (8) to (18.center);
\draw (1) to (8);
\draw (6) to (17.center);
\draw (12) to (20.center);
\draw (14) to (21.center);
\draw (4.center) to (14);
\draw (5.center) to (16);
\draw (3) to node[tick]{-} (12);
\end{pgfonlayer}
\end{tikzpicture}$$
Arithmetic from a GHZ/W-pair {#sec:arithmetic}
============================
Given a GHZ/W-pair, we can extract an arithmetic system. First, we establish some preliminary results.
Properties of GHZ-phases
------------------------
Below, all phases are GHZ-phases. When relying on plugging, we have the following:
\[thmdelta\_1\] $$\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, 1.5) {};
\node [style=none] (1) at (0.5, 1.5) {};
\node [style=dot] (2) at (0, 0.5) {};
\node [style=white dot] (3) at (0, -0.5) {$\psi$};
\node [style=none] (4) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (0) to (2);
\draw [bend right=30] (2) to (1);
\draw (2) to (3.center);
\draw (3.center) to (4);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ \stackrel{\delta_1}{=}\ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.75, 1.5) {};
\node [style=none] (1) at (0.75, 1.5) {};
\node [style=white dot] (2) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (3) at (0.75, 0.5) {$\psi$};
\node [style=dot] (4) at (0, -0.5) {};
\node [style=none] (5) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw [bend right] (2.center) to (4);
\draw [bend right] (4) to (3.center);
\draw (3.center) to (1);
\draw (4) to (5);
\end{pgfonlayer}
\end{tikzpicture}}$$
Plugging $\unit$ to one input: $$\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (-0.5, 1.25) {};
\node [style=none] (1) at (0.5, 1.5) {};
\node [style=dot] (2) at (0, 0.5) {};
\node [style=white dot] (3) at (0, -0.5) {$\psi$};
\node [style=none] (4) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (0) to (2);
\draw [bend right=30] (2) to (1);
\draw (2) to (3.center);
\draw (3.center) to (4);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=white dot] (3) at (0, 0) {$\psi$};
\node [style=none] (4) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (3.center);
\draw (3.center) to (4);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (-0.5, -0.5) {};
\node [style=none] (1) at (0.5, 1.5) {};
\node [style=dot] (2) at (0, -1) {};
\node [style=white dot] (3) at (0.5, 0) {$\psi$};
\node [style=none] (4) at (0, -1.5) {};
\node [style=white dot] (5) at (-0.5, 1) {$\psi$};
\node [style=dot] (6) at (-0.5, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (0) to (2);
\draw [bend right=30] (2) to (3.center);
\draw (1) to (3.center);
\draw (2.center) to (4);
\draw (5) to node[tick]{-} (6.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.25) {};
\node [style=none] (1) at (0.75, 1.5) {};
\node [style=dot] (2) at (0, -1) {};
\node [style=white dot] (3) at (0.75, 0) {$\psi$};
\node [style=none] (4) at (0, -1.5) {};
\node [style=white dot] (5) at (-1.5, 1) {$\psi$};
\node [style=white dot] (6) at (-0.75, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (6) to (2);
\draw [bend right=30] (5) to (6) (6) to (0);
\draw [bend right=30] (2) to (3.center);
\draw (1) to (3.center);
\draw (2.center) to (4);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (-0.75, 1.25) {};
\node [style=none] (1) at (0.75, 1.5) {};
\node [style=white dot] (2) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (3) at (0.75, 0.5) {$\psi$};
\node [style=dot] (4) at (0, -0.5) {};
\node [style=none] (5) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw [bend right] (2.center) to (4);
\draw [bend right] (4) to (3.center);
\draw (3.center) to (1);
\draw (4) to (5);
\end{pgfonlayer}
\end{tikzpicture}}$$ Plugging $\tickunit$ to one input of both sides: $$\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (-0.5, 1.25) {};
\node [style=none] (1) at (0.5, 1.5) {};
\node [style=dot] (2) at (0, 0.5) {};
\node [style=white dot] (3) at (0, -0.5) {$\psi$};
\node [style=none] (4) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (0) to node[tick]{-} (2);
\draw [bend right=30] (2) to (1);
\draw (2) to (3.center);
\draw (3.center) to (4);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=dot] (1) at (0, 1) {};
\node [style=dot] (2) at (0, 0.5) {};
\node [style=white dot] (3) at (0, -0.5) {$\psi$};
\node [style=none] (4) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1) (2) to node[tick]{-} (3);
\draw (3.center) to (4);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0.5, 0.5) {};
\node [style=none] (1) at (0.5, 1.5) {};
\node [style=dot] (5) at (0.5, 0) {};
\node [style=white dot] (2) at (0, -1) {};
\node [style=white dot] (3) at (-0.5, 0) {$\psi$};
\node [style=none] (4) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1);
\draw [bend right=30] (3.center) to (2.center);
\draw [bend right=30] (2.center) to node[tick]{-} (5.center);
\draw (2.center) to (4);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}}}\ {
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=dot] (1) at (0, 0.25) {};
\node [style=dot] (2) at (0, -0.25) {};
\node [style=none] (3) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1) (2) to node[tick]{-} (3);
\end{pgfonlayer}
\end{tikzpicture}}$$
$$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (-0.75, 1.5) {};
\node [style=none] (1) at (0.75, 1.5) {};
\node [style=white dot] (2) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (3) at (0.75, 0.5) {$\psi$};
\node [style=dot] (4) at (0, -0.5) {};
\node [style=none] (5) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (2);
\draw [bend right] (2.center) to (4);
\draw [bend right] (4) to (3.center);
\draw (3.center) to (1);
\draw (4) to (5);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (-0.5, 1.25) {};
\node [style=white dot] (1) at (-1.5, 1) {$\psi$};
\node [style=white dot] (2) at (-1, 0) {};
\node [style=dot] (3) at (0, -1) {};
\node [style=white dot] (4) at (1, 0) {};
\node [style=white dot] (5) at (0.5, 1.25) {$\psi$};
\node [style=none] (6) at (1.5, 1.5) {};
\node [style=none] (7) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (1) to (2) (2) to node[tick]{-} (0);
\draw [bend right=30] (5) to (4) (4) to (6);
\draw [bend right=30] (2) to (3) (3) to (4);
\draw (7) to (3);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (2) at (-1, 0) {};
\node [style=dot] (3) at (0, -1) {};
\node [style=white dot] (4) at (1, 0) {};
\node [style=white dot] (5) at (0.5, 1.25) {$\psi$};
\node [style=none] (6) at (1.5, 1.5) {};
\node [style=none] (7) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (5) to (4) (4) to (6);
\draw [bend right=30] (2) to node[tick]{-} (3) (3) to (4);
\draw (7) to (3);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (4) at (0, 0) {};
\node [style=white dot] (5) at (-0.5, 1.25) {$\psi$};
\node [style=none] (6) at (0.5, 1.5) {};
\node [style=dot] (7) at (0, -0.5) {};
\node [style=dot] (8) at (0, -1) {};
\node [style=none] (9) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (5) to (4) (4) to (6);
\draw (7) to node[tick]{-} (4) (8) to node[tick]{-} (9);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}}}\ {
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=dot] (1) at (0, 0.25) {};
\node [style=dot] (2) at (0, -0.25) {};
\node [style=none] (3) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1) (2) to node[tick]{-} (3);
\end{pgfonlayer}
\end{tikzpicture}}$$
\[thmdelta\_2\] $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ \stackrel{\delta_2}{=}\ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0)--(1);
\end{pgfonlayer}
\end{tikzpicture}}$$
$$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0) {};
\node [style=white dot] (3) at (-1, 1) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (1);
\draw [bend right=30] (3) to (2);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0)--(1);
\end{pgfonlayer}
\end{tikzpicture}}$$
Note that for $\whitemult$-phases we have: $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0) {$\frac{1}{\psi}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw (2.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ := \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (2.center);
\draw (2.center) to node[tick]{-} (1);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0) {};
\node [style=white dot] (3) at (1, 1) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (2.center);
\draw (2.center) to node[tick]{-} (1);
\draw [bend right=60] (2.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0) {};
\node [style=white dot] (3) at (1, 1) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw (2.center) to (1);
\draw [bend right=60] (2.center) to node[tick]{-} (3.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=small white dot,font=\footnotesize] (3) at (1, 1) {${\tickpsi}$};
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=60] (2.center) to (3.center);
\draw (1.center) to (2.center);
\draw (0.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}}$$ The particular choice of notation $\frac{1}{\psi}$ is justified below, and will play a key role in this paper.
\[thmdelta\_3\] $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=small white dot,font=\footnotesize] (2) at (0, 0.75) {$\psi$};
\node [style=small white dot,font=\footnotesize] (3) at (0, -0.5) {$\frac{1}{\psi}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ \stackrel{\delta_3}{=}\ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}}}\ {
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0)--(1);
\end{pgfonlayer}
\end{tikzpicture}}$$
Plugging $\unit$ into the input: $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=small white dot,font=\footnotesize] (2) at (0, 0.75) {$\psi$};
\node [style=small white dot,font=\footnotesize] (3) at (0, -0.5) {$\frac{1}{\psi}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0.75) {};
\node [style=white dot] (3) at (0, -0.5) {};
\node [style=white dot] (4) at (-1, 1.25) {$\psi$};
\node [style=white dot] (5) at (-1, 0) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (1.center);
\draw [bend right=30] (4) to (2) (5) to node[tick]{-} (3);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}}}\ {
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0)--(1);
\end{pgfonlayer}
\end{tikzpicture}}$$ Plugging $\tickunit$ into the input: $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=small white dot,font=\footnotesize] (2) at (0, 0.75) {$\psi$};
\node [style=small white dot,font=\footnotesize] (3) at (0, -0.5) {$\frac{1}{\psi}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (2);
\draw (2.center) to (3.center);
\draw (3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0.75) {};
\node [style=white dot] (3) at (0, -0.5) {};
\node [style=white dot] (4) at (-1, 1.25) {$\psi$};
\node [style=white dot] (5) at (-1, 0) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (2);
\draw (2.center) to (3.center);
\draw (3.center) to (1.center);
\draw [bend right=30] (4) to (2) (5) to node[tick]{-} (3);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{
{
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}}}\ {
\raisebox{5mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {$\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}\
\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) -- node[tick]{-} (1);
\end{pgfonlayer}
\end{tikzpicture}}$$
In a setting like ${\bf FHilb}_p$, where we ignore cancelable scalar multipliers, and provided that the scalars ${
\raisebox{-2.5mm}{\begin{tikzpicture}[scale=0.5]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {\footnotesize$\!\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}}}$ and ${
\raisebox{-2.5mm}{\begin{tikzpicture}[scale=0.5]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.5) {\footnotesize$\!\psi$};
\node [style=dot] (1) at (0, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}}}$ in the equation are cancelable, eqs. ($\delta_1$, $\delta_2$, $\delta_3$) simplify to: $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, 1.5) {};
\node [style=none] (1) at (0.5, 1.5) {};
\node [style=dot] (2) at (0, 0.5) {};
\node [style=white dot] (3) at (0, -0.5) {$\psi$};
\node [style=none] (4) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (0) to (2);
\draw [bend right=30] (2) to (1);
\draw (2) to (3.center);
\draw (3.center) to (4);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ \stackrel{\delta_1}{=}\ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.75, 1.5) {};
\node [style=none] (1) at (0.75, 1.5) {};
\node [style=white dot] (2) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (3) at (0.75, 0.5) {$\psi$};
\node [style=dot] (4) at (0, -0.5) {};
\node [style=none] (5) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw [bend right] (2.center) to (4);
\draw [bend right] (4) to (3.center);
\draw (3.center) to (1);
\draw (4) to (5);
\end{pgfonlayer}
\end{tikzpicture}}
\qquad \qquad \qquad
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, 0) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ \stackrel{\delta_2}{=}\ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0)--(1);
\end{pgfonlayer}
\end{tikzpicture}}
\qquad \qquad \qquad
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=small white dot,font=\footnotesize] (2) at (0, 0.75) {$\psi$};
\node [style=small white dot,font=\footnotesize] (3) at (0, -0.5) {$\frac{1}{\psi}$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (2.center) to (3.center);
\draw (3.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ \stackrel{\delta_3}{=}\ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0)--(1);
\end{pgfonlayer}
\end{tikzpicture}}$$ Examples of phases for which some of these simplified equations fail to hold are: $$\begin{tikzpicture}[scale=0.4]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-7, 1.5) {};
\node [style=none] (1) at (-3.25, 1.5) {};
\node [style=none] (2) at (3.25, 1.5) {};
\node [style=none] (3) at (7, 1.5) {};
\node [style=dot] (4) at (-6, 0.75) {};
\node [style=dot] (5) at (4.25, 0.75) {};
\node [style=dot] (6) at (-3.25, 0.5) {};
\node [style=dot] (7) at (7, 0.5) {};
\node [style=none] (8) at (-4.75, 0) {$=$};
\node [style=none] (9) at (5.5, 0) {$=$};
\node [style=white dot] (10) at (-7, -0.5) {};
\node [style=dot] (11) at (-3.25, -0.5) {};
\node [style=white dot] (12) at (3.25, -0.5) {};
\node [style=dot] (13) at (7, -0.5) {};
\node [style=none] (14) at (-7, -1.5) {};
\node [style=none] (15) at (-3.25, -1.5) {};
\node [style=none] (16) at (3.25, -1.5) {};
\node [style=none] (17) at (7, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (10);
\draw (11) to (15.center);
\draw (16.center) to (12);
\draw (14.center) to (10);
\draw (17.center) to node[tick]{-} (13);
\draw[bend right=45, looseness=0.75] (10) to (4);
\draw[bend right=45, looseness=0.75] (12) to node[tick]{-} (5);
\draw (1.center) to node[tick]{-} (6);
\draw (7) to (3.center);
\draw (2.center) to (12);
\end{pgfonlayer}
\end{tikzpicture}$$
Natural Number Arithmetic {#sec:natural-numbers}
-------------------------
Assume that we are given a GHZ/W-pair. In particular, we have two internal commutative monoids $(\mult, \unit)$ and $(\whitemult, \whiteunit)$. We will now consider the induced commutative monoids on elements: $$\left(
\raisebox{-3mm}{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 1.25) {$\psi$};
\node [style=none] (1) at (-0.5, 0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\raisebox{1.5mm}{,}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 1.25) {$\phi$};
\node [style=none] (1) at (-0.5, 0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}}
\right)
\mapsto
\raisebox{-4mm}{\begin{tikzpicture}[scale=0.8]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 1.25) {$\psi$};
\node [style=white dot] (1) at (0.5, 1.25) {$\phi$};
\node [style=dot] (2) at (0, 0.75) {};
\node [style=none] (3) at (0, 0.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2);
\draw (1) to (2);
\draw (2) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}}
\qquad\qquad
\left(
\raisebox{-3mm}{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 1.25) {$\psi$};
\node [style=none] (1) at (-0.5, 0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\raisebox{1.5mm}{,}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 1.25) {$\phi$};
\node [style=none] (1) at (-0.5, 0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}}
\right)
\mapsto
\raisebox{-4mm}{\begin{tikzpicture}[scale=0.8]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 1.25) {$\psi$};
\node [style=white dot] (1) at (0.5, 1.25) {$\phi$};
\node [style=white dot] (2) at (0, 0.75) {};
\node [style=none] (3) at (0, 0.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2);
\draw (1) to (2);
\draw (2) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}}$$ We will call $\mult$ applied to elements *addition *and $\whitemult$ applied to elements *multiplication*, for reasons that will become apparent shortly. Similarly, we call $\unit$ the *unit for addition *and $\whiteunit$ the *unit for multiplication*. By Theorem \[thmdelta\_1\] we have a distributivity law, up to a scalar, partially explaining our choices of the names addition and multiplication for the monoids:********
\[cor-dist\] $$\begin{tikzpicture}[scale=1]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-2.25, 1) {$\phi$};
\node [style=white dot] (1) at (-1.25, 1) {$\varphi$};
\node [style=white dot] (2) at (0.5, 1) {$\psi$};
\node [style=white dot] (3) at (1.25, 1) {$\phi$};
\node [style=white dot] (4) at (2.25, 1) {$\psi$};
\node [style=white dot] (5) at (3, 1) {$\varphi$};
\node [style=white dot] (6) at (-2.75, 0.5) {$\psi$};
\node [style=dot] (7) at (-1.75, 0.5) {};
\node [style=white dot] (8) at (1.25, 0.5) {};
\node [style=white dot] (9) at (2.25, 0.5) {};
\node [style=white dot] (10) at (-3.5, 0.25) {$\psi$};
\node [style=white dot] (11) at (-2.25, 0) {};
\node [style=none] (12) at (-0.25, 0) {$=$};
\node [style=dot] (13) at (1.75, 0) {};
\node [style=dot] (14) at (-3.5, -0.25) {};
\node [style=none] (15) at (-2.25, -0.5) {};
\node [style=none] (16) at (1.75, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7) to (11);
\draw (3) to (8);
\draw (11) to (15.center);
\draw (9) to (13);
\draw (10) to node[tick]{-} (14);
\draw (6) to (11);
\draw (2) to (8);
\draw (8) to (13);
\draw (1) to (7);
\draw (4) to (9);
\draw (0) to (7);
\draw (13) to (16.center);
\draw (5) to (9);
\end{pgfonlayer}
\end{tikzpicture}$$
Moreover, we can use these to do concrete arithmetic on the natural numbers. We start by defining an encoding for the natural numbers: $$\begin{tikzpicture}[scale=1]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-5.75, -1) {$0$};
\node [style=dot] (1) at (-3.75, -1) {};
\node [style=white dot] (2) at (-0.75, -1) {$n\!+\!1$};
\node [style=white dot] (3) at (1.25, -1) {$n$};
\node [style=white dot] (4) at (2.25, -1) {};
\node [style=none] (5) at (-4.75, -1.5) {$=$};
\node [style=none] (6) at (0.25, -1.5) {$=$};
\node [style=dot] (7) at (1.75, -1.5) {};
\node [style=none] (8) at (-5.75, -2) {};
\node [style=none] (9) at (-3.75, -2) {};
\node [style=none] (10) at (-0.75, -2) {};
\node [style=none] (11) at (1.75, -2) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4) to (7);
\draw (1) to (9.center);
\draw (3) to (7);
\draw (0) to (8.center);
\draw (7) to (11.center);
\draw (2) to (10.center);
\end{pgfonlayer}
\end{tikzpicture}$$ From hence forth, we shall assume we are working in a category with no non-trivial invertible (i.e. non-zero) scalars, such as ${\bf FHilb}_p$. Thus, we shall drop any invertible scalars. Furthermore, we shall assume scalar (i.) is invertible for all $n$ and scalar (ii.) is invertible for all $n \neq 0$: $$\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-1, 0.5) {$n$};
\node [style=white dot] (1) at (2, 0.5) {$n$};
\node [style=none] (2) at (-2, 0) {(i.)};
\node [style=none] (3) at (1, 0) {(ii.)};
\node [style=dot] (4) at (-1, -0.5) {};
\node [style=dot] (5) at (2, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (5);
\draw (0) to node[tick]{-} (4);
\end{pgfonlayer}
\end{tikzpicture}$$
That $\mult$ is the normal addition operation for these numbers follows immediately from their definition and associativity of $\mult$. We can also show that $\whitemult$ is the normal multiplication operation (noting first that the encoding of $1$ is $\whiteunit$, and hence the unit of $\whitemult$):
$$\raisebox{-8mm}{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 1.25) {$n$};
\node [style=white dot] (1) at (0.5, 1.25) {$m$};
\node [style=white dot] (2) at (0, 0.75) {};
\node [style=none] (3) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (2.center);
\draw (1.center) to (2.center);
\draw (2.center) to (3);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 1.75) {$\mdots$};
\node [style=none] (4) at (0.5, 1.25) {};
\node [style=white dot] (1) at (-0.5, 1.25) {$n$};
\node [style=white dot] (2) at (0, 0.75) {};
\node [style=none] (3) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4) to (2);
\draw (1) to (2);
\draw (2) to (3);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ \stackrel{\delta_1}{=}\ \
\raisebox{-8mm}{$\overbrace{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-1, 2) {$n$};
\node [style=white dot] (1) at (-0.5, 2) {};
\node [style=white dot] (2) at (-0.75, 1.5) {};
\node [style=white dot] (3) at (1, 2) {};
\node [style=white dot] (4) at (0.5, 2) {$n$};
\node [style=white dot] (5) at (0.75, 1.5) {};
\node [style=none] (6) at (0, 1.5) {$\ldots$};
\node [style=dot] (7) at (0, 0.75) {};
\node [style=none] (8) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2);
\draw (1) to (2);
\draw (3) to (5);
\draw (4) to (5);
\draw (7) to (2);
\draw (7) to (5);
\draw (7) to (8);
\end{pgfonlayer}
\end{tikzpicture}}^m$}
\ \ = \ \
\raisebox{-8mm}{$\overbrace{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.75, 1.5) {$n$};
\node [style=white dot] (1) at (0.75, 1.5) {$n$};
\node [style=none] (2) at (0, 1.25) {$\ldots$};
\node [style=dot] (3) at (0, 0.75) {};
\node [style=none] (4) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (3);
\draw (1) to (3);
\draw (3) to (4);
\end{pgfonlayer}
\end{tikzpicture}}^m$}
\ \ = \ \
\raisebox{-8mm}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 1.25) {$n m$};
\node [style=none] (1) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}}$$
The distributivity law stated earlier now translates into the normal distributivity of multiplication over addition in the natural numbers, up to a scalar.
Multiplicative inverses {#sec:rationals}
-----------------------
By Theorem \[thmdelta\_3\] we have that $\tick$ is also an inverse for $\whitemult$, up to a scalar:
\[cor-inverse\] $$\begin{tikzpicture}[scale=1]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-2.75, 0.5) {$\psi$};
\node [style=white dot] (1) at (-1.75, 0.5) {$\psi$};
\node [style=white dot] (2) at (1.5, 0.5) {};
\node [style=white dot] (3) at (0, 0.25) {$\psi$};
\node [style=white dot] (4) at (0.75, 0.25) {$\psi$};
\node [style=white dot] (5) at (-2.25, 0) {};
\node [style=none] (6) at (-1, 0) {$=$};
\node [style=dot] (7) at (0, -0.25) {};
\node [style=dot] (8) at (0.75, -0.25) {};
\node [style=none] (9) at (-2.25, -0.5) {};
\node [style=none] (10) at (1.5, -0.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to node[tick]{-} (5);
\draw (0) to (5);
\draw (3) to (7);
\draw (5) to (9.center);
\draw (4) to node[tick]{-} (8);
\draw (2) to (10.center);
\end{pgfonlayer}
\end{tikzpicture}$$
Hence, we have an encoding for the multiplicative inverses of the natural numbers: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0.5, 1) {$\frac{1}{n}$};
\node [style=none] (1) at (0.5, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\ \ \, \raisebox{6.5mm}{:=}\,
\overbrace{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0.75, 1.5) {};
\node [style=none] (1) at (1.25, 1.5) {$\ldots$};
\node [style=white dot] (2) at (1.75, 1.5) {};
\node [style=dot] (3) at (1.25, 1) {};
\node [style=none] (4) at (1.25, 0.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (3);
\draw (0) to (3);
\draw (3) to node[tick]{-} (4.center);
\end{pgfonlayer}
\end{tikzpicture}}^n
\, \raisebox{6.5mm}{=}\, \ \
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0.5, 1) {$n$};
\node [style=none] (1) at (0.5, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[tick]{-} (1.center);
\end{pgfonlayer}
\end{tikzpicture}$$ Thoughtout this paper, we shall assume any natural number occurring in the denominator is not equal to $0$. This allows us to encode positive fractions in the following form: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0.5, -1.25) {$n$};
\node [style=white dot] (1) at (2, -1.25) {$\frac{1}{m}$};
\node [style=white dot] (2) at (3.5, -1.25) {$\frac{1}{m}$};
\node [style=white dot] (3) at (5, -1.25) {$n$};
\node [style=white dot] (4) at (-1.25, -1.75) {$\frac{n}{m}$};
\node [style=none] (5) at (-0.25, -2) {$=$};
\node [style=white dot] (6) at (1.25, -2) {};
\node [style=none] (7) at (2.75, -2) {$=$};
\node [style=white dot] (8) at (4.25, -2) {};
\node [style=none] (9) at (-1.25, -2.75) {};
\node [style=none] (10) at (1.25, -2.75) {};
\node [style=none] (11) at (4.25, -2.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (8);
\draw (4) to (9.center);
\draw (8) to (11.center);
\draw (0) to (6);
\draw (3) to (8);
\draw (6) to (10.center);
\draw (1) to (6);
\end{pgfonlayer}
\end{tikzpicture}$$ where the second equality follows from associativity and commutativity of the GHZ-structure.
It should be noted that the construction of this encoding depends on the choice of numerator and denominator, and not just on the rational number being represented. Therefore, we should demonstrate that the actual point depends only on the number represented. We start by noting that, by corollary \[cor-inverse\], $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.75) {$n$};
\node [style=white dot] (1) at (1.5, 0.75) {$\frac{1}{n}$};
\node [style=white dot] (2) at (-1.75, 0.25) {$\frac{n}{n}$};
\node [style=white dot] (3) at (3.75, 0.25) {};
\node [style=none] (4) at (-0.75, -0) {$=$};
\node [style=white dot] (5) at (0.75, -0) {};
\node [style=none] (6) at (2.25, -0) {$=$};
\node [style=none] (7) at (-1.75, -0.75) {};
\node [style=none] (8) at (0.75, -0.75) {};
\node [style=none] (9) at (3.75, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (7.center);
\draw (3) to (9.center);
\draw (0) to (5);
\draw (5) to (8.center);
\draw (1) to (5);
\end{pgfonlayer}
\end{tikzpicture}$$ where we have ignored the scalar in corollary \[cor-inverse\], since it is cancellable for the points that we have used to encode the natural numbers.
We know that if $\frac{n}{m} = \frac{n'}{m'}$, there is a $p$ such that $n = n'p$ and $m = m'p$, so it follows easily that $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-5, 0.75) {\scriptsize$n$};
\node [style=white dot] (1) at (-3.5, 0.75) {$\frac{1}{m}$};
\node [style=white dot] (2) at (-2.25, 0.75) {\scriptsize$n'p$};
\node [style=white dot] (3) at (-0.75, 0.75) {$\frac{1}{m'p}$};
\node [style=white dot] (4) at (0.5, 0.75) {\scriptsize$n'$};
\node [style=white dot] (5) at (1.25, 0.75) {\scriptsize$p$};
\node [style=white dot] (6) at (2, 0.75) {$\frac{1}{m'}$};
\node [style=white dot] (7) at (3, 0.75) {$\frac{1}{p}$};
\node [style=white dot] (8) at (4.25, 0.75) {$\frac{n'}{m'}$};
\node [style=white dot] (9) at (5.75, 0.75) {$\frac{p}{p}$};
\node [style=white dot] (10) at (-6.25, 0.25) {$\frac{n}{m}$};
\node [style=white dot] (11) at (6.75, 0.25) {$\frac{n'}{m'}$};
\node [style=none] (12) at (-5.5, -0) {$=$};
\node [style=white dot] (13) at (-4.25, -0) {};
\node [style=none] (14) at (-2.75, -0) {$=$};
\node [style=white dot] (15) at (-1.5, -0) {};
\node [style=none] (16) at (0, -0) {$=$};
\node [style=white dot] (17) at (1.75, 0) {};
\node [style=none] (18) at (3.5, -0) {$=$};
\node [style=white dot] (19) at (5, -0) {};
\node [style=none] (20) at (6, -0) {$=$};
\node [style=none] (21) at (-6.25, -0.75) {};
\node [style=none] (22) at (-4.25, -0.75) {};
\node [style=none] (23) at (-1.5, -0.75) {};
\node [style=none] (24) at (1.75, -0.75) {};
\node [style=none] (25) at (5, -0.75) {};
\node [style=none] (26) at (6.75, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (7) to (17);
\draw (9) to (19);
\draw (4) to (17);
\draw (1) to (13);
\draw (17) to (24.center);
\draw (19) to (25.center);
\draw (8) to (19);
\draw (6) to (17);
\draw (5) to (17);
\draw (2) to (15);
\draw (10) to (21.center);
\draw (11) to (26.center);
\draw (0) to (13);
\draw (13) to (22.center);
\draw (3) to (15);
\draw (15) to (23.center);
\end{pgfonlayer}
\end{tikzpicture}$$
\[ex:invfac\] All the usual properties of fractions follow in a straightforward manner from the axioms of rational arithmetic that we have proved. For example, $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0.75, 1.5) {$\frac{n}{m}$};
\node [style=white dot] (1) at (2.25, 1.5) {$\frac{n'}{m'}$};
\node [style=white dot] (2) at (4.25, 1) {$\frac{n n'}{m m'}$};
\node [style=white dot] (3) at (1.5, 0.75) {};
\node [style=none] (4) at (3.25, 0.75) {$=$};
\node [style=none] (5) at (1.5, 0) {};
\node [style=none] (6) at (4.25, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (3);
\draw (1) to (3);
\draw (2) to (6.center);
\draw (3) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}$$ is immediate from the associativity commutativity of the GHZ-structure, and we also have: $$\raisebox{-8mm}{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=small white dot] (0) at (-0.5, 1.5) {$\frac{n}{m}$};
\node [style=small white dot] (1) at (0.5, 1.5) {$\frac{n'}{m'}$};
\node [style=dot] (3) at (0, 0.5) {};
\node [style=none] (4) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (3.center);
\draw (1.center) to (3.center);
\draw (3.center) to (4);
\end{pgfonlayer}
\end{tikzpicture}}
\ \stackrel{\delta_3}{=}\
\raisebox{-8mm}{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=small white dot] (0) at (-1.5, 2) {$\frac{n}{m}$};
\node [style=small white dot] (1) at (1.5, 2) {$\frac{n'}{m'}$};
\node [style=small white dot] (2) at (-0.5, 2) {$\frac{m'}{m'}$};
\node [style=small white dot] (3) at (0.5, 2) {$\frac{m}{m}$};
\node [style=white dot] (4) at (-1, 1) {};
\node [style=white dot] (5) at (1, 1) {};
\node [style=dot] (6) at (0, 0.5) {};
\node [style=none] (7) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (4.center);
\draw (2.center) to (4.center);
\draw (1.center) to (5.center);
\draw (3.center) to (5.center);
\draw (4.center) to (6.center);
\draw (5.center) to (6.center);
\draw (6.center) to (7);
\end{pgfonlayer}
\end{tikzpicture}}
\ = \
\raisebox{-8mm}{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=small white dot] (0) at (-1.5, 2) {\scriptsize$n m'$};
\node [style=small white dot] (1) at (1.5, 2) {\scriptsize$m n'$};
\node [style=small white dot] (2) at (-0.5, 2) {$\frac{1}{m m'}$};
\node [style=small white dot] (3) at (0.5, 2) {$\frac{1}{m m'}$};
\node [style=white dot] (4) at (-1, 1) {};
\node [style=white dot] (5) at (1, 1) {};
\node [style=dot] (6) at (0, 0.5) {};
\node [style=none] (7) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (4.center);
\draw (2.center) to (4.center);
\draw (1.center) to (5.center);
\draw (3.center) to (5.center);
\draw (4.center) to (6.center);
\draw (5.center) to (6.center);
\draw (6.center) to (7);
\end{pgfonlayer}
\end{tikzpicture}}
\ \stackrel{\delta_1}{=}\
\raisebox{-8mm}{\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=small white dot] (0) at (-0.5, 2) {\scriptsize$n m'$};
\node [style=small white dot] (1) at (0.5, 2) {\scriptsize$m n'$};
\node [style=small white dot] (2) at (0.75, 1) {$\frac{1}{m m'}$};
\node [style=dot] (3) at (0, 1.25) {};
\node [style=white dot] (4) at (0, 0.5) {};
\node [style=none] (5) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (3.center);
\draw (1.center) to (3.center);
\draw (3.center) to (4.center);
\draw (4.center) to (2.center);
\draw (4.center) to (5);
\end{pgfonlayer}
\end{tikzpicture}}
\ = \
\raisebox{-8mm}{
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=small white dot] (0) at (0, 1.5) {$\frac{n m' + m n'}{m m'}$};
\node [style=none] (1) at (0, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}}$$
where we used distributivity and ignored the scalars; this is fine as the scalars in corollary \[cor-dist\] are cancellable for all the points we are considering.
Additive inverses {#sec:additive-inverses}
=================
The last thing we need to actually produce a field of fractions is additive inverses. Suppose we have an involutive operation $\cross$ that is a phase for $\whitemult$: $$\begin{tikzpicture}[dotpic]
\node [style=none] (0) at (0,-1) {};
\node [style=white dot] (1) at (0,-0.25) {};
\node [style=none] (2) at (-0.5,0.5) {};
\node [style=none] (3) at (0.5,0.5) {};
\draw (0) to node[pauli z]{$\times$} (1) (1) to (2) (1) to (3);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=none] (0) at (0,-1) {};
\node [style=white dot] (1) at (0,-0.25) {};
\node [style=none] (2) at (-0.5,0.5) {};
\node [style=none] (3) at (0.5,0.5) {};
\draw (0) to (1) (1) to node[pauli z]{$\times$} (2) (1) to (3);
\end{tikzpicture}$$ Suppose further that $\{ \whiteunit,\ \whitecrossunit \}$ forms a plugging set: $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-6, 0.75) {};
\node [style=white dot] (1) at (-4.5, 0.75) {};
\node [style=white dot] (2) at (-2.5, 0.75) {};
\node [style=white dot] (3) at (-1, 0.75) {};
\node [style=none] (4) at (2, 0.75) {};
\node [style=none] (5) at (3.5, 0.75) {};
\node [style=square box] (6) at (-6, 0) {$f$};
\node [style=none] (7) at (-5.25, 0) {$=$};
\node [style=square box] (8) at (-4.5, 0) {$g$};
\node [style=none] (9) at (-3.5, 0) {$\wedge$};
\node [style=square box] (10) at (-2.5, 0) {$f$};
\node [style=none] (11) at (-1.75, 0) {$=$};
\node [style=square box] (12) at (-1, 0) {$g$};
\node [style=none] (13) at (0.5, 0) {$\Leftrightarrow$};
\node [style=square box] (14) at (2, 0) {$f$};
\node [style=none] (15) at (2.75, 0) {$=$};
\node [style=square box] (16) at (3.5, 0) {$g$};
\node [style=none] (17) at (-6, -0.75) {};
\node [style=none] (18) at (-4.5, -0.75) {};
\node [style=none] (19) at (-2.5, -0.75) {};
\node [style=none] (20) at (-1, -0.75) {};
\node [style=none] (21) at (2, -0.75) {};
\node [style=none] (22) at (3.5, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (14) to (21.center);
\draw (8) to (18.center);
\draw (4.center) to (14);
\draw (0) to (6);
\draw (6) to (17.center);
\draw (12) to (20.center);
\draw (16) to (22.center);
\draw (5.center) to (16);
\draw (2) to node[pauli z]{$\times$} (10);
\draw (1) to (8);
\draw (10) to (19.center);
\draw (3) to node[pauli z]{$\times$} (12);
\end{pgfonlayer}
\end{tikzpicture}$$ Then this will act as an additive inverse operation.
\[lem:pauli-z-homom\] $$\begin{tikzpicture}[dotpic]
\node [style=none] (0) at (0,-1) {};
\node [style=dot] (1) at (0,-0.25) {};
\node [style=none] (2) at (-0.5,0.5) {};
\node [style=none] (3) at (0.5,0.5) {};
\draw (0) to node[pauli z]{$\times$} (1) (1) to (2) (1) to (3);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=none] (0) at (0,-1) {};
\node [style=dot] (1) at (0,-0.25) {};
\node [style=none] (2) at (-0.5,0.5) {};
\node [style=none] (3) at (0.5,0.5) {};
\draw (0) to (1) (1) to node[pauli z]{$\times$} (2) (1) to node[pauli z]{$\times$} (3);
\end{tikzpicture}$$
$$\begin{tikzpicture}[dotpic]
\node [style=none] (0) at (0.5,1) {};
\node [style=none] (1) at (-0.5,1) {};
\node [style=dot] (2) at (0,0.25) {};
\node [style=none] (3) at (0,-0.5) {};
\draw (0) to node[pauli z]{$\times$} (2) (1) to node[pauli z]{$\times$} (2) (2) to (3);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, 1.25) {};
\node [style=none] (1) at (0.5, 1.25) {};
\node [style=white dot] (2) at (-1.25, 1) {};
\node [style=white dot] (3) at (1.25, 1) {};
\node [style=white dot] (4) at (-0.5, 0.5) {};
\node [style=white dot] (5) at (0.5, 0.5) {};
\node [style=dot] (6) at (0, 0) {};
\node [style=none] (7) at (0, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw[bend right] (2) to (4);
\draw (4) to (0.center);
\draw[bend right] (5) to (3);
\draw (4) to node[pauli z]{$\times$} (6);
\draw (5) to node[pauli z]{$\times$} (6);
\draw (6) to (7.center);
\draw (5) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, 1.25) {};
\node [style=none] (1) at (0.5, 1.25) {};
\node [style=white dot] (2) at (-1.25, 1) {};
\node [style=white dot] (3) at (1.25, 1) {};
\node [style=white dot] (4) at (-0.5, 0.5) {};
\node [style=white dot] (5) at (0.5, 0.5) {};
\node [style=dot] (6) at (0, 0) {};
\node [style=none] (7) at (0, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (6) to (7.center);
\draw (4) to (0.center);
\draw[bend right] (5) to node[pauli z]{$\times$} (3);
\draw (4) to (6);
\draw (5) to (1.center);
\draw[bend right] (2) to node[pauli z]{$\times$} (4);
\draw (5) to (6);
\end{pgfonlayer}
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=none] (0) at (0.5,1) {};
\node [style=none] (1) at (-0.5,1) {};
\node [style=dot] (2) at (0,0.5) {};
\node [style=white dot] (3) at (0,0) {};
\node [style=white dot] (4) at (0.5,0.5) {};
\node [style=none] (5) at (0,-0.5) {};
\draw (0) to (2) (1) to (2) (2) to (3) (3) to (5);
\draw [bend right] (3) to node[pauli z]{$\times$} (4);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=none] (0) at (0.5,1) {};
\node [style=none] (1) at (-0.5,1) {};
\node [style=dot] (2) at (0,0.5) {};
\node [style=white dot] (3) at (0,-0.25) {};
\node [style=white dot] (4) at (0.5,0.25) {};
\node [style=none] (5) at (0,-0.75) {};
\draw (0) to (2) (1) to (2) (2) to node[pauli z]{$\times$} (3) (3) to (5);
\draw [bend right] (3) to (4);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=none] (0) at (0.5,1) {};
\node [style=none] (1) at (-0.5,1) {};
\node [style=dot] (2) at (0,0.25) {};
\node [style=none] (3) at (0,-0.5) {};
\draw (0) to (2) (1) to (2) (2) to node[pauli z]{$\times$} (3);
\end{tikzpicture}$$
\[lem:pauli-z-zero\] $$\begin{tikzpicture}[dotpic]
\node [style=dot] (0) at (0,0.25) {};
\node [style=none] (1) at (0,-0.5) {};
\draw (0) to node[pauli z]{$\times$} (1);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=dot] (0) at (0,0.25) {};
\node [style=none] (1) at (0,-0.5) {};
\draw (0) to (1);
\end{tikzpicture}$$
$$\begin{tikzpicture}[dotpic]
\node [style=dot] (0) at (0,0.25) {};
\node [style=none] (1) at (0,-0.5) {};
\draw (0) to node[pauli z]{$\times$} (1);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=dot] (0) at (-0.5,0.5) {};
\node [style=dot] (1) at (0.5,0.5) {};
\node [style=dot] (2) at (0,0) {};
\node [style=none] (3) at (0,-0.5) {};
\draw[bend right] (0) to node[pauli z]{$\times$} (2) (2) to (1);
\draw (2) to (3);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=dot] (0) at (-0.5,0.5) {};
\node [style=dot] (1) at (0.5,0.5) {};
\node [style=dot] (2) at (0,0) {};
\node [style=none] (3) at (0,-0.5) {};
\draw[bend right] (0) to (2) (2) to node[pauli z]{$\times$} (1);
\draw (2) to node[pauli z]{$\times$} (3);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=dot] (0) at (0,0.5) {};
\node [style=none] (1) at (0,-0.5) {};
\draw (0) to node[pauli z,pos=0.3]{$\times$} node[pauli z,pos=0.7]{$\times$} (1);
\end{tikzpicture}
\ \ = \ \
\begin{tikzpicture}[dotpic]
\node [style=dot] (0) at (0,0.25) {};
\node [style=none] (1) at (0,-0.5) {};
\draw (0) to (1);
\end{tikzpicture}$$
\[lem:phi-minus-phi-inv\] $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (2) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (3) at (0.75, 0.5) {$\psi$};
\node [style=dot] (4) at (0, -0.5) {};
\node [style=none] (5) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (4);
\draw [bend right] (4) to node[pauli z]{$\times$} (3);
\draw (4) to (5);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (2) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (3) at (0.75, 0.5) {$\psi$};
\node [style=dot] (4) at (0, -0.5) {};
\node [style=none] (5) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (4);
\draw [bend right] (4) to node[pauli z]{$\times$} (3);
\draw (4) to node[pauli z]{$\times$} (5);
\end{pgfonlayer}
\end{tikzpicture}}$$
Follows immediately from Lemma \[lem:pauli-z-homom\] and the fact that $\cross$ is an involution.
\[thm:add-inv\] The cross is additive inverse: $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (1) at (0.75, 0.5) {$\psi$};
\node [style=dot] (2) at (0, -0.5) {};
\node [style=none] (3) at (0, -1.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw[bend right] (2) to node[pauli z]{$\times$} (1);
\draw[bend right] (0) to (2);
\draw (2) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 0.25) {$\psi$};
\node [style=white dot] (1) at (0.5, 0.25) {$\psi$};
\node [style=dot] (2) at (0, -0.5) {};
\node [style=dot] (3) at (1.5, -0.5) {};
\node [style=white dot] (4) at (0, -1) {};
\node [style=none] (5) at (1.5, -1.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (3) to (5.center);
\draw[bend right] (2) to node[pauli z]{$\times$} (1);
\draw[bend right] (0) to (2);
\draw (2) to (4);
\end{pgfonlayer}
\end{tikzpicture}}$$
Proof by plugging: Recall that
\(0) at (0,0.25) ; (1) at (0,-0.25) ; (0) to (1);
$= 1_{\mathbf{I}}$.
a) $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (2) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (3) at (0.75, 0.5) {$\psi$};
\node [style=dot] (4) at (0, -0.5) {};
\node [style=white dot] (5) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (4);
\draw [bend right] (4) to node[pauli z]{$\times$} (3);
\draw (4) to (5);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 0.25) {$\psi$};
\node [style=white dot] (1) at (0.5, 0.25) {$\psi$};
\node [style=dot] (2) at (0, -0.5) {};
\node [style=dot] (3) at (1.5, -0.5) {};
\node [style=white dot] (4) at (0, -1) {};
\node [style=white dot] (5) at (1.5, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (4);
\draw[bend right] (2) to node[pauli z]{$\times$} (1);
\draw[bend right] (0) to (2);
\draw (3) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}}$$
b) $$\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (2) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (3) at (0.75, 0.5) {$\psi$};
\node [style=dot] (4) at (0, -0.5) {};
\node [style=white dot] (5) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (4);
\draw [bend right] (4) to node[pauli z]{$\times$} (3);
\draw (4) to node[pauli z]{$\times$} (5);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ \stackrel{Lem.\ref{lem:phi-minus-phi-inv}}{=} \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (2) at (-0.75, 0.5) {$\psi$};
\node [style=white dot] (3) at (0.75, 0.5) {$\psi$};
\node [style=dot] (4) at (0, -0.5) {};
\node [style=white dot] (5) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right] (2.center) to (4);
\draw [bend right] (4) to node[pauli z]{$\times$} (3);
\draw (4) to (5);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ = \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 0.25) {$\psi$};
\node [style=white dot] (1) at (0.5, 0.25) {$\psi$};
\node [style=dot] (2) at (0, -0.5) {};
\node [style=dot] (3) at (1.5, -0.5) {};
\node [style=white dot] (4) at (0, -1) {};
\node [style=white dot] (5) at (1.5, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (4);
\draw[bend right] (2) to node[pauli z]{$\times$} (1);
\draw[bend right] (0) to (2);
\draw (3) to (5.center);
\end{pgfonlayer}
\end{tikzpicture}}
\ \ \stackrel{Lem.\ref{lem:pauli-z-zero}}{=} \ \
\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 0.25) {$\psi$};
\node [style=white dot] (1) at (0.5, 0.25) {$\psi$};
\node [style=dot] (2) at (0, -0.5) {};
\node [style=dot] (3) at (1.5, -0.5) {};
\node [style=white dot] (4) at (0, -1) {};
\node [style=white dot] (5) at (1.5, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (4);
\draw[bend right] (2) to node[pauli z]{$\times$} (1);
\draw[bend right] (0) to (2);
\draw (3) to node[pauli z]{$\times$} (5.center);
\end{pgfonlayer}
\end{tikzpicture}}$$
In the case of ${\bf FHilb}$, this operation is the Pauli Z gate multiplied by $-1$: $$\cross
\ \ = \ \
\left(\begin{array}{cc}
-1 & 0 \\
0 & 1
\end{array}\right)$$ In this case, we have $$\begin{tikzpicture}[dotpic]
\node [style=white dot] (0) at (0,0.5) {};
\node [style=none] (1) at (0,-0.5) {};
\draw (0) -- node[pauli z]{$\times$} (1);
\end{tikzpicture}
\ \ = \ \
-\sqrt{2} {|-\rangle}
\ \ = \ \
{|1\rangle} - {|0\rangle}$$ We can now naturally define: $$\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, -1) {};
\node [style=white dot] (2) at (0, 1) {$-\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (1);
\end{pgfonlayer}
\end{tikzpicture}
\ \ := \ \
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (1) at (0, -1) {};
\node [style=white dot] (2) at (0, 1) {$\psi$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to node[pauli z]{$\times$} (1);
\end{pgfonlayer}
\end{tikzpicture}$$ and, in particular, $$\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0.5, 1) {$-\frac{n}{m}$};
\node [style=none] (1) at (0.5, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
\ \ \, \raisebox{6.5mm}{:=}\,
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0.5, 1) {$\frac{n}{m}$};
\node [style=none] (1) at (0.5, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to node[pauli z]{$\times$} (1.center);
\end{pgfonlayer}
\end{tikzpicture}$$
Thus, we have reconstructed all of the axioms for the field of rational numbers. All of the expected identities involving $\cross$ then follow from the field axioms.
!-boxes and automation {#sec:automation}
======================
Graph rewriting is a computation process in which graphs are transformed by various *rewrite rules*. A rewrite rule can be thought of as a ‘directed graphical equation’. For example, the “specialness” equation from section \[sec:ghz-w\] could be expressed as a graph rewrite rule: $$L: \begin{tikzpicture}[dotpic,yshift=5mm]
\node [dot] (a) at (0,0) {};
\node [dot] (b) at (0,-1) {};
\draw [bend left] (a) to (b);
\draw [bend right] (a) to (b);
\draw (0,0.5) to (a) (b) to (0,-1.5);
\end{tikzpicture} \Rightarrow \
R: \begin{tikzpicture}[dotpic]
\draw (0,1) -- (0,-1);
\end{tikzpicture}$$
A graph rewrite rule $L \Rightarrow R$ can be applied to a graph $G$ by identifying a *matching*, that is, a monomorphism $m : L \rightarrow G$. The image of $L$ under $m$ is then removed and replaced by $R$. This process is called *double pushout (DPO) graph rewriting*. A detailed description of how DPO graph rewriting can be performed on the graphs described in this paper is available in [@DK].
It is also useful to talk not only about ‘concrete’ graph rewrite rules, but also *pattern graph* rewrite rules, which can be used to express an infinite set of rewrite rules. We form pattern graphs using *!-boxes* (or ‘bang-boxes’). These boxes identify portions of the graph that can be replicated any number of times. More precisely, the set of concrete graphs represented by a pattern graph is the set of all graphs (containing no !-boxes) that can be obtained by performing any sequence of these four operations:
- `COPY`: copy a !-box and its incident edges
- `MERGE`: merge two !-boxes
- `DROP`: remove a !-box, leaving its contents
- `KILL`: remove a !-box and its contents
For example, the following pattern graph represents the encoding of a natural number given in section \[sec:natural-numbers\]: $$\left\llbracket\ \
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=box vertex] (0) at (0, 0.75) {};
\node [style=white dot] (1) at (0, 0.75) {};
\node [style=dot] (2) at (0, 0) {};
\node [style=none] (3) at (0, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (3.center);
\draw (1) to (2);
\end{pgfonlayer}
\end{tikzpicture}\ \ \right\rrbracket =
\left\{\
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 0) {};
\node [style=none] (1) at (0, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}\ ,\ \
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (0, 0.75) {};
\node [style=dot] (1) at (0, 0) {};
\node [style=none] (2) at (0, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (2.center);
\draw (0) to (1);
\end{pgfonlayer}
\end{tikzpicture}\ ,\ \
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.5, 0.75) {};
\node [style=white dot] (1) at (0.5, 0.75) {};
\node [style=dot] (2) at (0, 0) {};
\node [style=none] (3) at (0, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (2);
\draw (2) to (3.center);
\draw (0) to (2);
\end{pgfonlayer}
\end{tikzpicture},\ \
\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=white dot] (0) at (-0.75, 0.75) {};
\node [style=white dot] (1) at (0, 0.75) {};
\node [style=white dot] (2) at (0.75, 0.75) {};
\node [style=dot] (3) at (0, 0) {};
\node [style=none] (4) at (0, -0.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2) to (3);
\draw (0) to (3);
\draw (3) to (4.center);
\draw (1) to (3);
\end{pgfonlayer}
\end{tikzpicture},\ \
\ldots\
\right\} :=
\left\{
\point{0},\ \point{1},\ \point{2},\ \point{3},\ \ldots
\right\}$$ Note how not only the vertices are duplicated, but also all of the edges connected to those vertices.
A pattern graph rewrite rule is a pair of pattern graphs with the same inputs and outputs. Furthermore, there is a bijection between the !-boxes occurring on the LHS and the RHS. When one of the four operations is performed to a !-box on the LHS, the same is performed to its corresponding !-box on the RHS.
We can rewrite (the natural numbers versions of) equations $\delta_1$, $\delta_2$, and $\delta_3$ as pattern graph rewrite rules. $$L:\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-0.5, 1.5) {};
\node [style=none] (1) at (0.5, 1.5) {};
\node [style=dot] (2) at (0, 0.5) {};
\node [style=white dot] (3) at (0, -0.5) {};
\node [style=none] (4) at (0, -1.5) {};
\node [style=white dot] (5) at (1, 0.5) {};
\node [style=box vertex,inner sep=1.5mm] (6) at (1, 0.5) {};
\node [style=dot] (7) at (1, 0) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (0) to (2);
\draw [bend right=30] (2) to (1);
\draw (2) to (3);
\draw (3) to (4);
\draw [bend right=30] (3) to (7);
\draw (7) to (5);
\end{pgfonlayer}
\end{tikzpicture}}
\Rightarrow\ \
R: \raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1.25, 1.5) {};
\node [style=white dot] (1) at (-0.25, 1.5) {};
\node [style=white dot] (2) at (0.25, 1.5) {};
\node [style=none] (3) at (1.25, 1.5) {};
\node [style=dot] (4) at (-0.25, 1) {};
\node [style=dot] (5) at (0.25, 1) {};
\node [style=white dot] (6) at (-0.75, 0) {};
\node [style=white dot] (7) at (0.75, 0) {};
\node [style=dot] (8) at (0, -1) {};
\node [style=none] (9) at (0, -1.5) {};
\node [style=box vertex,minimum height=2mm,minimum width=6mm] (10) at (0, 1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=30] (0) to (6.center);
\draw [bend right=30] (6.center) to (4.center);
\draw (4.center) to (1);
\draw [bend right=30] (6.center) to (8.center);
\draw [bend right=30] (8.center) to (7.center);
\draw [bend right=30] (5.center) to (7.center);
\draw [bend right=30] (7.center) to (3);
\draw (5.center) to (2);
\draw (8.center) to (9);
\end{pgfonlayer}
\end{tikzpicture}}
\qquad \qquad
L:\raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\node [style=white dot] (2) at (0, -0.5) {};
\node [style=white dot] (3) at (1, 1) {};
\node [style=dot] (4) at (1, 0.5) {};
\node [style=box vertex,inner sep=1.5mm] (5) at (1, 1) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (2.center);
\draw (2.center) to (1);
\draw [bend right=30] (2.center) to (4.center);
\draw (4.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}}
\Rightarrow\ \
R: \raisebox{-8mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=dot] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0)--(1);
\end{pgfonlayer}
\end{tikzpicture}}
\qquad \qquad
L:\begin{tikzpicture}[dotpic]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=white dot] (1) at (1.25, 1.5) {};
\node [style=box vertex, minimum width=9 mm, minimum height=2 mm] (2) at (1.75, 1.5) {};
\node [style=white dot] (3) at (2.25, 1.5) {};
\node [style=white dot] (4) at (0.5, 1) {};
\node [style=white dot] (5) at (1.5, 1) {};
\node [style=dot] (6) at (0.75, 0.5) {};
\node [style=dot] (7) at (1.75, 0.5) {};
\node [style=white dot] (8) at (0, 0) {};
\node [style=white dot] (9) at (0, -1) {};
\node [style=none] (10) at (0, -1.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (9) to (10.center);
\draw[bend right=15] (4) to (6);
\draw (8) to (9);
\draw (0.center) to (8);
\draw[out=30, in=258] (6) to (1);
\draw[bend right=15] (5) to (7);
\draw[bend right] (9) to node[tick]{-} (7);
\draw[bend right] (8) to (6);
\draw[bend right=15] (7) to (3);
\end{pgfonlayer}
\end{tikzpicture}
\Rightarrow\ \
R: \raisebox{-9mm}{\begin{tikzpicture}[scale=0.6]
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0, 1.5) {};
\node [style=none] (1) at (0, -1.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0)--(1);
\end{pgfonlayer}
\end{tikzpicture}}$$ These equations only apply to encodings of the natural numbers, not for arbitrary inputs $\psi$. However, we showed in sections \[sec:natural-numbers\], \[sec:rationals\], and \[sec:additive-inverses\] that even those weaker equations suffice to recover the usual identities for fraction arithmetic. Also note that the extra white vertices in $\delta_3'$ eliminate the case of $\frac{0}{0} \neq 1$.
So why bother expressing graphical identities as graph rewrite rules? Graph rewriting can be automated! [quantomatic]{} [@quanto] is a automatic graph rewriting tool developed by two of the authors. It is specifically designed to work with the kinds of diagram described in this paper and to perform pattern graph rewriting. It remains to be seen what new insights can be obtained by adding the new graphical identities derived in this paper to [quantomatic]{}.
Closing Remarks
===============
In previous work two of the authors showed that the main difference between the GHZ state and the W state, or more precisely, the induced GHZ structure and W structure, boils down to the value of the loop map of these CFAs: $$\frac{\mbox{GHZ}}{\mbox{W}}\ \ =\ \ \frac{\begin{tikzpicture}[dotpic,yshift=5mm]
\node [dot] (a) at (0,0) {};
\node [dot] (b) at (0,-1) {};
\draw [bend left] (a) to (b);
\draw [bend right] (a) to (b);
\draw (0,0.5) to (a) (b) to (0,-1.5);
\end{tikzpicture} = \
\begin{tikzpicture}[dotpic]
\draw (0,1) -- (0,-1);
\end{tikzpicture}
}{
\circl\ \begin{tikzpicture}[dotpic]
\node [dot] (a) at (0,0.5) {};
\node [dot] (b) at (0,-0.5) {};
\draw [bend left] (a) to (b);
\draw [bend right] (a) to (b);
\draw (0,1) to (a) (b) to (0,-1);
\end{tikzpicture}\ \ =
\begin{tikzpicture}[dotpic]
\node [dot] (a) at (0,0.7) {};
\node [dot] (b) at (0,-0.7) {};
\draw (0,1.2) to (a) (b) to (0,-1.2);
\draw (a) to [downloop] ();
\draw (b) to [uploop] ();
\end{tikzpicture}
}$$ In this paper, by focussing on the interaction of these two structures, we were able to establish a connection with the operations of basic arithmetic: $$\frac{\mbox{W}}{\mbox{GHZ}}=\frac{+}{\times}$$ More specifically, the diagrammatic language of these structures was sufficient to encode the positive rational numbers (and, with a minor extension, the whole field of rational numbers).
In the process of highlighting this encoding, we identified a surprising fact. The distributive law governing the interaction of addition and multiplication in arithmetic also captures the interaction of the GHZ-structure and W-structure. Future work includes exploiting this interaction in the study of multipartite quantum entanglement, which brings us back to the initial motivation for crafting a compositional framework to reason about multipartite states.
[^1]: Supported by EPSRC Advanced Research Fellowship EP/D072786/1, Office of Naval Research Grant N00014-09-1-0248 and EC FP6 STREP QICS. Some of this work was performed during a visit at IQOQI Vienna.
[^2]: Supported by a Clarendon scholarship
[^3]: Supported by a DTA scholarship
[^4]: Funded by DST, Govt. of India
|
---
abstract: 'Depth first search is a natural algorithmic technique for constructing a closed route that visits all vertices of a graph. The length of such route equals, in an edge-weighted tree, twice the total weight of all edges of the tree and this is asymptotically optimal over all exploration strategies. This paper considers a variant of such search strategies where the length of each route is bounded by a positive integer $B$ (e.g. due to limited energy resources of the searcher). The objective is to cover all the edges of a tree $T$ using the minimum number of routes, each starting and ending at the root and each being of length at most $B$. To this end, we analyze the following natural greedy tree traversal process that is based on decomposing a depth first search traversal into a sequence of limited length routes. Given any arbitrary depth first search traversal $R$ of the tree $T$, we cover $R$ with routes $R_1,\ldots,R_l$, each of length at most $B$ such that: $R_i$ starts at the root, reaches directly the farthest point of $R$ visited by $R_{i-1}$, then $R_i$ continues along the path $R$ as far as possible, and finally $R_i$ returns to the root. We call the above algorithm *piecemeal-DFS* and we prove that it achieves the asymptotically minimal number of routes $l$, regardless of the choice of $R$. Our analysis also shows that the total length of the traversal (and thus the traversal time) of piecemeal-DFS is asymptotically minimum over all energy-constrained exploration strategies. The fact that $R$ can be chosen arbitrarily means that the exploration strategy can be constructed in an online fashion when the input tree $T$ is not known in advance. Each route $R_i$ can be constructed without any knowledge of the yet unvisited part of $T$. Surprisingly, our results show that depth first search is efficient for energy constrained exploration of trees, even though it is known that the same does not hold for energy constrained exploration of arbitrary graphs.'
author:
- Shantanu Das
- Dariusz Dereniowski
- Przemysław Uznański
bibliography:
- 'batteries.bib'
title: 'Energy Constrained Depth First Search[^1]'
---
**Key Words:** DFS traversal, distributed algorithm, graph exploration, piecemeal exploration, online exploration
Introduction
============
Graph-theoretic problems in which one wants to cover the entire graph with one or more routes satisfying certain objective is a well established and long studied topic in many areas of computer science. Particular problems vary depending on the research area or potential applications, including the study of simple graph traversals like DFS or BFS, for algorithmic purposes, to complex transportation problems with many variations of traveling salesman problem (TSP), or pursuit-evasion games like the watchman problem, and finally distributed monitoring of networks using mobile agents.
For one possible application of our results, consider a mobile robot that needs to explore an initially unknown tree. We assume that the tree is edge-weighted and the weight of each edge denotes the length of that edge. Starting from a single vertex (the root) of the tree, the robot must traverse all edges of and return to its initial location. Upon visiting a vertex $v$ for the first time, the robot discovers the edges incident to $v$ and can choose one of them to continue the exploration. Provided that the robot can remember the visited vertices and edges, a simple depth first search (DFS) is an efficient algorithm for exploring the tree, achieving the optimal cost of twice the sum of the lengths of edges in the tree.
Consider a more interesting scenario, when the robot has a limited source of energy (e.g. a battery) which allows it to traverse a path of length at most $B$ (we say such a robot is *energy constrained*). Naturally, we assume that each vertex of the tree is at distance at most $B/2$ from the root, otherwise the tree cannot be fully explored. In this case, exploration is possible if the robot can recharge its battery whenever it returns back to the starting location. Thus, the exploration is a collection of routes of the robot, each of which starts and ends at the root, and has length at most $B$. We are interested in the minimum number of such routes needed (i.e. the number of times the robot has to recharge) to completely explore the tree.
This model of exploration may be of interest for several reasons. One obvious reason is related to the capabilities of the robot; it may have a restricted fuel tank capacity or perhaps a harsh or risky environment enforces a return to its home-base every so often. A robot that returns periodically to the root can inform about new discoveries — in this way the knowledge is accumulated gradually at the base-station while the algorithm progresses. This may, for example, reduce the risk of having no data in case of robot failure prior to the end of exploration. From a different point of view, this process may be seen as a *piecemeal learning*, that is, one in which it is possible to have a trade off between exploration and utilization; the two phases representing parts in which learning occurs (exploration) and part in which accumulated knowledge is used (utilization). Finally, having many restricted-length routes covering a tree instead of a single long route may be potentially applied in scenarios in which one wants to minimize exploration time by strategies using multiple robots. In fact, when the robots are incapable of refueling, we can use several robots to explore the tree, each robot traversing a path of length $B$. In that case, it is important to minimize the number of robots used as well as the total energy cost for exploration.
Note that the *piecemeal exploration* problem has been studied before not just for trees but also for arbitrary connected graphs. However those results were restricted to visiting vertices at depth of at most $B/2(1+\beta)$, for some $\beta > 0$, with the cost of exploration deteriorating sharply as $\beta$ approached zero. In this paper we would like to consider exploration strategies that completely visit all trees up to the maximum possible depth of $B/2$. No such exploration algorithm have been studied for either general graphs or special graphs such as trees. Simple strategies based on depth-first search (breadth first search) perform badly in the case of piecemeal exploration of arbitrary graphs. However as we show in this paper, the piecemeal version of depth-first search performs optimally in trees. This fact is surprising given the fact that for exploration by multiple open routes (routes that do not end at the root) depth-first strategies in trees can have an overhead of $\Omega(\log{n})$ [@DDK15].
**Related work:** There exists extensive literature on graph traversal and exploration, we survey here only the most relevant results on graph exploration by mobile agents. Exploration of general graphs having $n$ nodes and $m$ edges, by a single agent, has been studied in [@PanaiteP99] who gave an algorithm of $m+O(n)$ steps. For exploration by $k$ agents, [@FraigniaudGKP06] provides an exploration algorithm taking $O(D+n/\log k)$ steps in trees of $n$ nodes and height $D$. This algorithm turns out to be $O(k/\log k)$ competitive [@HigashikawaKLT14] (where competitiveness is the ratio of the number of steps of an algorithm over the optimal number of steps). Authors in [@BrassCGX11] give a $O(n/k+D^{k-1})$ time algorithm for tree exploration while [@DyniaKHS06] gives an algorithm for sparse trees with competitive ratio $O(D^{1-1/p})$, where $p$ is defined as the tree density. For some lower bound on exploration time, see [@DyniaLS07; @FraigniaudGKP06; @HigashikawaKLT14]. For other recent results on exploration time see e.g. [@BrassCGX11; @DisserMNSS16; @DobrevKM12; @MegowMS12; @OrtolfS14]. Other than optimizing time, exploration using little memory for the agents has also been studied, see e.g. [@AmbuhlGPRZ11; @DisserHK16].
None of the results mentioned above consider any energy limitation for the agents. The energy constrained exploration problem was first studied under the name of *Piecemeal Graph Exploration* [@BetkeRS95], with the assumption that the route length $B \geq 2(1+\beta)r$, where $r$ is the furthest distance from the starting node to other nodes, and $0<\beta<1$. That paper provided exploration algorithms for a special class of grid graphs with ‘rectangular obstacles’. Awerbuch et al. [@AwerbuchBRS99] showed that, for general graphs, there exists an energy constrained exploration algorithm with a total cost of $O(m+n^{1+o(1)})$. This has been further improved (by an algorithm that is a combination of DFS and BFS) to $O(m+n\log^2 n)$ in [@AwerbuchK98]. Finally [@DuncanKK06] provided an exploration algorithm for general unknown weighted graphs with total cost asymptotic to the sum of edge weights of the graph. Note that, as mentioned, all the above strategies require the length of each route to be strictly larger than the shortest return path from the starting vertex to the farthest vertex. In other words, these algorithms fail in the extreme cases when the height of the explored tree (or the diameter of the graph) is equal to half of the energy budget, which seem to be the most challenging cases.
The same tree exploration model as we study in this work has been considered in [@DDK15; @DyniaKS06] for unweighted trees and multiple agents, with one difference: each agent traverses a path of length at most $B$ such that the path starts at the root but may end at any node of the tree (in other words, agents do not have to return to the homebase). It has been shown in [@DDK15] that if the tree is not known in advance, then there exists an exploration algorithm (that minimizes the number of agents used) with competitive ratio of $O(\log B)$ and this is the best possible. On the other hand, it was shown that by allowing the route lengths to be a constant factor more than $B$, it is possible to explore the tree using the minimum number of agents [@DyniaKS06]. Distributed algorithms for energy constrained agents has been a subject of recent investigation, see e.g. [@AnayaCCLPV12; @AnayaCCLPV16; @BartschiC0DGGLM16]. Authors in [@CzyzowiczDMR16] also consider a model in which agents are allowed to transfer part of their energy to another agent. There are many studies on variants of the traveling salesman problem, including the $k$-TSP [@Arora98; @FredericksonHK78] related to the task of finding a bounded length route in a graph. Such results are out of scope for this paper.
**Our Results and Outline:** In this work we analyze a very natural process of partitioning a depth first search traversal ${R_{\textup{DFS}}}$ of a tree into a sequence ${\mathcal{S}}=(R_1,\ldots,R_k)$ of routes where each route $R_i$ has length at most $B$, starts and ends at the root of the tree (see Section \[sec:dfs\] for a formal definition). We prove that the number of routes $k$ is asymptotically optimal (Theorem \[thm:num-agents\]), that is, it is within a constant factor of the number of routes in any exploration strategy composed of routes of length at most $B$ that cover the entire tree. This fact, being intuitively expected for trees (although it does not hold in general graphs [@DuncanKK06]) turns out to be nontrivial. Our approach is to consider another parameter of an exploration strategy: the *cost* (see Section \[sec:results\]) defined as the sum of the lengths of all routes in an exploration strategy. In order to prove our main result, we argue, in Section \[sec:analysis\], that the cost of ${\mathcal{S}}$ is asymptotically optimal (see Theorem \[thm:cost\]). Then, in Section \[sec:results\] we argue that the fact that ${\mathcal{S}}$ has small cost implies that the number of routes in ${\mathcal{S}}$ is expectedly small.
We emphasize that the above claim holds independently of the choice of the initial depth first search traversal ${R_{\textup{DFS}}}$. The implications of this fact are twofold. First, it provides a theoretical insight into such a partitioning of a depth first search traversals into bounded-length segments. Second, for an exploration algorithm design it means that the routes $R_i$ may be constructed without knowing ${R_{\textup{DFS}}}$ in advance, or more precisely, the routes may be build in an online fashion based only on the knowledge of the subtree explored to date. This property makes our algorithm suitable for online exploration of unknown trees by energy constrained mobile agents.
Exploration strategies {#sec:preliminaries}
======================
In this work we consider edge-weighted rooted trees $T=(V(T),E(T),\omega\colon E(T)\to\mathbb{R}_+)$, with root $r$. We define a *route* $R$ as sequence of nodes, $R=(v_0,v_1,\ldots,v_l)$, where $v_i$ is a vertex of $T$ for each $i\in\{0,\ldots,l\}$, as follows:
1. $\{v_i,v_{i+1}\}\in E(T)$ for each $i\in\{0,\ldots,l-1\}$,
2. $v_0=v_l$ is the root $r$ of $T$.
Informally speaking, a route is a sequence of vertices forming a walk in $T$ that starts and ends at the root. We define the *length* of $R$ to be $${\ell(R)}=\sum_{i=1}^{l}\omega(\{v_{i-1},v_i\}).$$ We say that a vertex $v$ is *visited* (and edge $\{v_{i-1},v_i\}$ is traversed) by the route if $v=v_i$ for some $i\in\{1,\ldots,l\}$. We also say that the subtree of $T$ composed with all vertices visited by $R$ is *covered* by the route.
Given a tree $T$ and an integer $B$, we say that ${\mathcal{S}}=(R_1,\ldots,R_k)$ is a $B$-*exploration strategy* for $T$ (or simply *exploration strategy* if $B$ is clear from the context) if for each $i\in\{1 ,\ldots, k\}$, $R_i$ is a route in $T$ of length at most $B$, and each vertex of $T$ is visited by some route in ${\mathcal{S}}$. We write ${{\left|{\mathcal{S}}\right|}}$ to refer to the number of routes in ${\mathcal{S}}$, $k={{\left|{\mathcal{S}}\right|}}$.
Problem statement and DFS exploration {#sec:dfs}
=====================================
The formulation of the combinatorial problem, to which we refer as *energy constrained tree exploration*, we study in this work is as follows.
Energy Constrained Tree Exploration problem
: (${\textup{ECTE}}$)\
Given a real number $B>1$ and an edge-weighted rooted tree $T$ of height at most $B/2$ what is the minimum integer $k$ such that there exists a $B$-exploration strategy that consists of $k$ routes?
Our goal is to analyze a particular type of solution to this problem, namely, an exploration strategy that behaves like a depth first search traversal but adopted to the fact that route lengths are bounded by $B$. Let ${R_{\textup{DFS}}}=(v_0,v_1,\ldots,v_l)$ be a route in $T$ that covers the tree $T$ and performs a depth first search traversal of $T$. (Note that ${R_{\textup{DFS}}}$ is a route and thus we consider a depth first search traversal to have node repetitions.) For two vertices $u$ and $v$ of $T$, $d(u,v)$ denotes the distance between $u$ and $v$ understood as the sum of weights of the edges of the path connecting these vertices. We refer by ${\mathrm{PDFS}(T)}=(R_1,\ldots,R_k)$ (*Piecemeal Depth First Search*) to the following $B$-exploration strategy constructed iteratively for $i:=1,\ldots,k$ (see also Figure \[fig:strategy\] for an example):
1. let $j_0=0$ i.e. $v_{j_0} = v_0 = r$,
2. \[it:dfs2\] $R_i$ continues DFS exploration from where $R_{i-1}$ *stopped making progress* (from the node $v_{j_{i-1}}$) as long as for currently visited $v_p$: $$\label{eq:route-len}
d(r,v_{j_{i-1}}) + {\ell((v_{j_{i-1}},v_{j_{i-1}+1}, \ldots, v_p))} + d(v_p,r) \le B,$$
3. furthest $v_p$ (for $p \le l$) that satisfies condition from \[it:dfs2\] is denoted as $v_{j_i}$, the vertex where $R_i$ stopped making progress,
4. let $R_i= P_{i-1} \circ (v_{j_{i-1}},v_{j_{i-1}+1},\ldots,v_{j_i-1},v_{j_i}) \circ P_i^R$, where $P_{i-1}$ is the path from $r$ to $v_{j_{i-1}}$, and $P_i^R$ is the path from $v_{j_i}$ to $r$.
Such a strategy ${\mathrm{PDFS}(T)}$ is called a *DFS $B$-exploration*. We will say that the part of $R_i$ containing the subsequence $(v_{j_{i-1}},\ldots,v_{j_i})$ *makes progress* on the route ${R_{\textup{DFS}}}$.
![A route ${R_{\textup{DFS}}}$ and the corresponding ${\mathrm{PDFS}(T)}=(R_1,R_2,R_3)$ with $B=20$: (a) a depth first search traversal ${R_{\textup{DFS}}}=(a,b,c,b,d,e,d,f,d,b,a,g,a)$; (b)-(d) routes $R_1,R_2,R_3$ with lengths $18,16$ and $20$, respectively[]{data-label="fig:strategy"}](figs/fig-strategy.pdf)
We remark that different depth first search traversals ${R_{\textup{DFS}}}$ may result in different values of $k$ (different number of routes) in the resulting DFS $B$-exploration, although for a particular choice of ${R_{\textup{DFS}}}$ the corresponding ${\mathrm{PDFS}(T)}$ is unique. In the rest of the work we fix the route ${R_{\textup{DFS}}}$ arbitrarily and thus ${\mathrm{PDFS}(T)}$ refers to the unique DFS $B$-exploration strategy obtained from ${R_{\textup{DFS}}}$.
Our results {#sec:results}
===========
The following theorem provides the first main result of this work.
\[thm:num-agents\] Let $T$ be a tree and let the longest path from the root to a leaf in $T$ be at most $B/2$. It holds ${{\left|{\mathrm{PDFS}(T)}\right|}}\leq 10 {{\left|{\mathcal{R}}\right|}}$, where ${\mathcal{R}}$ is a $B$-exploration strategy that consists of the minimum number of routes.
The theorem refers to the number of routes in an exploration strategy. However, in order to analyze the behavior of ${\mathrm{PDFS}(T)}$, we will work with another parameter on which we will focus in the entire analysis in the subsequent section. For any $B$-exploration strategy ${\mathcal{S}}=(R_1,\ldots,R_k)$ of $T$ we will denote by ${\xi({\mathcal{S}})}$ the *cost* of ${\mathcal{S}}$ defined as $${\xi({\mathcal{S}})} = \sum_{i=1}^k {\ell(R_i)}.$$ We denote by ${\mathrm{COPT}(T)}$ an optimal solution with respect to the cost, that is, a $B$-exploration strategy whose cost is minimum over all $B$-exploration strategies. This strategy will serve as a reference point to prove asymptotic optimality (in terms of the number of routes) of the DFS exploration. More precisely, we will prove the following theorem.
\[thm:cost\] Let $T$ be a tree and let $B/2$ be greater than or equal to the longest path from the root to a leaf in $T$. It holds ${\xi({\mathrm{PDFS}(T)})}\leq 10\cdot{\xi({\mathrm{COPT}(T)})}$.
The proof is postponed to the next parts of the paper and we finish this section by concluding that Theorem \[thm:cost\] indeed implies Theorem \[thm:num-agents\].
We start with the following observation which relates the smallest possible number of routes in a $B$-exploration strategy and the minimum possible cost.
\[obs:opt\] Given $T$ and $B$, ${{\left|{\mathcal{R}}\right|}}\geq \left\lceil{\xi({\mathrm{COPT}(T)})}/B\right\rceil$, where ${\mathcal{R}}$ is a $B$-exploration strategy with minimum number of routes.
Each route of ${\mathcal{R}}$ is of length at most $B$. Thus, ${{\left|{\mathcal{R}}\right|}}\geq{\xi({\mathcal{R}})}/B$. By definition of ${\mathrm{COPT}(T)}$, ${\xi({\mathcal{R}})}\geq{\xi({\mathrm{COPT}(T)})}$, and since ${{\left|{\mathcal{R}}\right|}}$ is an integer, the claim follows.
Recall that ${R_{\textup{DFS}}}=(v_0,\ldots,v_l)$ is the depth first search traversal of $T$ used to obtain ${\mathrm{PDFS}(T)}$, and the $i$-th route $R_i$ in ${\mathrm{PDFS}(T)}=(R_1,\ldots,R_k)$ makes progress on the depth first search traversal by traversing the part of ${R_{\textup{DFS}}}$ that starts at $v_{j_{i-1}}$ and ends at $v_{j_i}$. By definition of ${\mathrm{PDFS}(T)}$, extending $R_i$ so that it makes progress with the walk $(v_{j_{i-1}},\ldots,v_{j_i},v_{j_{i+1}})$ would exceed its length to be more than $B$ for each $i<k$, i.e., ${\ell(R_i)}+2\omega(\{v_{j_i},v_{j_{i+1}}\})>B$. Consider a tree $T'$ obtained from $T$ by subdividing the edge $\{v_{j_i},v_{j_{i+1}}\}$ into two edges $\{v_{j_i},x_i\}$ and $\{x_i,v_{j_{i+1}}\}$ with weights $(B-{\ell(R_i)})/2$ and $\omega(\{v_{j_i},v_{j_{i+1}}\})-(B-{\ell(R_i)})/2$, respectively, for each $i\in\{1,\ldots,k-1\}$. (Hence the sum of the two weights of the new edges $\{v_{j_i},x_i\}$ and $\{x_i,v_{j_{i+1}}\}$ equals $\omega(\{v_{j_i},v_{j_{i+1}}\})$, the weight of the subdivided edge.) Note that the common nodes of $T$ and $T'$ (that is, the nodes of $T$) are visited by ${\mathrm{PDFS}(T)}$ and ${\mathrm{PDFS}(T')}$ in the same order, both ${\mathrm{PDFS}(T)}$ and ${\mathrm{PDFS}(T')}$ are $B$-exploration strategies and the length of each route in ${\mathrm{PDFS}(T')}$, except for the last one, is of length exactly $B$. The latter in particular implies $$\label{eq:no-routes-Tprime}
{{\left|{\mathrm{PDFS}(T')}\right|}}=\left\lceil\frac{{\xi({\mathrm{PDFS}(T')})}}{B}\right\rceil.$$ Note that $$\label{eq:cost-Tprime}
{\xi({\mathrm{COPT}(T)})} = {\xi({\mathrm{COPT}(T')})}$$ because, informally speaking, a strategy that minimizes the cost never reaches a node of degree two in order to return to previously visited node — thus, in particular, a traversal of $\{v_{j_i},x_i\}$ is immediately followed by a traversal of $\{x_i,v_{j_{i+1}}\}$ and vice versa.
From Equation , Theorem \[thm:cost\] applied to $T'$, Equation and Observation \[obs:opt\] (used in this order) we conclude that $${{\left|{\mathrm{PDFS}(T')}\right|}}=\left\lceil\frac{{\xi({\mathrm{PDFS}(T')})}}{B}\right\rceil \leq 10\cdot \left\lceil \frac{{\xi({\mathrm{COPT}(T')})}}{B} \right\rceil = 10\cdot \left\lceil \frac{{\xi({\mathrm{COPT}(T)})}}{B} \right\rceil \leq 10\cdot{{\left|{\mathcal{R}}\right|}},$$ and hence ${{\left|{\mathrm{PDFS}(T)}\right|}}={{\left|{\mathrm{PDFS}(T')}\right|}}$ completes the proof of Theorem \[thm:num-agents\].
Bounding the cost of ${\mathrm{PDFS}(T)}$ {#sec:analysis}
=========================================
Additional notation {#sec:additional-notation}
-------------------
When referring to subtrees, we consider them always in the context of their distance from the root. More specifically, we consider the *potential of a node* $v$, denoted by $\varphi(v)$, to be defined as $\varphi(v) = B/2 - d(r,v)$. If $u$ is the parent of $v$ in $T$, then we say that $u$ is the *higher* endpoint of $\{u,v\}$ and $v$ is the *lower* endpoint of the edge $\{u,v\}$; we also say that $\{u,v\}$ is a *downward edge* of $u$. For any subtree $T'$ of $T$, we define the *potential* of $T'$, denoted $\varphi(T')$, to be the potential of its root. Then, $2\varphi(T')$ is an upper bound on the total length of any route inside $T'$. We say that a route *reaches* a potential $x$ in some subtree if it reaches a vertex having potential $x$. Additionally, for any subtree $T'$ of $T$, we denote it *weight* to be $\omega(T') = \sum_{e \in E(T')} \omega(e)$, where $E(T')$ is the edge set of $T'$. In other words, the weight of $T$ to be the total weight of its edges. We denote by ${\ensuremath{T[v]}}$ the subtree of $T$ rooted at $v$ that contains $v$ and all its descendants, and by ${\ensuremath{T[e]}}$ the tree composed of the edge $e$ and ${\ensuremath{T[v]}}$ where $v$ is the lower endpoint of $e$.
We say that a subtree $T'$ of $T$ is *heavy* if $\omega(T') > \varphi(T')$, and otherwise we say that $T'$ is *light*. We extend this terminology to vertices and edges: a vertex $v$ or an edge $e$ is *heavy* if ${\ensuremath{T[v]}}$ or ${\ensuremath{T[e]}}$ is heavy, respectively. Additionally, by ${\ensuremath{\mathsf{heavydeg}}}(v)$ we denote the number of outgoing downward edges of $v$ that are heavy (note that if an edge is heavy, then both its endpoints are heavy as well). Observe that if $v$ is any vertex of $T$ and ${\ensuremath{T[v]}}$ is heavy, then one route is not enough to cover the entire ${\ensuremath{T[v]}}$ in any $B$-exploration strategy.
Adversarial DFS-exploration {#sec:adversarial}
---------------------------
When analyzing the cost of ${\mathrm{PDFS}(T)}$ we will use a recursive approach where the $B$-exploration of any subtree $T'$ would be defined by taking $B=2\varphi(T')$, the maximum size of route starting and ending at the root of $T'$. However the first agent to reach subtree $T'$ may have performed other explorations before entering $T'$. Therefore we need to use a slightly generalized DFS $B$-exploration, called a *$B'$-adversarial DFS $B$-exploration*, denoted by ${\mathrm{ADFS}_{B'}(T)}$, where the length of first route is bounded by $B'\leq B$. This is formally defined by replacing Equation for $i=1$ in condition \[it:dfs2\] in the definition of ${\mathrm{PDFS}(T)}$ by the following equation (see also Figure \[fig:adfs\]): $$\label{eq:route-len-new}
{\ell((v_{j_{0}},v_{j_{0}+1}, \ldots, v_p))} + d(v_p,r) \le B'.$$
![(a) a tree with a depth first search traversal $(a,b,c,b,d,b,a,e,f,e,g,e,a)$; (b) a $B'$-adversarial DFS $B$-exploration ${\mathcal{S}}$ with $B'=16$ and $B=20$ has three routes: $(a,b,c,b,a)$ (length $14\leq B'$), $(a,b,d,b,a,e,a)$ (length $18\leq B$) and $(a,e,f,e,g,e,a)$ (length $16\leq B$); ${\xi({\mathcal{S}})}=14+16+16=46$. Note that two routes $(a,b,c,b,d,b,a)$ and $(a,e,f,e,g,e,a)$ constitute a DFS $B$-exploration strategy with cost $34$[]{data-label="fig:adfs"}](figs/fig-adfs.pdf)
In other words, the length of the first route is bounded by $B'$ (Equation ) and the lengths of the remaining routes are bounded by $B$ (Equation for $i>1$).
For a given tree $T$, we define an *adversarial DFS exploration* of $T$, denoted by ${\mathrm{ADFS}_{}(T)}$, and defined to be an exploration strategy ${\mathrm{ADFS}_{B'}(T)}$ that maximizes the cost, across all possible values of $B'$: $$\label{eq:ADFS-def0}
{\xi({\mathrm{ADFS}_{}(T)})} = \max_{0 \le B' \le 2\varphi(T)} {\xi({\mathrm{ADFS}_{B'}(T)})}.$$
In the following analysis, it will be convenient for us to use arguments that rely on the fact that $B'$, for our purposes, takes only one of the finite values from $[0,2\varphi(T)]$. This is due to the above comment, namely, for ${\ensuremath{T[r]}}$, where $r$ is the root of $T$, we have that $B'=B$ and for any other subtree ${\ensuremath{T[v]}}$, the values of $B'$ interesting for us depend on the prefix of the route that starts at the root of $T$ and reaches $v$. Thus, a simple inductive argument allows us to conclude that the value of $B'$ depends on all possible DFS $B$-exploration strategies of $T$ (the number of those is finite). Hence, we denote by ${\mathcal{B}}$ the finite set that consists all values $B'=B-x$ such that there exists a route in $T$ of length $x$ that starts at the root of $T$ and ends at $x$. Thus, we can restate : $${\xi({\mathrm{ADFS}_{}(T)})} = \max_{B'\in{\mathcal{B}}} {\xi({\mathrm{ADFS}_{B'}(T)})}.$$ Note that it follows from the definition that $${\xi({\mathrm{ADFS}_{}(T)})} \geq {\xi({\mathrm{PDFS}(T)})}.$$ Intuitively, if $v$ is any node of the tree $T$, then ${\mathrm{ADFS}_{}({\ensuremath{T[v]}})}$ is the worst case scenario of how a DFS $B$-exploration may perform in ${\ensuremath{T[v]}}$ in terms of the cost; this worst case is understood as considering the worst possible ending point of the route that (in the entire tree $T$) precedes the considered strategy ${\mathrm{ADFS}_{}({\ensuremath{T[v]}})}$.
\[lem:opt-vs-dfs’\] If $v$ is any node of $T$ and $e_1,e_2,\ldots,e_k$ are all downward edges of $v$, then $$\label{eq:opt1} {\xi({\mathrm{COPT}({\ensuremath{T[v]}})})} = \sum_{1 \le i \le k} {\xi({\mathrm{COPT}({\ensuremath{T[e_i]}})})},$$ $$\label{eq:dfs1} {\xi({\mathrm{ADFS}_{}({\ensuremath{T[v]}})})} \le \sum_{1 \le i \le k} {\xi({\mathrm{ADFS}_{}({\ensuremath{T[e_i]}})})}.$$
Informally, equality in for an optimal solution follows from the fact that ${\mathrm{COPT}({\ensuremath{T[v]}})}$ has the freedom to pick the length of each route to be an arbitrary number in ${\mathcal{B}}$. Any strategies for ${\ensuremath{T[e_1]}},{\ensuremath{T[e_2]}}, \ldots, {\ensuremath{T[e_k]}}$ can be translated into strategy for ${\ensuremath{T[v]}}$: the latter one is constructed by simply concatenating the former exploration strategies. Similarly, if one takes an exploration strategy ${\mathrm{COPT}({\ensuremath{T[v]}})}$, then one can assume without affecting its cost that each of its routes has only two occurrences of the root: it is the first and last vertex of the route. But then, such a strategy ${\mathrm{COPT}({\ensuremath{T[v]}})}$ can be partitioned into the corresponding strategies for the trees ${\ensuremath{T[e_1]}},{\ensuremath{T[e_2]}}, \ldots, {\ensuremath{T[e_k]}}$.
We now prove . Consider an exploration strategy ${\mathrm{ADFS}_{}({\ensuremath{T[v]}})}$. Each route of this strategy is of length at most $2\varphi(T)$. Obtain an exploration strategy ${\mathcal{S}}$ by partitioning each route in ${\mathrm{ADFS}_{}({\ensuremath{T[v]}})}$ in such a way that the concatenation of all routes in ${\mathcal{S}}$ equals the concatenation of all routes in ${\mathrm{ADFS}_{}({\ensuremath{T[v]}})}$ and no route in ${\mathcal{S}}$ has $v$ as an internal vertex. (Thus, each route of ${\mathcal{S}}$ starts and ends at $v$.) Note that ${\xi({\mathcal{S}})}={\xi({\mathrm{ADFS}_{}({\ensuremath{T[v]}})})}$. Now, ${\mathcal{S}}$ can be partitioned into ${\mathcal{S}}_1,\ldots,{\mathcal{S}}_k$ such that ${\mathcal{S}}_i$ is a $b_i'$-adversarial DFS exploration strategy of ${\ensuremath{T[e_i]}}$ for some $b_i'\leq 2\varphi({\ensuremath{T[v]}})$, i.e., ${\mathcal{S}}_i={\mathrm{ADFS}_{b_i'}({\ensuremath{T[e_i]}})}$, for each $i\in\{1,\ldots,k\}$ and the concatenation of ${\mathcal{S}}_1,\ldots,{\mathcal{S}}_k$ gives ${\mathcal{S}}$. Thus, $${\xi({\mathrm{ADFS}_{}({\ensuremath{T[v]}})})} = {\xi({\mathcal{S}})} = \sum_{i=1}^k {\xi({\mathrm{ADFS}_{b_i'}({\ensuremath{T[e_i]}})})}.$$ To conclude the proof, observe that by the definition of adversarial exploration $${\xi({\mathrm{ADFS}_{b_i'}({\ensuremath{T[e_i]}})})} \leq {\xi({\mathrm{ADFS}_{}({\ensuremath{T[e_i]}})})},\quad i\in\{1,\ldots,k\}.\qedhere$$
The proof of Theorem \[thm:cost\] will follow from the following two results (and the fact that ${\xi({\mathrm{PDFS}(T)})}\leq{\xi({\mathrm{ADFS}_{}(T)})}$.)
\[lem:light\] For any tree $T$, if $T$ is light, then ${\xi({\mathrm{ADFS}_{}(T)})} < 2 \cdot {\xi({\mathrm{COPT}(T)})}.$
If $T$ is light, then observe that ${\mathrm{ADFS}_{}(T)}$ either consists of one route, in which case ${\xi({\mathrm{ADFS}_{}(T)})}={\xi({\mathrm{COPT}(T)})}$, or it contains at least two routes but then the second route, having length up to $2\varphi(T)$, will explore all remaining vertices of $T$ since $\varphi(T)\geq\omega(T)$ holds for a light tree. Thus, in the latter case ${\mathrm{ADFS}_{}(T)}$ has exactly two routes, one of them being strictly shorter than $2\omega(T)$, which gives ${\xi({\mathrm{ADFS}_{}(T)})} < 4\omega(T) \leq 2 \cdot {\xi({\mathrm{COPT}(T)})}$.
\[thm:induction\] If $T$ is heavy and $r$ is its root, then:
1. \[it:rec1\] if ${\ensuremath{\mathsf{heavydeg}}}(r) = 1$, then ${\xi({\mathrm{ADFS}_{}(T)})} < 10 \cdot {\xi({\mathrm{COPT}(T)})} - 8 \cdot \varphi(T)$,
2. \[it:rec2\] if ${\ensuremath{\mathsf{heavydeg}}}(r) \not=1$, then ${\xi({\mathrm{ADFS}_{}(T)})} < 10 \cdot {\xi({\mathrm{COPT}(T)})} - 16 \cdot \varphi(T)$.
In order to prove the above Theorem we will first define a special class of heavy trees called *Skinny Tree* which has the following property.
1. \[it:property\] **Skinny Tree Property:** *If the root $r$ of $T$ has heavy degree equal to one, then consider the longest path in $T$ that connects $r$ to such a $r'$ that each internal vertex of the path has heavy degree equal to $1$ (i.e. each edge of the path is heavy). We then require that each vertex of this path, except for $r'$, has at most one light edge incident to it.*
We can show (c.f. Section \[sec:rearrangement-appendix\] in the appendix) how to rearrange any tree to have the above property and we also show that:
\[lem:perturbe\] For any tree T, there exists $\varepsilon>0$ and corresponding tree ${T_{\varepsilon}}$ such that (i) ${T_{\varepsilon}}$ satisfies Property \[it:property\], and (ii) if Theorem \[thm:induction\] holds for ${T_{\varepsilon}}$ then Theorem \[thm:induction\] holds for $T$.
Due to above result, we can now focus on proving Theorem \[thm:induction\] for any tree $T$ with the above-mentioned property in the rest of the paper (Section \[sec:induction\]).
Proof of Theorem \[thm:induction\] for Skinny Trees {#sec:induction}
---------------------------------------------------
We will proceed by induction on the number of heavy edges in a tree. This is a valid approach since the parent of a heavy node is also heavy.
For the base case consider $T$ with no heavy edges. In particular we have that ${\ensuremath{\mathsf{heavydeg}}}(r)=0$. Denote downward edges at $r$ by $e_1,e_2,\ldots,e_{{l}}$. For each $i \in\{1,\ldots,{l}\}$, ${\ensuremath{T[e_i]}}$ is light and hence by Lemma \[lem:light\], ${\xi({\mathrm{ADFS}_{}({\ensuremath{T[e_i]}})})} < 2 \cdot {\xi({\mathrm{COPT}({\ensuremath{T[e_i]}})})}$. Thus in particular, by and and ${\xi({\mathrm{COPT}(T)})} \ge 2 \omega(T)$, $${\xi({\mathrm{ADFS}_{}(T)})} < 2 {\xi({\mathrm{COPT}(T)})} \le 10 {\xi({\mathrm{COPT}(T)})} - 16\omega(T) \le 10 {\xi({\mathrm{COPT}(T)})} - 16 \varphi(T).$$
For the induction step, we assume that Theorem \[thm:induction\] holds for all heavy proper subtrees of $T$. In what follows we consider two cases: when ${\ensuremath{\mathsf{heavydeg}}}(r)>1$ and ${\ensuremath{\mathsf{heavydeg}}}(r)=1$.
### Case of ${\ensuremath{\mathsf{heavydeg}}}(r)>1$ {#case-of-ensuremathmathsfheavydegr1 .unnumbered}
Let $e_1,\ldots,e_h$ be the heavy downward edges at $r$ and $e'_1, \ldots, e'_{{l}}$ be the light downward edges at $r$. By induction hypothesis (and precisely Theorem \[thm:induction\]\[it:rec1\]) we obtain $${\xi({\mathrm{ADFS}_{}({\ensuremath{T[e_i]}})})} < 10 \cdot {\xi({\mathrm{COPT}({\ensuremath{T[e_i]}})})} - 8 \varphi({\ensuremath{T[e_i]}})$$ for each $i\in\{1,\ldots,h\}$. Then, by of Lemma \[lem:opt-vs-dfs’\], by Lemma \[lem:light\], and by $\varphi(T) = \varphi({\ensuremath{T[e_i]}}))$ (used in this order): $$\begin{aligned}
{\xi({\mathrm{ADFS}_{}(T)})} & \leq & \sum_{i=1}^{h} {\mathrm{ADFS}_{}({\ensuremath{T[e_i]}})} + \sum_{i=1}^{{l}} {\mathrm{ADFS}_{}({\ensuremath{T[e_i']}})} \\
& < & \sum_{i=1}^{h} \big(10 {\xi({\mathrm{COPT}({\ensuremath{T[e_i]}})})} - 8 \varphi(T) \big) + \sum_{i=1}^{{l}} 2 {\xi({\mathrm{COPT}({\ensuremath{T[e'_i]}})})} \\
& \le & 10 \left( \sum_{i=1}^{h} {\xi({\mathrm{COPT}({\ensuremath{T[e_i]}})})} + \sum_{i=1}^{{l}} {\xi({\mathrm{COPT}({\ensuremath{T[e'_i]}})})} \right) - 8 \varphi(T) \cdot h \\
& \le & 10 {\xi({\mathrm{COPT}(T)})} - 16 \varphi(T).\end{aligned}$$ The last inequality is due to and $h={\ensuremath{\mathsf{heavydeg}}}(r)\ge 2$.
### Case of ${\ensuremath{\mathsf{heavydeg}}}(r)=1$ {#case-of-ensuremathmathsfheavydegr1-1 .unnumbered}
Let $r'$ be the closest descendant of $r$ in $T$ that is heavy and satisfies ${\ensuremath{\mathsf{heavydeg}}}(r') \neq 1$. Note that such a vertex always exists and $r'$ is unique. Let $P$ denote the path connecting $r$ to $r'$. Additionally, we denote by $e_1,e_2, \ldots, e_{{l}}$ all light edges incident to vertices in $V(P)\setminus\{r'\}$ in the *non-decreasing* order of their potentials. (We remark here that the subtree rooted at $r'$ has been covered by the base case of the induction and by the case when the heavy degree is greater than one.) Denote $\varphi_i = \varphi(e_i)$, $i\in\{1,\ldots,{l}\}$. Due to Lemma \[lem:perturbe\] (and more precisely by the fact that thanks to Lemma \[lem:perturbe\] we assume that in the tree $T$ all edges $e_1,\ldots,e_{{l}}$ have pairwise different potentials) we have: $$\varphi(r) > \varphi_{{l}} > \cdots > \varphi_2 > \varphi_1 > \varphi(r')=\varphi_0.$$ See Figure \[fig:hs\](a) that illustrates the path $P$ and placements of the edges $e_i$ and the corresponding potentials.
![(a) the path $P$ (although it may contain vertices of degree two, they are trivial from the point of view of the analysis, thus omitted in the picture); (b) illustration of $x_i$’s. Note that the first route (and possibly arbitrarily many more routes) always reaches $r'$[]{data-label="fig:hs"}](figs/fig-hs.pdf)
We take for brevity $$w_i = \omega({\ensuremath{T[e_i]}}) = \frac12{\xi({\mathrm{COPT}({\ensuremath{T[e_i]}})})} \text{\quad for\ } i \in \{1,\ldots,{l}\},\quad w_{0} = \frac12{\xi({\mathrm{COPT}({\ensuremath{T[r']}})})}.$$
Let $c$ be the number of routes in ${\mathrm{COPT}(T)}$, and we denote by $x_i$ the lowest potential $i$-th route reached on the path $P$ (we ignore potentials it reached in subtrees — see Figure \[fig:hs\](b)), where the routes are without loss of generality ordered so that $x_1 \le x_2 \le \cdots \le x_c.$
Consider $j\in\{1,\ldots,c\}$. Informally speaking, the first $j$ routes of ${\mathrm{COPT}(T)}$ need to cover all subtrees ${\ensuremath{T[e_i]}}$ such that the potential of the higher endpoint of $e_i$ is *strictly* smaller than $x_j$; otherwise some vertices would not be visited by ${\mathrm{COPT}(T)}$. The total weight of these subtrees is $\sum_{i : \varphi_i < x_j} w_i$. Observe, that the total length of all parts of an $i$-th route that do not belong to the path $P$ is at most $2x_i$. Thus, the above total weight of the above-mentioned subtrees satisfies $$\label{eq3}
\sum_{i : \varphi_i < x_j} w_i \le (x_1 + \cdots + x_{j-1}) \quad\text{for each } j\in\{1,\ldots,c+1\},$$ where we denote $x_{c+1} = +\infty$ for the sake of simplicity.
We are now interested in bounding the cost of ${\mathrm{ADFS}_{}(T)}$ on the path $P$ with respect to $\sum_{i=1}^c 2(\varphi(r)-x_i),$ that is, with respect to the cost of ${\mathrm{COPT}(T)}$ on the path $P$. To do this, we start by comparing $x_1,\ldots,x_c$ with $y_1,\ldots,y_d$ chosen by an appropriate greedy procedure: $$\label{eq4}
y_j = \min\big\{ y : \sum_{i : \varphi_i \le y} w_i > y_1 + \ldots + y_{j-1}\big\},$$ $$d = \min\left\{ i : y_1 + \cdots + y_i \ge w_0+w_1+\ldots+w_m \right\},$$ in other words, assigning $y_j$ to be the first value where is violated given only $y_1,\ldots,y_{j-1}$. (Notice that from the definition we have that always $y_j \in \{\varphi_0,\varphi_1,\ldots,\varphi_{{l}}\}$. Moreover, if $y_j > \varphi_0$, then $y_{j+1} > y_j$.) We obtain the following lemma which says that, across all sequences satisfying , $y_i$ takes maximal values:
\[lem:y-min\] It holds that $d \le c$ and $y_j \ge x_j$ for each $j\in\{1,\ldots,d\}$.
We proceed by induction on $j$.
For the inductive base, we have $x_1 = y_1 = \varphi(r')$. For the inductive step, assume the claim holds for all indices smaller than $j$. Suppose for a contradiction that $x_j > y_j$. Then, $$\sum_{i : \varphi_i \le y_j} w_i \le \sum_{i : \varphi_i < x_j} w_i$$ and by and the inductive assumption applied for all indices smaller than $j$, $$\sum_{i : \varphi_i < x_j} w_i\le y_1 + \ldots + y_{j-1}.$$ These two inequalities give a contradiction with $\eqref{eq4}$.
Observe that ${\mathrm{ADFS}_{}(T)}$ first traverses (in that order) some subset of light subtrees ${\ensuremath{T[e_i]}}$, whose indices we denote by $H \subseteq \{1,\ldots,{l}\}$, in a decreasing order of their indices. The above routes, none of which contains $r'$, will form the *first part of* ${\mathrm{ADFS}_{}(T)}$. Then, all vertices of ${\ensuremath{T[r']}}$ are visited (to those routes of ${\mathrm{ADFS}_{}(T)}$ we refer at the *second part of* ${\mathrm{ADFS}_{}(T)}$) and following that, remaining light subtrees ${\ensuremath{T[e_i]}}$ for $i \in H' = \{1,\ldots,{l}\}\setminus H$, in an increasing order of their indices (*third part of* ${\mathrm{ADFS}_{}(T)}$). Note that there may exist a route that has a non-empty intersection with a tree ${\ensuremath{T[e_i]}}$, $i\in H\cup H'$, and also contains $r'$ — this route belongs by definition to the second part of ${\mathrm{ADFS}_{}(T)}$.
Denote by $z_1,\ldots,z_p$ the lowest potentials reached by subsequent routes in ${\mathrm{ADFS}_{}(T)}$ on the path $P$ (ignoring as before the potentials they reach in subtrees), only in the first part of ${\mathrm{ADFS}_{}(T)}$ in the reversed order of entering $T$: $$\label{eq:zi-def}
\varphi(r) \ge z_p \ge \cdots \ge z_1 > \varphi(r').$$ We note that each route in the first part of ${\mathrm{ADFS}_{}(T)}$ visits subtrees with a continuous segment of indices from $H$, that is ${\ensuremath{T[e_i]}}$’s for $i \in H \cap\{i_1,\ldots, i_2\}$ for some integers $i_1,i_2$. We, due to the weight of a light tree, its vertices belong to at most two different routes.
\[lem:2step\] If for some $i,j$ there is $z_i > y_j$, then $z_{i+1} \ge y_{j+1}$.
If $y_{j+1} = y_j $ then the claim follows immediately from the fact that, by , $z_{i+1} \geq z_i \geq y_j = y_{j+1}$. Similarly, if $z_i \ge y_{j+1}$ then by we have $z_{i+1} \geq z_i \ge y_{j+1}$. Thus, assume that $y_{j+1} \ge z_i$ and $y_{j+1} > y_j$.
Let $a>b$ be indices such that $y_j = \varphi_a$ and $y_{j+1}= \varphi_b$. We have from the way $y_j$ is selected in : $$w_0 + \cdots + w_b > y_1 + \cdots + y_j \ge w_0 + \cdots + w_{b-1},$$ $$w_0 + \cdots + w_a > y_1 + \cdots + y_{j-1} \ge w_0 + \cdots + w_{a-1}.$$ Thus, $$y_j > w_{a+1} + \cdots + w_{b-1}.$$ This inequality, informally speaking, certifies that the total weight of all subtrees ${\ensuremath{T[e_s]}}$ with $s\in\{a+1,\ldots,b-1\}$ is smaller than $y_j$. By assumption $z_i>y_j$. By the definition of the sequence $z_1,\ldots,z_p$, the length of the $i$-th route in ${\mathrm{ADFS}_{}(T)}$ restricted to the subtrees ${\ensuremath{T[e_s]}}$ is at least $2z_i$. (Note that we are not using the fact that this route may avoid some subtrees ${\ensuremath{T[e_s]}}$ with $s\in\{a+1,\ldots,b-1\}$ as we analyze the first part of ${\mathrm{ADFS}_{}(T)}$ which ‘avoids’ each subtree ${\ensuremath{T[e_s]}}$ with $s\in\{a+1,\ldots,b-1\}\cap H'$.) Thus, the $i$-th route of ${\mathrm{ADFS}_{}(T)}$ visits the node of $P$ at potential $\varphi_b$ and hence $z_{i+1} \ge \varphi_b = y_{j+1}$ as required in the lemma.
We are now ready to bound the total cost of ${\mathrm{ADFS}_{}(T)}$ with relation to ${\mathrm{COPT}(T)}$. The cost of ${\mathrm{COPT}(T)}$ can be decomposed: $$\label{eq:opt-decomposed}
{\xi({\mathrm{COPT}(T)})}=O_{light} + O_{deep} + O_{path} + O_{flat},$$ where:
$O_{light}$ — is the cost restricted to light subtrees ${\ensuremath{T[e_i]}}$ with $i\in\{1,\ldots,{l}\}$,
$O_{deep}$ — is the cost restricted to the subtree ${\ensuremath{T[r']}}$,
$O_{path}$ — is the cost restricted to the path $P$ and the routes that do not contain $r'$, and
$O_{flat}$ — is the cost restricted to the path $P$ and routes that do contain $r'$.
Similarly, we express the cost of ${\mathrm{ADFS}_{}(T)}$ as a sum: $$\label{eq:dfs-decomposed}
{\xi({\mathrm{ADFS}_{}(T)})} = D_{light} + D_{deep} + D_{desc} + D_{flat} + D_{asc},$$ where
$D_{light}$ — is the cost of ${\mathrm{ADFS}_{}(T)}$ restricted to light subtrees ${\ensuremath{T[e_i]}}$ with $i\in\{1,\ldots,{l}\}$,
$D_{deep}$ — is the cost restricted to the subtree ${\ensuremath{T[r']}}$,
$D_{desc}$ — is the cost restricted to the path $P$ in the first part of ${\mathrm{ADFS}_{}(T)}$,
$D_{flat}$ — is the cost restricted to $P$ and the routes that contain $r'$ (i.e., the second part of ${\mathrm{ADFS}_{}(T)}$), and
$D_{asc}$ — is the cost restricted to the path $P$ in the third part of ${\mathrm{ADFS}_{}(T)}$.
By Lemma \[lem:light\], ${\xi({\mathrm{ADFS}_{}({\ensuremath{T[e_i]}})})} < 2 {\xi({\mathrm{COPT}({\ensuremath{T[e_i]}})})}$ for each $i\in\{1,\ldots,{l}\}$ and therefore $$\label{eq:Dlight}
D_{light} < 2\cdot O_{light}.$$
Denote by $s$ the smallest index such that $y_{s+1} > \varphi_0 = y_s$ and let $H=\{j_1,\ldots,j_q\}$, $j_1<j_2<\cdots<j_q$. Recall that $H$ is the set of indices $i$ such that ${\ensuremath{T[e_i]}}$ is covered in the first part of ${\mathrm{ADFS}_{}(T)}$. Since $z_{j_1}>y_s$, by iteratively applying Lemma \[lem:2step\] we obtain that $$z_{j_i} \geq y_{s+i-1} \textup{ for each }i\in\{1,\ldots,q\}.$$ Therefore, by Lemma \[lem:y-min\], $$\begin{aligned}
\label{eq:Ddesc}
\begin{aligned}
D_{desc} & = 2\sum_{i=1}^q(\varphi(T)-z_{j_i}) \leq 2\sum_{i=1}^q (\varphi(T)-y_{s+i-1}) \\
& \leq 2(\varphi(T) - \varphi_0) + 2\sum_{i=2}^q (\varphi(T)-x_{s+i-1}) \leq 2\omega(P) + O_{path}.
\end{aligned}\end{aligned}$$ By an analogous analysis, the same bound holds for the third part of ${\mathrm{ADFS}_{}(T)}$: $$\label{eq:Dasc}
D_{asc} \le 2\omega(P) + O_{path}.$$ By the inductive assumption we have $$\label{eq:Ddeep}
D_{deep} < 10 \cdot O_{deep} - 16 \varphi(r').$$ We also get the following bounds by analyzing how much each particular route can overlap with ${\ensuremath{T[r']}}$. The first one follows from an observation that each route having a non-empty intersection with ${\ensuremath{T[r']}}$ may have length restricted to ${\ensuremath{T[r']}}$ at most $2\varphi(r')$. Thus, there exist at least $O_{deep}/(2\varphi(r'))$ such routes in ${\mathrm{ADFS}_{}(T)}$ intersecting ${\ensuremath{T[r']}}$ and each such route contributes at least $2\omega(P)$ to $O_{flat}$. Hence, $$\label{eq:Oflat}
O_{flat} \ge \frac{\omega(P)}{\varphi(r')} \cdot O_{deep}.$$ As for an upper bound, there exist at most $\lceil D_{deep}/(2\varphi(r'))\rceil+1$ routes in ${\mathrm{PDFS}(T)}$ that contain $r'$ (note that the first such route may include no other vertices except for $r'$ from ${\ensuremath{T[r']}}$). Thus, $$\label{eq:Dflat}
D_{flat} \le 2\omega(P) \cdot \lceil D_{deep}/(2\varphi(r')) + 1 \rceil \le 4 \omega(P) + \omega(P) \cdot D_{deep}/\varphi(r').$$ Equations , and , used in that order, give us $$\begin{aligned}
D_{deep} + D_{flat} & \leq & 4\omega(P) + \left(\frac{\omega(P)}{\varphi(r')}+1\right)D_{deep} \\
& \leq & 4\omega(P) + \left(\frac{\omega(P)}{\varphi(r')}+1\right)(10\cdot O_{deep} - 16\varphi(r')) \\
& \leq & 4\omega(P) - 16\left(\omega(P)+\varphi(r')\right) + 10\cdot O_{deep} \left(\frac{\omega(P)}{\varphi(r')}+1\right) \\
& \leq & 10 O_{deep} + 10 O_{flat} - 12 \omega(P) - 16\varphi(r').\end{aligned}$$ Then, by , and we have $$D_{light} + D_{asc} + D_{desc} \le 2O_{light} + 4\omega(P) + 2 O_{path}.$$ The last two inequalities, , and $\varphi(r)=\omega(P)+\varphi(r')$ finally give $$\begin{aligned}
{\xi({\mathrm{ADFS}_{}(T)})} & \le & 10 O_{deep} + 10 O_{flat} + 2O_{light} + 2O_{path} - 8\omega(P) - 16\varphi(r') \nonumber \\
& \le & 10 {\xi({\mathrm{COPT}(T)})} - 8 \varphi(r), \nonumber\end{aligned}$$ which completes the inductive proof of Theorem \[thm:induction\].
Conclusions and open problems
=============================
Our strategy ${\mathrm{PDFS}(T)}$ achieves the asymptotically minimum number of routes and also minimizes the cost up to a small constant. In particular, we provided an upper bound of $10$ for the competitiveness of any online piecemeal exploration strategy. A trivial lower bound for the same problem is $3/2$ (Consider the tree with three branches of lengths $B/2$, $B/2$ and $B$, respectively, starting from the root: any online algorithm may cover the tree with 3 routes, while the optimal is 2 routes). This leaves a gap between the lower and upper bounds and the interesting open question is whether the strategy ${\mathrm{PDFS}(T)}$ is the best possible algorithm? Another open problem is to analyze similar strategies in other, more general, classes of graphs instead of trees.
**Appendix**
Tree rearrangement {#sec:rearrangement-appendix}
==================
This section is devoted to proving Lemma \[lem:perturbe\]. We start with an informal description providing a high level intuition that gives an overview of this section. Our first step (Section \[sec:Tprime-construction\]) is to construct a tree ${T_{\varepsilon}}$, $\varepsilon\in\mathbb{R}^+$, based on $T$ such that ${T_{\varepsilon}}$ satisfies property \[it:property\]. We will also need that ${T_{\varepsilon}}$ ‘resembles’ $T$ in the following way: a search strategy is valid for $T$ if and only if a ‘very similar’ strategy is valid for ${T_{\varepsilon}}$. To simplify the statements considerably, it will be convenient to encode strategies in an uniform way so that we can apply the same strategy for both trees, without going into the details of tedious but straightforward conversions between strategy for $T$ and strategy for ${T_{\varepsilon}}$. We thus define (Section \[sec:Tprime-epsilon\]) a collection ${\mathcal{C}}$ of all possible strategies (including adversarial ones and those that are not feasible for either $T$ or ${T_{\varepsilon}}$ because they contain routes that are too long or do not visit all vertices). Then in Section \[sec:Tprime-analysis\] we select the right value of $\varepsilon$. The value of $\varepsilon$ and the construction of ${T_{\varepsilon}}$ will ensure that a strategy in ${\mathcal{C}}$ is valid for $T$ if and only if it is valid for ${T_{\varepsilon}}$. We then finally provide the main result of this section (Lemma \[lem:perturbe\]) states that, again thanks to the choice of $\varepsilon$, if Theorem \[thm:induction\] holds for ${T_{\varepsilon}}$, then it holds for $T$, thus allowing us to restrict only to trees satisfying property \[it:property\].
The construction of ${T_{\varepsilon}}$ {#sec:Tprime-construction}
---------------------------------------
We now construct the tree ${T_{\varepsilon}}=(V(T)\cup X,E({T_{\varepsilon}}),\omega')$ based on $T=(V(T),E(T),\omega)$ and the construction depends on a parameter $\varepsilon>0$ that will be fixed later. We now impose only a condition on $\varepsilon$ that is needed for the construction itself to be valid: $$\varepsilon < \min\{\omega(e){\hspace{0.1cm}\bigl|\bigr.\hspace{0.1cm}}e\in E(T)\}.$$ Select an arbitrary vertex $u$ in $T$ with ${\ensuremath{\mathsf{heavydeg}}}(u)=1$ (denote by $\{u,v\}$ the heavy downward edge at $u$) and $d>1$ light downward edges $e_1,\ldots,e_d$ at $v$. Subdivide the edge $e$ (see Figure \[fig:perturbe\] for an illustration) by replacing it by a path $P=(u=v_0,v_1,\ldots,v_{d-1},v_d=v)$ with the following edge lengths: the first $d-1$ edges have length $\varepsilon/(d-1)$, i.e., $\omega'(v_i,v_{i+1})=\varepsilon/(d-1)$ for each $i\in\{0,\ldots,d-2\}$, and for the last edge we set $\omega'(\{v_{d-1},v\})=\omega(\{u,v\})-\varepsilon$. (Note that this preserves the distance between $u$ and $v$.) Then, the weight of each edge $e_i$ decreases by $\varepsilon$, $\omega'(e_i)=\omega(e_i)-\varepsilon$ and the higher endpoint of $e_i$ in ${T_{\varepsilon}}$ becomes $v_{i-1}$, $i\in\{1,\ldots,d\}$. (Note that this ensures that the distance between $u$ and the lower endpoint of $e_i$ or the distance between two children of $u$ is not greater in ${T_{\varepsilon}}$ than in $T$.) This construction allows us to assume (by permuting the edges $e_1,\ldots,e_d$ appropriately) that the DFS traversal of ${T_{\varepsilon}}$ visits the edges $e$ and $e_1,\ldots,e_d$ in the same order both in $T$ and in ${T_{\varepsilon}}$, ensuring that Condition \[it:P2\] is satisfied.
![(a) the node $u$ of $T$ with ${\ensuremath{\mathsf{heavydeg}}}(u)=1$ and light downward edges $e_1,\ldots,e_d$, $d=4$; (b) the corresponding edges in ${T_{\varepsilon}}$ (the “$-\varepsilon$” indicated that the weight of the edge is $\varepsilon$ smaller than the weight of the corresponding edge in $T$)[]{data-label="fig:perturbe"}](figs/fig-perturbe.pdf)
Since the vertex $v$ is selected arbitrarily, we repeat the above modification for each such vertex $v$ obtaining the final tree ${T_{\varepsilon}}$.
We will require the following conditions to be satisfied:
1. \[it:P1\] there are at most two downward edges at each vertex in ${T_{\varepsilon}}$ with ${\ensuremath{\mathsf{heavydeg}}}(v)=1$,
2. \[it:P2\] there exists a DFS traversal of ${T_{\varepsilon}}$ that visits the vertices in $V(T)$ in the same order as the DFS traversal that we have fixed for $T$ in this work,
3. \[it:P3\] ${\mathcal{S}}\in{\mathcal{C}}$ is feasible for $T$ if and only if ${\mathcal{S}}$ is feasible for ${T_{\varepsilon}}$.
Property \[it:P3\] will be proved in Lemma \[lem:classEps\] and we now note:
\[obs:P1P2\] For each $\varepsilon>0$, the tree ${T_{\varepsilon}}$ satisfies Conditions \[it:P1\] and \[it:P2\].
\[obs:P2ADFS\] The strategies ${\mathrm{ADFS}_{}(T)}$ and ${\mathrm{ADFS}_{}({T_{\varepsilon}})}$ visit the nodes in $V(T)$ in the same order.
Finding the right value of $\varepsilon$ {#sec:Tprime-epsilon}
----------------------------------------
We define a *potential route* as a following pair: $R'=(L,v)$, where $L=(l_1,\ldots,l_p)$ is a sequence of leaves and $v\in V({T_{\varepsilon}})$. Then, $R'$ *translates* to a route $R$ in $T$ as a concatenation of the following paths (in this order): the path from $r$ to $l_1$, the path from $l_i$ to $l_{i+1}$, $i=1,\ldots,p-1$, the path from $l_p$ to the closest ancestor $x$ of $v$ that belongs to $V(T)$ and finally the path from $x$ to $r$. The length of $R$ is $${\ell(R)}=d(r,l_1)+d(l_p,x)+d(x,r)+\sum_{1\leq i<p}d(l_i,l_{i+1}).$$ $R'$ translates to a route in ${T_{\varepsilon}}$ in the same way, except that take $x=v$, i.e., $v$ is not replaced by the ancestor. Then, a *potential strategy* is a sequence consisting of at most $j$ potential routes, $j\in\{1,\ldots,p\}$, where $p$ is the number of leaves in $T$.
Note that a potential strategy may not translate to a valid $B$-exploration strategy for $T$ or ${T_{\varepsilon}}$ because some nodes may not be explored and some routes may be too long. A potential strategy is *feasible* for $T$ (respectively ${T_{\varepsilon}}$) if it translates to a valid $B$-exploration strategy for $T$ (respectively ${T_{\varepsilon}}$). We denote by ${\mathcal{C}}$ a collection of all potential strategies. Clearly, the size of ${\mathcal{C}}$ is finite.
We conclude with the following:
\[obs:potential-strategies\] For any route $R$ that may appear in ${\mathrm{ADFS}_{}(T)}$, ${\mathrm{COPT}(T)}$, ${\mathrm{ADFS}_{}({T_{\varepsilon}})}$ and ${\mathrm{COPT}({T_{\varepsilon}})}$ there exists a potential route that translates to $R$. $\qed$
The analysis of ${T_{\varepsilon}}$ {#sec:Tprime-analysis}
-----------------------------------
In this section we argue that the construction of ${T_{\varepsilon}}$ ‘preserves’ the problem: the minimum costs of adversarial DFS explorations of both $T$ and ${T_{\varepsilon}}$, as well as ${\mathrm{COPT}(T)}$ and ${\mathrm{COPT}({T_{\varepsilon}})}$ remain close to each other for $\varepsilon$ small enough. Intuitively speaking, this follows from a ‘continuity argument’ formalized in the remaining part of this section.
For the tree ${T_{\varepsilon}}$ we define an interval denoted by ${\mathcal{I}}=(0,y)$, $y<\min\{\omega(e){\hspace{0.1cm}\bigl|\bigr.\hspace{0.1cm}}e\in E(T)\}$, such that for each $\varepsilon\in (0,y)$, Condition \[it:P3\] holds for ${T_{\varepsilon}}$. We now prove that this interval is well defined, that is, $y>0$. (Note that for $\varepsilon=0$, ${T_{\varepsilon}}$ and $T$ are the same.)
\[lem:classEps\] It holds ${\mathcal{I}}\neq\emptyset$.
First we argue that there exists $y>0$ such that ${T_{y}}$ fulfills Condition \[it:P3\]. We select $y$ based on the tree $T$ and the collection ${\mathcal{C}}$. Consider any $B'\in{\mathcal{B}}$. The number of potential strategies in ${\mathcal{C}}$ is finite, and hence the number of potential strategies in ${\mathcal{C}}$ that do not translate to feasible ones for $T$ (denote subset of those by ${\mathcal{C}}'$) is also finite. For each ${\mathcal{S}}\in{\mathcal{C}}'$, define its *deficiency* $x({\mathcal{S}})$ as follows: the $x({\mathcal{S}})$ is the maximum value such that either the length of the first route in ${\mathcal{S}}$ is $B'+x({\mathcal{S}})$ or the length of some other route in ${\mathcal{S}}$ in $T$ is $B+x({\mathcal{S}})$. Intuitively, ${\mathcal{S}}$ does not translate to a feasible $B'$-adversarial $B$-exploration strategy for $T$ because one of its routes exceeds the allowed length by $x({\mathcal{S}})$ and no route exceeds it by more than $x({\mathcal{S}})$. Take $$y:=\frac{1}{2n(m+1)}\cdot\min\left\{\min\{\omega(e){\hspace{0.1cm}\bigl|\bigr.\hspace{0.1cm}}e\in E(T)\},\min\{x({\mathcal{S}}'){\hspace{0.1cm}\bigl|\bigr.\hspace{0.1cm}}{\mathcal{S}}'\in{\mathcal{C}}'\}\right\},$$ where $m$ is the number edges in $T$. Since by definition, $x({\mathcal{S}})>0$ for each ${\mathcal{S}}\in{\mathcal{C}}'$, we obtain that $y>0$. Also, no route in any ${\mathcal{S}}\in{\mathcal{C}}'$ traverses an edge more than $2n$ times and hence the length of any route of ${\mathcal{S}}$ in ${T_{y}}$ decreases by at most $2ynm$ with respect to its length in $T$. This implies that some route $R$ of ${\mathcal{S}}$ has length in ${T_{y}}$ at least $$\label{eq:Bx}
\tilde{B}+x({\mathcal{S}})-2ynm\geq \tilde{B}+2yn(m+1)-2ynm>\tilde{B},$$ where take $\tilde{B}=B'$ if $R$ is the first route in ${\mathcal{S}}$ and $\tilde{B}=B$ otherwise. Therefore, ${\mathcal{S}}$ remains unfeasible in ${T_{y}}$. Since exploration strategy that is feasible in $T$ remains feasible in ${T_{y}}$ (recall that the length of each route is smaller in ${T_{y}}$ than in $T$), we have that Condition \[it:P3\] holds for ${T_{y}}$.
Finally, observe that substituting $y$ by any value $\varepsilon$ smaller than $y$ in the left hand side of keeps this equation true. Therefore, Condition \[it:P3\] is satisfied by ${T_{\varepsilon}}$ for each $\varepsilon\in(0,y)$, which completes the proof.
Before we state the main lemma of this section, we prove these technical bounds:
\[lem:pert-requirements\] For each $\varepsilon\in{\mathcal{I}}$ it holds:
1. \[it:per1\] ${\xi({\mathrm{ADFS}_{}(T)})}\leq{\xi({\mathrm{ADFS}_{}({T_{\varepsilon}})})}+4\varepsilon n^2$,
2. \[it:per2\] $\varphi({T_{\varepsilon}})\leq \varphi(T)$, and
3. \[it:per3\] ${\xi({\mathrm{COPT}({T_{\varepsilon}})})}\leq{\xi({\mathrm{COPT}(T)})}$.
By Observation \[obs:P2ADFS\], both ${\mathrm{ADFS}_{}(T)}$ and ${\mathrm{ADFS}_{}({T_{\varepsilon}})}$ visit the leaves of both trees in the same order. Consider any edge $\{u,v\}$ in $T$ such that $v$ is its lower endpoint. We have that there exists a *corresponding* edge $\{u',v\}$ in ${T_{\varepsilon}}$. Moreover, $\omega(\{u,v\})\leq\omega'(\{u',v\})+\varepsilon$. Since for each edge traversal in ${\mathrm{ADFS}_{}(T)}$, a traversal of the corresponding edge occurs in ${\mathrm{ADFS}_{}({T_{\varepsilon}})}$ due to Condition \[it:P3\], we obtain that if $t$ is the number of edge traversals in ${\mathrm{ADFS}_{}(T)}$, then $${\xi({\mathrm{ADFS}_{}(T)})} \leq {\xi({\mathrm{ADFS}_{}({T_{\varepsilon}})})} + t\varepsilon \leq {\xi({\mathrm{ADFS}_{}({T_{\varepsilon}})})} + 4n^2\varepsilon.$$ The latter inequality follows from bounding each route in ${\mathrm{ADFS}_{}({T_{\varepsilon}})}$ to have at most $2n$ edges, bounding the number of routes by $n$ and observing that an edge is traversed at most twice in each route.
Property \[it:per2\] is a direct consequence of the construction of ${T_{\varepsilon}}$: any path in $T$ corresponds to a path in ${T_{\varepsilon}}$ that connects the same common nodes and contains all common edges of the original path.
By construction of ${T_{\varepsilon}}$, the distance between any common nodes $u\in V(T)$ and $v\in V(T)$ is not greater in $T$ than in ${T_{\varepsilon}}$, which immediately gives \[it:per3\].
There exists $\varepsilon>0$ such that if Theorem \[thm:induction\] holds for ${T_{\varepsilon}}$, then Theorem \[thm:induction\] holds for $T$.
We will analyze condition \[it:rec1\] in Theorem \[thm:induction\] and the proof for \[it:rec2\] is analogous as we note at the end of the proof.
Define a parameter $\tau(n')$ to be the maximum number for which an inequality $${\xi({\mathrm{ADFS}_{}(T')})} \leq 10 \cdot {\xi({\mathrm{COPT}(T')})} - 8 \cdot \varphi(T') - \tau(n')$$ holds for each tree $T'$ on at most $n'$ nodes that satisfies Condition \[it:rec1\]. Since the number of such trees is finite and the number of potential strategies in ${\mathcal{C}}$ is finite for each tree $T'$, we obtain that $\tau(n')>0$. By Lemma \[lem:classEps\], there exists $\varepsilon\in{\mathcal{I}}$ such that $\varepsilon>0$ and $$\varepsilon<\frac{\tau(2n)}{4n^2},$$ $n={\left|V(T)\right|}$, such that ${T_{\varepsilon}}$ satisfies Condition \[it:P3\]. Suppose that Theorem \[thm:induction\]\[it:rec1\] holds for ${T_{\varepsilon}}$. Then, by Lemma \[lem:pert-requirements\] (in particular, \[it:per1\] of Lemma \[lem:pert-requirements\] is used to obtain the first inequality below and \[it:per2\] and \[it:per3\] are used to obtain the third inequality below) we get: $$\begin{aligned}
{\xi({\mathrm{ADFS}_{}(T)})} & < & {\xi({\mathrm{ADFS}_{}({T_{\varepsilon}})})} + 4\varepsilon n^2 \nonumber \\
& \le & 10 \cdot {\xi({\mathrm{COPT}({T_{\varepsilon}})})} - 8 \cdot \varphi({T_{\varepsilon}}) + 4\varepsilon n^2 - \tau(2n) \nonumber \\
& \le & 10 \cdot {\xi({\mathrm{COPT}(T)})} - 8 \cdot \varphi(T) + 4\varepsilon n^2 - \tau(2n) \nonumber \\
& < & 10 \cdot {\xi({\mathrm{COPT}(T)})} - 8 \cdot \varphi(T). \nonumber \end{aligned}$$ We can conduct the same argument for Theorem \[thm:induction\]\[it:rec2\] with the same value of $\varepsilon$. Hence we obtain that Theorem \[thm:induction\] holds for $T$.
[^1]: Research partially supported by National Science Centre (Poland) grant number 2015/17/B/ST6/01887.
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---
abstract: 'In present work the effective singlet-triplet model for $CuO_{2}$-layer in the framework of multiband p-d model of strongly correlated electrons is obtained. The resulting Hamiltonian has a form of generalized singlet-triplet t-t’-J model for p-type superconductors and form of usual t-t’-J model for n-type superconductors. In the mean field approximation in X-operator representation we derived equations for Gorkov type Green functions. The symmetry classification of the superconducting order parameter in the case of tetragonal lattice resulted in $d_{x^{2}-y^{2}}$- and $d_{xy}$-types of singlet pairing for both p- and n-type superconductors while s-type singlet pairing don’t take place. Also normal paramagnetic phase of effective singlet-triplet model was investigated and the Fermi-type quasiparticle dispersion over Brillouin zone, density of states and evolution of Fermi level with doping were obtained.'
address:
- ' L.V. Kirensky Institute of Physics, Siberian Branch of RAS, Krasnoyarsk, 660036, Russia'
- ' UNESCO Chair “New materials and technology”, Krasnoyarsk State Technical University, Krasnoyarsk, 660074, Russia'
author:
- 'Maxim M. Korshunov and Sergei G. Ovchinnikov'
title: 'Electronic Properties of the Effective Singlet-Triplet Model'
---
Introduction
============
Almost twenty years passed from the discovery of High-$T_{c}$ Superconductivity (HTSC) but for now there is no widely accepted theory of pairing mechanism in cuprates of p- and n-type. The necessary ingredient for discussing possible mechanisms of HTSC is the band structure of the fermion-like quasiparticles. However, it is a difficult subject for ab initio calculations due to the strong electron correlations. For this reason we will use a model approach and in order to get an agreement with experimental data we will start with a realistic multiband p-d model of transition metal oxides.
The Hubbard model is often used to study electronic structure of strongly correlated electron systems (SCES). To take into account the chemistry of metal oxides the Hubbard model is generalized to the p-d model, a simplest version of such a model has been proposed by Emery [@bib1] and Varma et al. [@bib2]. In this 3-band p-d model only $d_{x^{2}-y^{2}}$ Cu and $p_{\sigma}$ O orbitals are considered. One of the important features omitted in this model is asymmetry of n-type (electrons doped) and p-type (holes doped) cuprates. Point is that the spin-exciton concerned with singlet-triplet excitation of two-hole term occurs only in p-type systems, but not in n-type [@bib3]. Other feature is a non-zero occupancy of $d_{z^{2}}$ Cu orbitals [@bib4]. There also a dependence between $T_{c}$ and occupancy of $d_{z^{2}}$ was found. Hence more realistic model of $CuO_{2}$-layer has to include $d_{x^{2}-y^{2}}$- and $d_{z^{2}}$- orbitals on copper and $p_{x}$-, $p_{y}$-, $p_{z}$- orbitals on oxygen as well. Such model is the multiband p-d model that has been proposed by Gaididei and Loktev [@bib5]:
$$H_{p-d} =\sum_{r}H_{d} (r) +\sum_{i}H_{p} (i) +\sum_{<r,
i>}H_{pd} (r,i) +\sum_{<i, j>}H_{pp}(i,j), \label{eq1}$$
where $$\begin{aligned}
\fl H_{d}(r) = \sum_{\lambda, \sigma}\left[
\varepsilon_{\lambda}^{d} \,d_{\lambda r \sigma}^{+} \,d_{\lambda
r \sigma} +\frac{U_{\lambda }^{d} }{2} n_{\lambda r}^{\sigma} n_{\lambda r}^{\bar{\sigma}} \right. \nonumber \\
\lo- \left. \sum_{\lambda', \sigma'}\left( J_{\lambda
\lambda'}^{dd} d_{\lambda r \sigma}^{+} d_{\lambda r \sigma'}
d_{\lambda' r \sigma'}^{+} d_{\lambda' r \sigma} - \sum_{r'}
V_{\lambda \lambda'}^{dd} n_{\lambda r}^{\sigma}
n_{\lambda' r'}^{\sigma'} \right) \right], \\
\fl H_{p}(i) = \sum_{\alpha, \sigma} \left[
\varepsilon_{\alpha}^{p} p_{\alpha i \sigma}^{+} p_{\alpha i
\sigma} + \frac{U_{\alpha }^{p} }{2} n_{\alpha i}^{\sigma}
n_{\alpha i}^{\bar{\sigma}} \right. \nonumber \\
\lo- \left. \sum_{\alpha',\sigma'}\left( J_{\alpha \alpha'}^{pp}
p_{\alpha i \sigma}^{+} p_{\alpha i \sigma'} p_{\alpha' i
\sigma'}^{+} p_{\alpha' i \sigma} - \sum_{i'}V_{\alpha
\alpha'}^{pp} n_{\alpha i}^{\sigma} n_{\alpha' i'}^{\sigma'} \right) \right], \\
\fl H_{pd}(r,i) = \sum_{\alpha, \lambda, \sigma, \sigma'} \left[
\left( t_{\lambda \alpha}^{pd} p_{\alpha i \sigma}^{+} d_{\lambda
r \sigma} + H.c. \right) + V_{\alpha \lambda}^{pd} n_{\alpha
i}^{\sigma} n_{\lambda r}^{\sigma'} \right], \\
\fl H_{pp}(i,j) = \sum_{\alpha, \beta, \sigma} \left[ t_{\alpha
\beta}^{pp} p_{\alpha i \sigma}^{+} p_{\beta j \sigma} + H.c
\right].\end{aligned}$$ Here $r$ and $i$ are Cu and O sites, $\lambda={d_{x^{2}-y^{2}},
d_{z^{2}}}$ and $\alpha={p_{x}, p_{y}, p_{z}}$ are orbital indexes on given copper and oxygen site respectively, $\varepsilon^{d}$ and $\varepsilon^{p}$ are energies of $d_{x^{2}-y^{2}}$- and $d_{z^{2}}$- holes on copper and $p_{x}$-, $p_{y}$-, $p_{z}$- holes on oxygen. $U^{d}$, $U^{p}$ are intra-atomic Coulomb interactions, $t^{pd}$ is hopping integral for nearest neighbors copper-oxygen, $t^{pp}$ is hopping integral for oxygen-oxygen, $V^{dd}$, $V^{pp}$, $V^{pd}$ are inter-atomic Coulomb interactions and $J^{dd}$, $J^{pp}$ are exchange integrals. Abbreviation “H.c.” means Hermitian Conjugation and “$<r, i>$” denotes that sum runs only over indices $r \neq i$.
Simplest calculations of this model have been made by exact diagonalization of $CuO_{4}$ [@bib3] and $CuO_{6}$ [@bib6] clusters. It was shown that when $d_{z^{2}}$-orbitals are neglected then the triplet with energy $\varepsilon_{2T}$ lies above singlet with energy $\varepsilon_{2S}$ on about 2 eV. This lets us to neglect $d_{z^{2}}$-orbitals and we immediately come back to the 3-band model. However, at approach of $d_{z^{2}}$-orbital energy to $d_{x^{2}-y^{2}}$-orbital energy the difference $\varepsilon_{2T}-\varepsilon_{2S}$ decreases and, at the certain parameters, crossover of singlet and triplet take place. Similar results where obtained in [@bib7] by self-consistent field method and in [@bib8] by perturbation theory. All these facts give us cause for deeper investigation of processes concerned with two-particle triplet.
Formulation of Effective Model
==============================
In this paper we will use Hubbard operators (or so-called X-operators) $X_{f}^{p\,q} \equiv \left| p\right\rangle
\left\langle q\right| $ on site f instead of annihilation and creation operators because they are a good tool in the case of strong electron correlation. Also defining that $m\leftrightarrow
(p,q)$ numerate a quasiparticle described by the Hubbard operator $X^{p\,q}$, and $\gamma_{\lambda \sigma} (m)$ is a parameter of X-operator representation for the single-electron annihilation operator with orbital $\lambda$ and spin $\sigma$ we can have the next correspondence between annihilation (and creation operators) and Hubbard operators: $$a_{f \lambda \sigma} = \sum_{m} \gamma_{\lambda \sigma}(m)
X_{f}^{m}.$$ Hermitian conjugation of Hubbard operator indicated by cross: ${X_{f}^{m}}^{+}$.
To calculate quasiparticle band structure in SCES the Generalized Tight-Binding (GTB) method has been proposed [@bib9.1]. This method combines the exact diagonalization of a cell part of the Hamiltonian and the perturbation treatment of the intercell part in the X-operator representation. In paper [@bib9.2] the consequent development of GTB method for $La_{2}CuO_{4}$ with a $CuO_{6}$ cluster as elementary cell was given. The problem of non-orthogonality of nearest clusters’ molecular orbitals was solved in direct way by constructing explicitly Vanier functions on $d_{x^{2}-y^{2}}, d_{3z^{2}-r^{2}}, p_{x}, p_{y}, p_{z}$ - five orbitals’ initial basis of atomic states. In a new symmetric basis one-cell part of Hamiltonian are factorized allowing to symmety classification of all possible effective one-particle excitations in $CuO_{2}$ plane as transitions from n-th hole term to (n+1)-hole term. The X-operators are constructed in the Hilbert space that consists of a vacuum $n_{h}=0$ state $| 0
\rangle$, single-hole $| \sigma \rangle = \left\{ | \uparrow
\rangle, | \downarrow \rangle \right\}$ molecular orbital of $b_{1g}$ symmetry, two-hole singlet $| S \rangle \equiv |
\uparrow, \downarrow \rangle$ of $^{1}A_{1g}$ symmetry and two-hole triplet $| TM \rangle$ (where $M=+1,0,-1$) of $^{3}B_{1g}$ symmetry states. In the X-operator basis the multiband p-d model Hamiltonian is given by: $$\begin{aligned}
\fl H = \sum_{f} \left( \varepsilon_{1} \sum_{\sigma} X_{f}^{\sigma \sigma} + \varepsilon_{2S}X_{f}^{S S} + \varepsilon_{2T} \sum_{M} X_{f}^{TM TM} \right) \nonumber \\
\lo + \sum_{<f, g>, \sigma}\left[ t_{fg}^{0 0} X_{f}^{\sigma 0}
X_{g}^{0 \sigma} + 2 \sigma t_{fg}^{0 b} \left(
X_{f}^{\sigma 0} X_{g}^{\bar{\sigma} S} + X_{f}^{S \bar{\sigma}} X_{g}^{0 \sigma} \right) + t_{fg}^{b b} X_{f}^{S \bar{\sigma}} X_{g}^{\bar{\sigma} S} \right] \nonumber \\
\lo + \sum_{<f, g>, \sigma}\ t_{fg}^{a a} \left( \sigma \sqrt{2} X_{f}^{T0 \ \bar{\sigma}} - X_{f}^{T2\sigma \ \sigma} \right) \ \left( \sigma \sqrt{2} X_{g}^{\bar{\sigma} \ T0} - X_{g}^{\sigma \ T2\sigma} \right) \nonumber \\
\lo + \sum_{<f,\ g>,\ \sigma}\ t_{fg}^{a b} \left[ \left( \sigma
\sqrt{2} X_{f}^{T0 \ \bar{\sigma}} - X_{f}^{T2\sigma \ \sigma}
\right) \ \left( -\upsilon X_{g}^{0 \sigma} + 2\sigma \gamma_{b}
X_{g}^{\bar{\sigma} S} \right) + H.c. \right] \label{eq2}\end{aligned}$$ Here the energies $\varepsilon_{1}$, $\varepsilon_{2S}$ and $\varepsilon_{2T}$ are counted off from chemical potential $\mu$, and $t_{fg}$ indexes $0$, $a$ and $b$ relevant to quasiparticle in lower singlet ($0$), in higher singlet ($b$) and in higher triplet ($a$) Hubbard’s bands. The scheme of the levels and available quasiparticle excitations between them are presented in figure \[fig1\].
![\[fig1\]Quasiparticle excitations between state in full singlet-triplet model (excitations for only one spin projections are shown).](fig1.eps){width="70.00000%"}
The t-t’-J model is derived by exclusion the intersubband hopping between low (LHB) and upper (UHB) Hubbard subbands for the Hubbard model [@bib10] and for 3-band p-d model [@bib11]. We write the Hamiltonian in the form $$H=H_{0}+H_{1},$$ where the excitations via the charge transfer gap are included in $H_{1}$. Then we define an operator $$H\left(\epsilon \right)=H_{0}+\epsilon H_{1}$$ and make the unitary transformation $$\tilde{H} \left( \epsilon \right) =\exp{\left(-\rmi \epsilon
\hat{S} \right)} H\left( \epsilon \right) \,\exp{\left( \rmi
\epsilon \hat{S} \right)}.$$
Vanishing linear in $\epsilon$ component of $ \tilde{H} \left(
\epsilon \right) $ gives the equation for matrix $\hat{S}$:
$$H_{1} + \rmi \left[ H_{0} ,\hat{S} \right]=0. \label{eq3}$$
The effective Hamiltonian is obtained in second order in $\epsilon$ and at $\epsilon=1$ is given by: $$\tilde{H}=H_{0}+\frac{1}{2} \rmi \left[ H_{1}, \hat{S} \right].
\label{eq4}$$
It is convenient to express the matrix $\hat{S}$ in terms of X-operators [@bib12]. Thus, for the multiband p-d model (\[eq1\]) with electron doping (n-type systems) we obtain the usual t-J model: $$\fl H_{t-J}=\sum_{i, \sigma}\varepsilon_{1} X_{i}^{\sigma \sigma}
+ \sum_{<i, j>, \sigma} t_{ij}^{0 0} X_{i}^{\sigma 0} X_{j}^{0
\sigma} + \frac{1}{2} \sum_{<i, j>} J_{ij} \left( \vec{S}_{i}
\vec{S}_{j} - \frac{1}{4} n_{i} n_{j} \right),$$ where spin operators $\vec{S}_{f}$ and number of particles operators $n_{f}$ can be expressed in terms of Hubbard operators as follows: $$\vec{S}_{i} \vec{S}_{j} - \frac{1}{4} n_{i} n_{j} = \frac{1}{2}
\sum_{\sigma}\left( X_{i}^{\sigma \bar{\sigma}}
X_{j}^{\bar{\sigma} \sigma} - X_{i}^{\sigma \sigma}
X_{j}^{\bar{\sigma} \bar{\sigma}} \right).$$
For p-type systems effective Hamiltonian has the form of a singlet-triplet t-t’-J model [@bib12]: $$\tilde{H}=H_{0}+H_{t}+H_{J}, \label{eq5}$$ where $H_{0}$ (unperturbated part of the Hamiltonian), $H_{t}$ (kinetic part of $\tilde{H}$) and $H_{J}$ (exchange part of $\tilde{H}$) are given by the following expressions:
$$\begin{aligned}
\fl H_{0} = \sum_{i}\left( \varepsilon_{1} \sum_{\sigma} X_{i}^{\sigma \sigma} + \varepsilon_{2S} X_{i}^{S S} + \varepsilon_{2T} \sum_{M} X_{i}^{TM TM} \right), \\
\fl H_{t} = \sum_{<i, j>, \sigma} t_{ij}^{b b} X_{i}^{S \bar{\sigma}} X_{j}^{\bar{\sigma} S} + \sum_{<i, j>, \sigma} t_{fg}^{a a} \left( \sigma \sqrt{2} X_{i}^{T0 \ \bar{\sigma}} - X_{i}^{T2\sigma \ \sigma} \right) \left( \sigma \sqrt{2} X_{j}^{\bar{\sigma} \ T0} - X_{j}^{\sigma \ T2\sigma} \right) \nonumber \\
\lo + \sum_{<i, j>, \sigma} t_{ij}^{a b} 2 \sigma \gamma_{b} \left[ X_{i}^{S \bar{\sigma}} \left( \sigma \sqrt{2} X_{j}^{\bar{\sigma} \ T0} - X_{j}^{\sigma \ T2\sigma} \right) +H.c. \right], \\
\fl H_{J} = \frac{1}{2} \sum_{<i, j>} \left( J_{ij} + \delta
J_{ij} \right) \left( \vec{S}_{i} \vec{S}_{j} - \frac{1}{4} n_{i}
n_{j} \right) - \frac{1}{2} \sum_{<i, j>, \sigma} \delta J_{ij}
X_{i}^{\sigma \sigma} X_{j}^{\sigma \sigma}.\end{aligned}$$
The $J_{ij}$ is the exchange integral: $$J_{ij} = 4 \frac{ \left( t_{ij}^{0 b} \right)^{2}}{E_{ct}},
\label{eq6}$$ and $\delta J_{ij}$ is the correction to it (dependent on the triplet’s contribution): $$\delta J_{ij} = 2 \upsilon^{2} \frac{\left(t_{ij}^{a b}
\right)^{2}}{E_{ct}}. \label{eq7}$$ For the nearest neighbors ($i=0$, $j=1$) the estimation gives: $$\frac{\delta J_{ij}}{J_{ij}} \sim 10^{-2}.$$ Here $E_{ct}$ is the charge-transfer energy gap (similar to $U$ in the Hubbard model, $E_{ct} \approx 2 eV$ for cuprates).
Previously the motion of triplet holes and the simplest version of singlet-triplet model have been studied in [@bib13; @bib14]. But these authors used the 3-band p-d model [@bib1; @bib2] where the difference between energy of singlet and energy of triplet $\varepsilon_T-\varepsilon_S$ is about 2 eV. This fact leads to negligible contribution of singlet-triplet excitations in low-energy physics. But in multiband p-d model $\varepsilon_T-\varepsilon_S$ depends strongly on various model parameters, particularly on distance of apical oxygen ($p_z$-orbitals) from planar oxygen ($p_x$- and $p_y$-orbitals), energy of apical oxygen, difference between energy of $d_{z^2}$-orbitals and $d_{x^2-y^2}$-orbitals ($\varepsilon_{d_{z^2}}-\varepsilon_{d_{x^2-y^2}}$). For the realistic values of model parameters $\varepsilon_T-\varepsilon_S$ appears to be less or equal to 0.5 eV (see [@bib3] and [@bib9.2]). It is this small value of singlet-triplet splitting that let us believe that singlet-triplet excitations has a non-negligible contribution to band structure and also can give new superconducting pairing channel.
As easily can be noted the resulting Hamiltonian (\[eq5\]) is the generalization of t-J model to account of two-particle triplet state. But the inclusion of this triplet leads to such significant changes in Hamiltonian as renormalization of exchange integral (\[eq7\]) and appearing of $X_{i}^{\sigma \sigma}
X_{j}^{\sigma \sigma}$ term in $H_{J}$.
More significant feature of effective singlet-triplet model (\[eq5\]) is the asymmetry for n- and p-type systems. This fact is known experimentally. In particular, the holes suppress antiferromagnetism more strongly than electrons. It was observed in $La_{2-x}Sr_{x}CuO_{4}$ in compare with $Nd_{2-x}Ce_{x}CuO_{4}$ [@bib15]. For n-type systems the usual t-J model takes place while for p-type superconductors with complicated structure on the top of the valence band the singlet-triplet transitions plays an important role. In first case we have spin-fluctuation pairing mechanism (see review [@bib16]). In the second case in addition to spin-fluctuations described by $H_{J}$ we also have pairing mechanism due to singlet-triplet excitations. Really, let’s look at structure of $H_{t}$. There we can see terms like $X_{i}^{S\,\bar{\sigma}} X_{j}^{\bar{\sigma} \,T0} $ which can be identically written according to multiplication rule of Hubbard operators: $$X_{i}^{S \bar{\sigma}} X_{j}^{\bar{\sigma} T0} = X_{i}^{S
\bar{\sigma}} X_{j}^{\bar{\sigma} S} X_{j}^{S T0}.$$
Here we can see three processes: creation ($X_{i}^{S
\bar{\sigma}}$) and annihilation ($X_{j}^{\bar{\sigma} S}$) of the hole at different sites i and j, and spin-exciton $X_{j}^{S
T0}$. This spin-exciton can play a role of intermediate boson in superconducting pairing.
Green Functions for Effective Singlet-Triplet Model
===================================================
Processes described by Hamiltonian (\[eq5\]) are shown in the figure \[fig2\]. We have three Fermi-type excitations and hence three base root vectors [$\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$]{}. Corresponding basis of the Hubbard operators is: $$\left\{ X_{i}^{\bar{\sigma} S}, X_{i}^{\bar{\sigma} \ T0},
X_{i}^{\sigma \ T2\sigma} \right\} \Rightarrow \left\{ X_{i}^{1},
X_{i}^{2}, X_{i}^{3} \right\}.$$ The condition of basis completeness has a form: $$\sum_{\sigma} X_{f}^{\sigma \sigma} + X_{f}^{S S} + \sum_{M}
X_{f}^{TM \ TM}=1,$$ where $M=\left\{ 2\sigma, \ 0, \ 2\bar{\sigma} \right\}$.
![\[fig2\]Quasiparticle excitations between state in effective singlet-triplet model (excitations for only one spin projections are shown).](fig2.eps){width="60.00000%"}
In the introduced notations $H_{t}$ has the following form: $$H_{t} = \sum_{<i,j>, \sigma} \sum_{m,n} \gamma_{i j,
\sigma}(m,n)\,{X_{i \sigma}^{m}}^{+} X_{j\sigma}^{n}, \label{eq8}$$ where $m$ and $n$ are root vectors. Also, for further convenience, direct dependence on spin $\sigma$ is introduced. Matrix $\gamma_{ij,\,\sigma}(m,n)$ can easily be obtained from Hamiltonian (\[eq5\]): $$\gamma_{ij,\sigma} (m,n)=\left(
\begin{array}{ccc}
t_{ij}^{bb} & \frac{1}{\sqrt{2} } \gamma_{b} t_{ij}^{ab} &
-2\sigma \gamma_{b} t_{ij}^{ab} \\
\frac{1}{\sqrt{2} } \gamma_{b} t_{ij}^{ab} & \frac{1}{2}
t_{ij}^{aa} & -\sigma \sqrt{2} t_{ij}^{aa} \\
-2\sigma \gamma_{b} t_{ij}^{ab} & -\sigma \sqrt{2}
t_{ij}^{aa} & t_{ij}^{aa}
\end{array}
\right),$$
Equations of motion for Hamiltonian (\[eq5\]) can be expressed as: $$\rmi \frac{\rmd}{\rmd t} X_{f\sigma}^{p} =\left[ X_{f\sigma}^{p}
,H_{0} +H_{t} +H_{J} \right] =\Omega_{p} X_{f\sigma}^{p}
+L_{f\sigma}^{p}, \label{eq9}$$ where $ L_{f\sigma}^{p} = L_{f\sigma}^{p}(t)+L_{f\sigma}^{p}(J)$, $L_{f\sigma}^{p}(t) \equiv [X_{f\sigma}^{p},H_{t}]$, $L_{f\sigma}^{p}(J) \equiv [X_{f\sigma}^{p},H_{J}]$, and $\Omega_{p} \equiv [X_{f\sigma}^{p},H_{0}]$ is vector of one-electronic energies: $$\Omega_{p}=\left(
\begin{array}{l}
\varepsilon_{2S}-\varepsilon_{1} \\
\varepsilon_{2T}-\varepsilon_{1} \\
\varepsilon_{2T}-\varepsilon_{1}
\end{array}
\right).$$
Taking into account equation (\[eq8\]) and identity $\left[
A,BC\right] \equiv \left\{ A,B\right\} C-B\left\{ A,C\right\}$ we can right away get an expression for $L_{f\sigma}^{p}(t)$: $$L_{f\sigma}^{p}(t) = \sum_{g, \sigma'} \sum_{m,n} \gamma_{fg,
\sigma'}(m,n)\left( E_{f}^{\sigma \sigma'}(p,m) X_{g\sigma'}^{n}
- {X_{g \sigma'}^{m}}^{+} D_{f}^{\sigma \sigma'}(p,n) \right),$$ where $E_{f}^{\sigma \sigma'}(p,m) \equiv \left\{
X_{f\sigma}^{p}, {X_{f\sigma'}^{m}}^{+} \right\}$ is the neutral boson and $D_{f}^{\sigma \sigma'}(p,n) \equiv \left\{
X_{f\sigma}^{p}, X_{f\sigma'}^{n} \right\}$ is $2e$ charged boson:
$$\begin{aligned}
\fl E_{f}^{\sigma \sigma'}(p,m) = \delta_{\sigma \sigma '}
\left(\begin{array}{ccc} X_{f}^{SS} +X_{f}^{\bar{\sigma}
\bar{\sigma} } & X_{f}^{S T0} & 0 \\
X_{f}^{T0 S} & X_{f}^{T0 T0} + X_{f}^{\bar{\sigma} \bar{\sigma}
} & 0 \\
0 & 0 & X_{f}^{T2\sigma T2\sigma} + X_{f}^{\sigma \sigma}
\end{array}
\right) \nonumber \\
\lo + \delta_{\bar{\sigma} \sigma '} \left(\begin{array}{ccc}
X_{f}^{\bar{\sigma} \sigma} & 0 & X_{f}^{S T2\sigma} \\
0 & X_{f}^{\bar{\sigma} \sigma} & X_{f}^{T0 T2\sigma} \\
X_{f}^{T2\bar{\sigma} S} & X_{f}^{T2\bar{\sigma} T0} & 0
\end{array}
\right), \\
\fl D_{f}^{\sigma \sigma '}(p,n)=0.\end{aligned}$$
Vanishing of $D_{f}^{\sigma \sigma'}(p,n)$ is due to neglect of two-hole excitations $| 0 \rangle \rightarrow | S \rangle $ in the t-t’-J model. Nevertheless such excitations had their effect resulting in the superexchange interaction J.
It’s easy to rewrite expression for $L_{f\sigma}^{p}(t)$ in reciprocal space (or so-called k-space): $$L_{k\sigma}^{p}(t) = \sum_{q,\sigma'} \sum_{m,n} \gamma_{-q,
\sigma'}(m,n) E_{k-q}^{\sigma \sigma'}(p,m) X_{q \sigma'}^{n}.
\label{eq10}$$
Introducing for convenience operator: $$Y_{q \sigma} \equiv \frac{1}{2} J_{-q} X_{q}^{\sigma \sigma} +
\delta J_{-q} X_{q}^{\bar{\sigma} \bar{\sigma}}, \label{eq11}$$ we can now write down $L_{f\sigma}^{p}(J)$: $$\begin{aligned}
\fl L_{k\sigma}^{1}(J) = \frac{1}{2} \sum_{q} \left( X_{k-q}^{\bar{\sigma} S} Y_{q\sigma} + Y_{q\sigma} X_{k-q}^{\bar{\sigma} S} - \frac{1}{2} J_{-q} \left( X_{k-q}^{\sigma S} X_{q}^{\bar{\sigma} \sigma} + X_{q}^{\bar{\sigma} \sigma} X_{k-q}^{\sigma S} \right) \right), \\
\fl L_{k\sigma}^{2}(J) = \frac{1}{2} \sum_{q} \left( X_{k-q}^{\bar{\sigma} \ T0} Y_{q\sigma} + Y_{q\sigma} X_{k-q}^{\bar{\sigma} \ T0} - \frac{1}{2} J_{-q} \left( X_{k-q}^{\sigma \ T0} X_{q}^{\bar{\sigma} \sigma} + X_{q}^{\bar{\sigma} \sigma} X_{k-q}^{\sigma \ T0} \right) \right), \\
\fl L_{k\sigma}^{3}(J) = \frac{1}{2} \sum_{q}\left(
X_{k-q}^{\sigma \ T2\sigma} Y_{q\bar{\sigma}} + Y_{q
\bar{\sigma}} X_{k-q}^{\sigma \ T2\sigma} - \frac{1}{2} J_{-q}
\left( X_{k-q}^{\bar{\sigma} \ T2\sigma} X_{q}^{\sigma
\bar{\sigma}} + X_{q}^{\sigma \bar{\sigma}} X_{k-q}^{\bar{\sigma}
\ T2\sigma} \right) \right).\end{aligned}$$
For decoupling of equations on Green functions we will use the method of irreducible operators [@bib17]. This method based on linearization of equations of motion within Generalized Hartree-Fock Approximation (GHFA). Let us first introduce the irreducible operator: $$\overline{L_{k\sigma}^{p}} = L_{k\sigma}^{p} - \sum_{h}
C_{k\sigma}(p,h) X_{k\sigma}^{h} - \sum_{h} \Delta_{k\sigma}(p,h)
{X_{-k\bar{\sigma}}^{h}}^{+}, \label{eq12}$$ where $$\begin{aligned}
C_{k\sigma}(p,h) = \frac{\left\langle \left\{ L_{k\sigma}^{p}, {X_{k\sigma}^{h}}^{+} \right\} \right\rangle } {\left\langle \left\{X_{k\sigma}^{h}, {X_{k\sigma}^{h}}^{+} \right\} \right\rangle }, \\
\Delta_{k\sigma}(p,h) = \frac{\left\langle \left\{
L_{k\sigma}^{p}, X_{-k\bar{\sigma}}^{h} \right\} \right\rangle }
{\left\langle \left\{{X_{-k\bar{\sigma}}^{h}}^{+},
X_{-k,\bar{\sigma}}^{h} \right\} \right\rangle }. \label{eq13}\end{aligned}$$
Here $\Delta_{k\sigma}$ connected to the superconducting order parameter. Its symmetrical properties will be analyzed in next chapter.
Now we can write down expressions for $C_{k\sigma}(p,h)$ and $\Delta_{k\sigma}(p,h)$ in compact form: $$\begin{aligned}
\fl C_{k\sigma}(p,h) = \frac{1}{\left\langle E_{k=0}^{\sigma \sigma}(h,h) \right\rangle}
\sum_{q} \left\langle E_{q}^{\sigma \sigma}(h,h) \right\rangle \nonumber \\
\lo \times \left[ \gamma_{q-k,\sigma}(p,h)
\left\langle E_{-q}^{\sigma \sigma}(p,p) \right\rangle +
\delta_{p,h} \left\langle Y_{-q\sigma}(p,h) \right\rangle \right], \label{eq14} \\
\fl \Delta_{k\sigma}(p,h) = \frac{1}{\left\langle
E_{k=0}^{\bar{\sigma} \bar{\sigma}}(h,h) \right\rangle} \sum_{q}
\left[ - A_{q-k}^{(1)}(p,h)
B_{q\sigma}(h,p) - A_{q+k}^{(2)}(p,h) B_{q\sigma}(p,h) \right. \nonumber \\
\lo + \left. \sum_{m,n} \left( \gamma_{-q,\sigma}(m,n)
R_{q\sigma}^{(1)}(p,m;h,n) - \gamma_{-q,\bar{\sigma}}(m,n)
R_{q\sigma}^{+(2)}(p,m;h,n) \right) \right]. \label{eq15}\end{aligned}$$ Tensors $R_{q\sigma}^{(1)} (p,m;h,n)$ and $R_{q\sigma}^{+\,(2)}
(p,m;h,n)$ are presented in \[appendixA\].
Averaging-out in $C_{k\sigma}(p,h)$ were made in Hubbard-I approximation and all averages in $C_{k\sigma}(p,h)$ are as follows: $$\begin{aligned}
\fl \left\langle E_{k}^{\sigma \sigma'}(p,h) \right\rangle =
\delta_{\sigma \sigma'} \left(
\begin{array}{ccc}
\left\langle X_{k}^{SS} +X_{k}^{\bar{\sigma} \bar{\sigma} }
\right\rangle & 0 & 0 \\
0 & \left\langle X_{k}^{T0 T0} +X_{k}^{\bar{\sigma} \bar{\sigma
} } \right\rangle & 0 \\
0 & 0 & \left\langle X_{k}^{T2\sigma T2\sigma} +X_{k}^{\sigma
\sigma} \right\rangle
\end{array}
\right), \label{eq16} \\
\fl \left\langle Y_{q\sigma}(p,h) \right\rangle = \left(
\begin{array}{ccc}
\left\langle Y_{q\sigma} \right\rangle & 0 & 0 \\
0 & \left\langle Y_{q\sigma} \right\rangle & 0 \\
0 & 0 & \left\langle Y_{q\bar{\sigma} } \right\rangle
\end{array}
\right), \label{eq17}\end{aligned}$$ where $Y_{q\sigma}$ is determined by (\[eq11\]). Also the anomalous averages were introduced: $$B_{k\sigma}(p,h) \equiv \left\langle X_{-k\bar{\sigma}}^{p}
X_{k\sigma}^{h} \right\rangle. \label{eq18}$$ Direct calculations can reveal one useful property of this averages: $$B_{-k\bar{\sigma}}(h,p) = -B_{k\sigma}(p,h). \label{eq19}$$
At calculation of $ \left\langle \left\{ L_{k\sigma}^{3}(J),
X_{-k\bar{\sigma}}^{h} \right\} \right\rangle $ the terms of type $ \left\langle X_{k-q}^{\bar{\sigma} \ T2\sigma}
P_{k,q}^{\sigma}(h) \right\rangle$ appears where $ P_{k,
q}^{\sigma}(h) = \sqrt{N} [X_{q}^{\bar{\sigma} \sigma},
X_{-k\bar{\sigma}}^{h}]$ is a Fermi-like operator. These terms are responsible for 3-fermion excitations and in the Hubbard-I approximation they are equal to zero.
Tensors $R_{q\sigma}^{(1)}(p,m;h,n)$ and $R_{q\sigma}^{+\,(2)}(p,m;h,n)$ are presented in Appendix I. Shortly, each of their elements is equal to anomalous average $B_{q\sigma}$ or zero. Matrixes $A_{q-k}^{(1)}(p,h)$ and $A_{q+k}^{(2)}(p,h)$ consist of exchange integrals and has the following explicit form: $$\begin{aligned}
A_{q-k}^{(1)}(p,h) = \frac{1}{2} \left(
\begin{array}{ccc}
J_{q-k} & J_{q-k} & 2\delta J_{q-k} \\
J_{q-k} & J_{q-k} & 2\delta J_{q-k} \\
2\delta J_{q-k} & 2\delta J_{q-k} & 2\delta J_{q-k}
\end{array}
\right), \\
A_{q+k}^{(2)}(p,h) = \frac{1}{2} \left(
\begin{array}{ccc}
J_{q+k} & J_{q+k} & 0 \\
J_{q+k} & J_{q+k} & 0 \\
0 & 0 & 0
\end{array}
\right).\end{aligned}$$
We can now perform elementary check of the obtained $\Delta_{k\sigma}(p,h)$ and $C_{k\sigma}(p,h)$ by using fact that when moving energy of the triplet $TM$ to the infinity we have to obtain the usual t-J model [@bib18; @bib19]. Because in paper [@bib18] Hubbard operators were also used we will compare our gap and renormalization of spectrum with its results: $$\begin{aligned}
C_{k\sigma}^{t-J} = t_{k} (1-n_{\bar{\sigma}}) - \frac{J}{2} n_{\bar{\sigma}}, \\
\Delta_{k\sigma}^{t-J} = \frac{1}{1-n_{\sigma}} \sum_{q} \left(
2t_{q} - \frac{1}{2} \left( J_{k+q} + J_{k-q} \right) \right)
B_{q\sigma}^{t-J}.\end{aligned}$$ Neglecting all excitations to the triplet zone ($\alpha_{2}
\rightarrow 0$, $\alpha_{3} \rightarrow 0$, $\delta J_{q}
\rightarrow 0$, and $t_{fg}^{nm} \rightarrow 0$ if n or m is equal to a) coefficients and matrixes in singlet-triplet model becomes: $$\begin{aligned}
\gamma_{fg,\sigma}(m,n) = \delta_{m 1} \delta_{n 1} t_{fg}^{bb}, \\
\left\langle E_{f}^{\sigma \sigma'} \right\rangle = \delta_{p 1} \delta_{m 1} \delta_{\sigma, \sigma'} \left\langle X_{f}^{S S} + X_{f}^{\bar{\sigma} \bar{\sigma}} \right\rangle, \\
B_{q\sigma}(p,h) = \delta_{p 1} \delta_{h 1} \left\langle X_{-q\bar{\sigma}}^{1} X_{q\sigma}^{1} \right\rangle \equiv B_{q\sigma}, \\
\left\langle Y_{q\sigma}(p,h) \right\rangle = \delta_{p 1}
\delta_{h 1} \left\langle \frac{1}{2} J_{-q} X_{q}^{\sigma \sigma}
\right\rangle.\end{aligned}$$ Having made all these limiting transitions in (\[eq14\]) and (\[eq15\]) we derive: $$\begin{aligned}
\fl C_{k\sigma} \rightarrow \frac{1}{1-n_{\bar{\sigma}}} \sum_{q}(1-n_{\bar{\sigma}})
\left( t_{q-k}^{bb} (1-n_{\bar{\sigma}}) +
\frac{1}{2} J_{-q} (-n_{\bar{\sigma}})\right) \delta_{q,0} \nonumber \\
\lo = t_{k}^{bb} (1-n_{\bar{\sigma}}) - \frac{1}{2} J_{0} n_{\bar{\sigma}}, \\
\fl \Delta_{k\sigma} \rightarrow \frac{1} {1-n_{\sigma}} \sum_{q}
\left( 2t_{-q}^{bb} - \frac{1}{2} \left( J_{q-k} + J_{q+k}
\right) \right) B_{q\sigma}.\end{aligned}$$ This is exactly what we have in the t-J model. So the effective singlet-triplet model in the low energy limit (triplet energy tend to infinity) reduces to the t-J model.
Let’s return to decoupling of Green functions. In GHFA we have to put irreducible operator $\overline{L_{k\sigma}^{p}}$ to zero. In this case the equations of motion take the following form: $$\fl \cases{ \rmi \frac{\rmd}{\rmd t} X_{k\sigma}^{p} = \sum_{h}
\left[ \Omega_{p} \ \delta_{p,h} + C_{k\sigma}(p,h) \right]
X_{k\sigma}^{h} +
\sum_{h} \Delta_{k\sigma}(p,h) {X_{-k\bar{\sigma}}^{h}}^{+} \\
\rmi \frac{\rmd}{\rmd t} {X_{-k\bar{\sigma}}^{p}}^{+} = -
\sum_{h} \left[ \Omega_{p} \ \delta _{p,h} +
C_{-k\bar{\sigma}}^{+}(p,h) \right] {X_{-k\bar{\sigma}}^{h}}^{+} -
\sum_{h} \Delta_{-k\bar{\sigma}}^{+}(p,h) X_{k\sigma}^{h} }
\label{eq20}$$ Now we can easily write down a system of Gorkov type equations for normal and abnormal, and this system is closed: $$\fl \cases{ \sum_{h} \left[ \left( E-\Omega_{p} \right)
\delta_{p,h} - C_{k\sigma}(p,h) \right] \hat{G}_{k\sigma}(h,l) -
\sum_{h} \Delta_{k\sigma} (p,h) \hat{F}_{k\sigma}^{+}(h,l) =
\left\langle \left\{ X_{k\sigma}^{p}, {X_{k\sigma}^{l}}^{+} \right\} \right\rangle \\
\sum_{h} \left[ \left( E+\Omega_{p} \right) \delta_{p,h} +
C_{-k\bar{\sigma}}^{+}(p,h) \right] \hat{F}_{k\sigma}^{+}(h,l) +
\sum_{h} \Delta_{-k\bar{\sigma}}^{+}(p,h) \hat{G}_{k\sigma}(h,l)
= 0 }$$ Actually this is an equation for a matrix normal Green function $\hat{G}_{k\sigma}=\left\langle \left\langle X_{k\sigma}^{h}
\right.\left| {X_{k\sigma}^{l}}^{+} \right\rangle \right\rangle$ and matrix abnormal Green function $\hat{F}_{k\sigma}^{+}=\left\langle \left\langle
{X_{-k\bar{\sigma}}^{h}}^{+} \right.\left| {X_{k\sigma}^{l}}^{+}
\right\rangle \right\rangle$.
Introducing matrixes: $$\begin{aligned}
\hat{\Re}_{k\sigma} = \left( \Omega_{p} \delta_{p,h} + C_{k\sigma}(p,h)\right), \\
\hat{\Delta}_{k\sigma} = \Delta_{k\sigma}(p,h), \\
\hat{E}_{k}^{\sigma \sigma} = E_{k}^{\sigma \sigma}(p,h) \equiv
\sqrt{N} \left\{ X_{k\sigma}^{p}, {X_{k\sigma}^{l}}^{+} \right\},\end{aligned}$$ and solving equations we obtain expressions for $\hat{G}_{k\sigma}$ and $\hat{F}_{k\sigma}^{+}$: $$\cases{ \hat{F}_{k\sigma}^{+} = - \left[ E\hat{I} +
\hat{\Re}_{-k\bar{\sigma}}^{+}
\right]^{-1} \hat{\Delta}_{-k\bar{\sigma}}^{+} \hat{G}_{k\sigma} \\
\hat{G}_{k\sigma} = \left[ E\hat{I} -\hat{\Re }_{k\sigma}
+\hat{\Delta }_{k\sigma} \left( E\hat{I} +
\hat{\Re}_{-k\bar{\sigma}}^{+} \right) ^{-1}
\hat{\Delta}_{-k\bar{\sigma}}^{+} \right]^{-1} \left\langle
\hat{E}_{k}^{\sigma \sigma} \right\rangle / \sqrt{N} }
\label{eq21}$$ where $\hat{I}$ is identity matrix and N is number of vectors in k-space.
It can be seen now by analogy with the BCS theory of low-temperature superconductivity that $\Delta_{k\sigma}$ is the superconducting order parameter.
The system (\[eq21\]) is the set of matrix Green function and can be used to obtain energy spectrum and averages for any problem with defined basis of root vectors. In following chapters we will use system (\[eq21\]) to perform symmetry classification of order parameter and to investigate normal paramagnetic phase.
Symmetry Classification of Superconducting Order Parameter
==========================================================
An early suggestion that AF spin fluctuation could give rise to singlet $d_{x^{2}-y^{2}}$- wave pairing in p-type cuprate superconductors was made by Bickers, Scalapino and Scalettar [@bib20]. This suggestion has been supported by the FLEX approximation to the Hubbard model [@bib21] that is, however not valid in the case of $U \gg t$. In this limit of SCES the proper model is the t-J model. Exact diagonalization and quantum Monte-Carlo method results for small clusters have been discussed by Dagotto [@bib22]. For the infinite lattice the most adequate perturbation approach to the t-J model has been formulated in the X-operator representation because of the exact treatment of local constraint due to X-operators algebra. The mean-field solution [@bib23; @bib24; @bib25] of the t-J model and analysis of the self-energy correlations beyond the mean-field approximation by diagram technique [@bib26] and by high-order decoupling scheme [@bib19] has confirmed the $d_{x^{2}-y^{2}}$- pairing in the hole-doped system with typical $T_{c}(x)$ dependence. Latest experimental results appear to prove $d_{x^{2}-y^{2}}$- pairing not only in p-type systems but also in n-type systems (see review [@bib27]).
Let’s proceed to a symmetry classification of the order parameter $\Delta_{k\sigma}(p,h)$ in the effective singlet-triplet model considering case of square lattice. First, we have to break hopping and exchange integrals in two terms: $$\begin{aligned}
& & \gamma_{k}(p,h) = t(p,h) \omega_{k} + t'(p,h) \tilde{\omega}_{k}, \\
& & J_{k} = J \omega_{k} + J' \tilde{\omega}_{k}, \\
& & \delta J_{k} = \delta J \omega_{k} + \delta J'
\tilde{\omega}_{k},\end{aligned}$$ where $$\begin{aligned}
\omega_{k} & = & \sum_{\delta} \exp{\left(\rmi k\delta \right)} = 2\left( \cos k_{x} + \cos k_{y} \right), \\
\tilde{\omega }_{k} & = & \sum_{\delta'} \exp{\left( \rmi k
\delta' \right)} = 4 \cos k_{x} \cos k_{y},\end{aligned}$$ and non-primed values are concerned with nearest neighbor (figure \[fig3\](a), first coordination sphere) and primed values are concerned with next-nearest neighbor (figure \[fig3\](b), second coordination sphere).
![\[fig3\]Location of nearest neighbors (a) and next-nearest neighbors (b) on square lattice.](fig3.eps){width="70.00000%"}
Second, to make a classification we will distinguish following symmetry types:
s-type
------
In superconducting phase we have a constraint condition: $$\frac{1}{N} \sum_{k} \left\langle a_{k\sigma} a_{-k\bar{\sigma}}
\right \rangle = 0. \label{eq22}$$ Right side of this identity in the effective singlet-triplet model is equal to zero due to choose of basis of root vectors $\alpha_{i}$, i.e. because of absence of the transitions from lower to higher Hubbard bands. Moreover, condition (\[eq22\]) is satisfied only in case of p- and d-pairing but not for symmetric s-pairing. Hence in the singlet-triplet model the symmetric s-type singlet pairing is absent.
p-type
------
Triplet pairing of p-type is impossible because for realization of this symmetry there must be a ferromagnetic interaction (see e.g. [@bib18]) but in case of singlet-triplet model we have only antiferromagnetic exchange.
d-type
------
Singlet d-type pairing is forbidden neither by the constraint condition (\[eq22\]) nor by the type of interaction.
First consider $d_{x^{2}-y^{2}}$- pairing type. In this case there is a restriction: $$\sum_{q} \cos q_{x} B_{q\sigma}(p,h) = -\sum_{q} \cos q_{y}
B_{q\sigma}(p,h), \label{eq23}$$ and, as a consequence, $$\sum_{q} \sin q_{x} B_{q} = 0,$$ and $$\sum_{q} \sin q_{y} B_{q} = 0.$$
This immediately leads to the significant simplification of superconducting gap $\Delta_{k\sigma}(p,h)\equiv
\Delta_{k\sigma}^{(d_{x^{2} -y^{2}})}(p,h)$: $$\Delta_{k\sigma}^{(d_{x^{2}-y^{2}})}(p,h) = -\frac{2
\Delta_{\sigma}^{(d_{x^{2}-y^{2}})}(p,h)} {\sqrt{N} \left\langle
E_{k=0}^{\bar{\sigma} \bar{\sigma}}(h,h) \right\rangle} (\cos
k_{x} - \cos k_{y}). \label{eq24}$$ where $\Delta_{\sigma}^{(d_{x^{2}-y^{2}})}(p,h)$ is the impulse-independent part of the gap: $$\Delta_{\sigma}^{(d_{x^{2}-y^{2}})} \equiv \left(
\begin{array}{ccc}
J \Psi_{ \sigma}^{1,1} & J\left( \Psi_{ \sigma}^{1,2} +\Psi_{
\sigma}^{2,1} \right) /2 & \delta
J \Psi_{ \sigma}^{3,1} \\
J\left( \Psi_{ \sigma}^{1,2} +\Psi_{ \sigma}^{2,1} \right) /2 & J
\Psi_{ \sigma}^{2,2} & \delta
J \Psi_{ \sigma}^{3,2} \\
\delta J \Psi_{ \sigma}^{1,3} & \delta J \Psi_{ \sigma}^{2,3} &
\delta J \Psi_{ \sigma}^{3,3}
\end{array}
\right), \label{eq25}$$
$$\Psi_{\sigma}^{p,h} \equiv \sum_{q} (\cos q_{x} - \cos q_{y})
B_{q\sigma}(p,h). \label{eq26}$$
Impulse-independent $\Delta_{\sigma}^{(d_{x^{2}-y^{2}})}(p,h)$ includes exchange integrals only for nearest neighbors and therefore can exist only in boundaries of first coordination sphere.
Now lets consider $d_{xy}$- pairing type. The restriction made by this symmetry is as follows: $$\sum_{q} \cos q_{x} \cos q_{y} B_{q\sigma}(p,h) = 0. \label{eq27}$$ Also $$\sum_{q} \sin q_{x} \cos q_{y} B_{q} = 0,$$ and $$\sum_{q} \cos q_{x} \sin q_{y} B_{q} = 0.$$
The superconducting gap $\Delta_{k\sigma}(p,h) \equiv
\Delta_{k\sigma}^{(d_{xy})}(p,h)$ takes the form: $$\Delta_{k\sigma}^{(d_{xy})}(p,h) = -\frac{4
\Delta_{\sigma}^{(d_{xy})}(p,h)} {\sqrt{N} \left\langle
E_{k=0}^{\bar{\sigma} \bar{\sigma}}(h,h) \right\rangle} \sin
k_{x} \sin k_{y}. \label{eq28}$$ where impulse-independent part of the gap $\Delta_{\sigma}^{(d_{xy})}(p,h)$: $$\Delta_{\sigma}^{(d_{xy})} \equiv \left(
\begin{array}{ccc}
J' \Phi_{ \sigma}^{1,1} & J'\left( \Phi _{ \sigma}^{1,2} +\Phi_{
\sigma}^{2,1} \right) /2 &
\delta J' \Phi_{ \sigma}^{3,1} \\
J'\left( \Phi_{ \sigma}^{1,2} +\Phi_{ \sigma }^{2,1} \right) /2 &
J' \Phi_{ \sigma}^{2,2} &
\delta J' \Phi_{ \sigma}^{3,2} \\
\delta J' \Phi_{ \sigma}^{1,3} & \delta J' \Phi_{ \sigma}^{2,3}
& \delta J' \Phi _{ \sigma}^{3,3}
\end{array}
\right), \label{eq29}$$
$$\Phi_{\sigma}^{p,h} \equiv \sum_{q} \sin q_{x} \sin q_{y}
B_{q\sigma}(p,h). \label{eq30}$$
Coexistence of $d_{x^{2}-y^{2}}$- and $d_{xy}$- pairing types is forbidden by the group theory: in the considered case of tetragonal lattice symmetry $d_{x^{2}-y^{2}}$- and $d_{xy}$- types belongs to different irreducible representation (see review [@bib27]) so there must be concurrence between these types of pairing.
Note that triplet channel results in $\delta J$ in matrix (\[eq25\]) and $\delta J'$ in matrix (\[eq29\]). Obviously triplet channel gives the additional pairing.
It is very interesting that $d_{xy}$-type includes exchange integrals only for next-nearest neighbors (see equation (\[eq29\])) and therefore can exist only in second coordination sphere. Hence in nearest neighbor approximation we can have only order parameter of $d_{x^{2}-y^{2}}$-symmetry.
Actually equations (\[eq24\]) and (\[eq28\]) are equations for order parameters $\Psi_{\sigma}^{p,h}$ and $\Phi_{\sigma}^{p,h}$ respectively and should be solved self-consistently via Green function equations (\[eq21\]). When this is done we can obtain energy spectrum and phase diagram $T_{c}$ versus concentration $x$.
Other symmetry types
--------------------
The other symmetry types are not realized due to absence of corresponding combinations of trigonometrical functions in expression for superconducting gap (\[eq15\]).
Combination of previously analyzed types such as s+d is impossible because in the considered case of tetragonal lattice symmetry s- and d- types belongs to different irreducible representation (see review [@bib27] and references from there). So the order parameter symmetry must be of d-type or, more precisely, to be a concurrence of $d_{x^{2}-y^{2}}$ and $d_{xy}$ singlet pairing types. This conclusion is true not only for hole doped systems but also for electron doped systems because in the limit of infinite energy of singlet-triplet excitation (i.e. absence of excitations to the triplet states - this case corresponds to n-type cuprates) our equations take the form of t-J model ones and the gap symmetry remains.
Normal Paramagnetic Phase
=========================
In normal paramagnetic phase $\Delta_{k\sigma}=0$ and Green function equations (\[eq21\]) become essentially simpler: $$\cases{ \hat{G}_{k\sigma} = \left( E\hat{I} -\hat{\Re }_{k\sigma}
\right)^{-1} \left\langle \hat{E}_{k}^{\sigma
\sigma} \right\rangle / \sqrt{N} \\
\hat{F}_{k\sigma}^{+} = 0 }$$
By solving these equations one can obtain energy spectrum and self-consistently find Fermi level. Also using following definition for the spectral density in terms of one-electron annihilation operators $c_{k}^{+}$: $$A(k,\sigma,E) = -\frac{1}{\pi} \mathrm{Im} \left\langle
\left\langle c_{k \sigma}^{+} \right. \left| c_{k \sigma}
\right\rangle \right\rangle,$$ we can calculate density of states (DOS): $$N(E) = \frac{1}{N} \sum_{k \sigma} A(k,\sigma,E) = \frac{1}{N}
\sum_{k, \sigma, \lambda} \left( -\frac{1}{\pi} \mathrm{Im}
\left\langle \left\langle c_{k \sigma \lambda}^{+} \right.\left|
c_{k \sigma \lambda} \right\rangle \right\rangle \right),$$ where $\sigma$ is spin and $\lambda$ is the orbital index. In terms of Hubbard operators the expression for DOS has the following explicit form: $$\begin{aligned}
\fl N(E) = - \frac{1}{N} \sum_{k \sigma} \left[ \left(
\gamma_{x}^{2} + \gamma_{b}^{2} \right) \mathrm{Im} \left\langle
\left\langle {X_{k\sigma}^{1}}^{+} \right.\left| X_{k\sigma}^{1}
\right\rangle \right\rangle
+ \right. \nonumber \\
\lo + \left. \left( \gamma_{z}^{2} + \gamma_{a}^{2} +
\gamma_{p}^{2} \right) \left( \frac{1}{2} \mathrm{Im} \left\langle
\left\langle {X_{k\sigma}^{2}}^{+} \right. \left| X_{k\sigma}^{2}
\right\rangle \right\rangle + \mathrm{Im} \left\langle
\left\langle {X_{k\sigma}^{3}}^{+} \right.\left| X_{k\sigma}^{3}
\right\rangle \right\rangle \right) \right] \nonumber \\
\lo = \frac{1}{N} \sum_{k \sigma} \left[ \left( \gamma_{x}^{2} +
\gamma_{b}^{2} \right) \mathrm{Im}
\hat{G}_{k\sigma}^{\mathrm{a}}(1,1)
+ \right. \nonumber \\
\lo + \left. \left( \gamma_{z}^{2} + \gamma_{a}^{2} +
\gamma_{p}^{2} \right) \left( \frac{1}{2} \mathrm{Im}
\hat{G}_{k\sigma}^{\mathrm{a}}(2,2) + \mathrm{Im}
\hat{G}_{k\sigma}^{\mathrm{a}}(3,3) \right) \right].\end{aligned}$$ Here $\epsilon$ and $N_{0}(\epsilon)$ is the dispersion and DOS for non-interacting case, $\gamma_{m}$ is the coefficient of transformation $c_{k\sigma} = \sum\limits_{m} \gamma_{m}
X_{k\sigma}^{m}$, index “$\mathrm{a}$” of Green function denotes that $\hat{G}_{k\sigma}^{\mathrm{a}}$ is advance Green function (while all over the paper we have used retarded Green functions). There is simple relation between this two types: $\hat{G}_{k\sigma}(p,h) = - \hat{G}_{k\sigma}^{\mathrm{a}}(h,p)$.
The calculations described above where performed for High-$T_{c}$ superconductor $La_{2-x}Sr_{x}CuO_{4}$ with the following set of the model parameters (in eV):
---------------- ----------------------------------- ----------------------------- --------------------------- ----------------------------
$t_{pd}=1$, $\varepsilon(d_{x^{2}-y^{2}})=0$, $\varepsilon(d_{z^{2}})=2$, $\varepsilon(p_{x})=1.5$, $\varepsilon(p_{z})=0.45$,
$t_{pp}=0.46$, $t'_{pp}=0.42$, $U_{d}=9$, $U_{p}=4$, $V_{pd}=1.5$, $J_{d}=1$.
---------------- ----------------------------------- ----------------------------- --------------------------- ----------------------------
The same parameters where previously used in [@bib9.2] for undoped $La_{2-x}Sr_{x}CuO_{4}$. The energy dispersion, obtained there, proved to be in good agreement with experimental ARPES data.
Our results for slightly overdoped copper oxide (concentration of dopant $x=0.2$) are presented in figure \[fig4\].
![\[fig4\]Energy Dispersion (on the left) and Density of States (on the right) in normal paramagnetic phase of the effective singlet-triplet model for concentration of dopant $x=0.2$. Energy of the singlet sub-band and its DOS are marked by dash-dotted line. Energy of the triplet sub-bands and corresponding DOS are marked by straight and dotted lines. Dashed line indicates location of the Fermi level.](fig4.eps){width="100.00000%"}
Interesting case of coincidence of Fermi level and Van-Hove singularity is shown in figure \[fig5\].
![\[fig5\]Energy Dispersion (on the left) and Density of States (on the right) in normal paramagnetic phase of the effective singlet-triplet model for concentration of dopant $x=0.43$. All definitions are the same as in figure \[fig4\]. Distance between Van-Hove singularity and Fermi level is 0.074 eV](fig5.eps){width="100.00000%"}
It can be seen that singlet sub-band is very wide - almost 3 eV. It is a consequence of Hubbard I approximation in spectrum renormalization term $C_{k\sigma}(p,h)$ (equation (\[eq14\])). In more rigorous approximations with inclusion of spin correlators the energy bands should be more narrow (see [@bib28] and [@bib29]).
At the concentration $x=0.43$ Fermi level coincides with Van-Hove singularity in singlet sub-band. But if we will neglect hopping on second coordination sphere (nearest neighbor approximation) then we will get the flattening of dispersion in $(\pi,0)-(0,\pi)$ direction and shift of Van-Hove singularities to higher energies. This will result in coincidence of Fermi level and singularity in singlet sub-band at concentration $x=0.33$. It is typical value for the t-J model with nearest neighbor hopping t. It is also known that in t-t’-J model with the next-nearest neighbor hopping t’ the same shift in energies takes place so this phenomena is common both to t-t’-J model and the effective singlet-triplet model. Accepting the Van-Hove scenario of superconductivity, where the optimal doping correspond to coincident of Fermi level with Van-Hove singularity, we can clearly see that this should happened at $x=0.43$. Meanwhile the experimentally obtained optimal doping value for $La_{2-x}Sr_{x}CuO_{4}$ is $0.18$. So, either the Van-Hove scenario is not applicable or the model should be refined. This question could be answered only after the complete theory of superconductivity in the frame of the effective singlet-triplet model will be constructed.
Conclusion
==========
In present work we obtained the effective Hamiltonian of the singlet-triplet model for copper oxides. Being the generalization of the t-J model to account of two-particle triplet state resulting Hamiltonian has several important features. At first, the $X_{i}^{\sigma \sigma} X_{j}^{\sigma \sigma}$ term appear which can have a non-trivial contribution to superconducting pairing. Second, the account of a triplet leads to renormalization of exchange integral. And third, the singlet-triplet model is asymmetric for n- and p-type systems. For n-type systems the usual t-J model takes place while for p-type superconductors with complicated structure on the top of the valence band the singlet-triplet transitions plays an important role. The asymmetry of p- and n-type systems is known experimentally.
The analysis of the possible processes in the effective singlet-triplet model shows that besides spin-fluctuation superconducting pairing mechanism typical for the t-J model we have pairing due to singlet-triplet transitions. These transitions induce spin-exciton which can play a role of intermediate boson in superconducting pairing.
We have also performed a symmetry classification of superconducting order parameter. It was shown that in case of tetragonal lattice s-type singlet pairing doesn’t take place while the $d_{x^{2}-y^{2}}$- and $d_{xy}$-types of singlet pairing can exist. Moreover, there must be a concurrence between $d_{x^{2}-y^{2}}$- and $d_{xy}$-types. At the same time, $d_{xy}$-type can exist only within the second coordination sphere and $d_{x^{2}-y^{2}}$-type can exist only within the first. This fact lets us to take into account only $d_{x^{2}-y^{2}}$-type symmetry of order parameter in nearest-neighbors approximation.
As concerns n-type cuprates, the gap symmetry was unclear for a long time. Recently, phase sensitive tunnel experiments by Tsuei and Kirtley [@bib27] find an evidence for dominant $d_{x^{2}-y^{2}}$ symmetry in electron doped cuprates. This fact coincides with our results because in the limit of infinite energy of singlet-triplet excitation (i.e. absence of excitations to the triplet states - this case corresponds to n-type cuprates) our equations take the form of t-J model ones and the $d_{x^{2}-y^{2}}$-gap symmetry remain.
For normal paramagnetic phase we have obtained the energy dispersion over Brillouin zone and calculated density of states. Evolution of Fermi level with doping is also found. Both singlet and triplet excitations contributes to the density of states which leads to appearing of two Van-Hove singularities. At holes concentration $x=0.43$ the Fermi level crosses the first Van-Hove singularity corresponding to singlet sub-band. And at holes concentration $x=0.64$ the Fermi level crosses the second singularity corresponding to triplet sub-band.
Acknowledgements
================
The authors would like to thank N.M. Plakida and A.V. Sherman for discussions and useful comments.
This work has been supported by the RFBR grant 00-02-16110, by Krasnoyarsk Regional Science Foundation, grant 10F003C, by RFBR-“Enisey” grant 02-02-97705. INTAS support of research programme “Electronic and magnetic properties in novel superconductors: spin fluctuations vs electron-phonon coupling” at number 654 is also acknowledged.
Tensors $R_{q\sigma}^{(1)} (p,m;h,n)$ and $R_{q\sigma}^{+\,(2)}
(p,m;h,n)$ {#appendixA}
===============================================================
Tensors $R_{q\sigma}^{(1)} (p,m;h,n)$ and $R_{q\sigma}^{+\,(2)}
(p,m;h,n)$ where introduced as follows: $$\sqrt{N} \left\langle \lbrack X_{-k\bar{\sigma} }^{h},
E_{k-q}^{\sigma \sigma'}(p,m)] X_{q\sigma'}^{n} \right\rangle
\equiv \delta_{\sigma \sigma'} R_{q\sigma}^{(1)}(p,m;h,n) +
\delta_{\bar{\sigma} \sigma'} R_{q\bar{\sigma}}^{(2)}(p,m;h,n).$$ The “+” symbol in $R_{q\sigma}^{+(2)}(p,m;h,n)$ means that all averages $B_{q\sigma}(p,h)$ in it has a interchanged indices p and h.
[@l|l|lll]{} & & m=1 & m=2 & m=3\
& h=1 & $B_{q\sigma}(1,n)$ & 0 & 0\
p=1 & h=2 & 0 & $B_{q\sigma}(2,n)$ & 0\
& h=3 & $-B_{q\sigma}(3,n)$ & 0 & 0\
& h=1 & $B_{q\sigma}(2,n)$ & 0 & 0\
p=2 & h=2 & 0 & $B_{q\sigma}(2,n)$ & 0\
& h=3 & 0 & $-B_{q\sigma}(3,n)$ & 0\
& h=1 & 0 & 0 & $-B_{q\sigma}(1,n)$\
p=3 & h=2 & 0 & 0 & $-B_{q\sigma}(2,n)$\
& h=3 & 0 & 0 & 0\
[@l|l|lll]{} & & m=1 & m=2 & m=3\
& h=1 & $-B_{q\sigma}(n,1)$ & 0 & 0\
p=1 & h=2 & $B_{q\sigma}(n,2)$ & 0 & 0\
& h=3 & 0 & 0 & $B_{q\sigma}(n,1)$\
& h=1 & 0 & $-B_{q\sigma}(n,1)$ & 0\
p=2 & h=2 & 0 & $-B_{q\sigma}(n,2)$ & 0\
& h=3 & 0 & 0 & $B_{q\sigma}(n,2)$\
& h=1 & $B_{q\sigma}(n,3)$ & 0 & 0\
p=3 & h=2 & 0 & $B_{q\sigma}(n,3)$ & 0\
& h=3 & 0 & 0 & 0\
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|
---
abstract: 'It is proved that the finite Hilbert transform $T\colon X\to X$, which acts continuously on every rearrangement invariant space $X$ on $(-1,1)$ having non-trivial Boyd indices, is already optimally defined. That is, $T\colon X\to X$ cannot be further extended, still taking its values in $X$, to any larger domain space.'
address:
- 'Facultad de Matemáticas & IMUS, Universidad de Sevilla, Calle Tarfia s/n, Sevilla 41012, Spain'
- 'School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tas. 7001, Australia'
- 'Math.–Geogr. Fakultät, Katholische Universität Eichstätt–Ingolstadt, D–85072 Eichstätt, Germany'
author:
- 'Guillermo P. Curbera'
- Susumu Okada
- 'Werner J. Ricker'
title: |
Non-extendability of the finite\
Hilbert transform
---
Introduction and main result {#S1}
============================
The finite Hilbert transform $T(f)$ of $f\in L^1(-1,1)$ is the well known principal value integral $$(T(f))(t)=\lim_{\varepsilon\to0^+} \frac{1}{\pi}
\left(\int_{-1}^{t-\varepsilon}+\int_{t+\varepsilon}^1\right) \frac{f(x)}{x-t}\,dx ,$$ which exists for a.e. $t\in(-1,1)$ and is a measurable function. It has important applications to aerodynamics via the airfoil equation, [@cheng-rott], [@reissner], [@tricomi-1], [@tricomi], and to problems arising in image reconstruction; see, for example, [@katsevich-tovbis], [@sidky-etal].
For each $1<p<\infty$ the classical linear operator $f\mapsto T(f)$ maps $L^p(-1,1)$ continuously into itself; denote this operator by $T_p$. Tricomi showed that $T_p$ is a Fredholm operator and exhibited inversion formulae, [@tricomi-1], except for the case when $p=2$, [@tricomi §4.3]. For $T_2$ the situation is significantly different, as already pointed out somewhat earlier in [@sohngen p.44]. Partial operator theoretic results for $T_2$ on $L^2(-1,1)$ were obtained by Okada and Elliot, [@okada-elliot]; see also the references.
In [@curbera-okada-ricker] the finite Hilbert transform $T$ was studied when acting on suitable rearrangement invariant (r.i., in short) spaces $X$ on $(-1,1)$; see below for the relevant definitions. Actually, $T$ acts continuously on $X$ (denote this operator by $T_X$) precisely when the Boyd indices of $X$ are non-trivial, that is, when $0<\underline{\alpha}_X\le\overline{\alpha}_X<1$; see the Proposition below. This class of r.i. spaces is the largest and most adequate replacement for the $L^p$-spaces when undertaking a further study of the finite Hilbert transform $T$. This is due to the facts that $T\colon X\to X$ is injective if and only if the function $1/\sqrt{1-x^2}\notin X$ and (for the case of $X$ separable) that $T\colon X\to X$ has non-dense range if and only if $1/\sqrt{1-x^2}$ belongs to the associate space $X'$ of $X$. In terms of r.i. spaces the previous conditions can be phrased as follows: $T\colon X\to X$ *is injective if and only if $L^{2,\infty}(-1,1)\not\subseteq X$ and (for $X$ separable) $T\colon X\to X$ has a non-dense range if and only if $X \subseteq L^{2,1}(-1,1)$*, where $L^{2,1}(-1,1)$ and $L^{2,\infty}(-1,1)$ are the usual Lorentz spaces. Various types of inversion results of Tricomi for the operator $T_p$ (when $1<p<2$ and $2<p<\infty$) were extended to $T_X$ when the Boyd indices of $X$ satisfy the condition $0<\underline{\alpha}_X\le\overline{\alpha}_X<1/2$ or $1/2<\underline{\alpha}_X\le\overline{\alpha}_X<1$; see [@curbera-okada-ricker Theorems 3.2 and 3.3]. It was also shown that $T$ is necessarily a Fredholm operator in such r.i. spaces, [@curbera-okada-ricker Remark 3.4]. These results admit the possibility for a refinement of the solution of the airfoil equation; see [@curbera-okada-ricker Corollary 3.5]. Additional operator theoretic results concerning $T_X$ in r.i. spaces $X$ (e.g., compactness, order boundedness, integral representation, etc.) occur in the recent article [@curbera-okada-ricker2].
Regarding the possibility of extending the domain of $T_p$, with $T_p$ still maintaining its values in $L^p(-1,1)$, it was shown in [@okada-ricker-sanchez Example 4.21], for all $1<p<2$ and all $2<p<\infty$, that there is *no* larger Banach function space (B.f.s. in short) containing $L^p(-1,1)$ such that $T_p$ has an $L^p(-1,1)$-valued continuous extension to this space. This result was generalized in [@curbera-okada-ricker Theorem 4.7]. Namely, it is not possible to extend the finite Hilbert transform $T_X\colon X\to X$ for any r.i. space $X$ satisfying $$\label{a}
0<\underline{\alpha}_X\le\overline{\alpha}_X<1/2
\quad\textrm{or}\quad
1/2<\underline{\alpha}_X\le\overline{\alpha}_X<1.$$
The arguments used in [@curbera-okada-ricker] for establishing the above result do not apply to $T_X$ for r.i. spaces $X$ which fail to satisfy . In particular, they do not apply to $T_2\colon L^2(-1,1)\to L^2(-1,1)$. However, in [@curbera-okada-ricker] it was also established, via a completely different approach, that at least $T_2$ does not have a continuous $L^2(-1,1)$-valued extension to any larger B.f.s., [@curbera-okada-ricker Theorem 5.3].
Thus, the question of extendability of $T_X$ remains unanswered for a large sub-family of r.i. spaces which have non-trivial Boyd indices. Indeed, with the exception of $X=L^2(-1,1)$, this is the case for all those r.i. spaces $X$ satisfying $0<\underline{\alpha}_X\le1/2\le \overline{\alpha}_X<1$. In particular, this includes all the Lorentz spaces $L^{2,q}$ for $1\le q\le\infty$ with $q\not=2$. The aim of this note is to answer the above question for *all* r.i. spaces $X$ on which $T_X$ is continuous.
Let $X$ be a r.i. space on $(-1,1)$ with non-trivial Boyd indices. The finite Hilbert transform $T_X\colon X\to X$ has no continuous, $X$-valued extension to any genuinely larger B.f.s. containing $X$.
Preliminaries {#S2}
=============
In this paper the relevant measure space is $(-1,1)$ equipped with its Borel $\sigma$-algebra $\mathcal{B}$ and Lebesgue measure $m$ (restricted to $\mathcal{B}$). We denote by $L^0(-1,1)=L^0$ the space (of equivalence classes) of all $\mathbb{C}$-valued measurable functions, endowed with the topology of convergence in measure. The space $L^p(-1,1)$ is denoted simply by $L^p$, for $1\le p\le\infty$.
A *Banach function space* (B.f.s.) $X$ on $(-1,1)$ is a Banach space $X\subseteq L^0$ satisfying the ideal property, that is, $g\in X$ and $\|g\|_X\le\|f\|_X$ whenever $f\in X$, $g\in L^0$ and $|g|\le|f|$ a.e. The *associate space* $X'$ of $X$ consists of all $g\in L^0$ satisfying $\int_{-1}^1|fg|<\infty$, for every $f\in X$, equipped with the norm $\|g\|_{X'}:=\sup\{|\int_{-1}^1fg|:\|f\|_X\le1\}$. The space $X'$ is a closed subspace of the Banach space dual $X^*$ of $X$. The space $X$ satisfies the Fatou property if, whenever $\{f_n\}_{n=1}^\infty\subseteq X$ satisfies $0\le f_n\le f_{n+1}\uparrow f$ a.e. with $\sup_n\|f_n\|_X<\infty$, then $f\in X$ and $\|f_n\|_X\to\|f\|_X$. In this paper *all* B.f.s.’ $X$ are on $(-1,1)$ relative to Lebesgue measure and, as in [@bennett-sharpley], satisfy the Fatou property.
A *rearrangement invariant* (r.i.) space $X$ on $(-1,1)$ is a B.f.s. such that if $g^*\le f^*$ with $f\in X$, then $g\in X$ and $\|g\|_X\le\|f\|_X$. Here $f^*\colon[0,2]\to[0,\infty]$ is the decreasing rearrangement of $f$, that is, the right continuous inverse of its distribution function: $\lambda\mapsto m(\{t\in (-1,1):\,|f(t)|>\lambda\})$. The associate space $X'$ of a r.i. space $X$ is again a r.i. space. Every r.i. space $X$ on $(-1,1)$ satisfies $L^\infty\subseteq X\subseteq L^1$. Moreover, if $f\in X$ and $g\in X'$, then $fg\in L^1$ and $\|fg\|_{L^1}\le \|f\|_X \|g\|_{X'}$, i.e., Hölder’s inequality is available.
The family of r.i. spaces includes many classical spaces appearing in analysis, in particular the Lorentz $L^{p,q}$ spaces, [@bennett-sharpley Definition IV.4.1].
The dilation operator $E_t$ for $t>0$ is defined, for each $f\in X$, by $E_t(f)(s):=f(st)$ for $-1\le st\le1$ and zero in other cases. The operator $E_t\colon X\to X$ is bounded with $\|E_t\|_{X\to X}\le \max\{t,1\}$. The *lower* and *upper Boyd indices* of $X$ are defined, respectively, by $$\underline{\alpha}_X\,:=\,\sup_{0<t<1}\frac{\log \|E_{1/t}\|_{X\to X}}{\log t}
\;\;\mbox{and}\;\;
\overline{\alpha}_X\,:=\,\inf_{1<t<\infty}\frac{\log \|E_{1/t}\|_{X\to X}}{\log t} ,$$ [@bennett-sharpley Definition III.5.12]. They satisfy $0\le\underline{\alpha}_X\le \overline{\alpha}_X\le1$. Note that $\underline{\alpha}_{L^p}= \overline{\alpha}_{L^p}=1/p$.
For all of the above and further facts on r.i. spaces see [@bennett-sharpley], for example.
Proof of the Theorem {#S3}
====================
The proof follows the strategy devised to establish Theorems 4.7 and 5.3 in [@curbera-okada-ricker]. There the following space was constructed, namely $$\label{tx}
[T,X]:=\big\{f\in L^1: T(h)\in X, \;\forall |h|\le|f|\big\}$$ which, for a r.i. space $X$ satisfying $0<\underline{\alpha}_X\le \overline{\alpha}_X<1$, is a B.f.s. for the norm $$\label{TX-norm}
\|f\|_{[T,X]}:=\sup_{|h|\le|f|} \|T(h)\|_X,\quad f\in[T,X],$$ [@curbera-okada-ricker Proposition 4.5]. We point out that the proof of this fact uses, in an essential way, a deep result of Talagrand concerning $L^0$-valued measures. The space $[T,X]$ is the largest B.f.s. containing $X$ to which $T_X\colon X\to X$ has a continuous, linear, $X$-valued extension, [@curbera-okada-ricker Theorem 4.6]. Thus, in order to show that no genuine extension of $T_X$ is possible it suffices to show that $[T,X]\subseteq X$.
Fix $N\in{\mathbb N}$. Given $a_1,\dots,a_N\in{\mathbb C}$ and disjoint sets $A_1\dots,A_N$ in $\mathcal{B}$, define the simple function $$\phi:=\sum_{n=1}^N a_n\chi_{A_n}.$$
On $\Lambda:=\{1,-1\}^N$ consider the probability measure $d\sigma$, which is the product measure of $N$ copies of the uniform probability on $\{1,-1\}$. Define the bounded measurable function $F$ on $\Lambda$ by $$\sigma=(\sigma_1,\dots,\sigma_N)\in\Lambda \mapsto F(\sigma)
:=\bigg\| T_X\bigg(\sum_{n=1}^N\sigma_n a_n\chi_{A_n}\bigg)\bigg\|_X.$$
On the one hand, by an analogous argument as in the proof of Theorem 5.2 in [@curbera-okada-ricker] for the case of $L^2$, we have that $$\label{1}
\|F\|_{L^\infty(\Lambda)} \le
\sup_{|\theta|=1} \|T(\theta \phi)\|_X
\le\sup_{|h|\le|\phi|} \|T(h)\|_X= \|\phi\|_{[T,X]}.$$
On the other hand, an application of Fubini’s theorem yields $$\begin{aligned}
\label{2}
\|F\|_{L^\infty(\Lambda)}
& \ge \|F\|_{L^1(\Lambda)}\nonumber
\\ &= \int_{\Lambda}|F(\sigma)|\,d\sigma \nonumber
\\ &= \int_{\Lambda}\bigg\| \sum_{n=1}^N\sigma_n a_nT\left(\chi_{A_n}\right)\bigg\|_X\,d\sigma \nonumber
\\ &= \int_{\Lambda} \bigg(\sup_{\|g\|_{X'}=1} \int_{-1}^1 |g(t)|
\bigg| \sum_{n=1}^N\sigma_n a_nT\left(\chi_{A_n}\right)(t)\bigg|\,dt\bigg)d\sigma
\nonumber
\\ &\ge \sup_{\|g\|_{X'}=1} \int_{\Lambda} \bigg(\int_{-1}^1 |g(t)|
\bigg| \sum_{n=1}^N\sigma_n a_nT\left(\chi_{A_n}\right)(t)\bigg|\,dt\bigg)d\sigma
\nonumber
\\ & = \sup_{\|g\|_{X'}=1} \int_{-1}^1 |g(t)| \bigg( \int_{\Lambda}
\bigg| \sum_{n=1}^N\sigma_n a_nT\left(\chi_{A_n}\right)(t)\bigg|\,d\sigma\bigg)dt.
$$
Consider now the inner integral over $\Lambda$ in the last term of the previous expression. For $t\in(-1,1)$ fixed, set $$\beta_n:=a_nT\left(\chi_{A_n}\right)(t),\quad n=1.\dots,N.$$ It is known that the coordinate projections $$P_n:\sigma\in\Lambda\mapsto \sigma_n\in\{-1,1\}, \quad n=1,\dots,N,$$ form an orthonormal set, that is, $$\label{orto}
\int_\Lambda P_jP_k\,d\sigma=\int_\Lambda \sigma_j\sigma_k\,d\sigma
=\delta_{j,k},\quad j,k=1,\dots,N.$$ Then, for the inner integral in , we have $$\int_{\Lambda} \bigg| \sum_{n=1}^N\sigma_n a_nT\left(\chi_{A_n}\right)(t)\bigg|\,d\sigma
=
\int_{\Lambda} \bigg| \sum_{n=1}^N\beta_nP_n(\sigma)\bigg|\,d\sigma.$$ Apply the Khintchine inequality for $\{P_n\}_{n=1}^N$ yields $$\int_{\Lambda} \bigg| \sum_{n=1}^N\beta_nP_n(\sigma)\bigg|\,d\sigma
\ge \frac{1}{\sqrt2}
\bigg( \sum_{n=1}^N |\beta_n|^2\bigg)^{1/2}.$$ Accordingly, $$\label{3}
\int_{\Lambda} \bigg| \sum_{n=1}^N\sigma_n a_nT\left(\chi_{A_n}\right)(t)\bigg|\,d\sigma
\ge \frac{1}{\sqrt2}
\bigg( \sum_{n=1}^N |a_n|^2\left|T\left(\chi_{A_n}\right)(t)\right|^2\bigg)^{1/2}.$$ Then, from and , it follows that $$\begin{aligned}
\label{4}
\|F\|_{L^\infty(\Lambda)}
& \ge \frac{1}{\sqrt2} \sup_{\|g\|_{X'}=1} \int_{-1}^1 |g(t)|
\bigg( \sum_{n=1}^N |a_n|^2\left|T\left(\chi_{A_n}\right)(t)\right|^2\bigg)^{1/2}\,dt\nonumber
\\ & = \frac{1}{\sqrt2} \bigg\|
\bigg( \sum_{n=1}^N |a_n|^2\left|T\left(\chi_{A_n}\right)\right|^2\bigg)^{1/2}\bigg\|_X.
$$
We recall the following consequence of the Stein-Weiss formula for the distribution function of the Hilbert transform $H$ on ${\mathbb R}$ of a characteristic function, due to Laeng, [@laeng Theorem 1.2]. Namely, for $A\subseteq\mathbb{R}$ with $m(A)<\infty$ (where $m$ also denotes Lebesgue measure in ${\mathbb R}$), we have $$m(\{x\in A:\left| H(\chi_A)(x))\right|>\lambda\}) =
\frac{2m(A)}{e^{\pi\lambda}+1},\quad \lambda>0.$$ In particular, for any set $A\subseteq(-1,1)$ it follows, for each $\lambda>0$, that $$\begin{aligned}
m(\{x\in A:\left| T(\chi_A)(x)\right|>\lambda\})
=
m(\{x\in A:\left| H(\chi_A)(x)\right|>\lambda\})
=
\frac{2m(A)}{e^{\pi\lambda}+1}.
$$ That is, $$\label{extra}
m(\{x\in A:\left| T(\chi_A)(x)\right|>\lambda\})
=
\frac{2m(A)}{e^{\pi\lambda}+1},
\quad
A\in\mathcal{B},\;\; \lambda>0.$$ Set $\lambda=1$ and $\delta:=2/(e^{\pi}+1)<1$. For each $n=1,\dots,N$, define $$A_n^1:=\{x\in A_n:\left| T(\chi_{A_n})(x)\right|>1\}.$$ Then implies that $$\label{e}
m(A_n^1)= \frac{2m(A_n)}{e^{\pi}+1}=\delta m(A_n), \quad n=1,\dots,N.$$ Since the sets $A_1,\dots, A_N$ are pairwise disjoint, so are their subsets $A_1^1,\dots, A_N^1$. Note that $|T\left(\chi_{A_n}\right)(x)|>1$ for $x\in A_n^1$, for $n=1,\dots,N$. Thus, on $(-1,1)$ we have the pointwise estimates $$\begin{aligned}
\label{extra2}
\bigg( \sum_{n=1}^N |a_n|^2\left|T\left(\chi_{A_n}\right)\right|^2\bigg)^{1/2}
&\ge
\bigg( \sum_{n=1}^N |a_n|^2\left|T\left(\chi_{A_n}\right)\right|^2\chi_{A_n}\bigg)^{1/2}\nonumber
\\ & =
\sum_{n=1}^N |a_n|\left|T\left(\chi_{A_n}\right)\right|\chi_{A_n} \nonumber
\\ &\ge
\sum_{n=1}^N |a_n|\chi_{A_n^1}.
$$ Since $\|\cdot\|_X$ is a lattice norm, yields $$\label{7}
\bigg\| \bigg( \sum_{n=1}^N |a_n|^2\left|T\left(\chi_{A_n}\right)\right|^2\bigg)^{1/2}\bigg\|_X
\ge
\bigg\| \sum_{n=1}^N |a_n|\chi_{A_n^1}\bigg\|_X = \|\varphi\|_X,$$ where $\varphi$ is the simple function $$\varphi:=\sum_{n=1}^N a_n\chi_{A_n^1}.$$ From it follows that $$\label{extra4}
m(\{x\in(-1,1): |\varphi(x)|>\lambda\})
= \delta m(\{x\in(-1,1): |\phi(x)|>\lambda\}),
\quad \lambda>0.$$
Let $\widetilde X$ be a r.i. space on $(0,2)$ given by the Luxemburg representation for $X$, [@bennett-sharpley Theorem II.4.10], and $E_{\delta}\colon \widetilde X\to \widetilde X$ be the dilation operator for $\delta<1$ (as defined above), that is, $E_{\delta}(t)=f(\delta t)$, for $t>0$. For the decreasing rearrangements $ \phi^*$ and $\varphi^*$ of $\phi$ and $\varphi$, respectively, it follows from that $$\phi^*=E_{\delta}(\varphi^*).$$ Consequently, with $\|E_{\delta}\|$ denoting the operator norm of $E_\delta\colon \widetilde X\to\widetilde X$, we have $$\begin{aligned}
\label{8}
\|\phi\|_X=\|\phi^*\|_{\widetilde X}&
=\|E_{\delta}(\varphi^*)\|_{\widetilde X}\le \|E_{\delta}\|
\cdot \|\varphi^*\|_{\widetilde X}
=\|E_{\delta}\|\cdot \|\varphi\|_X.\end{aligned}$$ It follows, from , , and that $$\begin{aligned}
\|\phi\|_X & \le \|E_{\delta}\| \cdot \|\varphi\|_X
\\ & \le
\|E_{\delta}\| \cdot
\bigg\| \bigg( \sum_{n=1}^N |a_n|^2\left|T\left(\chi_{A_n}\right)\right|^2\bigg)^{1/2}
\bigg\|_X
\\ &
\le \sqrt2 \|E_{\delta}\| \cdot \|\phi\|_{[T,X]}.\end{aligned}$$ That is, there exists a constant $M>0$, depending exclusively on $X$, such that $$\begin{aligned}
\label{9}
M \|\phi\|_X \le\|\phi\|_{[T,X]},\end{aligned}$$ for *all* simple functions $\phi$.
In order to extend to all functions in $[T,X]$ fix $f\in[T,X]$. For every simple function $\phi$ satisfying $|\phi|\le |f|$ it follows from that $$M\|\phi\|_X \le \|\phi\|_{[T,X]} \le\|f\|_{[T,X]}.$$ Taking the supremum with respect to all such $\phi$ yields, via the Fatou property of $X$, that $f\in X$ and $$M\|f\|_X \le \|f\|_{[T,X]}.$$ In particular, $[T,X]\subseteq X$. Consequently, $[T,X]=X$ with equivalent norms. Thus, no genuine $X$-valued extension of $T_X\colon X\to X$ is possible.
For the Hilbert transform on ${\mathbb R}$ the following result is well known, [@boyd].
Let $X$ be a r.i. space on $(-1,1)$ and $T$ be the finite Hilbert transform $$(Tf)(x):=\frac1\pi\int_{-1}^1\frac{f(t)}{t-x}\,dt,\quad -1<x<1.$$ Then $T$ is bounded on $X$ if and only if $X$ has non-trivial Boyd indices.
The fact that $0<\underline{\alpha}_X\le \overline{\alpha}_X<1$ implies the continuity of $T\colon X\to X$ follows from Boyd’s theorem; see [@curbera-okada-ricker §3].
For the converse implication, we show that the proof of Theorem 3.7 in [@boyd] can be adapted to the case when $X$ is a r.i. space on $(-1,1)$.
Again let $\widetilde X$ be a r.i. space on $(0,2)$ given by the Luxemburg representation for $X$. Suppose that $T\colon X\to X$ is bounded. Without loss of generality, we may assume that $\widetilde X$ is a r.i. space on $(0,1)$ (by applying a contraction with a factor of $1/2$). Denote $T\colon X\to X$ by $T_X$.
*Step 1. [@boyd Lemma 2.2]*. Consider the positive operator $$(Su)(t):=\int_0^1\frac{u(s)}{s+t}\,ds,\quad 0<t<1.$$ We show that $S\colon\widetilde X\to \widetilde X$ boundedly (denoted by $S_{\widetilde X}$). Let $u\in \widetilde X$ be a non-negative and decreasing function in which case also $Su$ is non-negative and decreasing. Define $$f(x):=\chi_{(-1,0)}(x)u(-x),\quad x\in(-1,1).$$ Then $f^*(t)=u(t)$, for $0<t<1$. Note, for $0<x<1$, that $$(T_Xf)(x)=\frac1\pi\int_{-1}^0\frac{u(-t)}{t-x}\,dt=
-\frac1\pi\int_{0}^1\frac{u(t)}{t+x}\,dt=-\frac1\pi(Su)(x).$$ Let $$g(x):=\pi(T_Xf)(x)=-(Su)(x),\; 0<x<1,\quad\text{and}\quad g(x):=0, \; -1<x<0.$$ Then $g^*(t)=(Su)(t)$ for $0<t<1$, and $$|g(x)|\le\pi |(T_Xf)(x)|,\quad -1<x<1.$$ Thus $$\label{13}
\|Su\|_{\widetilde X}=\|g^*\|_{\widetilde X}=\|g\|_{X}\le \pi \|T_Xf\|_{X}\le
\pi \|T_X\|\|f\|_{X}=\pi \|T_X\|\|u\|_{\widetilde X}.$$ For a general $u\in \widetilde X$ observe that $|Su|\le S|u|$. Moreover, for $0<t<1$, the function $s\mapsto 1/(s+t)$ is non-negative and decreasing on $(0,1)$ and so the Hardy-Littlewood inequality [@bennett-sharpley Theorem II.2.2] implies that $S|u|\le Su^*$. Then applied to $u^*$ and the identity $\|u\|_{X}=\|u^*\|_{\widetilde X}$ imply that $\|Su\|_{\widetilde X}\le \pi \|T_X\|\|u\|_{\widetilde X}$. Hence, $S\colon \widetilde X\to\widetilde X$ boundedly.
Denote $S\colon \widetilde X'\to \widetilde X'$ by $S_{\widetilde X'}$. Fix $u\in\widetilde X$ and $g\in \widetilde X'$. Then $\langle S_{\widetilde X}u,g\rangle=\langle u,S_{\widetilde X'}g\rangle$. Similar arguments as above yield $$|\langle u,S_{\widetilde X'}g\rangle|\le \|S_{\widetilde X}u\|_{\widetilde X} \|g\|_{\widetilde X'}
\le \|S_{\widetilde X}\| \|u\|_{\widetilde X} \|g\|_{\widetilde X'}.$$ It follows that $S_{\widetilde X'}g\in \widetilde X'$, for $g\in \widetilde X'$, and $\|S_{\widetilde X'}\|\le \|S_{\widetilde X}\|\le\pi \|T_X\|$. So, $S_{\widetilde X'}$ is also a bounded operator.
*Step 2. [@boyd p.603]*. Consider, for a measurable function $f$ and $0<t<1$, the functions $$(Pf)(t)=\frac{1}{t}\int_0^tf(s)\,ds,
\quad \text{and}\quad
(P'f)(t)=\int_t^1 f(s)\,\frac{ds}{s},$$ whenever they are meaningfully defined. The claim is that the corresponding operators $P, P'\colon \widetilde X\to \widetilde X$ boundedly. Clearly they are positive operators.
Let $u\in \widetilde X$ with $u\ge0$. For $0<t<1$, it follows that $$\begin{aligned}
0\le \max&\Big\{(Pu)(t),(P'u)(t)\Big\} \le (Pu)(t)+(P'u)(t)
\\ &
=\int_0^t\frac{u(s)}{t}\,ds + \int_t^1\frac{u(s)}{s}\,ds
= \int_0^1u(s)\min\Big\{\frac{1}{t},\frac{1}{s}\Big\}\,ds
\\ &
= 2\int_0^1u(s)\min\Big\{\frac{1}{2t},\frac{1}{2s}\Big\}\,ds
\le 2\int_0^1\frac{u(s)}{t+s}\,ds
=
2(Su)(t).\end{aligned}$$ So, $Pu\le 2Su$ and $P'u\le 2Su$. Since $\widetilde X$ is a lattice, the above estimates together with the boundedness of $S\colon \widetilde X\to \widetilde X$ (see Step 1) imply that both operators $P, P'\colon \widetilde X\to \widetilde X$ boundedly.
Similar arguments, now applied to $u\in \widetilde X'$ with $u\ge0$, show that $Pu\le 2S_{\widetilde X'} u$ and $P'u\le 2 S_{\widetilde X'} u$. Since also $S_{\widetilde X'}$ is bounded, it follows that $P',P\colon \widetilde X'\to \widetilde X'$ boundedly.
*Step 3. [@boyd Theorem 3.1]*. Let $a$ be a measurable function on $(0,\infty)$. For a measurable function $f$ define $$\label{15}
(Af)(t):=\int_0^\infty a(s)f(st)\,ds,\quad t>0,$$ whenever it is meaningful to do so. Suppose that there exists $C>0$ such that $$\int_0^\infty |a(s)|\cdot\|E_s\|\,ds\le C,$$ where $E_s$ is the dilation operator $f\mapsto f(\cdot s)$ on $\widetilde X$ and $\|E_s\|$ is its operator norm. Then $A\colon \widetilde X\to \widetilde X$ (briefly $A_{\widetilde X}$) and $\|A_{\widetilde X}\|\le C$. To prove this fix $f\in \widetilde X$ and $g\in \widetilde X'$. Note, since $f$ is defined on $(0,1)$ that, for a given $t>0$, $f(st)$ is defined only when $0<st<1$; in other cases it is understood to be 0. Then $$\begin{aligned}
\langle A_{\widetilde X}(|f|),|g|)\rangle &=
\int_0^1|g(t)|\Big(\int_0^\infty|a(s)|\, |f(st)|\,ds\Big)\ dt
\\ & =
\int_0^\infty|a(s)|\Big(\int_0^1|g(t)|\, |f(st)|\,dt\Big)\,ds
= \int_0^\infty|a(s)|\langle|E_s(|f|)|,|g|\rangle\,ds
\\ &
\le \int_0^\infty|a(s)|\, \|E_s\|\|f\|_{\widetilde X}\|g\|_{\widetilde X'}\,ds
\le C\|f\|_{\widetilde X}\|g\|_{\widetilde X'}.\end{aligned}$$ Since $\widetilde X$ is a lattice and $|A_{\widetilde X}(f)|\le A_{\widetilde X}(|f|)$, it follows that $$\big|\langle A_{\widetilde X}(f),g)\rangle\big| \le C\|f\|_{\widetilde X}\|g\|_{\widetilde X'}.$$
Taking the supremum with respect to $g\in\widetilde X'$ satisfying $\|g\|_{\widetilde X'}\le1$, and using the fact that the closed subspace $X'\subseteq X^*$ is norming for $X$, [@bennett-sharpley Theorem I.2.9], it follows that $A_{\widetilde X}(f)\in \widetilde X$. Hence, $A_{\widetilde X}\colon \widetilde X\to \widetilde X$ and $\|A_{\widetilde X}(f)\|_{\widetilde X}\le C \|f\|_{\widetilde X}$ for $f\in \widetilde X$. That is, $\|A_{\widetilde X}\|\le C$.
For a measurable function $a$ on $(0,\infty)$ and $A$ as given by assume that there exists $C>0$ such that $$\int_0^\infty |a(s)|\cdot \|E'_s\|\,ds\le C,$$ where $E_s'$ denotes the dilation operator acting from $\widetilde X'$ to $\widetilde X'$. Then $A\colon \widetilde X'\to \widetilde X'$ boundedly (briefly $A_{\widetilde X'}$) and $\|A_{\widetilde X'}\|\le C$. Indeed, for $f\in \widetilde X$ and $g\in \widetilde X'$, the analogous calculations as above yield $$\langle|f|,A_{\widetilde X'}(|g|)\rangle\le C\|f\|_{\widetilde X} \|g\|_{\widetilde X'},$$ from which it follows that $|\langle f,A_{\widetilde X'}(g)\rangle|\le C\|f\|_{\widetilde X} \|g\|_{\widetilde X'}$. We can conclude that $\|A_{\widetilde X'}(g)\|_{\widetilde X'}\le C\|g\|_{\widetilde X'}$. Hence, $A_{\widetilde X'}$ is bounded and $\|A_{\widetilde X'}\|\le C$.
*Step 4. [@boyd Lemma 3.3(a)]*. Consider the operator $A$ defined in for a measurable function $a\ge0$ on $(0,\infty)$. Suppose that $A\colon \widetilde X\to \widetilde X$ (i.e., $A_{\widetilde X}$) is bounded and let $\|A_{\widetilde X}\|$ be its operator norm. Then $$\label{16}
\bigg(\int_0^sa(x)\,dx\bigg)\cdot \|E_s\| \le \|A_{\widetilde X}\|,
\quad s>0.$$
To see this let both $f\in \widetilde X$ and $g\in \widetilde X'$ be non-negative and decreasing. Then $$\begin{aligned}
\langle A_{\widetilde X}(f),g\rangle&=\int_0^1(A(f))(t)g(t)\,dt
=\int_0^1g(t)\Big(\int_0^\infty a(s)f(st)\,ds\Big)dt
\\ &
=\int_0^\infty a(s)\Big(\int_0^1g(t)(E_sf(t))\,dt\Big)\,ds
= \int_0^\infty a(s)\langle E_s(f),g\rangle\,ds.\end{aligned}$$ Since the function $s\mapsto \langle (E_sf),g\rangle$ is decreasing, for $s>0$ it follows that $$\begin{aligned}
\label{17}
\langle E_s(f),g\rangle\int_0^s a(x)\,dx
& \le
\int_0^s a(x)\langle E_x(f),g\rangle\,dx
\le \int_0^\infty a(x)\langle E_x(f),g\rangle\,dx
\\ & = \langle A_{\widetilde X}(f),g\rangle \nonumber
\le \|A_{\widetilde X}\| \|f\|_{\widetilde X} \|g\|_{\widetilde X'}.\nonumber\end{aligned}$$ Let $f\in\widetilde X$ and $g\in \widetilde X'$ be arbitrary. By the Hardy-Littlewood inequality, $$|\langle E_s(f),g\rangle|\le \langle E_s(f^*),g^*\rangle,\quad s>0.$$ Since $\|f\|_{\widetilde X}=\|f^*\|_{\widetilde X}$ and $\|g\|_{\widetilde X'}=\|g^*\|_{\widetilde X'}$, we arrive at with $f^*, g^*$ in place of $f,g$. Consequently $$\langle E_s(f),g\rangle\int_0^s a(x)\,dx
\le
\|A_{\widetilde X}\| \|f\|_{\widetilde X} \|g\|_{\widetilde X'}.$$ Taking the supremum over $g\in X'$ with $\|g\|_{\widetilde X'}\le1$ and over $f\in X$ with $\|f\|_{\widetilde X}\le1$, we arrive at .
Consider the operator $A$ in for $a\ge0$. Suppose that $A\colon \widetilde X'\to \widetilde X'$ (i.e., $A_{\widetilde X'}$) is bounded and let $\|A_{\widetilde X'}\|$ denote its operator norm. Then, with the notation $E_s'$ as in Step 3, we have $$\label{18}
\|E'_s\|\bigg(\int_0^sa(x)\,dx\bigg)\le \|A_{\widetilde X'}\|,
\quad s>0.$$ Indeed, similar arguments as above imply, for arbitrary $f\in \widetilde X$, $g\in\widetilde X'$, that $$\langle f,E'_s(g)\rangle\int_0^s a(x)\,dx
\le
\|A_{\widetilde X'}\| \|f\|_{\widetilde X} \|g\|_{\widetilde X'},$$ from which, as argued above, the claim follows.
*Step 5. [@boyd Theorem 3.4]*. Suppose that $P'\colon \widetilde X\to \widetilde X$, denoted by $P'_{\widetilde X}$, is bounded. Then $$\label{18}
\int_1^\infty \|E_s\|\frac{ds}{s}
\le 2\sqrt 2 \|P'_{\widetilde X}\|.$$ To verify we consider $Q:=(P'_{\widetilde X})^2$. Then $Q\colon \widetilde X\to\widetilde X$ (i.e., $Q_{\widetilde X}$) is bounded and $\|Q_{\widetilde X}\|\le \|P'_{\widetilde X}\|^2$. Let $f\in\widetilde X$. For $0<t<1$, we have via Fubini’s theorem that $$\begin{aligned}
(Q_{\widetilde X}f)(t)&=\int_t^1(P'_{\widetilde X}f)(s)\,\frac{ds}{s}=
\int_t^1\bigg(\int_s^1f(u)\frac{du}{u}\bigg)\frac{ds}{s}
\\ & =
\int_t^1f(u)\bigg(\int_t^u\frac{ds}{s}\bigg)\frac{du}{u}
= \int_t^1f(u)\log(u/t)\frac{du}{u}
\\ & =
\int_1^{1/t}f(vt)\log(v)\,\frac{dv}{v}
=\int_1^{\infty}f(vt)\log(v)\,\frac{dv}{v},\end{aligned}$$ because $f(vt):=0$ whenever $vt>1$. Applying Step 4, it follows for $s>1$, that $$\|E_s\|\bigg(\int_1^s\frac{\log(x)}{x}\,dx\bigg)
\le \|P'_{\widetilde X}\|^2,$$ which implies that $$\|E_s\|\le 2\|P'_{\widetilde X}\|^2/(\log(s))^2 .$$ Since, for $s>1$ we have $\|E_s\|\le1$, it follows that $$\|E_s\|\le \min\Big\{1,2\|P'_{\widetilde X}\|^2/(\log(s))^2\Big\},\quad s>1.$$ Let $a>1$ satisfy $\log a= \sqrt2 \|P'_{\widetilde X}\|$. Then $$\begin{aligned}
\int_1^\infty \|E_s\|\frac{ds}{s} &
\le
\int_1^a \frac{ds}{s} + \int_a^\infty \frac{2\|P'_{\widetilde X}\|^2}{(\log(s))^2}\frac{ds}{s}
\\ & = \log a + 2\|P'_{\widetilde X}\|^2/\log a =2\sqrt 2 \|P'_{\widetilde X}\|.\end{aligned}$$
Suppose now that $P'\colon \widetilde X'\to \widetilde X'$ (denoted by $P'_{\widetilde X'}$) is bounded. Then $$\label{20}
\int_1^\infty \|E'_s\|\frac{ds}{s} \le 2\sqrt 2 \|P'_{\widetilde X'}\|.$$ Indeed, let $Q_{\widetilde X'}=(P'_{\widetilde X'})^2$. The analogous calculation as above, for $g\in\widetilde X'$, gives $$(Q_{\widetilde X'})(t)=\int_0^\infty g(vt)a(v)\,dv.$$ Now use Step 4 (with $a(x):=(\log(x)/x)\chi_{(1,\infty)}(x)$) to deduce that $$\|E_s'\|\cdot \bigg(\int_1^s\frac{\log(x)}{x}\,dx\bigg)\le 2\|P'_{\widetilde X'}\|.$$ With this inequality we can proceed along the above lines to deduce .
*Step 6. [@boyd Theorem 3.4]*. Suppose that $P_{\widetilde X}\colon \widetilde X\to \widetilde X$ is bounded. Then $$\label{14}
\int_0^1 \|E_s\|\,ds
\le
2\sqrt 2 \|P_{\widetilde X}\|.$$ Indeed, for suitable functions $f,g$ we have $\langle P_{\widetilde X}f,g\rangle= \langle f,P'_{\widetilde X'}g\rangle$, from which it follows that $\|P_{\widetilde X}\|\ge\|P'_{\widetilde X'}\|$. Thus, we can apply Step 5 to $P'_{\widetilde X'}$ to obtain $$\int_1^\infty \|E'_s\|\frac{ds}{s}
\le
2\sqrt 2 \|P'_{\widetilde X'}\|.$$ Using the general fact, [@boyd Lemma 3.2(a)], that $$\int_1^\infty \|E'_s\|\frac{ds}{s} =
\int_0^1 \|E_s\|\,ds,$$ the inequality follows.
*Step 7.* Condition is condition (i) in Lemma 3.6(b) of [@boyd], which is equivalent to condition (iv) in the same Lemma; this is precisely $\underline{\alpha}_X>0$.
Condition is condition (i) in Lemma 3.6(a) of [@boyd], which is equivalent to condition (iv) in the same Lemma; this is precisely $\overline{\alpha}_X<1$.
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|
---
abstract: 'We develop, analyse and numerically simulate a model of a prototype, glucose-driven, rhythmic drug delivery device, aimed at hormone therapies, and based on chemomechanical interaction in a polyelectrolyte gel membrane. The pH-driven interactions trigger volume phase transitions between the swollen and collapsed states of the gel. For a robust set of material parameters, we find a class of solutions of the governing system that oscillate between such states, and cause the membrane to rhythmically swell, allowing for transport of the drug, fuel and reaction products across it, and collapse, hampering all transport across it. The frequency of the oscillations can be adjusted so that it matches the natural frequency of the hormone to be released. The work is linked to extensive laboratory experimental studies of the device built by Siegel’s team. The thinness of the membrane and its clamped boundary, together with the homogeneously held conditions in the experimental apparatus, justify neglecting spatial dependence on the fields of the problem. Upon identifying the forces and energy relevant to the system, and taking into account its dissipative properties, we apply Rayleigh’s variational principle to obtain the governing equations. The material assumptions guarantee the monotonicity of the system and lead to the existence of a three dimensional limit cycle. By scaling and asymptotic analysis, this limit cycle is found to be related to a two-dimensional one that encodes the volume phase transitions of the model. The identification of the relevant parameter set of the model is aided by a Hopf bifurcation study of steady state solutions.'
author:
- |
Lingxing Yao\
Department of Mathematics, Applied Mathematics and Statistics\
Case Western Reserve University\
Cleveland, OH 44106\
[ ]{}\
M. Carme Calderer
- |
Yoichiro Mori\
School of Mathematics\
\
Ronald A. Siegel\
Departments of Pharmaceutics and Biomedical Engineering\
University of Minnesota, Minneapolis, MN 55455
bibliography:
- 'gel.bib'
title: 'Rhythmomimetic drug delivery: modeling, analysis and numerical simulation'
---
34C12, 37G15, 74B20, 82B26, 94C45
[^1]
[polyelectrolyte gel, volume transition, chemical reaction, hysteresis, weak solution, inertial manifold, limit cycle, competitive dynamical system, multiscale.]{}
Introduction {#intro}
============
Research on drug delivery systems is focused on presenting drug at the right place in the body at the right time [@DrugDelivery]. To this end, we have introduced a prototype chemomechanical oscillator that releases GnRH in rhythmic pulses, fueled by exposure to a constant level of glucose [@dhanarajan2002autonomous; @dhanarajan2006; @Misra]. Experience with chemical and biochemical oscillators [@GrayScott], [@EpsPoj] and [@Goldbeter], and with electrical and mechanical relaxation oscillators [@Adronov], shows that rhythmic behavior can be driven by a constant rate stimulus, provided proper delay, memory and feedback elements are employed in device dynamics. The device consists of two fluid compartments, an external cell (I) mimicking the physiological environment, and a closed chamber (II), separated from (I) by a hydrogel membrane. Cell I, which is held at constant pH and ionic strength, provides a constant supply of glucose to cell II, and also serves as clearance station for reaction products. Cell II contains the drug to be delivered to the body, an enzyme that catalyzes conversion of glucose into hydrogen ions, and a piece of marble to remove excess hydrogen ions that would otherwise overwhelm the system. When the membrane is swollen, glucose flux into Cell II is high, leading to rapid production of hydrogen ions. However, these ions are not immediately released to Cell I but react, instead, with the negatively charged carboxyl groups of the membrane, which collapses when a critical pH is reached in Cell II, substantially attenuating glucose transport and production of ions. Subsequent diffusion of membrane attached H-ions increases again the concentration of negative carboxyl groups in the membrane, causing the gel to reswell, and so, the process is poised to repeat itself. Since drug release can only occur when the membrane is swollen, it occurs in rhythmic pulses that are coherent with the pH oscillations in Cell II and swelling oscillations of the membrane.
While rhythmic hormone release across the membrane is the ultimate goal of this research, a main purpose of this article is the study of the pH oscillations in Cell II. These oscillations correlate with those of ${{\textrm H}^+}$ concentration in the membrane, which determine its swelling state.
A polyelectrolyte gel is a mixture of polymer and fluid, the latter containing several species of ions and the polymer including electrically charged (negative) side groups. We model the gel as an incompressible, saturated mixture of polymer and fluid. The hydrogel (polyelctrolyte gel having water as the fluid solvent) of the experimental device is a relatively thin membrane that is laterally restrained by clamping justifying its treatment as a one-dimensional system. Moreover, the experimental apparatus is kept well-stirred at all times, allowing for further reduction to a time-dependent, spatially homogeneous system. The system consists of three coupled mechanical-chemical ordinary differential equations for the time evolution of the membrane thickness $L=L(t)$, the hydrogen concentrations $x (C_H^M)$ and $z (C_H^{II})$ inside the membrane and in the chamber, respectively. The polymer volume fraction $\phi(t)$ is related to $L(t)$ by the equation of balance of mass of the polymer, $L=\frac{\phi_0}{\phi}$, where $\phi_0$ denotes the corresponding volume fraction in the reference state of the membrane. The governing system also takes into account the algebraic constraint of electroneutrality. Analysis and simulation of gel swelling in higher dimensional geometries, in the purely mechanical case, have been carried out in [@chabaud-calderer14] and [@Suo].
The swelling force in the hydrogel consists of mixing, elastic and ionic terms [@English; @Flory]. The first two terms are derived from the Flory-Huggins energy of mixing and the neo-Hookean form of the elastic stored energy, respectively. The combination of Lagrangian elasticity and Eulerian mixing energies gives rise to the well known Flory-Rehner theory. The ionic force follows Van’t Hoff’s ideal law. The latter force depends on the degree of ionization due to acid-base equilibria between pendant ionizable groups in the hydrogel and free hydrogen ion, as described by a Langmuir isotherm, and the resulting Donnan partitioning of counterions and co-ions into the hydrogel. Taking all these factors into account, we call the model for swelling stress the Flory-Rehner-Donnan-Langmuir (FRDL) model.
A scaling analysis reveals that, of the many physical parameters of the model, five dimensionless parameter, $\mathcal A_i$, encode most mechanical and chemical properties of the system. These together with control parameters such as salt concentration ${C_{\textrm{\tiny{NaCl}}}}$, reaction rate constant of the marble ${k_{\textrm{\begin{tiny}mar\end{tiny}}}}$, degree of ionization of the polymeric side groups $\sigma_0$, reference polymer volume fraction $\phi_0$ and degree of polymer cross-linking $\rho_0$ are sufficient to fully describe the evolution properties of the system.
A study of the stationary states of the system shows the hysteretic behavior for appropriate parameter ranges that is consistent with laboratory experiments [@BakerSiegel1996; @FirestoneSiegel1988; @BhallaSiegel2014]. Moreover, we find that, for every set of parameters within the relevant range, there exists a unique steady state that is hyperbolic. We study the Hopf bifurcation of the steady state and numerically identify the parameter regions within which oscillations occur. We found these regions to be in good agreement with experiment.
Dimensionless parameter groups determine the relative time scales of the system, with the fast time corresponding to the swelling ratio of the membrane. In particular, this implies that solutions of the governing system remain arbitrarily close to those of a suitable two dimensional system, for most of the time. Although the original system has positive $\textrm C^1$-solutions with $\phi$ bounded away from 0 (and $\phi$ bounded away from $1$, as well), the two-dimensional restriction involves multivalued functions, leading to existence of weak solutions, with discontinuous $\phi$ and, consequently, the swelling ratio being discontinuous as well. The latter corresponds to a volume phase transition taking place that drives the permeability changes of the membrane. The Poincar[é]{}-Bendixon theory for plane systems applies to the current problem from which existence of a two-dimensional limit cycle follows. It turns out that the limit cycle is also the omega-limit set of positive semi-orbits of the full system.
In order to show existence of a limit-cycle of the three dimensional system, we must appeal to the theory of monotone dynamical systems. Specifically, we show that our three dimensional system is competitive with respect to a properly defined alternate cone, that results from the intersection of half-spaces. We also find that the Jacobian matrix of the system satisfies the properties of [*sign-stability*]{} and [*sign-symmetry*]{}, from which existence of a three-dimensional limit cycle follows. It turns out that our system is qualitatively analogous to that modeling the dynamics of the HIV in the regime where the concentrations of infected and uninfected T-cells and the virus follow a periodic holding pattern, away from the fully infected state represented by a hyperbolic steady state [@virus].
The outline of this paper is as follows. In section \[hormone-delivery\], we describe the prototype oscillator and outline its basis for operation. In section \[Parameter-Model\], we study the mechanical and chemical properties of the system. The assumptions of the model and the subsequent formulation of a precursory system of ordinary differential equations are presented in section \[chemo-mechanical model\]. In section \[scaling\], we describe the parameters of the system and explore its scaling properties, from which the system of ordinary differential equations that model the dynamics of the oscillator follows. The Hopf bifurcation analysis of the unique hyperbolic equilibrium point is presented in section \[numerical-simulations\]. In section \[inertial-manifold\], we construct an inertial manifold of the three-dimensional system and analyze the corresponding two-dimensional flow in the manifold, demonstrating existence of an asymptotically stable limit cycle. In section \[3D-limit-cycle\], we study the monotonicity properties of the system and prove the existence of a three-dimensional limit cycle. A multiscale, asymptotic analysis is developed in section \[multiscale\] that yields decay estimates of solutions as they approach the two-dimensional manifold, where hysteresis properties manifest themselves, including the can[à]{}rd feature of the system. We conclude with a discussion of the benefits and deficiencies of the present model.
Rhythmic Hormone Delivery: A Simple Experimental System {#hormone-delivery}
========================================================
A simplified schematic representation of the experimental oscillator is depicted in Figure 1 [@dhanarajan2002autonomous; @Misra]. The apparatus consists of two well stirred fluid compartments, or cells, separated by a sweallable hydrogel membrane. Cell I, which is meant to mimic the external physiological fluid environment, contains glucose in saline solution, with fixed glucose concentration maintained by introduction of fresh medium and removal of reaction products by flow, and fixed pH 7.0 enforced by a pH-stat servo (autotitrating burette). Cell II, which simulates the interior of the rhythmic delivery device, contains the hormone to be released (e.g. GnRH) and the enzymes glucose oxidase, catalase, and gluconolactonase, which catalyze conversion of glucose to gluconic acid. The latter rapidly dissociates into hydrogen ion (${\textrm H}^+$) and gluconate ion: $$\textrm{Glucose} +{\textrm O}_2 \stackrel{\textrm{\begin{tiny}\emph{enzymes}\end{tiny}}}{\longrightarrow} {\textrm H}^+ +{\textrm {Gluconate}}^- + \frac{1}{2}{\textrm O}_2 \textrm{ (I)}$$
\[Figure1\]
[{width="90mm"}]{}
Cell II also contains physiologic saline, which is exchanged with Cell I through the membrane, and a piece of solid marble. Marble is solid calcium carbonate, ${\textrm{ CaCO}}_3(\textrm{s}), $ which reacts with $\textrm{H}^+$ according to $$2{\textrm H}^+ + {\textrm{ CaCO}}_3(\textrm{s}) \longrightarrow
{\textrm{ Ca}}^{2+} + {\textrm{ CO}}_2\uparrow \,\, +{\textrm{
H}}_2\textrm{O} \textrm{ (II)}$$ The hydrogel membrane is clamped between Cells I and II. The degree of swelling of the membrane and permeabilities to glucose and GnRH, depend on the internal concentration of ${\textrm H}^+$ ions, through the reaction [@English; @grimshaw1990kinetics; @ricka-tanaka]. $$\vdash{\textrm{COO}}^- + {\textrm{H}}^+ \rightleftharpoons {\textrm{ }} \vdash{\textrm{COOH}} \textrm{ (III)}$$ At low ${\textrm{H}}^+$ concentration, the membrane is charged, swollen, and highly permeable to both glucose and GnRH, but permeability to these compounds is substantially attenuated at higher ${{\textrm H}^+}$ concentrations where the membrane has less charge and is relatively collapsed. The placement of the membrane between a source of fuel, glucose, and its converting enzymes creates a dynamic environment with competing effects. On the one hand, the enzyme reaction produces $ {{\textrm H}^+}, $ which affects the permeability of the membrane; it is removed from the system by reacting with marble, and also by diffusing out to the environment. This creates a negative feedback mechanism between the enzyme reaction and the membrane permeability to glucose. Under proper conditions, this arrangement can lead to oscillations.
When the membrane is ionized and swollen, glucose permeates from Cell I to Cell II and is converted to ${{\textrm H}^+},$ which diffuses back into the membrane, binds and neutralizes the $\vdash\textrm{COO}^-$ groups, and causes the hydrogel membrane to collapse. The membrane is now impermeable to glucose, and enzymatic production of ${{\textrm H}^+}$ is attenuated. Eventually the ${{\textrm H}^+}$ ions bound to the membrane diffuse into Cell I, where they are neutralized by the pH-stat and removed in the waste stream. The membrane then reionizes and reswells, and the system is primed to repeat the previous sequence of events.
In order to achieve sustained oscillations, a steady state in which flux of glucose, enzyme reaction rate, and flux of ${{\textrm H}^+}$ are balanced and equal at all times, must be avoided. As it will be seen, bistability, or hysteresis of membrane swelling response to ${{\textrm H}^+},$ provides a means for destabilizing such a steady state.
The reader is referred to experimental details and results in previous publications [@dhanarajan2002autonomous; @dhanarajan2006; @Misra].
A Lumped Parameter Model {#Parameter-Model}
=========================
A full mathematical description of the experimental system just described would require a detailed account of 1) diffusional and convective fluxes of solvent and solutes, 2) the spatial three-dimensional mechanics of the hydrogel membrane which, though constrained by clamps, exhibits swelling and shrinking which is both time and position dependent, 3) the kinetics of enzymatic conversion of glucose to ${{\textrm H}^+}$, and 4) the kinetics of binding and dissociation of ${{\textrm H}^+}$ with $\vdash\textrm{COO}^-$ groups. An accurate, verifiable model of this sort, which would require partial differential equations to describe intramembrane processes, does not yet exist. Here we simplify the problem by assuming that the membrane is a lumped, uniform element. All mechanical and chemical variables are homogenized to single values representing the whole membrane. We recognize that some potentially important consistencies are lost in this approximation. First, there will always be a difference in pH between Cells I and II, which will lead to intramembrane gradients in the chemical and mechanical variables. Second, self consistent boundary conditions, which would follow naturally from a PDE model, must be replaced by somewhat *ad hoc* assumptions.
As a second simplification, we assume that the enzyme reactions, and the process of distribution of the dominant background electrolyte, NaCl, between Cells I and II and the hydrogel membrane, are very fast compared to the other dynamical processes. These assumptions can be justified, respectively, by the excess of enzyme used in the experiments, and the fact that the capacity of the membrane for acidic protons (${{\textrm H}^+},$ or $\vdash$COOH) relative to Cells I and II, far exceeds its relative capacities for $\textrm{Na}^+$ and $\textrm{Cl}^-$. Many experimental studies with polyacidic hydrogels have confirmed that ${{\textrm H}^+}$ dynamics and poroelastic relaxations are much slower than those of NaCl [@FirestoneSiegel1988; @BakerSiegel1996; @BhallaSiegel2014].
Swelling of Hydrogels
---------------------
The membranes considered in this work are crosslinked networks of polymer chains, or hydrogels, which absorb substantial amounts of water. Depending on water content, or degree of swelling, the hydrogel will be more or less permeable to solutes such as glucose and ${{\textrm H}^+}$. Hydrogels have a long history of application in drug delivery and medicine due to their mechanical and chemical compatibility with biological tissues and their ability to store and release drugs in response to environmental cues [@Peppas].
In the present system, we utilize the hydrogen ions ${{\textrm H}^+}$ that are enzymatically generated from glucose to control hydrogel swelling and hence release of hormone. In polyacid hydrogels, swelling is controlled by degree of ionization, which results from dissociation of acidic side groups that are attached to the polymer chains. When NaCl is present in the aqueous fraction of the hydrogel, the ionization equilibrium is represented by $$\vdash\textrm{COOH} + {\textrm{ Na}}^+ + {\textrm{Cl}}^- \rightleftharpoons\,
\vdash{\textrm{COO}}^-{\textrm{ Na}}^+ + {\textrm{H}}^+ + {\textrm{Cl}}^-$$ Swelling of a polyacidic hydrogel results from three thermodynamic driving forces [@English; @Flory; @Katchalsky; @RickaTanaka]. First there is the tendency of solvent (water) to enter the hydrogel and mix with polymer in order to increase translational entropy. The mixing force also depends on the relative molecular affinity or aversion of the polymer for water compared to itself due to short range van der Waals, hydrogen bonding, and hydrophobic interactions. Second, there is an elastic force, which is a response to the change in conformational entropy of the polymer chains that occurs during swelling or shrinking. The third force is due to ionization of the acidic pendant groups, which leads to an excess of mobile counterions and salt inside the hydrogel compared to the external medium, promoting osmotic water flow into the hydrogel. The ion osmotic force acts over a much longer range than the direct polymer/water interaction.
In the present work, it is assumed that the hydrogel is uniform in composition when it is prepared. The hydrogel at preparation is taken as the reference state for subsequent thermodynamic model calculations. At preparation, the volume fraction of polymer in the hydrogel is denoted by $\phi_0$. Crosslinking leads to an initial density of elastically active chains, $\rho_0$. The initial density of ionizable acid groups, fixed to the polymer chains, is denoted by $\sigma_0$. Both densities have units mol/L, and are referred to the total volume of the hydrogel (polymer+water) at preparation.
Mechanics of Swelling
---------------------
The swelling state of a hydrogel is characterized by the principal stretches or elongation ratios $ (\alpha_x,\alpha_y,\alpha_z)$, and the volume fraction $\phi$ of the polymer. We let $0<\phi_0<1$ and $L_0>0$ denote the volume fraction of polymer and the thickness of the membrane, respectively, in the reference state. The equation of conservation of mass of polymer in the gel is $$\phi=\phi_0(\alpha_x\alpha_y\alpha_z)^{-1}. \label{mass-balance-equation}$$ The swelling ratio of the membrane relative to its undeformed reference state is $\phi_0/\phi=\alpha_x\alpha_y\alpha_z$, that is, the Jacobian of the deformation map from the reference to the deformed membrane. We assume that the elongation ratios and volume fraction vary with time. Since in the present system the hydrogel is a relatively thin membrane that is laterally restrained by clamping, we assume that the main swelling effect occurs along the thickness direction. Hence, we consider uniaxial deformations $$\alpha:=\alpha_x, \alpha_y=\alpha_z=1. \label{uniaxial}$$
A rigorous justification of the uniaxiality assumption on the membrane deformation may follow from a dimensional reduction analysis. However, a main drawback of the one-dimensional treatment is that it neglects undulating surface instabilities observed in gel free surfaces, which may ultimately hamper sustained oscillatory behaviour.
Denoting by $L=L(t)$ the thickness of the membrane at time $t\geq 0$, equation (\[mass-balance-equation\]) reduces to $$L\phi=L_0\phi_0. \label{mass-balance-1d}$$ In the model presented below, the three swelling forces, mixing, elastic and ionic, yield the uniaxial [*swelling stress*]{}, $$s=s_{\textrm{\begin{tiny} {mix} \end{tiny}}} +s_{\textrm{\begin{tiny} {\textrm elast}\end{tiny}}}+ s_{\textrm{\begin{tiny} {\textrm ion} \end{tiny}}}. \label{ras1}$$ which reflects the excess free energy density of the hydrogel relative to equilibrium, at a given state of swelling in a prescribed aqueous medium. At equilibrium, $s=0$. Let ${g_{\textrm{\begin{tiny} {mix}
\end{tiny}}}}$ and $ {{g_{\textrm{\begin{tiny}elast\end{tiny}}}}}$ denote the Flory-Huggins mixing and elastic energy densities with respect to deformed volume, respectively. The corresponding total energy quantities, per unit cross-sectional area, are $${{G_{\textrm{\begin{tiny}mix\end{tiny}}}}}= L{g_{\textrm{\begin{tiny} {mix}
\end{tiny}}}}(\phi), \quad
{{G_{\textrm{\begin{tiny}elast\end{tiny}}}}}=L{{g_{\textrm{\begin{tiny}elast\end{tiny}}}}}(\phi), \label{ElasticAndMixingEnergies}$$ A standard variational argument gives the dimensionless stress components as $${s_{\textrm{\begin{tiny} {mix} \end{tiny}}}}=-\frac{d{{G_{\textrm{\begin{tiny}mix\end{tiny}}}}}}{dL}, \quad {s_{\textrm{\begin{tiny} {elast} \end{tiny}}}}=-\frac{d{{G_{\textrm{\begin{tiny}elast\end{tiny}}}}}}{dL}.$$ Consequently, $$s_i=-(g_i-\phi\frac{\partial g_i}{\partial \phi}), \,\, i=\textrm{\scriptsize{mix}}, \textrm{\scriptsize{elast}}. \label{ras2}$$ The mixing free energy density of solvent (water) with the hydrogel chains is modeled according to the Flory-Huggins expression $${g_{\textrm{\begin{tiny} {mix}
\end{tiny}}}}= \frac{RT}{{v_{\textrm{\begin{tiny}w\end{tiny}}}}}[(1- \phi)\ln(1- \phi) + \chi\phi(1-
\phi)], \label{ras3}$$ where $R $ is the gas constant, $T$ is absolute (Kelvin) temperature, and ${v_{\textrm{\begin{tiny}w\end{tiny}}}}$ is the molar volume of water. The first term in the square brackets accounts for the translational entropy change of solvent molecules as they move from the external environment into the hydrogel, temporarily assuming that the bath is pure solvent. The second term accounts for short range contact interactions between polymer and solvent, summarized by the dimensionless parameter $\chi$, which is the molar free energy required to form solvent/polymer contacts, normalized by $RT$. Introducing (\[ras3\]) into (\[ras2\]) yields $${s_{\textrm{\begin{tiny} {mix} \end{tiny}}}}=-\frac{RT}{{v_{\textrm{\begin{tiny}w\end{tiny}}}}}[\ln(1-\phi)+\phi+\chi\phi^2].\label{ras4}$$ In the simplest form of Flory-Huggins theory $\chi$ is constant, but in many hydrogel systems, especially those that undergo sharp swelling/shrinking transitions, a pseudo-virial series in $\phi$ is used. In the present work the series is truncated at the linear term, taking the form [@ErmanFlory], [@Hirotsu], $$\chi=\chi_1+ \chi_2\phi. \label{ras5}$$ With $\chi_2>0$, polymer and solvent become increasingly incompatible as polymer concentration increases, with hydrogel shrinking, and mixing pressure decreasing.
We model the elastic energy density according to the Neo-Hookean expression $${{g_{\textrm{\begin{tiny}elast\end{tiny}}}}}=\frac{RT}{2}\rho_0(\frac{\phi}{\phi_0}) (\alpha_x^2+\alpha_y^2+\alpha_z^2-3-\ln\alpha_x\alpha_y\alpha_z). \label{ras6}$$ The logarithmic term accounts for change in entropy of localization of crosslinks in the hydrogel due to swelling. For the uniaxial swelling under consideration, $${{g^{\textrm{\begin{tiny}uni\end{tiny}}}_{\textrm{\begin
{tiny}elast\end{tiny}}}}}= RT\rho_0(\frac{\phi}{\phi_0})[(\frac{\phi_0}{\phi})^2-\ln\frac{\phi_0}{\phi}-1] \label{ras7}$$ which upon introduction into (\[ras2\]) yields $${s_{\textrm{\begin{tiny} {elast} \end{tiny}}}}=-RT\rho_0(\frac{\phi_0}{\phi}-\frac{\phi}{2\phi_0}).\label{ras8}$$ Summing (\[ras4\]) and (\[ras8\]) we obtain the Flory-Rehner expression for the nonionic component of the swelling stress (see also (\[equation-mechanics\]), below. We now derive the evolution equation of the membrane according to the minimum rate of dissipation principle.
Suppose that $\phi=\phi(t)$ and $L=L(t), \, t\geq 0$ satisfy equation (\[mass-balance-1d\]). Let us define the rate of dissipation function and the total energy as $$W=\frac{K}{2} (\frac{dL}{dt})^2\, {\textrm{\small{and}}} \, \, \, E={{G_{\textrm{\begin{tiny}mix\end{tiny}}}}}+{{G_{\textrm{\begin{tiny}elast\end{tiny}}}}},$$ respectively, with $K>0$ denoting a friction coefficient. Then the function $v=\frac{dL}{dt}$ that minimizes the Rayleighian functional $R= \dot E +W,$ among spatially homogeneous fields, satisfies the equation $${s_{\textrm{\begin{tiny} {mix} \end{tiny}}}}+ {s_{\textrm{\begin{tiny} {elast} \end{tiny}}}}-K\frac{dL}{dt}=0, \label{motion-equation}$$ with ${s_{\textrm{\begin{tiny} {mix} \end{tiny}}}}$ and ${s_{\textrm{\begin{tiny} {elast} \end{tiny}}}}$ as in (\[ras4\]) and (\[ras8\]), respectively, and ${{G_{\textrm{\begin{tiny}elast\end{tiny}}}}}$ and ${{G_{\textrm{\begin{tiny}mix\end{tiny}}}}}$ as in (\[ElasticAndMixingEnergies\]).
The proof results from the simple calculation $$\frac{dE}{dt}=\frac{d}{dt}\big({g_{\textrm{\begin{tiny} {mix}
\end{tiny}}}}+{{g_{\textrm{\begin{tiny}elast\end{tiny}}}}}\big), \quad
= -({s_{\textrm{\begin{tiny} {mix} \end{tiny}}}}+{s_{\textrm{\begin{tiny} {elast} \end{tiny}}}})v, \quad v=\frac{dL}{dt}. \label{crit-pointR}$$ So, the critical points $v=v(t)$ of $R$ satisfy equation (\[motion-equation\]), for all $t\geq 0$. Defining the [*permeability coefficient*]{} as $
{{K}_{\textrm{\begin{tiny}w\end{tiny}}}}= \frac{RT}{K\nu_w},$ and using equations (\[ras4\]) and (\[ras8\]), equation (\[motion-equation\]) becomes $$\frac{dL}{dt}= -K_w[ \ln(1-\phi)+\phi+\chi\phi^2 + \nu_w\rho_0(\frac{\phi_0}{\phi}-\frac{\phi}{2\phi_0})].\label{equation-mechanics}$$ Equation (\[motion-equation\]) should be expanded to include the force ${s_{\textrm{\begin{tiny} {ion} \end{tiny}}}}$ as in (\[ras1\]). The calculation of such a contribution is given next.
Chemical reactions
------------------
While the mixing and elastic terms represent important contributions to swelling pressure, the ionic term responds to ${{\textrm H}^+}$ concentration inside the membrane, and this is what forms the basis for the oscillator’s dynamic behavior. As described above, ionization of the hydrogel occurs by dissociation of pendant carboxylic acid groups. The fraction of these groups that are ionized, $f$, is modeled according to a Langmuir isotherm relation, $$f=\frac{{K_{\textrm{\begin{tiny}A\end{tiny}}}}}{{K_{\textrm{\begin{tiny}A\end{tiny}}}}+{C_{\textrm{\begin{tiny}H\end{tiny}}}}}.
\label{ras9}$$ where $ {C_{\textrm{\begin{tiny}H\end{tiny}}}}$ is the concentration of $ {{\textrm H}^+}$ in the aqueous portion of the hydrogel and $K_A$ is the dissociation constant of the carboxylic acid. The concentration of ionized groups, referenced to the aqueous portion of the hydrogel, is then given by $${C_{\textrm{\begin{tiny}A-\end{tiny}}}}=f\sigma_0(\phi/\phi_0), \label{ras10}$$ where $\sigma_0$ denotes the reference density of ionized groups fixed to the polymer chains. Letting ${C_{\textrm{\begin{tiny}AH\end{tiny}}}}$ denote the concentration of $H^+$ linked to carboxyl groups, the conservation of intra-membrane hydrogen ions is given by the equation $$L({C_{\textrm{\begin{tiny}H\end{tiny}}}}+{C_{\textrm{\begin{tiny}AH\end{tiny}}}})= L_0\sigma_0. \label{polymer-charge-concentration}$$ Combining equations (\[ras10\]) and (\[polymer-charge-concentration\]) yields $${C_{\textrm{\begin{tiny}AH\end{tiny}}}}= \frac{\phi}{\phi_0}(1-f)\sigma_0. \label{d4}$$ Ionization of hydrogel side-chains, plus the requirement for quasi-electroneutrality over any distance exceeding a few Debye lengths ($\sim 5$ nm in physiological systems) leads to a distribution of mobile ions in the hydrogel that differs from that in the external bath. Ignoring very small contributions from $ {{\textrm H}^+}$ and $ \textrm{OH}^{-} $ ions, a quasineutrality condition is $${C_{\textrm{\begin{tiny}Na\end{tiny}}}}-{C_{\textrm{\begin{tiny}Cl\end{tiny}}}}-{C_{\textrm{\begin{tiny}A-\end{tiny}}}}=0. \label{ras11}$$ where ${C_{\textrm{\begin{tiny}Na\end{tiny}}}}$ and $ {C_{\textrm{\begin{tiny}Cl\end{tiny}}}}$ are the concentrations, respectively, of $\textrm{Na}^{+}$ and $\textrm{Cl}^{-}$ in the aqueous portion of the hydrogel.
For simplicity, we assume that Cells I and II contain fully dissociated NaCl, with ion concentrations ${C_{\textrm{\begin{tiny}Na\end{tiny}}}}'={C_{\textrm{\begin{tiny}Cl\end{tiny}}}}'={C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'$. Ionic swelling stress in the membrane is modeled using van’t Hoff’s ideal law:
$${s_{\textrm{\begin{tiny} {ion} \end{tiny}}}}=RT({C_{\textrm{\begin{tiny}Na\end{tiny}}}}+{C_{\textrm{\begin{tiny}Cl\end{tiny}}}}-2{C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'). \label{ras12}$$
Derivation of this term from a free energy density expression is possible but not carried out here since the basis for van’t Hoff’s law is well understood.
It can be shown that diffusion of salt inside the hydrogel is very fast compared to diffusion of $ \textrm{H}^+$ and mechanical relaxation of swelling pressure. We may therefore assume that at any point in the hydrogel a Donnan quasi-equilibrium, where $ {C_{\textrm{\begin{tiny}Na\end{tiny}}}}=\lambda{C_{\textrm{\begin{tiny}Na\end{tiny}}}}' $ and $ {C_{\textrm{\begin{tiny}Cl\end{tiny}}}}=\lambda^{-1}{C_{\textrm{\begin{tiny}Cl\end{tiny}}}}'$, with $\lambda$ the Donnan ratio, determined by combining (\[ras9\])-(\[ras11\]), giving $$(1-\phi)(\lambda-\frac{1}{\lambda}){C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'
-\sigma_0(\phi/\phi_0)f=0.
\label{ras13}$$ With $\lambda$ in hand, (\[ras12\]) becomes $${s_{\textrm{\begin{tiny} {ion} \end{tiny}}}}=RT(\lambda+\frac{1}{\lambda}-2){C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}' .
\label{ras14}$$ Introducing (\[ras4\], \[ras5\], \[ras8\], \[ras14\]) into (\[ras1\]) gives $$\frac{{v_{\textrm{\begin{tiny}w\end{tiny}}}}s}{RT}=-[\ln(1-\phi)+\phi+(\chi_1+\chi_2\phi)\phi^2]-{v_{\textrm{\begin{tiny}w\end{tiny}}}}\rho_0(\frac{\phi_0}{\phi}-\frac{\phi}{
2\phi_0})+(\lambda+\frac{1}{\lambda}-2){v_{\textrm{\begin{tiny}w\end{tiny}}}}{C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}', \label{ras15}$$ which, combined with (\[ras13\]), determines the total swelling stress. Rewriting equation (\[equation-mechanics\]) taking the total stress into account leads to the force balance equation stated in (\[d1\]). Because (\[ras15\]) and (\[ras13\]) include elements of Flory-Rehner theory, and Donnan and Langmuir quasi-equilibria, we call these two equations the FRDL model for uniaxial swelling stress. The behavior of this model depends on the hydrogel parameters $\phi_0$, $\rho_0$, $\sigma_0$, $\chi_1$, $\chi_2$ and $p{K_{\textrm{\begin{tiny}A\end{tiny}}}}$, and the external salt concentration ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'$, which is assumed to be constant. These, together with relevant dimensionless parameter groups, determine the swelling and permeability properties of the hydrogel membrane.
Swelling equilibria
-------------------
Before pursuing dynamics, it is useful to examine homogeneous uniaxial swelling equilibria for a hydrogel in a bath of fixed $pH=-\log{C_{\textrm{\begin{tiny}H\end{tiny}}}}'$. In this case $ s=0
$ in (\[ras15\]) and ${C_{\textrm{\begin{tiny}H\end{tiny}}}}=\lambda{C_{\textrm{\begin{tiny}H\end{tiny}}}}'=\lambda 10^{-pH}$. Setting $p{K_{\textrm{\begin{tiny}A\end{tiny}}}}:=-\log
{K_{\textrm{\begin{tiny}A\end{tiny}}}}$, equation (\[ras13\]) becomes $$\begin{aligned}
&&10^{-(pH-p{K_{\textrm{\begin{tiny}A\end{tiny}}}})}(1-\phi){C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'\lambda^3+
(1-\phi){C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'\lambda^2-[10^{-(pH-p{K_{\textrm{\begin{tiny}A\end{tiny}}}})}(1-\phi){C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'\nonumber\\
&&\quad\quad\quad+\sigma_0(\frac{\phi}{\phi_0})]\lambda-(1-\phi){C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'=0.
\label{ras18} \end{aligned}$$ We discuss the solvability of equation (\[ras18\]) with respect to $\lambda$. Since the physical parameters contributing to the polynomial coefficients are positive and $\phi\leq1$, there is only one sign change in the descending polynomial coefficients, and Descartes’ rule of signs mandates a single positive real root. (Negative or complex values of $\lambda$ are physically meaningless as they would predict negative or complex ion concentrations inside the hydrogel.) Moreover, substituting $\lambda=\nu+1$ in (\[ras18\]) and rearranging terms, we obtain a cubic polynomial in $\nu$ that also has only one sign change, hence $\nu\geq 0$ and $\lambda\geq1 $. This makes physical sense, since for negatively charged hydrogel the internal cation concentration must exceed its external counterpart.
The first and third terms of (\[ras18\]) can vary over several orders of magnitude with pH, indicating that pH, which affects ionization, will have a strong influence on salt ion partitioning and hence ion swelling pressure. When $ pH<<p{K_{\textrm{\begin{tiny}A\end{tiny}}}}$, the polymer is uncharged, $f\to 0$ and $\lambda \to 1$, and swelling takes on a minimum value determined by the mixing and elastic stresses. When $ pH>> p{K_{\textrm{\begin{tiny}A\end{tiny}}}}$, all acid groups are ionized with $f\to 1$, and swelling is maximal with $\lambda $ substantially greater than $1$.
Figure \[Ronfig1\]a displays the effect of fraction ionized, $f,$ on uniaxial swelling ratio $\alpha$=$\chi_1 =0.48,$ $\chi_2 =0.60.$ This figure was generated using equation (\[ras15\]) with $ s=0$ and a version of (\[ras13\]).
[ ![The left, middle and right figures show the [*uniaxial swelling ratio*]{} $\alpha$, the [*fixed charge density*]{} $\sigma$ and the [*Donnan ratio*]{} $\lambda$ respectively, versus [*ionized fraction*]{} $f$, for different salt concentration.[]{data-label="Ronfig1"}](Ronfig1a.pdf "fig:") ![The left, middle and right figures show the [*uniaxial swelling ratio*]{} $\alpha$, the [*fixed charge density*]{} $\sigma$ and the [*Donnan ratio*]{} $\lambda$ respectively, versus [*ionized fraction*]{} $f$, for different salt concentration.[]{data-label="Ronfig1"}](Ronfig1b.pdf "fig:") ![The left, middle and right figures show the [*uniaxial swelling ratio*]{} $\alpha$, the [*fixed charge density*]{} $\sigma$ and the [*Donnan ratio*]{} $\lambda$ respectively, versus [*ionized fraction*]{} $f$, for different salt concentration.[]{data-label="Ronfig1"}](Ronfig1c.pdf "fig:") ]{}
As expected, swelling increases with decreasing salt concentration. For ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'= 250\textrm{mM}$, the membrane remains in its essentially collapsed state with $\alpha<1$. In this case hydrophobicity is the dominant force, and the effect of ion osmotic stress is relatively weak. For ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'= 155\textrm{mM}$ and $50\textrm{mM},$ an initially shallow relation between $\alpha$ and $f$ is punctuated by a rather sharp rise, indicating initial dominance of hydrophobicity that is overcome by ion osmotic stress at higher $f$. As ionic strength (salt concentration) decreases, the sharp rise occurs at lower ionization degree.
For ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'= 250\textrm{mM}$ and $155\textrm{mM},$ $ f$ uniquely determines $\alpha$. For ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'= 50\textrm{mM}, $ however, a range of bistability and hysteresis is observed. Over this range in $f,$ total stress $s$ vanishes at three values of $\alpha$, corresponding to two free energy minima and one maximum in between. The latter, which corresponds to the negative slope branch of the swelling curve between the turning points, is unstable, and need not be considered in the discussion of equilibria. Figure \[Ronfig1\]b shows the fixed charge density, $\sigma=(\phi/\phi_0)f\sigma_0$, as a function of $f$ for the three salt concentrations. In all cases, this quantity initially rises with increasing $f$, since $f$ is increasing but swelling does not change significantly. The rise is followed by a drop attributed to the sudden increase in swelling. Bistability is observed for ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}' = 50\textrm{mM} $ over the same interval of $f$ as before.
Figure \[Ronfig1\]c exhibits the calculated Donnan ratio $\lambda$ as a function of $f$ for the three salt concentrations. While these ratios provide information regarding the ion osmotic swelling force via (\[ras14\]), they also provide a link between external and internal pH. The curves are all nonmonotonic, with bistability for ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'= 50\textrm{mM}.$ The swings in $\lambda$ more or less follow those of $\sigma$ and can be explained in essentially the same way.
Figure \[Ronfig2\] displays the relationship between pH and swelling for the three salt concentrations, considering both internal and external pHs (dotted and continuous lines, respectively). These values are determined according to (\[ras13\]) and (\[ras15\]), with $s=0$, and $pH(int)=-\log{C_{\textrm{\begin{tiny}H\end{tiny}}}}$ and $pH(ext)=pH(int)+\log\lambda$. Evidently, increasing salt concentration leads to an alkaline shift (higher pH) at which the major transition in swelling occurs. With increasing salt, a larger degree of ionization and hence higher pH is needed to effect the transition. The three pairs of graphs exhibit qualitatively different behaviors. For ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'= 250\textrm{mM},$ both curves are monotonic and single valued, although internal pH is clearly lower than external pH, as expected based on Donnan partitioning of hydrogen ion. For ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}' =
50mM,$ both curves exhibit bistability. At the intermediate salt concentration, ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}' = 155\textrm{mM}$, the internal pH curve is single valued, while the external pH curve shows bistability. This difference can be attributed to the Donnan effect, which nudges external pH to higher values at low degrees of ionization and swelling. Of course, the real control variable is external pH.
We conclude that there are two potential mechanisms underlying bistability, one originating from the relation between fixed charge density and degree of swelling, and the other stemming from the Donnan effect. While in these studies we looked at qualitative behaviors as a function of ${C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}'$, these behaviors might also be observed by altering structural parameters of the hydrogel.
[![ versus $pH-pK_A$ at different salt concentrations. []{data-label="Ronfig2"}](Ronfig2.pdf "fig:")]{}
To simplify the notation, from now on, we will remove the ’prime’ symbol from the external salt concentration variable $C'_{\textrm{\tiny{NaCl}}}$ and agree in writing ${C_{\textrm{\tiny{NaCl}}}}$ instead.
Spatially homogeneous chemomechanical model
===========================================
[\[chemo-mechanical model\]]{} Now that the axially restricted equilibrium swelling properties of the hydrogel membrane have been explored, we introduce a lumped parameter model of the chemomechanical oscillator illustrated in Figure 1. Variants of this model have been presented previously [@dhanarajan2002autonomous; @dhanarajan2006]. The dynamic model is based on the following assumptions:
[**a.**]{} Membrane properties, including the fixed charge concentration, swelling state and ion concentrations inside the membrane are assumed to be homogeneous in space. Upon suppressing lateral swelling and shrinking, the swelling state of the membrane is determined by its thickness $L=L(t)$. This homogeneity assumption is made even though the membrane is subjected to a pH gradient. External pH is “homogenized” by averaging the constant $\textrm{H}^+$ concentration in Cell I, $C_H^I$ and the variable $\textrm{H}^+$ concentration in Cell II, $C_H^{II}$, that is, $pH=\log[(C_H^I+C_H^{II})/2]$.
[**b.**]{} Chemical potentials inside the membrane are determined by the 1-D FRDL equation of state for one-dimensional swelling, as described above. From these, the total swelling stress, including the elastic, mixing and ionic contributions, is given by (\[ras15\]), with $\lambda$ determined according to the electroneutrality condition (\[ras13\]). Electroneutrality is enforced by rapid exchange of ${\textrm{Na}}^+$ and ${\textrm{Cl}}^-$ between the membrane and Cells I and II. $\textrm{NaCl}$ concentrations in Cells I and II are assumed constant and equal [@Hirotsu].
[**[c.]{}**]{} Enzymatic conversion of glucose to gluconic acid is assumed to be instantaneous. This assumption is valid when the concentration of enzyme in Cell II is sufficiently high that transport of glucose into Cell II is rate limiting. Also, gluconate and bicarbonate, produced according to chemical reactions (I) and (II), respectively, are presumed to not perturb the system’s dynamics. The latter assumption is expected to hold best during early oscillations.
[**[(d)]{}**]{} The rate of change of fixed, negative charge concentration is controlled by the rate of transport of hydrogen ions, which reversibly bind to pendant carboxylates as they diffuse through the membrane [@grimshaw1990kinetics].
[**[(e)]{}**]{} Permeability of the membrane to glucose is expressed as ${K_{\textrm{\begin{tiny}G\end{tiny}}}^{\textrm{\begin{tiny}
0\end{tiny}}}}e^{-\beta\phi}$, as suggested by free volume theory [@LamazePeterlin1971], [@YasudaLamazePeterlin1971]. The parameters ${K_{\textrm{\begin{tiny}G\end{tiny}}}^{\textrm{\begin{tiny}
0\end{tiny}}}}$ and $\beta$ represent, respectively, the hypothetical permeability with vanishing polymer concentration, and the sieving effect of the polymer, which depends on both polymer chain diameter and radius of the diffusant (glucose in this case). For hydrogen ions or water, which are much smaller than glucose, the sieving factor is assumed to be negligible, and we simply multiply the respective permeability coefficients, ${K_{\textrm{\begin{tiny}h\end{tiny}}}}$ and ${{K}_{\textrm{\begin{tiny}w\end{tiny}}}}$, by $1- \phi$ to account for the aqueous space in the hydrogel that is available for transport of solutes.
In this model, diffusion of $\textrm{Na}^+$ and $\textrm{Cl}^-$ are regarded as instantaneous, while diffusion of $\textrm{H}^+$ is regarded as rate determining, even though the diffusion coefficient of $\textrm{H}^+$ is decidedly larger. This assumption is justified in part due to the reversible binding of $\textrm{H}^+$ to the pendant carboxylates, which does not occur with $\textrm{Na}^+$ and $\textrm{Cl}^-$, and partly due to the very low $\textrm{H}^+$ concentration, which qualifies it for “minority carrier” status. Changes in concentration of $\textrm{H}^+$ in the membrane are rapidly “buffered” by readjustments of $\textrm{Na}^+$ and $\textrm{Cl}^-$ concentrations, preserving electroneutrality.
Based on these assumptions, we write the following differential equations for $L$, the flux of hydrogen ions into the membrane, which then become either free intramembrane hydrogen ions or protons bound to pendant carboxyls, with respective concentrations ${C_{\textrm{\begin{tiny}H\end{tiny}}}}$ and ${C_{\textrm{\begin{tiny}AH\end{tiny}}}}$, and the flux of $\rm{H}^+$ to cell II: $$\begin{aligned}
\frac{dL}{dt}=&& -{{K}_{\textrm{\begin{tiny}w\end{tiny}}}}(1-\phi)[ln(1-\phi)+\phi +(\chi_1+\chi_2\phi)\phi^2 +{\nu_{\textrm{\begin{tiny}w\end{tiny}}}}\rho_0(\frac{\phi_0}{\phi}-\frac{\phi}{2\phi_0})\nonumber\\
&&\,\,-{\nu_{\textrm{\begin{tiny}w\end{tiny}}}}{C_{\textrm{\begin{tiny}NaCl\end{tiny}}}}(\lambda+\frac{1}{\lambda}-2)], \label{d1}\\
\frac{d}{dt}[L({C_{\textrm{\begin{tiny}H\end{tiny}}}}+{C_{\textrm{\begin{tiny}AH\end{tiny}}}})]&&=2{K_{\textrm{\begin{tiny}h\end{tiny}}}}{(1-\phi)}[\lambda\frac{({C_{\textrm{\begin{tiny}H\end{tiny}}}^{\textrm{\begin{tiny}
I\end{tiny}}}}+{C_{\textrm{\begin{tiny}H\end{tiny}}}^{\textrm{\begin{tiny}
II\end{tiny}}}})}{2}-{C_{\textrm{\begin{tiny}H\end{tiny}}}}],
\label{d2}\\
\frac{d}{dt}{C_{\textrm{\begin{tiny}H\end{tiny}}}^{\textrm{\begin{tiny}
II\end{tiny}}}}&&= \frac{A{K_{\textrm{\begin{tiny}G\end{tiny}}}^{\textrm{\begin{tiny}
0\end{tiny}}}}}{V}e^{-\beta\phi}{C_{\textrm{\begin{tiny}G\end{tiny}}}}-
\frac{A{K_{\textrm{\begin{tiny}h\end{tiny}}}}}{V}{(1-\phi)}(\lambda{C_{\textrm{\begin{tiny}H\end{tiny}}}^{\textrm{\begin{tiny}
II\end{tiny}}}}-{C_{\textrm{\begin{tiny}H\end{tiny}}}})-{k_{\textrm{\begin{tiny}mar\end{tiny}}}}{C_{\textrm{\begin{tiny}H\end{tiny}}}^{\textrm{\begin{tiny}
II\end{tiny}}}}, \label{d3}
\end{aligned}$$ where $V$ is the volume of Cell II and ${C_{\textrm{\begin{tiny}G\end{tiny}}}}$ is the glucose concentration in Cell I. Then, the governing system consists of equations (\[d1\])-(\[d3\]) together with the algebraic constraints (\[mass-balance-1d\]), (\[ras9\]), (\[d4\]) and (\[ras13\]). With an additional scaling argument, we obtain the equations to analyze.
Model scaling and the governing system
======================================
[\[scaling\]]{} In this section, we identify the relevant parameters of the system and their numerical values. This will allow us to rigorously derive a reduced model from the system of equations (\[d1\])-(\[d3\]). Five dimensionless groups give the system multiscale structure and properties, such as the role of the lower dimensional manifold discussed in section \[inertial-manifold\]. However, the full dynamics cannot be explained in terms of the dimensionless parameter groups only. We find that the range of the oscillatory behavior is further determined by a few individual parameters, such as ${C_{\textrm{\tiny{NaCl}}}}$, $\sigma_0$ and $\phi_0$, with the remaining ones held fixed. These values are in full agreement with the experiments and are also those used in the simulations of sections \[swelling-equilibria\] and \[numerical-simulations\].
For convenience, we relabel the variable fields of the problem and define dimensionless variables: $$\begin{aligned}
&& x={C_{\textrm{\begin{tiny}H\end{tiny}}}}, \,\, y={C_{\textrm{\begin{tiny}AH\end{tiny}}}}, \,\, z={C_{\textrm{\begin{tiny}H\end{tiny}}}^{\textrm{\begin{tiny}
II\end{tiny}}}}, \,\, h={C_{\textrm{\begin{tiny}H\end{tiny}}}^{\textrm{\begin{tiny}
I\end{tiny}}}}, \label{fieldsnew}\\
&&\bar x=\frac{x}{c}, \quad \bar y=\frac{y}{c}, \quad \bar z=\frac{z}{c}, \quad \bar L=\frac{L}{L_0}, \quad \bar t=\frac{t}{T}, \label{scaled-variables}\end{aligned}$$ where $T$ and $c$ are typical time and concentration variables, respectively. For now, we use the superimposed [*bar*]{} notation to represent dimensionless quantities. We choose $$T=\frac{L_0\phi_0}{{{K}_{\textrm{\begin{tiny}w\end{tiny}}}}}, \quad c={K_{\textrm{\begin{tiny}A\end{tiny}}}}.$$ Note that $T$ corresponds to the time scale of equation (\[d1\]), which will allow us to properly separate the dynamics of the membrane from that of the chemical reactions. The choice of $c$ will lead to a reduced model resulting from the simplification of equation (\[d2\]) as we shall see later in the section.
We now introduce five relevant dimensionless parameter groups consistent with the proposed scaling: $$\begin{gathered}
\mathcal A_1=\frac{A}{V}\frac{{C_{\textrm{\begin{tiny}G\end{tiny}}}}}{{K_{\textrm{\begin{tiny}A\end{tiny}}}}}{K_{\textrm{\begin{tiny}G\end{tiny}}}}^0 T, \quad \mathcal A_2=\frac{A}{V} T {K_{\textrm{\begin{tiny}H\end{tiny}}}}, \quad \mathcal A_3={k_{\textrm{\begin{tiny}mar\end{tiny}}}}T, \quad \mathcal A_4= 2\frac{{K_{\textrm{\begin{tiny}H\end{tiny}}}}}{K_w}\phi_0\frac{{K_{\textrm{\begin{tiny}A\end{tiny}}}}\phi_0}{\sigma_0}, \quad \mathcal A_5=\frac{{K_{\textrm{\begin{tiny}A\end{tiny}}}}\phi_0}{\sigma_0}. \label{A}\end{gathered}$$ The scaled form of equation (\[d4\]) combined with (\[ras9\]) is then $$\bar y=\mathcal A_5^{-1} \frac{\phi\bar x}{1+\bar x}, \,\,\, {\textrm{so}} \,\,\, \,
\bar x+\bar y=\bar x(1+ \mathcal A_5^{-1}\frac{\phi}{1+\bar x})=\mathcal A_5^{-1}\frac{\bar x\phi}{1+\bar x}+O(1). \label{x+y}$$ The latter approximation holds for the range of parameters of the problem, for which $
\mathcal A_5^{-1}\approx 3*10^4. $ Applying it to the scaled version of equation (\[d2\]), which together with (\[d1\]) and (\[d3\]), give the set of equations of the model that we analyse: $$\begin{aligned}
\frac{d\phi}{d\bar t}=&& {\mathcal R}_1,\label{dphi}\\
\frac{d\bar x}{d\bar t}=&&{\mathcal {A}}_4{\mathcal R}_2+ \mathcal A_5(1+\bar x)^2 {\mathcal R}_1, \label{dx}\\
\frac{d\bar z}{d\bar t}=&& \mathcal A_1{\mathcal R}_3, \label{dz}\\
&&\lambda=p(\phi)f+\sqrt{p^2(\phi)f^2+1},\quad f=(1+{\bar x})^{-1}, \label{electroneutrality-soln}\end{aligned}$$ where $$\begin{aligned}
{\mathcal R}_1(\phi,\lambda)=&& (1-\phi)\phi^2\big( H(\phi) -R(\lambda)\big), \quad {\mathcal R}_2(\phi, x, \lambda)=(1-\phi)(1+x)^2(\frac{\lambda}{2}\bar z-\bar x), \label {R2}\\
{\mathcal R}_3(\phi, x, z, \lambda)=&& e^{-\beta\phi} -\mathcal A_2{\mathcal {A}}_1^{-1}{(1-\phi)}(\lambda \bar z-\bar x)- \mathcal A_3{\mathcal {A}}_1^{-1} \bar z, \,\,\,\, \textrm{and} \label{R3}\\
H(\phi)= && \ln(1-\phi)+
\phi+(\chi_1+\phi\chi_2)\phi^2+\bar{\nu_{\textrm{\begin{tiny}w\end{tiny}}}}\bar\rho_0(\frac{\phi_0}{\phi}-\frac{\phi}{
2\phi_0}), \label{Hlambda}\\
R(\lambda)= && {\nu_{\textrm{\begin{tiny}w\end{tiny}}}}{C_{\textrm{\tiny{NaCl}}}}(\sqrt{\lambda}-\frac{1}{\sqrt{\lambda}})^2, \quad p(\phi)= \frac{\gamma\phi}{2(1-\phi)}, \label{Rlambda}\\
q(\phi)= &&1+\frac{1}{2\gamma_0}H(\phi), \quad \gamma=\frac{\sigma_0}{{C_{\textrm{\tiny{NaCl}}}}\phi_0}, \quad \gamma_0={v_{\textrm{\begin{tiny}w\end{tiny}}}}{C_{\textrm{\tiny{NaCl}}}}. \label{gamma-gamma0}\end{aligned}$$ The graph of $H(\phi)$ is shown in figure \[Hfig\]. Parameters of the problem, shown in tables (S2)-(S4) of the Supplementary Materials section, are of three main types: device specifications, hydrogel parameters and rate quantities. It should be noted that while some of these values either reflect the experimental conditions or are literature parameters for NIPA/MAA hydrogels [@PEN], some parameters were chosen to produce results that are in line with experimental observations. Specifically, the size of the marble component in Cell II was chosen with a sufficiently large surface area so as to provide the necessary reaction sites to remove excessive hydrogen ions from the system. For the given parameter values, we find that $$\mathcal A_1\approx 0.16*10^{1}, \,\, \mathcal A_2\approx 0.52* 10^{-2}, \,\, \mathcal A_3\approx 0.75*10^{-2}, \,\, \mathcal A_4\approx 0.33 *10^{-5}. \label{parameter-values}$$ We point out that $
0<\mathcal A_4<< 1, \quad \mathcal A_1^0:= A_1e^{-\beta\phi_0}= 0.016\,\,\,\, \textrm {and} \,\,\, \, \mathcal A_1^0>\mathcal A_2, \mathcal A_3. \,$ So, the first term of the right hand side of (\[dz\]) is of the order $\mathcal A_1^0$, since $$\mathcal A_1\mathcal R_3= \mathcal A_1^0 e^{-\beta(\phi-\phi_0)} + \dots \label{eqn-dz0}$$These parameter properties are relevant to obtain estimates of the solutions.
\[parameter-space\] We let $\mathcal P$ denote the set of parameters in tables (S2)-(S4) and with $${C_{\textrm{\tiny{NaCl}}}}\in [20*10^{-3}, 155*10^{-3}] \textrm{M}, \quad \sigma_0= [0.10, 0.40], \quad \phi_0=[0.1, 0.3]. \label{parameters}$$
Experiments have been carried out with values of ${C_{\textrm{\tiny{NaCl}}}}$ in the list $ \{350, 250, 150, 100, 50, 40, 25, 20\}*10^{-3}$M. However for salt concentration values above $155*10^{-3}$M, no oscillatory behaviour has been found.
From now on, we suppress the superimposed [*bar*]{} in the equations and assume that all variables are scaled. We now study the properties of the functions in the governing system.
\[Hphi\]
![[]{data-label="Hfig"}](Hphi.pdf "fig:") ![[]{data-label="Hfig"}](mZphi_zero "fig:")
Suppose that the parameters in $(\ref{Hlambda})$ belong to $\mathcal P$. Then the function $H(\phi)$ is non-monotonic in $(0,1)$ and has the following properties: $$\lim_{\phi\to 0^{+}}H(\phi)=\infty, \quad H(\phi)=O(\frac{1}{\phi}) \,\, {\textrm {and}}\, \,
\lim_{\phi\to 1^{-}}H(\phi)=-\infty, \quad H(\phi)=O(\log{(1-\phi)}).$$ Moreover, there exists a unique $\phi^*\in(0,1)$ such that $
H(\phi^*)=0.$ Furthermore, $H(\phi)$ has a unique local maximum, $\phi_{\textrm{\begin{tiny}{max}\end{tiny}}}$ and a unique local minimum, $\phi_{\textrm{\begin{tiny}{min}\end{tiny}}}$ in $(0,\phi^*)$, as shown in Figure \[Hphi\].
We represent the governing system (\[dphi\])-(\[dz\]) and denote the corresponding domain as $$\dot{\mathbf x}= \boldsymbol f({{\mathbf x}}) \quad \textrm{and} \quad \mathcal I=\{{{\mathbf x}}=(\phi, x, z): \, \phi\in(0, 1), x\in(0, \infty), z\in(0, \infty)\}, \label{I}$$ respectively, with the components of $\boldsymbol f$ given by the right hand side functions of the system. It is easy to check that $\boldsymbol f$ is continuously differentiable in the convex set $\mathcal I$, also the set of initial data of the problem. In order to denote solutions corresponding to specific initial data in $\mathcal I$, we write $$\begin{aligned}
&&\phi(t)= \phi(\phi^0, x^0, z^0)(t), \, \, x(t)= x(\phi^0, x^0, z^0)(t), \,\, z(t)=z(\phi^0, x^0, z^0)(t), \nonumber\\
&&\phi_M(t)= \phi(\phi^0, x^+(\phi^0), z^0)(t), \, \, x(t)= x(\phi^0, x^+(\phi^0), z^0)(t), \,\, z(t)=z(\phi^0, x^+(\phi^0), z^0)(t), \nonumber\end{aligned}$$ with the latter expression referring to restrictions to the two-dimensional manifold $\mathcal M$ defined in (\[Mminus\]).
### Time scales of the model
The relative sizes of the dimensionless parameters $\mathcal A_i$, in particular, the fact that $\mathcal A_4<< 1$ in equation (\[dx\]) indicates that $x$ is the slow field of the problem. Accordingly, we define the time scale associated with $x$ as $$\tau={\mathcal {A}}_4 \bar t= \frac{{\mathcal {A}}_4}{T} t, \quad \epsilon:={\mathcal {A}}_4, \quad \mu=\frac{\mathcal A_4}{\mathcal A_1}. \label{time-scales}$$ So, the original dimensionless time variable $\bar t$ corresponds to the [*fast* ]{} dynamics, whereas $\tau$ gives the [*slow* ]{} time, that now we take as reference. In the new time scale, the governing system reduces to $$\begin{aligned}
{\mathcal A}_4\frac{d\phi}{d\tau}=&& {\mathcal R}_1(\phi, \lambda)\label{phi333}\\
\frac{d\bar x}{d\tau}=&&{\mathcal R}_2(\phi,\bar x,\lambda)+ \frac{{K_{\textrm{\begin{tiny}A\end{tiny}}}}\phi_0}{\sigma_0\mathcal A_4}(1+\bar x)^2\mathcal R_1 , \label{x333}\\
\frac{{\mathcal A}_4}{{\mathcal A}_1}\frac{d\bar z}{d\tau}=&& {\mathcal R}_3(\phi, \bar x, \bar z,\lambda), \label{z333}\end{aligned}$$ together with equations (\[electroneutrality-soln\]). For our typical parameter values, $\mathcal A_4=0.33*10^{-5}$, $\frac{\mathcal A_4}{\mathcal A_1^0}=0.12*10^{-3} $ and $\frac{{K_{\textrm{\begin{tiny}A\end{tiny}}}}\phi_0}{\mathcal A_4\sigma_0}=10.$ This motivates identifying a reduced model consisting of equations (\[x333\])-(\[z333\]) together with the algebraic equation $
0={\mathcal R}_1(\phi, \lambda),$ and equations (\[electroneutrality-soln\]). We also point out that the scaling falls through when $\phi>0$ is arbitrarily small, that is, for highly swollen regimes. This follows from estimating equation (\[dphi\]) for $\phi>0$ small as well as the observation that $\lambda=O(1) $ and $R(\lambda)\approx 0$. Specifically, for given $\varepsilon>0$, and for $\phi\in(0, \varepsilon)$: $$\label{asymptotic}
p(\phi)=\gamma\phi+o(\varepsilon),\quad
\lambda=1+ o(\varepsilon), \quad R(\lambda)= o(\varepsilon), \quad H(\phi)={v_{\textrm{\begin{tiny}w\end{tiny}}}}\rho_0\frac{\phi_0}{\phi}+ o(\varepsilon).$$ This allows us to establish the following lemma:
Let $\tilde T>0$ be such that $\phi=\phi(t)$ and $\lambda=\lambda(t)$ satisfy equation (\[dphi\]) for $t\in [0,\tilde T]$. Then $$\frac{d\phi}{dt}= \frac{\nu_w\rho_0}{{\mathcal A}_4}O(\frac{1}{\phi})\,\, {\textrm{as}}\,\, \phi\to 0^+.$$
The behavior of the constitutive functions in (\[asymptotic\]) yields the following:
\[boundedness\] Solutions of (\[dphi\]) corresponding to initial data in $
\mathcal I$ have the property that $\phi$ remains bounded away from $\phi=0$ and $\phi=1$ for all time. Moreover, there exist positive numbers $t_{\textrm{\begin{tiny}m\end{tiny}}}$ and $t_{\textrm{\begin{tiny}M\end{tiny}}}$, depending on the initial data, such that $\phi(t_{\textrm{\begin{tiny}m\end{tiny}}})>0$ and $\phi({t_\textrm{\begin{tiny}M\end{tiny}}})>0$ are minimum and maximum values of $\phi$, respectively.
Let us first recall that $\phi^*$ denotes the largest value of $\phi$ such that $H(\phi^*)=0$. From the governing equation for $\phi$, we see that for initial data $\phi_i\geq \phi^*$, $\frac{d\phi}{dt}<0$. This proves the boundedness of the orbit away from $\phi=1$. Let us consider an orbit with initial data $\phi_i\in(0,{{\epsilon}})$ and such that $\frac{d\phi}{dt}<0$ at some $t>0$. Integrating the governing equation for $\phi$ while taking into account the estimates in (\[asymptotic\]) on its right hand side function yields $\phi(t)=\phi_i O(e^{\frac{t}{{v_{\textrm{\begin{tiny}w\end{tiny}}}}\rho_0\phi_0}}).$ This contradicts the assumption that $\frac{d\phi}{dt}<0$ at some $t>0$, and so proving the statement of the proposition.
Proposition (\[boundedness\]) states that the solution $\phi$ of the governing system remains in the interval $(0, 1)$, for all time of existence. However, due to the separation of time scales, only disconnected subintervals of $(0,1)$ are admissible. This is the topic of the next section.
The following lemma is used in the estimate of bounds of solutions. Let us rewrite equation (\[dz\]) as $$\frac{dz}{dt}+w(t)z=g(t)\, \, \textrm {\,with\,}\, w:= {\mathcal A}_2(\lambda(1-\phi)+ 1), \quad g:={\mathcal A}_1e^{-\beta\phi}+{\mathcal A}_2(1-\phi)x.\label{z-auxiliary}$$ The proof follows by integration of the previous auxiliary equation.
Suppose that $z_0>0$, $x=x(t)$, $\phi=\phi(t)$ and $\lambda=\lambda(t)$ are prescribed, continuous functions with $ t\in[0, \hat T]$, for some $\hat T>0$. Then, the solution of (\[dz\]) satisfies the equation $$z(t)= E^{-1}(t)\big(z_0+ \int_0^t E(s)g(s)\,ds\big), \,\, t\in [0, \hat T], \,z_0=z(0), \quad {\rm{and}}\,\, E(t):= \exp{\int_0^t}w(s)\,ds.$$
Next, we show that solutions of (\[dx\]) remain bounded away from $x=0$.
Suppose that $\phi=\phi(t)$ and $z=z(t)>0$ are prescribed continuous functions for $t\geq 0$, and such that $\phi(t)$ and $z(t)$ remain bounded away from $\phi=0$ and $z=0$, respectively. Then the solution $x=x(t)$ of equation (\[dx\]) corresponding to initial data $x(0)>0$ remains bounded away from $x=0$ for all $t>0$.
We argue by contradiction and assume that for a prescribed arbitrarily small $\epsilon>0$, there exits $t_1=t_1(\epsilon)>0$ such that $0\leq x(t)\leq x(t_1)\leq\varepsilon$, for $t\geq t_1$. A simple calculation using (\[electroneutrality-soln\]) gives $$\lambda(t_1)= p(\phi(t_1))+ \sqrt{p^2(\phi(t_1))+1}+ O(\varepsilon).$$ Hence, $\mathcal R_2(t_1)\geq (1-\phi(t_1))(1+x(t_1))^2(\frac{\lambda(t_1)}{2}z(t_1)-\varepsilon).$ So, $\frac{dx}{dt}(t_1)>0$ and bounded away from 0, and therefore, $x(t)$ cannot further decrease to 0.
Using the previous lemmas on boundedness of solutions, we can now state the following.
\[bounded-orbits\] Suppose that the parameters of the governing system belong to $\mathcal P$. Let $\{\phi=\phi(t), \,x=x(t)$, $z= z(t)\}$, $t\in [0, T)$, denote a solution of the system (\[dphi\])-(\[dz\]) and (\[electroneutrality-soln\]) corresponding to initial data in $\mathcal I$, and with $T>0$ representing the maximal time of existence. Then $\{\phi(t), x(t), z(t)\} $ are bounded. Moreover, the lower bounds are strictly positive and the upper bound of $\phi$ is strictly less than 1. Furthermore, $\mathcal I$ is invariant under the flow of the governing system.
Hopf bifurcation: a numerical study
====================================
[\[numerical-simulations\]]{} We numerically investigate the following properties of the solutions corresponding to the parameter set $\mathcal P$:
1. Uniqueness of the steady state solution and its stability.
2. Occurrence of Hopf bifurcation.
3. Non-monotonicity of the graph $x=x^+(\phi)$ in (\[lambdapm\]) and the reduced system (section (\[inertial-manifold\])).
We have seen that the third property guarantees the construction of closed orbits of the approximate two-dimensional system. This together with existence of a unique, unstable steady state provide sufficient conditions for the Poincare-Bendixon theorem to apply, from which existence of a limit cycle follows.
Steady state values of the variables $x$, $z$ and $\phi$ satisfy $$\begin{aligned}
&&x=\frac{\lambda}{2} z, \quad z= {\mathcal A}_1e^{-\beta\phi}({\mathcal A}_2(1-\phi)\frac{\lambda}{2}+{\mathcal A}_3)^{-1}, \nonumber\\
&& \frac{p}{\sqrt{q^2-1}}-1- \frac{{\mathcal A}_1}{2}\lambda e^{-\beta\phi}({\mathcal A}_2(1-\phi)\frac{\lambda}{2}+{\mathcal A}_3)^{-1}=0, \label{equilibrium-phi}
\end{aligned}$$ where the last equation follows from the previous expressions of $x$ and $z$, together with relations (\[lambdapm\]) and (\[gamma-gamma0\]). The second graph in Figure \[Hphi\] shows plots of the function on the left hand side of equation (\[equilibrium-phi\]) with the corresponding unique root, which after substitution into equations for $x$ and $z$, yields their steady state values. The computational results summarized next are in full agreement with those of laboratory experiments.
[[**1.**]{} For $ {C_{\textrm{\tiny{NaCl}}}}=0.155$ M:]{}
[**a.**]{}We found that for each $\phi_0=0.1, 0.15, 0.2, 0.295$, there is a unique physically relevant steady state, that is, with $\phi<\phi^*$. We point out that this result is valid for $\phi_0\in [0.1, 0.3]$. This is justified by the implicit function theorem applied to the left hand side function in (\[equilibrium-phi\]) with respect to each root. In this proof, we also use the fact that the slope of the tangent line to the graph at the intersection with the $\phi$-axis is non-horizontal, as shown in figure \[Hphi\] (Right).
[**[b.]{}**]{} The stability of the unique equilibrium solution is represented in the diagrams of figures \[Hopf-bifurcation\]. We find a region in parameters plane determined by the Hopf bifurcation curves, within which the equilibrium is unstable; outside, the solution is stable. The unstable equilibrium has a pair of complex conjugate eigenvalues with positive real part and a negative eigenvalue. Consequently, the existence of a Hopf bifurcation is guaranteed by the [*Hopf bifurcation theorem* ]{} [@Marsden1976] (section 3).
[**[c.]{}**]{} The graph $x=x(\phi)$ of the function (\[lambdapm\]) is found to be non-monotonic inside the regions determined by the Hopf bifurcation curves in figures \[Hopf-bifurcation\] and monotonic outside. Oscillatory behaviour is numerically found inside these regions.
[**2.**]{} We have also found that decreasing ${C_{\textrm{\tiny{NaCl}}}}$ gives a qualitative behavior analogous to decreasing $\phi_0$, that is, it promotes a single unstable stationary state of the system and the occurrence of Hopf bifurcation. In particular, the following behaviors have been found:
[**[a.]{}**]{} For ${C_{\textrm{\tiny{NaCl}}}}=0.145$M and $\phi_0=0.3$, the steady state solution corresponds to $\phi= 0.145$.
[**[b.]{}**]{} For ${C_{\textrm{\tiny{NaCl}}}}=0.148$M and $\phi_0=0.3$, the maximum and minimum values of $x=x(s)$ occur at $\phi=0.24$ and $\phi=0.26$. In both, these cases, a Hopf bifurcation has also been found.
[**[c.]{}**]{} Hopf bifurcations have also been found for ${C_{\textrm{\tiny{NaCl}}}}=0.125$M and ${C_{\textrm{\tiny{NaCl}}}}=0.115$M.
[**[d.]{}**]{} For $\phi_0=0.3$, ${C_{\textrm{\tiny{NaCl}}}}=0.155$M and $\sigma_0=0.28$, no Hopf bifurcation occurs. The simulations do not show oscillatory behaviour and there is at least one stable stationary state.
\[Hopf-bifurcation1\]
![Hopf bifurcation diagrams in different parameter planes. The red line denotes the instability threshold. []{data-label="Hopf-bifurcation"}](kmcs.pdf "fig:") ![Hopf bifurcation diagrams in different parameter planes. The red line denotes the instability threshold. []{data-label="Hopf-bifurcation"}](kmp0hb.pdf "fig:") ![Hopf bifurcation diagrams in different parameter planes. The red line denotes the instability threshold. []{data-label="Hopf-bifurcation"}](csp0.pdf "fig:")
Inertial manifold of the governing system
==========================================
[\[inertial-manifold\]]{} We now derive and study a reduced two-dimensional system such that, solutions of initial value problems of the original three-dimensional system remain arbitrarily close to those of reduced one, for most of the time of existence. Moreover, we shall see that, for the parameters of interest, the functions defining the 2-dimensional system are specified in two separate branches. So, it is necessary to extend the concept of solution giving it a weak interpretation as shown in the next section. Weak solutions also give a good physical description of the phase transition features of the model.
In order to identify the slow manifold, we consider the nullcline $$H(\phi)=R(\lambda)\label{HR}$$ of the original system. This corresponds to setting $\epsilon=0$ in (\[phi333\]), so that $\mathcal R_1\equiv 0$ while keeping $\mu$ fixed. Note that the right hand side of equation (\[x333\]) is simplified accordingly.
We now list the properties of the solutions of the equation (\[HR\]) that follow from the positivity of $R(\cdot)$ and the fact that $\lambda\geq 1$.
Let $H(\phi)$ and $R(\lambda)$ be as above. Then, solution pairs $(\phi, \lambda)$ of equation (\[HR\]) satisfy: $$0<\phi\leq \phi^* \,\, \textrm {so that}\,\, H(\phi)\geq 0 \quad \, \textrm{and} \quad \,
\lambda>1, \,\, \textrm {with } \, \,\lambda=1\, \, \textrm {when}\,\, \phi=\phi^*.$$
This allows us to characterize the nullcline (\[HR\]) as $$\begin{aligned}
\mathcal N=&&\{(\phi, \lambda, z)\in {\mathbf R}^3: 0<\phi<1, \, z>0, \lambda>1, \, H(\phi)=R(\lambda), \, {\textrm{and such that (\ref{electroneutrality-soln}) holds}}\}, \label{N}\\
\mathcal N:=&& \mathcal N^+\cup \mathcal N^{-},\,\,\,
\mathcal N^+= \{(\phi, \lambda, z)\in \mathcal N: H'(\phi)>0\}, \quad \mathcal N^-= \{(\phi, \lambda, z)\in \mathcal N: H'(\phi)\leq 0\}. \label{N+}\end{aligned}$$ Since relations (\[electroneutrality-soln\]) define $\lambda $ as a monotonic function of $x$, we can exchange the roles of $x$ and $\lambda$ in (\[N\])-(\[N+\]) at convenience.
Let us obtain an explicit representation of $\mathcal N$ and the governing equations of the reduced system. For this, we first solve equations (\[electroneutrality-soln\]) and (\[HR\]) using (\[Hlambda\]) and (\[Rlambda\]) and (\[gamma-gamma0\]) as follows $$\frac{\gamma\phi}{(1-\phi)} f= \lambda-\frac{1}{\lambda}, \quad \frac{H(\phi)}{\gamma_0}= \lambda+\frac{1}{\lambda}-2. \label{lambda-RH}$$ Addition and subtraction of these equations yields $$\lambda=q(\phi)+p(\phi) f, \quad
\frac{1}{\lambda}=q(\phi)- p(\phi) f, \label{lambda-p-q}$$ respectively, which, in turn, give the equation ${f_{+}}=\frac{1}{p}\sqrt{q^2-1},$ [for]{} $ q\geq 1$ and with $ H(\phi)\geq 0$. The corresponding values of the concentration $x$ and the Donnan ratio $\lambda$ are $$x^{+}= \frac{p(\phi)}{\sqrt{q^2(\phi)-1}} -1, \quad \lambda^{+}= q(\phi)+ \sqrt{q^2(\phi)-1}, \label{lambdapm}$$ respectively. Figures \[LdP-XdP\] present the graphs of these functions, for parameters in the class $\mathcal P$, showing their non-monotonicity. We label the critical points of $x^+(\phi)$ as $$0<\phi_s^1<\phi_s^2: \,\, \frac{dx^+}{d\phi}(\phi_s^i)=0, \, i=1,2, \quad {\textrm{and}}\quad x_s^i:=x^+(\phi_s^i),
\label{crit-points}$$ and consider the strictly increasing branches $(0,\phi_s^1)\cup(\phi_s^2, \phi^*)$. Let $\phi^+(x)$ denote the inverse of the restrictions of $x^+(\phi)$ to the monotonic branches, and define $$\begin{aligned}
\mathcal M_1:=
&& \{(\phi, x, z): \, z\geq 0, \, x=x^+(\phi), \, \phi\in(0, \phi_s^1)\}, \quad
\mathcal M_2
:=\{(\phi, x, z): \, z\geq 0, \, x=x^+(\phi), \, \phi\in(\phi_s^2, \, \phi^*)\}, \nonumber\\
\mathcal M:=&& \mathcal M_1\cup \mathcal M_2, \quad \mathcal M^-:=\{(\phi, x, z): \, z\geq 0, \, x=x^+(\phi), \, \phi\in(\phi_s^1, \, \phi^2_s)\}. \label{Mminus}\end{aligned}$$ That is, $\mathcal M$ consists of two branches where $x^+(\phi)$ increases monotonically. We point out that this graph is the cross section with respect to $z$ of the surface $H(\phi)=R(\lambda)$. This surface divides the whole space into upper and lower regions, with ${\mathcal R}_1>0$ and ${\mathcal R}_1<0$, respectively.
We now study the governing system restricted to $\mathcal M$. In terms of the original dimensionless time variable (which we still denote by $t$) and dependent variables, the governing system in $\mathcal M$ reduces to $$\frac{dx}{dt}= \mathcal A_4\phi(1-\phi)(1+\gamma\phi f^2)^{-1}\big(\frac{\lambda}{2} z-x\big), \quad \frac{dz}{dt}= \mathcal A_1e^{-\beta\phi} -\mathcal A_2{(1-\phi)}(\lambda z-x)-\mathcal A_3 z, \label{z-3d}$$ together with equations (\[lambdapm\]).
![ Plot of $\lambda=\lambda^+(\phi)$ and $x=x^+(\phi)$ from equations (\[lambdapm\]) that enter in the definition of $\mathcal M$ []{data-label="LdP-XdP"}](LdP_NM.pdf "fig:") ![ Plot of $\lambda=\lambda^+(\phi)$ and $x=x^+(\phi)$ from equations (\[lambdapm\]) that enter in the definition of $\mathcal M$ []{data-label="LdP-XdP"}](XdP_NM.pdf "fig:")
Suppose that the parameters of the problem belong to the class $\mathcal P$. Then $\mathcal M\cup \mathcal M^-$ is a two-dimensional invariant manifold of the three-dimensional system. Furthermore, the vector field of the two dimensional system is Lipschitz in $\mathcal M$. Moreover, $\mathcal N^-\subset \mathcal M.$
Let us consider initial data $(\phi_0, x_0, z_0)\in\mathcal M\cup\mathcal M^-$. It is easy to see that we can construct a solution of the three dimensional system belonging to $\mathcal M\cup\mathcal M^-$. So, by uniqueness, solutions with initial data satisfying $x_0=x^+(\phi_0)$, $\phi_0\in(0, \phi^*)$ belong to $\mathcal M\cup\mathcal M^-$, for all time of existence. Moreover, applying the same arguments as in Proposition \[bounded-orbits\], we find that $x(t) $ remains bounded away from $x=0$, and so $\phi(t)>\phi^{**}$, for all $t>0$. The last statement of the proposition follows by taking the derivative with respect to $\phi$ of the function $x=x^+(\phi)$ in equation (\[lambdapm\]), to give $$\frac{dx}{d\phi}=\frac{1}{\sqrt{q^2-1}}\big((1-\phi)^{-2}-\frac{pq}{q^2-1}H'(\phi)\big). \label{dxdphi}$$ So, $H'(\phi)<0$ implies that $\frac{dx}{d\phi}>0$, for $\phi<\phi^*$.
In order to study solutions with initial data outside $\mathcal M\cup \mathcal M^-$, we observe the following properties of the term $\mathcal R_1$: $$\mathcal R_1>0 \,\,\textrm{ for } \,\, x>x^+(\phi) \quad \textrm{and}\quad
\mathcal R_1<0 \,\,\textrm{ for } \,\, x<x^+(\phi). \label{R1-sign}$$
Suppose that the parameters of the system belong to the class $\mathcal P$. Then the following statements on the three-dimensional system hold:
1. Solutions with initial data such that $x^0\neq x^+(\phi^0)$ have the property that $|\mathcal R_1(\phi(t), x(t))|$ decreases with respect to $t$, for sufficiently large $t>0$. In particular, $|\mathcal R_1|$ is strictly decreasing for $\phi^0\in(0, \phi_s^1)\cup (\phi_s^2, \phi^*)$, for all $t>0$.
2. Let $\epsilon>0$ be sufficiently small. Then $\mathcal M$ is asymptotically stable, that is, there exists $(\hat\phi(t), \hat x(t), \hat z(t))\in \mathcal M$ such that, for sufficiently large $t$, solutions corresponding to initial data $(\phi^0, x^0, z^0)\in \mathcal I$ satisfy $$\phi(t, \epsilon)= \hat\phi(t)+ O(e^{-\frac{t}{\epsilon}}),\, \, x(t, \epsilon)= \hat x(t)+ O(e^{-\frac{t}{\epsilon}}),\,\, z(t, \epsilon)= \hat z(t)+ O(e^{-\frac{t}{\epsilon}}). \label{estimate}$$
Part 1 follows directly from the first equation in (\[R2\]), the properties of the functions $H(\phi)$ and $R(\lambda)$ and (\[dphi\]). The functions $(\hat\phi(t), \hat x(t), \hat z(t))$ are obtained as weak solutions of the two dimensional system. Their construction applies Definition \[weak-solution\], and estimate (\[estimate\]) follows from Theorem \[convergence\] on multiscale analysis of the system.
We characterize weak solutions of the two dimensional system. These correspond to hysteresis loops on the left graph of figure \[Hfig\]. One main ingredient in the construction is the theorem on continuation and finite time blow-up of solutions of ordinary differential equations together with the existence of unique, unstable, equilibrium point. This, together with the Poincar[é]{}-Bendixon theorem for two dimensional systems, leads to the existence of a limit cycle for such a system. The latter is also the $\omega$-limit set of the positive semi-orbits of the three dimensional system.
\[weak-solution\] A weak solution of the two dimensional system corresponding to initial data $(\phi^0, x^0, z^0)\in\mathcal M$ has the following properties:
1. There exists $0<\hat t$ such that $(\phi(t), x(t), z(t))$ is a classical solution for $t\in(0, \hat t)$.
2. $\phi(\hat t)=\phi_s^1$ (or $\phi_s^2$).
3. $\phi(t)$ is discontinuous at $\hat t$ experiencing a jump $[\phi(\hat t)]=\phi_s^2-\phi_s^1.$ (alternatively, $[\phi(\hat t)]=\phi_s^1-\phi_s^2.$)
4. The time derivatives of $\phi$ experience finite time blow up, that is, $\lim_{t\to \hat t}\frac{d\phi}{dt}(t)=+\infty$ (alternatively, $\lim_{t\to \hat t}\frac{d\phi}{dt}(t)=-\infty$).
5. $x(t)$ and $z(t)$ are continuous at $\hat t$ and their derivatives experience a jump discontinuity.
6. The solutions can be continued for $t>\hat t$ and are bounded.
We point out that $\mathcal M^{-}$ is also an invariant manifold of the two-dimensional system. In particular, it contains the unique stationary point, with eigenvalues forming a complex conjugate pair, with positive real part. The limit cycle of the 2 dimensional system found next is also the $\omega$-limit set of solutions with initial data in $\mathcal M^-$.
We point out that values of $\phi\in(\phi^1_s, \phi_s^2)$ are excluded from the range of solutions of the two-dimensional system. Indeed, in the next section, we construct weak solutions of the system such that $\phi$ experiences jump discontinuities, so as to avoid this interval. However, this interval is accessible to solutions of the full system, and, in particular, it contains the unique unstable equilibrium point. We will also show that this interval is covered in the fast time scale.
The following result follows from the uniqueness of classical solutions of the initial value problem of the system (\[dphi\])-(\[Rlambda\]).
We denote the weak solution as $\boldsymbol \psi_{\mathcal M, t}= (\phi_{\mathcal M}, x_{\mathcal M}(\phi), z_{\mathcal M}) $, where $\boldsymbol \psi_{\mathcal M, t} $ is the flow. Let $\boldsymbol\varphi_t$ denote the flow of the three dimensional system with initial data in $\mathcal I$.
For each set of initial data $(\phi^0, x^0, z^0)\in\mathcal M$, there is a a unique weak solution $(\phi_{\mathcal M}(t), x_{\mathcal M}(t), z_{\mathcal M}(t)) $ of the two dimensional system, that exists for all $t>0$. Moreover, the $\omega$-limit set $\omega(\pi^+)$ of a semi-orbit $\pi^+$ of the three dimensional system with initial data in $\mathcal I$ satisfies $
\omega(\pi^+)=\omega(\pi^+_{\mathcal M}),
$ where $\omega(\pi^+_{\mathcal M})$ is the $\omega$-limit set of the trajectories of the two dimensional system.
Without loss of generality let us assume that $(\phi(0), x(0), z(0)) \in \mathcal M_2$ and with $ \frac{1}{2}{\lambda(x(0), \phi(0))}z(0)-x(0)<0.$ Then, there is $\hat t>0$ such that $(0, \hat t)$ gives the maximal interval of existence of the classical solution of the 2-dimensional system. Two different situations may occur, according to the choice of initial data:
1. $\hat t=\infty$ in which case the solution is bounded and such that $x(t)\in \mathcal M_2$ for all time, or
2. $0<\hat t<\infty$, in which case $x(\hat t)= x_s^2$, and $\phi(\hat t)=\phi_s^2$.
In case (2), we further distinguish two cases, also according to initial data:
[**.**]{} $ \frac{1}{2}{\lambda(x(\hat t), \phi(\hat t))}z(t)-x(\hat t)=0$, in which case $\frac{dx}{dt}(\hat t)=0$. The solution can then be continued as in case 1, that is, as classical solution in $\mathcal M_2$, and, so it never reaches $\mathcal M_1$.
[**.**]{} $ \frac{1}{2}{\lambda(x(\hat t), \phi(\hat t))}z(t)-x(\hat t)<0$, and so $\frac{dx}{dt}(\hat t)<0$. Since $\lim_{t\to\hat t}\frac{dx}{d\phi}(t)=0$, then $\lim_{t\to\hat t}\frac{d\phi}{dt}(t)= -\infty$. We set $\phi(\hat t^+)=\phi_s^1$, so the jump condition $[\phi]= \phi_s^1-\phi_s^2$ is satisfied, $x(\hat t^-)= x(\hat t^+)=x_s^2$ and $z(\hat t^-)= z(\hat t^+)$. Note that at $ t=\hat t^+$, $(\phi, x, z)\in \mathcal M_1$.
To continue the solution for $t>\hat t$, we solve the initial value problem with initial data $(\phi(\hat t^+), x(\hat t^+), z(\hat t^+))$. It is easy to see that there is a time $\tilde t>\hat t $ such that $\lim_{t\to \tilde t^-} \phi(t) = \phi_s^1$ and $ \lim_{t\to \tilde t^-} x(t)=x_s^1$. In fact, it follows from the fact that $\phi$ and $x$ remain bounded away from 0, so that the values $\phi_s^1$ and $x_s^1$, respectively, can be reached, allowing the solution to be continued in $\mathcal M_2$. The fact that $\phi<\phi^*$ and that there are no equilibrium points in $\mathcal M_2$ guarantees the existence of a point of return on the trajectory $(\phi(t), x(t), z(t))$, and so the process can be continued.
Since the orbits $\pi^+$ and $\pi^+_{\mathcal M}$ are both bounded, the scaling with respect to $\mathcal A_4$ holds and so does the estimate $$|\phi(t)-\phi_{\mathcal M}(t)|=O(e^{-\frac{t}{\epsilon}}), \label{estimate1}$$ together with the analogous estimates for $x(t), z(t)$ which completes the proof of the theorem.
Since $\mathcal M$ is an invariant set that does not contain equilibrium points, the existence of a limit cycle for the two dimensional system follows from the Poincar[é]{}-Bendixon theorem stated next. Its corollary gives the stability of the limit cycle with respect to, both, the two-dimensional and the three-dimensional dynamics.
If $\pi^+$ is a bounded semiorbit and $\omega(\pi^+)$ does not contain any critical point, then either $ \pi^+=\omega(\pi^+),$ or $ \omega(\pi^+)={\bar\pi^+}/\pi^+$. In either case, the $\omega-limit$ set is a periodic orbit. Moreover, for a periodic orbit $\pi^0$ to be asymptotically stable it is necessary and sufficient that there is a neighborhood $G$ of $\pi^0$ such that $\omega(\pi(p))=\pi^0$ for any $p\in G$.
![The left figure shows orbits of the three dimensional system with the manifold $\mathcal M$. The figure on the right shows plots of orbits of the three dimensional system approaching a plane closed curve. It is the $\omega$-limit set of both systems and the limit cycle of the two dimensional system in the manifold $\mathcal M$. []{data-label="3d-curves"}](Aug27_14Osc-1 "fig:") ![The left figure shows orbits of the three dimensional system with the manifold $\mathcal M$. The figure on the right shows plots of orbits of the three dimensional system approaching a plane closed curve. It is the $\omega$-limit set of both systems and the limit cycle of the two dimensional system in the manifold $\mathcal M$. []{data-label="3d-curves"}](csol-crop "fig:")
The orbits of the two-dimensional system analyzed in this section, and in particular the limit cycle, correspond to the projection to the $x-z$ plane of the three-dimensional orbits shown in figure \[3d-curves\](b). Moreover, these two-dimensional orbits correspond to the leading terms in expansions (\[ilx\])-(\[ilz\]).
Next theorem shows that the limit cycle of the two dimensional system is also the limit cycle of the three dimensional one.
Suppose that the parameters of the system belong to the class $\mathcal P$. Then the two-dimensional system has an asymptotically stable limit cycle $\pi^+$ in $\mathcal M$. Moreover $\pi^+$ is also the $\omega$-limit set of positive semi-orbits of the three-dimensional system with initial data in $\mathcal I$.
Let us consider solutions with initial data in $\mathcal M$. For these initial data, we previously showed that the two-dimensional system admits bounded weak solutions that exist for all time. Moreover, the orbits of these solutions do not contain any equilibrium point. So, the existence of a limit cycle $ \pi^+$ in $\mathcal M$ follows directly from the Poincar[è]{}-Bendixon theorem. The asymptotic stability of $\pi^+$ follows from the boundedness of solutions and the absence of a stationary state in $\mathcal M$. To prove the last statement, let us consider solutions with initial data $(\phi^0, x^0, z^0)\in \mathcal I$ and such that $x^0\neq x^+(\phi^0)$. By an estimate as (\[estimate1\]), we can assert that sufficiently large $t$, $
|x(t)-x_M(t)|= O(e^{-Mt}), $ with $M>0$ as in (\[M1\]), and $x_M(t):=x_M(\phi^0, x^+(\phi^0), z^0)(t)$. This indicates that the solution of the three dimensional system approaches the two-dimensional manifold, for sufficiently large $t$. Since the only equilibrium point in the three-dimensional space is unstable, then the three-dimensional solution also approaches the limit cycle (figure \[3d-curves\](b)).
We observe that the solutions of the plane system, and in particular the limit cycle, are discontinuous with respect to $\phi$. In particular, $\phi$ is discontinuous at $t=\hat t$, with $[\phi(\hat t)]\neq 0$. The regularization of the solutions is done by [*connecting*]{} the discontinuity values of $\phi$ by a function $\bar\phi(\frac{t}{\mathcal A_4})$ that evolves according to the [*fast* ]{} dynamics presented in the next section. For sufficiently small $\mathcal A_4$, the solutions of the three-dimensional system remain in $\mathcal M$ for most of the time, emerging to the third dimension in order to connect the separate branches $\mathcal M_1$ and $\mathcal M_2$, in the fast time scale. This is developed in section \[multiscale\].
Competitive systems and three-dimensional limit cycle
=====================================================
[\[3D-limit-cycle\]]{} It is well known that for three dimensional systems, the compactness of a steady state free $\omega-$limit set of an orbit is not sufficient to prove the existence of a periodic orbit. That is, the Poincar[é]{}-Bendixon theorem in its original form does not apply. However, a three-dimensional generalization is available for [*competitive* ]{} (and [*cooperative*]{}) systems ([@Hirsh1], [@Hirsh2], [@Hirsh4], [@Smith-survey]). This concept is framed in terms of monotonicity properties of the vector field of the system with respect to a convex subspace of ${\mathbb R}^3$. The theorem is due to Hirsch and it is stated as follows:
\[PB-theorem-3d\] A compact limit set of a competitive or cooperative system in $ {\mathbf R^3}$ that contains no equilibrium points is a periodic orbit.
A dynamical system is called [*competitive*]{} in the positive cone ${\mathbb{R}^+}^{n} $ if $ \frac{\partial f_i}{\partial x_j}({{\mathbf x}})\leq 0, \ i\neq j, \, {{\mathbf x}}\in \mathcal I.$ This property holds in a general cone $\mathcal K$ (intersection of half-spaces), by requiring the following two properties:
\[sign-symmetry\] A $n\times n$ Jacobian matrix $(\nabla\boldsymbol f)$ is [*[sign-stable]{}*]{} in $\mathcal I$ if for each $i\neq j$, either ${\ensuremath{\frac{\partial f_i}{\partial x_j}}}\geq 0 $ or ${\ensuremath{\frac{\partial f_i}{\partial x_j}}}\leq 0 $, for all ${{\mathbf x}}\in\mathcal I$. It is [*[sign-symmetric]{}*]{} if ${\ensuremath{\frac{\partial f_i}{\partial x_j}}}({{\mathbf x}}){\ensuremath{\frac{\partial f_j}{\partial x_i}}}({{\mathbf y}})\geq 0 $ for all $i\neq j$ and for all ${{\mathbf x}}, {{\mathbf y}}\in\mathcal I$.
The three dimensional governing system (\[dphi\])-(\[electroneutrality-soln\]) is competitive with respect to the cone $
\mathcal K=\{(\phi, x, z): \phi<0, \, \, x>0, \,\, z<0\}.$
The proof involves identifying $\mathcal K$ and verifying definition (\[sign-symmetry\]), which follows from these two lemmas.
\[inequalities1-2\] The inequalities $
\phi(1-\phi)\frac{\partial\lambda}{\partial \phi}=\lambda \frac{fp}{\sqrt{1+p^2f^2}}\geq \lambda \,\, \textrm{and}\,\, \frac{\partial\lambda}{\partial x} <0
$ hold on trajectories $\boldsymbol \varphi_t$ corresponding to initial data in $\mathcal I$.
For $\lambda, p $ and $f$ as in (\[electroneutrality-soln\]) and (\[gamma-gamma0\]), simple calculations show that $$\begin{aligned}
&& \frac{2}{\gamma}(1-\phi)^2\frac{\partial\lambda}{\partial \phi}=\frac{f}{\sqrt{1+p^2f^2}}=\frac{f\lambda}{\sqrt{1+p^2f^2}}= \frac{2}{\gamma}\frac{(1-\phi)}{\phi}\frac{f p\lambda}{\sqrt{1+p^2f^2}},\nonumber\\
&&\frac{\partial\lambda}{\partial x}=\frac{p f'}{\sqrt{1+p^2f^2}}\lambda= -\frac{p}{(1+x)^2}\frac{\lambda}{\sqrt{1+p^2f^2}}> -\frac{p}{(1+x)^2}\lambda,
\end{aligned}$$ from which the stated inequalities follow.
The Jacobian matrix $\mathcal J$ of the three-dimensional system is sign-stable and sign-symmetric.
Let us calculate the Jacobian matrix $J:=\{a_{ij}\} $ at an arbitrary state $(\phi, x,z)$: $$\begin{aligned}
&&a_{12}= -(1-\phi)\phi^2R'(\lambda)\frac{\partial\lambda}{\partial x}, \quad
a_{21}= (1-\phi)(1+x)^2\frac{z}{2}\frac{\partial\lambda}{\partial\phi}-(1+x)^2(\frac{\lambda}{2}z-x),\\
&&a_{31}= -\beta e^{-\beta\phi}-\frac{{\mathcal A}_2}{{\mathcal A}_1}\big((1-\phi)z\frac{\partial\lambda}{\partial\phi}-(\lambda z-x)\big), \quad a_{13}=0,\\
&& a_{23}= (1-\phi)(1+x)^2\frac{\lambda}{2}, \quad a_{32}= -\frac{{\mathcal A}_2}{{\mathcal A}_1}(1-\phi)(z\frac{\partial\lambda}{\partial x}-1).
$$ Note that the off-diagonal elements of the matrix $J$ have the signs: $$\begin{aligned}
&&a_{12}>0, \, a_{13}=0, \, a_{23}>0, \, a_{32}>0, \, a_{21}>0,
\, a_{31}<0.\nonumber
$$ from which, sign-symmetry and sign-stability immediately follow. For this, let us write $$a_{21}=(1+x)^2\big((1-\phi)\frac{\partial\lambda}{\partial \phi}\frac{z}{2}-\frac{\lambda}{2}z+x\big)> \frac{x}{(1+x)^2}>0.$$ Note that $a_{23}>0$ and $ a_{31} <-\beta e^{-\beta\phi}-\frac{{\mathcal A}_2}{{\mathcal A}_1} x<0, $ where we have used the first inequality in lemma \[inequalities1-2\]. Finally, to check the sign of $a_{12}$, we differentiate $R(\lambda) $ in (\[Rlambda\]) to find $ R'(\lambda)= (\sqrt{\lambda}-\frac{1}{\sqrt{\lambda}})(\lambda^{-\frac{1}{2}}+ \lambda^{-\frac{3}{2}})$, from which the sign of $a_{12}$ follows, taking into account the second inequality in lemma \[inequalities1-2\]. The identification of the corresponding cone $\mathcal K$ is done in the Supplementary Material section.
One of the practical difficulties in the application of the Poincar[è]{}-Bendixon theorem to three-dimensional systems as compared to the two-dimensional counterpart is the verification that the $\omega$-limit set does not contain equilibrium points. The case that the system has a single unstable equilibrium point is the simplest one to treat. Existence of a three dimensional limit cycle, together with additional properties of competitive systems relevant to the current analysis are summarized next.
Let $\mathcal P$, $\mathcal M$ and $\mathcal I$ be as in definition (\[parameter-space\]), (\[Mminus\]) and (\[I\]), respectively. Then
1. The flow on the $\omega$-limit set of orbits of the three-dimensional system in $\mathcal I$ is topologically equivalent to the flow on the $\omega$-limit set of orbits of the two-dimensional system in $\mathcal M$.
2. Let ${\mathbf p }=(\Phi, X, Z)$ be the unique equilibrium point of the system. If ${\mathbf p }\neq \mathbf q \in \mathcal I$, then ${\mathbf p }\notin\omega(\mathbf q). $
3. The three-dimensional system has a limit cycle.
The first item follows from the fact that the two-dimensional system is Lipschitz, the competitiveness of the three dimensional system, and from the compactness of the $\omega$-limit set of both systems. The second statement follows from competitiveness and the fact that the equilibrium point is unique and hyperbolic. In fact, it is a consequence of the theorem that states that [*a compact limit set of a competitive or cooperative system cannot contain two points related by $<<$*]{} ([@Smith-survey], Theorem 3.2). Finally, property (3) is a consequence of (2), the compactness of the $\omega$-limit set and Theorem \[PB-theorem-3d\].
In the case that the unique equilibrium point ${\mathbf p }$ is hyperbolic, to prove that ${\mathbf p }\notin\omega(\mathbf q)$, $\mathbf q\in \mathcal D$, it is necessary to assume that the system is competitive. This is the case encountered in some applications such as in virus dynamics [@virus].
Multiscale Analysis
===================
[\[multiscale\]]{} The analysis carried out in the previous sections is mostly based on the time scale separation between the mechanical and chemical evolution components of the system. In turn, this is based on the relative sizes of the dimensionless parameters $\mathcal A_i$. The goal of this section is to find estimates for the solutions of the three dimensional system with respect to the fast time scale. Let us consider the time scales $\bar t$ and $\tau$ as in (\[time-scales\]), representing the [*fast*]{} and [*slow*]{} times, respectively. We consider the system of equations (\[phi333\])-(\[z333\]), together with (\[electroneutrality-soln\])-(\[R3\]). We propose the following solution ansatz $$\begin{aligned}
&&\phi(\bar t,\tau, {{\epsilon}})=\Phi(\tau,{{\epsilon}}) + {\tilde\phi}(\bar t,{{\epsilon}})=\big(\sum_{j=0}^N\Phi_j(\tau)+\mathcal E_0^1\big)+ \,\big(\sum_{j=0}^N{\tilde\phi}_j({{\bar t}})+\mathcal E_I^1\big), \label{ilphi} \\
&&x(\bar t,\tau,\epsilon)= X(\tau,\epsilon) + \tilde x(\bar t,\epsilon)= \big(\sum_{j=0}^NX_j(\tau)+\mathcal E_0^2\big)+ \,\big(\sum_{j=0}^N{\tilde x}_j({{\bar t}})+\mathcal E_I^2\big) \label{ilx}\\
&&z(\bar t,\tau,{{\epsilon}}) =Z(\tau,{{\epsilon}}) +{\tilde z}(\bar t,{{\epsilon}}) + \big(\sum_{j=0}^NZ_j(\tau)+\mathcal E_0^3\big)+ \,\big(\sum_{j=0}^N\bar z_j({{\bar t}})+\mathcal E_I^3\big) , \label{ilz}\end{aligned}$$ with ${{\epsilon}}=\mathcal A_4, \,\, \bar t=\frac{\tau}{{{\epsilon}}},$ and $\mathcal E_0^j, \mathcal E_I^j$, $j=1 ,2 ,3,$ denote error terms. We now consider $\frac{{\mathcal A}_4}{{\mathcal A}_1}$ fixed and $0<{{\epsilon}}<<1$. Equations for the terms in (\[ilphi\])-(\[ilz\]) are obtained as follows: [**[1]{}.**]{} Holding $\tau$ fixed, and letting ${{\epsilon}}\to 0$ yields equations for $\Phi, X, Z$. These are studied in section (\[inertial-manifold\]).
[**[2.]{}**]{} Holding $\bar t$ fixed, and letting ${{\epsilon}}\to 0$ yields equations for the initial layer term ${\tilde\phi}, {\tilde x}$ and ${\tilde z}$. Moreover, we seek solutions with the following asymptotic property $\{{\tilde\phi}(t,{{\epsilon}}), \, {\tilde x}(t,{{\epsilon}}), \, {\tilde z}(t,{{\epsilon}})\}=O(\exp(-ct)) \, \textrm{as} \, t\to \infty,$ where $c>0$ is a material dependent parameter. Note that since $X$ and $Z$ satisfy initial conditions at $\tau=0$, then ${\tilde x}=0={\tilde z},$ and it is necessary to calculate only the initial layer for $\tilde\phi$.
Solutions in the fast time scale $\bar t$
-----------------------------------------
To obtain equations for the initial layer terms, we substitute expressions (\[ilphi\])-(\[ilz\]) into the governing equations, fix ${{\bar t}}>0$ and take the limit ${{\epsilon}}\to 0$. In particular, this yields the limit $\tau=0$. Moreover, taking into account that ${\mathcal R}_1(\Phi, \Lambda)=-(1-\Phi)\Phi^2\big(R(\Phi)-H(\Lambda)\big)=0$, and linearizing about $(\Phi, \Lambda)$ gives the following equation for ${\tilde\phi}(\bar t,\tau,{{\epsilon}}):$ $$\label{M1}
\frac{d{\tilde\phi}}{d\bar t}= -M(\Phi,\Lambda){\tilde\phi}(\bar t,{{\epsilon}})+ o({\tilde\phi}), \quad
M(\Phi, \Lambda)
:= \frac{\partial {\mathcal R}_1}{\partial \phi}(\Phi, \Lambda)+ \frac{\partial {\mathcal R}_1}{\partial \lambda}(\Phi, \Lambda)\frac{\partial\lambda}{\partial\phi}.$$ We point out that $(\Phi, \Lambda, Z)$ solve the two-dimensional system. To calculate the right hand side of (\[M1\]), we take derivatives in equation (\[lambda-p-q\]), $\frac{\partial\lambda}{\partial \phi}(\Phi, X)= p'(\Phi) f(X) + q'(\Phi), \,\, R'(\Lambda)=\gamma_0(1-\frac{1}{\Lambda^2}).$ So, $$\begin{aligned}
M(\Phi, \Lambda)&&= -(1-\Phi)\Phi^2\big(H'(\Phi)-R'(\Lambda)(p'f(X) +q'(\Phi))\big)\nonumber\\
&&= -\frac{1}{2}(1-\Phi)\Phi^2\big(H'(\Phi)(1+\frac{1}{\Lambda^2}) -\frac{\gamma\gamma_0}{2(1-\Phi^2)}f(X)(1-\frac{1}{\Lambda^2})\big). \label{M2}\end{aligned}$$ We point out that according to the earlier observation that $(\Phi, \Lambda)$ are evaluated at $\tau=0$, it follows that the rate of decay of the fast component of the solution depends on the projection of the initial data on the slow manifold.
Consider initial data $(\phi^0, \lambda^0, z^0)\in\mathcal I$ and let $M^0:=M(\phi^0, \lambda^+(\phi^0), z^0)$. If $M_0>0$, then the solution of the linear equation (\[M1\]) satisfies $\tilde\phi({{\bar t}})=O(e^{-M^0 {{\bar t}}}), $ for ${{\bar t}}>0$ large. Otherwise, $\tilde\phi({{\bar t}})=O(e^{|M_0| {{\bar t}}}), $ for ${{\bar t}}>0$ large.
This lemma motivates the definition of [*slow*]{} and [*fast*]{} manifolds of the (three-dimensional) system. Let $$\begin{aligned}
&& \mathcal S= \{(\phi, \lambda, z): \lambda=\lambda^+(\phi), \, M>0\}, \quad \mathcal F= \mathcal I/\mathcal S, $$ with $\lambda^+$ as in (\[lambdapm\]). Note that the decomposition $\mathcal S\bigoplus\mathcal F$ corresponds to that in (\[ilphi\])-(\[ilz\]).
Note that a sufficient condition for $M^0>0$ is that $H'(\phi^0)\leq0$, that is $\phi^0\in \mathcal N^{-}$. So, the first inequality may still hold in the case that $H'(\phi^0)>0$, that is, for $\phi_0>\phi_{\textrm{\tiny{min}}}$ or $\phi_0<\phi_{\textrm{\tiny{max}}}$ in figure \[Hfig\]. This property is referred to as the reduced system having a [*can[à]{}rd*]{} structure [@Bold-canards2003; @Showalter-canards1991]. The case $M=0$ cannot be characterized in terms of linear stability.
\[convergence\] Let $(\phi^0, x^0, z^0)\in\mathcal I$ and suppose that the parameters of the system belong to the class $\mathcal P$. Then there exists $\epsilon_0>0$ such that for $\epsilon\in(0,\epsilon_0) $ the governing system has a unique $C^1$-solution $(\phi(t; \epsilon), x(t; \epsilon), z(t; \epsilon))\in\mathcal I$, for all $t>0$. Moreover $(\phi(\cdot), \lambda(\cdot), z(\cdot))\in \mathcal S\bigoplus\mathcal F $ and it admits the asymptotic expansions (\[ilphi\]), (\[ilx\]) and (\[ilz\]), with the property that $\mathcal E_0^k= O({{\epsilon}}^N), \,\, \mathcal E_I^k=O(e^{-M_0{{\bar t}}}), \,\, k=1,2,3,$ for sufficiently large ${{\bar t}}$.
The proof follows from linearization of the three-dimensional system with respect to solutions of the two dimensional one, subsequently transforming it into a system of integral equations, and applying Schauder’s fixed point theorem to it.
We consider the model obtained by further reducing the time scale, that is, setting the limiting problem in the [*slow* ]{} scale as $0= {\mathcal R}_1(\phi, \lambda), \,\,
\frac{d\bar x}{d\tau}={\mathcal R}_2(\phi,\bar x,\lambda), \,\,
0={\mathcal R}_3(\phi,\bar x, \bar z,\lambda),$ together with equations (\[electroneutrality-soln\]). This is consistent with taking $\frac{{\mathcal A}_4}{{\mathcal A}_1}=O({{\epsilon}}^{-3})$. This model yields that proposed by Siegel and Li [@li-siegel99], that sets a rely equation for the [*product*]{} of the reaction, in this case, the hydrogen ion concentration in the membrane.
\[oscillations-lingxing-april3\]
Concluding Remarks
==================
We have analyzed a [*lumped*]{} model for a chemomechanical oscillator suitable for rhythmic drug delivery. The model consists of a system of ordinary differential equations for the chemo-mechanical fields. For this system, we showed existence of periodic solutions, which correspond to experimentally and numerically observed oscillations. The tools of the analysis involve multiscale and dynamical systems methods, including the theory of competitive dynamical systems [@Hirsh1].
The membrane model, which ignores gradients of solute concentrations and swelling within the membrane, can be replaced by a distributed, PDE based system which, in addition to more accurately portraying the physical situation, can include self consistent, natural boundary conditions at the interfaces between the membrane and the two chambers. The results presented here will not be altered qualitatively, though there will be quantitative differences.
Overall, the model presented here dealt with the fundamental mechanisms underlying oscillatory behaviour of a table-top experimental device. It must be regarded as a first step, since complications associated with the buildup of gluconate ion in the system, which buffers and affects the dynamics of pH oscillations, have not been included. Also the effects of endogenous phosphate and bicarbonate buffering species would need to be included in a more comprehensive model, which would be of higher dimensionality, even in the lumped framework. These buffering effects are currently the main hurdle to develop an in-vivo device.
[^1]: M. Carme Calderer acknowledges the support of the National Science Foundation through the grants DMS-0909165 and DMS-GOALI-1009181. Ronald A. Siegel acknowledges support from the National Institutes of Health through grant HD040366.
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abstract: 'The effect of proportional transaction costs on systematically generated portfolios is studied empirically. The performance of several portfolios (the index tracking portfolio, the equally-weighted portfolio, the entropy-weighted portfolio, and the diversity-weighted portfolio) in the presence of dividends and transaction costs is examined under different configurations involving the trading frequency, constituent list size, and renewing frequency. Moreover, a method to smooth transaction costs is proposed.'
address:
- |
Johannes Ruf\
Department of Mathematics\
London School of Economics and Political Science
- |
Kangjianan Xie\
Department of Mathematics\
University College London
author:
- Johannes Ruf
- Kangjianan Xie
bibliography:
- 'TC.bib'
title: The Impact of Proportional Transaction Costs on Systematically Generated Portfolios
---
[^1]
Introduction
============
Although often neglected in portfolio analysis for sake of simplicity, transaction costs matter significantly for portfolio performance. Even small proportional transaction costs can have a large negative effect, especially when trades are made to rebalance the portfolio in a relatively high frequency. Hence, one should at least test the performance of a given portfolio when transaction costs are imposed, even if transaction costs are not explicitly taken into account while constructing the portfolio.
In this paper, we examine the effects of imposing transaction costs on systematically generated portfolios, for example, functionally generated portfolios. Such portfolios play a significant role in Stochastic Portfolio Theory; see [@MR1894767]. [@ruf2018generalised] and [@karatzas2018trading] demonstrate empirically that functionally generated portfolios outperform the market portfolio in the absence of transaction costs. To explore whether or to what extent this result still holds when transaction costs are imposed, we empirically examine the performance of portfolios under different configurations including trading frequency, transaction cost rate, constituent list size, and renewing frequency. For the diversity-weighted portfolio, we also propose a method to smooth transaction costs. [@wong2019information] indicates an alternative approach, namely to adjust the trading frequency based on certain information-theoretic quantities.
[@magill1976portfolio] are among the first to study the impact of proportional transaction costs in portfolio choice. We refer to [@MR3183924] and [@muhle2017primer] for an overview of the transaction cost literature that evolved afterwards. Most of this literature focuses on the case of one risky asset only. For a discussion of transaction costs in the presence of several risky assets, we refer to [@MR2380942], [@MR3032935], and [@MR3418824]. An empirical analysis of the effects of transaction costs is provided in [@stoll1983transaction], [@bajgrowicz2012technical], and [@olivares2018robust]. We follow up on this research by providing a systematic analysis of the impact of transaction costs on functionally generated portfolios.
The following is an outline of this paper. Section \[sec data\] proposes a framework of backtesting portfolio performance in the presence of transaction costs. In particular, Subsection \[subsec 2.1\] incorporates proportional transaction costs when rebalancing a portfolio and Subsection \[subsec 2.2\] provides some practical considerations and details when backtesting portfolio performance. Section \[sec E\] empirically examines the performance of several different portfolios under various configurations. A method to smooth transaction costs is also provided in Section \[sec E\]. Section \[sec C\] concludes.
Backtesting in the presence of transaction costs {#sec data}
================================================
Incorporating transaction costs into wealth dynamics {#subsec 2.1}
----------------------------------------------------
We shall study the performance of long-only stock portfolios that are rebalanced discretely. The market is not assumed to be frictionless; transaction costs are imposed when we trade in the market to rebalance the portfolios. The portfolios are constructed in such a way that their weights match given target weights after paying transaction costs. This construction is more rigid than the one in [@garleanu2013dynamic], for example, where the portfolio weights may deviate from the target weights.
To be more specific, consider a market with $d\geq2$ stocks. Denote the amount of currency invested in each stock by $\psi(\cdot)=(\psi_{1}(\cdot),\cdots,\psi_{d}(\cdot))'$ and the total amount invested in a portfolio by $$V(\cdot)=\sum_{i=1}^{d}\psi_{i}(\cdot)\geq0.$$ Furthermore, denote the portfolio weights by $\pi(\cdot)=(\pi_{1}(\cdot),\cdots,\pi_{d}(\cdot))'$. Note that $\psi_{i}(\cdot)=\pi_{i}(\cdot)V(\cdot)$, for all $i\in\{1,\cdots,d\}$.
Assume that trading stocks involves proportional transaction costs at a time-invariant rate ${\mathrm{tc}}^{\rm{b}}$ (${\mathrm{tc}}^{\rm{s}}$), with $0\leq{\mathrm{tc}}^{\rm{b}},{\mathrm{tc}}^{\rm{s}}<1$ for buying (selling) a stock. This means that the sale of one unit of currency of a stock nets only $\left(1-{\mathrm{tc}}^{\rm{s}}\right)$ units of currency in cash, while buying one unit of currency of a stock costs $\left(1+{\mathrm{tc}}^{\rm{b}}\right)$ units of currency.
Let us now consider how to trade the stocks in order to match the target weights when transaction costs are imposed. To begin, let us focus on trading at a specific time $t$. When rebalancing the portfolio at time $t$, we know the wealth $\psi(t-)$ invested in each stock and hence the total wealth of the portfolio $V(t-)=\sum_{i=1}^{d}\psi_{i}(t-)$ (exclusive of dividends). We also know the dividends paid at time $t-$, their total denoted by $D(t-)\geq0$.
Given target weights $\pi$, we require $\pi(t)=\pi$ after the portfolio is rebalanced at time $t$. After trading, the wealth $\psi(t)$ invested in each stock in the portfolio satisfies $$\label{eq psi}
\psi_{j}(t)=\pi_{j}(t)\sum_{i=1}^{d}\psi_{i}(t),\quad j\in\{1,\cdots,d\}.$$ We provide details about how to compute $\psi(t)$ later in this subsection.
As the portfolio needs to be self-financing, the amount of currency used to buy extra stocks should be exactly the amount of currency obtained from selling redundant stocks plus the dividends if there are any. This yields $$\label{eq Phi1}
\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}\left(\psi_{i}(t)-\psi_{i}(t-)\right)^{+}=\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\left(\psi_{i}(t-)-\psi_{i}(t)\right)^{+}+D(t-).$$ The total transaction costs imposed from trading stocks at time $t$ are computed by $$\label{eq TC}
{\mathrm{TC}}(t)={\mathrm{tc}}^{\rm{b}}\sum_{i=1}^{d}\left(\psi_{i}(t)-\psi_{i}(t-)\right)^{+}+{\mathrm{tc}}^{\rm{s}}\sum_{i=1}^{d}\left(\psi_{i}(t-)-\psi_{i}(t)\right)^{+}.$$ Therefore, the total wealth of the portfolio at time $t$, given by $V(t)=\sum_{i=1}^{d}\psi_{i}(t)$, satisfies $$V(t)=V(t-)+D(t-)-{\mathrm{TC}}(t).$$
### Method of computing $\psi(t)$ {#method-of-computing-psit .unnumbered}
In the following, we propose a method to compute $\psi(t)$, given $\psi(t-)$, $D(t-)$, and the target weights $\pi$. Throughout this section, we assume $$V(t-)>0,\quad D(t-)\geq0,\quad\sum_{i=1}^{d}\pi_{i}=1,\quad\pi_{j}\geq0,\quad\text{and}\quad\psi_{j}(t-)\geq0,$$ for all $j\in\{1,\cdots,d\}$.
To begin with, implies that $\psi(t)$ is of the form $$\label{eq Phi}
\psi_{j}(t)=cV(t-)\pi_{j}(t),\quad j\in\{1,\cdots,d\},$$ for some $c>0$. Note that if the market is frictionless, i.e., if ${\mathrm{tc}}^{\rm{b}}={\mathrm{tc}}^{\rm{s}}=0$, and if there are no dividends paid at time $t-$, i.e., if $D(t-)=0$, then $V(t)=V(t-)$ and $c=1$. When transaction costs are imposed, we shall use the constraint to determine $c$.
To make headway, define $$\label{eq Dt}
\widehat{D}=\frac{D(t-)+\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\psi_{i}(t-)\mathbf{1}_{\pi_{i}(t)=0}}{V(t-)}$$ and $$c_{j}=\frac{\pi_{j}(t-)}{\pi_{j}(t)}\mathbf{1}_{\pi_{j}(t)>0},\quad j\in\{1,\cdots,d\}.$$ Then dividing both sides of by $V(t-)$ yields $$\label{eq Psi1}
\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}\left(c-c_{i}\right)^{+}\pi_{i}(t)=\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\left(c_{i}-c\right)^{+}\pi_{i}(t)+\widehat{D}.$$
Note that the LHS of is a continuous function of $c$ and strictly increasing from 0 to $\infty$, as $c$ changes from $\min_{i\in\{1,\cdots,d\}}c_{i}$ to $\infty$. Moreover, the RHS of is a continuous function of $c$ strictly decreasing from $\infty$ to $\widehat{D}\geq0$, as $c$ changes from $-\infty$ to $\max_{i\in\{1,\cdots,d\}}c_{i}$, and equals $\widehat{D}$ afterwards, as $c$ changes from $\max_{i\in\{1,\cdots,d\}}c_{i}$ to $\infty$. Hence, both sides of as functions of $c$ must intersect at some unique point, i.e., a unique solution exists for . To proceed, define $$\label{eq D}
\widehat{D}_{j}=\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}\left(c_{j}-c_{i}\right)^{+}\pi_{i}(t)-\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\left(c_{i}-c_{j}\right)^{+}\pi_{i}(t),\quad j\in\{1,\cdots,d\}.$$ We are now ready to provide an expression for the unknown constant $c$.
\[lemma 2\] Recall that and imply $\widehat{D}\geq0$ and $\min_{i\in\{1,\cdots,d\}}\widehat{D}_{i}\leq0$. Hence, $$\label{eq j}
j=\underset{i\in\{1,\cdots,d\}}{\arg\max}\left\{\widehat{D}_{i};\widehat{D}_{i}\leq\widehat{D}\right\}$$ is well-defined. Then $$\label{eq c}
c=\frac{\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}c_{i}\pi_{i}(t)\mathbf{1}_{c_{i}\leq c_{j}}+\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\pi_{i}(t-)\mathbf{1}_{c_{i}>c_{j}}+\widehat{D}}{\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}\leq c_{j}}+\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}>c_{j}}}$$ solves uniquely.
By the definition of $\widehat{D}_{j}$ given in and by some basic computations, is equivalent to $$c=c_{j}+\frac{\widehat{D}-\widehat{D}_{j}}{\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}\leq c_{j}}+\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}>c_{j}}},$$ which implies $\mathbf{1}_{c_{i}\leq c}\geq\mathbf{1}_{c_{i}\leq c_{j}}$, for all $i\in\{1,\cdots,d\}$.
In the case $\max_{i\in\{1,\cdots,d\}}\widehat{D}_{i}\leq\widehat{D}$, we have $\mathbf{1}_{c_{i}\leq c_{j}}=1$, hence $\mathbf{1}_{c_{i}\leq c}\leq\mathbf{1}_{c_{i}\leq c_{j}}$, for all $i\in\{1,\cdots,d\}$. In the case $\max_{i\in\{1,\cdots,d\}}\widehat{D}_{i}>\widehat{D}$, define $$j'=\underset{i\in\{1,\cdots,d\}}{\arg\min}\left\{\widehat{D}_{i};\widehat{D}_{i}>\widehat{D}\right\}.$$ Then is equivalent to $$\begin{aligned}
c&=\frac{\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}c_{i}\pi_{i}(t)\mathbf{1}_{c_{i}<c_{j'}}+\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\pi_{i}(t-)\mathbf{1}_{c_{i}\geq c_{j'}}+\widehat{D}}{\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}<c_{j'}}+\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}\geq c_{j'}}}\\
&=c_{j'}+\frac{\widehat{D}-\widehat{D}_{j'}}{\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}<c_{j'}}+\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}\geq c_{j'}}},
\end{aligned}$$ which implies $\mathbf{1}_{c_{i}>c}\geq\mathbf{1}_{c_{i}>c_{j}}$, for all $i\in\{1,\cdots,d\}$. All in all, we have shown $\mathbf{1}_{c_{i}\leq c}=\mathbf{1}_{c_{i}\leq c_{j}}$, for all $i\in\{1,\cdots,d\}$.
Define next $$\begin{gathered}
\Pi^{\rm{b}}=\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}\leq c_{j}},\quad\Pi^{\rm{s}}=\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\pi_{i}(t)\mathbf{1}_{c_{i}>c_{j}},\\
\overline{\Pi}^{\rm{b}}=\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}c_{i}\pi_{i}(t)\mathbf{1}_{c_{i}\leq c_{j}},\quad\overline{\Pi}^{\rm{s}}=\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\pi_{i}(t-)\mathbf{1}_{c_{i}>c_{j}}.\end{gathered}$$ Hence, after inserting $c$ by into , the LHS of becomes $$\mathrm{LHS}=c\Pi^{\rm{b}}-\overline{\Pi}^{\rm{b}}=\frac{\Pi^{\rm{b}}\overline{\Pi}^{\rm{s}}-\Pi^{\rm{s}}\overline{\Pi}^{\rm{b}}+\Pi^{\rm{b}}\widehat{D}}{\Pi^{\rm{b}}+\Pi^{\rm{s}}},$$ and the RHS of becomes $$\mathrm{RHS}=\overline{\Pi}^{\rm{s}}-c\Pi^{\rm{s}}+\widehat{D}=\frac{\Pi^{\rm{b}}\overline{\Pi}^{\rm{s}}-\Pi^{\rm{s}}\overline{\Pi}^{\rm{b}}-\Pi^{\rm{s}}\widehat{D}}{\Pi^{\rm{b}}+\Pi^{\rm{s}}}+\widehat{D}=\mathrm{LHS}.$$ Therefore, $c$ defined by indeed solves .
In practice, we can apply both numerical and analytical methods to find the constant $c$. As suggested by , to find $c$ numerically, we can simply search for the minimum of the function $$c\mapsto\left|\left(1+{\mathrm{tc}}^{\rm{b}}\right)\sum_{i=1}^{d}\left(c-c_{i}\right)^{+}\pi_{i}(t)-\left(1-{\mathrm{tc}}^{\rm{s}}\right)\sum_{i=1}^{d}\left(c_{i}-c\right)^{+}\pi_{i}(t)-\widehat{D}\right|.$$ Alternatively, by determining the index $j$ given by , we can apply Proposition \[lemma 2\] to compute $c$ analytically.
If the analytical approach is implemented, we can speed up the algorithm by making the following observations. We expect the value of $c$ not to be far away from $1$, which is precisely the value in the case of no transaction costs and no dividends. As suggested by the proof of Proposition \[lemma 2\], the family $(\widehat{D}_{i})_{i\in\{1,\cdots,d\}}$ has the same ranking as $(c_{i})_{i\in\{1,\cdots,d\}}$. Therefore, we proceed by ranking all $c_i$’s in ascending order and comparing $\widehat{D}_{k}$ with $\widehat{D}$, where $$k=\underset{i\in\{1,\cdots,d\}}{\arg\max}\left\{c_{i};c_{i}\leq1\right\}.$$ If $\widehat{D}_{k}=\widehat{D}$, then $j=k$ and we are done. If $\widehat{D}_{k}>\widehat{D}$, then we repeatedly compute $\widehat{D}_i$ corresponding to a smaller $c_{i}<c_{k}$ each time until we find the exact index $j$. If $\widehat{D}_{k}<\widehat{D}$, then we simply go the other way around.
Proposition \[lemma 2\] is applied to determine the constant $c$ used in in order to compute $\psi(t)$. Note that, in this subsection, we take $\psi(t-)$ and $D(t-)$ as given. In the next subsection, we discuss how to compute $\psi(t-)$ and $D(t-)$ from the data.
Practical considerations {#subsec 2.2}
------------------------
For the preparation of the empirical study in the next section, we now introduce the method used to backtest the portfolio performance.
To begin with, assume that we are given the total market capitalizations and the daily returns for all stocks; denote these processes by $S(\cdot)=(S_{1}(\cdot),\cdots,S_{d}(\cdot))'$ and $r(\cdot)=(r_{1}(\cdot),\cdots,r_{d}(\cdot))'$, respectively. Assume that there are in total $N$ days. For all $l\in\{1,\cdots,N\}$, let $t_{l}$ denote the end of day $l$, at which the end of day total market capitalizations and the daily returns for day $l$ are available. Moreover, if we trade on day $l$, then we call day $l$ a trading day and the trade is made at time $t_{l}$.
Now focus on a specific trading day $l$ with $l\in\{1,\cdots,N\}$ and fix $i\in\{1,\cdots,d\}$ for the moment. In Subsection \[subsec 2.1\], given $\psi(t_{l}-)$ and $D(t_{l}-)$, as well as the target weights specified by the corresponding portfolio at time $t_{l}$, we have shown how to compute $\psi(t_{l})$. In the following, we show how to obtain $\psi(t_{l}-)$ and $D(t_{l}-)$.
The daily return $r_{i}(t_{l})$ includes the dividends of stock $i$ if there are any. We decompose the daily return $r_{i}(t_{l})$ into two parts: the dividend rate $r^{D}_{i}(t_{l})$ and the realised rate $r^{R}_{i}(t_{l})$. The dividend rate $r^{D}_{i}(t_{l})$ is computed as $$\label{eq rD}
r^{D}_{i}(t_{l})=\max\left\{1+r_{i}(t_{l})-\frac{S_{i}(t_{l})}{S_{i}(t_{l-1})},0\right\}$$ and yields the amount of dividends received at time $t_{l}$ for each unit of currency invested in stock $i$ at time $t_{l-1}$[^2]. The realised rate $r^{R}_{i}(t_{l})$ is computed as $$r^{R}_{i}(t_{l})=r_{i}(t_{l})-r^{D}_{i}(t_{l})$$ and yields the units of currency held in stock $i$ at time $t_{l}$ for each unit of currency invested in stock $i$ at time $t_{l-1}$.
The maximum is used in to make sure that the dividend rate is nonnegative. Indeed, occasionally the data may suggest $S_{i}(t_{l-1})(1+r_{i}(t_{l}))<S_{i}(t_{l})$. This can happen, for example, when company $i$ issues extra stocks at time $t_{l}$. In this case, we simply assume that there are no dividends paid at time $t_{l}$.
A special situation requires us to pay extra attention. A few times, some stock $i$ is delisted from the market at time $t_{l}$, for example, due to bankruptcy or merger. In this case, we still have data for $r_{i}(t_{l})$, but not for $S_{i}(t_{l})$. To deal with this situation, we assume that there are no dividends paid in stock $i$ at time $t_{l}$. As a result, we have $r^{D}_{i}(t_{l})=0$ and $r^{R}_{i}(t_{l})=r_{i}(t_{l})$ for such stock $i$. To close the position in stock $i$, we assume that one needs to pay transaction costs.
Without loss of generality, assume that there are $n\geq1$ days (including the trading day $l$) involved since the last trading day, i.e., the last trading day before $l$ is $l-n$. For all $k\in\{l-n+1,\cdots,l\}$, we compute $r^{D}(t_{k})$ and $r^{R}(t_{k})$ as above. In particular, if some stock $i$ in the portfolio is delisted from the market at time $t_{u}$, for some $u\in\{l-n+1,\cdots,l-1\}$, then we set $r^{R}_{i}(t_{v})=r^{D}_{i}(t_{v})=0$, for all $v\in\{u+1,\cdots,l\}$.
Then given $\psi(t_{l-n})$, we compute $$\psi_{i}(t_{l}-)=\psi_{i}(t_{l-n})\prod_{k=l-n+1}^{l}\left(1+r^{R}_{i}(t_{k})\right),\quad i\in\{1,\cdots,d\}.$$ Since all dividends paid between two consecutive trading days are only reinvested at time $t_{l}$, the total dividends available for reinvesting are computed by $$D(t_{l}-)=\sum_{i=1}^{d}\psi_{i}(t_{l-n})\sum_{k=l-n+1}^{l}r^{D}_{i}(t_{k})\prod_{u=l-n+1}^{k-1}\left(1+r^{R}_{i}(t_{u})\right).$$
Examples and empirical results {#sec E}
==============================
In this section, we analyze the performance of several portfolios empirically. The target weights are expressed in terms of the market weights $\mu(\cdot)=\big(\mu_{1}(\cdot),\cdots,\mu_{d}(\cdot)\big)'$ with components $$\mu_{j}(\cdot)=\frac{S_{j}(\cdot)}{\sum_{i=1}^{d}S_{i}(\cdot)},\quad j\in\{1,\cdots,d\}.$$ In Subsection \[sec div\], we also propose a method to smooth transaction costs.
We shall consider the largest $d$ stocks. We will vary the number $d$ between 100, 300, and 500. The constituent list (the list of the top $d$ stocks) is renewed either weekly, monthly, or quarterly. Whenever we renew the constituent list, we keep the $d$ stocks with the largest total market capitalizations at that time. We trade only these $d$ stocks afterwards until we renew the constituent list again. If any of these stocks stops to exist in the market due to any reason, we simply invest in the remaining stocks without adding a new stock to the list before we renew it next time. Note that renewing the constituent list implies trading to replace the old top $d$ stocks with the new top $d$ stocks. We trade with a specific frequency, which can be either daily, weekly, or monthly. For research on optimal trading frequency, we refer to [@MR3766056].
At time $t_{0}$, we take the transaction costs due to initializing a portfolio as sunk cost, i.e., we set ${\mathrm{TC}}(t_0)=0$. Moreover, we start a portfolio with initial wealth $V(t_0)=1000$. Note that unless otherwise mentioned, the logarithmic scale is used when plotting $V(\cdot)$ and ${\mathrm{TC}}(\cdot)$ for the purpose of better interpretability. To simplify the analysis, we impose a uniform transaction cost rate ${\mathrm{tc}}$ on both buying and selling the stocks, i.e., we set ${\mathrm{tc}}^{\rm{b}}={\mathrm{tc}}^{\rm{s}}={\mathrm{tc}}$.
For each example, we provide tables with the yearly returns, the excess returns (relative to the corresponding index tracking portfolio), the standard deviations of the yearly returns, the Sharpe ratios[^3], and the wealth and the cumulative transaction costs at the end of the investment period of the portfolios.
Data source {#data-source .unnumbered}
-----------
The data of the total market capitalizations $S(\cdot)$ and the daily returns $r(\cdot)$ is downloaded from the CRSP US Stock Database[^4]. This database contains the traded stocks on all major US exchanges. More precisely, we focus on ordinary common stocks[^5]. The data starts January 2$^{\rm{nd}}$, 1962 and ends December 30$^{\rm{th}}$, 2016.
The total market capitalizations are computed by multiplying the numbers of outstanding shares with the share prices, and are essential in determining the target weights. The daily returns include dividends but also delisting returns in case stocks get delisted (for example, the recovery rate in case a traded firm goes bankrupt).
Index tracking portfolio {#sec I}
------------------------
In this subsection, we introduce the index tracking portfolio. This portfolio is used to benchmark the performance of other portfolios studied in the following subsections. The index tracking portfolio has target weights $$\pi_{j}(\cdot)=\mu_{j}(\cdot),\quad j\in\{1,\cdots,d\}.$$ Note that this portfolio is rebalanced only when the constituent list changes or when dividends are reinvested.
The index tracking portfolio includes the effects of paying transaction costs and reinvesting dividends. In contrast, the capitalization index with wealth process $$\sum_{i=1}^{d}S_{i}(\cdot)\times\frac{1000}{\sum_{i=1}^{d}S_{i}(t_0)}$$ does not take transaction costs and dividends into consideration.
In the following, we examine the performance of the index tracking portfolio under different trading frequencies, renewing frequencies, as well as constituent list sizes $d$, when there are no transaction costs, i.e., when ${\mathrm{tc}}=0$, and when ${\mathrm{tc}}=0.5\%$ and ${\mathrm{tc}}=1\%$, respectively. These numbers are consistent with the transaction cost estimates in [@stoll1983transaction], [@keim1997transactions], and [@fong2017best].
### Varying the trading frequency {#varying-the-trading-frequency .unnumbered}
We fix the constituent list size $d=100$ and use monthly renewing frequency. Table \[tab 8\] shows the performance of the index tracking portfolio and the corresponding capitalization index under daily, weekly, and monthly trading frequencies, respectively. Note that the capitalization index does not depend on the trading frequency. As expected, with the same trading frequency, the portfolio performs worse under a larger transaction cost rate ${\mathrm{tc}}$. In addition, the portfolio outperforms the corresponding index, which implies that the dividends paid exceed the transaction costs imposed even if ${\mathrm{tc}}=1\%$. In Figure \[fg CI\_1\], the wealth processes of the daily traded index tracking portfolio and the corresponding capitalization index are plotted.
CI IT$^{d}_{0}$ IT$^{d}_{0.5}$ IT$^{d}_{1}$ IT$^{w}_{0}$ IT$^{w}_{0.5}$ IT$^{w}_{1}$ IT$^{m}_{0}$ IT$^{m}_{0.5}$ IT$^{m}_{1}$
--------------- ------- -------------- ---------------- -------------- -------------- ---------------- -------------- -------------- ---------------- --------------
Yearly return 8.84 10.30 10.09 9.89 10.30 10.10 9.90 10.27 10.08 9.89
Std 16.59 16.87 16.84 16.81 16.88 16.85 16.82 16.88 16.86 16.83
Sharpe ratio 0.22 0.30 0.29 0.28 0.30 0.29 0.28 0.30 0.29 0.28
Wealth 54.5 111.7 100.7 90.7 111.3 100.7 91.0 109.7 99.7 90.6
TC 2.7 5.0 2.5 4.7 2.3 4.3
: Yearly returns in percentage, standard deviations of yearly returns (Std), Sharpe ratios, and the wealth and the cumulative transaction costs (TC) in thousands at the end of the investment period of the index tracking portfolio (IT) and the corresponding capitalization index (CI) under different trading frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and monthly renewing frequency. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $d$, $w$, and $m$ indicate daily, weekly, and monthly trading frequencies, respectively.[]{data-label="tab 8"}
![The wealth processes of the index tracking portfolio (IT) and the corresponding capitalization index (CI) on logarithmic scale under different transaction cost rates ${\mathrm{tc}}$ with $d=100$, daily trading frequency, and monthly renewing frequency. The weekly and the monthly traded portfolio performs similarly to the daily traded portfolio under the same transaction cost rate ${\mathrm{tc}}$.[]{data-label="fg CI_1"}](CI_1.png){width="\textwidth"}
### Varying the renewing frequency {#varying-the-renewing-frequency .unnumbered}
Still fixing the constituent list size $d=100$, we now use daily trading frequency and vary the renewing frequency between weekly, monthly, and quarterly frequencies, respectively. As shown in Figure \[fg CI\_2\] and Table \[tab 9\], under the same transaction cost rate ${\mathrm{tc}}$, the less frequently the constituent list is renewed, the better the portfolio performs. As trades are made when we renew the constituent list, renewing more frequently will impose larger transaction costs, which impacts the performance of the portfolio to a higher degree. Additionally, the more frequently the constituent list is renewed, the more sensitive the portfolio is to a larger transaction cost rate ${\mathrm{tc}}$.
![The wealth processes of the index tracking portfolio (IT) and the corresponding capitalization index (CI) on logarithmic scale under different renewing frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and daily trading frequency. The performance of the weekly renewed portfolio when ${\mathrm{tc}}=0$ is similar to that of the quarterly renewed portfolio when ${\mathrm{tc}}=1\%$. The weekly and the quarterly renewed capitalisation index is not very different from the monthly renewed one.[]{data-label="fg CI_2"}](CI_2.png){width="\textwidth"}
CI$^{W}$ IT$^{W}_{0}$ IT$^{W}_{0.5}$ IT$^{W}_{1}$ CI$^{Q}$ IT$^{Q}_{0}$ IT$^{Q}_{0.5}$ IT$^{Q}_{1}$
--------------- ---------- -------------- ---------------- -------------- ---------- -------------- ---------------- --------------
Yearly return 8.85 10.14 9.73 9.33 8.82 10.34 10.20 10.06
Std 16.66 16.89 16.84 16.79 16.44 16.83 16.81 16.79
Sharpe ratio 0.22 0.29 0.27 0.25 0.22 0.31 0.30 0.29
Wealth 54.5 102.2 83.5 68.1 54.4 114.2 106.5 99.2
TC 4.2 7.2 2.0 3.8
: Yearly returns in percentage, standard deviations of yearly returns (Std), Sharpe ratios, and the wealth and the cumulative transaction costs (TC) in thousands at the end of the investment period of the index tracking portfolio (IT) and the corresponding capitalization index (CI) under different renewing frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and daily trading frequency. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $W$ and $Q$ indicate weekly and quarterly renewing frequencies, respectively.[]{data-label="tab 9"}
### Varying the constituent list size $d$ {#varying-the-constituent-list-size-d .unnumbered}
With daily trading and monthly renewing frequencies, we now backtest the performance of the index tracking portfolio under different constituent list sizes $d$. As shown in Figure \[fg CI\_3\] and Table \[tab 10\], the portfolio outperforms the corresponding index even with transaction cost rate ${\mathrm{tc}}=1\%$. The more stocks the constituent list contains, the better the portfolio performs.
![The wealth processes of the index tracking portfolio (IT) and the corresponding capitalization index (CI) on logarithmic scale under different constituent list sizes $d$ and transaction cost rates ${\mathrm{tc}}$ with daily trading and monthly renewing frequencies. For both the portfolio and the index, the wealth processes with $d=300$ are omitted. Everything else equal, they would lie between the plotted ones with $d=100$ and with $d=500$.[]{data-label="fg CI_3"}](CI_3.png){width="\textwidth"}
CI$^{300}$ IT$^{300}_{0}$ IT$^{300}_{0.5}$ IT$^{300}_{1}$ CI$^{500}$ IT$^{500}_{0}$ IT$^{500}_{0.5}$ IT$^{500}_{1}$
--------------- ------------ ---------------- ------------------ ---------------- ------------ ---------------- ------------------ ----------------
Yearly return 8.94 10.61 10.46 10.31 9.01 10.83 10.71 10.59
Std 16.14 16.57 16.55 16.53 16.15 16.61 16.60 16.58
Sharpe ratio 0.23 0.33 0.32 0.31 0.24 0.34 0.33 0.33
Wealth 58.7 132.5 123.1 114.3 60.6 147.2 139.0 131.1
TC 2.4 4.5 2.3 4.3
: Yearly returns in percentage, standard deviations of yearly returns (Std), Sharpe ratios, and the wealth and the cumulative transaction costs (TC) in thousands at the end of the investment period of the index tracking portfolio (IT) and the corresponding capitalization index (CI) under different constituent list sizes $d$ and transaction cost rates ${\mathrm{tc}}$ with daily trading and monthly renewing frequencies. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $300$ and $500$ indicate $d=300$ and $d=500$, respectively.[]{data-label="tab 10"}
Equally-weighted portfolio
--------------------------
This subsection examines the equally-weighted portfolio (see [@benartzi2001naive] and [@windcliff20041] for a discussion of this portfolio in the context of defined contribution plans, and [@demiguel2007optimal] for a deep study of its properties). Here, the target weights are given by $$\pi_{j}(\cdot)=\frac{1}{d},\quad j\in\{1,\cdots,d\}.$$
For each portfolio with a specific trading frequency, a specific renewing frequency, and a specific constituent list size $d$, we examine its performance when there are no transaction costs, i.e., when ${\mathrm{tc}}=0$, and when ${\mathrm{tc}}=0.5\%$ and ${\mathrm{tc}}=1\%$, respectively. As shown in the following, the equally-weighted portfolio outperforms the corresponding index tracking portfolio when there are no transaction costs. This well-behaved performance of the equally-weighted portfolio within a frictionless market is popular in the academic literature. However, the equally-weighted portfolio is very sensitive to transaction costs. Its performance is strongly compromised even with a small transaction cost rate ${\mathrm{tc}}=0.5\%$.
### Varying the trading frequency {#varying-the-trading-frequency-1 .unnumbered}
Let us fix $d=100$ and apply monthly renewing frequency. Figure \[fg EW\_1\] plots and Table \[tab 6\] summarises the wealth processes of the equally-weighted and the corresponding index tracking portfolio under different trading frequencies and transaction cost rates ${\mathrm{tc}}$. When there are no transaction costs, i.e., when ${\mathrm{tc}}=0$, the equally-weighted portfolio outperforms the corresponding index tracking portfolio under all three different trading frequencies. A similar observation is also provided in [@banner2018diversification]. In addition, the more frequently the portfolio is traded, the better it performs. Trading more frequently also allows to reinvest the dividends faster, which helps to enhance the portfolio performance.
![The wealth processes of the equally-weighted portfolio (EW) and the corresponding index tracking portfolio (IT) on logarithmic scale under different trading frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and monthly renewing frequency. Under the same transaction cost rate ${\mathrm{tc}}$, the weekly and the monthly traded index tracking portfolio performs similarly to the one traded daily.[]{data-label="fg EW_1"}](EW_1.png){width="\textwidth"}
EW$^{d}_{0}$ EW$^{d}_{0.5}$ EW$^{d}_{1}$ EW$^{w}_{0}$ EW$^{w}_{0.5}$ EW$^{w}_{1}$ EW$^{m}_{0}$ EW$^{m}_{0.5}$ EW$^{m}_{1}$
--------------- -------------- ---------------- -------------- -------------- ---------------- -------------- -------------- ---------------- --------------
Yearly return 11.10 9.19 7.31 10.94 9.82 8.72 10.53 9.81 9.10
Excess return 0.80 -0.9 -2.58 0.64 -0.28 -1.18 0.26 -0.27 -0.79
Std 16.83 16.65 16.48 16.93 16.81 16.69 17.00 16.91 16.83
Sharpe ratio 0.35 0.24 0.13 0.34 0.28 0.21 0.31 0.27 0.23
Wealth 168.2 64.0 24.3 153.8 87.7 50.0 123.7 86.1 59.9
TC 15.5 15.0 11.4 14.8 7.2 10.9
: Yearly returns and excess returns (with respect to the index tracking portfolio shown in Table \[tab 8\]) in percentage, standard deviations of yearly returns (Std), Sharpe ratios, and the wealth and the cumulative transaction costs (TC) in thousands at the end of the investment period of the equally-weighted portfolio (EW) under different trading frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and monthly renewing frequency. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $d$, $w$, and $m$ indicate daily, weekly, and monthly trading frequencies, respectively.[]{data-label="tab 6"}
When transaction costs are imposed, Figure \[fg EW\_1\] and Table \[tab 6\] suggest that under the same transaction cost rate ${\mathrm{tc}}$, the more frequently the portfolio is traded, the larger the decrease in portfolio performance is. The performance of the equally-weighted portfolio is strongly affected by transaction costs. Even with ${\mathrm{tc}}=0.5\%$, the corresponding index tracking portfolio outperforms the equally-weighted portfolio. However, slowing down trading helps to reduce the influence of transaction costs. Indeed, the performance of the monthly traded equally-weighted portfolio when ${\mathrm{tc}}=1\%$ is similar to that of the daily traded one when ${\mathrm{tc}}=0.5\%$. As shown in Figure \[fg EW\_2\], the cumulative transaction costs paid from a monthly traded equally-weighted portfolio when ${\mathrm{tc}}=1\%$ are smaller than that from a daily traded one when ${\mathrm{tc}}=0.5\%$.
![Cumulative transaction costs on logarithmic scale of the equally-weighted portfolio (EW) under different trading frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and monthly renewing frequency.[]{data-label="fg EW_2"}](EW_2.png){width="\textwidth"}
We now study the sensitivity of the Sharpe ratio with respect to the transaction cost rate ${\mathrm{tc}}$. Specifically, we compute the Sharpe ratios of the monthly traded equally-weighted and index tracking portfolio for ${\mathrm{tc}}\in\{0,0.01\%,0.02\%,\cdots,0.5\%\}$. As plotted in Figure \[fg EW\_3\], the Sharpe ratios of both the equally-weighted and the index tracking portfolio decrease as ${\mathrm{tc}}$ becomes larger. On the left hand side of the intersection when ${\mathrm{tc}}<0.22\%$, the equally-weighted portfolio has a higher Sharpe ratio. On the right hand side of the intersection when ${\mathrm{tc}}>0.22\%$, the inverse situation holds. This indicates that the equally-weighted portfolio depends more on transaction costs than the index tracking portfolio.
![Sharpe ratios of the equally-weighted portfolio (EW) and the index tracking portfolio (IT) under different transaction cost rates ${\mathrm{tc}}$ with $d=100$, monthly trading frequency, and monthly renewing frequency.[]{data-label="fg EW_3"}](EW_3.png){width="\textwidth"}
### Varying the renewing frequency {#varying-the-renewing-frequency-1 .unnumbered}
Now we examine the performance of the equally-weighted portfolio with $d=100$, daily trading frequency, and under weekly, monthly, and quarterly renewing frequencies, respectively. As shown in Figure \[fg EW\_4\] and Table \[tab 11\], under the same transaction cost rate ${\mathrm{tc}}$, the less frequently the constituent list is renewed, the better the portfolio performs. With ${\mathrm{tc}}=0.5\%$, the equally-weighted portfolio already performs worse than the corresponding index tracking portfolio. In particular, the portfolio with a more frequent renewing frequency is more sensitive to transaction costs. As studied in more detail in Subsection \[sec div\], the reason behind these observations is that trading on renewing days incurs extremely large transaction costs compared with trading on other days when the constituent list is not renewed. These large transaction costs paid on renewing days strongly impact the portfolio performance.
![The wealth processes of the equally-weighted portfolio (EW) and the corresponding index tracking portfolio (IT) on logarithmic scale under different renewing frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and daily trading frequency. For the index tracking portfolio, the wealth processes of the quarterly renewed one with ${\mathrm{tc}}=0$ and the weekly renewed one with ${\mathrm{tc}}=1\%$ are plotted. The omitted wealth processes of the index tracking portfolio lie between the plotted ones.[]{data-label="fg EW_4"}](EW_4.png){width="\textwidth"}
EW$^{W}_{0}$ EW$^{W}_{0.5}$ EW$^{W}_{1}$ EW$^{Q}_{0}$ EW$^{Q}_{0.5}$ EW$^{Q}_{1}$
--------------- -------------- ---------------- -------------- -------------- ---------------- --------------
Yearly return 10.62 8.20 5.83 11.21 9.47 7.76
Excess return 0.48 -1.53 -3.50 0.87 -0.73 -2.30
Std 16.95 16.71 16.50 16.82 16.65 16.50
Sharpe ratio 0.32 0.18 0.04 0.36 0.26 0.16
Wealth 129.8 37.8 11.0 177.6 73.9 30.8
TC 12.9 10.5 16.0 16.5
: Yearly returns and excess returns (with respect to the index tracking portfolio shown in Table \[tab 9\]) in percentage, standard deviations of yearly returns (Std), Sharpe ratios, and the wealth and the cumulative transaction costs (TC) in thousands at the end of the investment period of the equally-weighted portfolio (EW) under different renewing frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and daily trading frequency. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $W$ and $Q$ indicate weekly and quarterly renewing frequencies, respectively.[]{data-label="tab 11"}
The cumulative transaction costs of the equally-weighted portfolio of Table \[tab 11\] are shown in Figure \[fg EW\_5\]. Earlier on, the cumulative transaction costs are higher when weekly renewed than when monthly or quarterly renewed due to the large transaction costs associated with the renewal days. However, later on, the cumulative transaction costs of the weekly renewed portfolio are smaller. The reason is that the weekly renewed portfolio performs worse than the monthly or the quarterly renewed portfolio, hence the transaction costs imposed as a proportion of the portfolio wealth are also smaller.
![Cumulative transaction costs on logarithmic scale of the equally-weighted portfolio (EW) under different renewing frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and daily trading frequency.[]{data-label="fg EW_5"}](EW_5.png){width="\textwidth"}
### Varying the market size $d$ {#varying-the-market-size-d .unnumbered}
With daily trading and monthly renewing frequencies, Figure \[fg EW\_6\] plots and Table \[tab 7\] summarises the wealth processes of the equally-weighted and the corresponding index tracking portfolio under different constituent list sizes $d$. The more stocks the constituent list contains, the better the portfolio performs under the same transaction cost rate ${\mathrm{tc}}$. Again, its performance is reduced by transaction costs. Even with $d=500$ and ${\mathrm{tc}}=0.5\%$, the equally-weighted portfolio performs worse than the corresponding index tracking portfolio. In addition, the portfolio with a larger constituent list size $d$ is not necessarily more sensitive to transaction costs. Figure \[fg EW\_7\] plots the cumulative transaction costs generated by the portfolio of Table \[tab 7\].
![The wealth processes of the equally-weighted portfolio (EW) and the corresponding index tracking portfolio (IT) on logarithmic scale under different constituent list sizes $d$ and transaction cost rates ${\mathrm{tc}}$ with daily trading and monthly renewing frequencies. For the index tracking portfolio, the wealth processes of the one with $d=500$ when ${\mathrm{tc}}=0$ and the one with $d=100$ when ${\mathrm{tc}}=1\%$ are plotted. The omitted wealth processes of the index tracking portfolio lie between the plotted ones.[]{data-label="fg EW_6"}](EW_6.png){width="\textwidth"}
EW$^{300}_{0}$ EW$^{300}_{0.5}$ EW$^{300}_{1}$ EW$^{500}_{0}$ EW$^{500}_{0.5}$ EW$^{500}_{1}$
--------------- ---------------- ------------------ ---------------- ---------------- ------------------ ----------------
Yearly return 11.92 9.92 7.96 12.52 10.46 8.43
Excess return 1.31 -0.54 -2.35 1.69 -0.25 -2.16
Std 16.59 16.43 16.29 17.07 16.90 16.74
Sharpe ratio 0.41 0.29 0.17 0.43 0.31 0.19
Wealth 255.3 93.6 34.3 332.8 118.4 42.1
TC 21.8 20.5 27.3 24.8
: Yearly returns and excess returns (with respect to the index tracking portfolio shown in Table \[tab 10\]) in precenatge, standard deviations of yearly returns (Std), Sharpe ratios, and the wealth and the cumulative transaction costs (TC) in thousands at the end of the investment period of the equally-weighted portfolio (EW) under different constituent list sizes $d$ and transaction cost rates ${\mathrm{tc}}$ with daily trading and monthly renewing frequencies. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $300$ and $500$ indicate $d=300$ and $d=500$, respectively.[]{data-label="tab 7"}
![Cumulative transaction costs on logarithmic scale of the equally-weighted portfolio (EW) under different constituent list sizes $d$ and transaction cost rates ${\mathrm{tc}}$ with daily trading frequency and monthly renewing frequency.[]{data-label="fg EW_7"}](EW_7.png){width="\textwidth"}
Entropy-weighted portfolio {#sub ep}
--------------------------
In this subsection, we consider the entropy-weighted portfolio (see Section 2.3 in [@MR1894767] and Example 5.3 in [@karatzas2017trading]), which relies on target weights $$\pi_{j}(\cdot)=\frac{\mu_{j}(\cdot)\log\mu_{j}(\cdot)}{\sum_{i=1}^{d}\mu_{i}(\cdot)\log\mu_{i}(\cdot)},\quad j\in\{1,\cdots,d\}.$$
In the following, we examine the performance of the entropy-weighted portfolio under specific configurations when there are no transaction costs, i.e., when ${\mathrm{tc}}=0$, and when ${\mathrm{tc}}=0.5\%$. The performance of the entropy-weighted portfolio is less sensitive to transaction costs and is better when ${\mathrm{tc}}=0.5\%$, compared with that of the equally-weighted portfolio.
### Varying the trading frequency {#varying-the-trading-frequency-2 .unnumbered}
As before, when backtesting the portfolio under different trading frequencies, we set the constituent list size $d=100$ and apply monthly renewing frequency. Figure \[fg ETP\_1\] displays and Table \[tab 4\] summarises the wealth processes of the entropy-weighted and the corresponding index tracking portfolio under different trading frequencies. Compared with the equally-weighted portfolio summarised in Table \[tab 6\], the entropy-weighted portfolio performs worse (but still outperforms the corresponding index tracking portfolio) when there are no transaction costs, i.e., when ${\mathrm{tc}}=0$. However, opposite to the equally-weighted portfolio, the weekly and the monthly traded entropy-weighted portfolio still outperforms the corresponding index tracking portfolio when ${\mathrm{tc}}=0.5\%$.
![The wealth processes of the entropy-weighted portfolio (ET) and the corresponding index tracking portfolio (IT) on logarithmic scale under different trading frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and monthly renewing frequency. For both the entropy-weighted and the index tracking portfolio, the omitted wealth processes of Table \[tab 4\] lie between the plotted ones. The sum of the wealth process and of the cumulative transaction costs of the daily traded entropy-weighted portfolio when ${\mathrm{tc}}=0.5\%$ is also plotted. Note that the sum is below the wealth process of the daily traded entropy-weighted portfolio when ${\mathrm{tc}}=0$.[]{data-label="fg ETP_1"}](ETP_1.png){width="\textwidth"}
IT$^{d}_{0}$ ET$^{d}_{0}$ IT$^{d}_{0.5}$ ET$^{d}_{0.5}$ IT$^{w}_{0}$ ET$^{w}_{0}$ IT$^{w}_{0.5}$ ET$^{w}_{0.5}$ IT$^{m}_{0}$ ET$^{m}_{0}$ IT$^{m}_{0.5}$ ET$^{m}_{0.5}$
----- -------------- -------------- ---------------- ---------------- -------------- -------------- ---------------- ---------------- -------------- -------------- ---------------- ----------------
YR 10.30 10.53 10.09 9.97 10.30 10.50 10.10 10.12 10.27 10.40 10.08 10.11
ER 0.23 -0.12 0.21 0.03 0.14 0.03
Std 16.87 16.90 16.84 16.83 16.88 16.92 16.85 16.88 16.88 16.94 16.86 16.90
SR 0.30 0.32 0.29 0.28 0.30 0.31 0.29 0.29 0.30 0.31 0.29 0.29
W 111.7 125.1 100.7 94.6 111.3 123.1 100.7 101.7 109.7 116.9 99.7 100.8
TC 2.7 6.4 2.5 4.5 2.3 3.4
: Yearly returns (YR) and excess returns (ER) in precentage, standard deviations of yearly returns (Std), Sharpe ratios (SR), and the wealth (W) and the cumulative transaction costs (TC) in thousands at the end of the investment period of the entropy-weighted portfolio (ET) and the corresponding index tracking portfolio (IT) under different trading frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and monthly renewing frequency. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $d$, $w$, and $m$ indicate daily, weekly, and monthly trading frequencies, respectively.[]{data-label="tab 4"}
Over a large time horizon, the loss in the portfolio wealth resulting from paying transaction costs is usually higher than the cumulative transaction costs imposed. This is exhibited in Figure \[fg ETP\_1\], which also plots the sum of the wealth process and of the cumulative transaction costs of the entropy-weighted portfolio when ${\mathrm{tc}}=0.5\%$. Notice that the wealth process when ${\mathrm{tc}}=0$ is above this sum. Indeed, paying transaction costs not only takes money out of the portfolio, but also deprives the opportunity for making potential gains.
### Varying the renewing frequency {#varying-the-renewing-frequency-2 .unnumbered}
With $d=100$ and daily trading frequency, we now examine the performance of the entropy-weighted portfolio applying different renewing frequencies (renewed weekly, monthly, and quarterly, respectively). Figure \[fg ETP\_3\] displays and Table \[tab 2\] summarises the wealth processes of the entropy-weighted and the corresponding index tracking portfolio under different renewing frequencies. Similar to the equally-weighted portfolio, the less frequently the constituent list is renewed, the better the entropy-weighted portfolio performs. When transaction costs are imposed, its performance depends more on the renewing frequency. However, compared with the equally-weighted portfolio summarised in Table \[tab 11\], the performance of the entropy-weighted portfolio is less sensitive to transaction costs under the same renewing frequency.
![The wealth processes of the entropy-weighted portfolio (ET) and the corresponding index tracking portfolio (IT) on logarithmic scale under different renewing frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and daily trading frequency. For both the entropy-weighted and the index tracking portfolio, the omitted wealth processes of Table \[tab 2\] lie between the plotted ones.[]{data-label="fg ETP_3"}](ETP_3.png){width="\textwidth"}
IT$^{W}_{0}$ ET$^{W}_{0}$ IT$^{W}_{0.5}$ ET$^{W}_{0.5}$ IT$^{Q}_{0}$ ET$^{Q}_{0}$ IT$^{Q}_{0.5}$ ET$^{Q}_{0.5}$
--------------- -------------- -------------- ---------------- ---------------- -------------- -------------- ---------------- ----------------
Yearly return 10.14 10.31 9.73 9.50 10.34 10.58 10.20 10.11
Excess return 0.17 -0.23 0.24 -0.09
Std 16.89 16.93 16.84 16.84 16.83 16.86 16.81 16.81
Sharpe ratio 0.29 0.30 0.27 0.26 0.31 0.32 0.30 0.29
Wealth 102.2 111.3 83.5 73.9 114.2 129.0 106.5 101.6
TC 4.2 7.4 2.0 5.8
: Yearly returns and excess returns in percentage, standard deviations of yearly returns (Std), Sharpe ratios, and the wealth and the cumulative transaction costs (TC) in thousands at the end of the investment period of the entropy-weighted portfolio (ET) and the corresponding index tracking portfolio (IT) under different renewing frequencies and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and daily trading frequency. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $W$ and $Q$ indicate weekly and quarterly renewing frequencies, respectively.[]{data-label="tab 2"}
### Varying the market size $d$ {#varying-the-market-size-d-1 .unnumbered}
Applying daily trading and monthly renewing frequencies, we backtest the entropy-weighted portfolio under different constituent list sizes $d$ ($=100$, $300$, and $500$, respectively), as shown in Figure \[fg ETP\_4\] and Table \[tab 3\]. Similar to the equally-weighted and the index tracking portfolio, the more stocks the constituent list contains, the better the entropy-weighted portfolio performs. Compared with the equally-weighted portfolio, the entropy-weighted portfolio with the same $d$ depends less on transaction costs. In particular, with $d=500$ and ${\mathrm{tc}}=0.5\%$, the entropy-weighted portfolio still outperforms the corresponding index tracking portfolio.
![The wealth processes of the entropy-weighted portfolio (ET) and the corresponding index tracking portfolio (IT) on logarithmic scale under different constituent list sizes $d$ and transaction cost rates ${\mathrm{tc}}$ with daily trading and monthly renewing frequencies. For both the entropy-weighted and the index tracking portfolio, the omitted wealth processes of Table \[tab 3\] lie between the plotted ones.[]{data-label="fg ETP_4"}](ETP_4.png){width="\textwidth"}
IT$^{300}_{0}$ ET$^{300}_{0}$ IT$^{300}_{0.5}$ ET$^{300}_{0.5}$ IT$^{500}_{0}$ ET$^{500}_{0}$ IT$^{500}_{0.5}$ ET$^{500}_{0.5}$
--------------- ---------------- ---------------- ------------------ ------------------ ---------------- ---------------- ------------------ ------------------
Yearly return 10.61 10.87 10.46 10.43 10.83 11.16 10.71 10.75
Excess return 0.26 -0.03 0.33 0.04
Std 16.57 16.57 16.55 16.52 16.61 16.66 16.60 16.62
Sharpe ratio 0.33 0.34 0.32 0.31 0.34 0.36 0.33 0.34
Wealth 132.5 152.5 123.1 121.2 147.2 173.2 139.0 141.0
TC 2.4 6.2 2.3 6.4
: Yearly returns and excess returns in percentage, standard deviations of yearly returns (Std), Sharpe ratios, and the wealth and the cumulative transaction costs (TC) in thousands at the end of the investment period of the entropy-weighted portfolio (ET) and the corresponding index tracking portfolio (IT) under different constituent list sizes $d$ and transaction cost rates ${\mathrm{tc}}$ with daily trading and monthly renewing frequencies. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $300$ and $500$ indicate $d=300$ and $d=500$, respectively.[]{data-label="tab 3"}
Diversity-weighted portfolio and smoothing transaction costs {#sec div}
------------------------------------------------------------
One portfolio that draws much attention in Stochastic Portfolio Theory is the so-called diversity-weighted portfolio generated from the “measure of diversity” $$G_{p}(x)=\left(\sum_{i=1}^{d}x_{i}^{p}\right)^{1/p},\quad x\in\left\{(y_{1},\cdots,y_{d})'\in[0,1]^{d};\sum_{i=1}^{d}y_{i}=1\right\},$$ for some fixed $p\in(0,1)$. Without changing the relative ranking of the stocks, the function $G_{p}(\cdot)$ generates portfolio weights smaller (larger) than the corresponding market weights for stocks with large (small) market weights. This diversification property of $G_{p}$ is closely related to the implementation of relative arbitrage portfolios; see Section 7 in [@FK_survey] for details. Section 6.3 in [@MR1894767] provides a theoretical approximation of the diversity-weighted portfolio turnover. An empirical study of this portfolio using S&P 500 market data can be found in [@fernholz1998diversity] and Chapter 7 of [@MR1894767], as well as in Example 5 of [@ruf2018generalised].
In the following, we examine the performance of this portfolio and illustrate the tradeoff between trading with a higher frequency and paying transaction costs. To achieve this, we shall replace the market weights by a smoothed version, given by $$\overline{\mu}(\cdot)=\alpha\mu(\cdot)+(1-\alpha)\Lambda(\cdot)$$ with $\alpha\in(0,1)$. Here, the moving average process $\Lambda(\cdot)=(\Lambda_{1}(\cdot),\cdots,\Lambda_{d}(\cdot))'$ is given by $$\begin{aligned}
\Lambda_{j}(\cdot)=
\begin{cases}
\frac{1}{\delta}\int_{0}^{\cdot}\mu_{j}(t)\mathrm{d}t+\frac{1}{\delta}\int_{\cdot-\delta}^{0}\mu_{j}(0)\mathrm{d}t\quad&\text{on~}[0,\delta)\vspace{2mm}\\
\frac{1}{\delta}\int_{\cdot-\delta}^{\cdot}\mu_{j}(t)\mathrm{d}t\quad&\text{on~}[\delta,\infty)
\end{cases},\quad j\in\{1,\cdots,d\},
\end{aligned}$$ for a fixed constant $\delta>0$. This moving average process $\Lambda(\cdot)$ is also included in the portfolio generating function studied in [@schied2018model]. Then the target weights are given by $$\pi_{j}(\cdot)=\mu_{j}(\cdot)\left(\Xi_{j}(\cdot)-
\sum_{i=1}^{d}\mu_{i}(\cdot)\Xi_{i}(\cdot)+1\right),\quad j\in\{1,\cdots,d\},$$ where $$\Xi_{j}(\cdot)=\frac{\alpha\left(\overline{\mu}_{j}(\cdot)\right)^{p-1}}{\sum_{i=1}^{d}\left(\overline{\mu}_{i}(\cdot)\right)^{p}},\quad j\in\{1,\cdots,d\}.$$
To backtest the portfolio, we fix $d=100$, the renewing frequency to be quarterly, and the “diversity degree” $p=0.8$. Moreover, we compute the moving average process $\Lambda(\cdot)$ using a one-year window. To be more specific, with daily trading frequency, we set $\delta=250$; with weekly trading frequency, we set $\delta=52$. To compute $\Lambda(\cdot)$ under weekly trading frequency, we only use market weights $\mu$’s on the days when transactions are made.
### Varying the convexity weight $\alpha$ and the trading frequency {#varying-the-convexity-weight-alpha-and-the-trading-frequency .unnumbered}
In Table \[tab 5\], we summarise the wealth processes of the diversity-weighted and the corresponding index tracking portfolio under both daily and weekly trading frequencies and with three different choices for the convexity weight $\alpha$, when there are no transaction costs, i.e., when ${\mathrm{tc}}=0$, and when ${\mathrm{tc}}=0.5\%$ and ${\mathrm{tc}}=1\%$, respectively.
IT$^{w}_{0}$ IT$^{w}_{0.5}$ IT$^{w}_{1}$ $\alpha$ DW$^{d}_{0}$ DW$^{d}_{0.5}$ DW$^{d}_{1}$ DW$^{w}_{0}$ DW$^{w}_{0.5}$ DW$^{w}_{1}$
--------------- -------------- ---------------- -------------- ---------- -------------- ---------------- -------------- -------------- ---------------- --------------
0.2 10.36 10.20 10.03 10.36 10.20 10.05
Yearly return 10.34 10.20 10.06 0.6 10.43 10.18 9.93 10.42 10.23 10.03
1 10.54 10.11 9.68 10.51 10.24 9.96
0.2 0.03 0 -0.03 0.02 0.01 -0.01
Excess return 0.6 0.1 -0.01 -0.13 0.09 0.03 -0.03
1 0.2 -0.09 -0.38 0.18 0.04 -0.09
0.2 16.84 16.81 16.79 16.85 16.83 16.80
Std 16.85 16.83 16.81 0.6 16.84 16.81 16.77 16.86 16.83 16.80
1 16.84 16.79 16.74 16.87 16.83 16.79
0.2 0.31 0.30 0.29 0.31 0.30 0.29
Sharpe ratio 0.31 0.30 0.29 0.6 0.31 0.30 0.28 0.31 0.30 0.29
1 0.32 0.29 0.27 0.32 0.30 0.28
0.2 115.6 106.2 97.6 115.1 106.5 98.5
Wealth 123.4 108.3 94.2 0.6 119.8 105.4 92.8 118.8 107.8 97.8
1 126.2 101.6 81.7 124.3 108.3 94.2
0.2 2.1 4.0 2.0 3.7
TC 3.5 6.3 0.6 3.2 5.9 2.5 4.6
1 5.3 9.0 3.5 6.3
: Yearly returns and excess returns (with respect to the index tracking portfolio (IT) summarised here and in Table \[tab 8\]) in percentage, standard deviations of yearly returns (Std), Sharpe ratios, and the wealth and the cumulative transaction costs (TC) in thousands at the end of the investment period of the diversity-weighted portfolio (DW) under different trading frequencies, convexity weights $\alpha$, and transaction cost rates ${\mathrm{tc}}$ with $d=100$ and quarterly renewing frequency. The subscript $x$ corresponds to ${\mathrm{tc}}=x\%$ and the superscripts $d$ and $w$ indicate daily and weekly trading frequencies, respectively.[]{data-label="tab 5"}
We first consider the case when there are no transaction costs. Everything else equal, the daily traded diversity-weighted portfolio performs similarly to the weekly traded portfolio. Under either trading frequency, the smaller the convexity weight $\alpha$ is, the worse the portfolio performs. Generating the portfolio with a smaller $\alpha$ is somewhat alike to trading less frequently, as it assigns less weights on the volatile term $\mu(\cdot)$ and more weights on the stable term $\Lambda(\cdot)$ when constructing $\overline{\mu}(\cdot)$, and thus makes $\overline{\mu}(\cdot)$ less volatile.
Next, we consider the case with transaction costs. Under either daily or weekly trading frequency, a smaller convexity weight $\alpha$ tends to improve the portfolio performance when the transaction cost rate ${\mathrm{tc}}$ becomes larger. This can be useful, since decreasing $\alpha$ partially cancels out the effect of transaction costs. Moreover, when ${\mathrm{tc}}=1\%$, the daily traded portfolio with $\alpha=0.2$ performs similarly as the weekly traded portfolio with $\alpha=0.6$. This indicates that, instead of trading less frequently in order to avoid paying transaction costs, one can adjust the convexity weight $\alpha$ to reach a more favourable balance between trading frequently and paying transaction costs.
### Dynamic convexity weight $\alpha$ to smooth transaction costs {#dynamic-convexity-weight-alpha-to-smooth-transaction-costs .unnumbered}
Instead of fixing $\alpha$ throughout the investment period, we could adjust $\alpha$ dynamically to speed up or slow down trading. For example, given a baseline portfolio with constant convexity weight $\alpha_{0}$, we would choose $\alpha<\alpha_{0}$ ($\alpha>\alpha_{0}$) to trade less (more) in the next period if transaction costs paid in the last period are more (less) than a certain level. In the remaining part of this example, we fix daily trading frequency and dynamically adjust $\alpha(\cdot)$.
Let $M\geq4$ denote the total number of quarters in the investment period and let $t_{u}^{\mathrm{r}}$, for $u\in\{1,\cdots,M\}$, denote the trading days on which the constituent list is renewed. Moreover, set $t_{0}^{\mathrm{r}}=t_{0}$. On a specific renewing day $t_{u}^{\mathrm{r}}$, for $u\in\{1,\cdots,M\}$, let $\widetilde{{\mathrm{TC}}}(t_{u}^{\mathrm{r}})$ denote the averaged fictitious transaction costs relative to the wealth $V_{\alpha_{0}}(\cdot-)$ of the baseline portfolio paid in the previous period. More precisely, $\widetilde{{\mathrm{TC}}}(t_{u}^{\mathrm{r}})$ is computed as $$\widetilde{{\mathrm{TC}}}(t_{u}^{\mathrm{r}})=\frac{1}{\kappa_{u}}\sum_{t\in[t_{u-1}^{\mathrm{r}},t_{u}^{\mathrm{r}})}\min\left\{\frac{{\mathrm{TC}}_{\alpha_{0}}(t)}{V_{\alpha_{0}}(t-)},\xi\right\}.$$ Here, $\kappa_{u}$ is the number of trading days within the period $[t_{u-1}^{\mathrm{r}},t_{u}^{\mathrm{r}})$, ${\mathrm{TC}}_{\alpha_{0}}(\cdot)$ is computed by from the baseline portfolio, and $\xi$ is a predetermined level used to make the estimate more robust. On a trading day $t$, We regard ${\mathrm{TC}}_{\alpha_{0}}(t)/V_{\alpha_{0}}(t-)>\xi$ as “abnormal” transaction costs relative to $V_{\alpha_{0}}(t-)$. Such large costs appear, for example, when the constituent list is changing. The level $\xi$ is determined such that the days, on which “abnormal” transaction costs occur, only count for a small proportion of all trading days.
![Transaction costs ${\mathrm{TC}}_{\alpha_{0}}(\cdot)$ of the baseline portfolio relative to its wealth $V_{\alpha_{0}}(\cdot-)$, i.e., ${\mathrm{TC}}_{\alpha_{0}}(\cdot)/V_{\alpha_{0}}(\cdot-)$, paid when the constituent list is changed and unchanged, respectively, with $\alpha_{0}=0.6$, when ${\mathrm{tc}}=0.5\%$.[]{data-label="fg DIV_3"}](DIV_3.png){width="\textwidth"}
Figure \[fg DIV\_3\] shows the relative transaction costs ${\mathrm{TC}}_{\alpha_{0}}(\cdot)/V_{\alpha_{0}}(\cdot-)$ when the constituent list is changed and unchanged, respectively, with $\alpha_{0}=0.6$ and ${\mathrm{tc}}=0.5\%$. Transaction costs paid when the constituent list is changed are significantly larger than when the constituent list remains the same. The days when the constituent list is changed only account for less than $5\%$ of all trading days, i.e., $M/N<0.05$, where $N$ is again the total number of trading days.
We shall smooth the relative transaction costs ${\mathrm{TC}}(\cdot)/V_{\alpha_{0}}(\cdot-)$ by dynamically adjusting $\alpha(\cdot)$. Starting with $\alpha(t_{0})=\alpha_{0}$, the convexity weight $\alpha(\cdot)$ is piecewise constant and only updated on the renewal dates $t_{u}^{\mathrm{r}}$, for $u\in\{4,\cdots,M\}$. This reduces additional transaction costs incurred from updating $\alpha(\cdot)$. In particular, for all $u\in\{4,\cdots,M\}$, we set $$\alpha(t_{u}^{\mathrm{r}})=\max\left\{\min\left\{\alpha_{0}\left(1-\beta\times\overline{{\mathrm{TC}}}(t_{u}^{\mathrm{r}})\right),1\right\},0\right\}$$ with $$\overline{{\mathrm{TC}}}(t_{u}^{\mathrm{r}})=\frac{\widetilde{{\mathrm{TC}}}(t_{u}^{\mathrm{r}})}{\frac{1}{4}\sum_{\nu=u-3}^{u}\widetilde{{\mathrm{TC}}}(t_{\nu}^{\mathrm{r}})}-1.$$ Here, $\beta\geq0$ is a fixed non-negative constant that controls the sensitivity of $\alpha(\cdot)$. Hence, we compare the fictitious averaged transaction costs relative to $V_{\alpha_{0}}(\cdot-)$ within the most recent quarter to that of the past one year. The value $\overline{{\mathrm{TC}}}(\cdot)$ is positive (negative) if the baseline portfolio requires more (less) transaction costs in the most recent quarter than the last year. This will yield $\alpha(\cdot)<\alpha_{0}$ ($\alpha(\cdot)>\alpha_{0}$) and slow down (speed up) the trading within the next quarter.
Using a baseline portfolio with constant convexity weight $\alpha_{0}=0.6$ and assuming ${\mathrm{tc}}=0.5\%$, we now estimate the effects of a dynamic convexity weight $\alpha(\cdot)$ empirically. Moreover, we set the relative transaction cost level $\xi=10^{-5}$, as the fictitious relative transaction costs ${\mathrm{TC}}_{\alpha_{0}}(\cdot)/V_{\alpha_{0}}(\cdot-)$ are less than this level on more than 95% of all trading days. We examine the three cases $\beta\in\{0,0.05,0.1\}$. Note that $\beta=0$ yields $\alpha(\cdot)=\alpha_{0}$. With these choices of $\beta$, the portfolio with dynamic $\alpha(\cdot)$ performs similarly to the baseline portfolio; see column $V^{d}_{{\mathrm{tc}}\_0.5}$ in Table \[tab 5\] with $\alpha=0.6$.
The convexity weight process $\alpha(\cdot)$ corresponding to the sensitivity parameter $\beta$ is shown in Figure \[fg DIV\_5\]. As expected, $\alpha(\cdot)$ fluctuates more rapidly with a larger $\beta$. As mentioned before, increasing $\alpha$ speeds up trading and leads to more transaction costs, while decreasing $\alpha$ has the opposite effect. Choosing $\beta$ very large results in a portfolio far away from the baseline portfolio. This dependence on $\beta$ is illustrated in Figure \[fg DIV\_7\], which plots the total quadratic variation $\mathrm{QV}$ of relative transaction costs ${\mathrm{TC}}(\cdot)/V_{\alpha_{0}}(\cdot-)$, computed as $$\mathrm{QV}=\sum_{l=1}^{N}\left(\frac{{\mathrm{TC}}(t_{l})}{V_{\alpha_{0}}(t_{l}-)}-\frac{{\mathrm{TC}}(t_{l-1})}{V_{\alpha_{0}}(t_{l-1}-)}\right)^{2},$$ for different sensitivity parameters $\beta$. The total quadratic variation $\mathrm{QV}$ is a measure of volatility. Figure \[fg DIV\_7\] suggests that choosing $\beta\approx0.05$ minimises $\mathrm{QV}$.
![Convexity weight process $\alpha(\cdot)$ for different sensitivity parameters $\beta$ with $\alpha_{0}=0.6$, when ${\mathrm{tc}}=0.5\%$.[]{data-label="fg DIV_5"}](DIV_5.png){width="\textwidth"}
![Quadratic variation $\mathrm{QV}$ of relative transaction costs ${\mathrm{TC}}(\cdot)/V_{\alpha_{0}}(\cdot-)$ for different sensitivity parameters $\beta$ with $\alpha_{0}=0.6$, when ${\mathrm{tc}}=0.5\%$.[]{data-label="fg DIV_7"}](DIV_7.png){width="\textwidth"}
Conclusion {#sec C}
==========
In this paper, we empirically study the impact of proportional transaction costs on systemically generated portfolios. Given a target portfolio, we provide a scheme to backtest the portfolio using total market capitalization and daily stock return time series. Implementing this scheme, we examine the performance of several portfolios (the index tracking portfolio, the equally-weighted portfolio, the entropy-weighted portfolio, and the diversity-weighted portfolio), assuming various transaction cost rates, trading frequencies, portfolio constituent list sizes, and renewing frequencies.
As expected, everything else equal, a portfolio performs worse as transaction costs are higher and the portfolio renewing frequency of the underlying constituent list is higher. In the absence of transaction costs, trading under a higher frequency leads to better portfolio performance. However, in the presence of transaction costs, implementing a higher trading frequency can also result in larger transaction costs and reduce the portfolio performance significantly. Hence, trading under an appropriate frequency is necessary in practice. In addition, with or without transaction costs, a more diversified portfolio containing more stocks usually performs better.
The empirical results indicate that the equally-weighted portfolio performs well relative to the index tracking portfolio when there are no transaction costs. However, the performance of the equally-weighted portfolio is very sensitive to transaction costs. Although the entropy-weighted portfolio performs a bit worse than the equally-weighted portfolio (but still outperforms the index tracking portfolio) when there are no transaction costs, its performance depends much less on transaction costs, compared to the equally-weighted portfolio.
Last but not the least, we propose a method to smooth transaction costs. Without changing the trading frequency, this method is similar to altering the trading speed dynamically.
[^1]: We thank Camilo Garc[í]{}a, Johannes Muhle-Karbe, Soumik Pal, Vassilios Papathanakos, and Leonard Wong for many helpful discussions on the subject matter of this paper.
[^2]: The dividends computed from the dividend rate $r^{D}$ contain not only the actual stock dividends, but also other corporate actions. For example, AT&T, which dominated the telephone market for most of the 20$^{\mathrm{th}}$ century, was broken up into eight smaller companies in 1984. This lead to a significant drop in the stock price. In our analysis below, we assume that the investor obtained cash in exchange (instead of stocks in the newly established companies).
[^3]: To compute the Sharpe ratios of the portfolios and the indices, the one-year U.S. Treasury yields are used. The data of these yields can be downloaded from <https://www.federalreserve.gov>.
[^4]: See <http://www.crsp.com/products/research-products/crsp-us-stock-databases> for details.
[^5]: Those stocks in CRSP which have ‘Share Code’ 10, 11, or 12.
|
---
abstract: 'We report on a fiber-optics implementation of the Deutsch-Jozsa and Bernstein-Vazirani quantum algorithms for 8-point functions. The measured visibility of the 8-path interferometer is about 97.5%. Potential applications of our setup to quantum communication or cryptographic protocols using several qubits are discussed.'
author:
- 'E. Brainis,$^1$ L.-P. Lamoureux,$^2$ N. J. Cerf,$^2$ Ph. Emplit,$^1$ M. Haelterman,$^1$ and S. Massar$^{3,2}$'
title: |
Optical implementation of Deutsch-Jozsa and Bernstein-Vazirani\
quantum algorithms in eight dimensions
---
The last decade has seen the emergence of the field of quantum information processing. A particularly promising application is the concept of quantum algorithms, which allow certain problems such as factorization[@Shor] or searching[@Grover] to be solved much faster than on a classical computer. Another algorithm which we will be interested in here is Deutsch’s algorithm[@Deutsch], the first quantum algorithm ever discovered, which was later generalized by Deutsch and Jozsa[@DeutschJozsa] (DJ). The DJ algorithm discriminates between a constant or a balanced $N$-point binary function using one single quantum query, while a classical algorithm requires ${\cal O}(N)$ classical queries. It was later on adapted by Bernstein and Vazirani (BV) for efficiently querying a quantum database[@BV; @BS].
In the present paper, we report on a fiber-optics implementation of the DJ algorithm using standard telecom optical components and a single-photon detector. The DJ algorithm has already been implemented using NMR[@nmr] (see also [@nmr2] for a NMR implementation of the BV algorithm), table-top optics[@DJoptical] (optical demonstrations of other quantum algorithms also include Grover’s algorithm[@Kwiat; @Amsterdam]), molecular states[@molecular], and very recently using an ion trap[@trap]. However, our setup separates from these realizations (especially that of [@DJoptical]) on several major aspects. First, it relies on guided optics components, which makes it unnecessary to perform a precise alignment, and it is made robust against phase fluctuations by use of an autocompensation technique. Second, although it relies on linear optics, our realization is relatively efficient in terms of used optical resources compared to standard linear optical implementations of quantum computation. The central idea of such implementations consists in representing the basis states of a $N$-dimensional Hilbert space by $N$ optical paths so that unitary transformations are obtained by chaining linear optics components that make these paths interfere[@Reck; @CerfKwiat]. Such implementations seem, however, to be inherently inefficient since the space requirement (the number of optical components) and the time requirement both grow exponentially with the number $n$ of qubits (with $N=2^n$) [@fn]. In contrast, in our setup, the number of components is kept linear in $n$, while the time needed still grows exponentially. Note that any implementation of an algorithm involving an arbitrary $2^n$-point function (also called oracle) does in any case require exponential resources to simulate this function. Therefore, the linear optical implementation of quantum algorithms involving oracles can reasonably be made as efficient as any other implementation in this respect. For all these reasons, our experimental demonstration works with a 8-point (3-qubit) function and might probably be extended even further without fundamental difficulty, while today’s largest size optical demonstrator of the DJ algorithm involves a 4-point function[@DJoptical].
{width="17.5truecm"}
Let us start by recalling the principle of the DJ algorithm. At the core of the algorithm is the oracle which computes a function $f(x)$, where $x\in\{0,1\}^n$ is an $n$ bit string, and $f\in\{0,1\}$ is a single bit. The DJ problem is to discriminate whether $f$ is a constant or balanced function, while querying the oracle as few times as possible. A balanced function is such that the number of $x$’s on which $f(x)=0$ is equal to the number of $x$’s on which $f(x)=1$. Classically, $2^{n-1}+1$ queries are necessary in the worst case, whereas the DJ algorithm requires a single query as we shall see. In this algorithm, $n$ qubits are used, and the basis of the Hilbert space is chosen as $|x\rangle = |x_1 x_2 \ldots x_n\rangle$ where $x_i \in \{0,1\}$. The quantum oracle carries out the transformation $$|x\rangle |y\rangle \stackrel{\rm oracle}\longrightarrow |x\rangle
|y \oplus f(x)\rangle \ ,
\label{oracle1}$$ where $|y\rangle$ is an ancilla qubit. By choosing $|y\rangle=(|0\rangle - |1\rangle)/\sqrt{2}$, the action of the oracle simplifies into $$|x\rangle \stackrel{\rm oracle}\longrightarrow (-1)^{f(x)}|x\rangle \ .
\label{phaseoracle}$$ since $|y\rangle$ then remains unchanged.
The DJ algorithm starts with the system in the state $|0\rangle=|00 \ldots 0\rangle$. Next, a Hadamard transform $H$ is applied independently on each of the $n$ qubit. Using the definition $H |x\rangle = 2^{-n/2} \sum_{z\in
\{0,1\}^n} (-1)^{x \cdot z} |z\rangle $, where $x\cdot z =
\sum_i x_i z_i \bmod 2$ is the inner product of two $n$-bit strings, we see that the Hadamard transform acting on the initial state simply yields a uniform superposition of all states: $H |0\rangle =
2^{-n/2}\sum_{x} |x\rangle $. This state is then sent through the oracle whereupon it becomes $$\label{oracle}
2^{-n/2}\sum_x (-1)^{f(x)} |x\rangle \, .$$ The superposition principle allows the oracle to be queried on all input values in parallel. A second Hadamard transform is then carried out to obtain the state $$2^{-n} \sum_{x,z} (-1)^{x \cdot z + f(x)} |z\rangle \label{DJwavefunction}$$ which is finally measured in the $z$ basis. One easily deduces from Eq. (\[DJwavefunction\]) that when $f$ is constant, the probability of measuring $|0\rangle$ is one. In contrast, when $f$ is balanced, this probability is always zero, so the DJ algorithm can distinguish with certainty between these two classes of functions by querying the oracle a single time. The BV variant of this algorithm is also based on the transformation leading to Eq. (\[DJwavefunction\]). Suppose that the oracle is restricted to be of the form $f_j(x) = x \cdot j$ where $j\in \{ 0,1\}^n$ is an arbitrary $n$-bit string. The aim is to find the bit string $j$ with as few queries as possible. Classically one needs at least $n$ queries since each query provides one independent bit of information at most about $j$. Quantum mechanically, a single query suffices since using $f_j(x)$ in Eq. (\[DJwavefunction\]) shows that the measurement outcome is $z=j$ with probability.
Our all optical fiber (standard SMF-28) setup is illustrated in Fig. 1. Initially, a 3 ns light pulse is produced by a laser diode at $1,55~\mu$m, attenuated by an optical attenuator (Agilent 8156A), and then is processed through three unbalanced Mach-Zehnder (M-Z) interferometers with path length differences $\Delta_l$ ($l=1,2,3$) obeying $\Delta_3 = 2 \Delta_2 = 4 \Delta_1 = 15$ ns. Each MZ interferometer doubles the number of pulses so that, at the coupler C6, we get eight equally spaced pulses. This corresponds to the action of the Hadamard transform on the three input qubits in state $|0\rangle$. The pulses are then reflected by a Faraday mirror and, on their way back, are modulated by a phase modulator (Trilink) commanded by a pattern generator (Agilent 81110A) which selectively puts a phase shift of $0$ or $\pi$ on each pulse according to the 8-point function \[see Eq. (\[oracle\])\]. The pulses then pass back through the three M-Z interferometers, thereby realizing a second triple Hadamard transform, and are sent via a circulator to a single-photon detector (id Quantique id200), which completes the DJ or BV algorithm. The additional delay lines $L_1$, $L_2$, and $L_3$, which obey $L_3 > 2 L_2 > 4 L_1 > 8 \Delta_3$, ensure that the different outputs of the DJ or BV algorithm all reach the detector at different times. The delay lines $L_2$ and $L_3$ also contain an isolator so that the pulses are not transmitted on their way to the mirror. The photodetector was gated during 5 ns around the arrival time of each pulse by the pattern generator. The output of the detector was registered using a time to digital delay converter (ACAM-GP1) connected to a computer. All delays $\Delta_i$ and $L_i$ were chosen to be integer multiples of $\Delta_1$ within 0,2 ns. All electronic components were triggered by a pulse generator (Standford Research Inc. DG535). In order to maximize the visibility, polarization controllers were introduced in the long arm of each M-Z interferometer and in front of the polarization-sensitive phase modulator. Once optimized, the setup was stable for days.
This implementation of the DJ and BV algorithms differs from an earlier optical implementation of the DJ algorithm[@DJoptical] in several important aspects. First, we run the algorithm for $n=3$ qubits and, more importantly, we measure all 8 outcomes, which makes it possible to realize the BV algorithm as well (the previous implementation [@DJoptical] works with $n=2$ qubits and only measures the outcome $z=0$). Second, we operate at telecom wavelengths in optical fibers using a setup closely inspired from the “plug-and-play” quantum cryptographic system developed by Gisin and collaborators (see e.g. [@plugandplay]). For this reason, the present setup in a slightly modified version can be adapted to implement quantum cryptography using higher dimensional systems[@highdQcrypto] or to illustrate quantum communication complexity protocols[@BCW] over distances of a few kilometers. These potential applications will be discussed below. Third, the used resources are quite different in the two implementations: when scaled to a large $n$, the implementation of [@DJoptical] requires exponential time and exponentially many optical elements, whereas our implementation also requires exponential time but only a linear number of optical elements. This is because the $n$ qubits are realized as $2^n$ separate optical paths in [@DJoptical], whereas in our case they are represented as $2^n$ light pulses traveling in a single optical fiber, extending naturally the “time-bin” realization of qubits used in [@plugandplay]. The small number of optical elements in our setup therefore implies that it is relatively easy to increase the number of qubits $n$ while keeping the optical setup stable. As we shall see, a disadvantage of our setup is that the Hadamard transform can only be implemented with a probability of success of $1/2$. Since $2n$ Hadamard transforms are needed for the DJ algorithm with $n$ qubits, the resulting attenuation is $2^{-2n}$.
Let us now prove that our optical setup indeed realizes the DJ and BV algorithm. The quantum state describing the eight pulses at coupler C6 can be written as $$\begin{aligned}
|\psi\rangle \propto
\sum_{x=000}^{111}
\exp\left[ i \sum_{l=1}^3
(k x_l \Delta_l + \pi x_l ) \right] |\sum_{l=1}^3 x_l \Delta_l\rangle
%| x_1 \Delta_1 +
%x_2 \Delta_2 + x_3 \Delta_3 \rangle\end{aligned}$$ where $x$ stands for $(x_1,x_2,x_3)$ and $|p\rangle$ denotes a pulse located at position $p$. The three bits $x_1,x_2,x_3=0,1$ label whether the pulse took the short ($x=0$) or the long ($x=1$) path through each interferometer. The factor $\exp[i k \sum_l x_l \Delta_l]$ with $k$ being the wave number takes into account the phase difference between a pulse traveling along the short or long paths of the interferometers. The factor $\exp[i \sum_l \pi x_l ]$ takes into account the phase accumulated at the couplers: if the pulse takes the short path, it is transmitted at two couplers, whereas, if it takes the long path it is reflected twice. After reflection at the Faraday mirror and phase modulation, the pulses cross again the three M-Z interferometers and reach the photodetector in the state
$$\begin{aligned}
|\psi\rangle \propto \sum_{x,y, z}
(-1)^{f(x)}
\exp\left[ i \sum_{l=1}^3 \left(
k (x_l + y_l) \Delta_l
+ \pi ( x_l + y_l - y_l (z_l+z_{l+1}) + {z_l+z_{l+1}\over 2})
\right)
\right]
| \sum_{l=1}^3 ((x_l + y_l) \Delta_l + z_l L_l) \rangle\end{aligned}$$
where the bits $y_1,y_2,y_3$ equal $0$ or $1$ according to whether the pulse passed through the short or the long path of each of the interferometers on its way back, and the bits $z_1,z_2,z_3$ equal $0$ or $1$ according to whether or not the pulse exited each of the interferometer in the path containing the delay lines $L_1$, $L_2$ or $L_3$. Note that we put $z_4=0$. We have again taken into account the phases induced by transmission or reflexion at the couplers C1-C6. The final state contains 120 pulses, but we are only interested in the eight pulses such that $x_1+y_1 = x_2+ y_2 = x_3 + y_3 = 1$, which are those that exhibit 8-path interference. The other pulses are filtered out in the computer analysis (they correspond to different time bins). The final state then becomes $$\begin{aligned}
|\psi\rangle \propto
\sum_{z}
(-i)^{z_1} (-1)^{z_2 + z_3}
\sum_{x}
(-1)^{f(x_1,x_2,x_3)+
x_1 (z_1+z_2) + x_2 (z_2+z_3) + x_3 z_3}
| \Delta_1 + \Delta_2 + \Delta_3
+ z_1 L_1 + z_2 L_2 + z_3 L_3
\rangle \label{final}\end{aligned}$$
By relabeling the time bins according to the substitution $z_1\to z_1+z_2+z_3$ mod 2, $z_2 \to z_2+z_3$ mod 2, and $z_3 \to z_3$, this equation coincides (up to irrelevant phases and an overall normalization factor) with Eq. (\[DJwavefunction\]) with the 8 logical states $|z\rangle$ identified as specific time bins.
The setup thus realizes the DJ or BV algorithm, the main difference with an ideal algorithm being an extra attenuation by a factor of $2^{-7}$. A factor $2^{-3}$ originates from the couplers C2, C4, and C6 because each time a pulse passes through these couplers it only has a probability $1/2$ of exiting by the right path. Otherwise, it is absorbed by the isolators I1 and I2 or by the unconnected fiber pigtail at coupler C6. Another factor $2^{-3}$ is due to the filtering out of the 112 pulses produced on the way back that do not correspond to 8-path interferences. The remaining factor $2^{-1}$ is due to the coupler C7. This overall loss of 21 dB could be remedied by replacing the couplers C2, C4, C6, and C7 by optical switches which would direct the light pulses along the appropriate path. High speed, low loss optical switches are not available commercially at present, so we had to use couplers in the present experiment. However, we emphasize that this is a technological rather than a fundamental limitation.
In order to characterize the performances of our setup, we considered the $2^n$ oracles of the form $f_j(x)=x\cdot j$ and ${\overline f}_j(x)= x \cdot j + 1 {\rm ~mod~} 2$ (i.e., the oracles in the BV algorithm and their complements). For each oracle $f_j$ (or ${\overline f}_j$), we ran the algorithm 500,000 times and registered the number of counts in time bin $z$, denoted as $N_j(z)$ (or ${\overline N}_j(z)$). The algorithm gives constructive interference in the time bin $z=j$ for the oracle $f_j$ or ${\overline f}_j$, and destructive interference elsewhere. We then computed $$V_j(z) = {1 \over 2} \left (
{N_z(z) - N_j(z) \over N_z(z) + N_j(z) }
+ {{\overline N}_z(z) - N_j(z) \over {\overline N}_z(z) + N_j(z) }
\right)$$ for each pair of oracles with $j\ne z$, and calculated the visibility $V(z)$ in time bin $z$ by taking the average of $V_j(z)$ over all values of $j$. The measured visibilities $V(z)$ for 2 and 3 qubits are shown in Table I. Remarkably, they remain relatively high when going from 2 to 3 qubits in spite of the fact that 8 path interferences are involved. This is because the path differences are automatically compensated and only $n+1$ polarizations must be adjusted. It should therefore be relatively easy to go beyond $n=3$ without significantly decreasing the visibilities.
Because of the attenuation in our setup along with the quantum efficiency ($\simeq 10.5$ %) and the dark count probability ($\simeq 10^{-4}$ ns$^{-1}$) of our detector, the signal-to-noise ratio was not high enough to perform these visibility measurements in the single-photon regime. For this reason, in our experiment, approximatively 20 photons per pulse entered the phase modulator (oracle) in 4 dimensions, and approximatively 50 in 8 dimensions. However, minimal modifications should allow us to decrease the number of photons while keeping the signal-to-noise ratio constant. In particular, by reducing the pulse length or using two detectors instead of the coupler C7, it should be possible to operate in the single-photon regime in 4 (and possibly 8) dimensions. Moreover, as already mentioned, the Hadamard transform could be rendered deterministic by using fast low-loss optical switches instead of couplers, which would strongly reduce the losses.
z 1 2 3 4 5 6 7 8
-------------- ------- ------- ------- ------- ------- ------- ------- -------
$V(z)^{n=2}$ 98.4 97.4 98.5 98.6
$V(z)^{n=3}$ 96.78 97.99 97.68 97.32 97.33 97.56 97.37 97.28
: Measured average visibility $V(z)$ in the $z$th time bin for the DJ or BV algorithm with $n=2$ and $n=3$ qubits.
The present experiment can be extended in several ways. For example, one could implement the distributed Deutsch-Jozsa problem[@BCW] where two parties, which receive each as input a $2^n$-bit string (denoted as $f$ and $g$), must decide whether $f=g$ or $f$ differs from $g$ in exactly $2^{n-1}$ bits (they are promised that only one of these two cases can occur). The value of $n$ for which the gap between the classical and quantum algorithms sets in is unknown, but recent results suggest that it could be as early as $n=4$ [@B]. The distributed Deutsch-Jozsa algorithm could be realized with a slight modification of our setup in which the two parties, separated by by a few kilometers of optical fiber, would use each a phase modulator. This quantum protocol can also be easily adapted to multidimensional quantum cryptography, where two parties randomly choose their patterns $f$ and $g$. By publicly revealing part of $f$ and $g$, the parties can use the correlations between the measurement outcomes to establish a secret key. A detailed analysis shows that for n = 2, this exactly coincides with the 4-dimensional cryptosystem based on 2 mutually conjugate bases, which has been shown to present advantages over quantum cryptography in two dimensions[@multidimcrypto]. A final potential application of this setup is to test quantum non locality using the entanglement-based Deutsch-Jozsa correlations[@BCT]. An entangled state of $2^n$ time bins must be produced, for example using the source [@RMZG], and each party must then carry out phase modulation and a Hadamard transform (which we have demonstrated are easy to realize on time bin entangled photons). The correlations between the chosen phases and the measured time of arrival of the photons at each side should exhibit quantum non locality. It has recently been shown that these correlations are non local for $n=4$ [@B], and that they exhibit exponentially strong resistance to detector inefficiency for large $n$ [@M], which means here that the 3 dB losses at each Hadamard transform can in principle be tolerated.
In summary, we have demonstrated, by implementing the DJ and BV algorithms for 3 qubits, a simple and robust method for manipulating multidimensional quantum information encoded in time bins in optical fibers. We anticipate that our method will have wide applicability for quantum information processing and quantum communication using higher dimensional systems.
It is a pleasure to thank N. Gisin, W. Tittel, G. Ribordy, H. Zbinden, and all the members of [*GAP optique*]{} for very helpful discussions and for providing the single-photon detector. We acknowledge financial support from the Communauté Française de Belgique under ARC 00/05-251, from the IUAP programme of the Belgian government under grant V-18, and from the EU under project EQUIP (IST-1999-11053).
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abstract: 'We investigate the pinning and driven dynamics of vortices interacting with twin boundaries using large scale molecular dynamics simulations on samples with near one million pinning sites. For low applied driving forces, the vortex lattice orients itself parallel to the twin boundary and we observe the creation of a flux gradient and vortex free region near the edges of the twin boundary. For increasing drive, we find evidence for several distinct dynamical flow phases which we characterize by the density of defects in the vortex lattice, the microscopic vortex flow patterns, and orientation of the vortex lattice. We show that these different dynamical phases can be directly related to microscopically measurable voltage–current $V(I)$ curves and voltage noise. By conducting a series of simulations for various twin boundary parameters we derive several vortex dynamic phase diagrams.'
address: |
$1$. Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120\
$2$. Department of Physics, University of California, Davis, CA 95616
author:
- 'C. Reichhardt$^{1,2}$ C.J. Olson$^{1,2}$, and Franco Nori$^1$'
title: Dynamic Vortex Phases and Pinning in Superconductors with Twin Boundaries
---
Introduction
============
The understanding of vortex pinning and dynamics in high-Tc superconductors is of great interest for applications of superconductors which require strong pinning of vortices as well as the rich variety of behaviors that arise due to the competition of a static or driven elastic media with various forms of quenched disorder [@Blatter]. The physics of a vortex lattice interacting with disorder is relevant for a wide variety of condensed matter systems including charge-density-waves, driven Wigner crystals, magnetic bubble arrays, colloids, Josephson junction arrays and superconducting wire networks, as well as microscopic models of friction.
Twin boundaries are a very common defect found in $YBa_{2}Cu_{3}O_{7-x}$(YBCO) and their pinning properties have been extensively studied using Bitter decoration [@Decorations], torque magnetometry [@Gyorgy], magnetization [@Schwartzendruber; @Oussena; @Lairson; @Zhukov] transport [@Safar], magneto-optical imaging [@Duran; @Vlaskov; @Bishop; @Welp; @Olsson; @rinke], and theoretical studies [@review; @theory; @Groth]. Many of the earlier experiments on twinned YBCO samples found conflicting evidence for the role of twin boundaries in vortex pinning. In particular, the magneto-optical measurements by Duran [*et al*]{}. [@Duran] had shown that twin boundaries act as areas of reduced pinning that allow easy flux penetration, whereas studies by Vlasko-Vlasov [*et al*]{}. [@Vlaskov] found the twin boundaries to be barriers to flux motion. Further magneto-optical studies [@Bishop; @Welp; @Olsson; @rinke], systematic computer simulations [@Groth], and transport measurements [@Safar] have shown that these conflicting results can be resolved when the direction of the Lorentz force with respect to the twin boundary is considered. The twin boundary (TB) acts as an easy-flow channel when the Lorentz force is parallel to the twin, but acts as a strong barrier for forces perpendicular to the TB.
A very systematic simulational study, using samples with of the order of a million pinning sites, by Groth [*et al*]{}. [@Groth] of the angular dependence of the Lorentz force with respect to the twin boundary showed that, when the angle between the Lorentz force and the twin is large, a portion of the vortices get trapped inside the twin. This produces a pile-up effect leading to a higher density of vortices on one side of the twin in agreement with observations by several groups including, for example, Vlasko-Vlasov [*et al*]{}. [@Vlaskov], Welp [*et al*]{}. [@Welp], and Wijngaarden [*et al*]{}. [@rinke]. At lower angles between the Lorentz force and the twin, simulations [@Groth] show that the flux moves in channels along the twin boundary while some guided motion of vortices along the edge of the twin still occurs. At the lowest angles the flux flows most easily along the twin with a number of vortices escaping from the twin and forming a flame pattern flux profile in agreement with magneto-optical experiments [@Duran; @Vlaskov; @Crabtree; @rinke].
Recently interest in vortex systems has strongly focused on driven phases and dynamic phase transitions of vortices interacting with random or periodic defects in superconductors. The anisotropic pinning properties of twin boundaries as well as the possibility of tuning the strength of the twin boundary pinning make these defects quite distinct from random pinning or periodic pinning arrays, so that new dynamical phases can be expected to appear.
In systems containing random pinning, experiments using transport measurements [@danna-peak; @Beasley; @Andrei], voltage noise measurements [@Rabin; @danna], vibrating reed measurements [@Zhang], neutron scattering [@Yaron], and Bitter-decoration [@Pardo], as well as simulational work [@Koshelev; @ShortDriven], and work based on perturbation and/or elasticity theory [@Giamarchi] indicate that, at the depinning transition, the vortex lattice may disorder and undergo [*plastic flow*]{} in which vortices change nearest neighbors as mobile portions of the vortex lattice tear past pinned portions. At higher drives the vortex lattice may reorder and exhibit elastic or ordered flow. An intriguing question is whether specific types of plastic flow exist, and how they could be distinguished. Simulations with randomly placed pinning indicate the possible existence of at least two kinds of plastic flow. The first type consists of well-defined channels of mobile vortices flowing through the rest of the pinned vortex lattice [@Brass; @Jensen; @Olson; @Rivers]. A second type consists of intermittent or avalanching motion in which only a few vortices are mobile at any given time, but over time all the vortices take part in the motion so that well defined channels are not observed [@Jensen; @Olson; @Rivers].
Recent simulations using the time-dependent Ginzburg-Landau equations at $T = 0$ of vortices interacting with twin boundaries have suggested the possibility of the existence of three distinct flow phases which include two plastic flow phases and an elastic flow phase [@Crabtree]. Due to the nature of these simulations it was only possible to consider three different driving currents for each pinning parameter; so that $V(I)$ curves, voltage noise signals, and the evolution of the vortex order as a continuous function of increasing driving force could not be extracted, nor could the evolution of the flow phases with the system parameters be determined.
In order to examine the microscopic dynamics of vortices interacting with twin boundaries we have performed large scale molecular dynamics simulations for a wide variety of twin parameters which allow us to carefully compare the different kinds of plastic flow as a driving force is continuously increased. Our results in this work complement our previous simulational work on twin-boundaries [@Groth], where we considered only the case of very slow driving that occurs as a magnetic field is increased. In Ref. [@Groth] we considered flux-gradient-driven vortices and we focused on the magnetic flux front profiles and compared them to magneto-optical images. In this paper we focus on the microscopic aspects of current-driven, as opposed to flux-gradient-driven, vortex motion and structure as well as on transport measures.
Simulation
==========
We consider an infinite 2D slice in the $x$-$y$ plane of an infinitely long (in the $z$ direction) parallelepiped. We use periodic boundary conditions in the $x$-$y$ plane and simulate stiff vortices that are perpendicular to the sample (i.e, $ {\bf H} = H{\bf {\hat z}}$). These rigid flux lines can also be thought of as representing the “center of mass" positions of real, somewhat flexible vortices, and the pinning in the bulk as representing the average of the pinning along the length of the real vortex. For flexible vortices, the bulk pinning can be on the same order as the twin-boundary pinning even for large samples. We numerically integrate the overdamped equations of motion: $${\bf f}_{i} = {\bf f}_{i}^{vv} + {\bf f}_{i}^{vp} + {\bf f}_{i}^{vTB} +
{\bf f}_{d} = \ \eta{\bf v}_{i} .$$ Here, ${\bf f}_{i}$ is the total force on vortex $ i $, ${\bf f}_i^{vv}$ is the force on the $i$th vortex from the other vortices, ${\bf f}_{i}^{vp}$ is the force from the vortex pin interaction, ${\bf f}_{i}^{vTB}$ is the force from the vortex-twin interaction, and ${\bf f}_{d} $ is the driving force; $ {\bf v}_{i}$ is the net velocity of vortex $i$ and $ {\eta }$ is the viscosity, which is set equal to unity in this work. The interaction between vortex $i$ and other vortices is given by: $$\begin{aligned}
\ {\bf f}_{i}^{vv} = \ \sum_{j=1}^{N_{v}} \ f_{0} \ K_{1}\left(
\frac{|{\bf r}_{i} - {\bf r}_{j}|}{ \lambda}\right) \
{\bf {\hat r}}_{ij} \ .\end{aligned}$$ Here, ${\bf r}_{i}$ is the location of vortex $i$ and $ {\bf r}_{j}$ is the location of vortex $j$, $f_{0} = \Phi_{0}^{2}/8\pi^{2}\lambda^{3} $, $ \Phi_{0} = hc/2e$ is the elementary flux quantum, $ \lambda$ is the penetration depth, $N_{v}$ is the number of vortices, and $ {\bf {\hat r}}_{ij} = ({\bf r}_{i} - {\bf r}_{j})/|{\bf r}_{i} -
{\bf r}_{j}|$. The force between vortices decreases exponentially at distances greater than $ \lambda$, and we cut off this force for distances greater than $6\lambda$. A cutoff is also placed on the force for distances less than $ 0.1\lambda$ to avoid the logarithmic divergence of forces. These cutoffs have been found to produce negligible effects for the range of parameters we investigate here. For convenience, throughout this work all lengths are measured in units of $ \lambda$, forces in units of $f_{0}$, and fields in units of $\Phi_0/\lambda^{2}$.
To model pinning in the bulk, we divide our system into a $ 1000\times 1000$ grid where each grid element represents a pinning site. The pinning density $n_{p}$ is $ 496/\lambda^{2}$, which is within experimentally determined values. At each pinning site $(l,m)$ the pinning force $f_{l,m}^{thr}$ is chosen from a uniform distribution $[0,f_{p}]$, where $f_{p}$ is the maximum possible pinning force. If the magnitude of the force produced by the other vortices, driving force and twin boundaries acting on a vortex located on a pinning site $(l,m)$ is less than the threshold pinning force $f_{l,m}^{thr}$, the vortex remains pinned at the pinning site. If the force on the vortex is greater than $f_{l,m}^{thr}$, then the effective pinning force $f_{i}^{vp}$ drops to zero and the vortex moves continuously until it encounters a pinning site that has a threshold force greater than the net force on the vortex. The pinning therefore acts as a stick-slip friction force with the following properties $${\bf f}_{i}^{vp} \ =\ -\ {\bf f}_{i}^{net}, \ \ \ \ \ \ f_{i}^{net} < f_{l,m}^{thr}$$ and $${\bf f}_{i}^{vp} \ = \ 0, \ \ \ \ \ \ f_{i}^{net} > f_{l,m}^{thr} \ \ .$$
For the twin boundary pinning, we have considered a large number of models, all giving similar results. The simplest model that is most consistent with experiments is that of an attractive well containing stick-slip pinning with a different maximum threshold force $f_{p}^{TB}$ than that of the bulk pinning outside the TB, $ f_{p}$. This model of pinning is very similar to the one inferred from the measurements in [@Oussena] where the TB channel has strong depth variations. The ratio $f_{p}^{TB}/f_{p}$ is expected to vary as a function of temperature. In the case predicted for low $T$ [@Blatter] where $f_{p}^{TB}/f_{p} < 1$, the twin boundary acts as an easy flow channel for certain angles [@Groth]. On the other hand, at higher $T$, $f_{p}^{TB}/f_{p} > 1$, and the twin acts as a barrier to flux flow. This second case is the most similar to the simulations conducted in [@Crabtree] where the twin boundary was modeled as a line of parabolic pinning. In our simulations we can mimic the effects of temperature by varying the ratio of $f_{p}^{TB}/f_{p}$.
The twin boundary itself is modeled as an attractive parabolic channel with a width denoted by $ 2\xi^{TB}$. The force on the $i$th vortex due to the $k$th the twin boundary is $$\begin{aligned}
{\bf f}_{i}^{vTB} \; = \; f^{TB}
\left( \frac{ d_{ik}^{TB} }{ \xi^{TB} } \right) \
\Theta \! \left( \frac{ \xi^{TB} - d_{ik}^{TB} } { \lambda } \right)
\ {\hat {\bf r}}_{ik}\end{aligned}$$ where $ d_{ik}^{TB}$ is the perpendicular distance between the $i$th vortex and the $k$th twin boundary.
The driving force representing the Lorentz force from an applied current is modeled as a uniform force on all the vortices. The driving force is applied in the $x$-direction and is slowly increased linearly with time. We examine the average force in the $x$-direction $$V_{x} = \frac{1}{N_{v}}\sum_{i=1}^{N_{v}} \; {\bf v}_{i}\cdot{\bf {\hat x}} \ ,$$ as well as the average force in the $y$-direction $$V_{y} = \frac{1}{N_{v}}\sum_{i=1}^{N_{v}} \; {\bf v}_{i}\cdot{\bf {\hat y}} \ .$$ These quantities are related to macroscopically measured voltage-current [*V(I)*]{} curves.
We also measure the density of 6-fold coordinated vortices $P_{6}$. Strong plastic flow causes an increase in the number of defects and a corresponding drop in $P_{6}$, while elastic flow is associated with few or no defects. Another measure of order in the lattice is the average height of the first-oder peaks in the structure factor $S(k)$. $$S({\bf k}) = \frac{1}{L^{2}}\sum_{i,j}e^{i{\bf k}
\cdot
({\bf r}_{i} -{\bf r}_{j})} \ .$$
The defect density can also be correlated with the voltage noise power spectra $S(\nu)$. $$S(\nu) = \int V_{x}(t) \ e^{2\pi i\nu t} \ dt \ .$$ A vortex lattice that is flowing plastically should produce a large amount of voltage noise. To measure the quantity of noise produced, we integrated the noise power over one frequency octave [@Rabin; @danna].
Dynamic Phases
==============
In order to directly observe the nature of the vortex flow in the presence of twin boundaries, we have imaged the trajectories of the moving vortices as the driving current along the $x$-axis is increased. We find three types of vortex flow, which are shown in Fig. 1. There, and for three different applied driving forces, we show the vortex positions (dots) and the trajectories (lines) that the vortices follow when interacting with a twin boundary (dotted line) that acts as a strong pinning barrier for motion across the twin. Here $f_{p} = 0.02f_{0}$, $f^{TB}_{p} = 1.0f_{0}$, $f^{TB} = 0.15f_{0}$, with the twin boundary having a width of $0.5\lambda$. In Fig. 1(a) for the lowest drive, $f_{d} = 0.05f_{0}$, the vortex lattice is predominantly triangular, and [*aligned*]{} with the twin plane. The vortices that have struck the twin boundary are pinned, while the remaining vortices flow in an orderly fashion at a 45 degree angle from the $x$ axis, as seen in Fig. 1(b). The moving vortices do not [*cross*]{} the twin boundary but are instead [*guided*]{} so that the vortices do not move parallel to the direction of the applied driving force. We term this phase [*guided plastic motion*]{} (GPM), since vortex neighbors slip past each other near the twin boundary. The vortices trapped in the twin boundary remain permanently pinned in this phase. We also observe a build-up or a higher density of flux lines along one side of the twin boundary. This type of density profile has been previously observed in flux-gradient driven simulations and magneto-optical experiments.
At higher drives, as shown in Fig. 1(c,d) with $f_{d} = 0.35f_{0}$, there is a transition to a more disordered flow and the vortices start to [*cross*]{} the twin boundary. The overall vortex structure \[Fig. 1(c)\] is more disordered than it was at lower drives \[Fig. 1(a)\]. Unlike the guided plastic motion phase, the vortices pinned along the twin boundary are only temporarily trapped, and occasionally escape from the twin and are replaced by new vortices intermittently. The vortex trajectories shown in Fig. 1(d) also indicate that some vortex guiding still occurs. We label this phase the [*plastic motion*]{} (PM) phase. At even higher driving currents we observe a transition from the plastic flow phase to an [*elastic motion*]{} (EM) phase where the effect of the twin boundary becomes minimal, as shown in Fig. 1(e,f) for $f_{d} = 1.25f_{0}$. Here, the vortex lattice reorders \[Fig. 1(e)\], the vortices flow [*along*]{} the direction of the applied Lorentz force \[Fig. 1(f)\], and no build-up of the flux near the twin appears.
Current-Voltage Characteristics and Vortex Structure
====================================================
In order to quantify the phases illustrated in Fig. 1, we analyze the transverse $V_{y}$ and longitudinal $V_{x}$ average vortex velocities, as well as the six-fold coordination number $P_{6}$ and the average value of the first order peaks in the structure factor, $<S(k)>$, as a function of applied drive. As shown in Fig. 2, for drives less then the bulk pinning, $f_{d} < f_{p} = 0.02f_{0}$, the vortex lattice is pinned and $V_{y} = V_{x} = 0$. For low drives, $0.02f_0 < f_{d} < 0.17f_{0}$, the vortex velocities increase linearly with driving force, and $V_{x} \approx V_{y}$ indicating that the vortices are following the twin boundary by moving at a $45^{\circ}$ angle, as was shown in Fig. 1(b). The fraction of six-fold coordinated vortices, $P_{6} = 0.8$, remains roughly constant throughout the guided plastic motion phase. Above $f_{d}/f_{0} = 0.225$, two trends are observed. First, the [*longitudinal*]{} velocity $V_{x}$ continues to increase. This trend can be better seen in the inset of Fig. 2(a), which has a larger range of values for the vertical axis in order to monitor the linear growth over a wider range of velocities. Second, the [*transverse*]{} velocity $V_{y}$ flattens and then begins to decrease, indicating that the vortices have begun to move across the twin boundary.
The vortex lattice becomes slightly more disordered in this plastic flow phase as indicated by the drop in $P_{6}$ and the smaller drop in $<S(k)>$. As $f_{d}$ is increased further, $V_{y}$ gradually decreases, but remains finite as vortices cross the twin at an increasing rate. When $V_{y}$ approaches zero, near $f_{d}/f_{0} = 0.85$, the vortex lattice [*reorders*]{} as indicated by the increase in $P_{6}$ and $<S(k)>$. We note that the reordering transition in $P_{6}$ is considerably sharper than that typically observed in simulations with random pinning.
Noise Measurements
==================
An indirect experimental probe of the plastic vortex motion is the voltage noise produced by the flowing flux. During plastic flow the voltage noise is expected to be maximal. Indeed, in simulations with random pinning [@ShortDriven; @Rivers], large noise power was associated with the highly plastic motion of a disordered vortex lattice. Further, large noise is considered a signature for plastic flow in the peak effect regime. In order to compare the different plastic flow phases seen here with those observed for random pinning, we measure the noise power for each phase and plot the results in Fig. 3 along with the corresponding $V_{y}$ versus $f_{d}$ curve.
The noise power is relatively low in the GPM phase, increases to a large value in the PM phase, and then gradually decreases as the EM phase is approached. In the GPM regime, although tearing of the vortex lattice occurs at the boundaries between the pinned and flowing vortices, the vortex trajectories follow fixed channels and a large portion of the vortex lattice remains ordered. This very orderly vortex motion produces little noise. In the PM phase, the vortex lattice is highly disordered and the trajectories follow continuously changing paths so the corresponding voltage noise power is high. This difference in noise power between the static and changing channels for vortex flow agrees well with results obtained in systems with strong random pinning. In such systems, when the vortex flow follows fixed winding channels that do not change with time, low noise power is observed above the depinning threshold [@ShortDriven; @Rivers]. Similarly, when the pinning is weak and the vortices move in straight fixed lines, low voltage noise is observed [@ShortDriven; @Rivers]. This latter case agrees well with the result seen here in the GPM and EM phases, when the vortices follow straight paths and produce little noise power.
Dynamic Phase Diagrams
======================
To generalize our results to other parameters, we construct a phase diagram of the dynamic phases. We first measure the evolution of $V_{x}$, $V_{y}$ and $P_{6}$ as a function of driving force for varying $f_{p}^{TB}$ from $ 1.25f_{0}$ to $0.25f_{0}$. These are seen in Fig. 4. When the pinning strength $f_{p}^{TB}$ inside the twin increases, the width of the PM region grows, and the amount of disorder in the vortex lattice increases, as seen in the decrease of $P_{6}$. From the curves shown in Fig. 4, we construct a [*dynamic phase diagram*]{} which is plotted in Fig 5(a). We determine the transition between the guided plastic and plastic flow phase from the onset of disorder in $P_{6}$ and the downturn in $V_{y}$, whereas the plastic motion to elastic motion transition line is marked at the point when $P_{6}$ begins to plateau. The driving force $f_{d}$ at which both the GPM-PM and PM-EM transitions occur each grow linearly with $f_{p}$. In particular, the PM-EM transition roughly follows $f_{d} = f_{p}^{TB}$, indicating that the vortex lattice reorders once the pinning forces are overcome.
It might be expected that the transition out of the guided plastic motion phase would fall at $f_{d} = f_{p}^{TB}$, when the vortices are able to depin from the twin boundary. Since vortex interactions are important, however, in actuality the vortex density increases on one side of the twin while a lower vortex density appears on the other side. This localized flux gradient produces an additional force on the vortices at the twin boundary, depinning them at a driving force $f_{d} < f_{p}^{TB}$. The additional force from the flux-gradient is not spatially uniform, unlike the driving force, so some of the vortices will depin before others in a random manner. Once the applied driving force and the gradient force are large enough to start depinning vortices from the twin, the flux lines enter the plastic flow phase. The effect of the pinning on the vortices does not fully disappear until $f_{d} > f_{p}^{TB}$, however, which is seen in the existence of a finite $V_{y}$. We also note that there is a [*pinned phase*]{} where no vortex motion occurs when $f_{d} < f_{p}$.
By changing the vortex density we can examine the effects of changing the effective vortex-vortex interaction. In Fig. 5(b) we plot the phase diagram constructed from a series of simulations in which the vortex density is varied. As the vortex density decreases the GPM-PM and the PM-EM transition lines shift to higher drives. This is because lower values of $B$ (or $N_v$) increase the effective pinning force and shift the boundary to higher values of $f_d$.
Conclusion
==========
We have examined the dynamics of driven superconducting vortices interacting with twin boundary pinning. We find three distinct flow phases as a function of driving force. In the guided plastic motion phase, the partially ordered vortex lattice flows in stationary channels aligned with the twin boundary. In this phase the transverse and longitudinal velocities are equal and there is only a small amount of noise in the velocity signals. At higher drives, a flux gradient builds up along the twin and the vortices begin to cross the twin boundary intermittently. In this phase the vortex lattice is disordered and a large amount of voltage noise appears. The guiding effect of the twin gradually decreases for increasing drives and the vortex lattice reorders, producing an elastic flow phase. By conducting a series of simulations we have constructed phase diagrams both as a function of twin boundary pinning strength and as a function of the vortex density. The phase boundaries all shift linearly in driving force as the pinning strength increases. As the vortex density is lowered the width of the guided motion region increases, while the onset of the elastic motion phase is constant.
Twin boundaries correspond to one type of correlated pinning. Another type involves periodic arrays of pinning sites [@paps]. The dynamic phase diagrams of these structures with correlated pinning are also under current intense investigation.
[*Note Added:*]{} After completing this work we became aware of the experiments in Ref. [@Pastoriza] which measure both the longitudinal and transverse voltage signal for vortices driven in samples with unidirectional twin-boundaries. When the vortices are driven at 52 degrees with respect to the twin boundaries, at low temperatures the vortex motion deviates strongly from the direction of drive with a component moving along the twin boundary. Using this experimental set-up it should be possible to observe both the transverse and longitudinal vortex velocity as a function of applied current.
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H. Pastoriza, S. Candia and G. Nieva, to be published.
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---
abstract: 'Nonparametric correlations such as Spearman’s rank correlation and Kendall’s tau correlation are widely applied in scientific and engineering fields. This paper investigates the problem of computing nonparametric correlations on the fly for streaming data. Standard batch algorithms are generally too slow to handle real-world big data applications. They also require too much memory because all the data need to be stored in the memory before processing. This paper proposes a novel online algorithm for computing nonparametric correlations. The algorithm has $O(1)$ time complexity and $O(1)$ memory cost and is quite suitable for edge devices, where only limited memory and processing power are available. You can seek a balance between speed and accuracy by changing the number of cutpoints specified in the algorithm. The online algorithm can compute the nonparametric correlations 10 to 1,000 times faster than the corresponding batch algorithm, and it can compute them based either on all past observations or on fixed-size sliding windows.'
author:
- |
Wei Xiao\
\
bibliography:
- 'npcorr.bib'
title: An Online Algorithm for Nonparametric Correlations
---
Introduction
============
Robust statistics and related methods are widely applied in a variety of fields [@huber2011robust; @rousseeuw2005robust; @zaman2001econometric; @xiao2016robust]. Nonparametric correlations such as Spearman’s rank (SR) correlation and Kendall’s tau (KT) correlation are commonly used robust statistics. They are often used as a replacement of the classic Pearson correlation to measure the relationship between two random variables when the data contain outliers or come from heavy-tailed distributions. Applications include estimating the correlation structure of financial returns [@grothe2010estimating], comparing diets in fish [@fritz1974total], and studying the relationship between summer temperature and latewood density in trees [@franceschini2013divergence].
Nonparametric correlations have the following beneficial properties that standard Pearson correlation does not possess. First, nonparametric correlations can work on incomplete data (where only ordinal information of the data is available). Second, SR and KT equal 1 when $Y$ is a monotonically increasing function of $X$. Third, SR and KT are more robust against outliers or heavy-tailed errors. The latter two properties are demonstrated in Figure \[fig:comparison\]. Previous works have shown that the influence function of Pearson correlation is unbounded whereas the influence functions of SR and KT are both bounded [@croux2010influence; @devlin1975robust]. This fact proves that Pearson correlation lacks robustness. Furthermore, even though the Pearson correlation is the most efficient (in teams of asymptotic variance) for a normal distribution, the efficiency of ST and KT are both above 70% for all values of the population correlation coefficient [@croux2010influence].
One drawback of SR and KT compared with Pearson correlation is that they require more computational time. The computation of SR and KT requires sorting (finding rank) of the $X$ and $Y$ sequences, which is a very time-consuming step when the sample size $n$ is large. The minimum time complexities for batch algorithms for SR and KT are $O(n\log n)$ [@knight1966computer], whereas the time complexity for batch algorithm of Pearson correlation is $O(n)$.
In practice you sometimes would want to analyze correlation between variables in dynamic environments, where the data are streaming in. These environments include network monitoring, sensor networks, and financial analysis. A good algorithm should make it easy to incorporate new data and process the input sequence in a serial fashion. Such algorithms are called online algorithms in this paper. Online algorithms have interesting applications in various fields [@crammer2003ultraconservative; @oza2005online; @gama2010knowledge; @xiao2017online]. A standard online algorithm exists to compute Pearson correlation by using the idea of sufficient statistics [@gama2010knowledge]. The time complexity of this algorithm is $O(1)$, and its memory cost is also $O(1)$. However, because Pearson correlation is not robust against outliers, it is not the desirable method for some applications, for example, suppose you collect data from a huge sensor network of a complex system and you want to analyze the correlation in order to detect highly correlated sensor pairs. Outliers in sensor readings might occur because of noise, different temperature conditions, or failures of sensors or communication. Pearson correlation would not be robust enough to handle such an analysis.
This type of analysis demonstrates the need for an online algorithm for nonparametric correlations (such as SR and KT). However, the way the Pearson correlation is computed cannot be directly carried over to SR and KT. It cannot be carried over for SR because new data can change the ranks of all historical observations that were used to compute the correlation. For KT on the other hand, new data need to be compared with all historical data in the computation of the correlation. In order to compute SR and KT with streaming data exactly, it is necessary to keep all previous history in memory, which is impossible because the data streams can be unbounded in length.
This paper proposes an efficient online algorithm for SR and KT. The time complexity of this algorithm is $O(1)$, and its memory cost is also $O(1)$. Although the algorithm only approximately computes SR and KT, this paper shows through extensive simulation studies and real applications that the approximation is good enough for most cases. To the limit of the authors’ knowledge, the algorithm developed in this work is the first online algorithm for nonparametric correlation.
Online Algorithms for Nonparametric Correlations
================================================
Let $\{(x_i, y_i), i \geq 1\}$ denote the streaming inputs of two time series $x$ and $y$. At time $t$, the Pearson correlation ($r_{P}$), Spearman’s rank correlation ($r_{SR}$), and Kendall’s tau correlation ($r_{KT}$) computed based on all previous observations are defined as: $$\begin{aligned}
r_{P}\triangleq& \cfrac{\sum_{i=1}^{t}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{t}(x_i-\bar{x})^2\sum_{i=1}^{t}(y_i-\bar{y})^2}}\\
r_{SR}\triangleq& \cfrac{\sum_{i=1}^{t}(u_i - \bar{u})(v_i - \bar{v})}{\sqrt{\sum_{i=1}^{t}(u_i-\bar{u})^2\sum_{i=1}^{t}(v_i-\bar{v})^2}}\\
r_{KT}\triangleq& \cfrac{P-Q}{\sqrt{(P+Q+T)*(P+Q+U)}}\end{aligned}$$ where $u_i$ is the rank of $x_i$, $v_i$ is the rank of $y_i$, $P$ is the number of concordant pairs, $Q$ is the number of discordant pairs, $T$ is the number of ties only in $x$, and $U$ is the number of ties only in $y$.
The main idea in designing the online algorithms for nonparametric correlations is to coarsen the bivariate distribution of ($X$,$Y$). The coarsened joint distribution can represented by using a small matrix. Assume that both $X$ and $Y$ are continuous random variables. You provide $m_1^*$ and $m_2^*$ distinct cutpoints in ascending order for $X$ and $Y$ respectively, where both $m_1^*$ and $m_2^*$ are nonnegative integers. The cutpoints for $X$ are denoted as $(c^x_1$,…,$c^x_{m_1^*}){^{\mbox{\tiny {\sf T}}}}$, where $c^x_1<c^x_2<\ldots<c^x_{m_1^*}$. Similarly, the cutpoints for $Y$ are represented as $(c^y_1$,…,$c^y_{m_2^*}){^{\mbox{\tiny {\sf T}}}}$. Two default cutpoints are added for $X$: $c^x_0$ and $c^x_{m_1^*+1}$, where $c^x_0=-\infty$ and $c^x_{m_1^*+1}=\infty$. Cutpoints $\{c^x_i,\,i=0\ldots,m_1^*+1\}$ discretize $X$ into $m_1$ ranges, where $m_1=m_1^*+1$. The same is done for $Y$, and cutpoints $\{c^y_i,\,i=0\ldots,m_2^*+1\}$ discretize $Y$ into $m_2$ ranges, where $m_2=m_2^*+1$. The $m_1 \times m_2$ count matrix $M$ is then constructed, where $M[i,j]$ stores the number of observations that falls into the range $[c^x_{i-1}, c^x_{i})\times[c^y_{j-1}, c^y_{j})$. An example of an $M$ matrix is shown in Figure \[plot:count\_matrix\], where three cutpoints are chosen for $X$ and four cutpoints are chosen for $Y$. Using the count matrix $M$ has two advantages: first, instead of entire $(x_i, y_i)$ series (which maybe unbounded in length) being stored,the information is stored in a matrix of fixed size. Second, when $(x_i,y_i)$ are discretized and stored in $M$, they are naturally sorted, and fast algorithms exist to quickly compute $r_{SR}$ and $r_{KT}$ from $M$. This paper proves that the time complexity of the algorithms is $O(m_1m_2)$ for both Spearman’s rank correlation and Kendall’s tau correlation. Because both $m_1$ and $m_2$ are fixed integers, the algorithms for both Spearman’s rank correlation and Kendall’s tau have $O(1)$ time complexity and $O(1)$ memory cost. This makes the implementation of these algorithms quite attractive on edge devices, where only limited memory and processing power are available. In practice cutpoints for $X$ and $Y$ need to be chosen. One good choice for cutpoints are the equally spaced quantiles of the random variable. For example, to choose 9 cutpoints for $X$, we can use the sample quantiles of $X$ that correspond to the probabilities $0.1, 0.2, \ldots, 0.9$.
![Example of count matrix $M$ with three cutpoints of $x$ and four cutpoints of $y$.[]{data-label="plot:count_matrix"}](count_matrix){width="80.00000%"}
The preceding discussion assumes that both $X$ and $Y$ are continuous. When $X$ and $Y$ are discrete or ordinal, cutpoints can be selected so that each pair of consecutive cutpoints contains only one level of the random variable. When both $X$ and $Y$ are discrete or ordinal no information is lost by using $M$ to approximate the bivariate distribution of ($X$, $Y$), and the algorithm’s result is the exact nonparametric correlation between $X$ and $Y$.
The general online algorithm for SR and KT is given in Algorithm \[algo:online\_npcorr\]. To expedite the computation, not only is matrix $M$ tracked, but also its row sum $n_{\mathrm{row}} =
(n_{\mathrm{row}}[1],\ldots, n_{\mathrm{row}}[m_1]){^{\mbox{\tiny {\sf T}}}}$, its column sum $n_{\mathrm{col}} = (n_{\mathrm{col}}[1],\ldots, n_{\mathrm{col}}[m_2]){^{\mbox{\tiny {\sf T}}}}$, and its total sum $n=\sum_{i=1}^{m_1}\sum_{j=1}^{m_2}M[i,j]$. The algorithm is also designed to compute nonparametric correlation and return the result every $n_{\mathrm{gap}}$ new observations. The $n_{\mathrm{gap}}$ is a user-specified parameter, where $n_{\mathrm{gap}}\geq 1$. When the observation index $t$ mod $n_{\mathrm{gap}}$ is not equal to 0, only $M$, $n_{\mathrm{row}}$, $n_{\mathrm{col}}$, and $n$ need to be updated (Step 3–6), and the nonparametric correlation does not need to be computed (Step 8) in the iteration. When $t$ mod $n_{\mathrm{gap}}$ is equal to 0, it is necessary both to update $M$, $n_{\mathrm{row}}$, $n_{\mathrm{col}}$ and $n$, and to compute the nonparametric correlation. Unlike the step that computes the nonparametric correlation, the updating steps can be done very efficiently with time complexity $O(\max(\log(m_1),\log(m_2)))$.
Input: $\{(x_t, y_t)\}_{t=1}^{T}$ (streaming observations), $(c^x_1$,…,$c^x_{m_1}){^{\mbox{\tiny {\sf T}}}}$ (cutpoints for $x$), $(c^y_1$,…,$c^y_{m_2}){^{\mbox{\tiny {\sf T}}}}$ (cutpoints for $y$), $n_{\mathrm{gap}}$.
Step 8 in Algorithm \[algo:online\_npcorr\] is described in detail for SR and KT respectively in Algorithms \[algo:online\_SR\] and \[algo:online\_KT\], where the nonparametric correlations are quickly computed based on matrix $M$, $n_{\mathrm{row}}$, $n_{\mathrm{col}},$ and $n$. It is easy to verify that the time complexity of both algorithms is $O(m_1m_2)$ in linear proportion to the number of cells in matrix $M$.
Input: $M$, $n_{\mathrm{row}}$, $n_{\mathrm{col}}$, $n$, $m_1$, and $m_2$ \# iteratively compute the rank that corresponds to each row of $M$ $r = 0$ \# iteratively compute the rank that corresponds to each column of $M$. $r = 0$. $r^{*}_{\mathrm{row}}\leftarrow r_{\mathrm{row}} - (n+1)/2$; $r^{*}_{\mathrm{col}}\leftarrow r_{\mathrm{col}} -
(n+1)/2.$ $r^{*}_{\mathrm{row}}\leftarrow
r^{*}_{\mathrm{row}}/\sqrt{\sum_{i=1}^{m_1}n_{\mathrm{row}}[i]r^{*}_{\mathrm{row}}[i]^2}$; $r^{*}_{\mathrm{col}}\leftarrow
r^{*}_{\mathrm{col}}/\sqrt{\sum_{i=1}^{m_2}n_{\mathrm{col}}[i]r^{*}_{\mathrm{col}}[i]^2}.$ $corr \leftarrow (r^{*}_{\mathrm{row}}){^{\mbox{\tiny {\sf T}}}}M r^{*}_{\mathrm{col}}.$
Input: $M$, $n_{\mathrm{row}}$, $n_{\mathrm{col}}$, $n$, $m_1$, and $m_2$. Initialize $N$ as an $m_1$ by $m_2$ zero matrix. Compute $P$ (the number of concordant pairs): $P = \sum_{i=1}^{m_1}\sum_{j=1}^{m_2}M[i,j]*N[i,j]$ Compute $T$ (the number of ties only in x): $T = [\sum_{i=1}^{m_1}(n_{\mathrm{row}}[i]^2 - \sum_{j=1}^{m_2}M[i,j]^2)]/2$ Compute $U$ (the number of ties only in y): $T = [\sum_{j=1}^{m_2}(n_{\mathrm{col}}[j]^2 - \sum_{i=1}^{m_1}M[i,j]^2)]/2$ Compute $B$ (the number of ties in both x and y): $B=\sum_{i=1}^{m_1}\sum_{j=1}^{m_2}M[i,j]*(M[i,j] - 1)/2$ Compute $Q$ (the number of discordant pairs): $Q = (n+1)*n/2 - P- T - U - B$ $corr \leftarrow (P-Q)/\sqrt{(P+Q+T)*(P+Q+U)}$
The parameters $(m_1, m_2)$ control a tradeoff between the accuracy and efficiency of the online algorithms. The more cutpoints that are chosen for $x$ and $y$ (larger $m_1$, $m_2$), the more accurate the approximation of the bivariate distribution of $(x,y)$ with $M$ and a more accurate result is generally achieved. However, increasing $m_1$ and $m_2$ also decreases the speed of the online algorithm. Based on the extensive numerical studies of the next section, a rule-of-thumb choice of $(m_1,m_2)$ is $m_1=m_2=30$ for SR and $m_1=m_2=100$ for KT.
In practice, the proposed online algorithms for nonparametric correlations usually work well for the following reasons: First, when $(m_1,m_2)$ increase to infinity, the result of online algorithms converges to the true value. Second, for a reasonably large $(m_1,m_2)$, each cell of matrix $M$ represents only a very local area of the $(X, Y)$ distribution. Positive errors and negative errors can cancel each other out when summed together.
Algorithms \[algo:online\_SR\] and \[algo:online\_KT\] compute SR and KT over all past data. Frequently, you are interested in computing the statistics only over the recent past. Specifically, you might like to compute SR and KT with a sliding window of a fixed size. Algorithm \[algo:online\_npcorr\] can be easily modified to deal with such cases. The only change that is needed is to add some steps after step 6; in the added steps $M$, is updated by first finding the row index and the column index that correspond to the observation $(x_{t-n_{\mathrm{win}}}, y_{t-n_{\mathrm{win}}})$ and then decreasing the corresponding cell of $M$ by 1. Here $n_{\mathrm{win}}$ represents the size of the sliding windows. The details are left to the readers.
Simulation Studies
==================
In the following simulation studies, equal numbers of cutpoints are always chosen for both $X$ and $Y$, and all cutpoints are chosen as equally spaced quantiles of a standard normal distribution. We implement the batch and online nonparametric correlation algorithms in python. Users can download the package from https://github.com/wxiao0421/onlineNPCORR.git.
Simulation Study with Nonparametric Correlations Computed over All Past Observations
------------------------------------------------------------------------------------
This simulation evaluates the online algorithms for SR and KT (computed over all past observations) by comparing them with the corresponding batch algorithms. The $x_i$, $i=1,\ldots,T$ are generated from an independently and indentically distributed (iid) normal distribution $N(0,1)$. Let $y_i=(z_i+\sigma x_i)/\sqrt{\sigma^2+1}$, where $z_i$, $i=1,\ldots,T$ are iid $N(0,1)$ random variables, which are independent of $x_i$, $i=1,\ldots,T$. It is easy to verify that both $\{x_i\}_{i=1}^{T}$ and $\{y_i\}_{i=1}^{T}$ are iid $N(0,1)$ with a Pearson correlation coefficient of $\sigma/\sqrt{\sigma^2+1}$.
The result for SR is shown in Figure \[fig:sim1\_SR\]. All numbers are averaged over 10 replications. All results in subplots (a) and (b) are based on $n_{\mathrm{gap}}=1$. Subplot (a) compares the run times of the batch algorithm and the online algorithms with different numbers of cutpoints. It is clear that for the online algorithms, increasing the number of cutpoints decreases the speed of the algorithms. Furthermore, as the number of observations ($T$) increases, the differences between the run times for the batch algorithm and the online algorithm also increase dramatically. The SR algorithm with 20 cutpoints (online SR (20)) takes less than 10 seconds to run a $T=10^5$ case, whereas the batch algorithm takes more than 1,000 seconds. When $T$ increases to $10^6$, the batch algorithm becomes too slow to handle such cases (time complexity $O(T^2\log T)$), whereas it is easy to both numerically and theoretically prove that the run time of the online algorithm is proportional to $T$. Subplot (b) compares the L1 error of the estimated Spearman’s rank correlation from the online algorithm with different numbers of cutpoints (computed at $T$). The L1 error does not seem to increase with $T$, and it generally decreases with the number of cutpoints. For all cases, the L1 error is kept below 0.004. Subplot (c) compares the run times of the online algorithm (with 50 cutpoints) for $n_{\mathrm{gap}}=1$ and $100$. The increase in speed is 30-fold for $n_{\mathrm{gap}}=100$ for all $T$. Last, the batch algorithm is implemented very efficiently in C (using the Python Scipy package) whereas the online algorithm is Purely python code. So if the online algorithm is also implemented in C, you would likely see another 10 to 100-fold speed increase for the online algorithm.
\
\
The result for KT is shown in Figure \[fig:sim1\_KT\]. Subplot (a) compares the run times of the batch algorithm and the online algorithm with different numbers of cutpoints. The observed pattern is similar to that of SR, where the run times increase with the number of cutpoints in the online algorithm. Furthermore, as the number of observations ($T$) increases, the differences between the run times of the batch algorithm and online algorithm also increase dramatically. The KT online algorithm with 100 cutpoints (online KT (100)) takes approximately 10 seconds to run a $T=10^4$ case, whereas the batch algorithm takes more than 1,000 seconds. When $T$ increases to $10^5$, the batch algorithm becomes too slow to handle such cases, whereas the online algorithm can still finish the computation in a very short period of time. Subplot (b) compares the L1 error of the estimated Kendall’s tau correlation of the online algorithm with different numbers of cutpoints (computed at $T$). The L1 error does not seem to change much with $T$, and it decreases with the number of cutpoints. For all cases where the number of cutpoints is larger than 50, the L1 error is below 0.01. Subplot (c) compares the run times of the online algorithm under $n_{\mathrm{gap}}=1$ and $100$ (with 50 cutpoints). You see increases of approximately 60 to 70 times for $n_{\mathrm{gap}}=100$ and all $T$.
\
\
Simulation Study with Nonparametric Correlations Computed Based on Sliding Windows
----------------------------------------------------------------------------------
This simulation study compares the batch and online algorithms for SR and KT based on sliding windows. Generate $x_i$, $i=1,\ldots,T$ from an iid normal distribution $N(0,1)$. Let $y_i=(z_i+\sigma(i)
x_i)/\sqrt{\sigma(i)^2+1}$, where $z_i$, $i=1,\ldots,T$ are iid $N(0,1)$ random variables, which are independent with $x_i$, $i=1,\ldots,T$. Then $T=100,000$ and $n_{\mathrm{win}}=10,000$ for SR, and $T=10,000$ and $n_{\mathrm{win}}=1,000$ for KT. We choose $\sigma(i)=5[(i-m)/m]^2$, where $m=50,000$ for SR and $m=5,000$ for KT.
The results are shown in Figure \[fig:sim2\]. The SR online algorithm generates a very accurate estimate of Spearman’s rank correlation even when the number of cutpoints is small ($m_1=m_2=30$). The KT online algorithm seems to generate a very accurate estimate of the Kendall’s tau correlation when the absolute value of the correlation is small, but it seems to generate a more biased estimate when the absolute value of the correlation is large. This is because when $x$, $y$ are highly correlated, the $(x_i,y_i)$ pairs are likely to be concentrated on the diagonal of matrix $M$. This concentration leads to a poor approximation of the bivariate distribution of $(x,y)$ with matrix $M$, which leads to biased estimates of Kendall’s tau correlation. For the KT online algorithm we suggest keeping the number of cutpoints above 100 in order to achieve a more accurate result.
Application to Sensor Data Generated in Industrial plant
========================================================
This section uses the proposed online algorithms to compute nonparametric correlations based on sensor data that were generated in industrial plant from 2015 Prognostics and Health Management Society Competition [@phm]. The data contains sensor readings of 50 plants. For each plant, it provides sensor readings of four standard sensors S1-S4, and four control sensors R1-R4. We use the sensor readings of the first component in the first plant to do our experiment, where we compute nonparametric correlations based on sliding windows with window size 35,040 (corresponds to a one-year window).
First, we compute the nonparametric correlation between $R_1$ and $S_1$. $R_1$ contains 10 unique values and we choose 9 cutpoints so that each unique value has its own cell in matrix $M$. $S_1$ has 121 unique values, and we experiment on two methods to choose cutpoints for $S_1$. In the first method, we choose 120 cutpoints for $S_1$ so that each unique value of $S_1$ has its own cell in matrix $M$. In the second method, the cutpoints are chosen by first computing sample quantiles of $S_1$ at probabilities $0.05, 0.10, \ldots, 0.95$, and keeping only the unique values. This leads to choosing 19 cutpoints for $S_1$. The result is shown in Figure \[fig:R1vsS1\]. We refer the result of the online algorithm with 19 cutpoints for $S_1$ as online SR (approximate) and online KT (approximate), respectively. Because the returned nonparametric correlations will only approximately equal the true values. We see a 20-50 fold speed up for SR and 5-20 fold speed up for KT.
Then, we compute the nonparametric correlation between $R_3$ and $S_3$. We choose 7 cutpoints for $R_3$ and 11 cutpoints for $S3$ so that each unique value of $R_3$ and $S_3$ has its own cell in matrix $M$. The result is shown in Figure \[fig:R3vsS3\]. The online algorithm returns the same result as that of batch algorithm with a 20-40 fold speed up.
Conclusion
==========
This paper proposes a novel online algorithm for the computation of Spearman’s rank correlation and Kendall’s tau correlation. The algorithm has time complexity $O(1)$ and memory cost $O(1)$, and it is quite suitable for edge devices, where only limited memory and processing power are available. By changing the number of cutpoints specified in the algorithm, users can seek a balance between speed and accuracy. The new online algorithm is very fast and can easily compute the correlations 10 to 1,000 times faster than the corresponding batch algorithm (the number varies over the settings of the problem). The online algorithm can compute nonparametric correlations based either on all past observations or on fixed-size sliding windows.
|
---
abstract: 'The dependence of wakefield amplitude and phase on beam and plasma parameters is studied in the parameter area of interest for self-modulating proton beam-driven plasma wakefield acceleration. The wakefield sensitivity to small parameter variations reveals the expected level of shot-to-shot jitter of experimental results. Of all the parameters, the plasma density stands out, as the wakefield phase is extremely sensitive to this parameter. The study of large variations determines the effects that limit the achievable accelerating field in different parts of the parameter space: nonlinear elongation of the wakefield period, insufficient charge of the drive beam, emittance-driven beam divergence, and motion of plasma ions.'
author:
- 'K.V.Lotov, V.A.Minakov, A.P.Sosedkin'
title: 'Parameter sensitivity of plasma wakefields driven by self-modulating proton beams'
---
Introduction
============
The concept of proton driven plasma wakefield acceleration emerged five years ago [@NatPhys9-363; @PRST-AB13-041301]. The main motivation for using protons is that the energy content of state-of-the-art proton beams is sufficiently high to bring a substantial amount of electrons to TeV energy scale in a single plasma section. Since plasmas can support extremely strong accelerating fields [@RMP81-1229], this concept may open a path to the future of high-energy particle physics [@AWAKE].
In the initial proposal [@NatPhys9-363; @PRST-AB13-041301], the proton beam was assumed to be longitudinally compressed to the sub-millimeter length. That was necessary for driving the plasma wave with the wavelength of 1mm or shorter. However, as available proton beams have lengths of about 10cm, such strong compression seems unrealistic to obtain with conventional techniques [@IPAC10-4395]. Soon it was realized that the plasma wave can slice the initially long proton beam into the train of equally spaced micro-bunches which in turn resonantly drives the plasma wave [@PPCF53-014003; @PRL104-255003]. The slicing is caused by the saturated self-modulation instability [@EPAC98-806; @PRL104-255003; @PRL107-145003; @PRL107-145002].
Parameter, notation Value
------------------------------------------------------------------ ------------------------------------
Plasma density, $n_0$ $7 \times 10^{14}\,\text{cm}^{-3}$
Atomic weight of plasma ions, $M_i$ 85.5
Beam population, $N_b$ $3\times 10^{11}$
Beam length, $\sigma_{zb}$ 12cm
Beam radius, $\sigma_{rb}$ 0.02cm
Beam energy, $W_b$ 400GeV
Beam energy spread, $\delta W_b$ 0.35%
Beam angular spread, $\delta \alpha_b = \epsilon_b/\sigma_{rb}$, $4.5 \times 10^{-5}$
Seed location relative to beam center, $\xi_s$ 0cm
Plasma skin depth, $c/\omega_p$, 0.02cm
Wavebreaking field, $E_0=mc\omega_p/e$, 2.54GV/m
Interaction length, $L_\text{max}$ 10m
Maximum beam density, $n_{b0}$ $4\times 10^{12}\,\text{cm}^{-3}$
Beam emittance, $\epsilon_b$ $9\,\mu$mmrad
Beam normalized emittance, $\epsilon_{bn}$ 3.6mmmrad
: Parameters of the baseline AWAKE scenario and the notation.[]{data-label="t1"}
To test beam self-modulation and subsequent wakefield excitation, a proof-of-principle experiment named AWAKE was launched at CERN [@AWAKE; @IPAC13-1179; @TDR; @NIMA-740-48]. In the baseline experiment scenario, the 400GeV proton beam from the SPS synchrotron passes through the 10 meter long plasma section. Beam self-modulation is seeded by a short laser pulse co-propagating with the beam, which instantly ionizes a highly uniform rubidium vapor and produces the plasma of the same density as that of the neutral vapor [@Oz]. As the wakefield of the seed perturbation is substantially stronger than the shot noise [@PRST-AB16-041301], the beam self-modulates at first 4 meters and then excites the plasma wave and accelerates externally injected test electrons to the energy of about 2GeV. The drive beam and plasma parameters corresponding to the baseline scenario are given in Table \[t1\].
In this paper we numerically study how sensitive are the excited wakefields to variations of the parameters listed in the upper part of Table \[t1\]. The lower part values are provided for reference. We examine the wakefield amplitude in Section \[s2\] and the wakefield phase in Section \[s3\]. The wakefield response to small parameter variations characterizes the level of shot-to-shot jitter of experimental results. The study of large variations determines which of the effects can limit the accelerating fields and which parameters are to be improved for better performance.
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SPS (AWAKE) LHC-2009 LHC-nominal HERA TEVATRON RHIC
Beam energy (TeV) 0.4 3.5 7 0.92 0.98 0.25
Bunch length (cm) 12 9 7.5 8.5 50 70
Bunch population (units of $10^{11}$) 3 1.5 1.15 0.7 0.9 1.65
Peak current (A) 50 30 30 15 3.5 4.5
--------------------------------------- --------------- ------------- ---------------- --------- ------------- -------
The key beam parameter determining the wakefield behavior is the peak current or, equivalently, the field increment due to a single micro-bunch focused to $\sigma_{rb} \sim c/\omega_p$ [@PoP18-103101]. This parameter does not vary much in modern TeV-class proton accelerators (Table \[t2\]). Thus the qualitative results obtained for the SPS proton beam are also applicable to several other machines.
The simulations are made with the quasi-static axisymmetric 2d3v code LCODE [@PoP5-785; @PRST-AB6-061301; @IPAC13-1238; @LCODE]. We use cylindrical coordinates $(r, \varphi, z)$ with the $z$-coordinate measured from the plasma entrance and the co-moving coordinate $\xi = z-ct$ measured from the beam central plane.
Wakefield amplitude {#s2}
===================
When we analyze the wakefield amplitude, we refer to the maximum $\Phi_\text{max} (z)$ of the dimensionless wakefield potential $\Phi (z,\xi)$ on the axis: $$\label{e1}
\Phi (z,\xi) = \frac{\omega_p}{c E_0} \int_\xi^{\infty} E_z(z, \xi') \, d \xi',$$ where $E_z$ is the on-axis electric field. The maximum is taken over all values of $\xi$ for a fixed $z$. The reason for using this quantity is that the wakefield potential is more noise-resistant than $E_z$ itself. We also use the dimensional quantity $\Phi_m = E_0 \Phi_\text{max}$ to characterize the accelerating gradient.
Main results of the parameter scan are shown in Fig.\[f1-main\]. In each group of graphs we vary one parameter from Table \[t1\] and plot functions $\Phi_m (z)$.
. []{data-label="f2-density"}](f2-density.eps){width="223bp"}
The plasma density dependence of the wakefield amplitude \[Fig.\[f1-main\](a)\] is mainly determined by the scaling $E_0 \propto \sqrt{n_0}$. Namely, the higher the plasma density is, the stronger field this plasma can support. At high plasma densities, the beam is several plasma skin depths wide, and beam filamentation could develop [@PF30-252]. This phenomenon is substantially three-dimensional and cannot be simulated by the axisymmetric code. The exact beam width at which filamentation starts to dominate over self-modulation is not clear yet. We therefore plot the highest density curves by thinner lines to indicate that these curves may not be realistic.
The level of wakefield saturation can be understood from dependence of the dimensionless amplitude $\Phi_\text{max}$ on the dimensionless propagation distance (Fig.\[f2-density\]). We see that there is an optimum plasma density at which the dimensionless wakefield amplitude is maximal. At higher densities, the wakefield is limited by nonlinear elongation of the plasma wave period. The approximate theory of this effect [@PoP20-083119] gives the following expression for the maximum wakefield amplitude at the moment of optimum beam micro-bunching: $$\label{e2}
\Phi_\text{max} = \left( \frac{4 \beta \Delta E}{\alpha E_0} \right)^{1/3},$$ where $\Delta E$ is the wakefield increment due to one micro-bunch, and $\alpha \approx 0.1$ and $\beta \approx 0.5$ are numerical factors that depend on the beam radius. Formula (\[e2\]) gives $\Phi_\text{max} \approx 0.4$ for the baseline case in agreement with Fig.\[f2-density\]. Since the plasma wave is weakly nonlinear and the beam is wider than $c/\omega_p$, the contribution $\Delta E$ of a single micro-bunch is determined mainly by the beam density [@PoP18-103101]. Thus, $\Delta E$ changes almost not at all if we change the plasma density. The limiting field (\[e2\]) depends on the plasma density through $E_0$ and $\alpha$. Compared to the baseline variant ($n_0 = 7 \times 10^{14} \text{cm}^{-3}$), the higher density variant with $n_0 = 5 \times 10^{15} \text{cm}^{-3}$ has approximately 2.7 times higher $E_0$, twice bigger $\alpha$ (as the wave is almost one dimensional with $\alpha=3/16$), and 40% smaller expression (\[e2\]) for $\Phi_\text{max}$ in good agreement with Fig.\[f2-density\]. The wakefield amplitude at high densities is thus limited by the nonlinear saturation of wave growth.
The theory [@PoP20-083119] also specifies the number of micro-bunches coherently exciting the wakefield: $$\label{e3}
N_\text{eff} = \left( \frac{4 \beta E_0^2}{\alpha \Delta E^2} \right)^{1/3}.$$ Multiplied by the plasma wavelength $2\pi c/\omega_p$, this number gives the length $L_\text{eff}$ of the beam part that efficiently excites the wave. For the baseline variant $N_\text{eff} = 130$ and $L_\text{eff} = 16$cm, which is roughly the whole available beam. At smaller plasma densities the plasma wavelength is longer, the number $N$ of macro-bunches in the beam is smaller than (\[e3\]), the field growth along the beam does not come to saturation, and the maximum field is roughly $N \Delta E \propto \sqrt{n_0}$. The low-density curves in Fig.\[f2-density\] follow this scaling quite well, as the curve maxima for $n_0 = (5,3,2) \times 10^{14} \text{cm}^{-3}$ constitute a ratio of $\sqrt{5} : \sqrt{3} : \sqrt{2}$. The wakefield amplitude at low plasma densities is thus limited by the beam length, and the baseline case is located close to the border between plasma nonlinearity-limited and beam length-limited regimes of field excitation.
. Illustration of true (thick lines) and effective (thin lines) beam densities for normal (red) and elongated (blue) beams (d). []{data-label="f3-population"}](f3-population.eps){width="219bp"}
The dependence of the wakefield amplitude on the beam population, either for the fixed beam length \[Fig.\[f1-main\](b)\] or for the fixed peak current \[Fig.\[f1-main\](c)\], is also determined by interplay of the above two limitations. To demonstrate this, we plot the maximum wakefield amplitude versus varied parameters (Fig.\[f3-population\]). For the fixed beam length, $\Delta E \propto N_b$, and the formula (\[e2\]) gives the scaling $\Phi_m \propto \Phi_\text{max} \propto N_b^{1/3}$ \[thin solid line in Fig.\[f3-population\](a)\]. If the wakefield amplitude is limited by the beam length, then it is directly proportional to the number of particles driving the wave, and $\Phi_m \propto N_b$ (thin dotted line). The baseline case (always plotted in red) is in the transition region.
For the fixed beam current there is no simple scaling for the maximum amplitude in the nonlinearity-limited regime. For a uniform beam with a constant current we may expect no dependence on the beam length. Here the beam has a Gaussian-like density distribution, so the effective beam density is smaller than the peak density. The longer the beam, the closer the effective density to the peak density \[Fig.\[f3-population\](d)\]. As the average $\Delta E$ is proportional to the effective density, we observe a weak amplitude growth with the beam population in the nonlinearity-limited part of Fig.\[f3-population\](b). In the length-limited part, $\Phi_m \propto N_b \propto \sigma_{zb}$.
If we change the beam length keeping the beam population fixed \[Fig.\[f1-main\](d)\], then again the result follows the scaling (\[e2\]). Now $\Delta E \propto n_{b0} \propto \sigma_{zb}^{-1}$, and $\Phi_m \propto \sigma_{zb}^{-1/3}$. This scaling is shown in Fig.\[f3-population\](c) with a thin line. For small $\sigma_{zb}$ the beam length decreases faster than $N_\text{eff} \propto \sigma_{zb}^{2/3}$, thus short beams are in the length-limited regime with $\Phi_m\approx \text{const}$.
![ Real space portraits of beam fragments at the time of developed modulation ($z=3.6$m) for initial beam radius 0.05mm (a) and 0.2mm (b).[]{data-label="f4-divergence"}](f4-divergence.eps){width="214bp"}
As we can see from Fig.\[f1-main\](e), the baseline beam radius is the optimum one for the specified set of other parameters. This optimum appears due to two effects. At large beam radii the contribution $\Delta E$ of a micro-bunch drops as $\sigma_{zb}^{-1}$. The total wakefield drops accordingly, since wave excitation for $\sigma_{zb}>0.2$mm is in the length-limited regime. At small radii the beam has a large angular spread that causes fast beam divergence and reduction of the maximum wakefield amplitude [@PoP18-103101]. The diffuse character of beam divergence for small $\sigma_{zb}$ is illustrated with Fig.\[f4-divergence\] in comparison with the usual self-modulation.
The dependence of the wakefield amplitude on the initial angular spread \[Fig.\[f1-main\](f)\] is determined by emittance driven divergence. If the emittance is below some threshold value, it has no effect on the amplitude and affects only the rate of beam degradation at the full modulation stage [@PoP18-024501]. As the emittance exceeds the threshold, the wakefield amplitude steeply drops down. Thus the baseline variant has no safety margin for beam emittance, and a slightly higher emittance would cause substantial degradation of the wakefield.
The dependence of the maximum wakefield on the ion atomic weight \[Fig.\[f1-main\](g)\] is in line with results of Ref.[@PRL109-145005]. For the chosen beam parameters, reduction of the wakefield due to ion motion is substantial for atomic weights below 40, but there is no large safety margin. For improved beam parameters, ion motion may be an issue.
![ The maximum wakefield amplitude $\Phi_m$ versus the normalized propagation distance $z/\sqrt{W_b}$ for various beam energies. []{data-label="f5-energy"}](f5-energy.eps){width="191bp"}
The beam energy has a small effect on the wakefield amplitude \[Fig.\[f1-main\](h)\], but changes the time scale of the process. The theory [@PRL104-255003] predicts that the time scale of self-modulation is proportional to $\sqrt{W_b}$, and this scaling is nicely reproduced in simulations (Fig.\[f5-energy\]). Visible deviations from the common curve shape in Fig.\[f5-energy\] observed for lowest and highest energies are due to the emittance change that accompanies the energy variation. At low energies the amplitude is reduced by emittance-driven divergence, at high energies the amplitude decreases slower at the stage of beam degradation.
Reasonable beam energy spreads have no effect on the wakefield \[Fig.\[f1-main\](i)\].
![ (a) Length dependencies of the maximum wakefield amplitude $\Phi_m (z)$ for various positions $\xi_s$ of the beam leading edge (marked near the curves) and (b) the absolute maximum of the wakefield amplitude $\Phi_m$ (bar height) versus the position of the leading edge. The curve in fragment (b) shows the shape of uncut beam. []{data-label="f6-timing"}](f6-timing.eps){width="174bp"}
The wakefield amplitude also depends on the position $\xi_s$ of the ionizing laser pulse with respect to the beam central plane (Fig.\[f6-timing\]). This distance is controlled by time synchronization of laser and proton beams. As propagation velocities of both beams are close to the speed of light, this distance changes almost not at all during the development of self-modulation. Therefore the plasma response to the two beams (proton and laser ones) is the same as to the single proton beam with a hard leading edge located at $\xi_s$. We simulate the latter case and take the initial proton beam density in the form $$\begin{gathered}
\nonumber
n_b (r, \xi) = 0.5 \, n_{b0} \, e^{-r^2/2 \sigma_{rb}^2} \left[ 1 + \cos \left( \sqrt{\frac{\pi}{2}} \frac{\xi}{\sigma_{zb}} \right) \right], \\
-\sigma_{zb} \sqrt{2\pi} < \xi < \xi_s,\end{gathered}$$ which is close to Gaussian distribution and smoothly vanishes at $|\xi| = \sigma_{zb} \sqrt{2\pi}$. For negative $\xi_s$ the wakefield is expectedly small, as there is not enough charge in the intact part of the beam. The strongest wakefield is observed for $\xi_s \approx \sigma_{zb}$, as in this case the seed perturbation is strong enough and the most dense part of the beam undergoes self-modulation. A similar result was earlier obtained for self-modulated electron beams [@PoP19-063105]. A smaller wakefield amplitude for $\xi_s > \sigma_{zb}$ is due to smaller seed perturbations and later development of self-modulation \[Fig.\[f6-timing\](a)\].
To summarize the obtained results, we list sensitivities $S_a$ of the wakefield amplitude to variation of beam and plasma parameters in Table \[t3\]. We define $S_a$ as the proportionality coefficient between the relative variation of the absolute maximum of $\Phi_m$ and the small relative variation of a quantity $X$: $$\label{e4}
\delta \Phi_m / \Phi_m = S_a^X \delta X / X.$$ To calculate the seed location sensitivity, we take $\sigma_{zb}$ as $X$. As we can see, there are no parameters of the system the wakefield amplitude is extremely sensitive to.
Wakefield phase {#s3}
===============
The wakefield phase established after self-modulation is also of high importance as it shows the optimal location for the accelerated particles. A typical phase behavior is illustrated by Fig.\[f7-phase\]. Here we show how local maxima $\xi_m$ of the wakefield potential $\Phi$ moves in the co-moving frame during beam self-modulation and subsequent degradation of micro-bunches. The slope of these curves determines the local phase velocity $v_\text{ph}$ of the plasma wave: $$\label{e5}
v_\text{ph} = c \left(1 + \frac{d \xi_m}{dz} \right).$$ The interval $-14.1\,\text{cm} < \xi < -13.5\,\text{cm}$ is selected for presentation since this is the place where the bunch of test electrons must be injected to produce a narrow final energy spectrum [@AWAKE]. This figure explains the choice of the optimum injection parameters given in Ref.[@AWAKE]. The optimum electron energy 16MeV is that for which the electron velocity equals $v_\text{ph}$ at the stage of self-modulation ($z<4$m). The optimum injection place ($z \approx 4$m) corresponds to the length of full micro-bunching. The injection delay $\xi \approx 13.8$cm with respect to the laser pulse corresponds to the area where the phase line flattens out immediately after self-modulation has developed.
![ Co-moving coordinates $\xi_m$ of several local maxima of the wakefield potential versus the propagation distance $z$ for the baseline variant. The maximum used for sensitivity calculation is shown in red.[]{data-label="f7-phase"}](f7-phase.eps){width="220bp"}
![ A family of constant phase curves for various beam populations. Circles denote the points used for calculation of phase sensitivities. []{data-label="f8-phases"}](f8-phases.eps){width="230bp"}
Parameter $S_a$ $S_\varphi$ (4m) $S_\varphi$ (10m)
---------------------------------- ------- ------------------ -------------------
Plasma density 0.4 375 340
Beam population 0.7 -4.75 0.026
Beam length (fixed peak current) 0.3 -0.5 -0.19
Beam length (fixed population) -0.4 4.2 -0.25
Beam radius 0 2.5 -0.42
Beam angular spread -0.3 0.9 0.26
Atomic weight of plasma ions 0 0.85 0.1
Beam energy 0.07 1.9 0.38
Beam energy spread 0 0 0
Seed location 0.3 1.3 -0.39
: Sensitivity of wakefield amplitude and phase to variations of beam and plasma parameters.[]{data-label="t3"}
To quantify the sensitivity of wakefield phase to parameter variations, we introduce the phase sensitivity $S_\varphi$ as $$\label{e6}
\delta \xi_m \omega_p / c = S_\varphi^X \delta X / X,$$ where $\delta \xi_m$ is the shift of some constant phase point. The phase curves may change in a complicated way due to variations of beam or plasma parameters, as exemplified by Fig.\[f8-phases\]. For most curves, the strongest phase deviation is located near the amplitude maximum (at $z \sim 4$m), while at the end of the plasma section the phase shift is small. We therefore calculate $S_\varphi$ in two plasma cross-sections: $z=4$m and $z=10$m. To this end we take the local maximum of $\Phi (\xi)$ located at $\xi\approx -13.86$cm.
Phase sensitivities listed in Table \[t3\] display several noticeable features. The most flaring one is the high sensitivity to plasma density variations. As shown in Ref.[@PoP20-013102], a small density variation $\delta n_0$ results in the relative change $\delta \lambda_p / \lambda_p = - \delta n_0 / (2 n_0)$ of the plasma wavelength and the forward shift of the wakefield pattern by $L_\text{per} \delta n_0 / (2 n_0)$, where $L_\text{per}$ is the distance between the observation point and the area of wakefield formation. This distance was not defined strictly in Ref.[@PoP20-013102], and the phase sensitivity can be used to refine the definition. Here $L_\text{per} = 2 S_\varphi^{n_0} c/\omega_p \approx 14$cm, so the full distance to the seed laser pulse is the “effective” wakefield length if the beam self-modulates in the plasma of a detuned density. This contrasts to the case considered in Ref.[@PoP20-013102], where an already modulated beam enters a perturbed density plasma, and $L_\text{per}$ is roughly twice shorter.
The extreme phase sensitivity to plasma density variations makes it challenging to deterministically inject externally generated short electron bunches into the wakefield of self-modulating proton beam. Assume that the bunch must be placed into a certain accelerating bucket with a longitudinal precision of about $0.1\,c/\omega_p$. Then it straightforwardly follows that the plasma density must be held to a designated value within $0.1/S_\varphi^{n_0} \approx 0.03$%.
Another noticeable feature is that all other sensitivities are small at $z=10$m. Values less than unity means that even a twofold change of a parameter moves a fixed phase point by a small fraction of the wakefield period. Thus the wakefield structure established after partial destruction of micro-bunches is well phase locked to the seed perturbation.
At $z=4$m sensitivities to beam population, beam length at fixed current, and beam radius are high since these parameters determine the beam density $n_{b0}$ which in turn governs the theoretically predicted phase velocity $v_\text{ph}^\text{th}$ of the growing self-modulation mode [@PRL107-145003]: $$\label{e7}
1-\frac{v_\text{ph}^\text{th}}{c} \approx \frac{1}{2} \left( \frac{\xi}{z} \right)^{1/3} \left( \frac{n_{b0} m c^2}{2 n_0 W_b} \right)^{1/3}.$$ From (\[e7\]) we may expect $(S_\varphi^{N_b}, S_\varphi^{\sigma_{zb}}, S_\varphi^{\sigma_{rb}}, S_\varphi^{W_b})$ to relate as $(-2, 2, 1, 2)$. This reasonably agrees with the simulations except for the sensitivity to the beam energy which is roughly twice lower.
Note also the relatively high sensitivity to the atomic weight of plasma ions at $z=4$m. In comparison to the amplitude sensitivity this shows that ion motion manifests itself much stronger in phase shifts than in the amplitude change.
Discussion {#s4}
==========
The performed study has not only clarified the sensitivity of the excited wakefield to beam and plasma parameters (summarized in Table \[t3\]), but also helped to identify physical effects limiting the wakefield amplitude in different parts of the parameter space. The most important effect is the nonlinear elongation of the wakefield period [@PoP20-083119]. As long as this effect has not come into play, the wakefield amplitude is directly proportional to the number of protons driving the wakefield. Once the limit is achieved by improving some of the parameters, further growth of the accelerating field drastically slows down. Two other important effects are emittance driven divergence and motion of plasma ions. Both effects are characterized by some threshold values or, to be exact, surfaces in the multi-dimensional parameter space. If the threshold is crossed, the wakefield amplitude rapidly drops. If not, there is no influence on the wakefield amplitude.
The baseline parameter set of the AWAKE experiment (Table \[t1\]) falls exactly at the onset of nonlinear period elongation and close to thresholds of emittance driven divergence and ion motion. In other words, this is the point beyond which efforts in increasing beam charge or peak current will not result in the proportional increase of the wakefield amplitude. Also there is no much safety margin in beam emittance and ion weight. A way to stronger wakefields may probably be opened by operating at lower beam emittances and higher plasma densities, but this is a subject of a separate study.
Very high sensitivity of the wakefield phase to plasma density variations stimulates interest in injection methods based on selective trapping of electrons from an initially long (several wavelengths) beam [@PRL107-145003; @JPP78-455].
Acknowledgements
================
The authors are grateful to participants of AWAKE collaboration for stimulating discussions.
This work was supported by The Ministry of Education and Science of the Russian Federation, Siberian Supercomputer Center SB RAS, and RFBR grant 14-02-00294.
[18]{} A.Caldwell, K.Lotov, A.Pukhov, and F.Simon, Nature Phys. [**5**]{}, 363 (2009). K.V.Lotov, Phys. Rev. ST Accel. Beams [**13**]{}, 041301 (2010). E. Esarey, C. B. Schroeder, and W. P. Leemans, Rev. Mod. Phys. **81**, 1229 (2009). AWAKE Collaboration, Proton-driven plasma wakefield acceleration: a path to the future of high-energy particle physics (to appear in Plasma Physics and Controlled Fusion, 2014). G.Xia, A. Caldwell, Producing Short Proton Bunch For Driving Plasma Wakefield Acceleration. Proceedings of IPAC2010 (Kyoto, Japan), p.4395–4397. A. Caldwell, K. Lotov, A. Pukhov and G. Xia, Plasma Phys. Controlled Fusion [**53**]{}, 014003 (2011). N.Kumar, A.Pukhov, and K.Lotov, Phys. Rev. Lett. [**104**]{}, 255003 (2010). K.V.Lotov, Instability of long driving beams in plasma wakefield accelerators. Proc. 6th European Particle Accelerator Conference (Stockholm, 1998), p.806-808. A. Pukhov, N. Kumar, T. Tuckmantel, A. Upadhyay, K. Lotov, P. Muggli, V. Khudik, C. Siemon, and G. Shvets, Phys. Rev. Lett. **107**, 145003 (2011). C.B.Schroeder, C.Benedetti, E.Esarey, F.J.Gruener, and W. P. Leemans, Phys. Rev. Lett. **107**, 145002 (2011). P. Muggli, A. Caldwell, O. Reimann, E. Oz, R. Tarkeshian, C. Bracco, E. Gschwendtner, A. Pardons, K. Lotov, A. Pukhov, M. Wing, S. Mandry, J. Vieira, Physics of the AWAKE Project. Proceedings of IPAC2013 (Shanghai, China), p.1179-1181. AWAKE Collaboration, AWAKE Design Report: A Proton-Driven Plasma Wakefield Acceleration Experiment at CERN. CERN-SPSC-2013-013; SPSC-TDR-003. C.Bracco, E.Gschwendtner, A.Petrenko, H.Timko, T.Argyropoulos, H.Bartosik, T.Bohl, J.E.Mueller, B.Goddard, M.Meddahi, A.Pardons, E.Shaposhnikova, F.M.Velotti, H.Vincke, Nucl. Instr. Meth. A **740**, 48 (2014). E.Oz, P.Muggli, Nucl. Instr. Meth. A **740**, 197 (2014). K.V.Lotov, G.Z.Lotova, V.I.Lotov, A.Upadhyay, T.Tuckmantel, A.Pukhov, A.Caldwell, Phys. Rev. ST Accel. Beams 16, 041301 (2013). A. Caldwell and K. V. Lotov, Phys. Plasmas [**18**]{}, 103101 (2011). J. Beringer et al. (Particle Data Group), Phys. Rev. D **86**, 010001 (2012). K.V.Lotov, Phys. Plasmas **5**, 785 (1998). K.V.Lotov, Phys. Rev. ST Accel. Beams [**6**]{}, 061301 (2003). K.V. Lotov, A. Sosedkin, E.Mesyats, Simulation of Self-modulating Particle Beams in Plasma Wakefield Accelerators. Proceedings of IPAC2013 (Shanghai, China), p.1238-1240. `www.inp.nsk.su/~lotov/lcode`. R.Keinigs and M.E.Jones Phys. Fluids **30**, 252 (1987). K.V.Lotov, Phys. Plasmas 20, 083119 (2013). K.V.Lotov, Phys. Plasmas 18(2) (2011) 024501. J.Vieira, R.A.Fonseca, W.B.Mori, and L.O.Silva, Phys. Rev. Lett. **109**, 145005 (2012). J. Vieira, Y. Fang, W. B. Mori, L. O. Silva, and P. Muggli, Phys. Plasmas **19**, 063105 (2012). K.V.Lotov, A.Pukhov, and A.Caldwell, Phys. Plasmas 20(1), 013102 (2013). K.V.Lotov, J. Plasma Phys. 78(4), 455-459 (2012).
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abstract: 'We consider the problem of estimating a regression function in the common situation where the number of features is small, where interpretability of the model is a high priority, and where simple linear or additive models fail to provide adequate performance. To address this problem, we present GapTV, an approach that is conceptually related both to CART and to the more recent CRISP algorithm [@petersen:etal:2016], a state-of-the-art alternative method for interpretable nonlinear regression. GapTV divides the feature space into blocks of constant value and fits the value of all blocks jointly via a convex optimization routine. Our method is fully data-adaptive, in that it incorporates highly robust routines for tuning all hyperparameters automatically. We compare our approach against CART and CRISP and demonstrate that GapTV finds a much better trade-off between accuracy and interpretability.'
bibliography:
- 'crispier.bib'
title: 'GapTV - Appendix'
---
|
---
author:
- 'S. Hubrig'
- 'M. Schöller'
- 'N. V. Kharchenko'
- 'N. Langer'
- 'W. J. de Wit'
- 'I. Ilyin'
- 'A. F. Kholtygin'
- 'A. E. Piskunov'
- 'N. Przybilla'
- the MAGORI collaboration
date: 'Received date / Accepted date'
title: |
Exploring the origin of magnetic fields in massive stars:\
A survey of O-type stars in clusters and in the field[^1]
---
Introduction {#sect:intro}
============
Magnetic fields play an important role in astrophysical phenomena of the universe at various scales. In galaxies, dynamo models associated with various MHD instabilities occurring in the interstellar medium (ISM) are used to explain the formation of the galactic structure (e.g., Gomez & Cox [@GomezCox2004]; Bonanno & Urpin [@BonannoUrpin2008]). Magnetic fields play a role in the evolution of interstellar molecular clouds and in the star formation process, where the cloud collapse is probably taking place along the magnetic field lines (e.g. Alves et al. [@Alves2008]). They are also present at all stages of stellar evolution, from young TTauri stars and Ap/Bp stars to the end products: white dwarfs, neutron stars, and magnetars. On the other hand, role of magnetic fields in massive O-type stars and Wolf-Rayet (WR) stars remains unknown. No definitive magnetic field was ever detected in WR stars and presently only less than a dozen O stars have published magnetic fields. Also, the theories on the origin of magnetic fields in O-type stars are still poorly developed, mostly due to the fact that the distribution of magnetic field strengths in massive stars from the ZAMS to more evolved stages has not yet been studied.
In our study we focus on magnetic fields of massive stars observed in different environments: in open clusters at different ages and in the field. The results of our recent kinematic analysis of known magnetic O-type stars using the best available astrometric, spectroscopic, and photometric data indicates that the presence of a magnetic field is more frequently detected in candidate runaway stars than in stars belonging to clusters or associations (Hubrig et al. [@Hubrig2011b]). As the sample of stars with magnetic field detections is still very small, a study of a larger sample is urgently needed to confirm the detected trend by dedicated magnetic field surveys of O stars in clusters/associations and in the field. We were granted four nights in 2010 May with FORS2 at the VLT to survey magnetic fields in massive stars, but due to deteriorated weather conditions, only half of the granted time could be used for observations. Notwithstanding, the obtained results allow us to preliminarily constrain the conditions which enable the presence of magnetic fields and give first trends about their occurrence rate and field strength distribution. This information is critical for answering the principal question of the possible origin of magnetic fields in massive stars.
In the following, we present 41 new measurements of magnetic fields in 36 massive stars using FORS2 at the VLT in spectropolarimetric mode. Our observations and the obtained results are described in Sect. \[sect:observations\], and their discussion is presented in Sect. \[sect:discussion\].
Observations and results {#sect:observations}
========================
-------------- --------------- ------- --------------------
CPD$-$282561 SAO174826 10.09 O6.5 fp
CPD$-$472963 SAO220701 8.45 O4 III (f)
CPD$-$582620 CD$-$583529 9.27 O6.5 V ((f))
HDE303311 CPD$-$582652 8.98 O5 V
HD93129B CPD$-$582618B 7.16 O3.5 V ((f+))
HD93204 CPD$-$592584 8.48 O5 V ((f))
HDE303308 CPD$-$592623 8.14 O4 V ((f+))
HD93403 CPD$-$582680 7.30 O5 III (f) var sec
HD93843 CPD$-$592732 7.33 O5 III(f) var
HD105056 GS Mus 7.34 ON9.7 Ia e
HD115455 CPD$-$613575 7.97 O7.5 III ((f))
HD120521 CPD$-$576339 8.56 O8 Ib (f)
HD123590 CPD$-$614382 7.62 O7/8
HD125206 CPD$-$605298 7.94 O9.5 IV: (n)
HD125241 CPD$-$605300 8.31 O8.5 Ib (f)
HD130298 CPD$-$556191 9.29 O6.5 III (n)(f)
HD148937 CPD$-$477765 6.77 O6.5 f?p
HD328856 CPD$-$468218 8.50 O9.5III$^*$
HD152233 CPD$-$417718 6.59 O6 III: (f)p
HD152247 CPD$-$417732 7.20 O9.5 II-III
HD152249 HR6263 6.47 OC9.5 Iab
HD153426 CPD$-$386624 7.49 O9 II-III
HD153919 V884Sco 6.53 O6.5 Ia f+
HD154368 V1074Sco 6.18 O9.5 Iab
HD154643 CPD$-$346733 7.15 O9.5 III
HD154811 CPD$-$468416 6.93 OC9.7 Iab
HD156154 CPD$-$356916 8.04 O8 Iab (f)
HD156212 CPD$-$275605 7.95 O9.7 Iab
HD165319 BD$-$144880 7.93 O9.5Iab$^*$
HD315033 CPD$-$246163 8.90 B3$^*$
HD168075 BD$-$134925 8.73 O6 V ((f))
HD168112 BD$-$124988 8.55 O5 III (f)
HD169582 BD$-$094729 8.70 O6 I f
HD171589 BD$-$145131 8.21 O7 II (f)
HD175754 BD$-$195242 7.02 O8 II ((f))
HD187474 V3961Sgr 5.32 Ap$^*$
-------------- --------------- ------- --------------------
: List of O-type stars observed with FORS2. Spectral classifications are listed according to the Galactic O Star Catalogue (Ma[í]{}z-Apell[á]{}niz et al. [@maiz:2004]).[]{data-label="tab:objects"}
Notes: $^*$ indicates Spectral Types taken from SIMBAD.
Spectropolarimetric observations were carried out in 2010 May 20–23 in visitor mode at the European Southern Observatory with FORS2 mounted on the 8-m Antu telescope of the VLT. This multi-mode instrument is equipped with polarization analyzing optics, comprising super-achromatic half-wave and quarter-wave phase retarder plates, and a Wollaston prism with a beam divergence of 22$\arcsec$ in standard resolution mode[^2]. Polarimetric spectra were obtained with the GRISM 600B and the narrowest slit width of 0$\farcs$4 to achieve a spectral resolving power of $R\sim2000$. The use of the mosaic detector made of blue optimized E2V chips and a pixel size of 15$\mu$m allowed us to cover a large spectral range, from 3250 to 6215Å, which includes all hydrogen Balmer lines from H$\beta$ to the Balmer jump. The spectral types and the visual magnitudes of the studied stars are listed in Table \[tab:objects\].
A detailed description of the assessment of the longitudinal magnetic-field measurements using FORS2 is presented in our previous papers (e.g., Hubrig et al. [@hubrig04a; @hubrig04b], and references therein). The mean longitudinal magnetic field, $\left< B_{\rm z}\right>$, was derived using
$$\frac{V}{I} = -\frac{g_{\rm eff} e \lambda^2}{4\pi{}m_ec^2}\ \frac{1}{I}\
\frac{{\rm d}I}{{\rm d}\lambda} \left<B_{\rm z}\right>,
\label{eqn:one}$$
where $V$ is the Stokes parameter that measures the circular polarisation, $I$ is the intensity in the unpolarised spectrum, $g_{\rm eff}$ is the effective Landé factor, $e$ is the electron charge, $\lambda$ is the wavelength, $m_e$ the electron mass, $c$ the speed of light, ${{\rm d}I/{\rm d}\lambda}$ is the derivative of Stokes $I$, and $\left<B_{\rm z}\right>$ is the mean longitudinal magnetic field.
Longitudinal magnetic fields were measured in two ways: using only the absorption hydrogen Balmer lines or using the entire spectrum including all available absorption lines. The lines that show evidence for emission were not used in the determination of the magnetic field strength. The feasibility of longitudinal magnetic field measurements in massive stars using low-resolution spectropolarimetric observations was demonstrated by previous studies of O and B-type stars (e.g., Hubrig et al. [@hubrig:2006; @hubrig08; @hubrig09; @Hubrig2011a]). To check that the instrument was functioning properly, we observed the magnetic Ap star HD187474, which has a well studied longitudinal magnetic field, during the night of May 23 at rotation phase 0.66. HD187474 has a rotation period of 6.4yr and a longitudinal magnetic field ranging roughly from $-$2kG to 2kG. The measured value of the magnetic field, $\left< B_z\right>_{\rm all}= -1249\pm47$G, fits very well to the observations at the same phase presented by Landstreet & Mathys ([@LandstreetMathys2000]).
-------------- ----------- --------- ----- --------- ----- ---------
CPD$-$282561 55338.969 $-$381 122 $-$534 167 ND
CPD$-$472963 55337.094 $-$190 62 $-$154 96 ND
CPD$-$582620 55339.020 $-$53 71 $-$66 88
HDE303311 55337.131 $-$56 40 $-$19 61
HD93129B 55337.179 $-$49 44 $-$79 78
HD93204 55339.053 22 46 16 66
HDE303308 55339.078 122 54 137 96
HD93403 55339.116 39 41 $-$15 88
HD93843 55339.099 $-$157 42 $-$173 56 ND
HD105056 55337.208 $-$93 48 $-$156 63
HD115455 55337.230 4 47 13 64
HD120521 55337.257 25 44 $-$3 62
HD123590 55339.133 19 42 56 70
HD125206 55337.307 12 64 8 91
HD125241 55339.188 24 39 16 72
HD130298 55339.159 113 38 193 62 ND
HD148937 55336.307 $-$297 62 $-$293 85 CD
55337.285 $-$204 71 $-$225 103
55339.206 $-$290 85 $-$389 129
HD328856 55336.370 $-$173 53 $-$155 65 ND
55339.223 $-$149 48 $-$75 72
HD152233 55339.289 $-$74 52 $-$76 74
HD152247 55339.261 34 52 86 64
HD152249 55339.275 $-$22 51 $-$15 72
HD153426 55336.338 $-$27 53 $-$10 62
55339.246 $-$171 55 $-$275 70 ND
HD153919 55337.341 $-$213 68 $-$119 95 ND
HD154368 55339.324 $-$74 38 $-$77 63
HD154643 55339.340 110 34 121 52 ND
HD154811 55337.327 91 39 59 64
HD156154 55337.358 $-$118 38 $-$167 54 ND
HD156212 55337.377 $-$104 42 $-$51 63
HD165319 55339.306 $-$44 48 $-$38 74
HD315033 55337.402 $-$41 41 $-$36 52
HD168075 55339.389 17 42 3 65
HD168112 55336.418 $-$74 53 $-$66 74
55339.362 112 62 40 96
HD169582 55339.415 $-$87 58 $-$124 88
HD171589 55339.445 $-$62 140 $-$119 180
HD175754 55337.426 82 67 66 77
HD187474 55339.433 $-$1249 22 $-$1253 32 Ap star
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Although we were granted four nights for our survey, due to unfavorable weather conditions (snow and high humidity) only four stars could be observed during the first night, 14 stars during the second night, none during the third night, and 23 during the last night. Most of the targets were observed only once. The exceptions were the stars HD328856, HD153426, and HD168112, which we were able to observe twice. HD148937 was observed three times to assess the magnetic field variability over the rotation cycle. Apart from this star, which has a rotation period of seven days (Nazé et al. [@naze08]), no exact rotation periods are known for the other stars in our sample.
The results of our magnetic field measurements are presented in Table \[tab:fields\]. In the first two columns, we provide the star names and the modified Julian dates at the middle of the exposures. In Cols. 3 and 4 we present the longitudinal magnetic field $\left<B_{\rm z}\right>_{\rm all}$ using the whole spectrum and the longitudinal magnetic field $\left<B_{\rm z}\right>_{\rm hyd}$ using only the hydrogen lines. All quoted errors are 1$\sigma$ uncertainties. In Col. 5, we identify new detections by ND and in the case of HD148937 the confirmed detection is marked by CD.
Ten stars of our sample, CPD$-$282561, CPD$-$472963, HD93843, HD130298, HD148937, HD328856, HD153426, HD153919, HD154643, and HD156154, show evidence for the presence of a magnetic field.
Importantly, the strongest magnetic fields are detected in both Of?p stars CPD$-$282561 and HD148937. Walborn ([@walb73]) introduced the Of?p category for massive O stars displaying recurrent spectral variations in certain spectral lines, sharp emission or P Cygni profiles in He I and the Balmer lines, and strong C III emission lines around 4650Å. Only five Galactic Of?p stars are presently known: HD108, NGC1624-2, CPD$-$282561, HD148937, and HD191612 (Walborn et al. [@walb10]). Our observations of CPD$-$282561 reveal a magnetic field at 3.1$\sigma$ level using the whole spectrum and at 3.2$\sigma$ level using Balmer lines. The study of radial velocity variation of Levato et al. ([@Levato1988]) indicated the presence of variability of a few emission lines with a probable period of 17 days. Walborn et al. ([@walb10]) report that CPD$-$282561 undergoes extreme spectral transformations very similar to those of HD191612, on a timescale of weeks, exhibiting variable emission intensity of the C III $\lambda\lambda$ 4647-4650-4652 triplet. The detection of a mean longitudinal magnetic field $\langle$$B_z$$\rangle$=$-$254$\pm$81G in the Of?p star HD148937 using FORS1 at the VLT was previously reported by Hubrig et al. ([@hubrig08]). An extensive multiwavelength study of HD148937 was carried out by Nazé et al. ([@naze08]), who detected small-scale variations of 4686 and the Balmer lines with a period of 7days and an overabundance of nitrogen by a factor of 4 compared to the Sun. The periodicity of spectral variations in hydrogen and He lines was re-confirmed using additional higher resolution spectroscopic material indicating similarity to the other Of?p stars HD108 and HD191612 (Naze et al. [@Naze2010]). Our spectropolarimetric observations of this star indicate that the magnetic field is variable, but due to the low number of measurements it is not possible to verify the period deduced from spectroscopic observations. The magnetic field of this star was observed at 4.8$\sigma$, 2.9$\sigma$, and 3.4$\sigma$ levels on three different nights using all absorption lines.
The remaining three Of?p stars are located in the Northern hemisphere and cannot be reached with FORS2 at the VLT. To study magnetic fields in HD108 and HD191612 we recently used polarimetric spectra obtained with the SOFIN spectrograph installed at the Nordic Optical Telescope (Hubrig et al. [@Hubrig2010]). As a result, we detected a longitudinal magnetic field $\langle$$B_z$$\rangle$=$-$168$\pm$35G in the Of?p star HD108, which is in agreement with the longitudinal magnetic field measurement of the order of $-$150G recently reported by Martins et al. ([@Martins2010]). For the star HD191612 with a rotation period of 537.6d (Howarth et al. [@Howarth2007]) we measured a longitudinal magnetic field $\langle$$B_z$$\rangle$=450$\pm$153G at rotation phase 0.43 (Hubrig et al. [@Hubrig2010]). The only previously published magnetic field measurement for this star showed a negative longitudinal magnetic field $\langle$$B_z$$\rangle$=$-$220$\pm$38G at rotation phase 0.24 (Donati et al. [@donati:2006]), indicating a change of polarity over $\sim$100 days. No attempt has yet been made to measure the magnetic field of NGC1624-2. Clearly the recent results of magnetic field measurements in Of?p stars imply a tight relation between the observed properties of the Of?p star group and the presence of a magnetic field.
For the star CPD$-$472963 we achieved a 3.1$\sigma$ detection using all absorption lines. According to Walborn et al. ([@walb10]) this star belongs to the Ofc category, which consists of normal spectra with C III $\lambda\lambda$ 4647-4650-4652 emission lines of comparable intensity to those of the Of defining lines N III $\lambda\lambda$ 4634-4640-4642. The authors indicate that the Ofc phenomenon occurs primarily in certain associations and young clusters. However, the available kinematic and photometric data do not indicate cluster or association membership for CPD$-$472963. The origin of the magnetic field in this star is probably different compared to that of other magnetic O-type stars, as non-thermal radio emission, which is frequently observed in binary systems with colliding winds, was detected by Benaglia et al. ([@Benaglia2001]). On the other hand, the membership of CPD$-$472963 in a binary or multiple system has not been investigated yet. The authors suggest that the non-thermal radiation from this star possibly comes from strong shocks in the wind itself and/or in the wind colliding region if the star has a massive early-type companion. Both optical and radio observations reveal the presence of a second source separated by 5.
According to Walborn et al. ([@walb10]), also the star HD93843, with a 3.7$\sigma$ detection achieved using all absorption lines, belongs to the Ofc category. Prinja et al. ([@Prinja1998]) monitored the stellar wind of this star using IUE time series. They identified systematic changes in the absorption troughs of the and resonance lines with a repeatability of wind structures with a period of 7.1days. Noteworthy, the authors suggest the presence of a magnetic field as one of the possible mechanisms to explain the cyclical wind perturbation. On the other hand, three other stars of the Ofc category included in our survey, HD93204, HDE303308, and HD93403, do not show the presence of a magnetic field at a 3$\sigma$ level.
The star HD130298 with a longitudinal magnetic field observed at 3.1$\sigma$ level using the Balmer lines, is known as an object with a bow shock. Noriega-Crespo et al. ([@NoriegaCrespo1997]) used ISSA/IRAS archival spectra to identify stars surrounded by extended infrared emission at 60$\mu$m, which is a signature of wind bow shocks. The bow shocks are usually associated with runaway early-type stars with typical wind velocities of 500–3000kms$^{-1}$ and mass loss rates $\sim10^{-5}$ – $10^{-6}$ $M_{\sun}$yr$^{-1}$ (see e.g. Puls et al. [@Puls1996]).
------------------------- --------- ---- ----- ------------ ------ ------------ --------- --------- ------ ---------- ---------- ------
HD47839$^{\ast}$ 1021435 62 100 NGC2264 660 6.81 $-$3.84 $-$2.50 0.94 $-$2.70 $-$3.50 0.25
CPD$-$582620 2232461 44 1 Trumpler14 2753 6.67 $-$3.77 $-$2.20 2.37 $-$3.91 3.65 0.60
HDE303311 2232607 91 27 Trumpler14 2753 6.67 $-$2.91 2.64 2.40 $-$3.91 3.65 0.60
HD93129B 2232449 0 100 Trumpler14 2753 6.67 $-$9.80 10.86 1.96 $-$3.91 3.65 0.60
HD93204 2232588 75 0 Trumpler16 2842 6.90 $-$9.33 3.16 1.70 $-$11.10 4.02 0.42
HDE303308 2232708 19 0 Trumpler16 2842 6.90 $-$6.44 2.54 1.79 $-$11.10 4.02 0.42
HD105056 2396832 12 100 ASCC69 1000 7.91 $-$4.56 $-$1.81 0.87 7.91 $-$7.52 0.41
HD115455 2338543 90 72 Stock16 1640 6.78 $-$2.56 1.08 2.02 $-$3.31 $-$0.03 0.66
HD120521 2256830 0 100 Platais10 246 8.20 $-$5.55 0.40 1.39 $-$29.10 $-$10.73 0.40
HD123590 2345821 83 83 ASCC77 2200 6.99 $-$4.96 $-$2.45 0.99 $-$4.59 $-$1.49 0.47
HD328856 2074567 22 100 Hogg22 1297 6.70 0.78 $-$1.37 1.26 $-$0.84 $-$4.39 0.43
HD135240$^{\ast}$ 2351294 61 100 ASCC79 800 6.86 $-$3.00 $-$2.65 0.85 $-$2.67 $-$4.10 0.44
HD135591$^{\ast}$ 2351425 67 100 ASCC79 800 6.86 $-$3.00 $-$2.65 0.85 $-$2.67 $-$4.10 0.44
HD152233 1974789 13 100 NGC6231 1250 6.81 $-$4.11 $-$8.05 2.40 $-$0.39 $-$1.99 0.46
HD152247 1974811 95 100 NGC6231 1250 6.81 $-$0.55 $-$1.14 1.91 $-$0.39 $-$1.99 0.46
HD152249 1974812 84 100 NGC6231 1250 6.81 0.13 $-$0.81 1.36 $-$0.39 $-$1.99 0.46
HD153919 1877055 0 100 NGC6281 494 8.51 1.42 4.84 1.04 $-$2.96 $-$3.75 0.23
HD154368 1877523 88 100 ASCC88 1900 7.17 3.80 $-$2.08 1.05 2.89 $-$2.00 0.32
[*HD155806*]{}$^{\ast}$ 1783567 98 100 ScoOB4 1100 6.82 0.33 $-$2.02 0.80 0.46 $-$2.22 0.15
[*HD156154*]{} 1878846 82 100 Bochum13 1077 7.08 $-$0.88 $-$2.53 1.64 $-$0.28 $-$1.20 0.30
[*HD164794*]{}$^{\ast}$ 1593528 69 100 NGC6530 1322 6.67 1.92 $-$0.40 1.09 2.01 $-$1.81 0.51
HD315033 1593655 97 100 NGC6530 1322 6.67 2.33 $-$2.51 2.09 2.01 $-$1.81 0.51
HD167263$^{\ast}$ 1595683 97 100 SgrOB7 1860 6.64 1.56 $-$1.53 1.19 1.80 $-$1.20 0.26
HD168075 1407400 73 100 NGC6611 1719 6.72 2.81 $-$1.14 1.18 1.60 $-$0.35 0.41
HD168112 1407419 73 19 NGC6604 1696 6.64 1.67 $-$2.20 1.69 $-$0.36 $-$2.68 0.31
HD153426 1876676 0 0 Hogg22 1297 6.70(6.91) $-$0.21 $-$0.09 1.38 $-$0.84 $-$4.39 0.43
HD154643 1877779 0 0 ASCC88 1900 7.17(6.15) 3.66 $-$1.71 1.55 2.89 $-$2.00 0.32
HD169582 1323022 0 0 NGC6604 1696 6.64(6.74) $-$0.40 $-$1.63 1.14 $-$0.36 $-$2.68 0.31
HD171589 1409170 0 0 NGC6618 1814 7.78(6.52) 5.10 3.74 1.66 3.04 0.33 0.92
HD175754 1503597 0 0 ASCC93 2500 7.72(7.01) 1.49 2.15 1.01 $-$2.18 $-$1.80 0.55
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The two stars HD328856 and HD153426, both with magnetic field detections, were observed on two different nights, namely the first and the fourth night of our observing run. For HD328856 we obtained on these nights 3.3$\sigma$ and 3.1$\sigma$ level detections, respectively, using all absorption lines. Based on the photometric membership probability, this star is a member of the compact open cluster Hogg22 in the Ara region at an age of 5Myr and a distance of about 1300pc (see Sect. 3). On the other hand, its proper motions indicate that HD328856 is not fully co-moving with the other cluster members, deviating from the cluster mean proper motion by $\sim$2$\sigma$ (for more details on membership probabilities see Kharchenko et al. [@Kharchenko2004]). The observations of the star HD153426 revealed the presence of a mean longitudinal magnetic field at the 3.9$\sigma$ level using Balmer lines on the fourth night. The non-detection of the magnetic field for HD153426 on the first observing night can probably be explained by the strong dependence of the longitudinal magnetic field on the rotational aspect. HD153426 is a double-lined spectroscopic binary with unknown orbit parameters and was considered by de Wit et al. ([@deWit2005]) as a star in a newly detected cluster. Using the best presently available kinematic data on young open clusters, Schilbach & Röser ([@SchilbachRoeser2008]) suggested that HD153426 was ejected from the cluster Hogg22. Their back-tracing procedure indicates that the encounter time for HD153426, i.e. the time when the star was ejected, is about 8.1Myr, while the age of the cluster Hogg22 is only 5Myr.
The star HD153919 was observed only once, revealing the presence of a mean longitudinal magnetic field at the 3.9$\sigma$ level, using all absorption lines. The study of Ankay et al. ([@Ankay2001]) suggested that this star is a runaway X-ray binary, ejected from the OB association Sco OB1 about 2Myr ago due to the supernova of 4U1700-37’s progenitor. They considered this star as a companion to 4U1700-37, most likely a neutron star powered by wind accretion (e.g., Jones et al. [@Jones1973]). Since 4U1700-37 is a candidate for a low-mass black hole (Brown et al. [@Brown1996]), this system can be similar to the optical component (the O9.7 Iab supergiant) in the system Cyg X-1, for which the presence of a variable weak magnetic field was recently detected using a FORS1 spectropolarimetric time series over the orbital period of 5.6days (Karitskaya et al. [@Karitskaya2010]). Schilbach & Röser ([@SchilbachRoeser2008]) identified the origin of this field star in the cluster NGC6231 (the open cluster inside Sco OB1) at an age of about 6.5Myr and their back-tracing procedure indicates that the star was ejected from the cluster 1.1Myr ago.
The longitudinal magnetic field for the star HD154643 was observed at 3.2$\sigma$ level using all absorption lines. De Wit et al. ([@deWit2005]) characterise this star as a candidate runaway star associated with the young cluster Bochum13. However, Schilbach & Röser ([@SchilbachRoeser2008]) identified the origin of this field star in the cluster ASCC88 at an age of about 14.8Myr, ejected 1.4Myr ago.
The star HD156154, for which we achieved a 3.1$\sigma$ detection using all absorption lines, seems to be the only star with a high cluster membership probability among the presented O-type stars with detected magnetic fields (see Sect. 3). According to kinematic and photometric criteria belongs this star to the open cluster Bochum13 at an age of 12Myr and a distance of about 1kpc.
Discussion {#sect:discussion}
==========
A lot of effort has been put into the research of massive stars in recent years in order to properly model the effects of rotation, stellar winds, and surface chemical composition. However, possible paths for the formation of magnetic O-type stars were not analysed yet with modern theories for the evolution of single and binary stars. Clearly, the number of massive stars with detected magnetic fields is still small, and the available data are insufficient to prove statistically whether magnetic fields in massive stars are ubiquitous or appear in specific stars with certain stellar parameters and in a special environment. On the other hand, the observations of magnetic fields in massive stars accumulated over the last few years can be used to preliminarily constrain the conditions which enable the appearance of magnetic fields and give first trends about their occurrence and field strength distribution.
Since no longitudinal magnetic fields stronger than 300G were detected in our study (apart from the rather large field in the Of?p star CPD$-$282561), we confirm our previous conclusion (Hubrig et al. [@hubrig08]) that large-scale, dipole-like, magnetic fields with polar field strengths higher than 1kG are not widespread among O type stars. Our study presents the results of a magnetic field survey in 36 massive stars. Among them, 19 stars can be related to open clusters and associations at different age. The data on the cluster membership of these probable cluster O-type stars are presented in Table \[tab:clusters\]. To increase the significance of our statistic assessment, we present in the same table the data for an additional six probable cluster O-type stars, which have been studied during the last years by Hubrig et al. ([@hubrig08; @Hubrig2009; @Hubrig2011b]), marked for convenience by an asterisk. As database for the compilation of Table \[tab:clusters\] we used the All-sky Compiled Catalogue of 2.5 million stars (ASCC-2.5, 3rd version) of Kharchenko & Roeser ([@KharchenkoRoeser2009]). We note that for the calculation of kinematic membership probability only proper motions were used. According to Dias et al. ([@Dias2002]; Version 3.1 (24/11/2010)), one of the previously studied O-type stars, the star HD152408 (Hubrig et al. [@hubrig08]), is projected on the cluster Collinder316, which presents a large group of bright stars superposed on Trumpler24 at the age log$t$= 6.92. We have not included this star in Table \[tab:clusters\], as no membership criteria are discussed in this work. According to Mason et al. ([@Mason1998]) and Pourbaix et al. ([@Pourbaix2004])[^3], among the stars presented in Table \[tab:clusters\], six stars, HD47839, HD135240, HD152233, HD153919, HD154368, and HD167263, are members of spectroscopic binary systems, with orbital periods between 3.4 and 9247 days.
![ The positions of the stars studied for cluster membership in the colour-magnitude diagram. Different symbols indicate stars with different membership probabilities: Squares stand for stars with cluster membership probability larger than 60%, circles for stars with a probability between 14% and 60%, and triangles for membership probability between 1% and 14%. Non-members and runaway stars are marked by dots and crosses, respectively. One runaway star, HD171589, does not appear in this figure, as its colour and magnitude do not fit the presented parameter space (see Table \[tab:colours\]). The three stars with magnetic fields, HD155806, HD156154, and HD164794, with high cluster membership probabilities are denoted by filled squares. Isochrones for log$t=$6.6, 6.8, 7.0, and 7.2 are presented by solid lines and the zero-age main sequence is shown as a dashed line. []{data-label="fig:evol"}](cmd_clmem.eps){height="45.00000%"}
![ The age distribution of studied probable cluster members. The three stars with magnetic fields, HD155806, HD156154, and HD164794, are denoted by horizontal lines. []{data-label="fig:histo"}](logthisto.eps){width="45.00000%"}
From the inspection of kinematic and photometric membership probabilities, and following the membership criteria described by Kharchenko et al. ([@Kharchenko2004]), five stars in Table \[tab:clusters\], CPD$-$582620, HD93129B, HDE303308, HD105056, HD152233, have a rather low cluster membership probability. Two other stars, HD120521 and HD153919, are not kinematic members of the oldest clusters Platais 10 and NGC 6281, respectively, and should be regarded as field stars projected against the clusters by chance. Among the remaining stars, only in three stars, HD155806, HD156154, and HD164794, weak magnetic fields have been detected (more details on the kinematic study of HD155806 and HD164794 can be found in Hubrig et al. [@Hubrig2011b]). In Fig. \[fig:evol\] we display the positions of the stars listed in Table \[tab:clusters\] in the colour-magnitude diagram. The absolute magnitudes $M_{\rm V}$ and intrinsic colours $(B-V)_0$ were calculated using cluster distances and $E(B-V)$ values presented by Kharchenko et al. ([@Kharchenko2005a; @Kharchenko2005b]). The theoretical isochrones were calculated by the Padova group (Girardi et al. [@Girardi2002]) and the values for the zero-age main sequence were retrieved from Schmidt-Kaler ([@SchmidtKaler1982]). The absolute magnitudes and colours are presented in Table \[tab:colours\]. The uncertainty in the distance modulus is assumed to be 0.2. The positions of the magnetic stars with high cluster membership probabilities, HD156154, HD155806, and HD164794, do not reveal any sign of a specific distribution, which would hint at the origin of their magnetic fields at a certain evolutionary age.
-------------- ---------- ---------- -------
HD47839 $-$4.557 $-$0.266 0.007
CPD$-$582620 $-$4.239 $-$0.436 0.040
HDE303311 $-$4.632 $-$0.370 0.030
HD93129B $-$6.315 $-$0.285 0.020
HD93204 $-$5.381 $-$0.586 0.026
HDE303308 $-$5.756 $-$0.433 0.027
HD105056 $-$3.145 $-$0.143 0.011
HD115455 $-$4.563 $-$0.329 0.011
HD120521 $-$4.011 $-$0.277 0.019
HD123590 $-$5.017 $-$0.187 0.011
HD328856 $-$4.066 $-$0.282 0.038
HD135240 $-$4.942 $-$0.253 0.012
HD135591 $-$4.576 $-$0.261 0.004
HD152233 $-$5.317 $-$0.320 0.017
HD152247 $-$4.656 $-$0.276 0.018
HD152249 $-$5.471 $-$0.259 0.019
HD153919 $-$2.406 0.090 0.013
HD154368 $-$6.781 $-$0.047 0.006
HD155806 $-$5.214 $-$0.254 0.006
HD156154 $-$4.562 $-$0.242 0.020
HD164794 $-$5.603 $-$0.293 0.007
HD315033 $-$2.956 $-$0.203 0.030
HD167263 $-$6.125 $-$0.257 0.006
HD168075 $-$4.566 $-$0.319 0.035
HD168112 $-$5.545 $-$0.381 0.028
HD153426 $-$5.109 $-$0.548 0.013
HD154643 $-$5.775 $-$0.278 0.011
HD169582 $-$5.382 $-$0.474 0.028
HD171589 $-$8.036 $-$1.340 0.021
HD175754 $-$5.838 $-$0.379 0.011
-------------- ---------- ---------- -------
: Absolute magnitudes and intrinsic colours of stars studied for the cluster membership. []{data-label="tab:colours"}
In Fig. \[fig:histo\] we present the age distribution of the most probable cluster members. While the age of HD155806 and HD164794 is similar to the bulk of the studied cluster O-type stars, the star HD156154 is somewhat older at an age of $\sim$12Myr. HD155806 is classified as an Oe star, possibly representing the higher mass analogues of classical Be stars (e.g. Walborn [@walb73]). Only six members are suggested to belong to this group of stars (e.g. Negueruela et al. [@neg04]). The star HD164794 is a spectroscopic double-lined system with an orbital period of 2.4yr, known as emitting non-thermal radio-emission, probably associated with colliding winds (Nazé et al. [@Naze2010]). No specific information can be found in the literature about the luminous supergiant HD156154.
The available observations seem to indicate that the presence of a magnetic field is more frequently detected in field stars than in stars belonging to clusters or associations. It is generally accepted that the majority of massive stars form in star clusters and associations, and studies of kinematical properties of the massive star field population indicate that a major part of these stars can be traced back to their parent open clusters or associations (e.g. Schilbach & Roeser 2008). Pflamm-Altenburg & Kroupa ([@PflammAltenburgKroupa2010]) recently discussed in their work whether massive stars can form in isolation in the galactic field. According to de Wit et al. ([@deWit2005]), only a few per cent of all O-type stars can be considered as formed outside a cluster environment. Pflamm-Altenburg & Kroupa considered the two-step-ejection process, which presents the combination of the dynamical and the supernova ejection scenario with the result that massive field stars produced via this ejection process for the vast majority of cases cannot be traced back to their parent star clusters. These stars can be mistakenly considered as massive stars formed in isolation. While this can not be proven, the observed numbers of field O stars is consistent with this idea.
For the newly detected magnetic O-type stars, HD153426, HD153919, and HD154643, Schilbach & Roeser ([@SchilbachRoeser2008]) suggested that the three stars were ejected from the clusters Hogg22, NGC6231, and ASCC88, respectively. On the other hand, none of the four magnetic Of?p stars is known to belong to a cluster or an association. The study of the evolutionary state of HD108 and HD191612 indicates that both stars are significantly evolved (Martins et al. [@Martins2010]). Our kinematic study of the Of?p star HD148937 showed that it possesses a space velocity of 32kms$^{-1}$ with respect to the Galactic open cluster system, with the velocity component $U$=$-$26 directed opposite from the Galactic center and the velocity component $W$=$-$13 directed from the Galactic plane (Hubrig et al. [@Hubrig2011b]). These rather large velocities indicate that this star can be considered as a candidate runaway star.
It is striking that the major part of previously detected magnetic O-type stars are candidate runaway stars (Hubrig et al. [@Hubrig2011b; @Hubrig2011c]). Also in the sample of O-stars with magnetic fields detected in this work, four other stars, HD130298, HD153426, HD153919, and HD154643, are mentioned in the literature as candidate runaway stars. In the past, two mechanisms were discussed to explain the existence of runaway stars: In one scenario, close multibody interactions in a dense cluster environment cause one or more stars to be scattered out of the region (e.g. Leonard & Duncan [@LeonardDuncan1990]). For this mechanism, runaways are ejected in dynamical three- or four-body interactions. An alternative mechanism involves a supernova explosion within a close binary, ejecting the secondary due to the conservation of momentum (Zwicky [@Zwicky1957]; Blaauw [@Blaauw1961]). However, none of these scenarios consider the possibility how a massive star can acquire a magnetic field during the ejection process. Clearly, these findings generate a strong motivation to carry out a kinematic study of all stars previously surveyed for magnetic fields to search for a correlation between the kinematic status and the presence of a magnetic field.
Based on the still very limited magnetic surveys in massive stars, we cannot yet answer the question if O-type stars are magnetic in certain evolutionary states and in a specific environment. Open star clusters and associations are very useful laboratories to test star formation and stellar evolution. The ages of our subsample of three stars with magnetic fields do not contradict the idea, that it is drawn from the general distribution of cluster ages. We have to keep in mind though that we have a very small number statistics. It appears that our observations are consistent with the assumption that the presence of a magnetic field can be expected in stars of different classification categories. Although it was possible to recognize a few hot Of?p magnetic stars as being peculiar on the basis of their spectral morphology, prior to their field detection (Walborn [@walborn06]), the presence of a magnetic field can also be expected in stars of other classifications. Future magnetic field measurements are urgently needed to constrain the conditions controlling the presence of magnetic fields in hot stars, and the implications of these fields on their physical parameters and evolution.
NVK and AEP thank for support by DFG grant RO 528/10-1. AEP acknowledges support of the RFBR grant 10-02-91338.
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[^1]: Based on observations obtained at the European Southern Observatory, Paranal, Chile (ESO programme 085.D-0667(A)).
[^2]: The spectropolarimetric capabilities of FORS1 were moved to FORS2 in 2009.
[^3]: http//sb9.astro.ulb.ac.be/mainform.cgi
|
---
abstract: 'We show the results of microsecond resolution radio data analysis focused on flux measurements of single pulses of PSR B1133+16. The data were recorded at 4.85 GHz and 8.35 GHz with 0.5 GHz and 1.1 GHz bandwidth, respectively, using Radio Telescope Effelsberg (MPIfR). The most important conclusion of the analysis is, that the strongest single pulse emission at 4.85 GHz and 8.35 GHz contributes almost exclusively to the trailing part of the leading component of the pulsar mean profile, whereas studies at lower frequencies report that the contribution is spread almost uniformly covering all phases of the pulsar mean profile. We also estimate the radio emission heights to be around 1%–2% of the light cylinder radius which is in agreement with previous studies. Additionally these observations allowed us to add two more measurements of the flux density to the PSR B1133+16 broadband radio spectrum covering frequencies from 16.7 MHz up to 32 GHz. We fit two different models to the spectrum: the broken power law and the spectrum based on flicker noise model, which represents the spectrum in a simpler but similarly accurate way.'
author:
- |
K. Krzeszowski,$^1$ O. Maron,$^1$ A. Słowikowska,$^1$ J. Dyks$^2$ and A. Jessner$^3$\
$^1$Kepler Institute of Astronomy, University of Zielona Góra, Lubuska 2, 65–265, Zielona Góra, Poland\
$^2$Nicolaus Copernicus Astronomical Center, Rabiańska 8, 87-100 Toruń, Poland\
$^3$Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany\
date: Released 2014
title: 'Analysis of single pulse radio flux measurements of PSR B1133+16 at 4.85 and 8.35 GHz'
---
\[firstpage\]
pulsars: general – pulsars: individual: B1133+16
Introduction
============
Pulsar radio emission is still not explained in details but it is believed to originate close to the pulsar surface (@krzeszowski2009 and references therein). The analysis of 62 mean profiles of 23 pulsars at different frequencies regarding aberration and retardation effects yielded the estimation of the emission height being below 1500 km, but most probably being of the order of 500 km above the pulsar surface. The advance in theoretical understanding of the emission mechanism and conditions in the magnetosphere have been driven mainly by the observations. Recently many observations have been concentrated on observing single pulses which carry detailed information about the physics of radio emission. Analysis of single pulse high resolution time series can show a few interesting properties i.e. giant and bright pulses, subpulse drift, nulling, microstructure, etc. Observations of single pulses for most pulsars may be carried out mainly at lower frequencies because pulsars are weak radio sources at high frequencies which is clearly seen in their steep spectra. The pulsar spectrum in general can be described by a power law $S\propto\nu^\alpha$, where $\alpha$ is the spectral index. The average spectral index for 266 pulsars for frequency spread from 0.4 to 23 GHz is $\alpha= -1.8$ [@mkk+00].
--------------------- ---------------------------------------------------
BNAME B1133+16
JNAME J1136+1551
$P$ 1.188 s
$\dot{P}$ $3.73 \times 10^{-15}$ s s$^{-1}$
RA (J2000) 11$^\mathrm{h}$36$^\mathrm{m}$03$^\mathrm{s}$
DEC (J2000) 15$^\circ$51’04”
DM 4.86 pc cm$^{-3}$
RM 1.1 rad m$^2$
Age $5.04 \times 10^6$ Yr
Distance 350 $\pm$ 20 pc
Proper motion 375 mas yr$^{-1}$
Transverse velocity $631^{+38}_{-35}$ km s$^{-1}$
$B_\mathrm{surf}$ $2.13 \times 10^{12}$ G
$\dot{E}$ $8.8 \times 10^{31}~\mathrm{erg}~\mathrm{s}^{-1}$
--------------------- ---------------------------------------------------
: Basic properties of PSR B1133+16 [@Brisken2002; @manchester2005].[]{data-label="tab:props"}
PSR B1133+16 is a nearby middle–aged pulsar with one of the highest proper motion, and thus, one with the highest transverse velocity [@Brisken2002]. Basic properties of PSR B1133+16 are gathered in Table \[tab:props\]. Its faint optical counterpart (B=28.1 $\pm$ 0.3 mag) was firstly detected by [@Zharikov2008]. Recently [@Zharikov2013] detected the optical candidate of the pulsar counterpart on the GTC and VLT images that is consistent with the radio coordinates corrected for its proper motion. This source was also detected in X-rays by [@Kargaltsev2006] using the *Chandra* satellite with the flux of (0.8 $\pm$ 0.2) $\times 10^{-14}$ ergs cm$^{-2}$ s$^{-1}$ in the 0.5–8.0 keV range. For the X–rays fit the assumed hydrogen column density was $n_\mathrm{H} = 1.5 \times 10^{20}$ cm$^{-2}$. Low value of $n_\mathrm{H}$ and no $H_\alpha$ Balmer bow shock imply a low density of ambient matter around the pulsar. This pulsar has not been detected by the *Fermi* satellite.
In this paper in Sec. \[sec:observations\] we describe observational parameters and technical issues about the recorded data. Sec. \[sec:single.pulses\] covers analysis of the data. We present two different approaches for data analysis: mean profiles composed of pulses that their flux fall into specific intensity range as well as the phase position and flux of single pulses that are stronger than 20$\sigma$. In Sec. \[sec:emission.heights\] we discuss radio emission height estimations, while in Sec. \[sec:spectrum\] we present the radio spectrum of PSR B1133+16 and discuss different spectrum models. We conclude our results in Sec. \[sec:conclusions\].
Observations and data reduction {#sec:observations}
===============================
Our analysis is based on the Radio Telescope Effelsberg archival data. The observational settings and parameters are collected in Table \[tab:params\]. Observations were made using the 4.85 GHz and the 8.35 GHz receivers of the MPIfR 100 m radio telescope in Effelsberg. The receivers have circularly polarised feeds. Both receivers feature cryogenically cooled High-Electron-Mobility Transistor (HEMT) low noise input stages with typical system temperatures of 27 K for the 4.85 GHz receiver and 22 K for the 8.35 GHz receivers. A calibration signal can be injected synchronously to the pulse period for accurate measurements of pulsar flux densities. The Effelsberg radio telescope is regularly calibrated on catalogued continuum sources and we used the mean height of the injected calibration signal of 1.2 K for 4.85 GHz and 2.083 K for 8.35 GHz for the flux calibrations. The two intermediate frequency (IF) signals (bandwidths of 500 MHz for 4.85 GHz and 1.1 GHz for 8.35 GHz) from each receiver, one for left hand (LHC) and one for right hand (RHC) circular polarisation are detected in a broad–band polarimeter attached to the receiver, providing four output independent signals relating to the power levels of LHC, RHC, LHC$\cdot$RHC$\cdot \sin($LHC,RHC$)$, LHC$\cdot$RHC$\cdot
\cos($LHC,RHC$)$. No dedispersion was used before or after the detection due to low value of DM of PSR B1133+16. The dispersion broadening amounts to 178 $\mu$s and 78 $\mu$s for 4.85 GHz and 8.35 GHz, respectively, which is of the order of their sampling rates. The four detected power levels are linearly encoded as short pulses of a variable frequency of typically 2–3 MHz corresponding to the system noise level, but ranging up to a maximum of 10 MHz. These signals were brought down to the station building and using the EPOS backend, the four frequency encoded power levels were recorded synchronously to the pulse period in 1024 phase bins (=samples) per period. The average signal power for each phase bin was determined by simply counting the number of pulses of the supplied frequency encoded signal for the duration of the phase bin and then recording the counts on disk for later off-line processing. Only the data from LHC and RHC channels were used and added together to yield the detected total power for our analysis. The individual phase bins had durations of 200 $\mu$s for 4.85 GHz and 60 $\mu$s for 8.35 GHz. With the given system temperatures and the respective antenna efficiencies of 1.5 K/Jy and 1.2 K/Jy, we achieved a typical sensitivity (rms) of 30 mJy and 80 mJy per single pulse phase bin for the two frequencies, respectively.
Digitisation effects
--------------------
The aforementioned sensitivity corresponds to a fraction of $2-3\times 10^{-3}$ of the equivalent background (baseline) noise level of 16–18 Jy. At the same time, we recorded typically only 600 (4.85 GHz) down to 200 (at 8.35 GHz) frequency encoded counts per phase bin. As a result, one finds that the signal power is resolved in steps of 30 mJy and 80 mJy for the two receivers and that the rms noise fluctuations are barely resolved, amounting to a couple of counts at most (see Fig. \[fig:correction\]).
Weather and radio interference
------------------------------
Weather, especially clouds, changes the opacity of the atmosphere and as a consequence we found that fluctuations of the sky background were of the order of a fraction of a Kelvin on the short timescales (200 ms and 60 ms) that were used for the measurement window and the subtraction of the noise baseline. Both receivers were also affected by a very low level 100 Hz modulation ($10^{-3} - 10^{-4}$ of the baseline level) originating in the receiver’s power supply and cooling systems. With typical observed pulse component widths of 5–10 ms, we find that we cannot rule out that weak individual components of individual single pulse may be affected by the interference. However, averaging and other statistical flux estimates using sufficient numbers ($>$10) of single pulses would not be affected by weather induced noise fluctuations or power supply interference.
The above mentioned effects are visible in the data as approximately linear trends added to each of the recorded single pulses. One example of such slope and effects of the correction at 8.35 GHz is presented in Fig. \[fig:correction\]. The values of the slopes show no trends from pulse to pulse and they range from -2 to 2 mJy / bin and their distribution is a Gaussian–like with a zero mean. The behaviour of the system at 4.85 GHz is roughly the same as of 8.35 GHz data and the same correction routines were applied.
------------------- ------------- -------------
Date 2002 Feb 07 2004 Apr 26
Frequency 4.85 GHz 8.35 GHz
Bandwidth 0.5 GHz 1.1 GHz
Observing time 67 min. 120 min.
Number of pulses 3361 6029
Sampling time 200 $\mu$s 60 $\mu$s
Mean flux density 1.59 mJy 0.73 mJy
------------------- ------------- -------------
: Observation parameters.[]{data-label="tab:params"}
![Weak single pulse of PSR B1133+16 at 8.35 GHz with a visible linear trend (top panel) and after the correction (bottom panel). Mean profile (dashed line) is plotted for comparison. Horizontal dashed line is shown for reference.[]{data-label="fig:correction"}](correction.eps){width="84mm"}
Single pulses in microsecond resolution {#sec:single.pulses}
=======================================
Average profiles of PSR B1133+16 at 4.85 GHz and 8.35 GHz consist of two main components connected by a bridge of emission (Fig. \[fig:mean.profiles\]). The leading component is approximately five times stronger than the trailing one and they are separated by around 5$^\circ$. The duty cycle of this pulsar is around 3% at 4.85 GHz and 8.35 GHz.
![Mean profiles of PSR B1133+16 at 4.85 GHz (top panel) and 8.35 GHz (bottom panel). The profiles are aligned with respect to the maximum of the trailing component.[]{data-label="fig:mean.profiles"}](mean_profiles.eps){width="84mm"}
We performed an analysis of microsecond resolution radio data focused on flux measurements of single pulses. High resolution observations allowed us to investigate the microstructure of single pulse shapes. The single pulse at 8.35 GHz in the top panel of Fig. \[1133-203\] shows interesting features in the trailing component. On the other hand the bottom panel of Fig. \[1133-203\] is focused on the leading component of another single pulse profile at the same frequency. As it can be clearly seen, single pulse profiles have complex structures, which can be resolved only with high time resolution observations. There were numerous studies of the microstructure of PSR B1133+16 and other pulsars (e.g. @ferguson78 [@lange98]). It is reported that microstructure is most probably related to the emission process and is present in many pulsars with the fraction of pulses showing microstructure being of the order of 30% to 70% [@lange98].
![An example of a strong single pulse of PSR B1133+16 at 8.35 GHz with some interesting features in the trailing component (top panel) and the structure of the leading component of another strong single pulse profile at 8.35 GHz (bottom panel).[]{data-label="1133-203"}](203.eps "fig:"){width="82mm"} ![An example of a strong single pulse of PSR B1133+16 at 8.35 GHz with some interesting features in the trailing component (top panel) and the structure of the leading component of another strong single pulse profile at 8.35 GHz (bottom panel).[]{data-label="1133-203"}](789.eps "fig:"){width="84mm"}
Timescale constraints on geometrical parameters
-----------------------------------------------
The timescale of real-time flux variability (such as the micro- or nano–structure) can be interpreted in terms of the size of the emitting region, or in terms of the angular size of relativistically–beamed radiation pattern (@lange98, @crossley10). The outcome depends on the actual spatial and temporal structure of the emission region, and its relation to the angular scale of the radiated beam.
Let us consider a localised, and relativistically-outflowing source of radio emission, e.g. a cloud of charges with Lorentz factor $\gamma$ moving along a *narrow* bunch of magnetic field lines. Here *narrow* means that the spread of $B$-field direction within the emitting stream, as measured for different rotational azimuths, is smaller than the intrinsic size of the emission pattern $1/\gamma$. In the observer’s reference frame the charges move along a narrow bunch of trajectories with the radius of curvature $\rho$. If the source emits detectable radiation for a limited period of time $\Delta t_{\rm em}$ (as measured in our reference frame) an observed spike of radio flux has a width of $\tau\equiv\Delta t_{\rm obs}= \Delta t_{\rm em}(c-v)/c\simeq \Delta t_{\rm em}/(2\gamma^2)$ (since the outflowing source is nearly catching up with the emitted photons). If the source persists for a time sufficient to sweep its full $1/\gamma$ beam across the observer’s line of sight, then $\Delta t_{\rm em} \simeq \rho/(\gamma c)$, and the timescale is: $$\tau = \tau_{\rho} \simeq \rho/(2 c \gamma^3) = 1.7\times 10^{-9}{\rm s}\
\rho_8/\gamma_2^3
\label{taurho}$$ where $\rho_8=\rho/10^8$ cm and $\gamma_2 = \gamma/100$ [@jackson75]. During that time the point source moves up in pulsar magnetosphere by a distance of $c\Delta t_{\rm em} \simeq 10^6{\rm cm}\ \rho_8/\gamma_2$. For $\rho=10^8$ cm the observed timescale of the microstructure of $\tau \sim 100$ $\mu$s limits the Lorentz factor to the mildly relativistic value $\gamma \simeq 2.6$. For $\gamma=10$ the observed timescale implies $\rho \simeq 6\times 10^{9}$ cm, comparable to the light cylinder radius of PSR B1133$+$16 ($R_{\rm LC} = 5.67\times 10^9$ cm). Note that for the curvature radiation from a localised emitter, by definition one expects $\tau \simeq 1/\nu$, where $\nu$ is the observed frequency (a few GHz). That is, the pair of values $\rho$ and $\gamma$ must ensure that the curvature spectrum extends up to the frequency $\nu$. The observed time scale of the microstructure ($\sim\negthinspace10^{-4}$ s) is then too long to directly correspond to the rapid sweep of the elementary beam of the curvature radiation emitted by a small plasma cloud.
Another case is encountered when the emitting clouds extend considerably along $B$-field lines ($\Delta x \gg
\rho\gamma^{-3}$), or when there is a steady outflow of uniformly-distributed matter that emits radio waves. Again, let us first consider the *narrow stream* case, in which the internal spread of rotational azimuths $\Delta \phi_B$ of $\vec B$ is much smaller than the size of the relativistic beam ($\Delta \phi_B \ll 1/\gamma$). In such a case, projection of the beam on the sky results in an elongated stripe of width $1/\gamma$. The observed timescale is then determined by the speed of sightline passage through the $1/\gamma$ stripe, as resulting from the rotation of the neutron star: $$\begin{aligned}
\tau = \tau_{\rm rot} \simeq P\ (2\pi\gamma\sin\zeta\sin\delta_{\rm cut})^{-1} = \\
= 1.6\times 10^{-3}P\ (\gamma_2\sin\zeta\sin\delta_{\rm cut})^{-1},
\end{aligned}
\label{taurot}$$ where $\zeta$ is the viewing angle between the sightline and the rotation axis, and $\delta_{\rm cut}$ is the ‘cut angle’ between the sky-projected emission stripe and the path of the line of sight while it is traversing through the beam (see Fig. 2 in @dyks2012). @lange98 provide a similar timescale estimate (eq. 6 therein) which is valid only for $\delta_{\rm cut}=90^\circ$, i.e. for orthogonal passage of the sightline through the beam. For the parameters $\alpha=88^\circ$ and $\zeta = 97^\circ$ determined by @ganga99, the observed separation of components ($2\phi\simeq5^\circ$) implies $\delta_{\rm cut}=74^\circ$. Eq. (\[taurot\]) then gives $\gamma_2 \simeq
20$, which is smaller than the minimum Lorentz factor required for the curvature spectrum to extend up to the observed frequency of a few GeV.
In a thick stream case, the spread of rotational azimuths of $\vec B$ within the stream is much larger than the angular size of the stream ($\Delta \phi_B \gg 1/\gamma$). The emitted radiation can be considered (approximately) tangent to the local magnetic field, and to the charge trajectory in the observer’s frame. The observed timescale is then determined by the angular extent of the stream in the magnetic azimuth $\phi_m$, measured around the dipole axis. For the stream extending between $\phi_{m,1}$ and $\phi_{m,2}$ the timescale is equal to $\tau=\tau_{\phi m}=\phi_{2} -
\phi_{1}$, where the pulse longitudes $\phi_1(\phi_{m,1})$ and $\phi_2(\phi_{m,2})$ correspond to the moments when our sightline starts and stops probing the region of the stream. For known (or assumed) $\alpha$, $\zeta$, $\phi_{m,1}$ and $\phi_{m,2}$ the values of $\phi_1$ and $\phi_2$ can be calculated from eqs. (18) and (19) in [@dyks2010]. However, the timescale of $\tau\sim100$ $\mu$s corresponds to the angle $2\pi\tau/P = 0.03^\circ$ which requires $\gamma \ge 2\times 10^3$. Thus, for PSR B1133$+$16 the thick stream case may need to be considered only for $\gamma >$ a few thousands.
Single pulse flux distribution
------------------------------
We define bright pulses as those with energy of ten times greater than the mean flux of the pulsar. Looking closer at the flux density distribution (Fig. \[fig:fluxes.histogram\]) one can see that there are not many bright pulses in our time series, only around 0.9% at both frequencies.
![Flux density distribution (grey bars) and off–pulse intensity distribution (black outlined bars) of 4.85 GHz (top panel) and 8.35 GHz (bottom panel) data. Vertical dashed lines denote ten intensity ranges with a number of pulses that fall into the particular range. Mean profiles composed of single pulses that fall into a particular range are presented in Fig. \[fig:intensity.ranges\] for both frequencies.[]{data-label="fig:fluxes.histogram"}](flux_histogram_4.eps "fig:"){width="84mm"} ![Flux density distribution (grey bars) and off–pulse intensity distribution (black outlined bars) of 4.85 GHz (top panel) and 8.35 GHz (bottom panel) data. Vertical dashed lines denote ten intensity ranges with a number of pulses that fall into the particular range. Mean profiles composed of single pulses that fall into a particular range are presented in Fig. \[fig:intensity.ranges\] for both frequencies.[]{data-label="fig:fluxes.histogram"}](flux_histogram_8_corrected.eps "fig:"){width="84mm"}
{width="168mm"}
All flux measurements (Fig. \[fig:fluxes.histogram\]) for both frequencies were divided into ten equally sized intensity ranges [@nowakowski96]. The ranges were constructed as follows: the minimum and maximum flux measurements from each data set were taken as boundaries and all remaining pulse flux values were assigned to one of the ($S_\mathrm{max} - S_\mathrm{min}$) / 10 ranges of the widths of around 5 mJy and 3 mJy for 4.85 GHz and 8.35 GHz, respectively. The minimum and maximum flux values are $-$0.83 mJy and 53.11 mJy as well as $-$1.38 mJy and 27.15 mJy for 4.85 GHz and 8.35 GHz data, respectively. Negative flux values are introduced into the data because of observing system properties which were discussed in detail in Sec. \[sec:observations\] and stochastic noise properties in pulses with no detection. For the single pulses with flux falling into a particular range the average profiles were constructed. Normalised profiles for 4.85 GHz and 8.35 GHz data and respective mean profiles of first five intensity ranges are presented in Fig. \[fig:intensity.ranges\]. The number of pulses falling into specific intensity range is written in each panel and also shown in the Fig. \[fig:fluxes.histogram\] above the distributions. In the Fig. \[fig:intensity.ranges\] it is easily noticeable that the maximum of the intensity level averaged profile (solid line) moves towards later phases with respect to the maximum of the mean pulsar profile (dotted line) for both frequencies and the shift is increasing with frequency from 0.36$^\circ$ at 4.85 GHz to 0.55$^\circ$ at 8.35 GHz. It means that low intensity single pulses contribute mainly to the leading part of first component, whereas higher intensity single pulses contribute almost exclusively to its trailing part. On the other hand the second component is composed mainly of the lowest intensity single pulses — it is visible only in the first two intensity ranges. The shifts of pulsar components were also investigated by @mitra2007. They report that the stronger emission of B0329+54 comes earlier than the weaker emission with a delay of 1.5$^\circ$. This is opposite to what we have observed. Moreover, Mitra et al. investigated the effect only at one frequency, i.e. 325 MHz. Therefore, further investigations of more pulsars single pulses are highly recommended because the mechanism behind observed effect is not understood and one does not know how common it may be in other pulsars.
It is clearly visible in Fig. \[maxima\] that at 4.85 GHz and 8.35 GHz maximum flux values (denoted with dots) contribute almost only to the trailing edge of the first component of the mean profile, whereas studies at lower frequencies report that the contribution is spread almost uniformly covering all phases of the pulse mean profile [@kss+11]. A mean profile composed only of single pulses with SNR $>~20\sigma$ is plotted with a solid line, whereas the mean profile composed of all of the single pulses is plotted with a dotted line. The numbers of such pulses are 758 (23%) at 4.85 GHz and 407 (7%) at 8.35 GHz, respectively.
![Maximum flux positions for single pulses with SNR $> 20 \sigma$ at 4.85 GHz (top panel) and 8.35 GHz (bottom panel). The mean profile composed of all pulses is plotted with dotted line, whereas the mean profile composed of the pulses with SNR $> 20 \sigma$ is plotted with solid line. Both mean profiles and flux maxima of individual pulses are normalised with respect to their maximum values. The profiles are aligned with respect to the maximum of the trailing component at both frequencies.[]{data-label="maxima"}](maxima.eps){width="84mm"}
Emission heights {#sec:emission.heights}
================
In the case of 8.35 GHz data from Fig. \[fig:intensity.ranges\] (right hand side panels) one can read out the shift of 0.0018 seconds ($\Delta\phi~\approx$ 0.55$^\circ$ = 0.001 rad) between the longitude of the low-flux and high-flux emission. Ignoring (for a while) the curved shape of B-field lines, this shift can be translated to altitude difference: $\Delta r_\mathrm{em}~=~R_\mathrm{LC}~\Delta\phi/2~=~2.7~\times~10^7$ cm (independent of $\alpha$ and $\zeta$). However, the radius of curvature of B–field lines at the rim of polar cap of this pulsar is $\rho_\mathrm{B}~=~(4/3)(r_\mathrm{NS}~R_\mathrm{LC})^{1/2}~=~10^8$ cm. The upward shift of emission by $\Delta r_\mathrm{em}$, results then in a change of emission direction by $\Delta r_\mathrm{em}
/ \rho_\mathrm{B}$ $\approx$ 0.27 rad $\approx$ 15$^\circ$, which is much larger than the observed displacement of 0.5$^\circ$. Therefore, if the observed misalignment of low–flux and high–flux emission has anything to do with the real spatial shift of emission region, it must be dominated by the effect of curved B–field lines rather than the aberration–retardation shift. Unfortunately, in such a case a full information on geometry is needed to determine $r_\mathrm{em}$ or $\Delta r_\mathrm{em}$. The same logic applied to 4.85 GHz data (left hand side panels Fig. \[fig:intensity.ranges\]) yields the following results: $\Delta\phi~\approx~0.36^\circ~=~0.006$ rad, $\Delta
r_\mathrm{em}~\approx~1.8\times10^6$ cm and $\Delta r_\mathrm{em} / \rho_\mathrm{B}$ $\approx~10^\circ$ which is also much bigger than the measured shift of 0.36$^\circ$.
Using $\alpha = 88^\circ$ and $\beta = 9^\circ$ [@ganga99] and canonical formulae [@lorimer2005] we derived radio emission heights at 4.85 GHz and 8.35 GHz. While making the height estimates we associate the peak separation with two different sets of B–field lines: the last open field lines which have the standard magnetic colatitude $\sin\theta_\mathrm{pc}=(r_\mathrm{NS}/R_\mathrm{LM})^{1/2}$, and with the critical field lines, which have $\sin\theta_c=(2/3)^{3/4}(r_\mathrm{NS}/R_\mathrm{LC})^{1/2}$.
Simple calculations yield the estimations of emission heights at 4.85 GHz are 67$\times10^6$cm and 122$\times10^6$cm for the last open and critical magnetic field lines, respectively, whereas for stronger emission it is closer to the neutron star surface at 66$\times10^6$cm and 120$\times10^6$cm. Similarly, at 8.35 GHz, emission heights are 66$\times10^6$cm and 122$\times10^6$cm for the last open and critical field lines, respectively, whereas for stronger emission it is 65$\times10^6$cm and 119$\times10^6$cm. Our analysis shows that the emission region is located at a distance of around 1%-2% of the light cylinder radius from the pulsar surface which is consistent with earlier studies (e.g. @krzeszowski2009).
Radio spectrum {#sec:spectrum}
==============
In general pulsars have steep spectra with an average spectral index around $-1.8$ [@mkk+00]. Except for basic spectrum (that can be described with power law $S \propto \nu^{\alpha}$) there are two common types: spectrum with a break (described with two power laws) and spectrum with a turn–over (clearly visible maximum flux). We collected flux density measurements from different publications (Table \[table:fluxes\]). Our dataset covers a very wide radio frequency range from 16.7 MHz up to 32 GHz. The analysis of the data presented in this paper yielded the mean flux values of 1.59 mJy and 0.73 mJy at 4.85 GHz and 8.35 GHz, respectively, with an estimated error of 10% of the original value. The spectrum of PSR B1133+16 spanning the wide radio frequency range is shown in Fig. \[fig:spectrum\]. Each point denotes a measurement of mean flux density with respective uncertainties. We included the measurements by @kss+11 at 7 frequencies, ranging from 116.75 MHz to 173.75 MHz. The authors claim that the spectrum of PSR B1133+16 over their “reasonably wide frequency range” of 57 MHz is of a broken power law type with spectral indices of $\alpha_1=2.33\pm2.55$ and $\alpha_2=-3.8\pm2.24$. In Fig. \[fig:spectrum\] we have indicated their spectrum by a dashed line. We cannot confirm Karuppusamy’s spectral indices which were obtained from low frequency measurements covering only a small range of frequencies. However, we find that Karuppusamy’s measurements have a typical spread of flux values and thereby fit well into the overall spectrum as can be seen in Fig. \[fig:spectrum\].
{width="168mm"}
Our spectrum of PSR B1133+16 may be described by two power laws in the whole frequency range with $\alpha_1=-0.04~\pm~0.0001$ and $\alpha_2=-1.96~\pm~0.0001$ with a break frequency of $\nu_b=256~\pm~0.016$ MHz (${\rm
\chi_{red}^2=5.1}$).
To reproduce the spectrum of PSR B1133+16 covering over 32 GHz frequency range we have also fitted the flicker noise model proposed by @ljk+08 which is described by $${\rm S(\omega)=S_0\left(\frac{1+\omega^2\tau_e^2}{\tau_e^2}\right)^{n-1}\!\!\!\!\times e^{-i(n-1)atan(\omega\tau_e)},}
\label{equation:spectrum}$$ where S$_0$ is a scaling factor, ${\rm \omega=2\pi\nu}$, $\nu$ is an observing frequency, ${\rm \tau_e}$ is a characteristic time for the nano–burst decay time, and n is the exponent which constrains a combination of physical parameters of nano pulses (for details refer to @ljk+08). Our fitted parameters of ${\rm
S_0=3.39~\pm~0.77~Jy}$, ${\rm \tau_e=0.40~\pm~0.05~ns}$ and ${\rm n=0.118~\pm~0.022}$ are in good agreement with @ljk+08 result and our reduced ${\rm \chi_{red}^2=4.7}$ is comparable to the value for the broken power law fit. The model, apart from the scaling factor S$_0$, is based on only two physical parameters. The first one is the duration of a nano–pulse ($\tau_\mathrm{e}$) and the second one (n) is related to the geometry of the emission process. It is based on the assumption that the pulsar radio emission is in fact the superposition of many nano–pulses, which in case of PSR B1133+16 have duration of $\tau_\mathrm{e} = 0.51$ ns according to @ljk+08. However our fit gives even shorter nano–pulse duration of 0.4 ns. @ljk+08 report that for 12 pulsars 0.1 ns $<~\tau_\mathrm{e}~<$ 2.0 ns.
----------- --------- --------- ----------- ----------- --------- --------- ------------
Frequency Flux Error Reference Frequency Flux Error Reference
\[MHz\] \[mJy\] \[mJy\] \[MHz\] \[mJy\] \[mJy\]
16.7 310 50 \[1\] 156 580 58 \[1\]
16.7 1410 320 \[1\] 160 520 80 \[1\]
20 700 100 \[1\] 170 380 150 \[1\]
25 560 100 \[1\] 173.75 380 38 \[4\]
25 1900 230 \[1\] 196 560 260 \[1\]
34.5 900 90 \[1\] 270 790 190 \[1\]
39 740 107 \[1\] 275 475 85 \[1\]
39 620 90 \[2\] 341 320 60 \[5\]
53 1105 260 \[1\] 370 305 124 \[1\]
61 830 170 \[1\] 408 256 53 \[6\]
61 700 140 \[2\] 606 144 35 \[6\]
74 780 150 \[1\] 626 120 40 \[5\]
80 640 70 \[1\] 925 37 11 \[6\]
80 1000 500 \[1\] 1400 24.92 15.33 \[7\]
85 910 150 \[1\] 1408 32 5 \[6\]
85 770 130 \[2\] 1412 60 20 \[5\]
100 420 150 \[1\] 1606 51 15 \[6\]
102 1210 180 \[1\] 2300 7.07 3.24 \[7\]
102 1280 550 \[3\] 2700 8.7 0.6 \[7\]
102 1520 660 \[1\] 4850 2.7 1.07 \[7\]
102 1020 200 \[2\] 4850 2.4 0.5 \[5\]
111 770 190 \[1\] 4850 1.59 0.16 this paper
116.75 550 55 \[4\] 8350 0.73 0.07 this paper
130 720 72 \[4\] 8500 0.86 0.12 \[7\]
139.75 1000 100 \[4\] 10550 0.63 0.12 \[7\]
142.25 1000 100 \[4\] 14600 0.169 0.0507 \[7\]
147.5 580 58 \[4\] 14800 0.26 0.078 \[7\]
151 1100 170 \[1\] 24000 0.178 0.0534 \[7\]
151 805 75 \[1\] 32000 0.03 0.02 \[7\]
156 800 80 \[4\] 32000 0.055 0.06 \[8\]
----------- --------- --------- ----------- ----------- --------- --------- ------------
Conclusions {#sec:conclusions}
===========
The analysis of PSR B1133+16 single pulses is a process that needs a certain amount of care. First off all, it is important to take into account different observational and technical effects that can affect recorded data, especially with very high time resolution. The effects that are presented in this paper play huge role and alter the data significantly. Understanding of such effects and their influence on the data recording process is important for proper data reduction. Some of the effects are not visible in the mean profiles but only in single pulses data.
The mean profiles of PSR B1133+16 at 4.85 GHz and 8.35 GHz consist of two components (Fig. \[fig:mean.profiles\]). The second component is emitted almost exclusively by low intensity individual pulses. On the other hand, the first component is seen in single pulses regardless of their intensity except for the cases when it is not present at all. However, lower intensity emission contributes mostly to the leading part of the first component whereas higher intensity single pulses contribute mainly to its trailing part (Fig. \[fig:intensity.ranges\]) which was also reported by @maron2013. The results of analysis of 4.85 GHz and 8.35 GHz data are consistent with previous studies by [@nowakowski96] at 430 MHz but studies of B0329+54 [@mitra2007] show entirely opposite behaviour without full explanation. This inconsistency requires further studies of other pulsars single pulses to explain this effect.
We show, in contradiction to studies at lower frequencies by @kss+11, who report an almost uniform spread of single pulse maxima, that the maximum emission of B1133+16 single pulses at 4.85 and 8.35 GHz contributes almost exclusively to the trailing part of the leading component of the mean profile. Our result is consistent with the behaviour at 341, 626, 1412 and 4850 HMz mentioned by @kra+03 and extends the studies up to 8.35 GHz.
Radio emission arises close to the pulsar surface at the distances of around 65 stellar radii at frequencies of 4.85 GHz and 8.35 GHz. Weaker emission, which contributes to the leading part of the leading components, comes in earlier phases which suggests that originates in magnetosphere further from the pulsar surface than more energetic emission. Our calculations shows, that the difference of the emission heights for stronger and weaker emission is of the order of a few stellar radii, amounting to a change of 1%–2% of the emission height, which is consistent with previous estimations [@krzeszowski2009].
There are 60 mean flux measurements in the literature of PSR B1133+16 that are known to us. They span a very wide radio frequency range from 16.7 MHz up to 32 GHz. To reproduce the spectrum we fitted two different models: the broken power–law model and one based on flicker noise model of pulsar radio emission [@ljk+08]. Surprisingly, the model proposed by @ljk+08 is not widely used in the literature although it reproduces the pulsar spectrum comparably well to the power-law model. Future high time resolution observations might be useful to verify nano–pulse emission model.
Due to the fact, that the pulsar radio emission is weaker at higher frequencies, giant pulses are the ones that can allow us to study closely their structure. In our both data samples there is roughly one per cent of bright pulses that are at least ten times stronger than mean flux and their microstructure is clearly visible.
Acknowledgements {#acknowledgements .unnumbered}
================
The presented results are based on the observations with the 100–m telescope of the MPIfR (Max-Planck-Institut für Radioastronomie) at Effelsberg. This work has been supported by Polish National Science Centre grants DEC-2011/03/D/ST9/00656 (KK, AS), DEC-2012/05/B/ST9/03924 (OM) and DEC-2011/02/A/ST9/00256 (JD). Data analysis and figures were partly prepared using R [@rcite].
\[lastpage\]
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---
abstract: 'The Yoneda algebra of a Koszul algebra or a $D$-Koszul algebra is Koszul. ${{\cal K}}_2$ algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a ${{\cal K}}_2$ algebra would be another ${{\cal K}}_2$ algebra. We show that this is not necessarily the case by constructing a monomial ${{\cal K}}_2$ algebra for which the corresponding Yoneda algebra is not ${{\cal K}}_2$.'
author:
-
bibliography:
- 'bibliog.bib'
title: 'The Yoneda algebra of a ${{\cal K}}_2$ algebra need not be another ${{\cal K}}_2$ algebra'
---
-0.1in 8.5in 5.5in
.5in .5in
\[section\] \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{}
\[thm\][Definition]{} \[thm\][Notation]{} \[thm\][Example]{} \[thm\][Remark]{} \[thm\][Remarks]{} \[thm\][Note]{} \[thm\][Example]{} \[thm\][Problem]{} \[thm\][Question]{}
-.2in Thomas Cassidy$^\dagger$, Christopher Phan$^\ddagger$ and Brad Shelton$^\ddagger$\
$^\dagger$Department of Mathematics\
Bucknell University\
Lewisburg, Pennsylvania 17837\
\
$^\ddagger$Department of Mathematics\
University of Oregon\
Eugene, Oregon 97403-1222
Introduction
============
Let $A$ be a connected graded algebra over a field $K$. Correspondences between $A$ and its bigraded Yoneda algebra $E(A)= \bigoplus_{n,m} Ext_A^{n,m}(K,K)$ have been studied in many contexts (e.g. , [@LPWZ2], [@M-V] and [@Priddy]). In particular there are very interesting classes of algebras where $E(A)$ inherits good properties from $A$. Perhaps the most famous and intently studied of such classes of algebra is the class of Koszul algebras.
An algebra is Koszul [@Priddy] if its Yoneda algebra is generated as an algebra by cohomology degree one elements. Koszul algebras will always have quadratic defining relations and given such an algebra, $A$, the Yoneda algebra is isomoprhic to the quadratic dual algebra $A^!$. In particular, one has Koszul duality: If $A$ is Koszul then $E(A)$ is Koszul and $E(E(A)) = A$.
The following natural generalization of Koszul was introduced in [@CS-K2] and also investigated in [@Mauger] and [@Phan]. We write $E^n(A)$ for $\bigoplus_p Ext_A^{n,m}(K,K)$.
The graded algebra $A$ is said to be ${{\cal K}}_2$ if $E(A)$ is generated as an algebra by $E^1(A)$ and $E^2(A)$.
Koszul algebras are simply quadratic ${{\cal K}}_2$ algebras. ${{\cal K}}_2$ algebras share many of the nice properties of Koszul algebras, including stability under tensor products, regular normal extensions and graded Ore extensions, (cf. [@CS-K2]). Every graded complete intersection is a ${{\cal K}}_2$ algebra.
Another important class of algebras is the class of $D$-Koszul algebras introduced by Berger in [@Berger]. This is the class defined by: $Ext_A^{n,m}(K,K) = 0$ unless $m = \delta(n)$, where $\delta(2n) = nD$ and $\delta(2n+1) = nD+1$. These algebras arise naturally in certain contexts and all such $D$-Koszul algebras are easily seen to be ${{\cal K}}_2$. A remarkable theorem in [@GMMZ] states that if $A$ is $D$-Koszul algebra, then $E(A)$ is a ${{\cal K}}_2$ algebra, and furthermore, it is possible to regrade $E(A)$ in such a way that $E(A)$ becomes a Koszul algebra. In particular one gets a “delayed” duality: $E(E(A)) = E(A)^!$ and $E(E(E(A))) = E(A)$.
Based on the above theorem of [@GMMZ], Koszul duality, and calculations of many other ${{\cal K}}_2$-examples, it seems reasonable to hope that the Yoneda algebra of any ${{\cal K}}_2$ algebra would also be ${{\cal K}}_2$, perhaps even Koszul. Unfortunately, this is not always the case, and the purpose of this article is to exhibit an example of a ${{\cal K}}_2$ algebra for which the corresponding Yoneda algebra is not Koszul nor even ${{\cal K}}_2$. Our example has 13 generators and 9 monomial defining relations. We believe that such a monomial algebra cannot be constructed with fewer generators and relations.
We wish to thank Jan-Erik Roos for pointing out an error in an earlier version of this paper.
The algebras $A, E(A)$ and $E(E(A))$
=====================================
Let $K$ be a field. Let $\{m,n,p,q,r,s,t,u,v,w,x,y,z\}$ be a basis for a vector space $V$. We define $A$ to be the $K$-algebra $T(V)/I$ where $I$ is the ideal generated by this list of monomial tensors: $$R=\{mn^2p,\ n^2pqr,\ npqrs,\ pqrst,\ stu,\ tuvwx,\ uvwxy,\ vwxy^2,\ xy^2z\}.$$
The algebra $A$ is ${{\cal K}}_2$, but the algebra $E(A)$ is not ${{\cal K}}_2$.
We use the algorithm given in section 5 of [@CS-K2] to prove that $A$ is ${{\cal K}}_2$. From the set $R$ one can calculate that $S_1=\{m,n,p,q,r,s,t,u,v,w,x,y,z\}$, $S_2=\{mn^2,\ n^2pq,\ npqr,\ pqrs,\ st,\ tuvw,\ uvwx,\ vwxy,\ xy^2\}$, $S_3=\{pqr, \ vw\}$, $S_4=\{n^2\}$ and $S_5= \emptyset$. One easily verifies that for every $b\in S$ with minimal left annihilator $a$ we have either deg$(a)=1$ or $ab\in R$, and hence $A$ is ${{\cal K}}_2$.
Let $B=E(A)$. In what follows we consider only the cohomology grading on $B$. Following section 5 of [@CS-K2] we can construct a minimal projective resolution $P^\bullet$ of $_AK$ and see that the Hilbert Series of the algebra $B$ is $1+13t^2+9t^2+8t^3+4t^4+3t^5+t^6$.
It is possible (although laborious) to describe $B$ in terms of generators and relations and then construct a minimal resolution of $_BK$ and apply Theorem 4.4 of [@CS-K2] to show that $B$ is not ${{\cal K}}_2$. However $B$’s failure to be ${{\cal K}}_2$ is apparent already in $Ext_B^3(K,K)$ and consequently there is a more efficient way for us to illustrate this.
Let $\bar m$ and $\bar z$ denote the basis elements in $B_1$ dual to $m$ and $z$ in $A_1$. The vector space $B_2$ has a basis dual to the elements of the list of relations $R$. We will use $\alpha, \beta$ and $\gamma$ to denote the dual basis elements corresponding to the monomials $n^2pqr, stu$ and $vwxy^2$. From the maps in the resolution $P^\bullet$ one can see that $\bar m \alpha$, $\gamma\bar z$ and $\beta \gamma$ are nonzero in $B$, while $\bar m \alpha\beta$ and $\beta\gamma\bar z$ are each zero.
Recall that $Tor^B(K,K)$ can be calculated using the bar-complex [@PP] where ${\mathcal B}{ar}_i(K,B,K)=K\otimes_B\otimes B\otimes B_+\otimes \dots\otimes B_+\otimes K=B_+^{\otimes i}$. Let $\zeta=\bar m\alpha\otimes\beta \gamma\otimes\bar z\in B_+^{\otimes 3}$. The differential on the bar-complex gives us $\partial(\zeta)=\bar m\alpha\beta \gamma\otimes\bar z -\bar m\alpha\otimes\beta \gamma\bar z=0$. $\zeta$ is not in the image of $B_+^{\otimes 4}$ because $\partial (\bar m\otimes\alpha\otimes\beta \gamma\otimes\bar z)= \bar m\alpha\otimes\beta \gamma\otimes\bar z-\bar m\otimes\alpha \beta \gamma\otimes\bar z$ while $\partial (\bar m \alpha\otimes\beta \otimes\gamma\otimes\bar z)= -\bar m \alpha \otimes \beta \gamma\otimes\bar z+\bar m \alpha \otimes \beta \otimes\gamma \bar z$. Thus $\zeta$ represents a non-zero homology class in $Tor^B_3$.
In contrast the element $\bar m\alpha\otimes\beta\gamma=\partial(-\bar m\alpha\otimes\beta\otimes\gamma)$ represents zero in $Tor^B_2$ and $\bar m\alpha=\partial(\bar m\otimes\alpha)$ represents zero in $Tor^B_1$. Therefore under the co-multiplication map $$\Delta:Tor_3^B(K,K)\to Tor_2^B(K,K)\otimes Tor_1^B(K,K)\oplus Tor_1^B(K,K)\otimes Tor_2^B(K,K)$$ we have $\Delta(\zeta)=0$. This failure of $\Delta$ to be injective is equivalent to the multiplication map $$E^2(B) \otimes E^1(B)\oplusÊÊE^1(B) \otimes E^2(B) \to E^3(B)$$ not being surjective. Hence $E(B)$ is not generated by $E^1(B)$ and $E^2(B)$, and so $B$ is not a ${{\cal K}}_2$ algebra.
|
---
abstract: 'The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using the proper extension of the so-called spatial averaging principle introduced by G. Sell and J. Mallet-Paret. Moreover, the extra regularity of this manifold is also obtained.'
address: '${}^1$ University of Surrey, Department of Mathematics, Guildford, GU2 7XH, United Kingdom.'
author:
- 'Anna Kostianko${}^1$ and Sergey Zelik${}^1$'
title: 'Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions '
---
[^1]
Introduction {#s0}
============
It is believed that the dynamics generated by dissipative PDEs in bounded domains is typically finite-dimensional. The latter means that despite the infinite-dimensionality of the initial phase space, the limit dynamics, say, on the so-called global attractor can be effectively described by finitely many parameters which satisfy a system of ODEs – the so-called inertial form of the dissipative PDEs considered, see [@BV; @CV; @MirZe; @28; @tem] and references therein. This reduction clearly works when the underlying PDE possesses an [*inertial*]{} manifold (IM) that is a finite-dimensional invariant $C^1$-smooth manifold with exponential tracking property. Then the desired inertial form can be constructed just by restricting the considered PDE to the invariant manifold, see [@FST; @mik; @rom-man]. However, the existence of an inertial manifold requires rather strong [*spectral gap*]{} assumptions which are usually satisfied only for the parabolic equations in the space dimension one and, despite a big permanent interest and many results obtained in this direction, the finite-dimensional reduction for the case where the IM does not exist remains unclear and there are even some evidence that the dissipative dynamics may be [*infinite-dimensional*]{} in this case, see [@EKZ; @sell-counter; @Zel] and reference therein.
It is also known that the above mentioned spectral gap assumptions are sharp and cannot be relaxed/removed at least on the level of the abstract functional models associated with the considered PDE, see [@EKZ; @mik; @rom-man]. However, an IM may exist for some concrete classes of PDEs even when the spectral gap condition is violated. The most famous example is a scalar reaction-diffusion equation $$\label{0.RDE}
\Dt u=\Delta_x u-f(u)$$ on a 3D torus $x\in[-\pi,\pi]^3$. Here the spectral gap condition reads $$\label{0.gapRDE}
\lambda_{N+1}-\lambda_N>2L,$$ where $\lambda_1\le\lambda_2\le\cdots$ are the eigenvalues of the minus Laplacian on a torus enumerated in the non-decreasing order and $L$ is a Lipschitz constant of the nonlinearity $f$. The eigenvalues of the Laplacian are all natural numbers which can be presented as sums of 3 squares and by the Gauss theorem, there are no gaps of length more than 3 in the spectrum, so the spectral gap condition clearly fails if the Lipschitz constant $L$ is large enough. Nevertheless, the corresponding IM can be constructed (for all values of the Lipschitz constant $L$) using the so-called [*spatial averaging*]{} principle introduced in [@mal-par].
One more example is the 1D reaction-diffusion-advection problem $$\label{0.RDEA}
\Dt u=\partial^2_x u+\partial_x f(u)-g(u),\ \ x\in[0,\pi],\ u\big|_{x=0}=u\big|_{u=\pi}=0,$$ where the spectral gap condition is also not satisfied initially, but is satisfied after the proper change of the dependent variable $u$, see [@Zel1] and also [@rom-th; @rom-th1] where the Lipschitz continuous inertial form is constructed.
Although the spatial averaging principle has been used to get the IM for reaction-diffusion equations in some non-toroidal domains, see [@kwean], to the best of our knowledge, it has been never applied before to the equations different from the scalar reaction-diffusion ones. The aim of the paper is to cover this gap by extending the method to the so-called Cahh-Hilliard equation on a 3D torus. To be more precise, we consider the following 4th order parabolic problem: $$\label{0.CH}
\Dt u+\Delta_x(\Delta_x u-f(u))=0,\ \ \partial_n u\big|_{\partial\Omega}=\partial_n\Delta_xu\big|_{\partial\Omega}=0,\ \ u\big|_{t=0}=u_0,$$ where $\Omega$ is a bounded 3D domain and $f(u)$ is a given non-linear interaction function, see [@CH; @Ell; @No1] and the references therein concerning the physical background of this equation. We also assume that this function satisfies some standard dissipativity assumptions, so the associated semigroup possesses a global attractor $\mathcal A$ which is bounded in $H^2(\Omega)\subset C(\Omega)$, see e.g., [@CMZ; @MirZe; @tem] for more details. By this reason, without loss of generality, we may assume from the very beginning that the function $f:\R\to\R$ is [*globally*]{} bounded and is [*globally*]{} Lipschitz continuous with the Lipschitz constant $L$. It worth mentioning that this equation possesses a mass conservation law $$\label{0.int}
\frac d{dt}\<u(t)\>=0,\ \ \<u\>=\frac1{|\Omega|}\int_\Omega u(x)\,dx,$$ so we assume from now on that $\<u(t)\>=\<u(0)\>=0$. Thus, the natural phase space of the problem is $$\label{0.phase}
H^{-1}:=H^{-1}(\Omega)\cap\{\<u_0\>=0\}.$$ Note that the spectral gap condition for the IM existence for equation reads $$\label{0.CHgap}
\frac{\lambda_{N+1}^2-\lambda_N^2}{(\lambda_{N}^2)^{1/2}+(\lambda_{N+1}^2)^{1/2}}=\lambda_{N+1}-\lambda_N>L.$$ This condition is clearly satisfied for 1D domains only, in the 2D case it is still an open problem whether or not the spectral gaps of arbitrary width exist for any/generic domains $\Omega$ although it will be so for some special domains like 2D sphere or 2D torus. In these cases the construction of the IM is straightforward, see [@BM; @tem] for more details. However, it is extremely unlikely that the spectral gap condition is satisfied for more or less general 3D domains (in a fact, we know the only example of a 3D sphere where it is true). In particular, it obviously fails for the case of a 3D torus $\Omega=\mathbb T^3=[-\pi,\pi]^3$ (endowed by periodic boundary conditions), therefore, the problem of finding the IM for the 3D Cahn-Hilliard equation with periodic boundary conditions becomes non-trivial and to the best of our knowledge, has been not considered before.
The next theorem gives the main result of the paper.
\[Th0.main\] For infinitely many values of $N\in\mathbb N$ there exists an $N$-dimensional IM $\mathcal M_N$ for the Cahn-Hilliard problem with periodic boundary conditions which is a graph of a Lipschitz continuous function over the $N$-dimensional space spanned by the first $N$ eigenvectors of the Laplacian. Moreover, this function is $C^{1+\eb}$-smooth for some small $\eb=\eb(N)>0$ and the manifold possesses the so-called exponential tracking property, see Section \[s2\] for the details.
The paper is organized as follows.
In Section \[s1\], we consider the functional model related with the problem considered and prepare some technical tools which will be used later.
In Section \[s2\], for the reader convenience, we remind the invariant cone and squeezing property as well as give the proof of the IM existence theorem for our class of equations under the assumption that the cone and squeezing property are satisfied (following mainly [@Zel]).
In Section \[s3\], we reformulate the cone and squeezing property in a more convenient form of a single differential inequality and derive some kind of normal hyperbolicity (dominated splitting) estimates which are necessary to verify the smoothness of an IM.
In Section \[s3.5\], we verify that the constructed manifold is $C^{1+\eb}$-smooth if the nonlinearity is smooth enough. This improves the result of [@mal-par] even on the level of reaction-diffusion equations where only $C^1$-smoothness has been verified.
The abstract form of spatial averaging principle has been stated in Section \[s4\] and the existence of the IM is verified under the assumption that this principle holds.
Finally, in Section \[s5\], we verify this principle for the case of the Cahn-Hilliard equation on a 3D torus and finish the proof of the main Theorem \[Th0.main\].
Preliminaries {#s1}
=============
We consider the following equation: $$\label{main_eq}
\pt u + A^2 u + A F(u) = 0,\ \ u|_{t=0}=u_0,$$ where $A: D(A) \to H$ is a linear self-adjoint positive operator with compact inverse, $D(A)$ is the domain of the operator $A$, and non-linearity $F: H \to H$ is a globally Lipschitz with Lipschitz constant $L$ and globally bounded, i.e., $$\label{1.lip}
1.\ \ \|F(u)\|_H \le C, \ \ u\in H, \quad \quad 2.\ \ \|F(u_1) - F(u_2)\| \le L \|u_1 - u_2\|, \ \ u_1, u_2 \in H.$$ It is well known that problem is globally well-posed and generates a non-linear semigroup $S(t)$ in $H$.
From Hilbert-Schmidt theorem we conclude that the operator $A$ possesses the complete orthonormal system of eigenvectors $\{e_n\}_{n=1}^{\infty}$ in $H$ which corresponds to eigenvalues $\lambda_n $ numerated in the non-decreasing way: $$A e_n = \lambda_n e_n, \ \ \ 0<\lambda_1 \le \lambda_2 \le \lambda_3\le ...$$ and due to the compactness of $A^{-1}$, we have $\lambda_n \to \infty$ as $n \to \infty$.
Thus, we may represent $u$ in the form: $$u = \sum_{n=1}^{\infty}u_n e_n, \ \ u_n = (u,e_n).$$ Then, as usual, the normed spaces $H^s:= D(A^{s/2})$, $s\in \R_+$, is defined as follows $$H^s=\bigg\{u\in H:\ \|u \|_{H^s}^2 = \sum_{n=1}^{\infty}\lambda_n^{s}u_n^2<\infty\bigg\}.$$ For $s<0$ such defined space $H^s$ is not complete. Thus for negative $s$ we define $H^s$ as completion of $H$ with respect to corresponding norm $\|\cdot\|_{H^s}$. Let us introduce the orthoprojector to the first $N$ Fourier modes: $$P_N u:= \sum_{n=1}^N u_n e_n$$ and denote by $Q_N = Id - P_N$, $H_+:= P_N H$ and $H_-:= Q_N H$. Obviously, the following estimates are valid: $$\label{est_for_H_+/-}
\begin{cases}
(Au,u) \le \lambda_N \|u\|^2_H,\ u \in H_+; \\
(Au,u) \ge \lambda_{N+1} \| u\|^2_H,\ u \in D(A^{1/2})\cap H_-.
\end{cases}$$ Throughout the work we will use notations $u_+ := P_N u$ and $u_- := Q_N u$ for given element $u \in H$.
The next proposition collects the standard dissipativity and smoothing properties of the solution semigroup associated with equation , see [@hen; @Zel; @tem] for more details.
\[Prop1.trivial\] Let the non-linearity $F$ and operator $A$ satisfy the above assumptions. Then, problem is uniquelly solvable for any $u_0\in H^{-1}$ and, therefore, the solution semigroup $S(t):H^{-1}\to H^{-1}$ is well-defined. Moreover, the following properties hold for any solution $u(t)$ of problem :
1\. Dissipativity in $H^s$ for $s \in [-1,2]$: $$\label{diss}
\|u(t)\|_{H^s} \le C e^{-\gamma t}\|u(0)\|_{H^s} + R_*,$$ where $C$, $\gamma$ and $R_*$ are some positive constants which are independent of the solution $u$ and $t$;
2\. Smoothing property: $$\label{smooth}
\|u(t)\|_{H^2} \le C t^{-1} \|u(0)\|_H + R_0, \ \ t>0,$$ where $C$ and $R_0$ are independent of $u$ and $t$
3\. Dissipativity of the $Q_N$ component: $$\label{diss_Q_N}
\|Q_N u(t)\|_{H^{2-\kappa}} \le C e^{-\gamma t}\|Q_N u(0)\|_{H^{2-\kappa}} + R_\kappa,$$ for all $N \in \N$ and $\kappa \in (0, 3]$. Here $C$, $\gamma$ and $C_\kappa$ are independent of $N$, $u$ and $t$.
We are now ready to give the key definition of the paper, namely, to define the inertial manifold (IM) associated with the Cahn-Hilliard equation.
\[Def1.IM\] The set $\M\subset H$ to be called an inertial manifold for problem if the following conditions are satisfied:
1\. The set $\M$ is invariant with respect to the solution semigroup $S(t)$, i.e. $S(t) \M = \M$;
2\. It can be presented as a graph of a Lipschitz continuous function $\Phi: H_{+} \to H_{-} $: $$\M:= \{u_+ + \Phi(u_+), u_+ \in H_+\};$$
3\. The exponential tracking property holds, i.e., there exist positive constants $C$ and $\alpha$ such that for every $u_0 \in H^{-1}$ there is $v_0 \in \mathcal M$ such that $$\label{1.llip}
\| S(t)u_0 - S(t)v_0\|_{H^{-1}} \le C e^{-\alpha t}\| u_0 - v_0\|_{H^{-1}},\quad t\geq0.$$
As usual, to verify the existence of the IM, we will use invariant cones method. Namely, introduce the following quadratic form in $H^{-1}$: $$\label{quadratic}
V( \xi)=\|Q_N \xi \|^2_{H^{-1}}- \|P_N \xi \|^2_{H^{-1}},\ \xi \in H^{-1}$$ and set $K^+=K^+(N):=\{\xi\in H^{-1},\ \ V(\xi)\le0\}$ to be the associated cone.
\[Def1.con-squeez\] Let the above assumptions hold. We say that equation possesses the cone property (invariance of the cone $K^+$) if $$\label{cone_con}
\xi_1 - \xi_2 \in K^+ \Rightarrow S(t)\xi_1 - S(t)\xi_2 \in K^+, \text{ for all } t\ge 0,$$ where $\xi_1, \xi_2 \in H^{-1}$ and $S(t)$ is a solution semigroup associated with .
Analogously, we say that possesses the squeezing property if there exists positive $\gamma$ and $C$ such that $$\label{squeez_pr}
S(T)\xi_1- S(T)\xi_2 \not \in K^+ \Rightarrow \|S(t)\xi_1 - S(t)\xi_2\|_{H^{-1}} \le C e^{-\gamma t} \|\xi_1 - \xi_2\|_{H^{-1}},\ t\in [0, T].$$
Invariant cones, squeezing property and inertial manifolds {#s2}
==========================================================
The aim of this section is to remind the reader how to construct an IM based on the cone and squeezing property, see [@fen; @FST; @mal-par; @rom-man] for more details. So, the main result of the section is the following theorem.
\[th\_in\_man\] Let the non-linearity $F$ be globally Lipschitz and globally bounded, see and let, in addition, the solution semigroup $S(t)$ associated with equation satisfies the cone and squeezing properties and for some $N\in\mathbb N$, see Definition \[Def1.con-squeez\]. Then equation possesses an N-dimensional inertial manifold in the sense of Definition \[Def1.IM\].
[*Step 1.*]{} Let us consider the following boundary value problem: $$\label{aux_eq}
\pt u + A^2 u + A F(u) = 0, \ \ P_N u|_{t=0}=u_0^+, \ \ Q_N u |_{t= - T} = 0.$$ We claim that it has a unique solution for any $T>0$ and any $u_0^+\in H^+$. Indeed, introduce the map $G_T: H^+ \to H^+$ by the following rule: $$G_T(w) = P_N S(T)w, \ \ w\in H^+,$$ where $S(t)$ is a solution operator of problem . Obviously this map is continuous. We want to prove that this map is invertible. Indeed, let $u_1(t)$, $u_2(t)$ be two solutions of the problem (with different initial data $u_1(-T)$ and $u_2(-T)$ belonging to $H_+$). Then, their difference $v(t)=u_1(t)-u_2(t)$ lies at the cone $K^+$ at the moment $t = -T$. Thus, from the cone property we conclude that $$u_1(t) - u_2(t) \in K^+, \ \ t\in [-T, 0].$$ The next lemma is the main technical tool for verifying the one-to-one property.
\[Lem2.new\] Let the above assumptions hold. Then, the following estimate hold for the solutions $u_1(t)$ and $u_2(t)$: $$\label{2.con-lip}
\|u_1(-T)- u_2(-T)\|^2_{H^{-1}} \le C e^{\alpha T}\|P_N u_1(0)- P_N u_2(0)\|^2_{H^{-1}},$$ for some constants $C$ and $\alpha$ which are independent of $u_i$.
Since $v(t)\in K^+$, we have the estimate $$\label{2.con-est}
\|v_-(t)\|^2_{H^{-1}}\le\|v_+(t)\|^2_{H^{-1}}.$$ Multiplying now the equation for the difference $v$ by $A^{-1}v_+$ and using that the nonlinearity is globally Lipschitz, we have $$\frac d{dt}\|v_+(t)\|^2_{H^{-1}}+\|v_+(t)\|^2_{H^1}=(F(u_1)-F(u_2),v_+)\ge-L\|v_+\|^2_H-L\|v_+\|_H\|v_-\|_H.$$ Integrating this inequality over $s\in(t,0)$ and using the interpolation between $H^1$ and $H^{-1}$ together with estimate , we end up with $$\label{2.almost}
\|v_+(t)\|^2_{H^{-1}}\le\|v_+(0)\|^2_{H^{-1}}+ C\int_t^0\|v_+(s)\|^2_{H^{-1}}\,ds+\frac12\int_t^0\|v_-(s)\|^2_{H^1}\,ds.$$ To estimate the last term in the right-hand side, we multiply the equation for $v$ by $A^{-1}v_-$ and integrate over $s\in(t,0)$. With the help of again, this gives $$\begin{gathered}
\int_t^0\|v_-(s)\|^2_{H^1}\,ds=\frac12\(\|v_-(0)\|^2_{H^{-1}}-\|v_-(t)\|^2_{H^{-1}}\)+\int_t^0(F(u_1)-F(u_2),v_-)\,ds
\le\\\le\frac12\|v_+(0)\|^2_{H^{-1}}
+L\int_t^0\|v\|^2_H\le\frac12\|v_+(0)\|^2_{H^{-1}}+\frac12\int_t^0\|v_-(s)\|^2_{H^1}\,ds+C\int_t^0\|v_+(s)\|^2_{H^{-1}}\,ds.\end{gathered}$$ Inserting the last estimate into the RHS of , we finally arrive at $$\label{2.almost1}
\|v_+(t)\|^2_{H^{-1}}\le 2\|v_+(0)\|^2_{H^{-1}}+ C\int_t^0\|v_+(s)\|^2_{H^{-1}}\,ds$$ and the Gronwall inequality finishes the proof of the lemma.
We are now ready to finish the first step of the proof of the theorem. Indeed, since $u_1(t)$ and $u_2(t)$ were chosen arbitrary, then we conclude that the map $G_T: H_+ \to H_+$ is injective. Consequently, by the Brouwer invariance of domain theorem, $G_T(H_+)$ is open. Moreover, estimate guarantees that the sequence $w_n\in H_+$ is bounded if $G_T(w_n)$ is bounded. Then, since $H_+$ is finite-dimensional, $G_T(H_+)$ is also closed by compactness arguments. Thus, $G_T(H_+)=H_+$ and $G_T$ is a (bi-Lipschitz) homeomorphism on $H_+$. Therefore, $G_T(v) = u_0^+$ has a unique solution for all $u_0^+ \in H_+$. Then, obviously $u(t) = S(t+T)G_T^{-1}(u_0^+)$ solves and the first step is completed.
[*Step 2.*]{} Let $u_{T, u_0^+}$ be the solution of the boundary value problem . We claim that for all $t\le0$, there exists a limit $$\label{lim}
u_{u_0^+}(t) = \lim_{T \to \infty} u_{T,u_0^+}(t)$$ which solves problem with $T =\infty$.
Indeed, since solution of the problem starts from $u_-(-T) = 0$, according to Proposition \[Prop1.trivial\], we have: $$\label{Q_est}
\|Q_N u_{T, u_0^+}(t)\|^2_{H^{2-\kappa}} \le \tilde{C_\kappa}$$ for all $T \ge 0$ and $u_0^+ \in H$ and $\kappa\in(0,3]$. In particular, the choice $\kappa=3$ gives the control of the $H^{-1}$-norm.
Let us introduce the following notations $u_i(t) := u_{T_i, u_0^+}(t)$ and $v(t) = u_1(t) - u_2(t)$. Then we know that at the moment $t=0$ we have $v_+(0) = 0$ and consequently $v(0) \not \in K^+$. By the cone property $$\label{not_in_K}
v(t) \not \in K^+ \text{ for all } t \in [-T, 0] \text{, where }T = \min \{T_1, T_2\}.$$ Thus, using squeezing property we get: $$\label{sq_pr_act}
\|v(t)\|_{H^{-1}} \le C e^{- \gamma(t+T)}\|v(-T)\|_{H^{-1}}.$$ Due to , we have $\|v_+(-T)\|_{H^{-1}}\le \|v_-(-T)\|_{H^{-1}}$ and consequently thanks to , $$\label{2.estt}
\|v(t)\|_{H^{-1}} \le C_1 e^{- \gamma(t+T)}.$$ Thus, $u_{T_i, u_0^+}$ is a Cauchy sequence in $C_{loc}((-\infty,0), H^{-1})$. Consequently, there exists limit and $u_{u_0^+}$ is a backward solution of the problem .
[*Step 3.*]{} Let us define a set $ \mathcal{N} \subset C _{loc}(\R, H)$ as the set of all solutions of the problem obtained as a limit . Then, by the construction, $\mathcal{N}$ is invariant with respect to the solution semigroup $S(t)$, i.e. $$S(h)\mathcal{N} = \mathcal{N}, \ \ (S(h)u)(t) = u(t+h), \ \ h \in \R.$$ Consider function $\Phi: H_+ \to H_-$ acting by the rule: $$\Phi(u_0)= Q_N u(0), \ \ u(t) \in \mathcal{N}, \ \ P_N u(0) = u_0.$$ Indeed, according to Steps 1 and 2, the trajectory $u\in\mathcal N$ exists for any $u_0\in H_+$. Moreover, as not difficult to see by approximating the solutions $u_1,u_2\in\mathcal N$ by the solutions of the boundary value problem , that $$\label{in_K}
u_1(t) - u_2(t) \in K^+, \ \ t\in \R$$ for any $u_1,u_2\in\mathcal N$. Therefore, $\Phi(u_0)$ is well defined and Lipschitz continuous. Thus, it remains to note that manifold $$\mathcal{M} := \big\{u_0 + \Phi(u_0): u_0 \in H^+\big\}$$ is invariant with respect to $S(t)$ which is a direct consequence of the invariance of $\mathcal{N}$.
[*Step 4.*]{} In order to prove that $\mathcal{M}$ is a desired inertial manifold, it remains to show that exponential tracking property holds. Let $u(t)$ , $t \ge 0$, be a forward solution of the problem and $u_T(t) \in \mathcal{N}$, $T >0$, be the solution of which belongs to the manifold $\mathcal{M}$ such that $$P_N u (T) = P_N u_T (T).$$ Then, due to the cone property $u(t) - u_T(t) \not \in K^+$ as $t \in [0,T]$ and consequently $$\|P_N(u(t) - u_T(t))\|^2_{H^{-1}} \le \|Q_N (u(t) - u_T(t))\|^2_{H^{-1}}, \ \ t \in [0,T].$$ From we know that $\|Q_N u_T(t)\|_{H^{-1}}$ is uniformly bounded with respect to $T$. Since $u(t)$ is also bounded in $H^{-1}$, see Proposition \[Prop1.trivial\], we conclude that the functions $u_T(t)\in\mathcal M$ are uniformly bounded as $T \to \infty$ in the norm $H^{-1}$ for all fixed $t \in [0,T]$. On the other hand, using squeezing property we obtain: $$\label{exp_tr}
\|u(t) - u_T(t)\|_{H^{-1}} \le C e^{-\gamma t} \|u(0) - u_T(0)\|_{H^{-1}}, \ \ t\in [0,T].$$ Since the manifold $\mathcal M$ is finite-dimensional and $u_{T}(0)$ is uniformly bounded, we may assume without loss of generality that $u_{T_n}(0)\to\tilde u(0)$ for some $T_n\to\infty$ and $\tilde u(0)\in\mathcal M$. Then, according to and Lemma \[Lem2.new\], $u_{T_n}\to\tilde u\in\mathcal N$ in $C_{loc}(\R,H^{-1})$. Passing to the limit $n\to\infty$ at we see that $$\label{2.exptr}
\|u(t) - \tilde u(t)\|_{H^{-1}} \le C e^{-\gamma t} \|u(0) - \tilde u(0)\|_{H^{-1}}, \ \ t\ge0.$$ Thus, the exponential tracking is verified and the theorem is proved.
\[Rem2.smooth\] Note that, according to Lemma \[Lem2.new\] and the property , any two solutions $u_1,u_2\in\mathcal N$ satisfy the backward Lipschitz continuity $$\label{est_tech}
\|u_1(-T)-u_2(-T)\|_{H^{-1}}\le Ce^{\alpha T}\|u_1(0)-u_2(0)\|_{H^{-1}},\ \ T\ge0,$$ where the positive constants $C$ and $\alpha$ are independent of $u_1,u_2\in\mathcal N$ and $T\ge0$.
One more simple but important observation is that the above given proof uses the cone and squeezing property not for all $u_1,u_2\in H^{-1}$ but only for those which satisfy estimate . By this reason, we actually need to verify the cone and squeezing property only for $u_1,u_2\in H^{2-\kappa}$ for some $\kappa\in(0,3]$ such that $$\label{2.important}
\|Q_N u_i\|_{H^{2-\kappa}}\le C_\kappa,\ \ i=1,2,$$ where $C_\kappa$ is some constant depending only on $\kappa$. Indeed, this inequality is automatically satisfied for all trajectories involving into the construction of the set $\mathcal N$ and the associated inertial manifold $\mathcal M$ (due to estimate and the fact that $Q_Nu(-T)=0$ in the boundary value problem ). The exponential tracking a priori holds for all trajectories starting from $u(0)\in H^{-1}$, however, due to the smoothing property (see Proposition \[Prop1.trivial\]), it is sufficient to verify it only for $u(0)$ satisfying . This observation plays a crucial role in the construction of a special cut-off for the spatial averaging method, see below.
Invariant cones and normal hyperbolicity {#s3}
========================================
In this section, we reformulate the cone and squeezing property in a more convenient (at least for our purposes) form of a single differential inequality on the trajectories of and its equation of variations and state a number of technical results related with the invariant cones and exponential dichotomy/normal hyperbolicity for the trajectories of the corresponding equation of variations. These results will be used in the next section for establishing the smoothness of the IM. We further assume that the nonlinearity $F(u)$ is at least Gateaux differentiable and there exists a derivative $F'(u)\in\mathcal L(H,H)$ for any $u\in H$. Then, obviously $$\label{3.1}
\|F'(u)\|_{\mathcal L(H,H)}\le L$$ and we also assume that the following integral version of the mean value theorem holds: $$\label{3.2}
F(u_1)-F(u_2)=l_{u_1,u_2}(u_1-u_2),\ \ l_{u_1,u_2}:=\int_0^1F'(su_1+(1-s)u_2)\,ds.$$ Then, the difference $v(t)=u_1(t)-u_2(t)$ of any two solutions $u_1(t)$ and $u_2(t)$ of the Cahn-Hilliard problem solves the following linear equation: $$\label{3.dif}
\Dt v+A(Av+l_{u_1(t),u_2(t)}v)=0,$$ however, it will be more convenient for us to study more general linear equations $$\label{3.gen}
\Dt v+A(Av+l(t)v)=0,$$ where $l\in L^\infty(\R,\mathcal L(H,H))$ satisfies $\|l(t)\|_{\mathcal L(H,H)}\le L$ for all $t\in\R$.
\[Def3.diffcon\] We say that equation satisfies the strong cone condition (in a differential form) if there exist a positive number $\mu$ and a function $\alpha:\R\to\R$ such that $$\label{3.squeez}
0<\alpha_-\le \alpha(t)\le \alpha_+$$ and for any solution $v(t)$, $t\in[S,T]$, $S<T$, of problem , the following inequality holds: $$\label{3.dcone}
\frac d{dt}V(v(t))+\alpha(t)V(v(t))\le-\mu\|v(t)\|^2_H$$ for all $t\in[S,T]$. Here and below $V$ is the quadratic form defined by .
As easy to see from inequality , the above assumptions guarantee that the cone $K^+$ is invariant with respect to the evolution generated by equation . Moreover, as will be shown below, the squeezing property is also incorporated in our version of the strong cone condition (due to the strict positivity of the exponent $\alpha(t)$), but we first need to remind some elementary properties of the introduced strong cone condition. We start with reformulating it in a pointwise form applicable to the non-homogeneous form of equation .
\[Lem3.point\] The strong cone condition in the differential form is equivalent to the following condition: $$\label{8}
-2 (l(t)w,w_--w_+)-2(A w,w_--w_+)+\alpha(t)(\|w_-\|^2_{H^{-1}}-\|w_+\|^2_{H^{-1}})\le-\mu\|w\|^2_H$$ for all $t\in\R$ and $w\in H^1$.
Indeed, let condition be satisfied. Then, multiplying equation by the expression $A^{-1}(v_-(t)-v_+(t))$ and using equation , we have $$\label{3.pro}
\frac d{dt}V(v(t))=2(\Dt v(t),v_-(t)-v_+(t))=-2(Av(t),v_-(t)-v_+(t))-2(l(t)v(t),v_-(t)-v_+(t)).$$ Estimating the right-hand side of this inequality by with $w=v(t)$, we end up with the desired cone inequality . Vise versa, let the cone condition hold and let $w\in H^1$ and $t_0\in\R$ be arbitrary. Let us consider the solution $v(t)$ of equation satisfying $v(t_0)=w$. Using then the cone condition with $t=t_0$ and formula for the derivative of the quadratic form $V$, we end up with the desired inequality with $t=t_0$. Thus, the lemma is proved.
\[Cor3.non\] Assume that equation possesses the strong cone property in the sense of Definition \[Def3.diffcon\]. Then, for any solution $v(t)$ of the non-homogeneous equation $$\label{9}
\Dt v+A^2 v + A\big( l(t)v\big)+ A h(t) = 0$$ with $h\in L^\infty(\R,H^{-1})$ the following analogue of holds: $$\label{10}
\frac d{dt} V(v(t))+\alpha(t) V(v(t))\le -\mu\|v(t)\|^2_H - 2(h(t),v_-(t)-v_+(t)).$$
Indeed, analogously to , but using the non-homogeneous equation , we have $$\label{3.pro1}
\frac d{dt}V(v(t))=-2(Av(t),v_-(t)-v_+(t))-2(l(t)v(t),v_-(t)-v_+(t))-2(h(t),v_-(t)-v_+(t))$$ and estimating the terms in the right-hand side of this inequality with the help of with $w=v(t)$, we end up with the desired inequality .
At the next step, we show that the strong cone condition is robust with respect to perturbations of the cone and this will give us the main technical tool for proving the smoothness of the IM. Namely, for any $\eb\in\R$, we define $$\label{veb}
V_\eb(\xi):=\eb\|\xi\|^2_{H^{-1}}+V(\xi)=(1+\eb)\|\xi_-\|_{H^{-1}}^2-(1-\eb)\|\xi_+\|^2_{H^{-1}}=(1+\eb)V(\xi)+2\eb\|\xi_+\|^2_{H^{-1}}.$$
\[Lem3.robust\] Let equation satisfy the strong cone condition in the differential form. Then, there exists $\eb_0>0$ such that, for any $0<\eb<\eb_0$, the following inequalities $$\label{3.difeq}
\frac d{dt} V_\eb(v(t))+(\alpha(t)+\frac12\lambda_1\mu)V_\eb(v(t))\le 0,\ \ \frac d{dt} V_{-\eb}(v(t))+(\alpha(t)-\frac12\lambda_1\mu)V_{-\eb}(v(t))\le 0$$ hold for any solution $v(t)$ of equation .
Let us check the first inequality of . Multiplying equation by $2\eb A^{-1}v$ and using the Lipschitz continuity, we have $$\eb\frac d{dt}\|v(t)\|^2_{H^{-1}}\le 2\eb L\|v(t)\|^2_{H}.$$ Taking a sum of this inequality with , using and fixing $\eb>0$ to be so small that $$(2 L+\alpha_+\lambda_1^{-1})\eb\le\frac14\mu,$$ we end up with $$\label{3.triv}
\frac d{dt} V_\eb(v(t))+\alpha(t)V_\eb(v(t))\le -\frac34\mu\|v(t)\|^2_H.$$ Combining this inequality with the obvious estimate $$\label{3.vhm}
-\lambda_1^{-1}\|\xi\|^2_H\le V(\xi)\le \lambda_1^{-1}\|\xi\|^2_H$$ and assuming that $\eb\le\frac12$, we prove the first formula of .
Let us prove the second inequality of . Multiplying by $-4\eb A^{-1}v_+(t)$ and using that $(Av,v)\ge\lambda_N\|v\|^2$ if $v\in H_+$, we get $$-2\eb\frac d{dt}\|v_+(t)\|^2\le 4\eb(\lambda_N+L)\|v\|^2_H.$$ Multiplying now inequality by $(1-\eb)>0$, taking a sum with the last inequality and fixing $\eb>0$ in such way that $4\eb(\lambda_N+L)\le \frac14(1-\eb)\mu$, we end up with $$\label{3.triv1}
\frac d{dt} V_{-\eb}(v(t))+\alpha(t)V_{-\eb}(v(t))\le -\frac34(1-\eb)\mu\|v(t)\|^2_H.$$ Using the inequality again and assuming that $\eb<\frac14$, we end up with the desired second estimate of and finish the proof of the lemma.
The next corollary shows that the strong cone condition implies some kind of normal hyperbolicity in the sense that the trajectories outside of the cone squeeze stronger than the trajectories inside of the cone may expand.
\[Cor3.hyp\] Let the equation possesses the strong cone property, $T\in\R_+$ and $v(t)$, $t\in [-T,0]$ be a solution of problem . Then
1\) If $v(-T)\in K^+$, the whole trajectory $v(t)\in K^+$, $t\in[-T,0]$ and the following estimate holds: $$\label{11}
\|v(t)\|_{H^{-1}}^2\le Ce^{\bar\alpha(t)+\frac12\lambda_1 \mu t}\|v(0)\|^2_{H^{-1}},\ \ t\in[-T,0],$$ where $\bar \alpha(t):=\int_{t}^0\alpha(s)\,ds$ and the constant $C$ is independent of $T$, $t$ and $v$.
2\) If $v(0)\notin K^+$, the whole trajectory $v(t)\notin K^+$, $t\in[-T,0]$ and the following estimate holds: $$\label{12}
\|v(0)\|_{H^{-1}}^2\le Ce^{-\bar\alpha(t)+\frac12\lambda_1 \mu t}\|v(t)\|^2_{H^{-1}},\ \ t\in[-T,0],$$ where the constant $C$ is independent of $T$, $t$ and $v$.
Let $v(-T)\in K^+$. Then $V(v(-T))\le 0$ and from the cone condition we conclude that $V(v(t))\le0$ for all $t\ge-T$ and $v(t)\in K^+$. Integrating the second inequality of , we have $$V_{-\eb}(0)\le e^{-\bar \alpha (t)-\frac12\lambda_1\mu t}V_{-\eb}(t)$$ and, therefore, since $V(v(t))\le0$, we obtain $$\begin{gathered}
\eb\|v(t)\|^2_{H^-1}\le \eb\|v(t)\|^2_{H^{-1}}-V(v(t))=-V_{-\eb}(v(t))\le\\\le e^{\bar\alpha(t)+\frac12\lambda_1 \mu t}(-V_{-\eb}(v(0))\le (1+\eb)e^{\bar\alpha(t)+\frac12\lambda_1 \mu t}\|v(0)\|^2_H\end{gathered}$$ and estimate is proved.
Let now $v(0)\notin K^+$. Then, from the cone property we conclude that $V(v(t))\ge0$ for all $t\in[-T,0]$. Integrating the first inequality of , we get $$V_{\eb}(v(0))\le e^{-\bar\alpha(t)+\frac12\lambda_1\mu t}V_{\eb}(v(t))$$ and, therefore, using that $V(v(t))\ge0$, we deduce $$\begin{gathered}
\eb\|v(0)\|^2_{H^{-1}}\le \eb\|v(0)\|^2_{H^{-1}}+V(v(0))=V_{\eb}(v(0))\le\\\le e^{-\bar\alpha(t)+\frac12\lambda_1 \mu t}V_{\eb}(v(t))\le (1+\eb)e^{-\bar\alpha(t)+\frac12\lambda_1 \mu t}\|v(t)\|_{H^{-1}}^2\end{gathered}$$ and the corollary is proved.
The next corollary shows that the strong cone property in the differential form implies [*both*]{} cone and squeezing properties for the solutions of equation .
\[cor\_con\_sq\] Let the equation possess the strong cone property and let $v(t)$, $t\in[0,T]$ be a solution of . Then the following properties are valid:
1\. Cone property (invariance of the cone $K^+$): $$\label{cone_con1}
v(0)\in K^+ \Rightarrow v(t) \in K^+, \text{ for all } t\ge 0.$$
2\. Squeezing property: there exists positive $\gamma$ and $C$ such that $$\label{3.squeez_pr}
v(T) \not \in K^+ \Rightarrow \|v(t)\|_{H^{-1}} \le C e^{-\gamma t} \|v(0)\|_{H^{-1}}, t\in [0, T],$$ where the constants $\gamma$ and $C$ are independent of $v$ and $T$.
Indeed, the first assertion is an immediate corollary of inequality . To verify the squeezing property, it is sufficient to use estimate on the interval $[0,T]$ instead of $[-T,0]$. This together with inequality give $$\|v(t)\|_{H^{-1}}^2\le Ce^{-\int_0^t\alpha(s)\,ds-\frac12\lambda_1\mu t}\|v(0)\|^2_{H^{-1}}\le Ce^{-(\alpha_-+\frac12\lambda_1\mu)t}\|v(0)\|_{H^{-1}}^2$$ and the squeezing property is verified. Thus, the corollary is also proved.
We are now ready to return to the non-linear equation and state for it the analogous strong cone condition in the differential form.
\[Def3.n-cone\] Let the nonlinearity $F(u)$ satisfy the assumptions stated at the beginning of this section. We say that equation possesses a strong cone property in the differential form if there exist a positive constant $\mu$ and a bounded (Borel measurable) function $\alpha: H\to\R$ such that for every trajectory $u(t)\in H$, $t\in[0,T]$ of equation and every solution $v(t)$ of the equation of variations $$\label{3.eqvar1}
\Dt v+A(Av+F'(u(t))v)=0,$$ the following analogue of holds $$\label{3.dcone1}
\frac d{dt}V(v(t))+\alpha(u(t))V(v(t))\le-\mu\|v(t)\|^2_H$$ and the function $\alpha$ satisfies $$0<\alpha_-\le\alpha(u)\le\alpha_+<\infty.$$
The relation between the strong cone conditions for the non-linear equation and for the equation for the differences of its solutions is clarified in the following lemma.
\[Lem3.con-con\] Let equation possess the strong cone property in the sense of Definition \[Def3.n-cone\]. Then, for every two solutions $u_1(t)$ and $u_2(t)$, $t\in[T,S]$ of equation , the associated equation for the differences of solutions also possess the strong cone condition (in the sense of Definition \[Def3.diffcon\]) with the same constant $\mu$ and with the constant $\alpha$ satisfying $$\label{20}
\alpha(t)=\alpha_{u_1,u_2}(t):=\int_0^1\alpha(s u_1(t)+(1-s)u_2(t))\,ds.$$
Indeed, according to Lemma \[Lem3.point\], the strong cone condition for equation is equivalent to $$\begin{gathered}
-2(F'(u)w,w_--w_+)-2(Aw,w_--w_+)+\\+\alpha(u)(\|w_-\|^2_{H^{-1}}-\|w_+\|^2_{H^{-1}})\le-\mu\|w\|^2_H,\ \ \forall w\in H^1\end{gathered}$$ and every $u\in H$. Replacing $u=s u_1(t)+(1-s)u_2(t)$ in this inequality and integrating over $s\in[0,1]$, we end up with $$\begin{gathered}
\label{23}
-2(l_{u_1,u_2}(t)w,w_--w_+)-2(Aw,w_--w_+)+\\+\alpha_{u_1,u_2}(t)(\|w_-\|^2_{H^{-1}}-\|w_+\|^2_{H^{-1}})\le-\mu\|w\|^2_H,\ \ \forall w\in H^1\end{gathered}$$ which according to Lemma \[Lem3.point\] again is equivalent to the strong cone condition for equation and the lemma is proved.
We summarize the obtained results in the following theorem which can be considered as the main result of the section.
\[Th3.main\] Let the nonlinearity $F(u)$ be globally bounded and satisfy and and let also the associated equation possess the strong cone condition in the sense of Definition \[Def3.n-cone\]. Then, equation possess the $N$-dimensional Lipschitz continuous inertial manifold (see Definition \[Def1.IM\]).
Indeed, according to Lemma \[Lem3.con-con\], for any two solutions $u_1(t)$ and $u_2(t)$ of equation , the associated equation for differences of solutions possesses the strong cone condition. Then, due to Corollary \[cor\_con\_sq\], equation satisfies the cone and squeezing property. In particular, looking at the solution $v(t)=u_1(t)-u_2(t)$ of this equation, we see that equation possesses the cone and squeezing property in the sense of Definition \[Def1.con-squeez\]. Thus, Theorem \[th\_in\_man\] is applicable and gives the existence of the desired inertial manifold. This finishes the proof of the theorem.
\[Rem3.smooth\] Theorem \[Th3.main\] simplifies the verification of the conditions for the IM existence. Indeed, according to this theorem, we only need to check [*one*]{} differential inequality for the equation of variations . Note also that, analogously to Remark \[Rem2.smooth\], it is sufficient to verify all of the conditions for the trajectories $u$ satisfying only.
To conclude this section, we show that the classical spectral gap condition implies the strong cone condition and, therefore, guarantees the existence of the IM.
\[Prop3.spectral\] Let the nonlinearity $F(u)$ be globally bounded and the following spectral gap condition be satisfied for some $N\in\mathbb N$: $$\lambda_{N+1} - \lambda_N > L,$$ where $L$ is a Lipschitz constant of the non-linearity $F$. Then for every two solutions $u_1(t)$ and $u_2(t)$ of the equation , the associated equation for the differences possesses the strong cone condition with $$\label{3.am}
\alpha(t)=2\alpha := 2\lambda_{N+1} \lambda_N\ \ \text{and}\ \ \mu := 2(\lambda_{N+1} - \lambda_N - L) > 0.$$ Thus, equation possesses the Lipschitz IM.
Let $v(t)$ be a solution of . Then, multiplying equation for $v(t)$ first by $A^{-1}v_-(t)$, second by $-A^{-1} v_+(t)$ and taking sum of them we obtain: $$\begin{gathered}
\label{3.cone-est}
\frac{1}{2} \frac{d}{dt}V(v(t)) + \alpha V(v(t)) = ((\alpha A^{-1} -A)v_-,v_-)+ \\ + ((A - \alpha A^{-1})v_+, v_+) - (l_{u_1(t),u_2(t)}v, v_- - v_+).\end{gathered}$$ By the definition of $\alpha$ and $\mu$, we have $$\begin{gathered}
\label{3.one-est}
((A - \alpha A^{-1})v_+, v_+)=\sum_{n=1}^N(\lambda_n-\lambda_N\lambda_{N+1}\lambda_n^{-1})|v_n|^2\le\\\le \sum_{n=1}^N(\lambda_N-\lambda_{N+1})|v_n|^2=-(\lambda_{N+1}-\lambda_N)\|v_+\|^2_H.\end{gathered}$$ Analogously, $$\begin{gathered}
\label{3.dva-est}
((\alpha A^{-1} -A)v_-,v_-)=\sum_{n=N+1}^\infty(\lambda_N\lambda_{N+1}\lambda_n^{-1}-\lambda_n)|v_n|^2\le\\\le \sum_{n=N+1}^\infty(\lambda_N-\lambda_{N+1})|v_n|^2=-(\lambda_{N+1}-\lambda_N)\|v_-\|^2_H.\end{gathered}$$ Inserting these estimates to and using that $\|l_{u_1(t),u_2(t)}\|_{\mathcal L(H,H)}\le L$, we have $$\frac{1}{2} \frac{d}{dt}V(v(t)) + \alpha V(v(t)) \le -(\lambda_{N+1} -\lambda_N)\|v\|^2_H + L\|v\|^2_H = -\mu \|v\|^2_ H.$$ Thus, the strong cone condition for is verified and the proposition is proved.
Smoothness of the inertial manifolds {#s3.5}
====================================
The aim of this section is to obtain the extra smoothness of the function $\Phi: H_+ \to H_-$ which determine the inertial manifold $\mathcal{M}$ under the assumption that non-linearity $F$ is $C^{1+\delta}(H,H)$ for some $\delta \in (0,1)$, i.e., $$\label{3}
\|F(u_1)-F(u_2)-F'(u_1)(u_1-u_2)\|_H\le C\|u_1-u_2\|^{1+\delta}_H, \ \ u_1, u_2 \in H.$$ To be more precise, the main result of this section is the following theorem.
\[Th4.main\] Let the assumptions of Theorem \[Th3.main\] hold and let also the assumption on $F$ be valid for some $\delta>0$. Then the map $\Phi$ is Frechet differentiable and $C^{1+ \varepsilon}$-smooth for some $\varepsilon>0$, i.e, $$\label{28}
\|\Phi(u^1_+)-\Phi(u^2_+)-\Phi'(u^1_+)(u_+^1-u_+^2)\|_{H^{-1}} \le C\|u^1_+-u^2_+\|_H^{1+\eb}.$$
To verify , we first need to study the Frechet derivative $\Phi'(u^1_+)$. Following the definition of $\Phi$, it is natural to expect that this derivative is defined as follows: $$\label{29}
\Phi'(u^1_+)w_+:=\lim_{T\to\infty} Q_N w_T(0),$$ where $u_+^i\in H_+$, $i=1,2$, and $w_T(t)$, $T>0$, solves $$\label{30}
\pt w + A^2w + A F'(u^1(t))w = 0,\ \ w_+\big|_{t=0}=w_+,\ \ w_-\big|_{t=-T}=0.$$ Here and below $w_+ \in H_+$ and $u^i(t)$, $t\le0$, $i=1,2$, are the solutions of belonging to the inertial manifold $\mathcal M$ and satisfying $P_Nu^i(0)=u^i_+$.
[*Step 1.*]{} Well-posedness of $\Phi'(u^1(t))$. The existence of a solution for problem can be verified exactly as in Theorem \[th\_in\_man\] and to define the operator $\Phi'$, we only need to check the existence of the limit . Indeed, since the trajectory $w_T \in K^+$, according to Corollary \[Cor3.hyp\] and estimate , we get $$\label{31}
\|w_T(t)\|_{H^{-1}}^2\le Ce^{\bar\alpha(t)+\frac12\lambda_1 \mu t}\|w_+\|_{H^{-1}}^2.$$ Here $\bar\alpha(t)=\int_t^0\alpha(u^1(s))\,ds$ and $\mu$ is the same as in the strong cone inequality. Consider now another approximation $w_{T_1}(t)$, $T_1\ge T$, and their difference $w_{T,T_1}(t):=w_{T_1}(t)-w_T(t)$. This trajectory does not belong to the cone $K^+$ at $t=0$ and, therefore, it is not in $K^+$ for all $t\in[-T,0]$. Using now together with estimate of Corollary \[Cor3.hyp\], we end up with $$\begin{gathered}
\label{32}
\|w_{T,T_1}(0)\|^2_{H^{-1}}\le Ce^{-\bar\alpha(-T)-\frac12 \lambda_1 \mu T}\|w_{T,T_1}(-T)\|^2_{H^{-1}}\le\\\le C_1e^{-\bar\alpha(-T)-\frac12 \lambda_1 \mu T}(\|w_T(-T)\|^2_{H^{-1}}+\|w_{T_1}(T)\|^2_{H^{-1}})\le\\\le
C_2e^{- \lambda_1 \mu T}(\|w_T(0)\|^2_{H^{-1}}+\|w_{T_1}(0)\|^2_{H^{-1}})\le C_3e^{-\lambda_1\mu T}\|w_+\|^2_{H^{-1}}.\end{gathered}$$ Thus, $w_T(0)$ is a Cauchy sequence and the limit exists and, therefore, the operator $\Phi'(u^1(t))$ is well-defined. Moreover, according to , we have the following estimate for the limit function $w(t)$: $$\label{33}
\|w(t)\|_{H^{-1}}^2\le Ce^{\bar\alpha(t)+\frac12\lambda_1 \mu t}\|w_+\|^2_{H^{-1}},\ \ w_+\in H_+,\ \ t\le0.$$ [*Step 2.*]{} Estimate for the difference $v(t):=u^1(t)-u^2(t)$.
Let $u^1(t)$ and $u^2(t)$ be two trajectories on the inertial manifold defined by the limit which correspond to the initial data $u^1_+\in H_+$ and $u_2^+\in H_+$ respectively and $w_+: = u_1^+ - u_2^+$. Since $v(t)\in K^+$ then, due to estimate and the assumption that the exponent $\alpha(t)$ is globally bounded, we have $$\label{34}
\|v(t)\|_{H^{-1}}^2\le Ce^{-Kt}\|w_+\|^2_{H^{-1}},\ \ t\in \R_-.$$ Our aim at this step is to improve and to obtain the estimate which is analogous to . To this end, we note that the function $v$ solves the equation $$\label{35}
\Dt v+A^2v + A F'(u^1(t))v + A [l_{u_1(t),u_2(t)}-F'(u_1(t))]v(t)=0,$$ where $l_{u_1(t),u_2(t)}:=\int_0^1F'(u_1(t)+s v(t))\,ds$. Since $F$ satisfies and $v(t)\in K^+$, we have $$\label{36}
\|l_{u_1(t),u_2(t)}-F'(u_1(t))\|_{\Cal L(H,H)}\le C\|v(t)\|_{H}^{\delta}\le Ce^{-K\delta t/2}\|w_+\|^\delta_H, \ \ t\in(-\infty,0].$$ Thus, treating equation as a non-homogeneous problem in the form of with the right-hand side $h(t):=[l_{u_1(t),u_2(t)}-F'(u_1(t))]v(t)$ and according to and (see also ), we get that, for a sufficiently small $\eb>0$, the following estimate holds: $$\begin{gathered}
\label{37}
\frac d{dt}V_{-\eb}(v(t))+(\alpha(u^1(t))-\frac12\lambda_1\mu)V_{-\eb}(v(t))\le\\\le-\frac\mu4\|v(t)\|^2_H+C\|h(t)\|_H\|v(t)\|_H\le
(-\frac\mu4+Ce^{-K\delta t/2}\|w_+\|_H^\delta)\|v(t)\|_H^2\le 0\end{gathered}$$ if $-t\le\frac2{K\delta}\ln\frac {\mu}{4C \|w_+\|_H^{\delta}}$. Thus, since $v\in K^+$, analogously to , we end up with the following estimate: $$\label{38}
\|v(t)\|_{H^{-1}}^2\le Ce^{\bar\alpha(t)+\frac12\lambda_1 \mu t}\|w_+\|^2_{H^{-1}},\ \ w_+\in H_+,\ \ t\in[-T,0],\ \ T=\frac2{K\delta}\ln\frac {\mu}{4C\|w_+\|_H^{\delta}}$$ which differs from only by the presence of the lower bound for $t$.
[*Step 3.*]{} Applying the parabolic smoothing property. Up to the moment, we have obtained estimates and for the $H^{-1}$ norms of the functions $v(t)$ and $w(t)$ only, but we need to control more regular norms of these functions in the sequel. To get this control, we remind that, analogously to Proposition \[Prop1.trivial\], we have the following parabolic smoothing property for the solutions of problem : $$\|w(t+1)\|_{H^{2-\kappa}}\le C_\kappa \|w(t)\|_{H^{-1}},$$ where $\kappa>0$ is arbitrary and the constant $C_\kappa$ depends only on $\kappa$. Combining this estimate with , we get $$\label{4.33}
\|w(t)\|_{H^{2-\kappa}}^2\le C_\kappa e^{\bar\alpha(t)+\frac12\lambda_1 \mu t}\|w_+\|^2_{H^{-1}},\ \ w_+\in H_+,\ \ t\le0.$$ Analogously, applying the parabolic smoothing property to equation and using , we get $$\label{4.38}
\|v(t)\|_{H^{2-\kappa}}^2\le C_\kappa e^{\bar\alpha(t)+\frac12\lambda_1 \mu t}\|w_+\|^2_{H^{-1}},\ \ w_+\in H_+,\ \ t\in[-T,0]$$ with $T=\max\{0,\frac2{K\delta}\ln\frac {\mu}{4C\|w_+\|_H^{\delta}}-1\}$.
[*Step 4.*]{} Estimate for $\theta(t):= w(t)-v(t)$. This function solves the following equation: $$\label{39}
\pt\theta+A^2 \theta + A (F'(u^1(t))\theta)+ A h(t)=0.$$ Then, combining and , for sufficiently small $\eb>0$, we have $$\label{40}
\frac d{dt}V_\eb(\theta(t))+(\alpha(u^1(t))+\frac12\lambda_1\mu)V_\eb(\theta(t))\le C\|h(t)\|_H\|\theta(t)\|_H$$ and using that $\theta(0)\not\in K^+$, analogously to , we have $$\label{41}
\|\theta(0)\|^2_{H^{-1}}\le Ce^{-\bar\alpha(-T)+\frac12\lambda_1\mu T}\|\theta(-T)\|^2_{H^{-1}}+
C\int_{-T}^0e^{-\bar\alpha(t)+\frac12\lambda_1 \mu t}\|h(t)\|_H\|\theta(t)\|_H\,dt.$$ Assume now that $T=\max\{0,\frac2{K\delta}\ln\frac {\mu}{4C\|w_+\|_H^{\delta}}-1\}$ and using estimates , and (with $\kappa=2$) as well as , we finally arrive at $$\begin{gathered}
\label{42}
\|\theta(0)\|^2_{H^{-1}}\le Ce^{-\bar\alpha(-T)+\frac12 \lambda_1 \mu T}(\|v(-T)\|^2_{H^{-1}}+\|w(-T)\|^2_{H^{-1}})+\\+
C\int_{-T}^0e^{-\bar\alpha(t)+\frac12\lambda_1 \mu t}\|v(t)\|_H^\delta\|v(t)\|_H(\|w(t)\|_H+\|v(t)\|_H)\,dt\le
\\\le
Ce^{-\lambda_1\mu T}\|w_+\|^2_{H^{-1}}+C\|w_+\|_{H^{-1}}^{2+\delta/2}\int_{-T}^0e^{-(-\lambda_1\mu+K\delta/2) t}\,dt.\end{gathered}$$ Decreasing the exponent $\delta$ if necessary, we may assume that $-\lambda_1\mu+K\delta/2\le-\lambda_1\mu/2$ and therefore $$\label{43}
\|\theta(0)\|^2_{H^{-1}}\le Ce^{-\lambda_1\mu T}\|w_+\|^2_{H^{-1}}+C\|w_+\|_{H^{-1}}^{2+\delta/2}\le C\|w_+\|^{2(1+\eb)}_{H^{-1}}$$ for some $\eb=\eb(\delta,\mu)>0$. Thus, the desired estimate is proved and the theorem is also proved.
\[Rem4.good\] Applying the parabolic smoothing property to the equation for $\theta(t)$, it is not difficult to verify the stronger version of estimate , namely $$\label{4.28}
\|\Phi(u^1_+)-\Phi(u^2_+)-\Phi'(u^1_+)(u_+^1-u_+^2)\|_{H^{2-\kappa}} \le C_\kappa\|u^1_+-u^2_+\|_H^{1+\eb},$$ where $\kappa>0$ is arbitrary and $C_\kappa$ depends only on $\kappa$. Moreover, as follows from the proof, estimate is actually used for $u_1$ and $u_2$ satisfying only and can be replaced by $$\label{4.3}
\|F(u_1)-F(u_2)-F'(u_1)(u_1-u_2)\|_H\le C\|u_1-u_2\|^{\delta}_{H^{2-\kappa}}\|u_1-u_2\|_{H}, \ \ u_1, u_2 \in H^{2-\kappa},$$ for some $\kappa\in(0,2]$. As we will see below, assumption is much easier to verify in applications than the initial assumption which is more natural for the abstract theory.
Note also that the result of Theorem \[Th4.main\] is in a sense optimal since the typical regularity of the inertial manifolds is exactly $C^{1+\eb}$ for some small $\eb>0$. The further regularity ($C^2$ or more) requires essentially stronger spectral gap assumptions which are usually satisfied only in the case of small Lipschitz constant $L$, see [@kok; @Zel] for more details.
Spatial averaging: an abstract scheme {#s4}
=====================================
In this section, we adapt the method of spatial averaging developed in [@mal-par] to the class of abstract Cahn-Hilliard equations . To this end, we first need to introduce some projectors.
Let $N \in \N$ and $k > 0$ be such that $\lambda_N > k$. Then, $$P_{k,N}u:=\sum_{n:\, \lambda_n < \lambda_N -k}(u,e_n)e_n, \ \ Q_{k,N}u := \sum_{n:\, \lambda_n > \lambda_{N} +k}(u,e_n)e_n,$$ and $$R_{k,N}u := \sum_{n:\, \lambda_N - k \le \lambda_n \le \lambda_{N} +k}(u,e_n)e_n.$$ As has been observed in [@mal-par] (at least on the level of reaction-diffusion equations, see also [@Zel]), the spectral gap condition is actually used only for the control the norm of the “intermediate” part $R_{k,N}\circ F'(u)\circ R_{k,N}$ of the derivative $F'(u)$ where $k\sim L^2$. Moreover, if this intermediate part is close to the scalar operator then the spectral gap condition may be relaxed. The following theorem adapts this result to the case of the Cahn-Hilliard equations.
\[th\_manifold\_ex\] Let the function $F$ be globally Lipschitz with the Lipschitz constant $L$, globally bounded and differentiable and let the number $N$ be such that $$\label{est_middle_terms}
\|R_{k,N}\circ F'(u)\circ R_{k,N}v - a(u) R_{k,N}v\|_H \le \delta \|v\|_H, \ \ u, v \in H,$$ where $a(u)\in \R$ is a scalar depending on $u$ and $\delta < L$. Assume also that $$\label{assump}
\frac{\theta}{2} > \delta+\frac {2L k}{\lambda_N-k}+\frac{2L^2}{k-4L}+\frac{2 L^2\lambda_N }{(2 \lambda_N - k)k - 4L \lambda_{N}}, \ \ \lambda_N > 2L,$$ as before $\theta = \lambda_{N+1} - \lambda_N$ and $k$ is chosen in such a way that $$(2\lambda_N - k)k> 4L \lambda_{N},\ \ k >4L.$$ Then, equation possesses the strong cone property in the differential form and, consequently, there exists a Lipschitz N-dimensional inertial manifold for this equation.
Due to Theorem \[Th3.main\], we know that in order to prove the existence of inertial manifold it is sufficient to check the validity of the strong cone inequality for the equation of variations associated with the solution $u(t)$ of the Cahn-Hilliard equation .
To this end, we first need the following estimate for the norm of $P_{k,N}w$ in $H^{-1}$ (here and below $\alpha=\lambda_N\lambda_{N+1}$): $$\begin{gathered}
\label{P_N,k}
( (\alpha A^{-1} -A)w_+, w_+)=\sum_{n=1}^N(\lambda_N\lambda_{N+1}\lambda_n^{-1}-\lambda_n)|w_n|^2\ge\\\ge\sum_{n:\, \lambda_n<\lambda_N-k}(\lambda_N\lambda_{N+1}-\lambda_n^2)\lambda_n^{-1}|w_n|^2\ge\\\ge(\lambda_N^2-(\lambda_N-k)^2)\|P_{k, N} w\|^2_{H^{-1}}=(2 \lambda_N k - k^2)\|P_{k, N} w\|^2_{H^{-1}}\end{gathered}$$ and the similar estimate for the norm of $Q_{k,N}v$ in $H$ $$\begin{gathered}
\label{Q_N,k}
((A - \alpha A^{-1})w_-, w_-)=\sum_{n=N+1}^\infty(\lambda_n-\lambda_N\lambda_{N+1}\lambda_n^{-1})|w_n|^2\ge\\\ge\sum_{n:\,\lambda_n>\lambda_N+k}
(\lambda_n-\lambda_N\lambda_{N+1}\lambda_n^{-1})|w_n|^2\ge(\lambda_N+k-\lambda_N)\|Q_{k,N} w\|^2_H=k\|Q_{k,N} w\|^2_H.\end{gathered}$$ Let $w(t)$ be a solution of the equation of variations . Then, arguing analogously to the derivation of but using estimates and together with and , we get: $$\begin{gathered}
\label{b_est_l(t)}
\frac{1}{2}\frac{d}{dt}V(w(t)) + \alpha V(w(t)) = ((\alpha A^{-1} -A)w_-,w_-) + ((A - \alpha A^{-1})w_+, w_+)- \\ -(F'(u)w, w_- - w_+) \le -\frac{\theta}{2}\|v\|^2_H + \frac{1}{2}(((\alpha A^{-1} -A)w_-,w_-) + ((A - \alpha A^{-1})w_+, w_+)) - \\ - (F'(u)w, w_- - w_+) \le - \frac{\theta}{2}\|w\|^2_H - \frac{k}{2}\|Q_{k, N}w\|^2_H - \frac{(2 \lambda_N - k)k}{2} \|P_{k, N}w\|^2_{H^{-1}} - (F'(u)w, w_- - w_+).\end{gathered}$$ Estimating the last term, we have: $$\begin{gathered}
\label{est_l(t)v,v--v+}
(F'(u)w, w_- - w_+) = (R_{k,N}\circ F'(u)w, w_- - w_+) +\\+((P_{k,N} + Q_{k,N})\circ F'(u)w, w_- -w_+) = (R_{k,N}\circ F'(u) \circ R_{k,N}w, w_- - w_+) +\\+ (F'(u)w, Q_{k,N}w - P_{k,N}w) + (F'(u)\circ (Q_{k,N}w + P_{k,N}w), w_- - w_+) \ge \\ \ge (R_{k,N}\circ F'(u) \circ R_{k,N}w, w_- - w_+) - 2 L \|w\|_H (\|Q_{k,N}w\|_H +\|P_{k, N}w \|_{H})\ge\\\ge
(R_{k,N}\circ F'(u) \circ R_{k,N}w, w_- - w_+) - 2 L \|w\|_H (\|Q_{k,N}w\|_H + \lambda_N^{1/2}\|P_{k, N}w \|_{H^{-1}}).\end{gathered}$$ It would be convenient to define two more spectral projectors: $$\tilde{P}_{k,N}w := \sum_{n:\, \lambda_N - k \le \lambda_n \le \lambda_N}(w,e_n)e_n \ \ \ \text{ and }\ \ \ \tilde{Q}_{k,N}w := \sum_{n:\, \lambda_{N+1} \le \lambda_n \le \lambda_{N} + k}(w,e_n)e_n.$$ Then, obviously $$\begin{gathered}
\bigg|\lambda_N\|\tilde {P}_{k,N}v\|_{H^{-1}}^2-\|\tilde{P}_{k,N} v\|_H^2\bigg|\le\\\le
\sum_{n:\, \lambda_N - k \le \lambda_n \le \lambda_N}|\lambda_N-\lambda_n|\lambda_n^{-1}|w_n|^2\le k\|\tilde {P}_{k,N}v\|_{H^{-1}}^2\le \frac{k}{\lambda_N-k}\|\tilde {P}_{k,N}v\|_{H}^2.\end{gathered}$$ and, analogously, $$\begin{gathered}
\bigg|\lambda_N\|\tilde {Q}_{k,N}v\|_{H^{-1}}^2-\|\tilde{Q}_{k,N} v\|_H^2\bigg|\le\\\le \sum_{n:\, \lambda_{N+1} \le \lambda_n \le \lambda_{N} + k}|\lambda_N\lambda_n^{-1}-1||w_n|^2\le \frac{k}{\lambda_N+k}\|\tilde{Q}_{k,N} v\|^2_H.\end{gathered}$$ Then, using assumption , we obtain: $$\begin{gathered}
(R_{k,N}\circ F'(u) \circ R_{k,N}w, w_- - w_+) \ge a(u)(\|\tilde{Q}_{k,N}w\|^2_H - \|\tilde{P}_{k,N}w \|^2_H) - \delta\|w\|^2_H \ge \\
\ge \lambda_N a(u)(\|\tilde{Q}_{k,N}w\|^2_{H^{-1}} - \|\tilde{P}_{k,N}w \|^2_{H^{-1}}) - \delta\|w\|^2_H -\\-|a(u)|\(\frac k{\lambda_N-k}\|\tilde {P}_{k,N}v\|_{H}^2+\frac{k}{\lambda_N+k}\|\tilde{Q}_{k,N} v\|^2_H\)
\ge \lambda_N a(u) V(w(t)) -\\ - |a(u)|(\lambda_N\|P_{k,N}w\|^2_{H^{-1}} + \frac{\lambda_N}{\lambda_{N}+k}\|Q_{k,N}w\|^2_{H}) -
(\delta+\frac k{\lambda_N-k}|a(u)|) \|w\|^2_H.\end{gathered}$$ Substituting this result into and using obvious fact that $|a(u)| \le L + \delta \le 2L$, we get: $$\begin{gathered}
\label{5.est}
-(F'(u)w, w_- - w_+)\le -\lambda_N a(u) V(w) + 2 L( \lambda_{N}\|P_{k,N}w\|^2_{H^{-1}} + \|Q_{k,N}w\|^2_{H}) + \\
+2L\|w\|_H(\lambda_N^{1/2}\|P_{k,N}w\|_{H^{-1}} + \|Q_{k,N}w\|_{H})+(\delta+\frac {2L k}{\lambda_N-k}) \|w\|^2_H.\end{gathered}$$ Therefore, due to the Young inequality, $$\begin{gathered}
\label{5.est1}
- \frac{\theta}{2}\|w\|^2_H - \frac{k}{2}\|Q_{k, N}w\|^2_H - \frac{(2 \lambda_N - k)k}{2} \|P_{k, N}w\|^2_{H^{-1}} - (F'(u)w, w_- -w_+)\le\\\le-\(\frac\theta2-\delta-\frac {2L k}{\lambda_N-k}\)\|w\|^2_H-\(\frac{(2 \lambda_N - k)k}{2}-2L\lambda_{N}\) \|P_{k,N}w\|^2_{H^{-1}}-\(\frac\kappa2-2L\)\|Q_{k,N}w\|^2_H+\\+2L\|w\|_H\(\lambda_N^{1/2}\|P_{k,N}w\|_{H^{-1}}+ \|Q_{k,N}w\|_{H}\)\le\\\le
-\(\frac\theta2-\delta-\frac {2L k}{\lambda_N-k}-\frac{2L^2}{\kappa-4L}-\frac{2 L^2\lambda_N }{(2 \lambda_N - k)k - 4L \lambda_{N}}\)\|w\|^2_H=-\frac\mu2\|w\|^2_H,\end{gathered}$$ where $$\frac\mu2:=\frac\theta2-\delta-\frac {2L k}{\lambda_N-k}-\frac{2L^2}{\kappa-4L}-\frac{2 L^2\lambda_N }{(2 \lambda_N - k)k - 4L \lambda_{N}}.$$ Finally, inserting estimate into the right-hand side of and taking into account assumptions we see that the differential cone inequality is satisfied with the above $\mu$ and with $$\alpha(u)= 2\alpha - 2\lambda_N a(u)> 2\lambda_{N}(\lambda_{N+1} - 2L)> 0.$$ Thus, the desired strong cone condition is proved and due to Theorem equation possesses an $N-$dimensional inertial manifold.
The typical situation to apply the above proved theorem is when, for sufficiently small $\delta>0$ and any $k$ there exists an infinite sequence of $N\in\Bbb N$ such that $$\lambda_{N+1}-\lambda_N\ge\rho>0$$ ($\rho$ is independent of $N$ and $k$) such that the spatial averaging assumption hold for every such $N$. Then, for very large $N$, the main condition reads $$\label{5.spa-inf}
\frac\rho2>\delta+\frac{2L^2}{k-4L}+\frac{L^2}{k-2L}$$ and we see that it is indeed satisfied if $\delta$ is small enough (say, $\delta<\frac\rho4$) and $k=k(\rho,L)$ is large enough (say, $k=4L+\frac{12L^2}\rho$). This gives the existence of the desired inertial manifold for these large $N$s.
As in the case of reaction-diffusion equations, see [@mal-par; @Zel], estimate is too restrictive since the constant $\delta$ is uniform with respect to $u\in H$ and in applications it usually depends on the higher norms of $u$. Namely, similar to [@mal-par; @Zel], we give the following definition.
\[def\_spat\_av\] We say that the non-linearity $F:H\to H$ satisfies the [*spatial*]{} averaging condition if it is globally bounded, Lipschitz continuous, differentiable in the sense that the mean value theorem holds and there exist a positive exponent $\kappa$ and a positive constant $\rho$ such that for every $\delta > 0$, $R>0$ and $k >0$ there exists infinitely many values $N \in \N$ satisfying $$\label{5.agap}
\lambda_{N+1} - \lambda_N \ge \rho$$ and $$\label{5.spa}
\sup_{\|u\|_{H^{2-\kappa}}\le R}\biggl\{ \|R_{k,N}\circ F'(u)\circ R_{k,N}v - a(u) R_{k,N}v\|_H \biggr\}\le \delta \|v\|_H,$$ for some scalar multiplier $a(u) = a_{N,k,\delta}(u)\in \R$ which is assumed to be bounded Borel measurable as a function from $H$ to $\R$.
In this case, although we do not know how to construct the IM for the initial problem , it is possible to [*modify*]{} this equation outside of the absorbing ball in such a way that the new equation will possess the IM. Then, the obtained IM will still be invariant with respect to the solution semigroup $S(t)$ of the initial equation at least in the neighborhood of the global attractor $\Cal A$ and therefore will contain all of its non-trivial dynamics. By this reason, the IM for the modified equation is often referred as the IM for the initial problem , see e.g., [@FST; @tem] for more details.
To be more precise, according to the dissipative estimate with $s=2$ together with the smoothing property , the set $$\Cal B_2:=\big\{u\in H^2,\ \ \|u\|_{H^2}\le 2R_*\big\}$$ is an absorbing ball for the solution semigroup $S(t)$ associated with the Cahn-Hilliard equation. Let us introduce, following [@mal-par], the cut-off function $\varphi(\eta) \in C^{\infty}(\R)$ such that: $$\varphi(\eta) = 1 ,\ \ \eta\le(2 R_*)^2 \ \text{ and }\varphi(\eta)=\frac{1}{2}, \ \ \eta \ge R_1^2,$$ where $R_1 > 2R_*$ and : $$\label{phi_rest}
\varphi'(\eta)\le 0 \ \ \text{ and }\ \ \frac{1}{2}\varphi(\eta) + \eta \varphi'(\eta)> 0 ,\ \eta \in \R.$$ Thus, gives us the restriction $$\varphi(\eta)\ge \frac{2 R_*}{\sqrt{\eta}},\ \ \eta \ge 16R_*$$ and, therefore, $R_1 \ge 4 R_*$.
Finally, for every $N \in \N$, we introduce the following cut -off version of the problem : $$\label{main_eq_cut}
\pt u + A^2 u + A F(u) - A^2 P_N u + \varphi(\|A P_N u \|^2_H)A^2 P_N u = 0.$$ Then, on the one hand, by the construction of the cut-off function $\varphi(\eta)$, we see that equation coincides with inside the absorbing ball $\Cal B_2$ and, on the other hand, the following key result holds.
\[Th5.main\] Let the non-linearity $F$ satisfy the spatial averaging assumption for some $\kappa\in(0,2)$. Then, there exist infinitely many $N$s such that the strong cone condition is satisfied for the modified equation and, thus, for every such $N$, it possesses an N-dimensional Lipschitz continuous IM.
According to Theorem \[Th3.main\], we only need to verify the strong cone condition for the equation of variations associated with the modified equation : $$\label{5.var}
\pt w + A^2 w + A (F'(u(t))w) = A^2 P_N w - T'(u(t))A w,$$ where $T(u):= \varphi (\|A P_N u\|^2_H)A P_N u$ and $u(t)$ is a solution of . To this end, we need the following technical lemma originally proved in [@mal-par].
\[lem\_est\_T(u)\] Under the above assumptions the following estimate is valid: $$\label{est_T(u)}
( T'(u)v,v) \le \frac{1}{2} \lambda_N \|v\|^2_H + \frac{1}{2}(A v, v),\ \ \forall v \in H_+,\ \ u\in H.$$
For the sake of completeness we provide the simplified proof of the lemma following to [@Zel]. It is based on the following inequality: $$\label{5.CS}
2(v,y)(w,y) \ge \|y\|_H^2 ((v,w) - \|v\|_H \|w\|_H)$$ for any 3 vectors $v,w,y\in H$. To verify it, we rewrite in the equivalent form $$\(v-2\frac{(v,y)}{\|y\|^2_H}y,w\)\le\|v\|_H\|w\|_H$$ and this follows from the Cauchy-Schwartz inequality and the fact that the map $v\to v-2\frac{(v,y)}{\|y\|^2_H}y$ is a reflection with respect to the plane orthogonal to $y$ and, thus, is an isometry.
Using now and inequalities , we have that, for any $v \in H_+$, $$\begin{gathered}
(T'(u)v,v)= 2 \varphi'(\|Au_+\|^2_H)(A u_+, A v)(A u_+, v) + \varphi(\|A u_+\|^2_H)(A v,v) \le \\ \le - \varphi'(\|Au_+\|^2_H) \|Au_+\|^2_H (\|Av\|_H \|v\|_H - (A v, v)) + \varphi(\|Au_+\|^2_H) (A v, v) \le \\ \le \frac{1}{2} \varphi(\|Au_+\|^2_H) \lambda_N \|v\|^2_H + \frac{1}{2} \varphi(\|Au_+\|^2_H)(A v, v)\le \frac{1}{2} (\lambda_N \|v\|^2_H + (Av,v))
\end{gathered}$$ and the lemma is proved.
We are now ready to complete the proof of the theorem. Indeed, multiplying equation by $w_--w_+$ acting as in the proof of Theorem \[th\_manifold\_ex\] we come to the following equality: $$\begin{gathered}
\label{5.id}
\frac{1}{2} \frac d{dt}V(w(t)) + \alpha V(w(t)) = ((\alpha A^{-1} - A)w_-, w_-) + ((A - \alpha A^{-1}) w_+, w_+) +\\+ (T'(u(t))w_+,w_+) - (A w_+,w_+) - (F'(u(t))w,w_--w_+),
\end{gathered}$$ where $\alpha=\lambda_N\lambda_{N+1}$.
Then, using Lemma and , we get: $$\begin{gathered}
\label{same}
\frac{1}{2}\frac d{dt}V(w(t)) + \alpha V(w(t))\le -\frac{1}{2}(\lambda_{N+1} - \lambda_N )\|w_+\|^2_H +\frac{1}{2}((A - \alpha A^{-1})w_+,w_+)+\\ + ((\alpha A^{-1} - A)w_-, w_-) - (F'(u(t))w,w_--w_+)\le\\\le -\frac12(\lambda_{N+1}-\lambda_N)\|w\|^2_H+\frac{1}{2}((A - \alpha A^{-1})w_+,w_+)+\\ + \frac12((\alpha A^{-1} - A)w_-, w_-)- (F'(u(t))w,w_--w_+).\end{gathered}$$ Fix an arbitrary point $t \ge 0$ and assume first that $$\label{f_as}
\|A P_N u(t)\|_H \le R_1,$$ where $R_1$ is the same as in the definition of the cut-off function $\varphi$. Thus, we see that the structure of is exactly the same as the structure of . In addition, due to , we may assume without loss of generality that $$\label{5.kappa}
\|Q_N u(t)\|_{H^{2-\kappa}}\le 2R_\kappa,$$ see Remarks \[Rem2.smooth\] and \[Rem3.smooth\]. Therefore, assumption implies that $$\|u(t)\|_{H^{2-\kappa}}\le \lambda^{-\frac{\kappa}{2}}_1 \|P_N u(t)\|_{H^2} + \|Q_N u(t)\|_{H^{2 - \kappa}} \le \lambda^{-\frac{\kappa}{2}}_1 R_1 + 2 R_\kappa \le R,$$ where $R$ is independent of the choice of $N$. Hence, using spatial averaging assumption (with this value of the parameter $R$, sufficiently small $\delta$ and sufficiently large $k$ in order to satisfy ) and repeating word by word the proof of Theorem , we conclude that there exist a sequence of $N$s such that $$\label{5.good}
\frac{1}{2} \frac{d}{dt}V(w(t)) + \alpha(u(t)) V(w(t)) \le -\mu \|w(t)\|^2_H,$$ where $\alpha(u):=\lambda_N\lambda_{N+1}-\lambda_N a(u)$ and $\mu>0$ is independent of $u$, $N$ and $w$. Thus, we have verified the strong cone condition in the case when is satisfied.
Let us now consider the opposite case $$\label{5.big}
\|A P_N u(t)\|_H \ge R_1.$$ The situation here is much simpler. Indeed, instead of estimate , we may use better identity $(T'(u(t))w,w) = \frac{1}{2}(Aw_+,w_+)$. Then, using the Lipschitz continuity of $F$ and the fact that both $(\alpha A^{-1} - A)Q_N$ and $(A - \alpha A^{-1})P_N$ are negatively definite (see estimates and ), we transform identity as follows: $$\begin{gathered}
\frac{1}{2} \frac{d}{dt} V(w(t)) + \alpha V(w(t)) \le\\\le ((\alpha A^{-1} - A)w_-, w_-) + ((A - \alpha A^{-1}) w_+, w_+)+L\|w\|^2_H-\frac12(Aw_+,w_+)\le\\\le
L\|w\|^2_H-\frac12\alpha\|w_+\|^2_{H^{-1}}+((\alpha A^{-1} - A)w_-, w_-)\le\\\le
L\|w\|^2_H+\frac14\alpha\(\|w_-\|^2_{H^{-1}}-\|w_+\|^2_{H^{-1}}\)-\frac14\lambda_{N+1}\|w_+\|^2_H-\frac14\|w_-\|_{H^1}^2\le\\\le
\frac{1}{4} \alpha V(w(t)) + \(L-\frac{1}{4} \lambda_{N+1}\)\|w\|^2_H.\end{gathered}$$ Thus, if $ L < \frac{1}{4} \lambda_{N+1}-\mu$ ($\mu$ is the same as in ), we end up with estimate with $\alpha(u)=\frac34\lambda_N\lambda_{N+1}$. Therefore the desired strong cone estimate is verified for the case when is satisfied as well and the theorem is proved.
\[Cor5.smooth\] Let the assumptions of Theorem \[Th5.main\] hold and let, in addition, the nonlinearity $F$ be smooth in the sense that assumption hold for some $\kappa$ and $\delta>0$. Then, the inertial manifolds $\Cal M=\Cal M_N$ of the modified equations are $C^{1+\eb}$-smooth for some $\eb=\eb_N>0$.
Indeed, the statement of the corollary follows immediately from the proved theorem and Theorem \[Th4.main\], see also Remark \[Rem4.good\].
Inertial manifolds for the classical Cahn-Hilliard equation {#s5}
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In this concluding section, we apply the developed abstract theory to the classical Cahn-Hilliard equation in the 3D case endowed by periodic boundary conditions. First, we need to embed this equation into the functional model . To this end, keeping in mind the conservation law and our agreement that $\<u(t)\>=0$, we introduce the space $$H:=L^2(\Bbb T^3)\cap\{\<u\>=0\}$$ and the operator $A=-\Delta_x$ with the domain $D(A)=H^2(\Bbb T^3)\cap\{\<u\>=0\}$. Then, $H^s=H^s(\Bbb T^3)\cap\{\<u\>=0\}$, $s\in\R$, where $H^s(\Bbb T^3)$ is a Soblolev space of $2\pi$ periodic functions from $\R^3$ to $\R$. Moreover, any function $u\in H$ can be splitted into the Fourier series $$\label{6.f}
u(x)=\sum_{l\in\Bbb Z^3,\, l\ne0} u_l e^{i l.x},\ \ u_l=\frac1{(2\pi)^3}\int_{\Bbb T^3} u(x)e^{i l.x}.$$ Here and below $l.x:=\sum_{i=1}^3l_ix_i$ is a usual inner product in $R^3$ and $|l|^2:=l.l$. The eigenvalues of the operator $A$ are naturally parametrised by the points of the lattice $l\in\Bbb Z^3\backslash\{0\}$, i.e., $$\label{6.e}
A e_l=\lambda_l e_l,\ \ e_l=e^{i l.x},\ \ \lambda_l=|l|^2,\ \ l\in\Bbb Z^3\backslash\{0\}.$$ Thus, due to the Parseval equality, the norm in the space $H^s$, $s\in\R$ is given by $$\label{6.n}
\|u\|_{H^s}^2=\sum_{l\in\Bbb Z^3,\, l\ne0}|l|^{2s} |u_l|^2,\ \ u(x)=\sum_{l\in\Bbb Z^3,\, l\ne0} u_l e^{i l.x}.$$ We now return to equation . We assume that the non-linearity $f\in C^3(\R,\R)$ and is globally bounded together with its first derivative: $$\label{6.b}
1. \ |f(u)|+|f''(u)|\le K,\ \ 2.\ \ |f'(u)|\le L$$ for all $u\in\R$. As we have already mentioned, is not a big restriction since in the general case of dissipative non-linearities $f$ (e.g., $f(u)=u-u^3$), we usually have an absorbing ball in $H^2\subset C(\Bbb T^3)$, making the proper cut-off of the non-linearity outside of the absorbing ball if necessary, we may assume without loss of generality that is satisfied. Finally, we introduce the non-linearity $F: H\to H$ via $$\label{6.non}
F(u)(x):=f(u(x))-\<f(u)\>,\ \ u\in H,$$ where the last term is introduced in order to guarantee that $\<F(u)\>=0$ and, therefore, $F(u)\in H$.
Then, it is immediate to see that equation is equivalent to equation , so the desired functional model is constructed. At the next step, we check that the assumptions stated above for the abstract equation are satisfied in this concrete case. First, from , it is immediate to check that the non-linearity $F(u)$ is globally bounded and globally Lipschitz continuous in $H$. The next key proposition checks that the spatial averaging assumption is also satisfied.
\[Prop6.av\] Let the non-linearity $f\in C^3(\R,\R)$. Then, the spatial averaging assumption is satisfied for the operator $F$ for all $\kappa\in(0,\frac12)$ and $$a(u):=\<f'(u)\>.$$
Note that $F'(u)v=f'(u)v-\<f'(u)v\>$, $u,v\in H$ and $$R_{k,N}\circ F'(u)\circ R_{k,N}v=R_{k,N}\(f'(u)R_{k,N}v\),$$ so the spatial averaging assumption for our case [*coincides*]{} with the analogous assumption for reaction-diffusion equations. By this reason, the proof of the proposition follows word by word to the one given in [@mal-par; @Zel] for the case of reaction-diffusion equations. For the convenience of the reader, we sketch this proof below. It is based on the following non-trivial result from the number theory.
\[Lem6.sp\] Let $$\Cal C^k_N:=\{l\in\Bbb Z^3\,:\, N-k\le|l|^2\le N+k\},\ \ \Cal B_r:=\{l\in\Bbb Z^3\,:\, |l|\le r\}.$$ Then, for every $k>0$ and $r>0$, there exist infinitely many $N\in\Bbb N$ such that $$\label{6.spav}
\(\Cal C^k_{N}-\Cal C^k_N\)\cap \Cal B_r=\{0\}.$$
The proof of this lemma is given in [@mal-par].
We are now ready to verify the spatial averaging principle for the non-linearity $f$. To this end, we first note that according to the Weyl asymptotics $$\lambda_N\sim CN^{2/3},$$ so without loss of generality, we may replace the projector $R_{k,N}$ by the projector to the Fourier modes belonging to $\Cal C^k_N$ which (in slight abuse of notations), we also denote it by $R_{k,N}$, so we use below the definition $$(R_{k,N}v)(x):=\sum_{l\in\Cal C^k_N} v_l e^{il.x}.$$ Denote $w(x):=f'(u(x))$. Then, the multiplication $w(x)v(x)$ is a convolution in Fourier modes $$[wv]_m=\sum_{l\in\Bbb Z^3} w_{m-l}v_l$$ and, due to condition , $$R_{k,N}\((w-\<w\>) R_{k,N}v\)=R_{k,N}\(w_{>r}R_{k,N}v\),$$ where $w_{>r}(x):=\sum_{|l|>r}w_le^{il.x}$. Therefore, $$\|R_{k,N}\((w-\<w\>) R_{k,N}v\)\|_H\le\|\(w_{>r}R_{k,N}v\)\|_H\le \|w_{>r}\|_{L^\infty}\|v\|_H.$$ Furthermore, due to the interpolation, for $\kappa<1/2$, $$\|w_{>r}\|_{L^\infty}\le C\|w_{>r}\|_H^{1-\theta}\|w_{>r}\|_{H^{2-\kappa}}^\theta\le C r^{-(1-\theta)(2-\kappa)}\|w\|_{H^{2-\kappa}},$$ where $\theta=\frac3{2(2-\kappa)}$. Finally, using that $H^{2-\kappa}$ is an algebra for $\kappa<\frac12$ and that $f'\in C^2$, we have $$\label{6.est}
\|R_{k,N}\((w-\<w\>) R_{k,N}v\)\|_H\le C r^{-(1-\theta)(2-\kappa)}Q(\|u\|_{H^{2-\kappa}})\|v\|_H$$ for some monotone increasing function $Q$. Thus, the right-hand side of can be made arbitrarily small by increasing $r$, so estimate indeed holds with $$a(u)=\<w\>=\frac1{(2\pi)^3}\int_{\Bbb T^3}f'(u(x))\,dx.$$ Since the eigenvalues of $A$ are integers, assumption also holds with $\rho=1$ and the nonlinearity $F(u)$ satisfies indeed the spatial averaging assumption. Thus, the proposition is proved.
The next simple proposition shows that the map $F$ is smooth if $f$ is smooth.
\[Prop6.sm\] Let the nonlinear function $f\in C^2$ and satisfy . Then, the non-linear operator $F(u)$ defined by satisfies estimate for any $\delta\in[0,1]$ and any $\kappa\in(0,\frac12)$.
We first note that it is sufficient to verify for $\delta=0$ and $\delta=1$ only. For simplicity, we verify estimate for the first term $f(u(x))$ in the definition of the nonlinearity $F(u)$ only. The estimate for the remaining term $\<f(u)\>$ can be obtained analogously. Let first $\delta=0$ and $u_1,u_2\in H$. Then, since $f$ is globally Lipschitz continuous, $$\|f(u_1)-f(u_2)-f'(u_1)(u_1-u_2)\|_H\le\|f(u_1)-f(u_2)\|_H+\|f'(u_1)(u_1-u_2)\|_H\le 2L\|u_1-u_2\|_H$$ and estimate is verified for that case. Let now $\delta=1$ and $u_1,u_2\in H^{2-\kappa}$. Then, according to the integral mean value theorem $$\begin{gathered}
f(u_1)-f(u_2)-f'(u_1)(u_1-u_2)=\int_0^1[f'(u_1+(1-s)(u_1-u_2))-f'(u_1)]\,ds(u_1-u_2)=\\=
\int_0^1\int_0^1f''(s_1u_1+s_1(1-s)(u_1-u_2))\,ds\,ds_1(u_1-u_2)^2\end{gathered}$$ and, due to assumption and the embedding $H^{2-\kappa}\subset C$, $$\begin{gathered}
\|f(u_1)-f(u_2)-f'(u_1)(u_1-u_2)\|_H\le K\|(u_1-u_2)(u_1-u_2)\|_H\le\\\le K\|u_1-u_2\|_{L^\infty}\|u_1-u_2\|_H\le CK\|u_1-u_2\|_{H^{2-\kappa}}\|u_1-u_2\|_H.\end{gathered}$$ Therefore, estimate holds for this case as well and the proposition is proved.
Thus, all abstract assumptions from the previous sections are verified and we have proved the following theorem which is the main result of the paper.
Let the non-linear function $f\in C^3(\R,\R)$ and satisfy assumptions . Then, there exists an infinite sequence of $Ns$ such that the classical Cahn-Hilliard problem on a 3D torus $\Bbb T^3=[-\pi,\pi]^3$ possesses an $N$ dimensional inertial manifold containing the global attractor. Moreover, these inertial manifolds are $C^{1+\eb}$-smooth for some $\eb=\eb_N>0$.
Indeed, the existence of IMs follows from Theorem \[Th5.main\] and Proposition \[Prop6.av\] and its smoothness follows from Theorem \[Cor5.smooth\] and Proposition \[Prop6.sm\].
[99]{} A. Babin and M. Vishik, [*Attractors of evolution equations.*]{} Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992. A. Bonfoh, M. Grasselli and A. Miranville, [*Inertial manifolds for a singular perturbation of the viscous Cahn-Hilliard-Gurtin equation,*]{} Topol. Methods Nonlinear Anal. 35, no. 1, (2010), 155–185. J. Cahn and J. Hilliard, [*Free energy of a nonuniform system. I. Interfacial free energy,*]{} J. Chem. Phys., [**28**]{} (1958), 258–267. V. Chepyzhov and M. Vishik, [*Attractors for equations of mathematical physics.*]{} American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. L. Cherfils, A. Miranville and S. Zelik, [*The Cahn-Hilliard equation with logarithmic potentials,*]{} Milan J. Math. 79, no. 2, (2011), 561–596. A. Eden, V. Kalantarov and S. Zelik, [*Counterexamples to the Regularity of Mané Projections in the Attractors Theory*]{}, Russian Math. Surveys, [**68**]{}, no. 2, (2013), 199–226. C. Elliott, [*The Cahn-Hilliard model for the kinetics of phase separation. Mathematical models for phase change problems,*]{} 35–73, Internat. Ser. Numer. Math., 88, Birkhauser, Basel, 1989. N. Fenichel, [*Persistence and smoothness of invariant manifolds for flows.*]{} Indiana Univ. Math. J. [**21**]{} (1971/1972), 193–226. C. Foias, G. Sell, and R. Temam, [*Inertial manifolds for nonlinear evolutionary equations.*]{} J. Differential Equations [**73**]{}, no. 2, (1988), 309–353. D. Henry, [*Geometric theory of semilinear parabolic equations.*]{} Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. N. Koksch, [*Almost Sharp Conditions for the Existence of Smooth Inertial manifolds.*]{} in: Equadiff 9: Conference on Differential Equations and their Applications : Proceedings, edited by Z. Dosla, J. Kuben, J. Vosmansky, Masaryk University, Brno, 1998, 139–166. H. Kwean, [*An Extension of the Principle of Spatial Averaging for Inertial Manifolds.*]{} J. Austral. Math. Soc. (Series A) [**66**]{} (1999), 125–142. , [Inertial manifolds for reaction-diffusion equations in higher space dimensions.]{}, J. Mallet-Paret, G. Sell, and Z. Shao, [*Obstructions to the existence of normally hyperbolic inertial manifolds.*]{} Indiana Univ. Math. J., [**42**]{}, no. 3, (1993), 1027–1055. M. Miklavcic, [*A sharp condition for existence of an inertial manifold.*]{} J. Dynam. Differential Equations, [**3**]{}, no. 3, (1991), 437–456. A. Miranville and S. Zelik, [*Attractors for dissipative partial differential equations in bounded and unbounded domains.*]{} In: Handbook of differential equations: evolutionary equations. Vol. IV, 103–200, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008. A. Novick-Cohen. [*The Cahn-Hilliard equation: mathematical and modeling perspectives,*]{} Adv. Math. Sci. Appl., [**8**]{} (1998), 965–985. J. Robinson, [*Dimensions, embeddings, and attractors.*]{} Cambridge Tracts in Mathematics, 186. Cambridge University Press, Cambridge, 2011. A. Romanov, [*Sharp estimates for the dimension of inertial manifolds for nonlinear parabolic equations.*]{} Russian Acad. Sci. Izv. Math., [**43**]{}, no. 1, (1994), 31–47. A. Romanov, [*Finite-dimensionality of dynamics on an attractor for nonlinear parabolic equations.*]{} Izv. Math., [**65**]{}, no. 5, (2001), 977–1001. A. Romanov, [ *Finite-dimensional limit dynamics of dissipative parabolic equations.*]{} Sb. Math., [**191**]{}, no. 3–4, (2000), 415–429. R. Temam, [*Infinite-dimensional dynamical systems in mechanics and physics. Second edition.*]{} Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. , [Inertial manifolds and finite-dimensional reduction for dissipative PDEs]{}, , [**144A**]{}, (2014), 1245–1327. , [Inertial Manifolds for 1D convective reaction-diffusion equations]{}, submitted.
[^1]: This work is partially supported by the grant RSF 14-41-00044 of RSF and the grant 14-01-00346 of RFBR
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abstract: 'Decomposing a digraph into subdigraphs with a fixed structure or property is a classical problem in graph theory and a useful tool in a number of applications of networks and communication. A digraph is strongly connected if it contains a directed path from each vertex to all others. In this paper we consider multipartite tournaments, and we study the existence of a partition of a multipartite tournament with $c$ partite sets into strongly connected $c$-tournaments. This is a continuation of the study started in 1999 by Volkmann of the existence of strongly connected subtournaments in multipartite tournaments.'
author:
- 'Ana Paulina Figueroa[^1]Juan José Montellano-Ballesteros[^2]Mika Olsen[^3]'
title: 'Decomposition of balanced multipartite tournaments into strongly connected tournaments[^4]'
---
[**Keyword:** ]{} Oriented graphs, Multipartite tournaments, Decomposition, Strong connected digraphs
Introduction and definitions {#sec1}
============================
Decomposing a digraph into subdigraphs with a fixed structure or property is a classical problem in graph theory and a useful tool in a number of applications of networks and communication. For instance, finding a decomposition in strongly connected components has been used in compiler analysis, data mining, scientific computing, social networks and other areas. In this paper we consider multipartite tournaments, and we study the existence of a partition of the set of vertices of a multipartite tournament with $c$ partite sets, into strongly connected tournaments of order $c$. Observe that every partite set of the multipartite tournament has exactly one vertex in each strongly connected tournament of the partition. We can illustrate our result with the following situation: if all the vertices of any partite set has the same information, and any pair of vertices of different partite sets has different information, then the total information spread among all the vertices of the digraph can be distributed effectively using the partition into strongly connected tournaments since each strongly connected tournament possess one vertex of each partite set.
Let $c$- be a non-negative integer, a [*$c$-partite or multipartite tournament*]{} is a digraph obtained from a complete $c$-partite graph by orienting each edge. In 1999 [@sub] Volkmann developed the first contributions in the study of the structure of the strongly connected subtournaments in multipartite tournaments. He proved that every almost regular $c$-partite tournament contains a strongly connected subtournament of order $p$ for each $p\in\{3,4, \ldots, c-1 \} $. In the same paper he also proved that if each partite set of an almost regular $c$-partite tournament has at least $\frac{3c}{2}-6$ vertices, then there exist a strong subtournament of order $c$. In 2008 [@Almost-regular], Volkmann and Winsen proved that every almost regular $c$-partite tournament has a strongly connected subtournament of order $c$ for $c\geq 5$. In 2011 [@X] Xu et al. proved that every vertex of regular $c$-partite tournament with $c\ge16$, is contained in a strong subtournament of order $p$ for every $p\in \{3,4,\dots,c\}$. Finally, in 2016 [@amo], we proved that for every (not necessarily strongly connected) balanced $c$-partite tournament of order $n\ge6$, if the global irregularity of $T$ is at most $\frac{c}{\sqrt{3c+26}}$, then $T$ contains a strongly connected tournament of order $c$.
Let $T$ be a $c$-partite tournament of order $n$ with partite sets $\{V_i\}_{i=1}^c$. We call $T$ *balanced*, if all partite sets contain the same number of vertices and we denote by $G_{r,c}$ a balanced $c$-partite tournament satisfying that $|V_i|=r$ for every $1\le i\le c$. Throughout this paper $|V_i|=r$ for each $i\in [c]$. [As a [*partition of $G_{r,c}$ in maximal tournaments*]{} we will understand a spanning subdigraph of $G_{r,c}$ which is a set of $r$ pairwise vertex-disjoint tournaments of order $c$]{}.
Our main result gives sufficient conditions in terms of the minimum degree, the number of partite and its order to guarantee that a $r$-balanced $c$-partite tournament has a partition [in maximal tournaments such that each of its $r$ tournaments is strongly connected. Such a partition will be called a *strong partition*.]{}
We will follow almost all the definitions and notation of [@B-J-G]. The maximal independent sets of $T$ are called the partite sets of $T$. If $T$ is a $c$-partite tournament, $\Delta(T)=\max\{d^j(x):x\in V(T), j\in\{+,-\} \}$ and $\delta(T)=\min\{d^j(x):x\in V(T), j\in\{+,-\} \}$. Notice that $i_g(T)=\Delta(T)-\delta(T)$. Let $x\in V(T)$ and $i\in [c]$, the out-neighborhood of $x$ in $V_i$ is $N^+_i(x) =V_i\cap N^+(x)$; the in-neighborhood of $x$ in $V_i$ is $N^-_i(x) =V_i\cap N^-(x)$; $d^+_i(x) =|N^+_i(x) |$ and $d^-_i(x) = |N^-_i(x)|$. If $x\in V(T)$, $\Delta_V^+(x)= \max \{d_i^+(x): i\in [c] \}$; $\delta_V^+(x)= \min \{d_i^+(x): i\in [c] \}$); $\Delta_V^-(x)= \max \{d_i^-(x): i\in [c] \}$ and $\delta_V^-(x)= \min \{d_i^-(x): i\in [c] \}$. If $T$ is a $c$-partite tournament, the *maximum out-degree of $T$ with respect to the parts* is $\Delta_V^+(T) = \max \{d_i^+(x) : i \in [c] \hbox{ \& } x\in V(T)\}$ and the *minimum out-degree of $T$ with respect to the parts* is $\delta_V^+(T) = \min \{ d_i^+(x) : i \in [c] \hbox{ \& } x\in V(T)\}$. Analogously, we define $\Delta_V^-(T) = \max \{d_i^-(x) : i \in [c] \hbox{ \& } x\in V(T)\}$ and $\delta_V^+(T) = \min \{ d_i^-(x) : i \in [c] \hbox{ \& } x\in V(T)\}$. We will simply write $\Delta^+_V$, for example, instead of $\Delta^+_V(T)$ whenever it is [clear]{} in which $c$-partite tournament $T$ we are working on.
If $G_{c,r}$ is a balanced $c$-partite tournament, we define a new measure of irregularity called the irregularity restricted to the parts as $\mu (G_{r,c})= \max\{\Delta^+_V-\delta_V^+, \Delta^-_V-\delta^-_V \}$. Our main result has $\mu(G_{r,c})$ as a parameter.
Main Results
============
In this section we used Lemmas $1-4$ in order to proof our Main Result. The prove of these lemmas can be found in Section \[pruebas lemas\].
\[particion\] The number of partitions of $G_{r,c}$ [in maximal tournaments]{} is $(r!)^{c-1}$.
Let $x\in V_c$ and let $\mathcal{H}_k(x)$ be the set of vectors $(h_1,h_2, \ldots h_{c-1})\in \{0,1\}^{c-1}$ such that $h_i=1$ if $d_i^+(x)=r$, $h_i=0$ if $d_i^+(x)=0$ and $\sum_{i=1}^{c-1} h_i=k$.
\[L M(gi)\] Let $G_{r,c}$ be a balanced $c$-partite tournament and let $x\in V_c$. The number of maximal [tournaments of $G_{r,c}$]{} for which $x$ has out-degree $k$ is equal to $$\sum_{h\in \mathcal{H}_k(x)}\prod_{i=1}^{c-1}d_i^+(x)^{h_i}d_i^-(x)^{1-h_i}.$$
For each $x\in V(T)$ let $T^+(x)$ (resp. $T^-(x)$ ) be the number of maximal [tournaments]{} of $G_{r,c}$ for which $x$ has out-degree (resp. in-degree) at most $\lceil\frac{c-2}{4}\rceil$. The following Lemma provides an upper bound for $T^+(x)$. An analogous result for $T^-(x)$ can be obtained using similar arguments.
\[promedio\] Let $G_{r,c}$ be a balanced $c$-partite tournament such that $\delta(G_{r,c}) \geq \lfloor\frac{c-2}{4}\rfloor (r + \mu) + max \{\delta^+_V, \delta^-_V\}$. Then, for every $x\in V(G_{r,c})$, $$T^+(x) \leq \sum\limits_{k=0}^{\left\lfloor \frac{c-2}{4} \right\rfloor} {{c-1}\choose {k}} \left(\frac{d^+(x)}{d^-(x)}\right)^k \left(\frac{d^-(x)}{c-1}\right)^{c-1}.$$
Moreover, if $\lceil\frac{c-2}{4}\rceil < |\{i : d_i^+(x) = r\}| $, $T^+(x) = 0$.
\[suma\] Let $G_{r,c}$ be a balanced $c$-partite tournament, with $c\geq 10$, such that $\delta(G_{r,c}) \geq r(c-1)\left(\frac{c+6}{4(c+1)}\right)$. Then, for every $x\in V_c$, $$\sum\limits_{k=0}^{ \lfloor\frac{c-2}{4}\rfloor}{{c-1}\choose {k}} \left(\frac{d^+(x)}{d^-(x)}\right)^k <
\begin{cases}
{{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}\left(\frac{3c-2}{2c-4}\right) \left(\dfrac{d^+(x)}{d^-(x)}\right)^{ \lfloor\frac{c-2}{4}\rfloor} & \hbox{ if } d^+(x)\geq d^-(x); \\
{{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}\left(\frac{3c-2}{2c-4}\right) \left(\dfrac{d^-(x)}{d^+(x)}\right)^{ \lfloor\frac{c-2}{4}\rfloor} & \hbox{ if } d^+(x)<d^-(x) .
\end{cases}$$
Let $\P$ be the set of all the partitions of $G_{r,c}$ in maximal tournaments. For each partition in maximal tournaments, $P$ of $G_{r,c}$ let $\omega (P)$ be the number of vertices $x\in V(G_{r,c})$ such that $\delta_P(x) \leq \lfloor\frac{c-2}{4}\rfloor$ and let $\omega (G_{r,c}) = \sum\limits_{P \in \P} \omega(P)$.
\[main\] Let $G_{r, c}$ be a balanced $c$-partite tournament, with $c\geq 10$, $\alpha=\frac{2\Delta (G_{r,c})}{r(c-1)}$ and $\beta=\frac{2\delta (G_{r,c})}{r(c-1)}$. $G_{r, c}$ has a strong partition if
$$\frac{\omega(G_{r,c})}{((r-1)!)^{c-1} rc} \geq {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}} \left(\frac{3c-2}{2c-4}\right) \left(\frac{\alpha}{\beta}\right)^{ \lfloor\frac{c-2}{4}\rfloor}\left(\frac{r}{2}\right)^{c-1}\left(\alpha^{c-1} + \beta^{c-1}\right)$$
and $$\delta (G_{r,c}) \geq \max\left\{r(c-1)\left(\frac{c+6}{4(c+1)}\right),\left\lfloor\frac{c-2}{4}\right\rfloor (r+\mu) + \max\{\delta_V^+,\delta_V^-\}\right \}.$$
Let $G_{r, c}$ be such that $\delta (G_{r,c}) \geq \max\{ r(c-1)\frac{c+6}{4(c+1)}, \lfloor\frac{c-2}{4}\rfloor(r+\mu) + \max\{\delta^+_V,\delta^-_V \}\}$ and suppose there is no strong partition. For each $x\in V(G_{r,c})$, let $F^+(x)$ (resp. $F^-(x)$ ) be the number of partitions $P$ of $G_{r,c}$ for which $d_P^+(x) \leq \lfloor\frac{c-2}{4}\rfloor$ (resp. $d_P^-(x) \leq \lfloor\frac{c-2}{4}\rfloor$). If follows that $\sum\limits_{x\in V(G_{r,c}) } (F^+(x)+ F^-(x)) = \omega (G_{r,c})$ and, by an average argument, we can notice that there exists $x_0\in V(G_{r,c})$ such that $$\label{m1} F^+(x_0) + F^-(x_0) \geq\frac{\omega (G_{r,c})}{rc}.$$ Notice that each maximal tournament is a member of $((r-1)!)^{c-1}$ partitions $P$ of $G_{r,c}$ in maximal tournaments (using the same argument as in the proof of Lemma 1). Then, $F^+(x_0) = ((r-1)!)^{c-1} T^+(x_0) $ and $F^-(x_0) = ((r-1)!)^{c-1} T^-(x_0)$. Thus, by (\[m1\]), $$\label{m2}
T^+(x_0) + T^-(x_0) \geq \frac{\omega(G_{r,c})}{((r-1)!)^{c-1}rc }.$$
Assume, w.l.o.g, $d^+(x_0)\geq d^-(x_0)$. Since $\delta (G_{r,c}) \geq \lfloor\frac{c-2}{4}\rfloor(r+\mu) + \max\{\delta^+_V,\delta^-_V \}$, by Lemma \[promedio\], $$T^+(x_0)\leq \sum\limits_{k=0}^{\left\lfloor \frac{c-2}{4} \right\rfloor} {{c-1}\choose {k}} \left(\frac{d^+(x_0)}{d^-(x_0)}\right)^k \left(\frac{d^-(x_0)}{c-1}\right)^{c-1}.$$
By hypothesis, $\delta (G_{r,c}) \geq r(c-1)\frac{c+6}{4(c+1)}$, then by Lemma \[suma\]
$$T^+(x_0)< {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}\left(\frac{3c-2}{2c-4}\right) \left(\dfrac{d^+(x_0)}{d^-(x_0)}\right)^{ \lfloor\frac{c-2}{4}\rfloor} \left(\frac{d^-(x_0)}{c-1}\right)^{c-1},$$ and since $\frac{d^+(x_0)}{d^-(x_0)}\leq \frac{\Delta(G_{r,c})}{\delta(G_{r,c})}= \frac{\alpha}{\beta}$ $$T^+(x_0)< {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}\left(\frac{3c-2}{2c-4}\right) \left(\dfrac{\alpha}{\beta}\right)^{ \lfloor\frac{c-2}{4}\rfloor} \left(\frac{d^-(x_0)}{c-1}\right)^{c-1}.$$
Analogously, since $d^+(x_0)\geq d^-(x_0)$, by Lemmas \[promedio\] and \[suma\], we can see that $$T^-(x_0) < {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}\left(\frac{3c-2}{2c-4}\right) \left(\dfrac{\alpha}{\beta}\right)^{ \lfloor\frac{c-2}{4}\rfloor} \left(\frac{d^+(x_0)}{c-1}\right)^{c-1}.$$
Thus, by (\[m2\]), $$\frac{\omega(G_{r,c})}{((r-1)!)^{c-1}rc} < {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}\left(\frac{3c-2}{2c-4}\right) \left(\dfrac{\alpha}{\beta}\right)^{ \lfloor\frac{c-2}{4}\rfloor} \left(\left(\frac{d^-(x_0)}{c-1}\right)^{c-1}+ \left(\frac{d^+(x_0)}{c-1}\right)^{c-1}\right).$$
Finally, since $d^+(x_0) + d^-(x_0) = r(c-1) = \Delta(G_{r,c}) + \delta(G_{r,c})$, with $\Delta(G_{r,c}) \geq d^+(x_0) \geq d^-(x_0)\ge\delta(G_{r,c})$, we see that $$\left(\dfrac{d^-(x_0)}{c-1}\right)^{c-1} + \left(\dfrac{d^+(x_0)}{c-1}\right)^{c-1}\leq \left(\dfrac{\delta(G_{r,c})}{c-1}\right)^{c-1} + \left(\dfrac{\Delta(G_{r,c})}{c-1}\right)^{c-1},$$ and $ \left(\dfrac{\delta(G_{r,c})}{c-1}\right)^{c-1} + \left(\dfrac{\Delta(G_{r,c})}{c-1}\right)^{c-1} =\left(\frac{\beta r}{2}\right)^{c-1} + \left(\frac{\alpha r}{2}\right)^{c-1} = \left(\frac{r}{2}\right)^{c-1}\left(\alpha^{c-1} + \beta^{c-1}\right). $
Therefore, $$\frac{\omega(G_{r,c})}{((r-1)!)^{c-1} rc} < {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}\left(\frac{3c-2}{2c-4}\right) \left(\frac{\alpha}{\beta}\right)^{ \lfloor\frac{c-2}{4}\rfloor}\left(\frac{r}{2}\right)^{c-1}\left(\alpha^{c-1} + \beta^{c-1}\right)$$ and the result follows.
\[unmalo\] Let $G_{r, c}$ be a balanced $c$-partite tournament, with $c\geq 10$, $\alpha=\frac{2\Delta (G_{r,c})}{r(c-1)}$ and $\beta=\frac{2\delta (G_{r,c})}{r(c-1)}$. Then, $G_{r, c}$ has a strong partition if
$$\frac{1}{r} \geq\sqrt{c}\left(\frac{9}{7}\right)\left(\frac{2}{3^\frac{3}{4}}\right)^{c-1} \left(\frac{\alpha}{\beta}\right)^{ \lfloor\frac{c-2}{4}\rfloor}\left(\alpha^{c-1} + \beta^{c-1}\right)$$
and $$\delta (G_{r,c}) \geq \max\left\{r(c-1)\left(\frac{c+6}{4(c+1)}\right),\left\lfloor\frac{c-2}{4}\right\rfloor (r+\mu) + \max\{\delta_V^+,\delta_V^-\}\right \}.$$
Let $G_{r, c}$ be a balanced $c$-partite tournament, with $c\geq 10$, and with no strong partition. For each partition $P$ of $G_{r,c}$, $\omega (P) \geq 1$, because every tournament of order $c$ that is not strongly connected has minimum degree at most $\lfloor\frac{c-2}{4}\rfloor$ and since $P$ is not a strong partition, at least one of its tournaments is not strongly connected. Thus, by Lemma \[particion\], $\omega(G_{r,c}) \geq (r!)^{c-1}$. By the Main Theorem,
$$\frac{(r!)^{c-1}}{((r-1)!)^{c-1} rc} \geq {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}} \left(\frac{3c-2}{2c-4}\right) \left(\frac{\alpha}{\beta}\right)^{ \lfloor\frac{c-2}{4}\rfloor}\left(\frac{r}{2}\right)^{c-1}\left(\alpha^{c-1} + \beta^{c-1}\right).$$ Simplifying this inequality we obtain that $$\label{sinomega}
\frac{1}{rc} \geq {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}} \left(\frac{3c-2}{2c-4}\right) \left(\frac{\alpha}{\beta}\right)^{ \lfloor\frac{c-2}{4}\rfloor}\left(\frac{1}{2}\right)^{c-1}\left(\alpha^{c-1} + \beta^{c-1}\right)$$ We will prove by induction that $$\label{bound}
p(c): {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}} \leq \left(\frac{9}{7\sqrt{c}}\right)\left(\frac{4}{3^\frac{3}{4}}\right)^{c-1} \left(\frac{2c-4}{3c-2}\right)$$ Base cases holds:
$c$ ${{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}$ $\leq$ $\left(\frac{9}{7\sqrt{c}}\right)\left(\frac{4}{3^\frac{3}{4}}\right)^{c-1} \left(\frac{2c-4}{3c-2}\right)$
----- ------------------------------------------------ --------------------------------------------------------------------------------------------------------------------
10 36 36.65
11 45 62.3
12 55 105.05
13 66 180.72
For the inductive step we will prove that $p(c)$ implies $p(c+4)$.
$$\begin{array}{lcl}
{{c+3}\choose{\lfloor\frac{c-2}{4}\rfloor}+1}&=&\frac{c(c+1)(c+2)(c+3)}{(c+2-\lfloor\frac{c-2}{4}\rfloor)(c+1-\lfloor\frac{c-2}{4}\rfloor)(c-\lfloor\frac{c-2}{4}\rfloor)(\lfloor\frac{c-2}{4}\rfloor+1)} {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}\\
& \leq & \frac{4^4}{3^3}\frac{3c(3c+3)}{(3c+13)(3c+5)}{{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}}\\
&\leq & \frac{4^4}{3^3}\frac{3c(3c+3)}{(3c+13)(3c+5)}\left(\frac{9}{7\sqrt{c}}\right)\left(\frac{4}{3^\frac{3}{4}}\right)^{c-1} \left(\frac{2c-4}{3c-2}\right) \\
&=&\left(\frac{4}{3^\frac{3}{4}}\right)^{c+3}\left(\frac{9}{7\sqrt{c}}\right)\frac{3c(3c+3)(2c-4)}{(3c+13)(3c+5)(3c-2)}
\end{array}$$
To complete the induction it is enough to prove that, for $c\geq 10$, $$\left(\frac{1}{\sqrt{c}}\right)\frac{3c(3c+3)(2c-4)}{(3c+13)(3c+5)(3c-2)}\leq \left(\frac{1}{\sqrt{c+4}}\right) \left(\frac{2c+4}{3c+10}\right).$$ This is a consequence of the fact that the real function $$f(c)= \left(\frac{1}{\sqrt{c}}\right)\frac{3c(3c+3)(2c-4)}{(3c+13)(3c+5)(3c-2)}-\left(\frac{1}{\sqrt{c+4}}\right) \left(\frac{2c+4}{3c+10}\right)$$ has no roots for positive $c\geq 10$ and $f(10)<0$, which can be proved by computer.
Therefore, $p(c+4)$ holds. By \[sinomega\] and \[bound\], the result follows.
Notice that if $G_{r,c}$ is regular, $\alpha = \beta = 1$ and since $\delta (G_{r,c}) = \frac{r(c-1)}{2},$ we have the following corollary.
\[regular\] Let $G_{r, c}$ be a regular balanced $c$-partite tournament, with $c\geq 10$. $G_{r, c}$ has a strong partition if
$$\frac{1}{r}\geq\sqrt{c}\left(\frac{18}{7}\right)\left(\frac{2}{3^\frac{3}{4}}\right)^{c-1} .$$
Let $C(r)$ be the number of partite sets such that if $c\geq C(r)$, every balanced multipartite tournament $G_{c,r}$ satisfying the minimum degree condition of the Main Theorem has a strong partition. By the Main Theorem, $C(r)$ is bounded for each non negative integer $r$. In the following table we illustrate the upper bounds of $C(r)$ for regular balanced multipartite tournament given by Corollary \[regular\].
$r$ 2 3 5 10 100
-------------- ---- ---- ---- ---- -----
$C(r) \leq $ 26 30 35 40 60
Proofs of the lemmas {#pruebas lemas}
====================
[**Proof of Lemma \[particion\].** ]{}
Let $G_{r,c}$ be a balanced $c$- partite tournament and let $V_1, \ldots V_c$ its partite sets. Let $V_1 = \{x_1, \dots, x_r\}$ and, given a partition of $G_{r,c}$ in maximal tournaments, let $\tau_1, \tau_2, \ldots \tau_r$ be its set of tournaments. W.l.o.g, assume that for every partition of $G_{r,c}$, the vertex $x_i$ of $V_1$ is a vertex of $\tau_i$. Then, every partition corresponds to a permutation of the vertices in each $V_j$ [(for $2\leq j \leq c$)]{} choosing the $i$-th member of the permutation of $V_j$ to be in $\tau_i$. Since there are $r!$ permutations of each $V_j$, $j\in \{2, 3, \ldots, c \}$, the result follows. ${\ \rule{0.5em}{0.5em}}$\
**Proof of Lemma \[L M(gi)\]**
Let $x\in V(G_{r,c})$. A maximal tournament containing the vertex $x$ with out-degree $k$ can be constructed choosing a vertex for each part $V_i$ for $1\le i\le c-1$ in the following way. Given $\textbf{h}=(h_1,h_2,\dots,h_{c-1})\in \mathcal{H}_k(x)$, we choose an out neighbor of $x$ from $V_i$ if and only if $h_i=1$. The number of maximal tournaments constructed in this way for a fixed $\textbf{h}$ is $\prod_{i=1}^{c-1}d_i^+(x)^{h_i}d_i^-(x)^{1-h_i} $. Therefore, for each $\textbf{h}\in \mathcal{H}_k(x)$, there are $d_i^+(x)^{h_i}d_i^-(x)^{1-h_i}$ ways to construct such a maximal [tournaments]{} and the result follows. ${\ \rule{0.5em}{0.5em}}$\
[**Proof of Lemma \[promedio\].** ]{}
Let $x \in V(G_{r,c})$. Assume w.l.o.g that $x\in V_c$, let $M(d_1^+(x),d_2^+(x),\dots,d_{c-1}^+(x);k)$ be the function that calculates the number of maximal [tournaments]{} for which $x$ has out-degree $k$. Then by Lemma \[L M(gi)\], $$M(d_1^+(x),d_2^+(x),\dots,d_{c-1}^+(x);k)=
\sum_{h\in \mathcal{H}_k(x)}\prod_{i=1}^{c-1}
d_i^+(x)^{h_i}d_i^-(x)^{1-h_i}$$ and $$T^+(x) = \sum\limits_{k=0}^{\lfloor\frac{c-2}{4}\rfloor} M(d_1^+(x), d_2^+(x) \ldots d_{c-1}^+(x);k).$$
To give an upper bound of $T^+(x)$ we need to extend the definition of $M(d_1^+(x),d_2^+ (x)\dots , d_{c-1}^+(x); k)$ to the real numbers, as follows:
Let $g_1,g_2, \ldots,g_{s}$ be real numbers such that $0 \le g_i \leq r$, we define $$M(g_1, \dots , g_s; k)=\sum_{h\in \mathcal{H}_{k,s}}\prod_{i=1}^{s} g_i^{h_i}(r-g_i)^{1-h_i},$$
where $\mathcal{H}_{k,s}$ is the set of $s$-vectors $(h_1,h_2, \ldots,h_s)\in \{0,1\}^s$ such that if $g_i=r$, $h_i=1$; if $g_i=0$, $h_i=0$, and $\sum_{i=1}^sh_i=k$. For the sake of readability, in what follows $M(g_1, \dots , g_s; k)$ can be denoted as $M(g_{[s]}; k).$ In order to prove the Lemma \[promedio\], we will prove the following general version:
\[general\] Let $g_1, g_2, \dots , g_{c-2}, g_{c-1},q$ be real numbers such that $0\leq g_i \leq r$. Let $p_r=|\{i\in \{1,\ldots,c-3\}: g_i=r \}|$, $p_0=|\{i\in \{1,\ldots,c-3\}: g_i=0 \}|$ and $t=c-3-p_r-p_0$ and $p_r+1\leq q.$ Let $\Gamma= max\{ g_i\}_{i\in[c-1]}$ and $\gamma = min \{g_i\}_{i\in[c-1]}$. If $\sum\limits_{i\in [c-1]} g_i \geq q(r+\Gamma-\gamma) + \gamma $, then $$\sum\limits_{k=0}^{q} M(g_{[c-1]}; k) \leq \sum\limits_{k=0}^{q} {{c-1}\choose {k}} (\epsilon)^k (r-\epsilon)^{c-1-k}$$ for $\epsilon = \sum\limits_{i\in [c-1]} \frac{g_i}{c-1}$.
Assume that $g_1,g_2, \ldots, g_{c-1}$ are ordered in the following way:
- $g_{c-1}= \Gamma$, $g_{c-2}= \gamma$,
- for every $i \in [t]$, $0< g_i < r$,
- for every $t+1 \leq i \leq t+p_r$, $g_i = r$,
- for every $t+p_r + 1\leq i \leq t+p_r + p_0 = c-3$, $g_i = 0$.
\[afirmacion\] For every $I\subseteq [t]$ with $|I| = t-(q-p_r) +1$, $\sum\limits_{i\in I} \frac{g_i}{r-g_i}\geq (q-p_r)$.
$$\centering \begin{array}{lcl}
\sum\limits_{i\in [c-1]} g_i & = &\sum\limits_{i\in I} g_i + \sum\limits_{i\in [t]\setminus I} g_i + \sum\limits_{i= t+1}^{t+p_r} g_i + \sum\limits_{i = t+ p_r +1}^{c-3} g_i + (g_{c-2} + g_{c-1})\\
&=& \sum\limits_{i\in I} g_i + \sum\limits_{i\in [t]\setminus I }g_i + rp_r + (g_{c-2} + g_{c-1}),
\end{array}$$ then, $$\centering \begin{array}{lcl}
\sum\limits_{i\in I} g_i &=& \sum\limits_{i\in [c-1]} g_i - \sum\limits_{i\in [t]\setminus I }g_i - rp_r - (g_{c-2} + g_{c-1}).\end{array}$$
Let $Q\geq g_i $ for $i\in [t]\setminus I$. Since $\sum\limits_{i\in [c-1]} g_i \geq q(\Gamma+ r-\gamma) + \gamma$,
$$\begin{array}{lcl}
\sum\limits_{i\in I} g_i &\geq &q(\Gamma+ r-\gamma) + \gamma - Q(q-p_r-1) - rp_r - \Gamma - \gamma \\
& = & q(r-\gamma) + (q-1)(\Gamma -Q) + p_r (Q -r).
\end{array}$$ If $\Gamma < r$, then $p_r = 0$, $Q\leq \Gamma$ and therefore, $\sum\limits_{i\in I} g_i \geq q(r-\gamma)$. If $\Gamma = r$, since $Q\leq r-1$ and $|I| = t-(q-p_r) +1$, $\sum\limits_{i\in I} g_i \geq q(r-\gamma) + (q-1-p_r)(r-Q) \geq q(r-\gamma).$ In both cases, $\sum\limits_{i\in I} g_i \geq q(r-\gamma)$ and since for every $i \in [c-1]$, $g_i \geq \gamma$, $$\sum\limits_{i\in I} \frac{g_i}{r-g_i} \geq \sum\limits_{i\in I} \frac{g_i}{r-\gamma}\geq \frac{q(r-\gamma)}{(r-\gamma)}\geq q$$ and the claim follows.
\[supremo\] For every $I\subseteq [t]$ such that $|I| = t-(q-p_r) +1$, we have that $$\sum\limits_{k=0}^{q} M(g_{[c-1]}; k) \leq \sum\limits_{k=0}^{q} M(g'_{[c-1]}; k)$$ where $g'_{c-2} = g'_{c-1} = \frac{g_{c-2} + g_{c-1}}{2}$; and for $i\in [c-3]$, $g'_i = g_i$;
Observe that for every $k \geq 0$, $$M(g_{[c-1]}; k) =\sum_{j=0}^2 M(g_{[c-3]}; k-j)M(g_{c-2}, g_{c-1}; j).$$ Therefore, for every $q\geq p_r+1$, $$\begin{array}{lcl}
\sum\limits_{k=0}^{q} M(g_{[c-1]}; k) &=& M(g_{[c-3]}; q)M(g_{c-2}, g_{c-1}; 0)\\
& + &M(g_{[c-3]}; q-1)\left[M(g_{c-2}, g_{c-1}; 0)+M(g_{c-2}, g_{c-1}; 1)\right] \\
&+& \sum\limits_{k=0}^{q-2} M(g_{[c-3]}; k)\left[M(g_{c-2}, g_{c-1}; 0)+M(g_{c-2}, g_{c-1}; 1)+M(g_{c-2}, g_{c-1}; 2)\right].
\end{array}$$ Notice that for every pair $x, y \in \mathbb{R}$, such that $0\leq x, y\leq r$,
1. $M(x, y; 0)+M(x, y; 1)+M(x, y; 2) = (r-x)(r-y) + x(r-y) + (r-x)y + xy= r^2$;
2. $M(x, y; 0)+M(x, y; 1)= (r-x)(r-y) + x(r-y) + (r-x)y = r^2 - xy$ y
3. $M(x, y; 0)= r^2 - r(x+y) + xy$.
Since $g_{c-2}+ g_{c-1} = g'_{c-2}+ g'_{c-1}$, we have that $$\begin{array}{lcl}
\sum\limits_{k=0}^{q} M(g'_{[c-1]}; k) - \sum\limits_{k=0}^{q} M(g_{[c-1]}; k)&= &M(g_{[c-3]}; q-1)\left[g_{c-2}g_{c-1}- g'_{c-2}g'_{c-1} \right] \\
&+& M(g_{[c-3]}; q)\left[ g'_{c-2}g'_{c-1} - g_{c-2}g_{c-1} \right]\\
&=& \left( g'_{c-2}g'_{c-1} - g_{c-2}g_{c-1}\right)\left[M(g_{[c-3]}; q) - M(g_{[c-3]}; q-1) \right].
\end{array}$$
Since $g'_{c-2}g'_{c-1} \geq g_{c-2}g_{c-1} $, $\label{cond}\sum\limits_{k=0}^{q} M(g_{[p+2]}; k)\leq \sum\limits_{k=0}^{q} M(g'_{[p+2]}; k)$ if and only if $$\label{cond1}
M(g_{[c-3]}; q-1)\leq M(g_{[c-3]}; q).$$
Let denote by $\mathcal{H}^{c-3}_{q-1} \subseteq \{0,1 \}^{c-3}$, the set of vectors ${\bf h}=(h_1,h_2,\dots,h_{c-3})$ such that if $g_i=r$, $h_i=1$; if $g_i=0$, $h_i=0$, and $\sum_{i=1}^{c-3} h_i=q-1$ .
For each ${\bf h}=(h_1,\ldots. h_{c-3})\in \mathcal{H}^{c-3}_{q-1} $ let $$\label{defF}
F({\bf h}) = \{(h'_1,\dots, h'_{c-3})\in \mathcal{H}^{c-3}_{q} : h_i \leq h'_i \hbox{ for } i \in [c-3]\}$$ [**Claim 2.1**]{} For each ${\bf h}=(h_1,\ldots. h_{c-3})\in \mathcal{H}^{c-3}_{q-1}$.
$$\sum\limits_{\textbf{h'}\in F({\bf h})} \prod\limits_{i=1}^{c-3} g_i^{h'_i} (r-g_i)^{1-h'_i} \geq (q-p_r) \prod\limits_{i=1}^{c-3} g_i^{h_i} (r-g_i)^{1-h_i}.$$
Let $\textbf{h}=(h_1,\dots, h_{c-3})\in \mathcal{H}^{c-3}_{q-1}$ and recall that for every $t+1 \leq j \leq t+p_r$, $g_j = r$; and for every $t+ p_r + 1\leq j \leq t+p_r + p_0 = p$, $g_j = 0$. Therefore, for every $t+1 \leq j \leq t+p_r$, $h_j = 1$ and for every $t+ p_r + 1 \leq j \leq c-3$, $h_j = 0$. By definition, $\sum\limits_{i = 1}^{c-3} h_i = q-1$, therefore w.o.l.g we may assume that $j \in[q-1- p_r]$; $h_i = 1$ and that $q-p_r \leq j \leq t$, $h_i = 0$. Hence, $$\label{unos}
(h'_1,\dots, h'_{c-3})\in F(\textbf{h}) \text{ if and only if } h'_i = 1 \text{ for } i\in[q-1-p_r], \text{ and } \sum\limits_{j=q-p_r}^{t} h'_j =1.$$
Therefore, by \[defF\] and \[unos\],
$$\frac{\sum\limits_{\textbf{h'}\in F(\textbf{h})} \prod\limits_{i=1}^{c-3} g_i^{h'_i} (r-g_i)^{1-h'_i}}{ \prod\limits_{i=1}^{c-3} g_i^{h_i} (r-g_i)^{1-h_i}}= \sum\limits_{\textbf{h'}\in F(\textbf{h})} \frac{\prod\limits_{i=1}^{c-3} g_i^{h'_i} (r-g_i)^{1-h'_i}}{ \prod\limits_{i=1}^{c-3} g_i^{h_i} (r-g_i)^{1-h_i}}=\sum\limits_{j= q-p_r}^{t} \frac{g_j}{r-g_j}.$$
By the hypothesis, $\sum\limits_{j= q-p_r}^{t} \frac{g_j}{r-g_j}\geq (q-p_r)$, then $$\frac{\sum\limits_{\textbf{h'}\in F(\textbf{h})} \prod\limits_{i=1}^{c-3} g_i^{h'_i} (r-g_i)^{1-h'_i}}{ \prod\limits_{i=1}^{p} g_i^{h_i} (r-g_i)^{1-h_i}}\geq \left( q -p_r\right)$$ and the Claim 2.1 follows.
Observe that for every $\textbf{h'}\in \mathcal{H}^{c-3}_{q} $ there are exactly $q-p_r$ elements $\textbf{h}\in \mathcal{H}^{c-3}_{q-1} $ such that, $\textbf{h'} \in F(\textbf{h})$. Therefore, $$\sum\limits_{\textbf{h}\in H^{c-3}_{q-1}} \left(\sum\limits_{\textbf{h'}\in F(\textbf{h})} \prod\limits_{i=1}^{c-3} g_i^{h'_i} (r-g_i)^{1-h'_i}\right) = (q-p_r) \sum\limits_{\textbf{h'}\in H^{c-3}_{q}} \prod\limits_{i=1}^{c-3} g_i^{h'_i} (r-g_i)^{1-h'_i}.$$ On the other hand, by Claim 2.1, $$\sum\limits_{\textbf{h}\in H^{c-3}_{q-1}}\left( \sum\limits_{\textbf{h'}\in F(\textbf{h})} \prod\limits_{i=1}^{c-3} g_i^{h'_i} (r-g_i)^{1-h'_i} \right)\geq \sum\limits_{\textbf{h}\in H^{c-3}_{q-1}} \left( q - p_r \right)\prod\limits_{i=1}^{c-3} g_i^{h_i} (r-g_i)^{1-h_i}$$ implying that $$\sum\limits_{\textbf{h'}\in H^{c-3}_q} \prod\limits_{i=1}^{c-3} g_i^{h'_i} (r-g_i)^{1-h'_i}\geq \sum\limits_{\textbf{h}\in H^{c-3}_{q-1}} \prod\limits_{i=1}^{c-3} g_i^{h_i} (r-g_i)^{1-h_i}$$ which is equivalent to (\[cond1\]). Therefore Claim 2 follows.${\ \rule{0.5em}{0.5em}}$\
By Claim \[afirmacion\] and Claim \[supremo\] we can conclude that $$\sum\limits_{k=0}^{q} M(g_{[c-1]}, k) \leq \sum\limits_{k=0}^{q} M(g'_{[c-1]}, k)$$ for $g_i = g'_i$, with $i\in [c-3]$; and $g'_{c-2} = g'_{c-1} = \frac{g_{c-2} + g_{c-1}}{2}$.
Let $\Gamma' = max\{ g'_i\}_{i\in [c-1]}$ and $\gamma'= min \{g'_i\}_{i\in[c-1]}$. Since $g_{c-1} \geq g'_{c-1} = g'_{c-2} \geq g_{c-2}$, observe that $\Gamma \geq \Gamma'$ $\gamma\leq \gamma'$. Therefore $\sum\limits_{i\in [c-1]} g'_i \geq q(\Gamma+ r-\gamma) + \gamma$, as $\sum\limits_{i\in [c-1]} g'_i = \sum\limits_{i\in [c-1]} g_i \geq q(\Gamma+ r-\gamma) + \gamma $ and $q\geq 1$.
We can iterate this process and find $g''_{1}, \dots , g''_{c-1}$ such that $max\{ g''_i\}_{i\in[c-1]} = min\{ g''_i\}_{i\in[c-1]}$, then, for every $i,j\in [c-1]$, $g''_i = g''_j$. Hence, for every $i\in [c-1]$, $g''_i = \frac{\sum\limits_{i\in [c-1]} g''_i}{c-1} = \alpha$, and for each $q\geq k \geq 0$, $$M(g''_{[c-1]}; k) = \sum\limits_{\textbf{h}\in \mathcal{H}^{c-1}_k} \prod\limits_{i=1}^{c-1} (g''_i)^{h_i} (r-g''_i)^{1-h_i} =$$ $$\sum\limits_{\textbf{h}\in \mathcal{H}^{c-1}_k} \prod\limits_{i=1}^{c-1} \alpha^{h_i} (r-\epsilon)^{1-h_i} =\sum\limits_{\textbf{h}\in \mathcal{H}^{c-1}_k} \alpha^k (r-\epsilon) ^{c-1-k}= {{c-1}\choose {k}} \alpha^k (r-\epsilon) ^{c-1-k}$$ and the Proposition \[general\] follows.
Let $x\in V(G_{r,c})$ and $q= \lfloor\frac{c-2}{4}\rfloor$. Suppose, w.o.l.g., $x\in V_c$. Let $\Gamma = \Delta^+_V(x) = d^+_{c-1}(x)$; $\gamma = \delta^+_V(x) = d^+_{c-2}(x)$; $p_r=|\{i\in \{1,\ldots,c-3\}: d^+_i(x)=r \}|$, $p_0=|\{i\in \{1,\ldots,c-3\}: d^+_i(x)=0 \}|$ and $t=c-3-p_r-p_0$. Observe that if $\Gamma < r$ then $p_r = 0$, and if $\Gamma = r$, the vertex $x$ has out-degree at least $p_r + 1$ in every maximal tournament. If $ \lfloor\frac{c-2}{4}\rfloor < p_r +1$, $T^+(x) = 0$. Thus, we can suppose that $q \geq p_r+1$. Since $\mu \geq \Delta^+_V(x) - \delta^+_V(x) = \Gamma - \gamma$ and $ max \{\delta^+_V, \delta^-_V\} \geq \delta^+_V(x) = \gamma$, $\sum\limits_{i\in [c-1]} d^+_i(x) \geq \delta(G_{r,c}) \geq q(r+\Gamma-\gamma) + \gamma$. By Proposition \[general\], $$T^+(x) = \sum\limits_{k=0}^{\lfloor\frac{c-2}{4}\rfloor} M(d^+(x)_{[c-1]}; k) \leq \sum\limits_{k=0}^{\lfloor\frac{c-2}{4}\rfloor} {{c-1}\choose {k}} (\epsilon)^k (r-\epsilon)^{c-1-k}$$ for $\epsilon = \sum\limits_{i\in [c-1]} \frac{d_i^+(x)}{c-1} = \frac{d^+(x)}{c-1}$ and the lemma follows. ${\ \rule{0.5em}{0.5em}}$\
[**Proof of Lemma \[suma\].** ]{} Let $G_{r,c}$ be a balanced $c$-partite tournament such that $\delta(G_{r,c}) \geq r(c-1)\frac{c+6}{4(c+1)}$, and let $x\in V(G_{r,c})$. In what follows, let $p = d^+(x)$ and $m=d^-(x)$. Observe that since $\delta(G_{r,c}) \geq r(c-1)\frac{c+6}{4(c+1)}$ and $r(c-1) = p+m$, $p \geq (p+m)\frac{c+6}{4(c+1)}$. Multiplying the [previous]{} inequality by $(c+1)$ and adding $-p\frac{c+6}{4}$ we obtain that $p\frac{3c-2}{4}\geq m\frac{c+6}{4}$. Since $\frac{c+6}{4}\geq \lfloor\frac{c-2}{4}\rfloor+2$ and $\frac{3c-2}{4}\leq c-1- \lfloor\frac{c-2}{4}\rfloor$ it follows that
$$\label{pvsm}
p\left(c-1-\left\lfloor\frac{c-2}{4}\right\rfloor\right) \geq m \left(\left\lfloor\frac{c-2}{4}\right\rfloor+2\right)$$
[**Claim.**]{} $\sum\limits_{k=0}^{ \lfloor\frac{c-2}{4}\rfloor}{{c-1}\choose {k}} \left(\frac{p}{m}\right)^k < {{c-1}\choose {\lfloor\frac{c-2}{4}\rfloor}} \left(\frac{p}{m}\right)^{\lfloor\frac{c-2}{4}\rfloor} \frac{p(c-1-\lfloor\frac{c-2}{4}\rfloor)}{p(c-1-\lfloor\frac{c-2}{4}\rfloor) - m(\lfloor\frac{c-2}{4}\rfloor+1)}$
For each integer $q\geq 0$, let $g(q) = \sum\limits_{k=0}^q {{c-1}\choose {k}} \left(\frac{p}{m}\right)^k$. Observe that, $$\begin{array}{lcl} g(q+1) = 1 + \sum\limits_{k=1}^{q+1} {{c-1}\choose {k}} \left(\frac{p}{m}\right)^k
&=& 1 + \sum\limits_{k=0}^{q} {{c-1}\choose {k+1}} \left(\frac{p}{m}\right)^{k+1} \\
& = & 1 + \frac{p}{m} \sum\limits_{k=0}^{q} {{c-1}\choose {k+1}} \left(\frac{p}{m}\right)^{k} \\
& = & 1 + \frac{p}{m} \sum\limits_{k=0}^{q} {{c-1}\choose {k}}\frac{c-1-k}{k+1} \left(\frac{p}{m}\right)^{k} \\
\end{array}$$ Notice that for each $k\leq q$, $\frac{c-1-q}{q+1}\leq \frac{c-1-k}{k+1}$ therefore
$$\begin{array}{lcl}
g(q+1) &\geq& 1 + \frac{p}{m} \sum\limits_{k=0}^{q} {{c-1}\choose {k}}\frac{c-1-q}{q+1} \left(\frac{p}{m}\right)^{k} \\
&=& 1+ \frac{p}{m}\frac{c-1-q}{q+1} \sum\limits_{k=0}^{q} {{c-1}\choose {k}} \left(\frac{p}{m}\right)^{k} \\
& >& \frac{p}{m}\frac{c-1-q}{q+1}g(q).
\end{array}$$
On the other hand, $g(q+1)= g(q) + {{c-1}\choose {q+1}} \left(\frac{p}{m}\right)^{q+1}$, and therefore $$g(q) + {{c-1}\choose {q+1}} \left(\frac{p}{m}\right)^{q+1}> \frac{\frac{p}{m}(c-1-q)}{q+1} g(q).$$
For $\lfloor\frac{c-2}{4}\rfloor = q$, since ${{c-1}\choose { q+1}} = {{c-1}\choose { q} }\frac{c-1-q}{q+1}$, multiplying the inequality by $m(q+1)$, adding $-m(q+1)g(q)$ and dividing by $p(c-1-q)-m(q+1)$ (which by (\[pvsm\]) is positive ), we obtain $${{c-1}\choose {q}} \left(\frac{p}{m}\right)^{q} \frac{p(c-1-q)}{p(c-1-q) - m(q+1)}> g(q)$$ and from here the claim follows.
By the Claim, it only remains to prove that
$$\left(\frac{p}{m}\right)^{\lfloor\frac{c-2}{4}\rfloor} \frac{p(c-1-\lfloor\frac{c-2}{4}\rfloor)}{p(c-1-\lfloor\frac{c-2}{4}\rfloor) - m(\lfloor\frac{c-2}{4}\rfloor+1)} <
\begin{cases}
\left(\frac{3c-2}{2c-4}\right) \left(\dfrac{p}{m}\right)^{ \lfloor\frac{c-2}{4}\rfloor} & \hbox{ if } p\geq m; \\
\left(\frac{3c-2}{2c-4}\right) \left(\dfrac{m}{p}\right)^{ \lfloor\frac{c-2}{4}\rfloor} & \hbox{ if } p<m.
\end{cases}$$
If $p\geq m$ then,
$$\begin{array}{lcl}
p\left(c-1-\left\lfloor\frac{c-2}{4}\right\rfloor\right) - m\left(\left\lfloor\frac{c-2}{4}\right\rfloor+1\right) &\geq& p\left(c-1-\left\lfloor\frac{c-2}{4}\right\rfloor\right) - p\left(\left\lfloor\frac{c-2}{4}\right\rfloor+1\right)\\
& =& p\left(c-2-2\left\lfloor\frac{c-2}{4}\right\rfloor\right)
\end{array}$$ thus, $$\frac{p(c-1-\lfloor\frac{c-2}{4}\rfloor)}{p(c-1-\lfloor\frac{c-2}{4}\rfloor) - m(\lfloor\frac{c-2}{4}\rfloor+1)} \leq \frac{c-1-\lfloor\frac{c-2}{4}\rfloor}{c-2-2\lfloor\frac{c-2}{4}\rfloor}\leq \frac{3c-2}{2c-4}.$$
For the case when $m>p$, let us suppose on the contrary, that $$\left(\frac{p}{m}\right)^{\lfloor\frac{c-2}{4}\rfloor} \frac{p(c-1-\lfloor\frac{c-2}{4}\rfloor)}{p(c-1-\lfloor\frac{c-2}{4}\rfloor) - m(\lfloor\frac{c-2}{4}\rfloor+1)} \geq \left(\frac{3c-2}{2c-4}\right) \left(\dfrac{m}{p}\right)^{ \lfloor\frac{c-2}{4}\rfloor}.$$ Multiplying by $ \left(\frac{\frac{1}{p}}{\frac{1}{p}}\right) \left(\dfrac{m}{p}\right)^{ \lfloor\frac{c-2}{4}\rfloor}$ both sides of the inequality we obtain that $$\label{eqmp}\frac{c-1-\lfloor\frac{c-2}{4}\rfloor}{(c-1-\lfloor\frac{c-2}{4}\rfloor) - \frac{m}{p}(\lfloor\frac{c-2}{4}\rfloor+1)} \geq \left(\frac{3c-2}{2c-4}\right)\left(\dfrac{m}{p}\right)^{ 2\lfloor\frac{c-2}{4}\rfloor}.$$
On the one hand, since $\frac{p}{m}\geq \frac{ \lfloor\frac{c-2}{4}\rfloor+2}{c-1- \lfloor\frac{c-2}{4}\rfloor}$ it follows that $\frac{m}{p}\leq \frac{ c-1- \lfloor\frac{c-2}{4}\rfloor}{\lfloor\frac{c-2}{4}\rfloor+2}$ and therefore,
$$\begin{array}{lcl}
(c-1-\lfloor\frac{c-2}{4}\rfloor) - \frac{m}{p}(\lfloor\frac{c-2}{4}\rfloor+1) &\geq& (c-1-\lfloor\frac{c-2}{4}\rfloor) - \frac{ (c-1-\lfloor\frac{c-2}{4}\rfloor)(\lfloor\frac{c-2}{4}\rfloor+1)}{\lfloor\frac{c-2}{4}\rfloor+2} \\
&=& (c-1- \lfloor\frac{c-2}{4}\rfloor)(1-\frac{\lfloor\frac{c-2}{4}\rfloor+1}{\lfloor\frac{c-2}{4}\rfloor+2} )= \frac{c-1- \lfloor\frac{c-2}{4}\rfloor}{\lfloor\frac{c-2}{4}\rfloor+2}.
\end{array}$$
Then, $$\frac{c-1-\lfloor\frac{c-2}{4}\rfloor}{(c-1-\lfloor\frac{c-2}{4}\rfloor) - \frac{m}{p}(\lfloor\frac{c-2}{4}\rfloor+1)} \leq \frac{c-1-\lfloor\frac{c-2}{4}\rfloor}{\frac{c-1- \lfloor\frac{c-2}{4}\rfloor}{\lfloor\frac{c-2}{4}\rfloor+2}}=\left\lfloor\frac{c-2}{4}\right\rfloor+2$$ and therefore, using inequality [(\[eqmp\])]{}, $$\left\lfloor\frac{c-2}{4}\right\rfloor+2\geq \left(\frac{3c-2}{2c-4}\right) \left(\dfrac{m}{p}\right)^{ 2\lfloor\frac{c-2}{4}\rfloor} \geq \frac{3}{2} \left(\dfrac{m}{p}\right)^{ 2\lfloor\frac{c-2}{4}\rfloor}.$$
Since $c\geq 10$, it follows that $\frac{m}{p}\leq 1.28$.
On the other hand, since $\frac{m}{p}> 1$ it follows that $$\frac{c-1-\lfloor\frac{c-2}{4}\rfloor}{(c-1-\lfloor\frac{c-2}{4}\rfloor) - \frac{m}{p}(\lfloor\frac{c-2}{4}\rfloor+1)} \geq \left(\frac{3c-2}{2c-4}\right) \frac{m}{p}.$$ Multiplying both sides of the inequality by $\frac{(c-1-\lfloor\frac{c-2}{4}\rfloor) - \frac{m}{p}(\lfloor\frac{c-2}{4}\rfloor+1)}{c-1-\lfloor\frac{c-2}{4}\rfloor} = 1 - \frac{m}{p}\left(\frac{\lfloor\frac{c-2}{4}\rfloor+1}{c-1-\lfloor\frac{c-2}{4}\rfloor}\right) $ we obtain that $$1 \geq \left(\frac{3c-2}{2c-4}\right) \frac{m}{p} - \left(\frac{3c-2}{2c-4}\right) \left(\frac{m}{p}\right)^2 \left(\frac{\lfloor\frac{c-2}{4}\rfloor+1}{c-1-\lfloor\frac{c-2}{4}\rfloor}\right)$$ and therefore $$\left(\frac{3c-2}{2c-4}\right) \left(\frac{m}{p}\right)^2 \left(\frac{\lfloor\frac{c-2}{4}\rfloor+1}{c-1-\lfloor\frac{c-2}{4}\rfloor}\right) - \left(\frac{3c-2}{2c-4}\right) \frac{m}{p} +1 \geq 0.$$ Since $\frac{\lfloor\frac{c-2}{4}\rfloor+1}{c-1-\lfloor\frac{c-2}{4}\rfloor} \leq \frac{c+2}{3c-2}$ we see that $$\left(\frac{3c-2}{2c-4}\right) \left(\frac{m}{p}\right)^2 \left(\frac{c+2}{3c-2}\right) - \left(\frac{3c-2}{2c-4}\right) \frac{m}{p} +1 = \left(\frac{c+2}{2c-4}\right) \left(\frac{m}{p}\right)^2 - \left(\frac{3c-2}{2c-4}\right) \frac{m}{p} +1 \geq 0.$$
Thus, using the quadratic formula, $\frac{m}{p} \geq \frac{4c-8}{2(c+2)}$ or $\frac{m}{p}\leq \frac{2c+4}{2(c+2)}= 1$. Since $m> p$ it follows that $\frac{m}{p} \geq \frac{4c-8}{2(c+2)}$ and since $c\geq 10$, $\frac{m}{p}\geq \frac{16}{12} = 1.33$ which is a contradiction because $\frac{m}{p}\leq 1.28$. From here, the result follows. ${\ \rule{0.5em}{0.5em}}$
Final remarks
=============
The problem of finding interesting sufficient conditions in order to guarantee a strong partition of multipartite tournaments [in general]{} seems to be much more complicated and probably would need different techniques. Let $\mathcal{A}(r,c)$ be the set of $r$-balanced $c$-partite tournaments with no strong partition and $\Omega(r,c)=\min \{\omega(G_{r,c}): G_{r,c}\in \mathcal{A}(r,c)\}$. Notice that better lower bounds of $\Omega(r,c)$ leads, by the Main Theorem, to better sufficient conditions for having strong partitions. For now, our corollaries assume that for every partition of a multipartite tournament of $\mathcal{A}(r,c)$ there exists exactly one vertex with in-degree or out-degree at most $\left\lfloor\frac{c-2}{4}\right\rfloor$, which we think its not a realistic approximation, and thus one may find a better way to estimate $\Omega(r,c)$.
[9]{}
J. Bang-Jensen, G. Gutin, [ Digraphs: Theory, Algorithms and Applications]{}, Springer, London, 2001.
A.P. Figueroa, J. J.Montellano Ballesteros, M. Olsen, Strong subtournaments and cycles of multipartite tournaments, Discrete Math **339** (2016), 2793–2803.
L. Volkmann, Strong subtournaments of multipartite tournaments, Australas J Combin **20** (1999), 189–196.
L. Volkmann, S. Winzen, Almost regular c-partite tournaments contain a strong subtournament of order c when $c \geq 5$, Discrete Math **308** (2008), 1710–1721.
G. Xu, S. Li, H. Li, Q. Guo, Strong subtournaments of order image containing a given vertex in regular $c$-partite tournaments with $c\ge 16$, Discrete Math **311** (2011), 2272–2275.
[^1]: Departamento de Matemáticas ITAM, México. email: apaulinafg@gmail.com
[^2]: Instituto de Matemáticas, UNAM, México, email: juancho@im.unam.mx
[^3]: Departamento de Matemáticas Aplicadas y Sistemas, UAM-C, México. email:olsen@correo.cua.uam.mx
[^4]: Research supported by PAPIIT-México under project IN104915.
|
---
abstract: |
A word-to-word function is continuous for a class of languages $\cV$ if its inverse maps $\cV$\_languages to $\cV$. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes.
Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. Previous algebraic studies of transducers have focused on the structure of the underlying input automaton, disregarding the output. We propose a comparison of the two algebraic approaches through two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses?
address:
- University of Oxford
- 'IRIF, Université Paris Diderot'
- Université de Lille
author:
- Michaël Cadilhac
- Olivier Carton
- Charles Paperman
bibliography:
- 'bib.bib'
title: Continuity and Rational Functions
---
\[toc\]
Introduction
============
The algebraic theory of regular languages is tightly interwoven with fundamental questions about the computing power of Boolean circuits and logics. The most famous of these braids revolves around $\vA$, the class of *aperiodic* or *counter-free* languages. Not only is it expressed using the logic $\text{FO}[<]$, but it can be seen as the basic building block of $\ACz$, the class of languages recognized by circuit families of polynomial size and constant depth, this class being in turn expressed by the logic $\text{FO}[\text{arb}]$ (see [@straubing94] for a lovely account). This pervasive interaction naturally prompts to lift this study to the functional level, hence to *rational functions*. This was started in [@cadilhac-krebs-ludwig-paperman15], where it was shown that a subsequential (i.e., input-deterministic) transducer computes an $\ACz$ function iff it preserves the regular languages of $\ACz$ by inverse image. Buoyed by this clean, semantic characterization, we wish to further investigate this latter property for different classes: say that a function $f\colon A^* \to B^*$ is $\cV$\_continuous, for a class of languages $\cV$, if for every language $L \subseteq B^*$ of $\cV$, the language $f^{-1}(L)$ is also a language of $\cV$. Our main focus will be on deciding $\cV$\_continuity for rational functions; before listing our main results, we emphasize two additional motivations.
First, there has been some historical progression towards this goal. Noting, in [@pin-sakarovitch82], that inverse rational functions provide a uniform and compelling view of a wealth of natural operations on regular languages, Pin and Sakarovitch initiated in [@pin-sakarovitch85] a study of regular-continuous functions. It was already known at the time, by a result of Choffrut (see [@berstel79 Theorem 2.7]), that regular-continuity together with some uniform continuity property *characterize* functions computed by subsequential transducers. This characterization was instrumental in the study of [@reutenaeur-schutzenberger95], who already noticed the peculiar link between uniform continuity for some distances on words and continuity for certain classes of languages. This link was tightened by Pin and Silva [@pin-silva05] who formalized a topological approach and generalized it to rational relations. More recently [@pin-silva11], the same authors made precise the link unveiled by , and developed a fascinating and robust framework in which language continuity has a topological interpretation (see the beginning of Section \[sec:contapp\], as we build upon this theory). Pin and Silva [@pin-silva17] notably proposed thereafter a study of functions for which continuity for a class is propagated to subclasses. In addition, Daviaud et al. [@DaviaudRT16; @DaviaudJRV17] recently explored continuity notions in the spirit of Choffrut’s characterization to study weighted automata and cost-register automata.
Second, the interweaving between languages, circuits, and logic that was alluded to previously can in fact be formally stated (see again [@straubing94; @tesson-therien07]). A central property towards this formalization is the correspondence between “cascade products” of automata, stacking of circuits, and nesting of formulas, respectively. Strikingly, these operations can all be seen as inverse rational functions [@tesson-therien07]. These operations are intrinsic in the construction of complex objects: languages, circuits, and formulas are often given as a sequence of simple objects to be composed (see, e.g., [@schneider04 Section 5.5]). We remark that a sufficient condition for the result of the composition to be in some given class (of languages, circuits, or logic formulas), is that each rational function be continuous for that class. Hence deciding continuity allows to give a sufficient condition for this membership question *without* computing the result of the composition, which is subject to combinatorial blowup.
Here, we report on three questions, the first two relating continuity to the main other algebraic approach to transducers, while allowing a more gentle introduction to the evaluation of *profinite words* by transducers:
- When is the transducer *structure* (i.e., its so-called *transition monoid*) impacting its continuity? The results of [@reutenaeur-schutzenberger95] can indeed be seen as the starting point of two distinct algebraic theories for rational functions; on the one hand, the study of continuity, and on the other the study of the transition monoid of the transducer (by disregarding the output). This latter endeavor was carried by [@filiot-gauwin-lhote16]. Loosely speaking, we show, in Section \[sec:struct\]:
Let $\cV$ be a variety of languages among $\vJ$, $\vR$, $\vL$, $\vDA$, $\vA$, $\vCom$, $\vAb$, $\vGnil$, $\vGsol$, or $\vG$.
- The statement “Any rational function *structurally* in $\cV$ is continuous for $\cV$” holds for $\cV \in \{\vA, \vGsol, \vG\}$ and does not otherwise;
- The statement “Any rational function continuous for $\cV$ is *structurally* in $\cV$” holds for $\cV \in \{\vGnil, \vGsol, \vG\}$ and does not otherwise.
- What is the impact of *variety inclusion* on the inclusion of the related classes of continuous rational functions? When the focus is solely on the structure of the transducer, there is a natural propagation to superclasses; when is it the case for continuity? We show, in Section \[sec:inc\]:
Let $\cV$ and $\cW$ be two different varieties of languages among $\vJ$, $\vR$, $\vL$, $\vDA$, $\vA$, $\vCom$, $\vAb$, $\vGnil$, $\vGsol$, or $\vG$. The statement “all rational functions continuous for $\cV$ are continuous for $\cW$” holds only when one of these properties is verified:
- $\cV, \cW \in \{\vGnil, \vGsol, \vG\}$ and $\cV \subseteq \cW$;
- $\cV = \vAb$ and $\cW = \vCom$;
- $\cV = \vDA$ and $\cW = \vA$.
- When is $\cV$\_continuity decidable for rational functions? We show, in Section \[sec:dec\]:
Let $\cV$ be a variety of languages among $\vJ$, $\vR$, $\vL$, $\vDA$, $\vA$, $\vCom$, $\vAb$, $\vGsol$, or $\vG$. It is decidable, given an unambiguous rational transducer, whether it realizes a function continuous for $\cV$.
This constitutes our main contribution; note that the case $\vGnil$ is left open.
Preliminaries
=============
We assume some familiarity with the theory of automata and transducers, and concepts related to metric spaces (see, e.g., [@berstel79; @pin] for presentations pertaining to our topic). Apart from these prerequisites, for which the notation is first settled, the presentation is self-contained.
We will use $A$ and $B$ for alphabets, and $A^*$ for words over $A$, with $1$ the empty word. For each word $u$, there is a smallest $v$, called the *primitive root* of $u$, such that $u=v^c$ for some $c$; if $c=1$, then $u$ is itself *primitive*. We write $|u|$ for the length of a word $u \in A^*$ and $\cts(u)$ for the set of letters that appear in $u$. For a word $u \in A^*$ and a language $L \subseteq A^*$, we write $u^{-1}L$ for $\{v \mid u\cdot v \in L\}$, and symmetrically for $Lu^{-1}$, these two operations being called the left and right quotients of $L$ by $u$, respectively. We naturally extend concatenation and quotients to relations, in a component-wise fashion, e.g., for $R \subseteq A^* \times A^*$ and a pair $\rho \in A^* \times A^*$, we may use $\rho^{-1}R$ and $R\rho^{-1}$. We write $L\comp$ for the complement of $L$. A *variety* is a mapping $\cV$ which associates with each alphabet $A$ a set $\cV(A^*)$ of regular languages closed under the Boolean operations and quotient, and such that for any morphism $h\colon A^* \to B^*$ and any $L \in \cV(B^*)$, it holds that $h^{-1}(L) \in \cV(A^*)$. Reg is the variety that maps every alphabet $A$ to the set $\text{Reg}(A^*)$ of regular languages over $A$. Given two languages $K, L \subseteq A^*$, we say that they are $\cV$\_separable if there is a $S \in \cV(A^*)$ such that $K \subseteq S$ and $L \cap S = \emptyset$.
### Transducers. {#transducers. .unnumbered}
A transducer $\tau$ is a 9-tuple $(Q, A, B, \delta, I, F, \lambda, \mu, \rho)$ where $(Q, A, \delta, I, F)$ forms an automaton (i.e., $Q$ is a state set, $A$ an input alphabet, $\delta \subseteq Q \times A \times Q$ a transition set, $I \subseteq Q$ a set of initial states, and $F \subseteq Q$ a set of final states), and additionally, $B$ is an output alphabet and $\lambda\colon I \to B^*, \mu\colon \delta \to B^*, \rho\colon F\to B^*$ are the output functions. We write $\Tif{\tau, q, q'}$ for $\tau$ with $I := \{q\}$ and $F := \{q'\}$, adjusting $\lambda$ and $\rho$ to output $1$ if they were undefined on these states. Similarly, $\Tif{\tau, q, \nop}$ is $\tau$ with $I := \{q\}$ and $F$ unchanged, and symmetrically for $\Tif{\tau, \nop, q}$. For $q \in Q$ and $u \in A^*$, we write $q.u$ for the set of states reached from $q$ by reading $u$. We assume that all the transducers and automata under study have no useless state, that is, that all states appear in some accepting path.
With $w \in A^*$, let $t_1t_2\cdots t_{|w|} \in \delta^*$ be an accepting path for $w$, starting in a state $q \in I$ and ending in some $q' \in F$. The output of this path is $\lambda(q)\mu(t_1)\mu(t_2)\cdots\mu(t_n)\rho(q')$, and we write $\tau(w)$ for the set of outputs of such paths. We use $\tau$ for both the transducer and its associated partial function from $A^*$ to subsets of $B^*$. Relations of the form $\{(u, v) \mid v \in \tau(u)\}$ are called *rational relations*.
The transducer $\tau$ is *unambiguous* if there is at most one accepting path for each word. In that case $\Tif{\tau, q, q'}$ is also an unambiguous transducer for any states $q, q'$. When $\tau$ is unambiguous, it realizes a word-to-word function: the set of functions computed by unambiguous transducers is the set of *rational functions*. Further restricting, if the underlying automaton is deterministic, we say that $\tau$ is *subsequential*. If $\tau$ is a finite union of subsequential rational functions of disjoint domains, we say that $\tau$ is *plurisubsequential*.
### Word distances, profinite words. {#word-distances-profinite-words. .unnumbered}
For a variety $\cV$ of regular languages, we define a distance between words for which, intuitively, two words are close if it is hard to separate them with $\cV$ languages. Define $d_\cV(u, v)$, for words $u, v \in A^*$, to be $2^{-r}$ where $r$ is the size of the smallest automaton that recognizes a language of $\cV(A^*)$ that separates $\{u\}$ from $\{v\}$; if no such language exists, then $d_\cV(u, v) = 0$. It can be shown that this distance is a *pseudo-ultrametric* [@pin Section VII.2]; we make only implicit and innocuous use of this fact.
The complete metric space that is the completion of $(A^*, d_{\text{Reg}})$ is denoted $\pro{A^*}$ and is called the *free profinite monoid*, its elements being the *profinite words*, and the concatenation being naturally extended. By definition, if $(u_n)_{n>0}$ is a Cauchy sequence, it should hold that for any regular language $L$, there is a $N$ such that either all $u_n$ with $n > N$ belong to $L$, or none does. For any $x \in A^*$, define the profinite word $x^\omega = \lim x^{n!}$, and more generally, $x^{\omega-c} = \lim x^{n!-c}$. That $(x^{n!})_{n>0}$ is a Cauchy sequence is a starting point of the profinite theory [@pin Proposition VI.2.10]; it is also easily checked that $x^{c \times \omega} = \lim x^{c\times n!}$ is equal to $x^\omega$ for any integer $c \geq 1$. Given a language $L \subseteq A^*$, we write $\clos{L} \subseteq \pro{A^*}$ for its closure, and we note that if $L$ is regular, $\clos{L}\comp = \clos{L\comp}$ and for $L'$ regular, $\clos{L \cup L'} = \clos{L} \cup \clos{L'}$, and similarly for intersection (see [@pin Theorem VI.3.15]).
### Equations. {#equations. .unnumbered}
For $u, v \in \pro{A^*}$, a language $L \subseteq A^*$ *satisfies the (profinite) equation* $u = v$ if for any words $s, t \in A^*$, $[s\cdot u\cdot t \in \clos{L} \Leftrightarrow s \cdot v \cdot t \in \clos{L}]$. Similarly, a class of languages satisfies an equation if all the languages of the class satisfy it. For a variety $\cV$, we write $u =_\cV v$, and say that $u$ is equal to $v$ in $\cV$, if $\cV(A^*)$ satisfies $u = v$. For a partial function $f$, $f(u) =_\cV f(v)$ means that either both $f(u)$ and $f(v)$ are undefined, or they are both defined and equal in $\cV$.
Given a set $E$ of equations over $\pro{A^*}$, the class of languages |defined| by $E$ is the class of languages over $A^*$ that satisfy all the equations of $E$. Reiterman’s theorem shows in particular that for any variety $\cV$ and any alphabet $A$, $\cV(A^*)$ is defined by a set of equations (the precise form of which being studied in [@gehrke-grigorieff-pin08]).
### More on varieties. {#more-on-varieties. .unnumbered}
Borrowing from Almeida and Costa [@almeida-costa17], we say that a variety $\cV$ is *supercancellative* when for any alphabet $A$, any $u, v \in \pro{A^*}$ and $x, y \in A$, if $u\cdot x =_\cV v \cdot y$ or $x \cdot u =_\cV y \cdot v$, then $u =_\cV v$ and $x = y$. This implies in particular that for any word $w \in A^*$, both $w\cdot A^*$ and $A^* \cdot w$ are in $\cV(A^*)$. We further say that a variety $\cV$ *separates words* if for any $s, t \in A^*$, $\{s\}$ and $\{t\}$ are $\cV$\_separable.
Our main applications revolve around some classical varieties, that we define over any possible alphabet $A$ as follows, where $x, y$ range over all of $A^*$, and $a, b$ over $A$:\
- $\vJ$, $(xy)^\omega \cdot x = y \cdot (xy)^\omega = (xy)^\omega$
- $\vR$, $(xy)^\omega \cdot x = (xy)^\omega$
- $\vL$, $y \cdot (xy)^\omega = (xy)^\omega$
- $\vDA$, $x^\omega\cdot z \cdot x^\omega = x^\omega$ for all $z \in \cts(x)^*$
- $\vA$, $x^{\omega +1} = x^\omega$
------------------------------------------------------------------------
- $\vCom$, $ab = ba$
- $\vAb$, $ab = ba$ and $a^\omega = 1$
- $\vGnil$, the languages rec. by nilpotent groups
- $\vGsol$, the languages rec. by solvable groups
- $\vG$, the languages rec. by groups
The varieties included in $\vA$ are called *aperiodic varieties* and those in $\vG$ are called *group varieties*. Precise definitions, in particular for the group varieties, can be found in [@straubing94; @pin-weil96]; we simply note that in group varieties, $x^\omega$ equals $1$ for all $x \in A^*$. All these varieties except for $\vAb$ and $\vCom$ separate words, and only $\vDA$ and $\vA$ are supercancellative. They verify:
### On transducers and profinite words. {#on-transducers-and-profinite-words. .unnumbered}
For a profinite word $u$ and a state $q$ of an unambiguous transducer $\tau$, the set $q.u$ is well defined; indeed, with $u = \lim u_n$, the set $q.u_n$ is eventually constant, as otherwise for some state $q'$, the domain of $\Tif{\tau, q, q'}$ would be a regular language that separates infinitely many $u_n$’s.
A transducer $\tau\colon A^* \to B^*$ is a *$\cV$\_transducer*,[^1] for a variety $\cV$, if for some set of equations $E$ defining $\cV(A^*)$, for all $(u = v) \in E$ and all states $q$ of $\tau$, it holds that $q.u = q.v$. A rational function is *$\cV$\_realizable* if it is realizable by a $\cV$\_transducer.
### Continuity. {#continuity. .unnumbered}
For a variety $\cV$, a function $f\colon A^* \to B^*$ is $\cV$\_continuous[^2] iff for any $L \in \cV(B^*)$, $f^{-1}(L) \in \cV(A^*)$. We mostly restrict our attention to rational functions, and their being computed by transducers implies that they are countably many. We note that much more Reg\_continuous functions exist, in particular uncomputable ones:
There are uncountably many Reg\_continuous functions.
Consider a strictly increasing function $g\colon \bbn \to \bbn$. Define $f\colon \{a\}^* \to \{a\}^*$ by $f(a^n) = a^{g(n)!}$. Recall that any regular language over a unary alphabet is a finite union of languages of the form $a^i(a^j)^*$. Moreover, we have that $f^{-1}(a^i(a^j)^*)$ is finite when $i \not\equiv 0 \bmod j$, and cofinite otherwise, thus $f$ is Reg\_continuous. There are however uncountably many functions $g$, hence uncountably many Reg\_continuous functions $f$.
Continuity is a formal notion of “functions being compatible with a class of languages.” An equally valid notion could be to consider classes of functions that contain the characteristic functions of the languages, and closed under composition; it turns out that the largest such class coincides with the class of continuous functions. Indeed, writing $\chi_L\colon A^* \to \{0, 1\}$ for the characteristic function of a language $L \subseteq A^*$:
Let $\cV$ be a variety such that $\{1\} \in \cV(\{0, 1\}^*)$. Let $\cF$ be the *largest* class of functions such that:
1. For any alphabet $A$, $\cF \cap \{0, 1\}^{A^*} = \{\chi_L \mid L \in \cV(A^*)\};$
2. $\cF$ is closed under composition.
It holds that $\cF$ is well defined and that it coincides with the class of $\cV$\_continuous functions.
We say that a class of functions is *good* if it verifies properties $(1)$ and (2). We show that the $\cV$\_continuous functions form a good class, and that any good class is included in the $\cV$\_continuous functions. This implies that there *is* a largest good class, and that it coincides with the class of $\cV$\_continuous functions, as claimed.
Clearly, the class of $\cV$\_continuous functions is closed under composition. Now consider a $\cV$\_continuous function $f \colon A^* \to \{0, 1\}$. By continuity, $L = f^{-1}(\{1\})$ is in $\cV(A^*)$, since by hypothesis $\{1\} \in \cV(\{0,1\}^*)$. Hence $f = \chi_L$ for some $L \in \cV(A^*)$, concluding this step.
Let $f\colon A^* \to B^*$ be in a good class, and let $L \in \cV(B^*)$; we ought to show that $f^{-1}(L)$ is in $\cV(A^*)$. It holds that: $$\begin{aligned}
f^{-1}(L) & = f^{-1}(\chi_L^{-1}(1))\\
& = (\chi_L \circ f)^{-1}(1)\\
& = g^{-1}(1)\enspace. \tag{with $g = \chi_L \circ f$}
\end{aligned}$$ Note that $\chi_L$ is by hypothesis in the good class, and it being closed under composition, $g$ also belongs to the good class. Since $g \in \{0, 1\}^{A^*}$, there is a $L' \in \cV(A^*)$ such that $g = \chi_{L'}$. This implies that $f^{-1}(L) = L'$, and it thus belongs to $\cV(A^*)$.
Continuity: The profinite approach {#sec:contapp}
==================================
We build upon the work of Pin and Silva [@pin-silva05] and develop tools specialized to rational functions. In Section \[sec:preslem\], we present a lemma asserting the equivalence between $\cV$\_continuity and the “preservation” of the defining equations for $\cV$. In the sections thereafter, we specialize this approach to rational functions. As noted in [@pin-silva05], it often occurs that results about rational functions can be readily applied to the larger class of Reg\_continuous functions; here, this is in particular the case for the Preservation Lemma of Section \[sec:preslem\].
Our main appeal to a classical notion of continuity is given by the:
\[thm:ucont\] Let $f\colon A^* \to B^*$. It holds that $f$ is $\cV$\_continuous iff $f$ is uniformly continuous for the distance $d_\cV$.
Consequently, if $f$ is Reg\_continuous then it has a unique extension to the free profinite monoids, written $\ext{f}\colon \pro{A^*} \to \pro{B^*}$. The salient property of this mapping is that it is continuous in the *topological sense* (see, e.g., [@pin]). For our specific needs, we simply mention that it implies that for any regular language $L$, we have that $\ext{f}^{-1}(\clos{L})$ is closed (that is, it is the closure of some set).
The Preservation Lemma: Continuity is preserving equations {#sec:preslem}
----------------------------------------------------------
The Preservation Lemma gives us a key characterization in our study: it ties together continuity and some notion of preservation of equations. This can be seen as a generalization for functions of the notion of equation satisfaction for languages. We will need the following technical lemma that extends [@pin Proposition VI.3.17] from morphisms to arbitrary Reg\_continuous functions; interestingly, this relies on a quite different proof.
\[lem:pinext\] Let $f\colon A^* \to B^*$ be a Reg\_continuous function and $L$ a regular language. It holds that $\ext{f}^{-1}(\clos{L}) = \clos{f^{-1}(L)}$.
First note that $f^{-1}(L) \subseteq \ext{f}^{-1}(\clos{L})$, and that the right-hand side of this inclusion is closed. Hence $\clos{f^{-1}(L)} \subseteq \ext{f}^{-1}(\clos{L})$.
For the converse inclusion, first write $D = f^{-1}(B^*)$, a regular language by hypothesis. We have that $\ext{f}^{-1}(\clos{L}) = (\ext{f}^{-1}(\clos{L}\comp))\comp \cap \clos{D}$, and similarly, $f^{-1}(L) = (f^{-1}(L\comp))\comp \cap D$. This latter equality implies that $\clos{f^{-1}(L)} = \clos{f^{-1}(L\comp)}\comp \cap \clos{D}$, since $f^{-1}(L\comp)$ and $D$ are regular.
Hence the inclusion to be shown, that is, $\ext{f}^{-1}(\clos{L}) \subseteq \clos{f^{-1}(L)}$, is equivalent to: $$\begin{aligned}
(\ext{f}^{-1}(\clos{L}\comp))\comp \cap \clos{D} \subseteq
\clos{f^{-1}(L\comp)}\comp \cap \clos{D}\enspace,\\
\intertext{or equivalently,}
\clos{f^{-1}(L\comp)} \cup \clos{D}\comp \subseteq
\ext{f}^{-1}(\clos{L}\comp) \cup \clos{D}\comp
\enspace.
\end{aligned}$$
The inclusion to be shown is thus implied by $\clos{f^{-1}(L\comp)} \subseteq \ext{f}^{-1}(\clos{L}\comp)$, that is, since $L$ is regular, by $\clos{f^{-1}(L\comp)} \subseteq \ext{f}^{-1}(\clos{L\comp})$. As in the proof of the converse inclusion, the right-hand side being closed, this inclusion holds.
Let $f\colon A^* \to B^*$ be a Reg\_continuous function and $E$ a set of equations that defines $\cV(A^*)$. The function $f$ is $\cV$\_continuous iff for all $(u = v) \in E$ and words $s, t \in A^*$, $\ext{f}(s\cdot u \cdot t) =_\cV \ext{f}(s \cdot v \cdot t)$.
Suppose $f$ is $\cV$\_continuous. Let $u, v \in \pro{A^*}$ such that $u =_\cV v$, and $s, t \in A^*$. Since by $\cV$\_continuity $f^{-1}(B^*) \in \cV(A^*)$, either both $s\cdot u\cdot t$ and $s \cdot v\cdot t$ belong to the closure of this language, or they both do not. The latter case readily yields the result, hence suppose we are in the former case.
By definition, $u = \lim u_n$ and $v = \lim v_n$ for some Cauchy sequences of words $(u_n)_{n> 0}$ and $(v_n)_{n>0}$. Since $s \cdot u \cdot t =_\cV s \cdot v \cdot t$, the hypothesis yields that $d_\cV(s\cdot u_n\cdot t, s\cdot v_n\cdot t)$ tends to $0$. By Theorem \[thm:ucont\], $f$ is uniformly continuous for $d_\cV$, hence $d_\cV(f(s\cdot u_n\cdot t), f(s\cdot v_n\cdot t))$ also tends to $0$ (note that both $f(s\cdot u_n\cdot t)$ and $f(s\cdot v_n\cdot t)$ are defined for all $n$ big enough). This shows that $\ext{f}(s\cdot u\cdot t) =_\cV \ext{f}(s\cdot v\cdot t)$.
Suppose that $f$ preserves the equations of $E$ as in the statement. Let $L \in \cV(B^*)$, we wish to verify that $L' = f^{-1}(L) \in \cV(A^*)$, or equivalently by definition, that $L'$ satisfies all the equations of $E$. Let $(u = v) \in E$ be one such equation, and $s, t \in A^*$; we must show that $s\cdot u\cdot t \in \clos{L'} \Leftrightarrow s \cdot v \cdot t \in
\clos{L'}$.
Suppose $s \cdot u \cdot t \in \clos{L'}$. Since $f$ is Reg\_continuous, it holds that $\ext{f}(s\cdot u \cdot t) \in \clos{L}$ (observe that $\ext{f}(s\cdot u\cdot t)$ is indeed defined). By hypothesis, $\ext{f}(s\cdot u \cdot t) =_\cV \ext{f}(s\cdot v \cdot t)$; now since $L \in \cV(B^*)$, it must hold that $\ext{f}(s\cdot v \cdot t) \in \clos{L}$. Taking the inverse image of $\ext{f}$ on both sides, it thus holds that $s \cdot v \cdot t \in \ext{f}^{-1}(\clos{L})$, and Lemma \[lem:pinext\] then shows that $s \cdot v \cdot t \in \clos{L'}$. As the argument works both ways, this shows that $s\cdot u\cdot t \in \clos{L'} \Leftrightarrow s \cdot v \cdot t \in
\clos{L'}$, concluding the proof.
Continuity can be seen as preserving *membership* to $\cV$ (by inverse image); this is where the nomenclature “$\cV$\_preserving function” of [@pin-silva17] stems from. Strikingly, this could also be worded as preserving *nonmembership* to $\cV$:
A Reg\_continuous total$\,$[^3] function $f\colon A^* \to B^*$ is $\cV$\_continuous iff for all $L \subseteq A^*$ that do *not* belong to $\cV(A^*)$, $f(L)$ and $f(L\comp)$ are not $\cV$\_separable.
We rely on a characterization due to Almeida [@almeida99 Lemma 3.2]: two languages $K$ and $L$ are $\cV$\_separable iff for all $u \in \clos{K}, v \in \clos{L}$, it holds that $u \neq_\cV v$.
Suppose $f$ is $\cV$\_continuous, and let $L \subseteq A^*$ be a language outside $\cV(A^*)$. There must be two profinite words $u, v \in \pro{A^*}$ such that $u =_\cV v$, $u \in \clos{L}$ and $v \in \clos{L}\comp$. By $\cV$\_continuity and the Preservation Lemma, $\ext{f}(u) =_\cV \ext{f}(v)$, and moreover, $\ext{f}(u) \in \clos{f(L)}$ and $\ext{f}(v) \in \clos{f(L\comp)}$. The characterization above thus implies that $f(L)$ and $f(L\comp)$ are not $\cV$\_separable.
Let $L \in \cV(B^*)$, we show that $f^{-1}(L) \in \cV(A^*)$. Suppose for a contradiction that this is not the case. Then $f(f^{-1}(L))$ and $f(f^{-1}(L)\comp)$ are not $\cV$\_separable. The former is included in $L$, and the latter equal to $f(f^{-1}(L\comp)) \subseteq L\comp$. In particular, this means that $L$ and $L\comp$ are not $\cV$\_separable, which is a contradiction since $L \in \cV(B^*)$.
The profinite extension of rational functions
---------------------------------------------
The Preservation Lemma already hints at our intention to see transducers as computing functions from and to the free profinite monoids. Naturally, if $\tau$ is a rational function, its being Reg\_continuous allows us to do so (by Theorem \[thm:ucont\]). For $u = \lim u_n$ a profinite word, we will write $\tau(u)$ for $\ext{\tau}(u)$, i.e., the limit $\lim \tau(u_n)$, which exists by continuity. In this section, we develop a slightly more combinatorial approach to this evaluation, and address two classes of profinite words: those expressed as $s\cdot u\cdot t$ for $s, t$ words and $u$ a profinite word, and those expressed as $x^\omega$ for $x$ a word.
Recall that for a transducer state $q$ and a profinite word $u$, $q.u$ is well defined. As a consequence, if $s$ and $t$ are words and $\tau$ is unambiguous, then there is at most one initial state $q_0$, one $q \in q_0.s$ and one $q' \in q.u$ such that $q'.t$ is final, and these states exist iff $\tau(s\cdot u\cdot f)$ is defined. Thus:
\[lem:eval\] Let $\tau$ be an unambiguous transducer from $A^*$ to $B^*$, $s, t \in A^*$ and $u \in \pro{A^*}$. Suppose $\tau(s \cdot u \cdot f)$ is defined, and let $q_0, q, q'$ be the unique states such that $q_0$ is initial, $q \in q_0.s$, $q' \in q.u$, and $q'.t$ is final. The following holds: $$\tau(s \cdot u \cdot t) = \Tif{\tau, \nop, q}(s) \cdot \Tif{\tau, q, q'}(u)
\cdot \Tif{\tau, q', \nop}(t)\enspace.$$
Let us now turn to the evaluation of $\omega$-terms:
\[lem:omega\] Let $\tau$ be an unambiguous transducer from $A^*$ to $B^*$ and $x \in A^*$. If $\tau(x^\omega)$ is defined, then there are words $s, y, t \in B^*$ such that: $$\tau(x^\omega) = s\cdot y^{\omega - 1} \cdot t\enspace.$$
Consider a large value $n$; we study the behavior of $x^{n!}$ on $\tau$. There is an initial state $q_0$, a state $q$, and a final state $q_1$ such that $x^{n!}$ is accepted by a path going from $q_0$ to $q$ reading $x^i$, from $q$ to $q$ reading $x^k$ with $k < n$, and from $q$ to $q_1$ reading $x^j$. Thus the accepting path for any word of the form $x^{m!}, m > n$ is similar to the one for $x^{n!}$: from $q_0$ to $q$, looping $(m! - n!)/k+1$ times on $q$, and then from $q$ to $q_1$. Let thus $s = \Tif{\tau, q_0, q}(x^i)$, $z = \Tif{\tau, q, q}(x^k)$, and $t = \Tif{\tau, q, q_1}(x^j)$. It then holds that $\tau((x^k)^{m!}) = s\cdot z^{m! - (n!/k) + 1} \cdot t$. Letting $c = n!/k - 1$, this shows that $\tau((x^k)^\omega) = s \cdot z^{\omega - c} \cdot t$. Now on the one hand, $(x^k)^\omega = x^\omega$, and on the other hand, we similarly have that $z^{\omega - c} = y^{\omega - 1}$ by letting $y = z^c$. We thus obtain that $\tau(x^\omega) = s \cdot y^{\omega - 1} \cdot t$.
These constitute our main ways to effectively evaluate the image of profinite words through transducers. Their use being quite ubiquitous in our study, we will rarely refer to these lemmas nominally.
The Syncing Lemma: Preservation Lemma applied to transducers
------------------------------------------------------------
We apply the Preservation Lemma on transducers and deduce a slightly more combinatorial characterization of transducers describing continuous functions. This does not provide an immediate decidable criterion, but our decidability results will often rely on it. The goal of the forthcoming lemma is to decouple, when evaluating $s \cdot u \cdot t$ (with the notations of the Preservation Lemma), the behavior of the $u$ part and that of the $s, t$ part. This latter part will be tested against an *equalizer* set:
Let $u, v \in \pro{A^*}$. The |equalizer set| of $u$ and $v$ in $\cV$ is: $$\equ_\cV(u, v) = \{ (s, s', t, t') \in (A^*)^4 \mid s\cdot u \cdot t
=_\cV s' \cdot v \cdot t'\}\enspace.$$
\[rk:almeida\] The complexity of equalizer sets can be surprisingly high. For instance, letting $\cV$ be the class of languages defined by $\{x^2 = x^3 \mid x
\in A^*\}$, there is a profinite word $u$ for which $\equ_\cV(u, u)$ is undecidable. On the other hand, equalizer sets quickly become less complex for common varieties; for instance, Lemma \[lem:apeq\] will provide a simple form for the equalizer sets of aperiodic supercancellative varieties.
Let $R, S \subseteq A^* \times B^*$. The |input synchronization| of $R$ and $S$ is defined as the relation over $B^* \times B^*$ obtained by synchronizing the first component of $R$ and $S$: $$R \sync S = \{(u, v) \mid (\exists s)[ (s, u) \in R \land (s, v) \in
S]\} \;\big(= S \circ R^{-1})\enspace.$$
Naturally, the input synchronization of two rational functions is a rational relation.
Let $\tau$ be an unambiguous transducer from $A^*$ to $B^*$ and $E$ a set of equations that defines $\cV(A^*)$. The function $\tau$ is $\cV$\_continuous iff:
1. $\tau^{-1}(B^*) \in \cV(A^*)$, and
2. For any $(u = v) \in E$, any states $p, q$, any $p' \in p.u$, and any $q' \in q.v$, and letting $u' = \Tif{\tau,p,p'}(u)$ and $v' = \Tif{\tau, q, q'}(v)$: $$(\Tif{\tau, \nop, p} \sync \Tif{\tau, \nop, q}) \times (\Tif{\tau, p', \nop} \sync
\Tif{\tau, q', \nop}) \subseteq \equ_\cV(u', v')\enspace.$$
We rely on the Preservation Lemma, since $\tau$ is Reg\_continuous.
Suppose that $\tau$ is $\cV$\_continuous, and apply the Preservation Lemma. This shows the first point to be proven. For the second, we use the notation of the statement. Let $(s, s', t, t') \in (\Tif{\tau, \nop, p} \sync \Tif{\tau, \nop, q}) \times
(\Tif{\tau, p', \nop} \sync \Tif{\tau, q', \nop})$. This implies that there are words $x, y \in A^*$ such that:
- $s = \Tif{\tau, \nop, p}(x), s' = \Tif{\tau, \nop, q}(x)$;
- $t = \Tif{\tau, p', \nop}(y), t' = \Tif{\tau, q', \nop}(y)$.
By Lemma \[lem:eval\], we have that $\tau(x \cdot u
\cdot y) = s \cdot u' \cdot t$ and $\tau(x \cdot v \cdot y) = s' \cdot v'
\cdot t'$. The Preservation Lemma then asserts that $s \cdot u' \cdot t =_\cV
s' \cdot v' \cdot t'$, showing that $(s, s', t, t') \in \equ_\cV(u', v')$.
Let $(u = v) \in E$ and $x, y \in A^*$. We must show that $\tau(x\cdot u \cdot y) =_\cV \tau(x \cdot v \cdot y)$. Since $\tau^{-1}(B^*) \in \cV(A^*)$, either $\tau$ is defined on both $x\cdot u\cdot y$ and $x \cdot v \cdot y$, or on neither; in this latter case, the equality is verified by definition. We thus suppose that both values are defined. This implies that there are states $p,q,p',q'$ as in the statement, and using the same notation, letting $s, s', t, t'$ just as above, the hypothesis yields that $s \cdot u' \cdot t =_\cV s' \cdot v' \cdot t'$, showing the claim.
A profinite toolbox for the aperiodic setting
---------------------------------------------
In this section, we provide a few lemmas pertaining to our study of aperiodic continuity. We show that the equalizer sets of aperiodic supercancellative varieties are well behaved. Intuitively, the larger the varieties are, the more their nonempty equalizer sets will be similar to the identity. For instance, if $s \cdot x^\omega =_\vA x^\omega$, for words $s$ and $x$, it should hold that $s$ and $x$ have the same primitive root. We first note the following easy fact that will only be used in this section; it is reminiscent of the notion of *equidivisibility*, studied in the profinite context by Almeida and Costa [@almeida-costa17].
\[lem:dance\] Let $u, v$ be profinite words over an alphabet $A$ and $\cV$ be a supercancellative variety. Suppose that there are $s, t \in A^*$ such that $u \cdot t =_\cV s \cdot v$, then there is a $w \in \pro{A^*}$ such that $u =_\cV s \cdot w$ and $v =_\cV w \cdot t$. If moreover $u = v$ and $\cV$ is aperiodic, then $u =_\cV s \cdot u \cdot t$.
Let $u = \lim u_n$. From $u \cdot t =_\cV s \cdot v$, and the fact that $s\cdot A^* \in \cV(A^*)$ by supercancellativity, we obtain that for $n$ large enough, $u_n \cdot t \in s \cdot A^*$. Since $u$ is nonfinite, $|u_n| > |s|$ for $n$ large enough, in which case $u_n = s \cdot w_n$ for some sequence $(w_n)_{n > 0}$. Let $w \in \pro{A^*}$ be a limit point of this sequence, that exists by compacity (this is an essential property of the free profinite monoid, see, e.g., [@pin Theorem VI.2.5]). It holds that $u = s \cdot w$. Replacing $u$ by this value in the equation of the hypothesis, we thus have that $s \cdot w \cdot t =_\cV s \cdot v$, and since $\cV$ is supercancellative, that $v =_\cV w \cdot t$.
For the last point, with $u = v$, we iterate the previous construction on $w$, since in that case, $u =_\cV w \cdot t =_\cV s \cdot w$. This provides a sequence $w = w_1, w_2, w_3, \ldots$ such that $u =_\cV s^n\cdot w_n =_\cV w_n \cdot t^n$. Taking a limit point $x$ of $(w_n)_{n>0}$, it thus holds that $u =_\cV s^\omega \cdot x =_\cV x \cdot t^\omega$, showing, by aperiodicity, that $u =_\cV s \cdot u =_\cV u \cdot t$.
\[lem:apeq\] Let $u, v$ be profinite words over an alphabet $A$ and $\cV$ be an aperiodic supercancellative variety. Suppose $\equ_\cV(u, v)$ is nonempty. There are words $x, y \in A^*$ and two pairs $\rho_1, \rho_2 \in (A^*)^2$ such that: $$\equ_\cV(u, v) = \Big(\Id\cdot \big((x^*, x^*)\rho_1^{-1}\big)\Big)
\times \Big(\big(\rho_2^{-1}(y^*, y^*)\big)\cdot
\Id\Big)\enspace.$$
Let us first establish the property for $u = v$. Assume that there are *nonempty primitive* words $x, y$ such that $x \cdot u \cdot y =_\cV u$; we show the statement of the lemma with these $x$ and $y$, and $\rho_1 = \rho_2 = (1, 1)$. Note that $x^\omega \cdot u \cdot y^\omega =_\cV u$, hence, since $x^{\omega+1} = x^\omega$ and similarly for $y$, it holds that $x \cdot u =_\cV u \cdot y =_\cV u$. This and the fact that $\cV$ is supercancellative show the right-to-left inclusion.
For the left-to-right inclusion, let $s, s', t, t'$ be such that $s\cdot u \cdot t =_\cV s' \cdot u \cdot t'$. Since $\cV$ is supercancellative, this implies that the equation also holds if common prefixes of $s$ and $s'$ and common suffixes of $t$ and $t'$ are removed. We may thus assume that we are in two possible situations, by symmetry:
1. Suppose $s' = t' = 1$, that is, $s \cdot u \cdot t =_\cV u$, and that $s, t$ are nonempty. By the same token as above, this shows that $s \cdot u =_\cV u \cdot t =_\cV u$. In particular, this implies that: $$s^{|x|} \cdot u \cdot t^{|y|} =_\cV x^{|s|} \cdot u \cdot
y^{|t|}\enspace,$$ which implies, since $\cV$ is supercancellative, that $s^{|x|} = x^{|s|}$ and $t^{|y|} = y^{|t|}$. As $x$ and $y$ are primitive, this shows that $s \in
x^*$ and $t \in y^*$. (Note that this holds even if one of $s$ or $t$ is empty.)
2. Suppose $s = t' = 1$, that is, $u \cdot t =_\cV s' \cdot u$, and that $s', t$ are nonempty. By Lemma \[lem:dance\], we have that $u =_\cV s' \cdot u \cdot t$, and we can appeal to the previous situation, showing that $s' \in x^*$ and $t \in y^*$.
(The cases where one of $s, t$ is empty, in the first point, or one of $s', t$ is empty, in the second, are treated similarly. Note that it is not possible for both $s$ and $s'$ to be nonempty, since that would imply that they start with different letters, falsifying the assumed equation by supercancellativity.)
We assumed that the $x, y$ existed, we ought to show the other cases verify the claim. The two situations above show that if $\equ_\cV(u,u)$ is nonempty, then such $x, y$ exist, although without the guarantee that they be nonempty. Now if $x \cdot u =_\cV u$ and there are no nonempty $y$ such that $x \cdot u \cdot y =_\cV u$, this implies that there are no nonempty $y$ such that $u \cdot y =_\cV u$. Consequently, in the above case, $t = t' = 1$, and the analysis stands. This concludes the proof for the case $u = v$.
We will reduce the case $u \neq v$ to this one. Indeed, suppose that $s \cdot u \cdot v = s' \cdot v \cdot t'$. Again, by stripping away common prefixes and suffixes, we are faced with two cases:
1. Suppose $s' = t' = 1$, that is, $s \cdot u \cdot t =_\cV v$. It holds that $\equ_\cV(u, v) = \equ_\cV(u, s \cdot u \cdot t)$, hence $(m, m',
n, n') \in \equ_\cV(u, v)$ iff $(m, m'\cdot s, n, t \cdot n') \in
\equ_\cV(u, u)$, and the result follows.
2. Suppose $s = t' = 1$, that is, $u \cdot t =_\cV s' \cdot v$. By Lemma \[lem:dance\], there is a profinite word $w$ such that $u =_\cV s'
\cdot w$ and $v =_\cV w \cdot t$, hence $(m, m', n, n') \in \equ_\cV(u, v)$ iff $(m \cdot s', m', n, t \cdot n') \in \equ_\cV(w, w)$, concluding the proof.
\[lem:apeqomega\] Let $x, y$ be words. For every aperiodic supercancellative variety $\cV$, it holds that $\equ_\cV(x^\omega, y^\omega) = \equ_\vA(x^\omega, y^\omega)$.
The inclusion from right to left is clear, since all equations true in hold in .
In the other direction, let us write $u = x^{|y|}$ and $v = y^{|x|}$; we have that $u^\omega = x^\omega$ and $v^\omega = y^\omega$. Suppose $s \cdot u^\omega t =_\cV s' \cdot v^\omega \cdot t'$. In particular, since $\cV$ is supercancellative, this means that $s \cdot u^n$ is a prefix of $s' \cdot v^n$, or vice-versa, depending on whether $|s| > |s'|$ or the opposite. This implies that $u \cdot u\subp = v\subs \cdot v$ for some prefix $u\subp$ of $u$ and suffix $v\subs$ of $v$. Hence (by, e.g., [@lothaire97 Proposition 1.3.4]) $u$ and $v$ are conjugate. Their respective primitive roots are thus conjugate (by [@lothaire97 Proposition 1.3.3]); writing $z\cdot z'$ and $z' \cdot z$ for them, we have that $u^\omega = (z\cdot z')^\omega$ and $v^\omega = (z'\cdot z)^\omega$.
Thus the equation above reads: $s \cdot (z \cdot z')^\omega \cdot t =_\cV s' \cdot (z' \cdot z)^\omega \cdot
t'$. As in the proof of Lemma \[lem:apeq\], removing the common prefixes and suffixes (which we can do both in $\cV$ and $\vA$), we are left with two possibilities:
- Suppose $s' = t' = 1$, that is, $s \cdot (z \cdot z')^\omega \cdot t =_\cV
(z' \cdot z)^\omega$. The same argument as in Lemma \[lem:apeq\] shows that $s \in z' \cdot (z\cdot z')^*$ and $t \in (z \cdot z')^*\cdot z$, and hence the equation holds in $\vA$ too;
- Suppose $s = t' = 1$, that is, $(z \cdot z')^\omega \cdot t =_\cV s'
\cdot (z' \cdot z)^\omega$. Similarly, as $\cV$ is supercancellative and aperiodic, this shows that $s' \in z \cdot (z' \cdot z)^*$ and $t \in (z
\cdot z')^*\cdot z$, and the equation holds in $\vA$ too, concluding the proof.
For two aperiodic supercancellative varieties $\cV$ and $\cW$, we could further show that if both $\equ_\cV(u, v)$ and $\equ_\cW(u, v)$ are nonempty, then they are equal, for any profinite words $u, v$. It may however happen that one equalizer set is empty while the other is not; for instance, with $u = (ab)^\omega$ and $v = (ab)^\omega\cdot a \cdot (ab)^\omega$, the equalizer set of $u$ and $v$ in $\vDA$ is nonempty, while it is empty in $\vA$.
Intermezzos {#sec:intermezzos}
===========
We present a few facts of independent interest on continuous rational functions. Through this, we develop a few examples, showing in particular how the Preservation and Syncing Lemmas can be used to show (non)continuity. In a first part, we study when the structure of the transducer is relevant to continuity, and in a second, when the (non)inclusion of variety relates to (non)inclusion of the class of continuous rational functions.
Transducer structure and continuity {#sec:struct}
-----------------------------------
As noted by [@reutenaeur-schutzenberger95 p. 231], there exist numerous natural varieties $\cV$ for which any $\cV$\_realizable rational function is $\cV$\_continuous. Indeed:
\[prop:transtocont\] Let $\cV$ be a variety of languages closed under inverse $\cV$\_realizable rational function. Any $\cV$\_realizable rational function is $\cV$\_continuous. This holds in particular for the varieties $\vA, \vGsol,$ and $\vG$.
This is due to a classical result of Sakarovitch [@sakarovitch79] (see also [@pin-sakarovitch85]), stating, in modern parlance, that a variety $\cV$ is closed under *block product* iff it is closed under inverse $\cV$\_realizable rational functions (note that there has been some fluctuation on vocabulary, since wreath product was used at some point to mean block product). That $\vA, \vGsol,$ and $\vG$ are closed under block product is folklore.
This naturally fails for all our other varieties, since they are not closed under inverse $\cV$\_realizable rational functions. For completeness, we give explicit constructions in the proof of the:
For $\cV \in \{\vJ, \vL, \vR, \vDA, \vAb, \vGnil, \vCom\}$, there are $\cV$\_realizable rational functions that are not $\cV$\_continuous.
We devise simple counter examples with $A = \{a, b\}$.
Recall that $A^*a \notin \vR(A^*) \cup \vCom(A^*)$. The minimal unambiguous two-state transducer $\tau$ that erases all of its input except for the last letter is a $(\vJ \cap \vCom)$\_transducer; indeed, $a$ acts in the same way as $b$ and they are idempotent on the transducer. However, $\tau^{-1}(a) = A^*a$.
Consider the *Dyck language* $D$ over $A$; this is the (nonregular) language of well-parenthesized expressions where $a$ is the opening and $b$ the closing parenthesis. Write $D^{(k)}$ for the Dyck language where parentheses are nested at most $k$ times, for instance $D^{(0)} = 1, D^{(1)}=(ab)^*$ and $D^{(2)} = (a(ab)^*b)^*$. These languages bear great importance in algebraic language theory, as they separate each level of the *dot-depth hierarchy* [@brzozowski-knast78]. It holds in particular that $D^{(1)} \notin \vDA(A^*)$.
Let $\tau$ be the rational function that removes the first letter of each block of $a$’s and each block of $b$’s; naturally, $\tau$ is $\vL$\_realizable. However, $\tau^{-1}(D^{k-1}) = D^k$, showing not only that $\tau$ is not continuous for , but also not continuous for *any* level of the dot-depth hierarchy.
Consider the two-state transducer $\tau$ where $a$ loops on both states, and a $b$ on one state goes to the other. When $a$ is read on the first state, it produces a $x$, while all the other productions are the identity. This is an $\vAb$\_transducer. However, $\tau(aba) = xba \neq baa = \tau(baa)$, hence it is not $\vAb$\_continuous by the Preservation Lemma, since $aba =_\vAb baa$. For $\vGnil$, let $L$ be the language over $\{a, b, x\}$ with a number of $x$ congruent to $0$ modulo $3$. It can be shown that $\tau^{-1}(L) \notin \vGnil(A^*)$, intuitively since this language needs to differentiate between those $a$’s that are an even number of $b$’s away from the beginning of the word, and those which are not.
The converse concern, that is, whether all $\cV$\_continuous rational functions are $\cV$\_realizable, was mentioned by [@reutenaeur-schutzenberger95] for $\cV = \vA$.
\[prop:conttrans\] For $\cV \in \{\vJ, \vL, \vR, \vDA, \vA, \vAb, \vCom\}$, there are $\cV$\_continuous rational functions that are not $\cV$\_realizable.
Let $A = \{a\}$, a unary alphabet. Consider the transducer $\tau$ that removes every second $a$: its minimal transducer not being a $\vA$\_transducer, it is not $\vA$\_realizable (this is a property of subsequential transducers [@reutenaeur-schutzenberger95]). However, all the unary languages of $\cV$ are either finite or co-finite, and hence for any $L \in \cV(A^*)$, $\tau^{-1}(L)$ is either finite or co-finite, hence belongs to $\cV(A^*)$.
Over $A = \{a, b\}$, define $\tau$ to map words $w$ in $aA^*$ to $(ab)^{|w|}$, and words $w$ in $bA^*$ to $(ba)^{|w|}$. Clearly, $a$ and $b$ cannot act commutatively on the transducer. Now $\tau(ab) =_\vCom \tau(ba)$, and moreover $\tau(x^\omega) =_\vAb (ab)^\omega =_\vAb 1 = \tau(1)$, hence $\tau$ is continuous for both $\vAb$ and $\vCom$ by the Preservation Lemma.
We delay the positive answers to that question, namely for $\vGnil, \vGsol, \vG$, to Corollary \[cor:grpconttotrans\] as they constitute our main lever towards the decidability of continuity for these classes.
Variety inclusion and inclusion of classes of continuous functions {#sec:inc}
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In this section, we study the consequence of variety (non)inclusion on the inclusion of the related classes of continuous rational functions. This is reminiscent of the notion of *heredity* studied by [@pin-silva11], where a function is $\cV$\_hereditarily continuous if it is $\cW$\_continuous for each subvariety $\cW$ of $\cV$. Variety noninclusion provides the simplest study case here:
Let $\cV$ and $\cW$ be two varieties. If $\cV \not\subseteq \cW$ then there are $\cV$\_continuous rational functions that are not $\cW$\_continuous.
Let $L \in \cV(A^*)$ be such that $L \notin \cW(A^*)$. Define $f\colon A^* \to A^*$ as the identity function with domain $L$. Clearly, as $f^{-1}(K) = K \cap L$, the function $f$ is $\cV$\_continuous. However, $f^{-1}(A^*) = L \notin \cW(A^*)$ and $A^* \in \cW(A^*)$, thus $f$ is not $\cW$\_continuous.
The remainder of this section focuses on a dual statement:
*If $\cV \subsetneq \cW$, are all $\cV$\_continuous rational functions $\cW$\_continuous?*
### The group cases
We first focus on group varieties. Naturally, if 1. $\cV$\_continuous rational functions are $\cV$\_realizable and 2. $\cW$\_realizable rational functions are $\cW$\_continuous, this holds. Appealing to the forthcoming Corollary \[cor:grpconttotrans\] for point 1 and Proposition \[prop:transtocont\] for point 2, we then get:
For $\cV, \cW \in \{\vGnil, \vGsol, \vG\}$ with $\cV \subsetneq \cW$, all $\cV$\_continuous rational functions are $\cW$\_continuous. This however fails for $\cV = \vAb$ and for any $\cW \in \{\vGnil, \vGsol, \vG\}$.
It remains to show the case $\cV = \vAb$. This is in fact the same example as in the proof of Proposition \[prop:conttrans\], to wit, over $A = \{a, b\}$, the rational function $\tau$ that maps $w \in aA^*$ to $(ab)^{|w|}$, and words $w \in bA^*$ to $(ba)^{|w|}$. Indeed, we saw that this function is continuous for $\vAb$, but it holds that $\tau(a) = ab$ on the one hand, and $\tau(b^\omega a) = (ba)^\omega ba =_\cW ba$, but $ab \neq_\cW ba$. The Preservation Lemma then shows that $\tau$ is not continuous for $\cW$.
\[prop:abtocom\] All $\vAb$\_continuous rational functions are $\vCom$\_continuous.
Indeed, if $u =_\vAb v$ with $u, v$ words, then $u =_\vCom v$, since these varieties separate the same words. As $\vCom$ is defined using equations on words, this directly shows the claim by the Preservation Lemma.
### The aperiodic cases
We now turn to aperiodic varieties. For lesser expressive varieties, the property fails:
\[prop:apnon\] For $\cV \in \{\vJ, \vL, \vR\}$ and $\cW \in \{\vL, \vR, \vDA, \vA\}$ with $\cV \subsetneq \cW$, there are $\cV$\_continuous rational functions that are not $\cW$\_continuous.
Define $\tau\colon \{a\}^* \to \{a, b\}^*$ to be the rational function that changes every other $a$ to $b$; that is, $\tau(a^{2n}) = (ab)^n$, and $\tau(a^{2n+1})=(ab)^n\cdot a$. Note that naturally, over a single letter, $a\cdot (aa)^\omega = (aa)^\omega \cdot a = a^{\omega+1}$. Now $\tau(a^\omega) = (ab)^\omega$ and $\tau(a^{\omega+1})= (ab)^\omega\cdot a$, and since these two profinite words are equal in $\vJ$ and $\vR$, the Preservation Lemma shows that $\tau$ is continuous for both $\vJ$ and $\vR$. However, these two profinite words are not equal in $\vL, \vDA,$ and $\vA$, showing that $\tau$ is continuous for none of those varieties.
The remaining case, that is, showing the existence of a $\vJ$\_continuous rational function that is not $\vR$\_continuous is done symmetrically, with the function mapping $a^{2n}$ to $(ab)^n$ and $a^{2n+1}$ to $b\cdot (ab)^n$.
Any $\vDA$\_continuous rational function is $\vA$\_continuous.
First note that both $\vDA$ and $\vA$ satisfy the hypotheses of Lemma \[lem:apeq\]. Consider a $\vDA$\_continuous rational function $\tau\colon A^* \to B^*$. By the Syncing Lemma, to show that it is $\vA$\_continuous, it is enough to show that 1. $\tau^{-1}(B^*) \in \vA(A^*)$, and 2. That some input synchronizations of $\tau$, based on equations of the form $x^\omega =_\vA x^{\omega + 1}$, belong to an equalizer set of the form (by Lemma \[lem:omega\]): $$\equ_\vA(\alpha \cdot y^\omega \cdot \beta,\; \alpha' \cdot z^\omega \cdot
\beta') = \{(s, s', t, t') \mid (s\cdot \alpha,\; s' \cdot \alpha', \;\beta\cdot
t,\; \beta'\cdot t') \in \equ_\vA(y^\omega,\; z^\omega)\}\enspace.$$
Applying the Syncing Lemma on $\tau$ for the variety $\vDA$, we get that point 1 is true, since $\tau^{-1}(B^*) \in \vDA(A^*)$. Similarly, point 2 is true since $x^\omega = x^{\omega+1}$ is an equation of $\vDA$, and Lemma \[lem:apeqomega\] implies that the equalizer set of the equation above is the same in $\vDA$ and $\vA$.
However, this property does not hold beyond *rational* functions:
There are nonrational functions that are continuous for both $\vDA$ and Reg but are not $\vA$\_continuous.
Define $f\colon \{a\}^* \to \{a, b\}^*$ by $f(a^{2n}) = (ab)^n$ and $f(a^{2n+1}) = (ab)^n\cdot a \cdot (ab)^n$. We first have to check that $f$ is indeed Reg\_continuous. Given a regular language $L$, we define a pushdown automaton over $\{a\}^*$ that recognizes $f^{-1}(L)$; since all unary context-free languages are regular, by Parikh’s theorem, this shows the claim. If the input is of the form $a^{2n}$, then the pushdown automaton may check that $(ab)^n \in L$, by simulating the automaton for $L$. If the input is of the form $a^{2n+1}$, then the pushdown automaton can guess the middle position of the input, and accordingly check that $(ab)^n \cdot a \cdot (ab)^n
\in L$, again using the automaton for $L$. This concludes the construction.
The function $f$ being Reg\_continuous, it has an extension $\ext{f}$ to the free profinite monoids. As in Proposition \[prop:apnon\], checking that $f$ is $\vDA$\_continuous amounts to check that $\ext{f}(a^{\omega}) =_\vDA \ext{f}(a^{\omega+1})$. The left-hand side being $(ab)^\omega$ while the right-hand side is $(ab)^\omega
\cdot a \cdot (ab)^\omega$, this holds. However, these two profinite words are equal in $\vDA$ but not in $\vA$, hence this function is not $\vA$\_continuous, again appealing to the Preservation Lemma.
Deciding continuity for transducers {#sec:dec}
===================================
Deciding continuity for group varieties {#sec:group}
---------------------------------------
showed in [@reutenaeur-schutzenberger95] that a rational function is $\vG$\_continuous iff it is $\vG$\_realizable. Since this is proven effectively, it leads to the decidability of $\vG$\_continuity. In Proposition \[prop:transtocont\], we saw that the right-to-left statement also holds for $\vGsol$; we now show that the left-to-right statement holds for all group varieties $\cV$ that contain $\vGnil$. As in [@reutenaeur-schutzenberger95], but with sensibly different techniques, we show that $\cV$\_continuous transducers are plurisubsequential. The Syncing Lemma will then imply that such transducers are $\cV$\_transducers. Both properties rely on the following normal form:
\[lem:nf\] Let $\tau$ be a transducer. An equivalent transducer $\tau'$ can be constructed by adjoining some codeterministic automaton to $\tau$ so that for any states $p, q$ of $\tau'$: $$\Big[(\exists x, y) \big[\emptyset \;\neq\; (\Tif{\tau', p, \nop} \sync
\Tif{\tau', q, \nop}) \;\subseteq\; (x, y) \cdot \Id\big]\Big] \Rightarrow p =
q\enspace.$$ Alternatively, the “dual” property can be ensured, adjoining a deterministic automaton to $\tau$, so that for any states $p, q$ of $\tau'$: $$\Big[(\exists x, y) \big[\emptyset \;\neq\; (\Tif{\tau', \nop, p} \sync
\Tif{\tau', \nop, q}) \;\subseteq\; \Id \cdot (x, y)\big]\Big] \Rightarrow p =
q\enspace.$$
[Lemma \[lem:nf\]]{} We show the first property, its dual being proved similarly. We construct the required transducer in three steps.
For any state $q \in Q$, write $R_q = \{w \mid q.w \in F\}$. Let $M_{q,q'}$ be a deterministic automaton recognizing the *reverse* of $R_q \cap R_{q'}$ (that is, the language consisting of the reverse of each word therein), and define $M'$ as the Cartesian product of all $M_{q,q'}$’s. Finally, let $M$ be the reverse of $M'$.
Define now $\tau'$ to be the Cartesian product of $\tau$ and $M$, with the initial states being the Cartesian product of that of $\tau$ and $M$, and likewise for the final states. The output function of $\tau'$ is that of $\tau$. Generalizing the notation $R_\bullet$ to $\tau'$ and $M$, we have:
\[fact:rp\] For any states $p, p'$ of $\tau'$, $R_p \cap R_{p'} \neq \emptyset \Rightarrow
R_p = R_{p'}$.
[Fact \[fact:rp\]]{} Let $p = (q, r)$ and $p' = (q', r')$, and assume $R_p \cap R_{p'} \neq
\emptyset$. Note that $R_p = R_q \cap R_r$ and $R_{p'} = R_{q'} \cap
R_{r'}$. Since their intersection is nonempty and $M'$ is codeterministic, this shows that $r = r'$. Now as $R_r$ contains a word of $R_q \cap
R_{q'}$, the state in $M$ corresponding to the automaton $M_{q,q'}$ is final in $M_{q, q'}$, showing that $R_r \subseteq R_q \cap R_{q'}$. We thus see that $R_p = R_q \cap R_r = R_r$, and similarly, $R_{p'} = R_r$, showing the claim.
Write simply $\tau$ for the result of Step 1. In this step, we make sure that outputs are produced “as soon as possible,” a process known as *normalization* (e.g., [@lothaire02 Section 1.5.2]) that we sketch for completeness. For every state $q$, write $\pi_q$ for the longest string such that $\Tif{\tau, q, \nop}(A^*) \subseteq \pi_q\cdot B^*$. Now define the new output function $(\lambda', \mu', \rho')$ by letting: $$\mu'(1, q) = \mu(1, q) \cdot \pi_q,\quad
\mu'(q, a, q') = \pi_q^{-1}\mu(q, a, q')\cdot \pi_{q'},\quad
\mu'(q, 1) = \pi_q^{-1}\mu(q, 1)\enspace.$$ Writing $\tau'$ for $\tau$ equipped with the output function $\mu'$, it holds that for no state $q$, there is a letter $b \in B^*$ such that $\Tif{\tau, q,
\nop}(A^*) \subseteq b\cdot B^*$.
Write again $\tau$ for the result of the previous step. Naturally, $\tau$ still verifies Fact \[fact:rp\]. Consider two states $p, q$ such that there are $x, y \in B^*$ verifying $\emptyset \neq (\Tif{\tau, p, \nop} \sync \Tif{\tau, q, \nop}) \subseteq (x,
y)\cdot \Id$. The first part of this assumption implies that $R_p \cap R_q \neq \emptyset$, and thus, by Fact \[fact:rp\], $R_p = R_q$. In other words, $\Tif{\tau, p, \nop}$ and $\Tif{\tau, q, \nop}$ have the same domain. The second part of the assumption thus indicates that every production of $p$ (of $q$) starts with $x$ (with $y$), and Step 2 asserts that $x = y = 1$. Hence $\Tif{\tau, p, \nop}$ and $\Tif{\tau, q, \nop}$ actually compute the same function. We can thus merge them into a single state without changing the function realized. Repeating this operation results in a transducer with the required property.
\[lem:nftocV\] Let $\cV$ be a variety of group languages that contains $\vGnil$. For any $\cV$\_continuous unambiguous transducer $\tau$, the transducer obtained by applying the dual of Lemma \[lem:nf\], then applying its first part, is a plurisubsequential $\cV$\_transducer.
Write $\tau'$ for the result of the dual part of Lemma \[lem:nf\] on $\tau$, and $\tau''$ for the result of the first part of Lemma \[lem:nf\] on $\tau'$. We show that $\tau''$ is a plurisubsequential $\cV$\_transducer. For clarity, we first assume that it is plurisubsequential and show that it is a $\cV$\_transducer, then we show that it is indeed plurisubsequential.
Consider an equation $u =_\cV v$, a state $q$ of $\tau''$, and let $p = q.u$ and $p' = q.v$. We show that $p=p'$, concluding this point. We rely on the Syncing Lemma, since $\tau''$ is $\cV$\_continuous; it ensures in particular that: $$\begin{aligned}
(\Tif{\tau'', \nop, q} \sync \Tif{\tau'', \nop, q}) \times
(\Tif{\tau'', p, \nop} \sync \Tif{\tau'', p', \nop}) \subseteq
\equ_\cV(u', v') \quad\text{with } u' = \Tif{\tau'', q, p}(u), v' =
\Tif{\tau'', q, p'}(v)\enspace.\label{eqn:p}
\end{aligned}$$ Let $(s, s, t_1, t_2)$ be in the left-hand side. It holds that $s\cdot u' \cdot t_1 =_\cV s \cdot v' \cdot t_2$, thus $u' \cdot t_1 =_\cV v' \cdot t_2$ (here and in the following, we derive equivalent equations by appealing to the fact that the *free group* is embedded, in a precise sense, in $\cV$ [@robinson95 § 6.1.9]). Now consider another tuple $(s', s', t_1', t_2')$ again in the left-hand side of Equation (\[eqn:p\]). It also holds that $u' \cdot t_1' =_\cV v' \cdot t_2'$, hence we obtain that $t_1\cdot t_2^{-1} =_\cV t_1'\cdot t_2'^{-1}$. This is in turn equal in $\cV$ to some $\alpha\cdot \beta^{-1}$ such that $\alpha$ and $\beta$ are words that do not share the same last letter. This shows that $t_1 = \alpha \cdot t$ and $t_2 = \beta \cdot t$ for some word $t$, and similarly for $t_1'$ and $t_2'$. More generally: $(\Tif{\tau'', p, \nop} \sync \Tif{\tau'', p', \nop}) \subseteq (\alpha,
\beta) \cdot \Id$, and the normal form of Lemma \[lem:nf\] thus shows that $p=p'$.
In either $\tau'$ or $\tau''$, call a triple a states $(p, q, q')$ a *fork* on $a$ if from $p$, the transducer can go to $q$ and $q'$ reading one $a$, and there is a path from $q$ to $p$ reading only $a$’s:
Dually, a triple $(q, q', p)$ is a *reverse fork* on $a$ if the transducer can go from $q$ and $q'$ to $p$ reading one $a$, and there is a path from $p$ to $q$ that reads only $a$. In both cases, the fork is *proper* if $q \neq q'$. We rely on two facts to show plurisubsequentiality:
\[fact:nofork\] There are no proper forks or reverse forks in $\tau''$.
\[fact:allom\] For any state $p$ of $\tau''$ and any letter $a$, it holds that $p \in p.a^\omega$.
Before proving the facts, we show how they imply the plurisubsequentiality of $\tau''$. Consider a state $p$ in $\tau''$ and a letter $a$. As $p \in p.a^\omega$ by Fact \[fact:allom\], there is a cycle of $a$’s on $p$. Call $q$ the first state of that cycle. Next, let $q'$ be such that $(p, a, q')$ is a transition of $\tau''$. Clearly, $(p, q, q')$ forms a fork, hence by Fact \[fact:nofork\], $q = q'$. Thus $\tau''$ is plurisubsequential. We conclude with the proofs of the two facts:
[Fact \[fact:nofork\]]{} We show the following statement: For any $\vG$\_continuous transducer $\kappa$, the result $\kappa'$ of Lemma \[lem:nf\] has no proper reverse forks. This shows similarly that applying the dual of Lemma \[lem:nf\] on a $\vG$\_continuous transducer results in a transducer with no proper forks. Since going from $\tau'$ to $\tau''$ cannot introduce new proper forks (as $\tau'$ is adjoined a codeterministic automaton in Lemma \[lem:nf\]), this shows the fact.
Consider a reverse fork $(q, q', p)$ on $a$ in $\kappa'$. As $p$ can be reached from both $q$ and $q'$ reading $a$, the product $P = \Tif{\kappa', q, \nop} \sync \Tif{\kappa', q', \nop}$ is nonempty. Write $x = \Tif{\kappa', q', p}(a), y = \Tif{\kappa', p, q}(a^n), z = \Tif{\kappa', q,
p}(a)$, for some $n$ such that $y$ is defined. We let $h$ be the longest common suffix of $x$ and $z$, and $x = x'\cdot h$ and $z = z' \cdot h$.
As $\kappa$ is $\vG$\_continuous, let us apply point 2 of the Syncing Lemma on $\kappa'$, the equation $(a^\omega = 1)$, from the pair of states $(q', q')$ to $(q, q')$. With $Z = \Tif{\kappa, \nop, q'} \sync \Tif{\kappa, \nop, q'}$, a nonempty subset of the identity, it holds that $Z \times P \subseteq \equ_\cV(x(yz)^{\omega - 1}y, 1)$. We write, in the following, $\nu^{-1}$ for $\nu^{\omega-1}$, to convey the fact that $\nu^{-1}$ is the inverse of $\nu$ in $\cV$; that is: $\nu\cdot \nu^{-1} =_\cV 1$ (this analogy naturally carries further, since for instance, $(\nu\eta)^{-1} =_\cV \eta^{-1}\cdot \nu^{-1}$). Let $(s, s, u, u') \in Z \times P$, then: $$\begin{aligned}
s \cdot u' & =_\cV s \cdot x\cdot (yz)^{\omega - 1} \cdot y \cdot u \tag{By the
Syncing Lemma}\\
& =_\cV s \cdot x \cdot z^{-1}\cdot y^{-1} \cdot y \cdot u\\
& =_\cV s \cdot x \cdot z^{-1} \cdot u\\
& =_\cV s \cdot (x'h) \cdot (z'h)^{-1} \cdot u =_\cV s \cdot x'
\cdot z'^{-1} \cdot u\enspace.
\end{aligned}$$ By cancellation, this shows that $u' =_\cV x' \cdot z'^{-1} \cdot u$. Since $u'$ is a word and $x'$ and $z'$ do not share a common suffix, there is a word $w$ such that $u = z'\cdot w$ (this is true in the *free group*, which is embedded, in a precise sense, in $\cV$ [@robinson95 § 6.1.9]). This implies that $u' =_\cV x' \cdot w$ and shows that $P \subseteq (z', x')\cdot \Id$, hence that $q = q'$ by the construction of Lemma \[lem:nf\].
[Fact \[fact:allom\] and Lemma \[lem:nftocV\]]{} This relies on Fact \[fact:nofork\]. Let $u, v$ be such that both $\Tif{\tau'', \nop, p}(u)$ and $\Tif{\tau'', p, \nop}(v)$ are defined. As $\tau''(u\cdot v)$ is defined, the Preservation Lemma asserts that $\tau''(u\cdot a^\omega \cdot v)$ should also be defined. This implies that for $n$ large enough, there is an accepting path for the word $u \cdot a^{n!} \cdot v$. This path reaches some state $p'$ after reading $u$, reads $a^{n!}$ reaching a state $q$, and accepts $v$ from $q$. The path from $p'$ to $q$ contains a cycle of $a$, hence if $p'$ ($q$) is not in the cycle, this creates a proper reverse fork (proper fork). Thus by Fact \[fact:nofork\], both $p'$ and $q$ are in the same cycle that reads $a^{n!}$; for $n$ large enough, $n!$ is a multiple of the length of each cycle, this shows that $p' = q$ and that $p' \in p'.a^\omega$. This also implies that $\Tif{\tau'', \nop, p'}(u)$ and $\Tif{\tau'', p', \nop}(v)$ are both defined, and since $\tau''$ is unambiguous, that $p = p'$, concluding the proof.
As an immediate corollary:
\[cor:grpconttotrans\] For $\cV \in \{\vGnil, \vGsol, \vG\}$, any $\cV$\_continuous rational function is $\cV$\_realizable.
Let $\cV$ be a variety of group languages that includes $\vGnil$ and that is closed under inverse $\cV$\_realizable rational functions. It is decidable, given an unambiguous transducer, whether it realizes a $\cV$\_continuous function. This holds in particular for $\vGsol$ and $\vG$.
Lemma \[lem:nftocV\] together with Proposition \[prop:transtocont\] shows that a transducer is $\cV$\_continuous iff its equivalent transducer effectively computed by Lemma \[lem:nf\] and its dual is a $\cV$\_transducer. This latter property being testable, the result follows.
Deciding continuity for aperiodic varieties
-------------------------------------------
We saw in Section \[sec:struct\] that the approach of the previous section cannot work: there is no correspondence between continuity and realizability for aperiodic varieties. Herein, we use the Syncing Lemma to decide continuity in two main steps. First, note that all of our aperiodic varieties are defined by an infinite number of equations for each alphabet. The Syncing Lemma would thus have us check an infinite number of conditions; our first step is to reduce this to a finite number, which we stress through the forthcoming notion of “pertaining triplet” of states. Second, we have to show that the inclusion of the second point of the Syncing Lemma can effectively be checked. This will be done by simplifying this condition, and showing a decidability property on rational relations.
We will need the following technical result in combinatorics on words in the proof of the forthcoming Lemma \[lem:apred\]:
\[lem:uvxyst\] Let $u, v, x, y, s, t \in A^*$ be words verifying:
1. $u\cdot v,\; x\cdot y \in s^*$;
2. $v\cdot u,\; y\cdot x \in t^*$;
3. $s$ and $t$ are primitive.
There exist $z, z' \in A^*$ such that:
1. $s = z\cdot z'$ and $t = z' \cdot z$;
2. $u,\; x \in s^*\cdot z$;
3. $v,\; y \in t^*\cdot z'$.
Write $u = s^c\cdot z$ and $v = z'\cdot s^{c'}$ such that $s = z\cdot z'$. It follows that $v\cdot u \in (z'\cdot z)^*$, and since $z' \cdot z$ is primitive, it holds that $t = z' \cdot z$. We can do the same with $x$ and $y$, letting $x = s^{\dot c} \cdot \dot z$ and $y = \dot z' \cdot s^{\dot c'}$, with $s = \dot z \cdot \dot z'$. The same reasoning then shows that $t = \dot z' \cdot \dot z$. Since $s$ has precisely $|s|$ conjugates (by [@lothaire97 Proposition 1.3.2]), it holds that $\dot z = z$ and $\dot z' = z'$, and the properties of the Lemma follow.
\[def:pert\] A triplet of states $(p, q, q')$ is *pertaining* if there are words $s,
u, t$ and an integer $n$ such that:
where $\cdot$ means “any word.” Further, a pertaining triplet is *empty* if, in the above picture, $\beta = \beta'\beta'' = 1$ and *full* if both words are nonempty; it is *degenerate* if only one of $\beta$ or $\beta'\beta''$ is empty.
It is called “pertaining” as the second point of the Syncing Lemma elaborates on properties of such a triplet, in particular, since $u^\omega = u^{\omega+1}$ is an equation of . The following characterization of $\vA$\_continuity is then made *without appeal* to equations or profinite words:
\[lem:apred\] A transducer $\tau\colon A^* \to B^*$ is $\vA$\_continuous iff all of the following hold:
1. $\tau^{-1}(B^*) \in \vA(A^*)$;
2. For all full pertaining triplets $(p, q, q')$, there exist $x, y \in B^*$ and $\rho_1, \rho_2 \in (B^*)^2$ such that $\Tif{\tau, \nop, p} \sync \Tif{\tau, \nop, q} \subseteq \Id\cdot \big((x^*,
x^*)\rho_1^{-1}\big) \quad\text{and}\quad
\Tif{\tau, p, \nop}
\sync \Tif{\tau, q', \nop} \subseteq \big(\rho_2^{-1}(y^*, y^*)\big)\cdot
\Id;$
3. For all empty pertaining triplets $(p, q, q')$ it holds that $(\Tif{\tau, \nop, p} \sync \Tif{\tau, \nop, q}) \cdot (\Tif{\tau, p, \nop}
\sync \Tif{\tau, q', \nop}) \subseteq \Id;$
4. No pertaining triplet is degenerate.
[Lemma \[lem:apred\]]{} Suppose $\tau$ is \_continuous, and let us appeal to the Syncing Lemma. Point 1 is then immediate. Point 2 is a direct consequence of the second point of the Syncing Lemma and of Lemma \[lem:apred\].
We shall now check point 3, by contradiction. Let $(p, q, q')$ be an empty pertaining triplet; we use the notations of Definition \[def:pert\]. Then by functionality of $\tau$, it holds that $\tau(s \cdot u^\omega \cdot t) = x_1 \cdot x_2$ and $\tau(s \cdot u^{\omega+1} \cdot t) = y_1 \cdot y_2$. By the Preservation Lemma, and since $s \cdot u^\omega \cdot t =_\vA s \cdot u^{\omega+1} \cdot t$, it should hold that $x_1\cdot x_2 = y_1 \cdot y_2$, proving point 3.
Point 4 is proven using similar ideas as point 3: with $(p, q, q')$ a degenerate pertaining tuple, and using the same notations as above, either the production of $s \cdot u^\omega \cdot t$ going through $p$ is a not a finite word while the production of $s \cdot u^{\omega+1} \cdot t$ through $q, q'$ is, or vice-versa. In both cases, it is not possible for these productions to be equal in $\vA$, hence if such a case happens, $\tau$ cannot be $\vA$\_continuous.
We again rely on the Syncing Lemma, the first point of which being verified by hypothesis. Let $u^\omega = u^{\omega+1}$ be an equation of $\vA$ with $u$ a word; the set of such equations defines $\vA(A^*)$. Let $p, q, p', q'$ be states such that $p' \in p.u^\omega$ and $q' \in q.u^{\omega+1}$, and let $s, t$ be words with $p,q \in q_0.s$ and $p'.t, q'.t \in F$. To conclude and apply the Syncing Lemma, we need to show that: $$\begin{aligned}
\Tif{\tau, \nop, p}(s) \cdot \Tif{\tau, p, p'}(u^\omega) \cdot \Tif{\tau,
p', \nop}(t) =_\vA \Tif{\tau, \nop, q}(s) \cdot \Tif{\tau, q,
q'}(u^{\omega+1}) \cdot \Tif{\tau, q', \nop}(t)\enspace.\label{eqn:m}
\end{aligned}$$ (This is a direct consequence of the way profinite words are evaluated in a transducer, as per Lemma \[lem:eval\].)
Consider a large number $N = n!$, so that $p' \in p.u^N$ and $q' \in
q.u^{N+1}$. With a large enough $N$, there must be two states $P$ and $Q$, and integers $i, j$ with $i + j = N$, such that:
- $P \in p.u^i$, $p' \in P.u^j$, and $P \in P.u^N$ (i.e., $P$ is “between” $p$ and $p'$, and belongs to a loop);
- $Q \in q.u^i$, $q' \in Q.u^{j+1}$, and $Q \in Q.u^N$.
(That such a pair exists can easily be seen on the product automaton of $\tau$ by itself: The path from $(p, q)$ to $(p', q'')$ with $q' \in q''.u$ reading $u^N$ must go twice through the same pair of states $(P, Q)$, and this pair respects the above requirements.)
Now define the following words:
\#1\#2
- $\gamma = \Tif{\tau, P, \nop}(u^j \cdot t)$,
- $\gamma' = \Tif{\tau, Q, \nop}(u^{j+1} \cdot t)$.
Using the same reasoning as Lemma \[lem:omega\], and the unambiguity of $\tau$, Equation (\[eqn:m\]) is equivalent to: $$\alpha\cdot \beta^{\omega - N} \cdot \gamma =_\vA \alpha' \cdot
\beta'^{\omega - N} \cdot \gamma'\enspace.$$ Naturally, since $\alpha\cdot \beta^{\omega - N} \cdot \gamma =_\vA
\alpha\cdot\beta^\omega\cdot\gamma$, and similarly for the right-hand side, Equation (\[eqn:m\]) is equivalent to: $$\alpha\cdot \beta^\omega \cdot \gamma =_\vA \alpha' \cdot
\beta'^\omega \cdot \gamma'\enspace.$$
To make use of the hypotheses of the present Lemma, define $Q'$ to be in $Q.u$ and such that $Q \in Q'.u^{N-1}$: $(P, Q, Q')$ is thus pertaining. The situation is then:
Since by hypothesis this triplet cannot be degenerate, either both of $\beta$ and $\beta'$ are empty, or none are. Suppose they are both empty, then the hypothesis on empty triplets shows that: $$\Tif{\tau, \nop, P}(s \cdot u^i) \cdot \Tif{\tau, P, \nop}(u^{N+j} \cdot t) =
\Tif{\tau, \nop, Q}(s \cdot u^i) \cdot \Tif{\tau, Q', \nop}(u^{N+j} \cdot
t)\enspace.$$ The left-hand side evaluates to $\alpha \cdot \gamma$. Since $\Tif{\tau, Q', \nop}(u^{N+j} \cdot t) = \Tif{\tau, Q', Q}(u^{N-1}) \cdot
\Tif{\tau, Q, \nop}(u^{j+1} \cdot t) = \gamma'$, the right-hand side evaluates to $\alpha' \cdot \gamma'$, and Equation (\[eqn:m\]) is thus verified.
Let us thus suppose that both $\beta$ and $\beta'$ are nonempty. We divide $\beta'$ into $b_1b_2$ such that $b_1 = \Tif{\tau, Q, Q'}(u)$ and $b_2 = \Tif{\tau, Q', Q}(u^{N-1})$. Now let $x, y \in B^*$ and $\rho_1, \rho_2 \in (B^*)^2$ be the (pairs of) words provided by point 2 for the triplet $(P, Q, Q')$. Define $L = \Id \cdot \big((x^*, x^*)\rho_1^{-1}\big)$ and $R = \big(\rho_2^{-1}(y^*, y^*)\big) \cdot \Id$. For any $k \geq 1$, and letting $\eta = s \cdot u^{i+k\times N}$ and $\eta' = u^{k \times N +j}\cdot t$, it holds by hypothesis that:
- $(\Tif{\tau, \nop, P}(\eta), \Tif{\tau, \nop, Q}(\eta)) = (\alpha \cdot
\beta^k,\; \alpha' \cdot \beta'^k) \in L$;(a)
- $(\Tif{\tau, P, \nop}(\eta), \Tif{\tau, Q', \nop}(\eta)) = (\beta^k
\cdot \gamma,\; b_2\cdot \beta'^{k-1}\cdot\gamma') \in R$.(b)
Let us first emphasize an easy property of $L$ and $R$:
\[fact:pre\] If $(w\cdot w', w \cdot w'') \in L$ with $|w'|, |w''| > |x|$, then $(w', w'') \in L$. Moreover, if $(w, w') \in L$, then $w$ is a prefix of $w'$ or vice-versa.
Similarly, if $(w'\cdot w, w'' \cdot w) \in R$ with $|w'|, |w''| > |y|$, then $(w', w'') \in R$. Moreover, if $(w, w') \in R$, then $w$ is a suffix of $w'$ or vice-versa.
[Fact \[fact:pre\]]{} We only show this for $L$, the case for $R$ being similar.
For the first part of the statement, the hypothesis ensures the existence of a word $z$, integers $n', n''$, and two prefixes $x', x''$ of $x$ such that $w \cdot w' = z \cdot x^{n'} \cdot x'$ and $w \cdot w'' = z \cdot x^{n''} \cdot x''$. If $w$ is a prefix of $z$, the property is easy to verify. In the other cases, $w = z\cdot x^n\cdot \chi$ for some integer $n < n', n''$ (strictness coming from the hypothesis) and $x = \chi\chi'$. Hence $w' = \chi'\cdot x^{n'-n-1} \cdot x'$ and $w'' = \chi'\cdot x^{n''-n-1} \cdot x'$, and thus both belong to $L$. The case of $R$ is similar.
For the second part of the statement, $w$ and $w'$ start with a common word $z$, then some repetitions of $x$, and a prefix of $x$. Clearly, one has to be a prefix of the other.
We first focus on the consequences of (a). First, since either $\alpha\cdot\beta^k$ is a prefix of $\alpha'\cdot\beta'^k$ or vice-versa, it holds that either $\alpha$ is a prefix of $\alpha'\cdot\beta'^k$, or $\alpha'$ a prefix of $\alpha\cdot\beta^k$, for some $k$. Suppose for instance that $\alpha' = \alpha\cdot\beta^c \cdot \beta\subp$, with $\beta = \beta\subp \cdot \beta\subs$; the other case will be treated later. Appealing to Fact \[fact:pre\], for $k$ big enough, factoring out $\alpha'$ yields that $((\beta\subs\beta\subp)^{k-c-1}\beta\subs, \beta'^k) \in L$. Hence $(\beta\subs\beta\subp)^*$ and $\beta'^*$ share common prefixes of unbounded length, implying that $\beta\subs\beta\subp$ and $\beta'$ are powers of a same primitive word $z_1$ (by, e.g., [@lothaire97 Proposition 1.3.5]).
Now similarly focusing on (b), we obtain that $\gamma$ is a suffix of $\beta'^k\cdot\gamma'$ or $\gamma'$ is a suffix of $\beta^k\cdot\gamma$, for some $k$. Suppose for instance that $\gamma = \beta'\subs \cdot \beta'^{c'} \cdot \gamma'$, with $\beta' = \beta'\subp \cdot \beta'\subs$, again delaying the other case. It follows, just as above, that $\beta$ and $\beta'\subs\beta'\subp$ are powers of a same primitive word $z_2$. Noting that $(\eta^c)^\omega = \eta^\omega$, for any $\eta$, Equation (\[eqn:m\]) is thus equivalent to: $$\alpha \cdot z_2^\omega \cdot \beta'\subs\cdot\beta'^{c'} \cdot \gamma'
=_\vA \alpha\cdot \beta^c \cdot \beta\subp \cdot z_1^\omega \cdot
\gamma'\enspace.$$
Lemma \[lem:uvxyst\] indicates that there exist words $z, z'$ such that $z_1 = z\cdot z'$, $z_2 = z' \cdot z$, and $\beta'\subs \in z_2^*\cdot z', \beta\subp \in z' \cdot z_1^*$. By eliminating $\alpha$ and $\gamma'$ we thus obtain that there are some integers $n_1, n_2$ such that Equation (\[eqn:m\]) is equivalent to $z_2^\omega\cdot z' \cdot z_1^{n_1 \times c'} =_\vA z_2^{n_2 \times c} \cdot
z' \cdot z_1^\omega$, which clearly holds as both sides evaluate to $(z' \cdot z)^\omega \cdot z'$.
We made two suppositions: $\alpha'$ is a prefix of $\alpha$, and $\gamma$ is a suffix of $\gamma'$. The case where $\alpha$ is a prefix of $\alpha'$ and $\gamma'$ a suffix of $\gamma$ is entirely symmetric. Let us keep our supposition on $\alpha'$ and assume that $\gamma'$ is a suffix of $\gamma$; the last remaining case is similar to this one.
Let us thus write $\gamma' = \betab\subs \cdot \beta^{c'} \cdot \gamma$, with $\beta = \betab\subp\cdot\betab\subs$. We then obtain, factoring out $\gamma'$ this time, that $(\beta^{k-c-1}\betab\subp, \beta'^k) \in R$. This implies that $(\betab\subs\betab\subp)$ and $\beta'$ are powers of the same primitive word, which can only be $z_1$. Writing $z_2$ for the primitive root of $\beta$, Lemma \[lem:uvxyst\] shows the existence of words $z, z'$ such that $z_1 = z \cdot z'$, $z_2 = z' \cdot z$, and $\beta\subp \in z_2^*\cdot z', \betab\subs \in z \cdot z_2^*$. By eliminating $\alpha$ and $\gamma$, we similarly obtain that Equation (\[eqn:m\]) is equivalent, for some $n_1, n_2$, to $z_2^\omega =_\vA z_2^{n_1 \times c} \cdot \beta\subp \cdot z_1^\omega \cdot
\betab\subs \cdot z_2^{n_2 \times c'}$. Then both sides evaluate to $z_2^\omega$, hence Equation (\[eqn:m\]) holds.
We show that the transducer of Proposition \[prop:conttrans\] is $\vA$\_continuous. Let $\tau$ be:
First, the function is total, hence the first point of Lemma \[lem:apred\] is verified. Second, there are no empty nor degenerate pertaining triplets, hence the third and fourth points are verified. Now the full pertaining triplets are $(p, p, p), (p, p, q), (q, q, q),$ and $(q, q, p)$. We check that the pertaining triplet $(p, p, q)$ verifies the second condition of Lemma \[lem:apred\], the other cases being similar or clear. The first half of the condition is immediate. Now $\Tif{\tau,p,\nop} \sync \Tif{\tau,q,\nop} = \{(a^{\lfloor n+1/2\rfloor},
a^{\lfloor n/2\rfloor}) \mid n \geq 0\}$ which verifies the condition.
We now show that the property of Lemma \[lem:apred\] is indeed decidable:
\[prop:dectrans\] It is decidable, given a rational relation $R \subseteq A^* \times A^*$, whether there is a word $x \in A^*$ and a pair $\rho \in
(A^*)^2$, such that $R \subseteq \Id \cdot \big((x^*, x^*)\rho^{-1})$.
We rely on the classical result that it is decidable whether a rational relation is included in the identity [@sakarovitch09 p. 650].
We first tackle a related, simpler decision problem: Given a rational relation $R \subseteq (A^* \times A^*)$ and a word $x \in A^*$, check whether $R \subseteq \Id \cdot (x^*, x^*)$. Write $f\colon A^* \to A^*$ for the function that removes the longest suffix in $x^*$ of its argument, and note that $f$ is a rational function. Closure under inverse and composition of rational relations implies that $R' = \{(f(u), f(v)) \mid (u, v) \in R\}$ is a rational relation computable from $R$. We have that $R' \subseteq \Id$ if and only if $R \subseteq \Id \cdot (x^*, x^*)$, hence the decision problem at hand is equivalent to checking whether $R' \subseteq \Id$, which is decidable.
We now reduce the main decision problem to the previous one. To do so, let us suppose that such an $x$ and $\rho$ exist, and search for them; if this search fails, no such $x$ and $\rho$ exist.
First note that we may suppose that one component of $\rho$ is the empty word. Indeed, write $x' = x\rho_1^{-1}, x''=x\rho_2^{-1}$. Assume $x'$ is also a prefix of $x''$ (the symmetric case being similar). We may thus write $x = x'y$ and $x'' = x'z$, and have that $R \subseteq \Id\cdot\big((yx')^*, (yx')^*z\big)$, showing that $x := yx'$ and $\rho := (1, z^{-1}yx')$ fit the requirements.
We thus suppose that $\rho_2$ is empty (the symmetric case being similar). We first check that $R \subseteq \Id$. If this is not the case, we are provided with a pair $(u, v) \in R \setminus \Id$ (again by [@sakarovitch09 p. 650]). All the suffixes of $u$ provide us with candidates for $x\rho_1^{-1}$; we go through all these candidates $x'$ (including the empty word).
We then verify that all pairs $(u, v) \in R$ are such that $u$ ends with $x'$ (this is decidable, e.g., since $R \cap \big((A^*u)\comp \times A^*\big) = \emptyset$ is decidable [@berstel79 Proposition 2.6, Proposition 8.2]). If this is not the case, then $x'$ cannot be $x\rho_1^{-1}$. Otherwise, let us write $R' =
R(x',1)^{-1}$, a rational relation.
We now check again that $R' \subseteq \Id$; if it is the case, we are done, otherwise, we are given a pair $(u, v) \in R' \setminus \Id$. Naturally, this implies that $u$ is a prefix of $v$, or vice-versa. In the former case, write $v = u\cdot z$; it should hold that $z \in x^*$, and this provides us with candidates for $x$: all the possible roots $z'$ of $z$. We may now test that one such $z'$ starts with $x'$, and check whether $R \subseteq \Id\cdot(z'^*, z'^*)$ using the above decision problem. If this holds, then there do exist an $x$ and a $\rho$ verifying $R \subseteq \Id\cdot\big((x^*, x^*)\rho^{-1}\big)$. Moreover, if such words exist, this procedure will find them.
\[rk:guillon\] In general, the problem of deciding, given a rational relation $R$ and a *recognizable* relation $K$, whether $R \subseteq \Id \cdot K$, is undecidable. Indeed, testing $R \cap \Id = \emptyset$ is undecidable [@berstel79], and equivalent to testing: $$R \subseteq \Id \cdot \big((A^+ \times \{1\}) \cup (\{1\} \times A^+) \cup
\bigcup_{a \neq b \in A} (a\cdot A^* \times b\cdot A^*)\big)\enspace,$$ the right-hand side being of the form $\Id \cdot K$.
It is decidable, given an unambiguous transducer, whether it realizes an $\vA$\_continuous function.
This is a consequence of Lemma \[lem:apred\]: Given a transducer, one can list all its pertaining triplets, and whether they are empty, full, or degenerate. For full pertaining triplets, the property of Lemma \[lem:apred\] is checked with Proposition \[prop:dectrans\] and the same Proposition applied on the reverse of the transducer. The property for empty triplets can be checked since the inclusion of a rational relation in $\Id$ is decidable.
The rest of this section focuses on conditions *à la* Lemma \[lem:apred\] for $\vJ$, $\vR$, $\vL$, and $\vDA$. In each of these cases, we define the proper notion of “pertaining” and rewrite the conditions of Lemma \[lem:apred\] to match the defining equations. Since the proofs are simple variants of that of Lemma \[lem:apred\], we omit them; we note that in each case, the conditions are effectively verifiable.
### The case of $\vJ$
We use a different set of equations to define $\vJ$, that can easily be proved to be equivalent to the one given in the preliminaries. Specifically, $\vJ$ is defined over any alphabet $A$ by the set of equations $x^\omega = y\cdot
x^\omega \cdot z$, with $y, z \in \cts(x)$. The definition of “pertaining” then reads as follows:
For two alphabets $C, D$, a quadruplet of states $(p, q, q', q'')$ is *$(C, D)$\_pertaining* if there are words $s, u, t$ with $\cts(u) = C$, words $z, z' \in C^*$, and an integer $n$ such that:
and moreover $\cts(\beta) \cup \cts(\beta') = D$. The pertaining quadruplet is *empty* if $D = \emptyset$; it is *full* if $\cts(\beta) = \cts(\beta') \neq \emptyset$, and *degenerate* otherwise.
A transducer $\tau\colon A^* \to B^*$ is $\vJ$\_continuous iff all of the following hold:
1. $\tau^{-1}(B^*) \in \vJ(A^*)$;
2. For all full $(C, D)$\_pertaining quadruplet $(p, q, q', q'')$, it holds that: $$\begin{aligned}
\Tif{\tau, \nop, p} \sync \Big(\Tif{\tau, \nop, q} \cdot \big(\eps, \Tif{\tau,
q, q'}(C^*)\big) \cdot \Tif{\tau, q', q'}\Big) & \subseteq \Id \cdot (D^*,
D^*)\enspace\text{and}\\
\Tif{\tau, p, \nop} \sync \Big(\Tif{\tau, q', q'} \cdot \big(\eps,
\Tif{\tau, q', q''}(C^*)\big) \cdot \Tif{\tau, q'', \nop}\Big) & \subseteq
(D^*,
D^*)\cdot \Id\enspace;
\end{aligned}$$
3. For all empty $(C, D)$\_pertaining quadruplet $(p, q, q', q'')$, it holds that: $$(\Tif{\tau, \nop, p} \cdot \Tif{\tau, p, \nop}) \sync
\Big(\Tif{\tau, \nop, q} \cdot \big(\eps, \Tif{\tau, q, q'}(C^*)\big)
\cdot \Tif{\tau, q', q'} \cdot \big(\eps, \Tif{\tau, q', q''}(C^*)\big)
\cdot \Tif{\tau, q'', \nop}\Big) \subseteq \Id\enspace;$$
4. No pertaining quadruplet is degenerate.
### The case of $\vR$
Again, we slightly diverge from the usual equations for $\vR$, as presented in the Preliminaries. Indeed, $\vR$ is also defined, over any alphabet $A$, by $x^\omega = x^\omega \cdot y$ with $y \in \cts(x)$. We turn to the definition of “pertaining:”
For an alphabets $C$, a triplet of states $(p, q, q')$ is *$C$\_pertaining* if there are words $s, u, t$ with $\cts(u) = C$, words $z \in C^*$, and an integer $n$ such that:
The pertaining triplet is *empty* if, in the above picture, $\beta = \beta' = 1$; it is *full* if none of $\beta, \beta'$ is empty, and *degenerate* otherwise.
A transducer $\tau\colon A^* \to B^*$ is $\vR$\_continuous iff all of the following hold:
1. $\tau^{-1}(B^*) \in \vR(A^*)$;
2. For all full $C$\_pertaining triplets $(p, q, q')$, there exist $x \in
B^*$ and $\rho \in (B^*)^2$ such that both inclusions hold: $$\begin{aligned}
\Tif{\tau, \nop, p} \sync \Tif{\tau, \nop, q} & \subseteq \Id\cdot
\big((x^*, x^*) \rho^{-1}\big)\enspace,\\
\Tif{\tau, p, \nop}
\sync \Big(\Tif{\tau, q, q} \cdot \big(\eps, \Tif{\tau, q, q'}(C^*)\big)
\cdot \Tif{\tau, q', \nop}\Big) & \subseteq \big(\cts(x)^*, \cts(x)^*\big)
\cdot \Id\enspace;
\end{aligned}$$
3. For all empty $C$\_pertaining triplets $(p, q, q')$, it holds that: $$(\Tif{\tau, \nop, p} \cdot \Tif{\tau, p, \nop}) \sync
\Big(\Tif{\tau, \nop, q} \cdot \big(\eps, \Tif{\tau, q, q'}(C^*)\big)
\cdot \Tif{\tau, q', \nop}\Big) \subseteq \Id\enspace;$$
4. No pertaining triplet is degenerate.
Note that the $x$ of Proposition \[prop:dectrans\] can be effectively found. The case of $\vL$ can be simply seen as the reversal of the previous case.
### The case of $\vDA$
Similarly, we use a slightly less standard equational definition of $\vDA$. Indeed, $\vDA$ is also defined, over any alphabet $A$, by $x^\omega = x^\omega
\cdot y \cdot x^\omega$ with $y \in \cts(x)$. The definition of “pertaining” reflects these equations:
For an alphabet $C$, a triplet of states $(p, q, q')$ is *$C$\_pertaining* if there are words $s, u, t$ with $\cts(u) = C$, a word $z \in C^*$, and an integer $n$ such that:
Further, a pertaining triplet is *empty* if, in the above picture, $\beta = \beta' = \beta'' = 1$; it is *left-empty* if only $\beta'$ is empty, *right-empty* if only $\beta''$ is empty, *full* if none of $\beta, \beta', \beta''$ is empty, and *degenerate* in the other cases.
A transducer $\tau\colon A^* \to B^*$ is $\vDA$\_continuous iff all of the following hold:
1. $\tau^{-1}(B^*) \in \vDA(A^*)$;
2. For all full $C$\_pertaining triplets $(p, q, q')$, there exist $x, y \in B^*$ and $\rho_1, \rho_2 \in (B^*)^2$ such that these three inclusions hold: $$\begin{aligned}
\Tif{\tau, \nop, p} \sync \Tif{\tau, \nop, q} & \subseteq \Id\cdot \big((x^*,
x^*)\rho_1^{-1}\big)\enspace,\\
\Tif{\tau, p, \nop}
\sync \Tif{\tau, q', \nop} & \subseteq \big(\rho_2^{-1}(y^*, y^*)\big)\cdot
\Id\enspace,\\
\Tif{\tau, q, q'}(C^*) &\subseteq \cts(x\cdot
y)^*\enspace;
\end{aligned}$$
3. For all empty $C$\_pertaining triplets $(p, q, q')$ it holds that: $$(\Tif{\tau, \nop, p} \sync \Tif{\tau, \nop, q}) \cdot (\eps, \Tif{\tau, q,
q'}(C^*)) \cdot (\Tif{\tau, p, \nop} \sync \Tif{\tau, q', \nop}) \subseteq
\Id\enspace;$$
4. For all right-empty $C$\_pertaining triplets $(p, q, q')$, there exist $x, y \in B^*$ and $\rho_1, \rho_2 \in (B^*)^2$ such that: $$\Tif{\tau, \nop, p} \sync \Tif{\tau, \nop, q} \subseteq \Id\cdot \big((x^*,
x^*)\rho_1^{-1}\big) \quad\text{and}\quad
\Tif{\tau, p, \nop} \sync \big((\eps, \Tif{\tau, q, q'}(C^*))\cdot
\Tif{\tau, q', \nop}\big) \subseteq
\big(\rho_2^{-1}(y^*, y^*)\big)\cdot \Id\enspace;$$
5. For all left-empty $C$\_pertaining triplets $(p, q, q')$, there exist $x, y \in B^*$ and $\rho_1, \rho_2 \in (B^*)^2$ such that: $$\Tif{\tau, \nop, p} \sync \big(\Tif{\tau, \nop, q} \cdot (\eps, \Tif{\tau, q, q'}(C^*))\big) \subseteq \Id\cdot \big((x^*,
x^*)\rho_1^{-1}\big) \quad\text{and}\quad
\Tif{\tau, p, \nop} \sync \Tif{\tau, q', \nop} \subseteq
\big(\rho_2^{-1}(y^*, y^*)\big)\cdot \Id\enspace;$$
6. No pertaining triplet is degenerate.
Deciding Com- and Ab-continuity
-------------------------------
The case of $\vCom$ and $\vAb$ is comparatively much simpler, in particular because these varieties are defined using a finite number of equations for each alphabet. However, the argument relies on different ideas:
For $\cV = \vCom, \vAb$, it is decidable, given an unambiguous transducer, whether it realizes a $\cV$\_continuous function.
We apply the Syncing Lemma. Its first point is clearly decidable. We reduce its second point to decidable properties about semilinear sets (see, e.g., [@ginsburg66]). We also rely on the notion of Parikh image, that is, the mapping $\pkh\colon A^* \to \bbn^A$ such that $\pkh(w)$ maps $a \in A$ to the number of $a$’s in the word $w$.
Since every $\vAb$\_continuous function is $\vCom$\_continuous (Proposition \[prop:abtocom\]), the conditions to test for $\vAb$\_continuity are included in those for $\vCom$\_continuity—this can also be seen as a consequence of the fact that if $u, v$ are words, $\equ_\vAb(u, v) = \equ_\vCom(u,v)$.
Let $\tau\colon A^* \to B^*$ be a given transducer. Consider an equation $ab = ba$ and four states $p, p', q, q'$ of $\tau$. Write $u = \Tif{\tau, p, p'}(ab)$ and $v = \Tif{\tau, q, q'}(ba)$. We ought to check, by the Syncing Lemma, the inclusion in $\equ_\vCom(u, v) = \{(s, s', t, t') \mid s\cdot u \cdot t =_\vCom s' \cdot v
\cdot t'\}$ of some input synchronization. Now this set is the set of $(s, s', t, t')$ such that $\pkh(s\cdot u \cdot t) = \pkh(s' \cdot v \cdot t')$, and is thus defined by a simple semilinear property. The input synchronizations themselves, e.g., $\Tif{\tau, \nop, p} \sync \Tif{\tau, \nop, q}$, are rational relations, and their component-wise Parikh image is thus a semilinear set. Since the inclusion of semilinear sets is decidable, the inclusion of the second point of the Syncing Lemma is also decidable.
For $\vAb$, we should additionally check the equations $a^\omega = 1$. The reasoning is similar. Consider three states $(p, p', q)$, and write $x \cdot u^{\omega-1} \cdot y$ for $\Tif{\tau, p, p'}(a^\omega)$. By commutativity and the fact that $u^{\omega-1}$ acts as an inverse of $u$ in the equations holding in $\vAb$, we have that $(s, s', t, t') \in \equ_\vAb(x\cdot u^{\omega-1} \cdot y, 1)$ iff $s\cdot t =_\vAb s'\cdot u \cdot t'$. This again reduces the inclusion of the second point of the Syncing Lemma to a decidable semilinear property.
Discussion
==========
We presented a study of continuity in functional transducers, on the one hand focused on general statements (Section \[sec:contapp\]), on the other hand on continuity for classical varieties. The heart of this contribution resides in decidability properties (Section \[sec:dec\]), although we also addressed natural and related questions in a systematic way (Section \[sec:intermezzos\]). We single out two main research directions.
First, there is a sharp contrast between the genericity of the Preservation and Syncing Lemma and the technicality of the actual proofs of decidability of continuity. To which extent can these be unified and generalized? We know of two immediate extensions: 1. the generic results of Section \[sec:contapp\] readily apply to Boolean algebras of languages closed under quotient, a relaxation of the conditions imposed on varieties, and 2. the varieties $\vG_p$ of languages recognized by $p$-groups can also be shown to verify Proposition \[prop:transtocont\] and Lemma \[lem:nftocV\], hence $\vGp$\_continuity is decidable for transducers. Beyond these two points, we do not know how to show decidability for $\vGnil$ (which is the *join* of the $\vGp$), and the surprising complexity of the equalizer sets for some Burnside varieties (e.g., the one defined by $x^2 = x^3$, see the Remark on page ) leads us to conjecture that continuity may be undecidable in that case, hence that no unified way to show the decidability of continuity exists.
Second, the notion of continuity may be extended to more general settings. For instance, departing from regular languages, it can be noted that every recursive function is continuous for the class of recursive languages. Another natural generalization consists in studying $(\cV, \cW)$\_continuity, that is, the property for a function to map $\cW$\_languages to $\cV$\_languages by inverse image. This would provide more flexibility for a sufficient condition for cascades of languages (or stackings of circuits, or nestings of formulas) to be in a given variety.
### Acknowledgment. {#acknowledgment. .unnumbered}
We are deeply indebted to Shaull Almagor, Jorge Almeida (in particular for the Remark on page ), Luc Dartois, Bruno Guillon (in particular for the Remark on page ), Ismaël Jecker, and Jean-Éric Pin for their insightful comments and kind help.
The first and third authors are partly funded by the DFG Emmy Noether program (KR 4042/2); the second author is funded by the DeLTA project (ANR-16-CE40-0007).
[^1]: The usual definition of $\cV$\_transducer is based on the so-called transition monoid of $\tau$, see, e.g., [@reutenaeur-schutzenberger95]; the definition here is easily seen to be equivalent by [@almeida99 Lemma 3.2] and [@cadilhac-krebs-ludwig-paperman15 Lemma 1].
[^2]: A note on terminology: There has been some fluctuation on the use of the term “continuous” in the literature, mostly when a possible incompatibility arises with topology. In [@pin-silva17], the authors use the term “preserving” in the more general context of functions from monoids to monoids. In our study, we focus on word to word functions, in which the natural topological context provides a solid basis for the use of “continuous,” as used in [@pin-silva05; @cadilhac-krebs-ludwig-paperman15].
[^3]: In all the varieties we are interested in, one can easily modify any partial function into a total function while preserving its continuity properties.
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